NUMBER WAR GAMES I: Number Concept and Relationships

Teaching Mathematics Facts Using Card Games

Children, all over the world, love to play games. I have successfully used games for initial teaching and remedial mathematics instruction, particularly, for learning arithmetic facts (addition, subtraction, multiplication, and division), comparison of fractions, and comparing and combining integers. An ordinary deck of playing cards, a pair of Dice and Dominos are good tools for teaching arithmetic, particularly, number conceptualization and simple arithmetic facts.  However, using cards from an ordinary deck assume number concept, in thier use and Dominos and Dice only teach subitizing. Whereas, A set of Visual Cluster Cards helps children to learn all the components of Nubmer Concept: learning visual clustering (generalization of subitizing), decomposition/recomposition, comparison of numbers, and their relationships.

Playing cards are used for playing games all over the world. Every culture has developed playing cards and games related to them. The games and their complexity vary from simple to complex and from simple comparison to strategies.  The number and type of games played using an ordinary deck of playing cards abound. Games, using playing cards, are enjoyed by all—from children to adults.  

One of the most popular games children play is called the Game of War. This is a family of card games. These games, under various names, are played by children and adults all over the world. I have adopted many of these games for teaching mathematics concepts and reinforcing them.  I call these games: ‘The Number War Games.’ 

I have designed a special set of playing cards called: Visual Cluster CardsTM for playing these games.  Visual Cluster Cards are without numbers on them. When children use these cards, within few days, they learn the most important component of number: decomposition/ recomposition. Through decomposition/ recomposition, they acquire the 45 sight-facts (addition and then subtraction facts of numbers up to 10). Visual Cluster Cards are better suited for these games[2].  

Visual Cluster Cards are modified ordinary deck of cards, in their design and in number. They are of two types: With face cards and without face cards. Both types of Visual Cluster cards have several arrangements of clusters for numbers such as: 0 (one blank card), 3 (two clusters), 8 (two), 9 (four), wild card (two), and 10 (two). All other numerals (1, 2, 4, 5, 6, and 7 have one card in each suit (spade, club, diamond, and heart).  There are 60 cards (the deck without face cards) in this deck.  The other deck of Visual Cluster cards includes face cards, in addition to all the other cards. The blank card represents zero and the wild card as a variable–assuming the value the context and the player assigns. In this deck, all face cards represent 10 (a good option when working on numbers up to 10) or the jack represents 11, queen represents 12, and king represents 13. 

In both decks, the cluster of objects (pips, icons) represents the numeral and the color (black = positive, red = negative) of the pips represent the number: e.g., five of clubs represents the number +5. Whereas, 5 of hearts represents, the number, 5. Thus, in both decks, when working with integers, red cards represent negative numbers, and black cards represent positive numbers.  As one can see, numeral is a representation of quantity and number is a directed numeral (it has a direction and quantity). Up to fifth grade, we do not make a differentiation between numeral and number. However, once the children beginn to learn about integers, we need to differentiate between numeral and numeber.

Children learn the quantity (numeral), number (positive and negative numerals) represented by the cards by observation (by sight), ultimately without counting. Since, children derive and learn the relationship between numbers up to ten by sight, these facts are called sight factsThere are a total of 45 sight facts[3]. Sight facts are like sight words. A child should master these 45 sight facts by the end of Kindergatrten.

The ordinary Game of War is played by children all over the world. My game begins in the same way as the Game of War. It is played essentially the same way and is easy to learn. Before, they play the game, however, it is important that children become familiar with the deck of Visual Cluster cards, particularly, the patterns of visual clusters on each card.  

Visual Cluster cards have clusters of objects displayed on the card. For example, there are five diamonds displayed in the middle in a particular pattern–a pattern that encourages decomposition/recomposition (see below).

An arrangement of this type is called a visual cluster[4]. The particular arrangement above is the visual cluster for five. It will be called the numeral 5 up to fifth grade. Later, it will be called numeral 5 and number +5.

Because of the patterns of pips, on individual visual cluster cards, they can be recognized, without counting, visualized, and then committed to memory with ease. The special nature of the visual pattern of a cluster of pips, representing the quantity, on a card helps a child to form a vivid image of that quantity, therefore, the numeral/quantity represented by the card, in their minds. Each Visual Cluster card is organized according to a particular cluster. This helps players to recognize the size of collections (up to 10) without counting. This also helps children to integrate: (a) orthographic image (5) of the numeral (when it is formalized in writing), (b) the auditory form (f-i-v-e), and, (c) the quantity represented by the cluster. This integration is called “numberness.” In this particular case, this integration is called “fiveness.” Writing should begin when a child can recognize the cluster representing a numeral instantly.

Children who are not able to form and hold these clusters in their minds and are, therefore, unable to recognize the size of a collection of objects by observation, have not conceptualized number, yet. This lack of integration of these three elements is a symptom and the manifestation of dyscalculia. Research supports this observation and shows that, in such a case, children  have difficulty in learning number concept, number relationships, particularly addition and subtraction facts and other higher concepts, and later operations on numerals and numbers (i.e., integers, etc.). These children keep on counting on fingers or on number line to find the sums and differences of even two small numbers. They also have great difficulty in automatizing arithmetic facts.  

The following games not only help children to conceptualize number but also help them to master arithmetic facts.  These games are highly motivating to children. 

There are several games, in this series, that are variations of other popular card games, such as “Go Fish.”  If you use or are aware of any card games that relate to number and number relationships, I would love to hear about them (

Game One:  Visual Clustering and Comparison of Numbers

(For children age 3 to those who are having difficulty mastering arithmetic facts)

Objective:  To teach number concept—numberness, decomposition/ recomposition, and sight facts.

The game can be played between two or three players.  However, it is most effective between two players.

Materials:  Take a deck of Visual Cluster cards including jokers (joker can assume any number value, according to context). In the case of Visual Cluster Cards with face cards, each card’s value is the number of objects displayed by the visual cluster on the card (e.g., Ace = 1 and the blank card = 0).  For example, the four of diamonds, clubs, spades or hearts will be known as number/numeral four. 

Each face card, jack, queen, and king is initially given the value of ten.  The ace represents number one.  The joker can assume any value and can be different each time it is used. When children know the teen’s numbers, then you can introduce: jack = 11, queen = 12, King = 13.  

How to Play: 

  1. The whole deck is divided into two equal piles of cards (if two players).  
  2. Each child gets one pile of cards.  One can also distribute the cards equally by counting out loud (This teaches children sequence of numbers and their location on the sequence of numbers. This increases number vocabulary–lexical enries for number) . Each person keeps the cards face down.  
  3. When the game begins, each person turns a card face up.  The bigger value card wins. For example, one has the three of hearts (value 3), and the other person has the seven of diamonds (value 7). The seven of diamonds wins. The winner collects all the displayed cards and puts them underneath his/her pile. (When playing this game with integers, three of hearts represents -3 and 7 of diamonds will represent -7).
  4. If both players have the same value cards (for example, one has the five of hearts, and the other has the five of spades), they declare war: “I declare war.” 
  5. Each player puts three cards face down on each sound of the word, in succession, saying I (for the first card) declare (for the second card), and war (for the third card). Then each player displays a fourth card face up.  The bigger valued fourth card wins. If they match again, the same process is repeated.
  6. The winner collects all cards and places them underneath his/her pile.  
  7. The first person with an empty hand loses. 

This game is appropriate for pre-K, Kindergarteners, and other children who have not mastered number concept. Number conceptualization is dependent on five interconnected skills: (i) Having a large number vocabulary, (ii) one-to-one correspondence with sequence, (iii) visual clustering (extension of subitizing)—recognizing a cluster of objects up to five by observation (without counting) is called subitizing and recognizing up to 10 objects is called visual clustering, (iv) decomposition/recomposition, and (v) ordering.  This game develops all of these prerequisite skills and many more.  Children with a lack of number concept have great difficulty in learning arithmetic facts and can derive them only by sequential counting. Which is a very inefficient strategy. Initially, for a short while, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize the value of cards. In a game, children have the opportunity of comparing almost five hundred pairs of numbers. 

Game Two: What Makes This Number

(For children age 3 to those who are having difficulty mastering arithmetic facts)

Objective: To master addition sight facts

Materials:  Same as above

How to Play: 

  1. The whole deck is divided into two equal piles of cards.  
  2. Each child gets a pile of cards.  The cards are kept face down. 
  3. Each person displays one card face up.  Each one finds two numbers whose sum is their card. For example, one has the three of hearts (value 3) and, therefore, gives two sight facts: 1 + 2 = 3, 2 + 1 = 3. The other has the seven of diamonds, the sight facts are: 1+ 6, 2+ 5, 3 +4, 4+3, 5+2, 6+1. The one with more sight facts wins. If the child, with the bigger number, cannot produce all the sight facts, the other player gets a chance and if he/she can give all the sight facts, he/she wins. In general, the person who is able to produce all the sight facts correctly and has the bigger number wins. The winner collects all the displayed cards and puts them underneath his/her pile. 
  4. If both players have the same number of sight facts, there is war.  For example, one has the five of hearts (value 5) and gives all the sight facts and the other has five of clubs (value 5) and gives all the sight facts. Or, one has five of diamonds and gives three sight facts only, and the other has nine of clubs (value 9) and gives three sight facts only, they declare war. 
  5. Each one puts three cards face down. Then each one displays another card face up. The bigger number of sight facts wins. 
  6. The winner collects all the cards and places them underneath his/her pile.  
  7. The first person with an empty hand loses. 

This game is appropriate for children who have not been introduced to sight facts or have not mastered/automatized simple addition facts. 

Initially, children will count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the sums. Within a few weeks, they can master all the 45 sight facts[5]. Initially, the game can be played with dominos or with a deck of cards of numbers up to five.

This series of posts will continue. In future editions, number games relating to other operations (inlcusding algebraic operations) will be included. Next few games will be on arithmetic operations.

[1]Copyright 2008 with Mahesh Sharma. 

[2]Visual Cluster Cards are available from Center for Teaching/Learning of Mathematics ($15 per deck plus $4.00 for shipping and handling).

[3]Number Conceptualization by Sharma (2008).  

[4]Same as above.

[5]The list of sight facts and how to teach them is included in How to Teach Number Concept Using Visual Cluster cards (Sharma, 2017).  Also see the post on Sight Words and Sight Facts on this Blog.

NUMBER WAR GAMES I: Number Concept and Relationships


Mahesh C. Sharma 

Children’s lower achievement—lack of arithmetic competence and difficulties in learning mathematics concepts, procedures and skills—and learning problems are often explained in terms of memory deficits (short-term, working, and long-term), problems with pre-requisite skills in learning mathematics (e.g., ability to follow directions), and information processing issues. In most studies, these factors are implicated alone as well as in combinations. 

Behavioral and neurological evidence from the last 30 years of research has demonstrated the complex network of skills involved in reading, writing and math achievement. Researchers have illustrated the innate processing differences among students with and without disabilities in different aspects of skill learning (e.g., reading, language, etc.). However, their role is less explored and understood in the area of mathematics learning difficulties.  

When a mathematics problem is posed, before linguistic, quantitative and spatial processes are applied in solving the problem, children and adults alike go through a combination of conscious and unconscious processing and decision-making protocols. A set of skills: processing skills, executive functions (working memory and flexibility of thought) are called upon to make sense of the information in the problem and then solving it.

A. Working Memory

Mathematics learning is a complex phenomenon, both in content and involvement of learning processes. One of the important aspects of this learning is holding and manipulating information during reading a definition, solving a problem, or seeing connection between different pieces of information.  This takes place in the memory complex, particularly, in the working memory. How do we keep everything in mind when solving a simple or complex mathematics problems? When you read a word problem, listen to the teacher or have a conversation about a concept or problem–how does our brain hold onto all that information? It takes place in the working memory

Unlike long-term memory (that is where what you have mastered resides—arithmetic facts, procedures, etc.), working memory is not about remembering the facts, formulas, and procedures already learned. Instead, it’s about holding together the current information (received from short-term memory, brought from long-term memory, generated by reflecting on these, and by visualizing) in our mind so we can learn, make decisions and solve problems. 

Working memory (WM) iscalled the workbench/sketch-pad, working band-width of the mind.  Working memory allows us to store useful bits of information for a few seconds and use that information across different brain areas to help solve problems, plan or make decisions.[1]Working memory has a very high correlation with learning, in general, and mathematics, in particular. Working memory skills are frequently utilized in almost every area of mathematics for holding information in the mind temporarily while simultaneously performing specific operations, in order to comprehend the problem, manipulate the information, and possibly produce a correct answer. 

Working memory is a key aspect of mathematical way of thinking (creating ideas, discerning patterns, seeing relationships, making connections, modeling ideas, etc.).Much of our mathematics learning depends on working memory. Think of the last time you followed a higher mathematics class. In the beginning, you might have kept up fine. But eventually it became harder and harder to understand what the teacher was saying. Even though you tried your best to pay attention, you left feeling confused and frustrated. The concepts and procedures being discussed required your working memory to process too much new and old information at the same time. As a result, the system became overwhelmed and broke down. This happens to our students, with and without poor working memory, in most of our mathematics classes.

One leading hypothesis contends that working memory works by far-flung brain areas firing synchronously. When two areas are on the same brain wavelength, communication is tight, and working memory functions seamlessly.  This is particularly important in mathematics concept/ procedures/problems. Most concept involves various parts of the brain concurrently: quantitative (counting, sequential procedural steps in left hemisphereand spatial information(e.g., integration of algebraic and geometrical concepts in right hemisphere)),discreteand continuous information (e.g., visual cluster and number line in learning number concept), linguisticand conceptualpart of a mathematics idea (e.g., solving a word problem). In such problems, there are higher demands made on the working memory to relate and integrate different components of the information.

Working memoryinvolves not only in receiving, retaining and manipulating mathematics information in auditory and visual processing, but also in monitoring attentionmental concentration(inhibition control), organization, andreasoning—all skills closely related to learning and achievement in mathematics. 

Researchers are trying to understand why this ability is poor in some children and fades as we age and whether we can improve it and slow, or reverse, that decline. Studies have examined the degree of overlap between executive functions and processing speed at different preschool age points; and (2) determine whether executive functions uniquely predicts children’s mathematics achievement after accounting for individual differences in processing speed.

For example, when a child with slow processing speed sees the letters that make up the word house, she may not immediately know what they say. She has to figure out what strategy to use to understand the meaning of the group of letters in front of her. It is not that she cannot read. It is just that a process that is quick and automatic for other kids her age takes longer and requires more effort for her. 

Similarly, when a student, with or without processing deficit) sees 8×7, she may not know what it is. She has to figure it out. Without the appropriate language and efficient strategies, she may take longer and can easily forget. Let us assume, she even had the language associated with it (8 groups of 7 or 8 sevens)—counting 8 groups of 7 takes a long time and possibility of error. In both cases, she may appear to have poor processing speed. 

When a student has poor processing speed, it is important to help student acquire easily accessible, and efficient strategies so that she can take action right away.  For example, when a student cannot recall a multiplication fact, she can decompose one of the factors (8×7), 7 in this case into 5 and 2 and then apply decomposition/recomposition (8×7 = 8×5 + 8×2 = 40 +16 = 56.). However, these strategies must be based on strong conceptual base and easy to use.

 Research demonstrates that age-related increases in processing speed result in increases in working memory as faster processing may facilitate the formation of connections between the current incoming information from short-term memory and resident information in the working and long-term memory spaces. The formation of more connections between different elements of the mathematics curricula at any grade level, in turn, results in age-related increases in working memory, processing, fluid reasoning and crystalized learning. 

B.  Processing Speed

It is a cognitive ability that could be defined as the time it takes a person to do a mental task. It is related to the speed in which a person can understand, react, and respond to the information they receive, whether it is visual (words, letters, numbers, symbols, representation), auditory(language—words, instructions, questions, expressions), orkinesthetic(psycho-motoric—touch, movement, concrete manipulation). Only through the response—the time it takes, the nature of the response one determines whether the slow response is due to organic reasons or it is lack of knowledge. 

Many times, deficits in processing speed and learning and attention issues coexist. Slow processing speed is not a learning or attention issue on its own. But it can contribute to learning and attention issues like ADHD, dyslexia, dyscalculia and auditory processing disorder. It can also impact executive functioning skills—working memory, inhibition control, organization, and flexible thinking.

However, having slow processing speed has nothing to do with how smart a student is—just how fast can he/she take in and use the information. 

For example, giving too many instructions with advanced and highly complex content with multiple directions may produce slower processing speed.  For example, the problem

In a collection of 4 consecutive even integerstwice the sum of first two consecutive even integers is 3 more than the sum of the last two consecutive integers.” 

contains several concepts, specialized language, and multiple steps. It is difficult to hold this information in the working memory to process the relationship between numbers and then relationship between the phrases. It is not uncommon such situations that a student in spite of having no processing speed deficit problems may have difficulty processing and expressing it. 

Whereas, when a student is given the problem: 

Find the least common multiple of 12 and 20.

First, the student will not respond for some time, as the answer is not forth coming, and then will respond: “I do not know.” When you insist that they try, most students, in this case, focus on “least” first and, then “common” and then “multiple,” therefore, they give a wrong answer.  

The wrong answer is neither because of poor processing, poor working memory, or any other cognitive deficits.  It is purely because of the language of mathematics. The order of words (syntax) of the phrase and their relationship are quite demanding.  Before introducing this concept and the procedure for finding it, I always give a parallel statement: 

John is an intelligent, handsome, tall boy.  

What is the relationship of these words with each other? Where do you start?  What do you understand? Students invariably say: 

“John is boy, then John is a tall boy, then John is a handsome tall boy, and then last: John is an intelligent, handsome, tall boy.”

Here, they can concretize it and can visualize the problem; therefore, they can process it.  

Processing Speedis the ability to quickly and correctly scan the information, discriminate visual and auditory components, receive it, and sequence tasks to be performed, perform the immediate relevant mental tasks, and communicate the received information in the appropriate form, contextually. At the lowest level, it involves short-term visual and auditory memory, attention, and visual motor coordination. It requires the student to plan and carry out some instructions given by the problem quickly and efficiently. Research shows that when reading ability is controlled for, arithmetic ability is best predicted by processing speed, with short-term memory accounting for no further unique variance.

Most mathematics tasks rely heavily on visual processing and in the initial stages of learning mathematics concepts (number concepts and number relationships) are dependent on fine-motor skills when children interact and use concrete materials and in the process develop language and conceptual schemas.  Later on, processing tasks involve receiving and representing the information visually—either as a diagram, writing equation(s), language and mathematics expression(s), and executing procedures. 

Students with poor fine motor skills processing problems do slightly better when there is less of a motor component required in learning and problem solving, whereas, the student with strength in processing in tactile/ kinesthetic modality can compensate for other processing deficits by relying on and using fine motor skills. 

Research suggests that we integrate both visual (non-verbal activity—watching lips and facial expressions) and auditory cues when processing speech. Similarly, in learning mathematics, multiple cues (language, visual representations, tactile, and auditory) are used in learning. Therefore, multisensory approaches to instruction have proven beneficial because such instruction provides more cues upon which to build a representation. 

The following examples demonstrate the importance of the integration of these phenomena in mathematics learning.  Let us consider few examples of such mathematics tasks, at different levels of the mathematics curriculum, to demonstrate the involvement of these processes.

  • Find the sum: 8 + 6

To find this sum, the student goes through a script (consciously and subconsciously) and asks himself: 

  • Do I know the answer? Searches the long-termmemory store for the answer. “No!”

The child with math anxiety or previous unsuccessful experiences gives up. Others try. 

  • Do I know how to find the answer? Searches long-termmemory for a strategy. “Yes!”

“I can use Cuisenaire rods.  Places brown (= 8) and dark green rods (= 6) end-to-end. It is more than ten. I place 10-rod parallel to the two rods.  I can place the 4-rod in the empty space and it becomes 10 +4.  So, 8 + 6 = 10.” 

  • Do I know any other way of finding the answer? “Yes!” 

“I can use Empty Number Line(ENL). I begin with 8 on the number line and 2 to make it 10. The 2 came from 6 so I add the remaining 4 to 10 to get 10. So, 8 + 6 = 10.” 

  • Can I do it without any materials?  “Yes!”

“The bigger number is 8. I will make it 10. I know how to make ten. I need 2 more to make 8 as 10. I will break 6 into 2 and 4. Then, I have 8 + 2 = 10, and then I add 10 + 4 = 14. So, 8 + 6 =14.” 

It should be noted that the strategies at all levels (Concrete, pictorial, visualization, and abstract) use a script and the script provides cues to hold and manipulate the information in the mind’s eye.  They provide the student actions to take rather than feel over-whelmed and feel defeated. The Cuisenaire rods, ENL, and decomposition/recomposition help in all aspects of learning.  

Most of this information and activity is being processed in the working memory. The child produces the result at each stage and communicates it to him/herself or to others. When this process is repeated, each time it leaves a trace of this action—as a residue in the long-term memory, and ultimately the fluency in constructing it is achieved. Then, the fact and the related strategy go to the long-term memory and get connected with other facts in the old “fact-file” and the “strategies file.” At the same time, a new “file” is also opened. This new fact is at the “constructed stage.” It needs reinforcement—reconstruction and practice. The presence of information in multiple files results in flexibility of thought. Then, the information is processed much faster.  Even the processing of new related facts gets easier. 

This process reinforces the previous knowledge (making ten, sight facts, teens’ facts, decomposition/ recomposition of number, following a sequence of steps, etc.) and also develops the stamina for constructing facts.  This also gives the student a possibility of trying other strategies.  

On the other hand, if the student derives the answer by counting:  “I start after 8 and count 6 numbers.  9-10-11-12-13-14.  8 + 6 = 14.” The student arrived at the answer. But, after finding the answer, the answer is forgotten.  No relationships are built; no patterns are discerned. The task is repeated next time when the sum of  8 + 6 is asked for. No reinforcement of working memory and long-term memory takes place. There is very mental  processing or flexibility of thought is developed. No mathematical way of thinking is developed or reinforced.  This kind of process does not help the student to become a better learner.  The deficit remains. 

In the absence of an efficient strategy, the student, because of poor strategy—such as counting, takes much longer and to the observer this delay looks like a case of slow processing. And the counting process does not leave any residue in the working memory, so no connections are made. 

Most times, the slow processing is misdiagnosed as a result of lack of knowledge both of the content and the absence of efficient strategies. It does not mean there are no cases of slow processing, but these are far fewer than reported.

Efficient strategies help develop and strengthen their ability in learning processes—memory system, processing, organization, and flexibility of thought.

To find this quotient, one decides to apply the process of executing standard long-division procedure.  Writes the problem in procedural form:

Learning long division procedure is important as once, learned properly, later in algebra, a similar problem—division of a polynomial by a binomial can be understood and executed. The division of a polynomial by a binomial

requires similar cognitive skills although with a different content. Processing and executing the long-division procedure involves: 

(a)Content retrieval

  • The steps of the procedure to be executed—divide, multiply, subtract, bring-down, repeatedly applied; 
  • Concepts —place value, multiplication, subtraction, spatial orientation/space organization
  • Language—how many groups, is it about right, less than, more than, what is left; and arithmetic facts from long-term memory; 

(b) Actions:

  • Holding and manipulating new and retrieved information in the working memory space; 
  • Integrating retrieved information with new information being generated in the process in the working memory; 
  • Expressing it orally (asking questions to execute the procedure—how many time does 21 divide into 45, etc.);  
  • Recording (in writing) – the outcome of these steps on paper; 
  • Making decisions, and, 
  • Evaluating the impact of these decisions on each step and on the outcome—is the product about right, does the answer make sense. 

Long division procedure/algorithm is a complex process and most students have difficulty learning, executing, and applying it, particularly, children with poor fact mastery. And, indirectly, those children with poor working memory and processing issues. The possibility of things going wrong is very high because of any of the memory, thinking, and processing deficits. 

All multi-step procedures, thus, place heavy demands on the working memory(to hold partial solutions, intermediate steps, and facts in the working memory), processing speed(several complex tasks to be executed fairly quickly and sequentially in order not to lose the information), flexibility of thought(multiple tasks to be performed presented in multiple forms, decisions to be made and evaluated for usage and efficiency). This creates problems for students with limited working memory capacityslow processing speed andrigidity in thought and action

Except the simplest of problems—involving one-step action or primary concepts, every other concept and similar issues affect procedure in arithmetic, geometry, and algebra. For example, solving multistep equations, writing proofs in geometry, solving a word problemwhere language processing and several concepts and operations involve the interaction of memory systems, processing speed, and executive functions. 

The underlying causes of the breakdown of memory systems and processing deficits are unclear.  However, some researchers suggest a slow articulation rate as a cause. 

Explanation: Slow articulation makes increased decay of information during recall and interaction. 

Others suggest limited space in the working memory.

Explanation:This prohibits a learner to hold the information in the working memory, thereby inability in manipulating the information. This, in turn, may prohibit formation of connections between the incoming information from the short-term memory and long-term memory. These preclude seeking, observing, and inability in extending patterns between quantities and shapes. That limits the ability for learning mathematics concepts. This is particularly so as mathematics is the study of patterns in quantity and shapes and their relationships. 

Others offer an explanation in terms of slow speed of item identification. 

Explanation:This creates difficulty in retrieving relevant information stored in long-term memory to the working memory. 

General processing speed is also related to measures of short-term memory. 

Another factor that is responsible in heightening the effects of deficits in working memory, slow processing speed is not new to learning, but has acquired higher significance.  It is the prevalence of distractions and shorter attention spans. They continue to be a growing challenge for students.

Explanation: Children, today, are exposed to a lot of technology — iPhones, iPads, and, therefore, higher degree of vicarious socialization. It is changing the architecture and the behavioral manifestation of the brain, so much so that applying themselves to a task that requires concentration is becoming very difficult.  It is part of the reason, that math-tutoring programs have become increasingly popular in the last decade. 

The key to math success includes teaching a critical combination of both understanding concepts along with improving working memory, processing speed, and flexibility of thought, on one hand, and drills and repetition exercises, on the other, to increase confidence and aptitude.

Children with arithmetic difficulties have problems specifically in automating basic arithmetic facts that may stem from a general speed-of-processing deficit.  Strategies such as counting and lack of decomposition/ recomposition may further slow down the processing speed. Moreover, the demands of understanding and meanings to be derived from mathematics terms, symbols, equations, and inequalities place heavier demands on the working memory, processing speed, and flexibility of thought than reading process and comprehension. That is one of the reasons, processing speed plays a very big role in mathematics achievement. 

Achieving mathematics understanding of concepts and procedures is more demanding than reading comprehension.  In the case of reading, some meaning and a level of comprehension can be derived just from context, but understanding of a mathematical idea cannot be derived just from context.   

Children’s information processing, particularly its speed is a driving mechanism in cognitive development that supports gains in executive processes—working memory, inhibitory control, organization, flexible thinking, and associated cognitive abilities. Accordingly, individual differences in early executive task performance and their relation to mathematics may reflect, at least in part, underlying variation in children’s processing speed and therefore achievement. Processing Speed, working Memory, and fluid Intelligence (Fry and Hale 1996) are highly correlated with each other and are essential elements for learning. Deficiency in anyone of these skills may have impact on learning. Slow processing speed may have impact on many of the mathematics processes. Slow processing speed is not a learning or attention issue on its own. 

Slow processing speed impacts learning mathematics at all stages. It can make it harder for young children to master the basics of mathematics language, writing numbers and symbols, understanding number and related numbersense. For example, let us consider, finding and learning the multiplication fact: 8×7= ?

In the absence of automatic recall, a student plans to construct it using a strategy based on decomposition/recomposition, The student decides to break the problem as: 8 ×7 = 8 ×5 + 8 ×2 (because he knows the sight facts of number 7 and the distributive property of multiplication over addition) = 40 + 16 (he knows the tables of 5 and 2) = 40 + 10 + 6 (he knows his teen’s numbers) = 50 + 6 = 56 (he knows his place value of two-digit numbers). 

As one can see, by task analysis, one can determine the component primary concepts: recognition of related sight facts (including the sight facts of 10), decomposition/recomposition of numbers, decomposition/ recomposition of teens’ numbers, place value of numbers, and tables of 5 and 2. Mastery of these component skills makes finding and mastering multiplication facts/tables is easier. Slow processing, in such a situation does not debilitate student progress in learning arithmetic facts. The same process works in intervention/remediation situations with older students. Using the same efficient strategies, they can acquire the ability to master arithmetic facts quickly and accurately. 

There is research and training efforts to improve auditory and visual processing to mitigate the deficits as related to reading and language development. However, none exist to minimize the impact of processing deficits for improving mathematics learning. We believe that first we need to differentiate the poor performance in mathematics due to poor processing or poor strategies. We should differentiate between appearance of poor processing and true deficit in processing. 

We have been working with children with and without learning disabilities and some with processing deficits and the choice of instructional materials, questioning processes, and training in visualization. We have observed that improvement in strategies helps children improve their processing speed, working memory, and flexibility of thinking. These, in turn, also have impact on executive function that is essential for learning mathematics. 

Training in executive function and processing does have impact in all areas of learning. For example, studies show that infant attention skills are significantly related to preschool executive function at age three and even later. Higher attention span in infancy may serve as an early marker of later executive function, processing speed, and attention to learning. 

Two very powerful tools, we have used to improve working memory and processing speed are: Efficient and effective Concrete/pictorialmodels, and developing scriptsfor streamlining tasks.  Concrete models should be selected with following characteristics in mind: color, shape, size, pattern, and generalizability.  Similarly, understanding the trajectory of the development of a concept and the related task analysis must develop scripts for task implementation. 

  • Auditory Processing Speed

Auditory process is making sense of information received from the auditory channels. When there is delay or deficit in processing this information it is called Auditory Processing Disorder (APD).  It is also called Central Auditory Processing Disorder (CAPD), and Specific Learning Disability/ Disorder with impairment in listening. It is quite common in Dyslexia and dyscalculia

Increasing evidence suggests that some children with developmental dyslexia and/or dyscalculia exhibit a deficit not only at the segmental level of phonological processing but also, by extension, at the supra-segmental level. Thus these children when confronted with mathematics language in the context of conceptual development and word problems exhibit some of the difficulties related to processing speed. 

The APD brain requires 2 to 5 times longer registering a speech sound. This is particularly so when the context and the complex and specialized vocabulary creates even more problems. The normal rate of speech is too fast for the APD brain to perceive and process all of the information heard. The result is a person who appears to have a long delay between what they hear and their response to it.  When they do respond, the response may be inappropriate or may clearly indicate that they did not comprehend the information heard. They cannot accurately repeat auditory information. Parts are missing. It is not effective to give them spoken instructions because they require lots of repetition and redirection. Nor, it is not enough to by-pass the auditory channel altogether.  It should be multi-sensory. For example, in a word-problem situation, ask the students to read the problem out loud and supplement it by asking a great deal of scaffolded probing questions.  Create a script and ask them to repeat the script as apply the instructions from the script. 

In reading fluently, processing speed plays a keyrole. The goal of improving reading rate and fluency is to positively impact reading comprehension; however, it is unclear how fast students with learning disabilities (LD) need to read to reap this benefit. There is a point of diminishing return for students who are dysfluent readers. In other words, it is important to determine where the linear relation between reading rate and comprehension breaks down for LD students: the rate at which getting faster no longer contributes clearly to reading comprehension improvement. For dysfluent readers, improving reading rate improves comprehension only in the bands between 35 and 75 words correct per minute in second grade and between 40 and 90 words correct in fourth grade. Reading at faster rates reveals no clear advantage for reading comprehension, beyond that point. 

In mathematics the role of processing speed is more complex; more demanding than in acquiring language and reading skills. For example, demands made on fluency while reading a word problem is sufficient about what one reads in the band between 50 and 90 words.  However, because of specialized vocabulary and syntax, skills needed to understand the content of word problems, the level of reading comprehension, is much higher.  Similarly, when it comes to constructing or producing a fact or a formula, and executing a standard procedure, we need speed, therefore, a higher level of processing speed is needed. 

On the other hand, when we are solving a problem we are not necessarily looking for speed, there we need to discern and discover patterns and relationships. For that, we need strong working memory and fluidity of thought. Mathematics learning (integration of language, concepts, and procedures), therefore, is a partnership between processing speed and working memory.  These two together give us fluid intelligence or flexibility of thought.  

Mastering arithmetic facts, formulas, and sequential steps involved in executing mathematics procedures calls for high level of processing speed, e.g., What are the factors of 48? It calls for knowing the multiplication facts that result in 48. It calls for automatization of multiplication facts. In the absence of this facility, one should know certain pre-skills: (a) divisibility tests, (b) short division, and (c) place value. 

The divisibility tests of 2 and 3 (the first prime number is 2 and it divides 48 as it is even number; 3 divides 48 as the sum of its digits is 12; 4 divides 48 as 40 is made up of 40 and 8, and both are divisible by 4; since 2 and 3 divide 48, therefore, 6 divides 48. And, since, the only number between 6 and 8 is 7, and that does not divide 48.  We have found all the factors of 48. 

Knowing the short division process facilitates the process of dividing by 2, 3, 4, and 6. In addition to this, one should know the organization of these numbers on the number line as follows.

48:  1, 2, 3, 4, 6,  8,      12,    16,     24,                       48.

Writing the factors in their appropriate place on the number line shows that there are no factors of 48 between 24 and 48; between 16 and 24, between 8 and 12, and the only number between 6 and 8 is 7 and that does not divide 48. It also is very convenient when we are looking for the greatest common factors (GCF) of two numbers. For example, to find the greatest common factor of 48 and 40, we have:

Once, can easily notice the common factors {1, 2, 4,and 8} and the greatest common factor (8). 

Another effective strategy for finding GCF and least common multiple (LCM) of two numbers is the use of prime factorization.

GCF (48 and 40) = 2×2×2 = 8 (the product of the prime factors in the first column)

LCM (48 and 40)= 2×2×2×6×5 = 240. (the product of factors in the left-most column and the last row. As one can see, LCM has both 48 and 40 as its factors, it is a multiple of both, and it is the least such multiple.

These efficient strategies and visual representation aid in discerning patterns, making connections, improving processing speed and the working memory. Therefore, development of mathematical way of thinking. With enough practice of efficient strategies and developed mathematical way of thinking, one can make up for slow processing speed and working memory deficits and in the process improve both of them. 

In the absence of appropriate processing speed one may not arrive at a fact easily and efficiently. A student may take two hours to do math homework that takes others only 20 minutes. This means that the studentoften does poorly on tests even though she may know the material. The student knows the procedures, but is not able to follow the multi-step directions, suchas executing the long-division algorithm, solving a system of simultaneous linear equations, adding two fractions with different denominators, and applying the order of operations in a complex numerical or algebraic expression. This is especially so when there is not much time to get the task done? 

While there are many possible reasons for these struggles, slow processing speed may be a factor. Having slow processing speed has nothing to do with how smart kids are—just how fast they can take in and use information. It may take kids who struggle with processing speed a lot longer than other kids to perform tasks, both school-related and in daily life. 

Lack of ability, fluency in decomposing and recomposing numbers, and poor processing speed interfere with learning number relationships. 

Comprehending mathematics concept involves integrating three actions—recognizing grapheme (the orthographic mathematics symbols), the idea it represents (concept), and the meaning and action associated with it (knowing the associated skill or procedure). For example, recognizing a cluster of objects (quantity—5 objects in a cluster), giving it a number name (numberness—five), and writing its orthographic image (5) results in number concept. Again, practice with instructional materials such as: Visual Cluster cards, Dominos, dice, Cuisenaire rods, TenFrames, etc. facilitate this integrative process and increase processing speed. 

Processing, integrating, and communicating these three items (whether in written, oral, or action forms) takes time, thus affecting learning. Many teachers, particularly interventionists/special educators, with good intensions of helping students and aware of their deficits, try to short change it by giving “cooked” procedures with short cuts. Many interventionists seek simplistic instructional/interventional programs that lack this integrative process. In the math education and special education circles this has created the “Math wars”—whether it is important to give students just simplistic mathematics procedures or they should have the conceptual understanding and leave the procedures to technology-assisted methods. 

This clash in opinion about mathematics instruction is most often expressed as a divide between “back-to-basics” and “reform ordiscovery math.”  Advocates of back-to-basics tend to think of math as a subject that is fact-focused, with emphasis on memorization of these facts, fluency and speed, teacher-centered and test-heavy. These are precisely opposite descriptions to learning concepts and procedures with understanding and applications. That approach is concept-driven, learner-centered, problem-rich, involves explorations and experimentation, and emphasizes learning generalizable strategies.  Just like we settled a similar debate in reading education by recognizing that decoding/encoding (fluency) and comprehension are both necessary elements in a “balanced” approach to teaching reading. 

Fortunately, the demands of a technological knowledge-based society call for the two sides to be closer to each other.  There are signs of that. On one side, few educators would now say that learning math is just a matter of memorizing facts and rules, similarly, most educators now recognize that mastery of basic facts and rules facilitates higher-order thinking and seeing patterns and connections. On this count, for the most part, both sides have compatible intentions. 

However, challenges arise when discussions shift from what to teach to how to teach children, with or without learning disabilities. This is where the debate between “direct instruction” and “inquiry learning” continues. Educators are often asked to choose between teacher-driven explanations of isolated topics and learner-driven explorations of whole concepts in rich settings.

Each side seems to have a compelling argument for its view. Proponents of direct instruction assert that mathematics is well defined and unambiguous, and so it should be delivered efficiently and with fidelity. They argue that it is ridiculous to expect high school students to “discover” concepts that eluded all but the best-prepared minds until quite recently. Reform advocates counter by noting that learning is not about “acquiring” objects of knowledge. Understanding cannot pass from teachers’ instructions to learners’ minds. Rather, learning is about building understandings from personal experiences. For instance, when asked for a quick definition of “number,” most people respond in terms of counting. That interpretation is limiting, and learners who haven’t incorporated additional meanings (for example, size, distance, location) are disadvantaged as early as upper elementary. Or, when they see multiplication only as “groups of,” “repeated subtraction,” or “an array.” Each of one of these linguistic and conceptual models is limited in scope. The “repeated subtraction” model does not work if both factors are fractions.  The “groups of” model does not work when dealing with multiplication of algebraic numbers and expressions.  The array model does not work when the factors are not discrete quantities. Whereas, when students by the end of third grade are not shown that all of these models can be generalized into “the area of a rectangle” model, they have difficulty conceptualizing multiplication of fractions, decimals, and algebraic expressions, etc.  

The raw materials for enriched understandings are found in personal action and interpretation of experiences and when the experiences are diverse, multi-faceted, and effective. It is the role of the teacher to provide opportunities for these rich interactions with language, concepts, materials, and human resources. That is why reform advocates contend that presenting mathematics as a purified or standardized form of knowledge risks making it meaningless while suppressing curiosity and motivation.

Mathematical knowledge is not an “object” or collection of facts that a teacher can simply hand off to students, but neither is it a web of concepts that are self-evident or inherently embedded in experience; mathematics is a complex, evolving combination of what we might call “principles” and “logics.” Mathematics knowledge, mathematics understnading, and mathematical way of thinking has to be constructed and practiced. Teachers, effective teaching, and efficient teaching resources are the bridging processes between experiences and the outcomes—mathematics way of thinking, mathematics content, and process outcomes in attitudes and interest. There are four prspectives and approaches to learning mathematics, where we combine basic principles into more abstract and more powerful principles using combinations of these perspectives: 

  • Through concrete experimentation, working on projects, and problem solving, we observe, develop, and relationships between ideas. Concrete experimentation helps build visaulization–a key ingredient in strengthening working memory and processing.
  • By discerning and extending patterns and regularities, we make conjectures. Conjectures invite us to establish them in to elements of mathematical way of thinking. They build intuition and store of mathematics content knowledge.
  • By reasoning analogically (such as noticing that different physical processes, such as movement in a straight line, elapsed time and growth can all be interpreted in terms of addition), we connect ideas and understanding. Using analogous thinking from multiple settings, we see realtionships and willingness to explore. Processes, such as analogies, metaphores, and similies build understanding and comprehension.
  • Using various logic—deductive and inductive reasoning, through sequenced chains of argument (such as, if a < b, and b< c, then a< c; one is less than two and two is less than three; so one must be less than three).

An effective teacher will offer exercises and activities that channel learners’ attentions to relevant principles and encourage appropriate use of different logics by systematically juxtaposing clusters of such experiences.

We need to use most effective and appropriate approaches to teaching mathematics to all children, with and without learning disabilities. An analogy might be drawn from research into how people move from novice to mastery stage; what actions and processes are involved. For example, how people learn to play chess at competitive levels. Intuition might suggest that playing game after game is key, but cognitive scientists have found players advance more quickly by doing exercises that offer incremental challenges but that do not overwhelm working memory. The tasks that are within their capacity and yet moderately challenging. Tasks that build new insights, skills, and stamina. Both automaticity and strategy can be effectively developed through, for instance, playing a mini-game with just a few pieces, analyzing a single position in-depth and studying sequences of moves by master players. Such conclusions aren’t specific to chess. Research on expert performance across domains consistently reveals that people learn best when they are engaged in a way that doesn’t exceed the limits of their working memories and are exposed to moderately challenging experiences. The task must fall in the Zone of Proximal Development (ZPD). This represents a very specific zone of difficulty, which looks different for every student.

Specifically, the ZPD is the area between comfort and frustration — a student has not completely mastered the material yet, nor are they frustrated with its difficulty. The teacher with her own skills, insights, and adaptive technology helps each student accesses this key area, maximizing learning efficiency and continuing that process. Of course, choice of instructional materials is the key here. Even during the time, the studnet is using the adaptive technology, the teacher must ask a great deal of questions to make sure the student is understanding the language and concepts behind the tasks they are involved with; pushing buttons, midlessly is not the role of adaptive technology.

Such research highlights that each side of the math wars is correct about something. You cannot have one without the other. Math is not just a collection of facts just to be memorized; some of it has to be memorized by doing it, using it. It is a complex system. Each part of it is related to other parts of the system. Skills and concepts in algebra and geometry give rise to coordinate geometry, trigonometry, calculus, and probability and statistics. Mathematics understanding is constructed. Mathematics is the study of patterns in quantity and spatial relationships and their integrations. Mathematics needs to be presented in a curriculum that is less cluttered, yet connected and more clear, but meaningful. We need to emphasize “non-negotiable concepts and skills” at each grade and age level for all children.[2]

Traditionalists are right to argue for carefully structured learning situations and ample supervised and independent practice. And reformists are right to insist that powerful learning is more likely when practice is embedded in rich and personally meaningful situations and efficient strategies. Together, these insights point to the need for fine-grained analyses of mathematical concepts, skilled design of learning situations to help learners notice and make connections. It calls for constant awareness of students’ evolving understanding, skills, attitudes, and flexibility to adapt teaching approaches depending on the content and how students are progressing. At the same time, we should be cognizant of the learning differences and their nature and how that impact learning.

Improved model of teaching for students having problems in learning mathematics not just with dyscalculia and/or dyslexia, but also children with processing speed can be built on math learning that focuses on engagement, when teachers check in with students regularly to ensure they understand and provide appropriate support and practice. In this method the teacher presents only one concept at a time, only enough that a student’s working memory can handle and ensure every student understands before they move on. And, then make sure appropriate connections with other concepts, procedures, and processes are made.

This involves: (a) selecting a key concept to learn; (b) task analysis for unwinding the fine details of the concept (taking it apart with students and then putting it back together again); (c) engaging students in effective concrete models and strategies for learning the concept; (d) helping students observe patterns and make conjectures; (e) plan and execute supervised and supported practice; (f) provide success at each junction; (f) make conncetions with other cocnepts and procedures–both vertically and horizontally, for example, how multiplication manifests itself at different grade levels and how each model of multiplicaiton is related with different modesl of division; and, (g) check in with students or “notice” whether they understand the relationship between the strategies and their success.  This process helps students become better learners and acquire meaningful and appropriate mathematics content.

[1]  See several posts on this blog about working memory and mathematics learning.

[2]  See posts on Non-negotiable Mathematics Skillsat Different Grade levels in this Blog. 







Professor Emeritus and Past-President, Cambridge College

President, Center for Teaching/Learning of Mathematics

Framingham, MA 

April 12, 2019

Organized by

Continuing Professional Education 

Framingham State University

Framingham, MA

Mathematics As a Second Language

God wrote the universe in the language of mathematics Galileo

Mathematics is truly a second language for all children. It is a complex language. It has its own alphabet, symbols, vocabulary, syntax, and grammar. It has its own structure.  Numeric and operational symbols are its alphabet; number and symbol combinations are its words; and equations and mathematical expressions are the sentences of this language. 

Acquiring proficiency in mathematics and solving mathematics word problems means learning this language well. For all grade levels

This workshop will cover:

  • Building a Mathematics Vocabulary
  • Use of Syntax and Voice in Mathematics Language
  • The Role of Language in Developing Conceptual Schemas
  • Developing Translation Ability—Solving Word Problems
  • The Role of Questions in Acquiring Mathematics Language
  • Understanding Instructions to Mathematics Problems
  • Teaching Mathematics Language

In studying the nature of mathematics disabilities, professionals have looked at the problem from several different perspectives. Psychologists and neuro-psychologists have examined the nature of mathematics learning problems from the perspective of the learner characteristics that relate to the underlying processes and mechanisms involved in mathematics skills.  Special educators and teachers have looked at the problem from the perspective of making modifications in teaching based on the nature of the disability. On the other hand, mathematics educators have looked for causes of children’s difficulties by focusing on the nature of the mathematics content itself. To improve mathematics instruction and learning and to address children’s difficulties in learning and achieving in mathematics, we need to consider and integrate what we know from the following: 

  • Thenature of mathematics content and learning
  • The learner characteristics and skill sets,
  • The teaching models those are effective for particular content and skills for particular groups of children.

A considerable amount of research in the area of mathematics learning disability has been conducted by extending the hypotheses and results from the field of reading and reading disability.  The reason for this is found in the key similarities that exist between reading and mathematics learning processes.  In particular, the similarities are in

         a) Symbols—in both reading and mathematics, children must recognize and comprehend the message being conveyed by the words or numerical and operation symbols;

         b) Vocabulary—for children to understand the symbols in both reading and mathematics, it is necessary to learn the vocabulary associated with each area.  In effect, they must develop the language whether it is mathematical language or native language;

         c) Skill hierarchy—both reading and mathematics contain hierarchies of skills.  Children show progress to attain mastery in these skills through several intermediate stages: intuitive, concrete, pictorial, abstract, applications, and communication (Sharma, 1979a);

         d) Readiness for learning—both reading and computation skills have prerequisite skills.  Readiness for certain mathematical skills can be viewed from at least four theoretical points of view—cognitive, neurological, instructional, and ontological;

         e) Sensory processes—since children must rely on input in both mathematics and reading, it is important that one pays attention to the modalities (visual, auditory, and tactile) of information processing;

         f) Decoding—decoding and encoding processes are crucial to a child’s success in mathematics and reading. When children encounter an event involving numerical quantities or spatial information, they must be able to express (decode) these numerical and spatial relationships in symbolic notations; and

         g) Comprehension—as in reading, the four factors—word recognition, comprehension, rate and accuracy—are important, similarly, in mathematics, the four factors of basic facts – recognition, comprehension, rate and accuracy – constitute good computational ability. 

         These and other similarities between reading and mathematics have given rise to comparable postulates in both disciplines. The common elements can help us understand some of the issues in mathematics disabilities.

         However, to assume that the same processes underlie the acquisition of reading skills and learning mathematics concepts, skills, and processes would be erroneous. There are many substantial differences. The differences lie in the nature of mathematics content, concepts, and in the diversity of skills and procedures involved in learning mathematics. To understand the nature of learning problems in mathematics means understanding the nature of these processes.

  • Linguistic component, 
  • Conceptual component, and 
  • Procedural component



MATHEMATICS         —->                           ENGLISH

(Forming number stories from mathematics expressions.)

ENGLISH                     —->                        MATHEMATICS

(Translating word problems into mathematical expressions.)

In relating any two languages, it is important to understand the interplay of three elements: 

  1. Vocabulary{words, terms, and phrases, such as: multiplication, product, sum, quotients, least common denominator, rational number, proportional reasoning, etc.} and Symbols {= -, x, +, %, ( ), etc.} 

To develop vocabulary, the teacher should, on one side of the board develop and define, by the help of students, terms when talking about vocabulary: For example:  Product

Product:The outcome/result of a multiplication operation. What is the product of 3 and 4? This is a mathematical sentence written in English language.  Now, students translate it into mathematics symbol form: 3×4, 3Ÿ4, 3(4), (3)4, (3)(4).

Sum: The outcome/result of an addition operation.  What is the sum of 2 and 4? 2 + 4.  

Rational number: What is a rational number?  

rational numberis a numberthat can be written as a ratio: a/b, where aand bare integers(a whole number, its opposite, and 0), b ≠ 0, and and bare relatively prime(the greatest common factorof aand bis 1). 

Note 1: The number of terms involved in the definition of rational numbers. The concept of rational numbers will not be understood if the phrases and terms involved in the definition are not understood and mastered. 

Note 2: The words, terms, and expressions that connote the mathematics ideas behind them do not help children to remember the mathematics idea. The words, terms, expressions, and symbols are language containers for concepts.  

  • Syntax: The order of wordsor order of operationsused in mathematical expressions (e.g., difference of a and b (a –b) is not the same as difference of b and a (b – a)).  

Note: The syntax in mathematics language is governed by strict rules.  

For example, 

How would you read?  27 ÷ 3

The expression is written as “3, procedural division operation sign, and then 27,” but is read as “27 divided by 3,” “3 divides into 27,”  “how many groups of 3 are there in 27?” “How many 3s can fit into 27?” 

Many students misread and translate division problems incorrectly.

 can also be written as: 27 ÷ 3 and is read as: “27 divided by 3” and as a quotient, it will be read as: “twenty-seven thirds.” And, as a ratio, it will be read as: “ratio and 27 and 3.”

  • Translation: To conceptualize mathematics ideas and solving word problems, students need competence in “two-way” translation: English to Mathematics as well as Mathematics to English. In learning and teaching mathematics we are constantly moving from one language to the other. For example: If we have:  17 − 9 = ?  How can we write this as an English sentence? Teacher should generate one or two statements and then, ask students to write more statements with their partners. 

(a) 17 subtract 9 equals what? 

Let students work together and then share out. Record them on the board. Discuss them and then reorganize them according to the pattern. For example:

(b) 17 minus 9 equals what?  

(c) 17 take away 9 equals what?

(d) 17 decreased by 9 equals what?

Here the syntax is direct: number, operation, number.

(e)If 9 is taken away from 17, what do you have? 

(f) 9 less than 17 is what?

(g) Subtract 9 from 17 is what? Or, What is when we subtract 9 from 17?

Here the syntax is indirect: the order of numbers is changed without changing the meaning.  The placement of the question does not change the syntax of mathematics operation. 

(h) What is the difference between 17 and 9?

(i)17 is how much more than 9? 

(j) 9 is how much less than 17?  

Here the syntax is more complex than the other two patterns. It is the beginning of algebraic reasoning.

Similarly, the teacher should spend some time writing statements for addition, multiplication, and division expressions.

Now let’s do the reverse translation – let’s go from English language to mathematical expressions.

Example: 8 decreased by 5 translates into 8 −5 as an arithmetical expression.

Example: 9 increased by 7 translates to 9 + 7.

Example: 12 increased by 6 translates to 12 + 6

At this point, the teacher should define terms such as: Arithmetical expression, numerical expression, algebraic expression, equation, inequality,etc.  On one side of the board, with the help of students, develop and define:  Arithmetical expressionis a mathematical expression that involves numbers, arithmetic operations, and arithmetic symbols.  It becomes an algebraic expressionwhen the expression involves numbers, operations, symbols and variables.

Have students try the next few with a partner and then share back out to the class.  Collect them and then discuss the relationship between words and mathematics counter parts. 

Now consider, 

Example: 8 less than 15 translates to 15 –8.  

This one might be hard for some students to see. Devote some time on the order of words.  Relate it back to the list previously made that showed the different orders and wordings.  Also bring attention to which number is being operated on, and which number is being operated at by what operation. In the expression: “8 less than 15” 15 (operand) is being operated on by the number 8 (operator) and the operationis subtraction. So, we begin with the 15 (operand, it will come first) and the 8 is the operator so 8 is subtracted from 15.

Example: 2 more than 32 translates into 23 + 2.  Here: 32 (operand), 2 (operator), and addition (operation). 

Example:15 decreased by xtranslates into 15− x.  

This expression introduces a variable and therefore becomes an Algebraic Expression. A variable (we defined already) is a symbol that may take different values or roles according to the situation/context. Now the teacher, shuld begin with some concrete examples. For example, a concrete example of a variable that might help students is:

Who sits in this seat?  (Point to a student chair.)  

Today it is student J’s seat, but next week it might be student M and next month it might be student A.  The person to whom this seat belongs varies on the day or week or month. 

However, who sits in this seat? Point to the teacher’s chair. Only the teacher sits on it.  No one else is allowed to sit in that chair. Therefore, this chair is not variable but constant.

Once the idea of variables and the translation process is understood, one can introduce the concept of equations. 

Example:  Write, “5 more than 2x is decreased by 3 less than twice a number” into a mathematics expression.

This is an example of a compound expression.  Two algebraic expressions are connected by the phrase “decreased by.”  Let us take each expression separately.  The expression “5 more than 2x” is: 2x + 5, and the expression “3 less than twice a number” is: 2y 3, where yis the number. Combining the two, we have: (2x + 5) – (2y – 3).  

         An equation is also a compound statement.  It joins two algebraic expressions.  A mathematics expression is called an equation when two algebraic expressions are equated. 

II.  Language Difficulties, Dyslexia, and Mathematics Learning

The nature of reading and language disability sheds light on a child’s mathematics learning. Some children are late in learning to read and have persistent difficulty in remembering spellings in contexts without any issues of intelligence or lack of opportunity to practice.  Some of them experience other difficulties also—long struggles over learning to tell time (on non-digital clock), uncertainty over left and right, confusion over times and dates, inability to recite without stumbling, any but the easiest arithmetic tables. These issues spill over into the area of mathematics. 

A.  Dyslexia

A child of average ability who is late at learning to read and has special difficulty over spelling, and if, in addition, shows confusion over getting things into spatial and temporal order, whether in language or arithmetic, has a distinctive disability called dyslexia. A person with this disability (or group of disabilities) has difficulty in receiving, comprehending, and/or producing language.

         Dyslexia, which affects three to four boys to every girl, has been associated with slow speech development, speech and language difficulties, delay in motor development, sequencing problems (usually the inability to remember the days of the week or the months of year is noticeable), impairments in temporal or spatial awareness, and visual perceptual deficits.

         Most dyslexics are late in learning to read and have considerable difficulty in learning to spell.  Most dyslexics remain slow readers, and although some speeding up is possible, any task, which calls for the processing of symbolic material or speed, is likely to cause them trouble. Dyslexics have difficulty with phonology—with the remembering and ordering of speech sounds.  Problems over left and right also persist in their work.

         Learning mathematics for a dyslexic is just like learning Latin as a second language (L2). Since, it is a dead language, it is mostly learned by writing and in written form. It is not orally supported, as it is not spoken outside the classroom.  Research has shown that individuals with dyslexia have difficulty in reading not only in their native language but also in a second language (L2). The considered L2, however, has always been a language acquired through exposure to both written and oral forms. A recent study examined the case of Italian adolescents reading in Latin as an L2, which is the special case of a dead language with very limited use of orality. As the learning of Latin is mainly based on the acquisition of grammar, this study also examined the relationship between grammatical proficiency and reading ability in Latin. Results suggested that, compared with control peers, students with dyslexia had difficulty in reading words and non-words in Latin. Interestingly, in spite of Latin being learnt mainly through written language, the extent of their difficulty was no larger than they encountered when reading in their native language. Also, despite the fact that students with dyslexia showed relatively less severe difficulties with Latin grammar (as compared to reading), this did not support them when reading Latin words, unlike typical readers. The theoretical and educational implications of these results of this study are profound when it comes to learning mathematics as a second language. 

         Mathematics calls for many different and some similar kinds of abilities. In general, dyslexics tend to be slow at certain basic aspects of mathematics—learning and recalling arithmetic facts, particularly multiplication tables, adding up columns of figures, etc. –but once they have understood the symbols, they may be quite creative. At the same time, based on their profile, some dyslexics can be quite strong in certain aspects of mathematics.  It is important to distinguish dyslexic children form slow learning children. 

         For individuals with dyslexia, learning mathematical concepts and vocabulary and the ability to process and use mathematical symbols can be impeded by problems similar to those that interfered with their acquisition of the written language. Too frequently and too readily, individuals with dyslexia who have difficulty with mathematics are misdiagnosed as having dyscalculia – literally trouble with numberness, knowing number relationships, calculating, and a neurologically based disability. Around 40 percent of dyslexics have difficulty with basic mathematics. Some dyslexics are only numerically dyslexic—having difficulty only in the numerical aspects of mathematics, but this can also be most embarrassing.

         Difficulties with math for dyslexics can be identified by the following symptoms:

  • The dyslexic may have a problem with numbers and calculations involving adding, subtracting, and timetables.
  • He may be confused by similar—looking mathematical signs: + and ×; –, :, ¸and = ; < (less than) and > (greater than).
  • He may not grasp that the words ‘difference’, ‘reduce’ ‘take away’ and ‘minus’ all suggest ‘subtraction’.
  • He may understand the term ‘adding’, yet be confused if asked to ‘find the total or sum’.
  • The dyslexic may reverse numbers, and read or write 17 for 71, or 2/3 as 3/2.
  • He may transpose numbers i.e., 752 to 572 or some other arrangement of digits.
  • He may have a difficulty with mental arithmetic.
  • He may have a problem with telling the time.

         Individuals with dyslexia may have problems with the language of mathematics and the concepts associated with it. These include spatial and quantitative references such as before, after, between, one more than, and one less than. Mathematical terms such as numerator and denominator, prime numbers and prime factors, and carrying and borrowing may also be challenging.

         In many ways, some of the symptoms of dyscalculia closely parallel the behaviors exhibited by students with language dysfunctions; therefore, many neurologists believe that dyscalculia does not exist as a separate dysfunction but is a manifestation of a brain lesion which is causing language and mathematics dysfunctions simultaneously. This view has some relevance, as mathematics is also a language—the language of quantity and space. Some mathematics problems are extensions of language difficulties such as alexia—an inability to read, and/or agraphia—an inability to write.  For example, half of the students with difficulties in mathematics also have difficulties in spelling. 

         The terms dyscalculia and acalculia have been used in the literature interchangeably.  Dyscalculia is the lack of or delay in the development of numberness, number relationships, calculations and other related mathematics difficulties, whereas acalculia is the loss of these abilities because of insult or injury to some specific part or regions of the brain. We will use the term dyscalculia referring to both, except when the problem relates to pure dyscalculia—difficulty with numberness, number relationships and outcome of numerical operations. 

         In contrast to what neurologists think, several recent researchers have argued that dyscalculia should be considered as separate from language related problems and different from the more general category of learning disabilities and even from learning disabilities in mathematics. In other words, dyscalculia or acalculia can exist independently of all other learning problems in mathematics and language.  That is, a number of individuals may manifest dyscalculia and no other learning disabilities.  Some others may exhibit difficulty in mathematics and even dyscalculia or acalculia in the presence of dyslexia. 

         Not all dyslexics have problems with mathematics, and not all dyscalculics have difficulty with reading and other language skills. Of course, there are many differences based on strengths and weaknesses of dyslexics and dyscalculics. Pure dyscalculia (the difficulty in conceptualizing number, number relationships, and outcome of numerical operations) and dyslexia are independent. Dyscalculia and dyslexia mutually influence each other when we consider dyscalculia as poor performance in mathematics including the failure of the number and calculation mechanisms. For many dyslexics, the difficulties that affect their reading and spelling also cause problems with mathematics. 

         There is some correlational evidence emerging between the coincidence of dyslexia and dyscalculia. The International Dyslexia Association has suggested that more than 40% of dyslexics have some difficulty with numbers or number relationships. In several studies, they found that almost 51% of dyscalculics also show signs of dyslexia. Of those whom do not have mathematics difficulties, about 11% excelled in mathematics.  The remaining have the same mathematical abilities as those who do not have learning difficulties.         

         Arithmetic and mathematics learning deficits can be caused by a variety of factors, sometimes because of reading difficulties and at other times with no connection to reading difficulties. Children with specific arithmetic difficulties and children with combined arithmetic-and-reading difficulties represent two different underachievement subtypes whose problems may be underpinned by qualitatively different cognitive and neuropsychological deficits. This is consistent with evidence from clinical and experimental studies suggesting that children of normal intelligence who experience difficulties with arithmetic can be divided into specific and combined subtypes. 

         The existence of a group of arithmetic-and-reading difficulty individuals shows that some arithmetic difficulties result from difficulties with reading. On the other hand, a number of children with normal reading scores obtain low scores on the arithmetic test, which shows that not all arithmetic difficulties can be attributed to a general deficit in language-related processing.  In addition, there exists a substantial group of poor readers of normal intelligence whose performance on the arithmetic test is higher. Therefore, mathematics learning problems in general and even the types of problems exhibited by dyscalculics, acalculics, and or dyslexics are not homogeneous in nature.  

         Because of the heterogeneous nature of mathematics learning problems, language related problems in mathematics fall into three major domains:  

  1. Mathematics problems related to and originating from language processing difficulties, 
  2. Mathematics problems that have the same basis as the reading problems because of the underlying learning mechanism, such as sequencing, visual-perceptual integration, working memory, organization, spatial orientation, etc. 
  3. Mathematics problems that originate from the combination of language and reasoning deficits. 

III. Mathematics Problems Related to Language 

Since formal and informal language plays varying roles in learning concepts and applying mathematics, some of the difficulties emanate from the interaction of the systems responsible for number, calculations, procedures, and language. In the case of language, it may be vocabulary, syntax, the ability to translate from mathematics to language and from language to mathematics, and reading.

         Many children with language related problems do not have problems in mathematics. However, since some of the same prerequisite skills are involved in both language acquisition and mathematics learning–at least in the early years, the coincidence of dyslexia and dyscalculia is not uncommon.  Many dyslexics can solve computational and spatial problems easily. They have difficulty with only language related problems in mathematics as they do not have the facility to receive, comprehend, and produce the quantitative and spatial language (words, symbols and expressions) properly. They are not able to solve problems that have heavy language involvement. Research studies have shown that many elementary-age students who perform poorly in mathematics also have basic language deficits.

         Dyslexics’ language related problems in mathematics are of two kinds: primary and secondary.  Primary problems are directly contributed to mathematics by language difficulties in reading, spelling, etc.  Dyslexic children sometimes also manifest problems in arithmetic that are of a secondary nature.  That is, since the dyslexic child is provided extra instructional experiences in reading and language areas, his experiences in and exposure to arithmetic may be limited. As a result, he may begin to do poorly in arithmetic. Consequently, in many cases, once the child’s primary problem in reading and other language areas is remedied, he may begin to do well in arithmetic unless the failure in arithmetic has by then affected the child’s self-esteem.  Then the problem translates into mathematics anxiety.  

         Mathematics anxiety is a person’s negative emotional response to consistent failure in mathematics.  The unsuccessful experiences in mathematics create negative feelings for anything mathematical—from mild distaste and aversion to strong hatred for mathematics.  The symptoms of this anxiety are the same as symptoms for general anxiety—fidgetiness, dilation of pupil, restlessness, sweating of palms, etc. Mathematics anxiety, up to about age ten, is only a symptoms of mathematics difficulties or disabilities. Up to this point, it may not have been internalized, and as a result, it may not have yet affected the child’s self-esteem.  Once anxiety has been internalized, it begins to be a causative factor.   Because of mathematics anxiety, the person avoids mathematics and does not do well. 

         Mathematics anxiety is a problem of secondary type. Both dyscalculics and dyslexics can suffer from it.  Sometimes even a student without any disability may do poorly in arithmetic and mathematics and may develop anxiety. The nature of math anxiety—global or specific, depends on whether it has an emotional or cognitive base. If the basis of mathematics anxiety is emotional, we will have to significantly improve that child’s self-esteem by providing successful experiences so that he may begin to participate in quantitative experiences and do better in arithmetic.

         For some students, their mathematics disability or difficulty is driven by problems with language.  The very same difficulties that the child experiences in reading and other language concepts interfere in learning mathematical concepts. In mathematics, however, the language problems are confounded by the inherently difficult terminology, some of which children hear nowhere outside of the mathematics classroom, and the language based logic of mathematics.

         The way that numbers are represented linguistically is significant. For example, twelve hundred and one thousand two hundred appear to be handled differently in the brain. Twelve hundred is understood as the product of twelve and a hundred (whether consciously or not), one thousand two hundred as the sum of one thousand and two hundred. Asking for the solution to fifteen hundred plus one hundred most frequently brings the response sixteen hundred, but one thousand five hundred plus one hundred is one thousand six hundred. Although the underlying numerical values are the same, the brain appears to process the numbers differently according to their linguistic representation. 

         Unlike language, where one engages only with words and their combinations, in the case of mathematics one uses symbols, words, and their combinations. In mathematics, words appear as operators and as operand. This differentiation has a major impact on a child’s ability to learn mathematics.   

         Operation (mathematical symbols +, —, =, ´, ÷, ( ),  <, £, , etc.): Each symbol is packed with concept and meaning. In mathematics, both symbols and words act as operators (mathematical operations) that are to be performed on quantity, words, and space. Examples: Solve for x: 3x + 7 = 11. Find the product of .34 and 67. Divide the circle into halves. The area of the circle is given by: A = , where r is the radius of the circle. 

         Operand:Number (quantity) and shapes are the arguments and objects of the operators. Example: “Find the square root of 144.”  “Find the product of 23 and 15.”  “Circle the figure that represents half of the circle.”  “Circumscribe the square in the diagram.”  “Integrate the function sin (3x + 5).” 

         To complicate matters, sometimes there is a combination of the two.  For example, find the square root of the sum of 19 and 6.  

         Operator: Mathematical symbols andnumbers, both can act as operators.  Fro example, 3 increased by 7, here number ‘3’ is the operand, ‘+ or increased by,’ is an operation, and 10 is 1o an operator. 

         Difficulty in understanding the operatoroperation, and operandcan be a problem in learning mathematics that is different from the simple but related problems of the dyslexic. For many students, it is difficult to separate the operator, operation, and the operand.  For example, when it comes to integer problems such as: -3 +7, they cannot determine the roles of signs  ‘-’ and ‘+’ before 7 as operator, operation, or operand.  

Students face the same issues when several operations are involved in an expression or an equation.  For example, in simplifying the expression: 4y(-2y+3y), using the distributive property of multiplication over addition, many students have difficulty deciding which signs are being used as operands, operations, or operators. Examples of this type abound in mathematics, more once operations have been introduced. 

1.  Lexical Entries (Naming, Labeling and Language Containers)

One of the major characteristics of dyslexics that they share with dyscalculics is the inability to recall arithmetic facts at an automatized level because of a smaller number of lexical entries for numbers.  The reading skill is acquired by creating images of letters, words, combination of words, objects, and ideas in our minds.  They are like names and labels or language containers for words, numbers, combination of words and numbers, combination of numbers (language containers), and information and concepts.  This naming process facilitates the recall of immediate knowledge. Just as it is possible to build lexical entries for words and word-parts as well as for single letters, so, too, it seems reasonable to suppose that there are lexical entries, which represent numbers, combinations of numbers (facts), and symbols. It is becoming evident that for number conceptualization of numbers up to 9 or 10, we form visual clusters (lexical entries for numbers—numberness) in our brain. It is reasonable to assume that there are lexical entries for arithmetic facts also. When the number of lexical entries is small, dyslexics face a major problem of labeling (tasks of visual and auditory discrimination). In the absence of lexical entries, the student constructs the facts each time he encounters them.

         Most dyslexics are less proficient than mental age matched controls on tests of object naming and slower on rapid automatized naming of visual and verbal stimuli. The reading disabled children are less accurate in labeling the objects and have particular difficulty with low frequency and polysyllabic words. For example, young dyslexic children, aged between 7 and 9 years, are no different from the matched controls in tasks that do not require labeling.  They are appreciably weaker than the controls in word analysis tasks and labeling, and in some of these tasks, they are weaker than the poor readers believed not to be dyslexics.  This disability affects mathematics learning. They may understand the logic of arithmetic operations, but they show difficulty and inability to perform simple calculations because they cannot recall the needed facts automatically. This also applies to naming geometric shapes and describing terms in arithmetic and mathematics.

         It seems dyslexics cannot rapidly access verbal labels and arithmetic facts as they have problems in retrieval from long-term memory or even from working memory. They have difficulty holding intermediate steps in calculations in their minds as they have problems with short-term memory function and with the retrieval of stored memory traces. There seems to exist a strong association between mathematics performance and response time on rapid automatized naming.

         Because dyslexics have smaller number of lexical entries, they have problems in most common areas of arithmetic: difficulty in memorizing and recalling simple addition and subtraction facts, difficulty in learning multiplication tables by rote, and difficulty arising from uncertainty over sequencing and direction both in space and time. This inability to retain complex information in the memory system over time gets in the way of learning mathematics normally.  Accordingly, one would expect a lesser range of immediate knowledge of facts and information by memory on the part of a dyslexic person. They have fewer number facts available to them than do non-dyslexics: thus, if the question is, “What is 6 ×7?” a non-dyslexic of suitable age and ability can respond “42” in one response whereas many dyslexics can reach the answer only by working it out. Most non-dyslexics are aware intuitively of the difference between situations when they can instantly give the answer (72 – 9 = 63 or 72 ÷ 9 = 8) to a calculation and when they need to work this out (127 × 23 = 2921).

         The non-dyslexic will learn, for instance, that 8 ×7 = 56 after a relatively small number of exposures to stimuli whereas the dyslexic, because of his slowness at naming and labeling, will be unable to make use of the presented learning opportunities.  Consequently, he needs more exposures to stimuli before these stimuli take on symbolic significance. The dyslexic’s slowness results in longer naming times, which has consequences for the recall of digits, facts, terms, formulas and therefore creates difficulties in learning and using calculation systems.       

         The development of lexical entries is dependent on two factors: the underlying prerequisite skills necessary and the amount, type, and quantity and quality of early training in number conceptualization. This is expected to vary considerably among individuals with dyslexia since the amount and type of compensatory early training and the level of mastery of prerequisite skills are likely to be different.  

         This does not mean dyslexics or dyscalculics cannot learn arithmetic and other mathematics facts and recall them fast; they just need special methods and efficient strategies. Ordinarily, for automaticity and faster recall we focus on rehearsal—increasing the number and frequency of exposures.  The non-dyslexic will learn, for instance, that 6 ×7 = 42 and automatize this fact, whereas the dyslexics, because of their slowness at naming will construct the fact and will have to pay attention to other intermediate information. This construction may take the form of laborious counting or recall of the sequence: 6, 12, 18, 24, 30, 36, and 42. The construction of facts distracts and dilutes the cumulative effect of the exposure to stimuli.  The habits of construction of facts carry into higher mathematics.  When they are older, these children have difficulty memorizing even the simplest of formulas in algebra and geometry. 

         The longer naming time problem, if not treated at the appropriate age, has consequences later on for the recall of digits and facts, a difficulty with calculations, and generating immediate knowledge for use in problem solving. The extent to which number-sense and mathematical conceptualization are impaired and these facts (lexical entries) and appropriate procedures are missing can be expected to vary considerably.     

         Many of the problems that dyslexics face in arithmetic can be put right with proper teaching; and in particular, the use of concrete instructional materials may help to generate the appropriate lexical entries.  Just as a multi-sensory and special approaches, such as the Orton-Gillingham reading program, help in the building of lexical entries for letters, combination of letters, and words, and teach efficient strategies so can the visible and tangible presence of say, seven objects in a visual cluster, the mark on paper,‘7’, the written letters ‘seven’ and the sound ‘seven’ jointly contribute to the formation of the same lexical entry for the number 7. 

         In some cases, better and extra exposures and more meaningful stimuli that match students’ mathematics learning personalities are needed.  For example, many dyslexics may also have sequencing difficulty, finding addition or subtraction facts by sequential ‘counting up’ and ‘counting down’ are difficult for them. Memorizing the tables in the usual sequential order and teaching them in the usual way may not work for them. Simply providing exposure to this counting process will not result in a successful experience. We have found that breaking the 100 (10 x 10) multiplication facts grid into meaningful chunks based on clearly identified patterns helps dyslexics to learn multiplication tables faster and to have better recall (How to Teach Arithmetic Facts Easily,Sharma 2005). For instance, the two hundred facts of addition and multiplication are reduced to almost half once we introduce the idea of commutative property of addition(If I know 6 + 7, then automatically I know 7 + 6.) and multiplication (If I know 6 ×7, then I automatically know 7 ×6.).  Similarly, if we know 10 facts of doubles (1 + 1, 2 + 2,  …, 9 + 9, and 10 + 10), then we know 18 facts of near doubles (1 + 2, 2 + 1, 2 + 3, 3 + 2, …, 8 + 9, 9 + 8, 10 + 9, 9 + 10); if we know the sum of the two numbers that make 10, then we know the sums that are near tens, and so on.  With these patterns and efficient strategies, we have found almost all children are able to memorize arithmetic facts with fluency.   

2. Mathematics Disability Subtypes 

Acquiring the skill of reading, in itself, is a complex matter. The reading in a specialized content area such as mathematics, is even more demanding. Many children have difficulty mastering it. However, extensive research in this area has clearly identified a core set of skills. For example, phonological decoding deficits—processes in which grapheme-to-phoneme conversion rules are applied to ‘sound out’ a word’s spoken representation, have been identified as core symptoms of reading disabilities (RD). Reading aloud of novel words is achieved by phonological decoding. These core deficits are evident across the various subtypes of RD that have been described. Understanding these core deficits leads diagnosticians to identify RD and special educators to provide effective remediation.   Understanding RD helps curriculum planners and classroom teachers to design programs and provide preventive instruction.         

         The reason it is possible to identify a set of core skills and related symptoms, as an explanation of RD is that reading skills are well defined. We know what constitutes fluent reading and when to expect its presence with mastery. Unfortunately, no such core symptoms and skills in the case of mathematics disabilities (MD) have yet been identified, as research on mathematics disability is less well developed than RD research. There are several reasons for the absence of defining core skills in order to diagnose MD. 

  • First, we do not have any agreement about what constitutes the core skills in mathematics. There is no one particular skill that is at the core of every mathematical operation. Despite a general agreement on the wider contours of mathematics concepts to be mastered by children in elementary school, there is no agreement on the specific nature and type of skills, the level of mastery and fluency, and the timetable for achieving the skills for mathematics achievement.  
  • Second, because of the cumulative nature of mathematics, we are not able to identify the core skills. Unlike the key basic processes that underlie reading achievement, mathematical achievement is cumulative and comprehensive throughout and beyond the elementary school years, with quantitative and qualitative changes occurring within and across grade levels. Almost 30% of curriculum material at each grade is new or expanded substantially.
  • Third, after acquisition of the key skill (number conceptualization), learning and applying mathematics depend on a diverse set of skills.  These skills are spread over several different domains of functions.  

The mastery of arithmetic and mathematics skills, concepts, and procedures and poor math achievement, therefore, is linked to several factors and skills: language (native and mathematics), memory, visuospatial skills, affective, prerequisite skills such as: sequencing, pattern recognition, and/or executive skills. 

         For reasons mentioned earlier, mathematics disability (MD) is emerging as a collection of subtypes that cluster around several problem areas: (a) related to number, (b) related to language, (c) conceptual/ procedural, (d) visuo-spatial, (d) related to executive function, and (e) affective/behavioral.  

Because of the complexity of mathematics skills and their varied nature, it is highly unlikely that MD subtypes will share a unifying core deficit. As mentioned earlier, phonological processing/lexical labeling, have been associated with computational math skills in children, in earlier grades, with poor math achievement. However, additional factors, other than reading, also influence MD outcome. For example, language-specific difficulties in children with MD and RD have been reported relative to children with MD only. Many children with only MD outperform their peers with both MD and RD on exact arithmetic tasks, whereas both groups demonstrate comparable difficulty on estimation tasks.

         Math performance levels are also linked to executive function skills (working memoryinhibitionorganization, and flexibility of thought).  Different components and aspects of executive functions appear to account for some of the variability in children’s math performance levels, with strong contributions of poor inhibition and poor working memory, particularly as it relates to visualization. Visualization takes place in the working memory. Therefore, working memory plays a critical role in mathematics learning and performance. For example, in working out a problem, a simple calculation, 12 × 8, may be involved as a subsidiary problem that needs to be resolved mentally before we can solve the main problem.  If the student knows the fact, there is no digression. When he does not have the fact automatized, he has to construct it.  If he must construct, he digresses. That construction takes place in the working memory space or on paper.  To do so in the mind, the student has to keep several pieces of information in his mind: 10 × 8 is 80 and 2 × 8 is 16, so 12 × 8 is 80 plus 16, which is equal to 96.  Therefore, 12 × 8 = 96.  The student has to mentally manipulate this information, which requires visualization. Therefore, working memory deficits in children with learning disabilities, including children with reading or math difficulties, also exist. There is consistency across reports that both reading and executive skills are associated with math achievement levels. However, there is not enough information to explain the extent to which these cognitive and neuro-psychological correlates underlie one or more specific MD subtypes.

3.  Mathematics Problems Related to Reading

One of the obvious connections between dyslexia and mathematics difficulties is reading. Yet, many children with reading problems may not have problems in acquiring mathematical concepts. This is particularly so with straight computational and procedural aspects of mathematics and where the instructions are straightforward. However, mathematics problems involving language, particularly reading, pose problems for them, for they may not have the facility to comprehend the words and expressions properly. 

         For this reason, some children with poor reading skills are also below average in arithmetic skills. Below-average performance can exist in different areas, from very simple language-based symbolic conceptualization to complex problem solving such as word problems, making conjectures, writing definitions and proofs in algebra and geometry, as well as communicating mathematics. They frequently perform below average on the arithmetic subtests of Wechsler Intelligence Scale for Children (WISC).   

         When developmental dyscalculics who were good readers and those who were poor readers are compared, the good readers misread signs, align rows and columns inappropriately, and miss entire calculation steps. The poor readers avoid unfamiliar words, word problems, and operations; they have problems with tables and in recalling appropriate calculation procedures. We find specific areas of difficulties in which failure in mathematics is coupled with reading disability. These students have 

  • Difficulty with the vocabulary and terminology of mathematics, understanding directions and explanations or translating word problems,
  • Difficulty with irrelevant information included in the word problem or out of sequence information,
  • Trouble learning or recalling concepts, definitions and meanings of abstract terms,
  • Difficulty reading texts to direct their own learning and communicating mathematics, including asking and answering questions,
  • Lack of information concerning mathematical facts due to the failure of the child to make normal school progress (since, the child with reading problem may be taken out of the mainstream class or placed in special classes where the emphasis is on reading progress, the child may not get enough instruction in mathematics and therefore has limited exposure to mathematics), and 
  • Emotional blocking due originally to reading disability but eventually extended to mathematics.      

         The demands on reading in mathematics extend far beyond story problems. They include reading equations, mathematics conceptualization, definitions, etc. Research in the area of problem solving demonstrates that there is clear relationship between reading of algebraic symbols, instructions, and concepts and performance accuracy.  In our remedial work in mathematics (Sharma, 1980, 1988 & 2004) with children and adults, we have found that many students do not read and comprehend the vocabulary of algebra, nor do they read and comprehend an expression such as 3x + 7 > 2(x + 5) fluently and accurately. Although the symbols themselves are not phonetic, each symbol does represent a vocabulary word whose meaning must be understood (Lerner, 1993).  

IV. Role of Instructions in Mathematics Learning and Problem Solving

Mathematics concepts and problem solving subtests (or even computational subtests) on standardized national, state, local, and classroom tests and examinations contain sentences which must be read, comprehended, and followed by the test taker in answering the problems.  Understanding these instructions is the key to student success on any assessment.  From the outset, one can say that mathematics instructions are easier to understand when one knows the mathematics content, properly.  However, the processes of reading, comprehending, understanding, and executing instructions are the key to answering questions and problem solving.

Each problem or a set of problems in a mathematics textbook has a set of instructions written in English but uses special terms and symbols. Many of our students do not read or understand instructions in these computational and/or word problems. They decide what to do from the context or information gleaned about the problem by superficial reading of the problem. Sometimes, it happens because they donot understand the instructions as the words and phrases are not familiar to them. Other times, they can read them, but they do not have the conceptual schemas invoked by these words or phrases. Some times, they do not have the ability to execute them, as they have not mastered the procedure invovled in the problem. And, other times, they do not comprehend them, as the instructions are not clear. 

Writing effcient and effective instructions and explaining instructions and their role are the mark of a good teacher and a textbook. This does not mean all instructions should be non-technical or overtly simple. It is that classroom instructional strategies should have definite emphasis on developing, explaining, and processes of understaning and executing instructions involved in problem solving.

Reading, comprehending, and understanding instructions, and then executing these insructions properly requires that, in their lessons, teachers emphasize and help students master the three major components of a mathematics idea. Students need to have:

  • The mastery of mathematics language in order to be able to read, comprehend and conceptualize the problem (The use of language is to create ideas, receive and communicate ideas);
  • The presence of and facility in recognizing, and relating the language to appropriate conceptual schemas(arithmetical, algebraical, geometrical—definitions, diagrams, formulas, and relationship); and,
  • The ability and facility in executing appropriate procedures(involved explicitly and implicitly in the language of the instructions and in the problem) in an effective and efficient manner. 

A mathematical idea is received or constructed by a student through language, explorations using concrete and visual (pictorial—iconic and diagrams) models, or through discussions and problem solving.Solving problems and discussions solidify new and old learning and help integrate them. Teacher’s instructional approach, langauge usage, and setting of insructional activities facilitate and accelerate this learning process.  The quality of language and questions used by the teacher are the most important factors in learning mathematics and problem solving. 

Language creates language containersin the student’s mind for mathematics ideas. These, in turn, help create and hold conceptual schemasfor these mathematics ideas. Conceptual schemas and language help us derive, construct, develop procedures, and the ability to apply and execute these procedures and algorithms. Instuctions are the conencting links/bridges between these different components of mathematics ideas. They play an important role in learning mathematics and solving problems. 

Understanding and mastery in execution of instruction is dependent, first, on the emphasis and distribution of these components in the lesson and then on the clarity of the given instructions in the problem. In the absence of clear instructions or lack of understanding of the instructions, students often ask teachers and tutors: 

What am I supposed to do here?” 

How do I solve this problem?” 

What formula or operation should I use?” 

Can you tell me whether it is ‘division’ or ‘multiplicaiotn’?” Etc.

Others ask: 

“Why do we have these instructions in a mathematics problem? Just tell us straight what to do?”

“Why do they hide the instruction in the problem, I can’t even find them?” “Why can’t they make them clear?” 

“Why can’t you just tell us what to do?”  

Many teachers (regular and sepecial educators) and tutors out of exasperation or exisgency provide the formula, operation, or other related information without explaining.

These questions sound simple, but many of our students’ mathematics difficulties can be traced to misunderstanding or lack of understanding of these instructions. The casue of their difficulty goes directly to the heart of mathematics teaching. Before students can solve problems, they need to understand the instruction(s); they need to connect the language with the problem. 

A.  What is a Mathematical Instruction?

Each mathematics problem is a mathematical expression or a collection of mathematics expressions. A mathematical expressionis a combination ofnumbersand/orvariablesterms(words, phrases) andsymbols(knowns and unknowns, simple and complex) in the form ofexpressions,equationsinequalitiessystemsorformulas.For example, anequationis the outcome of equating two mathematical expressions. 

Each word problem is interplay between native, academic, and mathematics languages.  In word problems, we have words, phrases, and expressions that provide information and instructions to translate words into mathematics expression(s), construct a mathematical problem, and to solve that problem(s). 

The words and their combination in mathematics, and in instructions, from the persepctive of their functions, fall in several categories: some words in the instructions are used as:

  • Identifiers (i.e., The shape in the diagram is called…; the digit in the hundered’s place in the number 45,678.12 is ___. );
  • Verbs (i.e., multiply the fractions: ⅘ and½; differentiate the function …;reciprocate the fraction …;find the square root of ….; locate the point √2 on number line; reduce the fraction …. to the lowest term; it implies that 2 is an even number; deduce that every square is a rectangle;  prove that 8 is not a prime number; show that 7 is a prime number; determine the nature and number of factors of square numbers; etc.);
  • Concepts/nouns (i.e., place-value, arithmetic sequnece, multiplication, ratio and proportion, exponential function, addition, etc.);
  • Qualifiers/adjectives(i.e., 123 is a 3-digit number; 24, and 2n are even numbers, where nis an integer; √(n)is an irrational number, for any non-square, positive, whole number number; the equation: y = mx + bis called the slope-intercept form of a linear equation; the least common multiple of 8 and 12 is 24; the greatest common factor of 8 and 12 is 4; y = x2is a continuous function for all x; etc.);
  • Objects/noun (i.e., triangle, quadratic formula, parabola, focus of a conic section, square-root symbol, etc.);
  • Outcome of operations (i.e., sum, difference, product, quotient, ratio, differential coefficient, square root, etc.);
  • Cognitive and mathematics thinking functions (i.e., compare, analyze, relate, recognize the pattern, extend the pattern, make a conjecture, conclude, arrange, organize, focus, visualize, manipulate the information in the mind’s eye, spatial orientation/space organization, logical connectives: all integers…, every squareis…; if and then, if and only if, etc.). 

Theverbsand thinking functionsin the problems make demands or give commands to do something. These commands, or requests, are called the “instructions.” Before a student could answer the question(s) in the problem, the student must understand the role, meaning and purpose of these instructions. 

When a student encounters mathematics problems, including word problems. 

Thefirst stepis to read the words, terms, and phrases. That is purely a reading skill combined with recognition of mathematics symbols. Success in this step is dependent on reading skills and knowledge of academic language. Some knowledge on the part of mathematics teachers about the key elements of the reading process and manifest difficulties is important, so that a teacher can decide whether she can help the student or the support of reading/special/support teacher is warranted. However, not giving no word problems to solve to students with reading difficutlies is not the answer. Language and reading gets better with support and practice.

The second step is to know the meanging and role of all the words in the problem, including logical connectives invovled. This cannot happen without comprehending the conceptual meanings behind them and their interrelationships. This step requires the mastery of academic langauge and mathematics language—particularly the knowledge of related conceptual schemas. For example, multiplication does not just getting the product of two numbers by counting or memorizing.  It means that it is: “repeated addition,” “groups of,” “an array,” or “the area of a rectangle.”

After successfully reading, comprehending, and understanding the text of the problem, the third stepis to identify the unknowns and knowns in the problem. What is being asked in the question? This gives rise to the identificaiton or defining of the variable(s). At the elemetntary level, it may involve only the identification of the operation or relationship asked for. 

The fourth stepis to identify the type and nature of the problem.  For example, some part of the instruction may ask for constructing or articualting a relationship between knowns and unknowns resulting in a formula, equation, inequality, table, pattern, or graph. This may also result a main problem and subsidiary problems. The solution of the subsidiary problems may answer the main question asked in the problem. 

Other times, the instruction may call for executing a procedure or a formula or forming and solving an equation. The words and symbols or collection of words and symbols in the instruction or the problem invite the student to translate them to develop and form mathematical expressions (e.g., geomerical, algebriac and numerical) and, then to perform operation(s) or action(s) on them—from simple to complex, from single step to multi-steps to answer the question(s) posed in the problem. 

Many students and teachers may focus only on this last step, by disregardig the earlier steps.  That robs students of forming new concepts and relating new concepts with old concepts. 

B.  Why are instructions such a challenge for many students? 

Understanding instructions is learning and mastering the language of mathematics.  Therefore, in any approach to helping students to understand instructions, the emphasis should be on understanding the role of language in mathematics problem solving.

In this context, we as mathematics educators have to ask: Could the instructions in the problem be given differently? Better, easier language, or more succinctly?  As in desgining test items, we ask for validity and reliabiity of the content, we need to use the same criteria in the case of writing instructions for problems.

A teacher/tutor must always ask:  

What do the instructions mean for the student in this problem?  

Have we achieved that goal?

We need to constantly ask our students what do they do and understand when they read an instruction to a problem. 

And the student has to ask: 

What are they asking me to do in this problem? 

Do I understand the meaning of the words and expressions in the problem?

What do I know here?  

What do I not know in this problem?  

Can I find an entry point to the problem? 

But, just focusing on the instructions is not the answer.  We need to go to the root—the development of the language of mathematics during our teaching. 

Many children do not read the instructions, as they do not understand them. One of the reasons is the language and unfamiliar phrases used in the instruction.  Most students determine such a task based on the context. They think:

We have been solving problems like this in this chapter.  This problem must use the same method.” 

The simple solution to these kinds of problems by changing the difficult terms with simpler ones has limited implications. This decision solves the problem of instructions to some extent. For example, many questions in mathematics are preceded by the term “evaluate.”  Most people never use this term in their day-to-day conversations, or even by mathematics teachers during their instruction.  Rather than using “evaluate,” in a problem, we can use “find the value of.” This expression is easier, but the larger problem of not understanding instruction due to lack of mastery of content remains. 

Every discipline and field of study has technical and specific terms, words, and expressions to describe ideas, concepts, and procedures. To be competent in the field, it is important to know them and use them well. For example, on a test for tenth graders in 2002, as part of the Massachusetts Comprehensive Assessment System (MCAS), the term “represent” appeared 17 times and in one question it appeared five times (released items).  Several places, it was the only term that could do justice to the question. When I asked students to explain or define the term “represent” few students could define it clearly and that too only in a non-mathematical context. It will be better, if instructors/parents helped the children to understand the instructions first, before they help them to attack problems.

Once, while I was tutoring a high school student in algebra, I asked her to read the following problem: 4p + q(q-3)

         Evaluate:  where p = 3and q = 7

(This was one of the problems on the exercise set given to the students on a test.)

Question:  What does the word “evaluate” mean here in the problem?

Answer:    I do not know.

Question:  Have you ever seen this word before?

Answer:    Maybe?

Question:  Can you guess the meaning of this word?

Answer:  I think it means, “subtract”? No, no!  It means “addition” as there is an addition sign. I think it means, “solve” as this is an equation.  Am I right? But, wait. I do not really understand what are they asking – the question has already given me the values of  “p” and “q”.  It is alreeady solved. I am really confused.”

These types of answers are repeated quite frequently in every mathematics class and in every tutoring session.  How can the student be expected to solve these types of problems if the instructions are not clear to him/her?  It is very important to have the instructions made clear in order for the student to work through the problems effectively.  This means that one should emphasize the language (vocabularysyntax, and translationof mathematical terms into mathematical symbols), and understanding the meaning of words, both linguistically and conceptually—mathematically.  

Some times, the difficulty arises as questions are embedded in the problem. Such problems elicit different levels of complexity of thinking according to the words (particularly the verbs) used. A particular verb elicits a specific level of thinking. Some only ask for recognition of information. Some prompt students to do analysis. Some want them to develop a conjecture, hypothesis, pattern, a relationship, or a thesis. Some call for synthesis of ideas. Each word expects different level of engagement and a type and level of action from the learner.  

For example, the question, ‘Is this a polygon?’ requires from a student a yes or no answer. The question, “Which one of these figures is a polygon?” This question requires the student to analyze the shapes, to separate them into two types of figures (polygon or non-polygon) and then to compare the catgoriezed figures (what is common to the identified group).  

Whereas, the question, “Describe, why is this figure called a polygon?’ elicits a different level of language production. 

The question: “Determine which one of these statements is true?” (a) “Every rectangle is a square.” (b) “Every square is a rectangle.” (c) “Both statements are true.” “Justify your answer.” The instructions of this type prompt students to talk further, elaborate, make connections, and even elicit questions from each other. The answer calls for the integration of academic language, mathematics langauge, and deductiv reasoning. It is asking for a lot. That is what creates the difficulty.

C.  Types of Instructions:  Explicit and Implicit

Instructions in mathematics problems are of two types: explicitor implicit.In explicit instructions, the student is clearly instructed on what to do. Action or operation is already determined by the problem. In the case of mathematics problems where the instructions are explicit, most students find it easier to determine what to do.  

The demands/commands in explicit instructions are quite clear(e.g., find theaverage of the data; multiply the numbers; write the equation for a line in slope-intercept form; etc.). In implicitinstructions, the demands are indirect and sometimes hidden in the problem (e.g., What is the value of the ☐in the equation: 9 + 4 = ☐+ 5; find the maximum area under the curve with the conditions given; find the dimensions of a rectangle, if the length is 3 more than twice the width and the perimeter is 96 cm; Given two angles of the traingle, find the third angle; etc.). 

When these demands/commands are explicit, a student knows what is being asked in the problem, but when they are not explicit, many studentsare at a loss. They give up easily. Many teachers construct problems with explicit instructions only, as a result students think, if the answer is not easily forthcoming, it must be an impossible problem.  It is important that chidlren experience a range and different types of problems, with explicit to implicit instruction from the very beginning of their schooling. 

When the instructions are explicit, probability of a student solving the problem is increased. When they know the content and when the instructions are explicit,students know what to do and can solve the problem. For example, 

  • Multiply the numbers 1.2 and 1.3,compared with the instruction: find the product of 1.2 and 1.3.
  • Multiply the binomials (2a + 3)and (3a + 4) using ‘FOIL’ or distributive property of multiplication, compared with the instruction: find the quadratic expression with binomials (2a + 3) and (3a + 4) as its facotrs.
  • Find the value of the function f(x) = (2x + 3)(7x + 5)for x = 3, compared with the instruction: find f(a), if f(x) = (2x + 3)(7x + 5).
  • Differentiate the function: f(x) = (2x + 3)(7x + 5) at x = 2, compared to the instruction find f(a) for the function: f(x) = (2x + 3)(7x + 5).
  • Differentiate the function f(x) = 2x3sin(5x) by parts, compared to the instruction find df/dx for the function: f(x) = 2x3sin(5x).
  • Subtract 7 from 10, compared with find the difference of 7 and 10. 

The instruction, in the first part, in each of these problems, is familiar and direct.  It is simple. It is almost an order to execute the operation. It is in common language. If the student knows the procedure associated with the term, she can execute it.  

In the second part, the instruction is clear, but the language, in each problem is not common. It is technical.  It has a specific meaning in the context of mathematics. For example, in the first problem, ‘find the product’ may not be present or recent in the student’s mind. It is specific to the operation of multiplication.  And, since it may not be familiar to some children, it becomes difficult. This siuation happens when mathematics being taught in the classroom is taught just procedurally.  

In (b) the instruction in the second part is related with several concepts, although the procedure for answering the problem is the same.  In (c) the instruction, on the surface, is clear if the student is familiar with these kinds of problems, otherwise it is unclear.  In (e) the instruction is clear if the student knows the process of differentiating by parts. Similarly, in (f) the first part is straight forward, whereas, the second part, many adults write it incorrectly. 

If the instructions in the problem are explicit, and if students know the meaning of terms in the question—product, factors, quadratic expression, function, differentiation, or differentiation by parts, they can provide the answer to these specific problems. In the case of problems with explicit instructions, the success is dependent on knowing the content.  The hindrance is not in the instructions. If the teacher has focussed on teaching the curriculum only from the persepctive of procedures, the children in that classroom will have more problems with these kinds of problems.  

         Many students have difficulty understanding instructions because they do not know the content of the problem—the vocabulary, the concept, and executing procedures. It is not just one or the other. Knowing the meaning of words in the problem is not enough.  For example, 

  • using the long division procedure, find the quotient of 7.25 divided by .025,
  • Find the product of fractions: 3½ and 2¾, or 
  • find the greatest common factor of numbers 6, 48, and 54. 

In (a) the students may know what the words division and long division, however, they may not be able to find the quotient, if they do not know the process of long division, particularly when decimal numbers are invovled in the dividend and divisor,

In (b), they may know the meaning of the product, but may not know how to multiply two mixed fractions.  In (c), they may know the meaning of the term the greatest common factor of a set of numbers, but may not have the procedure for finding the greatest common denominator of three numbers. Here, the vocabularyis known, but the problem may be with not knowing the appropriate concept and/orprocedurein the context of the problem. 

         A similar situation may exist when instructions are implicit.  In many implicit instructions, the information is assumed; the information is indirectly given or embedded in the problem. 

Some problems may be made complicated by items, which ask students to supply information, which are not stated in the problem but are nonetheless necessary to solve the problem.  

  • Fred is 63 inches tall.  What else must you know to find out how much he   has grown in the past year? 
    •   How much did he weigh a year ago?
    •   How tall will he be next year?
    •   How old is he this year?
    •   How tall was he last year?

(b) What is the measurement of the smallest angle of a triangle if the two angles of the triangle are 70°and 80°? 

Many word problems do not ask for the operation, algorithm, or procedure directly, but they are embedded in the problem. For example, 

         (c) Simplify the expression: -4y{xy3-2xy3+23(y3x +2)} +7xy3;

(d) John practices the piano 1.5 hours each day. His coach said: he needs to practice at least 30 hours before the next concert.  At least, how many days of practice does he need to be prepared for the next concert? 

In these examples, the instructions are indicated either by mathematical symbols or by words. The knowledge of the symbols, their role in the context, and the conceptual schema embedded in the words and phrases are the key for understanding and responding to the instructions correctly and efficently.  In other words, instructions are implicit and knowledge of the content is needed. Students who have had practice with these types of problems are more likely to answer correctly than students who have not practiced a particular type of problem.  Here the vocabularyis simple, but the conceptis hidden and, therefore, important to know to resolve the problem. 

         Just as we need to understand how to read a map before we can use it, before we can solve a problem, we need to understand the instructions. Students cannot solve problems if the instructions are not clear.  Understanding instructions is dependent on students’ mastery of the language of mathematics(vocabulary, language containers, syntax, and translation from English to mathematics and from mathematic to English), the content of mathematics(conceptual and procedural aspects), the mathematical way of thinking(organizing, classifying, seeing patterns, reasoning, critical thinking and communication skills).  

Sometimes, even problems with explicit instructions are difficult for many children. For example, even though the reading level of the instructions and the concept in the problem may be at a lower level, still many students particularly those with reading problems, with limited academic langauge, and/or without an appropriate mathematics vocabulary, find themselves at a disadvantage in word problems.  Similarly, some students may infer the procedure involved in the problem and could solve the problem if they knew how to execute or apply that procedure in the context of that problem.  Others, for example, even if they were able to read the instructions, they may have difficulty in understanding the instructions that lead to the particular concept and/or procedure, as they may not have the conceptual schema behind the words, therefore, may not be able to solve the problem.  

Ability to read the problem is a necessary conditon for solving the problem, but it is not a sufficient condition. A student may be able to read the problem without any difficulty, but may not be able to translate the technical words into mathematics concepts and procedures because of poor mathematics vocabulary and lack of conceptual schemas. Therefore, may not be able to arrive at the procedure or strategy to be applied. Only the proper mathematics language and its understanding will lead to the construction of cocneptual schemas and efficent concpetual schemas lead to procedures. 

D.  Role of Questioning in Understanding Mathematics Instructions

Following instructions, first, is a task in reading: vocabulary, comprehending, and understanding. The mathematics language plays a big role in it and learning how to read instructions and following them is an important part of mathematics learning. Once the reading task is performed, then, cognitively, it is connecting the vocabulary with the concepts, procedures, and mathematical way of thinking.  

Vocabulary for mathematics ideas, concepts, and procedures should emerge through discussion and experimentation and then formalized, connected to what is already known rather than transpalnted by giving vocabulary lists to memorize. Because words should result from a need to describe our world—this is where they gain their power.Therefore, the type and the number of questions we ask in the mathematics classroom determine how the students are going to do on mathematics tasks. 

Questions elicit different levels of complexity of thinking according to the words used (especially verbs). Particular verbs elicit a specific level of thinking to prompt: analysis of data and ideas; developing conjectures, hypotheses, and then a thesis about the problem; synthesizing different types of problems, strategies, and procedures; type of thinking—recognition, comparing, contrasting, and constructing; or, calling for levels of engagements from the learner. For example:

The question “Is this a polygon?” requires a student to say either yes or no. In contrast, the request “Describe why this shape is called a polygon” elicits a different level of language production. Instructions of this type prompt students to elaborate, make connections, and even form questions. Thus, questions and the art of questioning are critical to learning. Questions commence a cascade of actions in the brain:

  • Questions instigate language;
  • Language instigates models;
  • Models instigate thinking;
  • Thinking instigates understanding;
  • Understanding produces competent performance;
  • Competent performance is the basis of long lasting high self-esteem; and
  • High self-esteem contributes to motivation for learning and engagment. 

The type and number of questions we ask in the mathematics classroom determine how successful students will be in mathematics. Mathematics language plays a critical role, and learning how to read instructions and following them is an important part of mathematics learning. Students should also know the type of questions that will appear on a test, and teachers should feel comfortable giving them such information since it will focus students’ study efforts. 

For example, in the case of multiple-choice questions, they will be required to identify rather than generate information.  Although identification formats are generally less difficult than producing information, multiple choice tests often focus on the identification of a large amount of less important information.  These formats have different implications for language requirements in test taking.  

         Standardized tests have become a permanent feature of education.  A student should know there are great varieties of test item formats on standardized achievement tests.  There are also varieties of difficulty levels in different types of questions. To understand the instructions, a sufficient level of reading comprehension on the part of students is required.  Formats used for mathematics tests are usually relatively straightforward, but they could also vary.  To succeed on tests, it is important to know these formats; the knowledge of mathematics content alone is not enough.  Because teacher made tests may not follow the same format and structure, the teacher should check the manuals of the standardized tests to determine whether she can help her students on the reading part or the structure of the test items.  She should also know how content questions are formulated on these tests. Similarly, some of the test items may require the use of charts and graphs.  It is important that students have experience with these before taking the test.  

E.  Strategies for Improving the Understanding of Instructions 

Good teaching in mathematics, at the elementary level, requires that students be taught key number concepts associated with computation. They should have:

  • practiced arithmetic facts to the automatization level with efficient and effective strategies (arithmetic facts are best derived using decomposition/recompsotion strategies);  
  • know a concept in its different models(e.g., multiplication as: repeated addition, groups of, an array, and area of a rectangle);
  • and have applied computational procedures in a variety of different formats(e.g., division as partial quotient, long division, and short division).  

Students with learning problems in mathematics and in special education settings are presented with a restricted range of mathematics with the mistaken belief that they experience less complex material to minimize confusion and frustration.  That kind of strategy works initially but in the long term it is detrimental to their progress and their development as learners.  It does not provide exposure to meaningful mathematics and limits their cognitive development.  

The goal of special education support for children should be two-fold:  to help them improve cognition and to expose them to meaningful content, in meanginful ways. For that purpose, we can begin with simpler language, simple instructional models and with a narrow range of the content and setting, but then we need to increase the complexity of language, range of material and models, and content.  All students benefit from learning the range of problem formats and terminology and sufficient exposure to the rich vocabulary of mathematics. 

         Again, effective teaching strategies, which employ a large vocabulary and a variety of formats, are the most helpful practices to acquire flexibility of thought. In addition, knowledge of key vocabulary terms and use of multiple visual representations of mathematical information are important for a conceptual understanding of mathematical ideas. 

         In light of this, when students are confronted with an item in an unfamiliar format or context, teachers should encourage them to use scripts such as: “What is the key information here?” “Do I know a related word or expression?” “How else could one ask this question?”  “Can I think of another example for this?”  “Do I remember another problem like this?”  “Will I able be to solve this problem if I substitute simple numbers?” 

         Students should practice answering questions by replacing vocabulary they use less commonly and rewriting them in formats that are more familiar.  Once stated in more familiar terms, the student is more likely to answer the question correctly.  For example, consider the following:

         Which set has both odd and even numbers that are not square numbers as its members?

  1. {9, 11, 15, 3, 5}
  2. {6, 10, 4, 2, 8}
  3. {6, 10, 7, 5, 8}
  4. {25, 49, 225, 144, 9, 400}

This problem could potentially confuse students who are aware of odd and even numbers but uncertain of the meaning of the word set or square numbers. Or, they may have difficulty in understnading the phrase: both odd and even that are not square numbers.” 

If a teacher shows students how to actively reason through each item and to temporarily set aside unfamiliar terms, they should be able to see that one of the four answer choices has both odd and even numbers and therefore “c” is the best answer. Once, we have used proper reasoning to find the correct answer, we could introduce the term setand square numbers

We could also expand the idea by giving a few more examples of this type.  Using this problem, we can extend the discussion by asking: “What kind of numbers are in the set described in the option ‘d’?” and, then extending it to further discussion: “Can a square number have digits 2, 3, 7, and 8 in the one’s place?” Using the area definition of multiplication and identifying the product as the area of the rectangle and the sides as the factors, one can conncet the nature of numbers—even, odd, square, prime, and composite and properties of operations—zero property, comutative, associateive and distributive. 

When students do not read or understand instructions for a problem, they cannot show what they may know.  Over the past decade, teaching approaches have changed in the mathematics classroom. Some have resulted in improved learning and some have contributed to the detriment of learning.  One approach that has recently been neglected is time on task. The more time spent on being directly engaged in learning mathematics language, the better students will understand the instructions in problems involving mathematics language. It is important to increase the amount of actual time on task for students to work on mathematics language, in context of problem solving.  Time on task is the amount of real time spent on teaching and learning mathematics language and the related instructions in the context of word problems.

Another critical variable is the amount of contentmastered raher than just covered.  If the content is not covered, students will not have the opportunity to learn enough information, but if it is not mastered, they will not be able to apply it.  In such a situation, working on comprehension of instructions is ineffective.  Further, if the content is covered too rapidly, students may not have the opportunity to master the information sufficiently. Teachers need to develop a clear scope and sequence.  This should be planned at the beginning of the year, not during the year.  During the year, one can make adjustments to the content.  In countries where students achieve higher levels in mathematics have an unwritten pedagogy—they plan the scope and seuqence for a time that is at least one to month less than the school year.  The last two months are devoted to review, reinforcement, practice, and integration of content. Time to teach skills to understand instructions should be included in each lesson, and this should begin early in year.  Including test-taking strategy instruction with examples from standardized tests enables students to practice and apply them throughout the year. Since standardized tests are typically administered in the spring, training for these tests should take place several times a year. Intensive practice should take place prior to the administration of tests.

Throughout the school year, students should be taught to extract meaning from a variety of graphic displays and tables in mathematics contents (e.g., relief maps, topographic maps, weather maps, maps of ocean currents, timeline of historic events, scientific tables and charts, population charts and maps, and other graphic displays) and answer questions based on these displays in a multiple choice format.  Newspapers and the Internet are a good source of this information. Additionally, when they read passages describing mathematics content, students should use the same strategies found in reading comprehension tests.  These are valuable exercises because students can come to understand that the strategies they use to answer reading tests can be used for readings wherever they occur.   

All of these strategies relating to mastering mathematics content and solving problems depend on children practicing the three components of mathematics—linguistic, conceptual and procedural.  Instructionas are only label on the package involving mathematics contet.  If students do not know the content, instructions cannot be blamed for lower acheivment. 

F.  Examples of Instruction to Mathematics Problems

The following is an attempt to identify key vocabulary words, expressions, and symbols used as part of the instructions, generally used in mathematics texts, tests and examoination (in most cases few examples are used):

  • About/Approximately/Rounding: (a) About how many miles is 66.5 million feet? (b) The value of the number √(145) is close to what integer in value? (c) Nate says:“The value of the fraction ⅛ is about.13 when approximated to the hundredth place.”  Is he right? Did he round to the hundredth’s place correctly? (d) What will be the value of , if rounded to the tenth’s place? (e) Is rounding is same as approximation?
  • Add/subtract/multiply/divide:What is the value of , (a) if we add the other numbers of the set: {, 1, 8, 5 and 34}. (b) if we multiply other members of the set?  (c) What is the smallest quotient, if we divide any two members of the set {1, 6, 5, and 30}
  • Apply:(a) Apply the graphing method for solving the set of equations: 3x + 4y = 12and 4x + y = 29. What does mean to solve this system of equations? (b) Which of the following shows an application of the distributive/ associative/commutative property? (c) Apply any of the Prime Factoriazation methods to find the Greatest Common Factor (Least Common Multiple) of 24 and 40.                                        
  • Assume:(a) Assume that this triangle is equilateral. (b) Assume that the numbers mn in the fraction m/n are prime. Is the fraction, expressed in the lowest term? (c) What is the value of n, if we assume that the line passing through the points P(n, 5) and Q(2,7) is horizontal? 
  • Compare:Compare the following numbers:  and .24. Write a number relationship between these two numbers.
  • Compute/Calculate/Perform the operation: (a) Compute 35.2 ÷.574. (b) Which number in the box makes the number sentence (15 – 3) × (2 +3) = ÿ× 5 true? (Choices: 5, 15, 12, 30). (c) Perform the indicated operation in the following calculation(s) ….
  • Conclude:What pattern do you see in the data? What do you conclude from the result you derived from the data?  Write your pattern as a relationship between the two variables? When you compare your pattern relationship, with this equation: y = mx + b? What do you conclude by the slope in your equation? 
  • Consider: Consider that this pentagon is a regular figure, what does the term ‘regular’ indicate here? What is a regular triangle called?
  • Compare and contrast:  Compare and contrast the members of the set by their properties: {2, .2, 2%, , ½, (.2)−2, and 22}. 
  • Decide: (a) Decide which is the largest number in the set: {2, , .2, 2%, , ½, (.2)−2, and 22}.  (b) Decide which is the smalleest number in the set: {2, .2, 2%, , ½, (.2)−2, and 22}.
  • Describe:Describe the pattern that can be used to predict the height of the bounces of a ball that bounces back half as much as the previous bounce.
  • Determine: (a) Determine the relationship among the values of the coins from the following clues ….. (b) How can you determine if a rectangular array can be built for an expression …..? 
  • Distinguishbetween: (a) An even numberand an odd number; (b) a prime number and a non-prime number; (c) a polygonand a non-polygon, (d) an integerand a rational number; (e) a continuous functionand a non-continuous function;  …
  • Envision/visualize/picture/think:(a)Envisionyou rotated the diagram (rectangle, a square, an equilateral triangle, and a regular hexagon) by 90°clockwise. What will the figure look like after the rotation?; (b) What amount of rotation (and about what axis of rotation, or what point) will tranform the first diagram to the second diagram?  
  • Estimate:(a) What is the best estimate of how many more times Cathy jumped than Wilson? (b) Which arrow on the radio dial below is closest to 96.3?  (c)What is the closest degree measure of he angle formed between the hour and the minute hands of a clock at 3:40 PM?  (d) Which graph below most likely shows the outcome? (e) Three friends plan to equally share the cost of a video game that costs $38.89 including tax.  Which is the best estimateof the amount each will have to pay? (e) Using estimation, decide which sticker below has the greatestperimeter. (f) The value of  is closest to … (g) 2Ö5 is between what whole numbers. 
  • Evaluateeach expression: (a) 3xy2+ 5x2y -4x2y2, where x = −2and y =−.5.
  • Explain/express:Explain your reasoning in your words why a prime number has odd number of factors.
  • Extrapolate: (a) From the data given extrapolate the nature of the graph. (b) Assuming that her income and expenses continue to grow at approximately the same rate, estimate her income and expenses for the month of may.  Explain or show how you found your estimates. 
  • Find the value of: (a) 3xy2+ 5x2y -4x2y2, where x = ]−2and y =−.5.
  • Generalize: (a) Use the sequence of numbers 1, 3, 7, 15, 31, 63, …to find the general pattern/formula/expression.  
  • Graphing: (a) Graph/plot on a number line/coordinate plane. (b) Construct two line graphs using the given data. (c) Draw a circle with radius 5 and center (3, 4).
  • How many/long/much/much more/much less: (a) How many millimeters of iodine are in 1,000 ml of solution? (b) How many times greater is the surface area of the cube with side 2 inches and the cube with side one inch? (c) How long will each column of names be? (d) How long will take him to travel this distance? (d) How do a and b compare? 
  • Identify:  (a) Identify the reciprocal of .25. (b) Identify the inverse of the function f(x) = 3x + 4. (c) Identify the property of the equality used in the equation: 3(x + y) = 3x + 3y. Identify the shape that is: 
  • Interpret the graph: A graph is given.
  • Interpret the definition:  In what ways the definition of a prime number: “A whole number is called prime, if its only factors are 1 and itself”  differs from “A whole number is called prime, if it has exactly two factors, namely I and itself.” Which definition is accurate? Why 1 is not a prime number? 
  • Model the information:  Show the distributive property of multiplication over addition and subtraction using the area of a rectangle definition of multiplication. 
  • Name: (a) Name one of the shapes you chose.  Make a list of four different things that describe this shape.  (b) Name another one of the shapes you chose. Make a list of four different things that describe this shape. (c) Name the last shape you chose.  Make a list of four different things that describe this shape.  
  • Notice the list of numbers/formula/diagram/data
  • Observe the following information and:
  • Pattern: (a) When we multiply 37 by multiples of 3, we see a pattern. 37 × 3 = 111; 37 × 6 = 222; 37 × 9 = 333; 37 × 12 = 444; … If the pattern continues this way, then 37 × 21 =  ? (b) What is the next number in the pattern below?
  • Predict: (a) Predict the chances of getting a red balls out of the container that contains 3 red balls and 7 balls of different colors.  (b) Predict the height of the fifth bounce. (c) is it more likely that …. (c) Which is the BEST way for Bridgett to show this information? 
  • Prove: Which of the following statements gives enough additional information about the figure above to prove that DABC is similar to DDEC.  
  • Rewrite each expression in a simpler form:  (a) 48/128 (b)  (x-2)/(x-2)(x-3).
  • Remember: (a) Remember a polygon has more than three sides. Define a quadrilateral as a polygon. (b) Using estimation, decide which sticker below has the greatest perimeter. (Remember: Perimeter is the distance around a figure.) 
  • Represent:  (a) Represent this point on the coordinate graph. Do these points represent a circle? (b) Which graph below most likely represents Ms. Hall’s class on Tuesday? (c) Which point represents the intersection between the lines: 3x + 4 y = 7and 4x + 3x = 7
  • Show/describe:Show or describe how you found your answer.  (a) Use pictures, numbers, or words to show or explain how you found your answer.  (b) Use pictures, numbers, or words to show or explain how you know. (c) Which shows a slide of Y. (d) Show how to build rectangular arrays, if possible, for each of the following expressions using the math tiles.
  • Simplify: (a) Simplify the expression  … (b) Simplify the numerical expression …. (c) Use the expression 2x – 3(5x –8)to answer the question: What could be the first step in simplifying the given expression? 
  • Solve:(a) (Direct instruction) Solve the following equation for x ..(b)  (Indirect instruction) If 4 + 2 (3x – 4) = 8, then 3x – 4 equals…. (c) 24   ×3is the same as …. (d) Find allthe values of xthat satisfy the following equation. (e) The expression 4 x2+ 2x – 6 – x(3 – x)is equivalent to ….
  • Summarize:  Write a proof of the Pythagoras theorem for a right triangle.
  • Suppose that (see assume that): 
  • Tell:  (a) Tell whether each statement is true or false. (b) Which number sentence tells how much milk is in all the glasses? 
  • True:  (a) Which of the following statements is always/sometimes/never true?
  • Use: (a) Use the information in the scatter plot/graphic method/equation/process/table/chart to answer the question. (b) Use the balance scale to answer the following question. (c) Use t−2, 4). (b) Write a rule for the table shown below. (c) Write four different number sentences that follow these rules. Each number sentence must show a different way of getting the number 42.  Each number sentence must contain at least two different operations. Use each of the four operations at least once. An example is shown below.  You may not use this example as one of your four number sentences.  Example: (8 ¸4) + 44 − 4 = 42. (d) Write a number sentence to show how much money Ralph spent for stamps.  Be sure to include the answer in your number sentence.  

Numeracy & Literacy Analogies

The fields of mathematics and reading both have basic skills.  In both subjects, all students must master all of the basic skills to work productively.  But, while the skills necessary to read well are well known since the publishing of the Report of the National Reading Panel[1]in the year 2000, the equivalent skills to learn mathematics are less well understood.  The table below puts the two side by side to help teachers see the parallels.

Number Concept – an understanding of the concept of number in language, in orthographic symbols, and in visual clusters (create 3 part Venn diagram)1.     Phonemic Awareness – an understanding of the sounds in their language and how they form words
Decomposition/Recomposition– the ability to manipulate numbers to see number relationships and fluently solve unfamiliar problems using numbers2.     Decoding – the ability to figure out unfamiliar words, and to learn to read them fluently
Language of Numbers – mastery of the words and phrases used to describe numerical operations3.     Vocabulary – mastery of an adequate number of words to understand text passages
Fluency – Automatic knowledge of basic arithmetic facts without counting using: a) sight facts, and b) strategies4.     Fluency – transforming vocabulary into sight vocabulary through practice to automaticity
Understanding – the ability to understand the questions in a problem, apply appropriate facts and strategies to solve them, and explain the solution to others5.    Comprehension– the ability to understand the direct meaning of text, and also its implications and intention, and finally the ability to perform analysis on text
Communication – the ability to explain to others: a) the choice of numerical processes, concepts and procedures in solving problems; and b) explaining the nature of the solution.  C) These concepts and procedures may be expressed concretely, orally, pictorially or symbolically.6.     Writing – the ability to explain ideas to others so that they understand: a) the meaning of the text; and b) the implication of the text. C) This ability may be expressed in outline, expository, story-telling or other forms.


Center for Teaching/Learning of Mathematics 

CT/LM has programs and materials to assist teachers, parents, therapists, and diagnosticians to help children and adults with their learning difficulties in mathematics. We conduct regular workshops, seminars,and lectureson topics such as:                        

            1. How does one learn mathematics?This workshop focuses on psychology and processes of learning mathematics—concepts, skills, and procedures. The role of factors such as: Cognitive development, language, mathematics learning personality, pre-requisite skills, conceptual models, and key developmental milestones (number concept, place value, fractions, integers, algebraic thinking, and spatial sense) in mathematics learning. Participants learn strategies to teach their students more effectively.

            2. What are the nature and causes of learning problems in mathematics?  This workshop focuses on understanding the nature and causes of learning problems in mathematics.  We examine existing research on diagnosis, remedial and instructional techniques in dealing with these problems. Participants become familiar with diagnostic and assessment instruments for learning problems in mathematics. They learn strategies for working more effectively with children and adults with learning problems in mathematics, such as: dyscalculia and math anxiety. 

            3. Content workshops.  These workshops are focused on teaching key mathematics milestone concepts and procedures. For example, How to teach arithmetic facts easily and effectively?  How to teach fractions more effectively?  How to develop the concepts of algebra easily? Mathematics As a Second Language.In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a very simple concept from arithmetic and take it to the algebraic level. 

            4. What to look for in a results-oriented mathematics classroom: This is a workshop for administrators and teachers to understand the key elements necessary for an effective mathematics classroom.

            We offer individual diagnosisand tutoring servicesfor children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia. We provide:

            1. Consultation with and training for parents and teachers to help their children cope with and overcome their anxieties and difficulties in learning mathematics, including dyscalculia.

            2. Consultation services to schools and individual classroom teachers to help them evaluate their mathematics programs and teaching and help design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics.

            3. Assistance for the adult studentwho is returning to college and has anxiety about his/her mathematics.

            4. Assistance in test preparation (SSAT, SAT, GRE, GMAT, MCAS, etc.)

            5. Extensive array of mathematics publications to help teachers and parents to understand how children learn mathematics, why learning problems occur and how to help them learn mathematics better.

The Math Notebook (TMN)

Articles in TMN address issues related to mathematics learning problems, diagnosis, remediation, and techniques for improving mathematics instruction.  They translate research into practical and workable strategies geared towards the classroom teacher, parents and special needs teachers/tutors. Topics covered range from K through College mathematics instruction.

Selected Back Issues of The Math Notebook:

  • Children’s Understanding of the Concept of Proportion – Part 1 and 2 (double)
  • A Topical Disease in Mathematics: Mathophobia  (single)
  • Pattern Recognition and Its Application to Math  (double)
  • Mathematics Problems of the Junior and Senior High School Students  (double)
  • Mathematically Gifted and Talented Students  (double)
  • Types of Math Anxiety  (double)
  • Memory and Mathematics Learning  (double)
  • Problems in Algebra – Part 1 and Part 2 (special)
  • Reversal Problems in Mathematics and Their Remediation  (double)
  • How to Take a Child From Concrete to Abstract  (double)
  • Levels of Knowing Mathematics  (double)
  • Division: How to Teach It  (double)
  • Soroban: Instruction Through Concrete Learning  (double)
  • Mathematics Culture  (double)
  • Mathematics Learning Personality  (double)
  • Common Causes of Math Anxiety and Some Instructional Strategies  (double)
  • On Training Teachers and Teaching Math  (double)
  • Will the Newest “New Math” Get Johnny’s Scores Up?  (double)
  • Dyslexia, Dyscalculia and Some Remedial Perspectives For Mathematics Learning Problems (special)
  • Place Value Concept:  How Children Learn It and How To Teach It  (special)
  • Cuisenaire Rods and Mathematics Teaching  (special)
  • Authentic Assessment in Mathematics  (special)


FOCUS on Learning Problems in Mathematics

FOCUS has been an interdisciplinary journal. For the last thirty years, the objective of FOCUS was to make available the current research, methods of identification, diagnosis and remediation of learning problems in mathematics.  It published original articles from fields of education, psychology, mathematics, and medicine having the potential for impact on classroom or clinical practice.  Specifically, topics include reports of research on processes, techniques, tools and procedures useful for addressing problems in mathematics teaching and learning:  descriptions of methodologies for conducting, and reporting and interpreting the results of various types of research, research-based discussions of promising techniques or novel programs; and scholarly works such as literature-reviews, philosophical statement or critiques.  The publications in Focus have real contribution in the field of mathematics education, learning problems in mathematics and how to help children and adults in dealing with their mathematics difficulties.

Selected back issues ofFocus:

Volume 3, Numbers 2 & 3: Educational Psychology and Mathematical Knowledge

Volume 4, Numbers 3 & 4: Fingermath: Pedagogical Implications for Classroom Use

Volume 5, Number 2: Remedial and Instructional Prescriptions for the Learning Disabled Student in Mathematics

Volume 5, Numbers 3 & 4: Mathematics Learning Problems and Difficulties of the Post Secondary Students

Volume 6, Number 3: Education of Mathematically Gifted and Talented Children

Volume 6, Number 4: Brain, Mathematics and Learning Disability

Volume 7, Number 1: Learning Achievement:  Implications for Mathematics and Learning Disability

Volume 7, Numbers 3 & 4: Using Errors as Springboards for the Learning of Mathematics

Volume 8, Numbers 3 & 4: Dyscalculia

Volume 9, Numbers 1 & 2: Computers, Diagnosis and Teaching (Part One and Two)

Volume 11, Numbers 1 & 2: Visualization and Mathematics Education

Volume 11, 3 (1989): Research on Children’s Conceptions of Fractions

Volume 12, Numbers 3 & 4: What Can Mathematics Educators Learn from Second Language Instruction?

Volume 13, Number 1: Students’ Understanding of the Relationship between Fractions and Decimals

Volume 14, Number 1: The Psychological Analysis of Multiple Procedures

Volume 15, Numbers 2 & 3: Vygotskian Psychology and Mathematics Education

Volume 17, Number 2: Perspective on Mathematics for Students with Disabilities

Volume 18, Numbers 1-3: Gender and Mathematics:  Multiple Voices

Volume 18, Number 4: The Challenge of Russian Mathematics Education: Does It Still Exist?

Volume 19, Number 1: Components of Imagery and Mathematical Understanding

Volume 19, Number 2: Problem-Solution Relationship Instruction: A Method for Enhancing Students’ Comprehension of Word Problems

Volume 19, Number 3: Clinical Assessment in Mathematics: Learning the Craft

Volume 20, Numbers 2 & 3: Elements of Geometry in the Learning of Mathematics

Volume 22, Numbers 3 & 4: Using Technology for the Teaching and Learning of Mathematics

Volume 23, Numbers 2 & 3: Language Issues in the Learning of Mathematics

Volume 28, Number 3 & 4: Concept Mapping in Mathematics

Index of articles in Focus on Learning Problems in Mathematicsfrom Volume 1 to 30 available on request. 

Individual issue                                                                                      $ 15.00

Double issue                                                                                             $ 20.00

Each Volume (four issues)                                                                     $ 30.00  

Whole set of 30 volumes                                                                        $400.00

Math Notebook: Single issue ($3.00); Double issue ($6.00); Special issue ($8.00)

Other Publications

   Dyslexia and Mathematics Language Difficulties             $15.00

   How to Master Arithmetic Facts Easily and Effectively    $15.00

   Guide for an Effective Mathematics Lesson                      $15.00

   How to Teach Fractions Effectively                                   $15.00

   Math Education at Its Best: Potsdam Model                      $15.00

   How to Teach Number to Young Children                         $15.00

   Dyscalculia                                                                       $15.00

   How to Teach Subtraction Effectively and Easily               $12.00

  Literacy&Numeracy:Comprehension and Understanding $12.00

   The Questioning Process: A Basis for an Effective Lesson   $12.00

   The Games and Their Uses in Mathematics Learning      $15.00        

   Visual Cluster cards without numbers                              $12.00        


    How Children Learn: Numeracy                                              $30.00

(An interview with Professor Sharma on his ideas about how children learn mathematics)

    How To Teach Place Value                                               $30.00

(Strategies for teaching place value effectively)

Numeracy DVDs

(Complete set of six for $150.00 and individual for $30.00)

         1.  Teaching arithmetic facts,

            2.   Teaching place value, 

           3.   Teaching multiplication, 

           4.   Teaching fractions,

           5.   Teaching decimals and percents, and

            6.  Professional development: Teachers’ questions  

Most children have difficulty in mathematics when they have not mastered the key mathematics milestones in mathematics. The key milestones for elementary grades are: Number conceptualization and arithmetic facts (addition and multiplication), place value, fractions and its correlates—decimal, percent, ratio and proportion. These videos and DVDs present strategies for teaching these key mathematics milestone concepts. They apply Prof. Sharma’s approach to teaching numeracy. These were videotaped in actual, regular classrooms in the UK.   

Please mail or fax order to (add 20% extra for postage and handling):


754 Old Connecticut Path, Framingham, MA 01701

508 877 4089 (T) 508 788 3600 (F)

Mahesh Sharma

Professor Mahesh Sharma is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Framingham, Massachusetts, USA and Berkshire Mathematics in Reading, England. Berkshire Mathematicsfacilitates his work in the UK and Europe. 

       He is the former President and Professor of Mathematics Education at Cambridge College. Where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. Internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics.  

       He is an author, teacher and teacher-trainer, researcher, consultant to public and private schools, as well as a public lecturer. He has been the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children. 

       He provides direct services of evaluation and tutoring for students (children as well as adults) who have learning disabilities such as dyscalculia or face difficulties in learning mathematics. Professor Sharma works with teachers and school administrators to design strategies to improve mathematics curriculum and instruction for all.

Contact Information:

Mahesh C. Sharma

Center for Teaching/Learning of Mathematics

754 Old Connecticut Path

Framingham, MA 01701

Mathematics Blog




The next whole day workshop, in the series on professional development in mathematics education and learning problems for teachers, at Framingham State University, will be held on May 17, 2019.

The topic of the workshop is: Learning Problems in Mathematics, including dyscalculia. The workshop is open for classroom teachers, special educators, inteventionists, tutors, special education administrators, and parents.

For more information, please call Anne Miller at 508 215 5837.

To register go to:


Professor Mahesh Sharma’s next workshop in the series on improving mathematics education at Framingham State University on April 12, 2019 is on Mathematics as a Second Language. The focus will be on helping students develop mathematics language and its role in mathematics conceptualization and problem-solving. For more information and future workshops:

Professor Mahesh Sharma’s next workshop in the series on improving mathematics education at Framingham State University on April 12, 2019 is on Mathematics as a Second Language. The focus will be on helping students develop mathematics language and its role in mathematics conceptualization and problem-solving. For more information and future workshops:

Professor Mahesh Sharma’s workshop on Algebra to Arithmetic is on March 15, 2019 at Framingham State University. It is open to teachers from fifth grade to tenth grade. Here is the link:

Professor Mahesh Sharma’s workshop on Algebra to Arithmetic is on March 15, 2019 at Framingham State University. It is open to teachers from fifth grade to tenth grade. Here is the link:

Summer Slide and Regression

The summer is over for almost two months.  For most schools, the new academic year started with enthusiasm and new vigor. During these last two months, I have visited school systems in several states. I have visited classrooms from early childhood to Kindergartens, from first to fifth grades, and middle school math classes to AP Calculus classes. I have met teachers in workshops and courses.  As a tutor and diagnostician, I have seen struggling students and also gifted and talented students in mathematics. This article is motivated by these school visits and the work with these students and teachers. This article is not about summers passed; it is about how to prepare for future summers and the fall openings of schools.

Every year when schools reopen, teachers spend an inordinate amount of time bringing students to the grade level so that they can begin with the grade level curriculum. Many students never reach that level or the level of mastery they had achieved before the summer as reported by the previous grade teacher. Teachers believe that their students learned the material in their classes as most of them passed the required tests. They claim that their students should know the material from the previous grade. But, it is common knowledge that many students have forgotten a substantial amount of the material due to the summer inactivity. The achievement gap, for many, increases every year.

This loss in learning is neither unique nor new to American education. It is a well-documented phenomenon of our education that students’ summer regression of learned material from the previous year has enormous impact on their future work. The phenomenon is popularly known as “summer slide.” It does not mean that children in other countries do not forget what they learned during the previous year.  They do. But, the amount that an average American student forgets is significantly more.

Recent research indicates that summer vacation can cost students up to two months of learning. Longitudinal researchshows that although low-income children make as much progress in reading during the academic year as middle-income children do, the poorer children’s reading skills slip away more during the summer months. Researchers shows that two-thirds of the 9th grade reading achievement gap can be explained by summer regression due to unequal access to summer learning opportunities during elementary school. The same situation is true about students’ mathematics achievement.

Research shows students lose more learning in mathematics than reading. The summer loss of learning in mathematics is alarming. The summer achievement gap in mathematics is not just a function of student background; most groups of students regress significantly except the high performing students.  However, students’ summer slide in mathematics is a complex phenomenon.

Reasons for Summer Slide in Mathematics
There are several reasons for this significant regression in mathematics.

First, in mathematics, many more children leave elementary grades without appropriate grade level content mastery—concepts, mastery of arithmetic facts, and place-value. For example, second graderswithout the mastery of addition and subtraction facts and place value up to thousands; fourth graders without mastering multiplication and division facts and place-value up to hundredths. They answer questions and solve problems relating to addition/subtraction, and simple multiplication/division in the classroom and on tests merely by counting (on fingers, on a number line, objects, or marks on a paper) without the real mastery of facts. Parents and teachers alike see this level of performance as the evidence of the mastery of this material. But this is not true mastery of addition, subtraction, multiplication, or division.

Students see addition only as ‘counting up,’ subtraction as ‘counting down.’  Multiplication, to them, is ‘skip counting up’ and division is ‘skip counting down.’ They do not have fluency in and efficient strategies for arriving at arithmetic facts. With this limited understanding of concept, students need a great deal of repetition (e.g., with flash cards) to achieve some level of fluency at a heavy cost of time and without making connections between numbers. They lack numbersense that can be used for efficient problem solving and building higher order thinking. This is a poor background for future arithmetic and mathematics. This understanding of fundamental concepts is not adequate for mastering concepts such as fractions, integers and higher mathematics.

Answers arrived at by counting leave little residue in the memory system of the outcome (number relationships or strategies). By counting strategies, no lasting number relationships are formed in the mind. In order to arrive at the answer, the counting process has to be repeated each time. Such partial-level mastery of skills is easily forgotten when not in use. Summer regression is more prevalent in the case of students with this level of mastery, irrespective of their SES backgrounds.

True mastery of facts (e.g., arithmetic facts) means: (a) understanding the concept (having language containersand conceptual schemas[1]supported by the appropriate, precise language)[2], (b) having efficient, effective, and elegant[3]strategies for arriving at facts, (c) acquiring accuracyand fluency, and (d) abilityand flexibilityto applyand communicateit.

Second, many schools (private and public) assign children readings (fiction and non-fiction) during the summer months. These readings rarely include books with mathematics content. And many libraries seldom display any books on mathematicians, mathematical way of thinking, ways of learning mathematics, or interesting events in mathematics developments. There are many books for school children, at all levels, with interesting mathematics content that can be included in summer reading.

What is even more important is that when teachers and schools decide to assign some summer mathematics review, it does not become a longer version of the regular homework. It is therefore important to consider what should be in the summer review and how it should be done.

There are key developmental milestones in mathematics learning (number concept, number relationships, place value, fractions, integers, and algebraic thinking), and important specific mathematics content related thinking skills students should learn and master. Summer review should focus only on reviewing and reinforcing important and efficient strategies related to these key concepts.

Apart from these developmental milestones, there are certain non-mathematical prerequisite skills that are essential for mathematics learning. These are: sequencing—ability to follow sequential directions, spatial ability—spatial orientation/space organization, pattern recognition, visualization, estimation, deductive andinductive reasoning. These skills help children learn mathematics better and are essential for mathematical way of thinking[4].

Third, most parents read and children see them reading. And many regularly read to their children. Sometimes parents even discuss their readings with other members of the family, including children. The ubiquitous presence of books with adults encourages children to get interested in books. More children, therefore, are inclined to get interested in reading.

Mathematics content is rarely the topic of discussion in family gatherings. If parents, out of fear of mathematics or lack of mastery, do not discuss mathematics with their children, they can play board and thinking games. Many of the mathematics skills are best learned through playing games and toys. When families play with games and toys the pre-requisite skills for mathematics learning and even direct mathematics skills are developed. Summer is a good time to do that, but they should also be part of children’s activities throughout the academic year.

Fourth, many parents and schools organize summer visits for children to places (historical monuments and interesting locations, museums, parks, libraries, etc.). Many of these visits have a limited focus on quantitative aspects. With planning these visits have the possibility of multiple types of rich experiences for children involving fun, history, culture, geography, literacy and numeracy. Parents and schools should, therefore, make an extra effort to include visits that also focus on science, technology, engineering and mathematics (STEM) content.

Socio-Economic Status and Summer Slide
Summer slide is present in all SES groups, but it is almost non-existent in high-performing students from any background and those who engage in some organized review and learning during summer. However, children in lower SES groups may lose more mathematics learning during summer months than their higher SES peers. Children performing at lower levels in mathematics in all SES groups forget mathematics almost equally.

In most urban schools, because of fewer resources, less prepared teachers, larger classes, and less involvement from parents, regular mathematics instruction is not adequate during the academic year. Children in these schools are exposed to simplistic strategies rather than efficient, effective, and elegant strategies in mathematics instruction. Children are exposed to limited and less challenging mathematics concepts, procedures, and applications. In such situations, the use of higher order mathematical thinking skills is limited. There are lowered expectations in class and limited homework is required of students. Expectations are also lower for special needs students in spite of smaller classes, extra support and resources.

Role of Integration of Language, Concepts, and Procedures in the Retention of Information/Learning
Instruction that lacks key elements of effective mathematics pedagogy may have long-term effect on student capacity for learning. For example, every mathematics idea consists of three components: linguistic,conceptual, and procedural. Children who have been taught to integrate these components during instruction using efficient models, rich questioning[5], and solving meaningful problems acquire a higher level of mastery. They show no or little regression and learn new concept easier and effectively.

In many schools, there is less emphasis on the development of language of mathematics (vocabulary, syntax,and translation from math to English and from English to math). Many teachers rush to teaching procedures in mathematics classes. When only few questions are asked in the classroom and inefficient models and limited language are used to teach new concepts and procedures, then students are less engaged and concepts and procedures are not integrated. In the absence of these principles, there is lower level of mastery and, therefore, more regression in student learning.

Mathematics language acts as container for holding mathematics concepts, procedures, and strategies. Without the language containers, it is difficult to retain and communicate the learned information. Student response to questions helps them integrate the new information with the existing information, therefore, it is possible to retain it. Under these conditions, learning is retained longer.

Role of Conceptual Schemasin Retention of Learning
Effective and efficient instruction models make the concepts and procedures transparent and show the congruence between the concrete, pictorial and abstract concepts easier for children. They are easier to visualize.  For example, using Cuisenaire rods and BaseTen equipment for constructing the area model can help children to connect the concept and procedure of multiplication from whole numbers to fractions to decimals to algebraic expressions easier. Strategies derived through these materials and models have the potential to be effective, efficient and elegant that help students to make better connections between concepts and learn and retain better.

The presence of rich and large math vocabulary and strong conceptual models are antidotes to summer slide.

Role of Expectations in Learning and Retention
Many suburban parents and schools have higher expectations from administrators, teachers and students alike. They select demanding curricula, better instructional materials, effective and appropriate professional development for teachers, more resources, and intentional, timely interventions to help students with learner differences. There, students and teachers devote more time on mathematics instruction, and, to some extent, are able to make up for the limited language of mathematics taught and even possible ineffective teaching that is responsible for most of the summer slide.

Strategies for Reducing Summer Slide
1. Intentional Focus on Mathematics
In the last decade, educators and schools have focused on boosting literacy skills among low-income children in the hope that all children read well by the third grade. But the early-grade math skills of these same low-income children have not received the similar attention. Many high-poverty kindergarten classrooms don’t teach enough math and the lessons on the subject are often too basic—based only on sequential counting. While this kind of instruction may challenge children with no previous exposure to math, it is often not engaging enough for the growing number of kindergarteners with some math skills.

In 2016, only about 40 percent of fourth-graders scored at a proficient level on a nationwide math assessment. Just 26 percent of Hispanic students and 19 percent of African-American children tested at the proficient level in fourth-grade math. Proficient students, generally, have an appropriate level of mastery as mentioned above. With such a level of mastery, one can make connections—a prerequisite to retention. Only a few high performers show significant summer slide; however, the lower third of the performers show significant regression.

2. Summer School
To counter summer slide, many school systems plan summer school.  A typical summer school program is the mathematics review of procedures. A great deal of content is covered in a very short time to recover credits or satisfy credit hours. Most of these programs fail to develop efficient and effective strategies in learning key developmental milestones in mathematics. Further, programs are so fast-paced that the possibility of making connections is rare. They are also not integrative.

Summer school can be an answer to the problem of summer slide with the right mixture of the elements of effective instruction. Intervention programs, including summer school, should focus on (a) tool building, (b) mastering key developmental milestones of mathematics concepts, (c) developing mathematics language containers, (d) learning efficient strategies (using effective concrete models) that are generalizable, and (e) refrain from undue emphasis on procedures.

3. Teacher Training and Professional Development
The key to reducing the achievement gap and summer slide is the quality teaching during the year. Adequate investments in quality professional development of teachers and administrators in improve teaching are at the core of any effort in narrowing achievement gap. Effective teachers are the real solution to the problem of summer slide. Some teachers need crucial classroom support to acquire better classroom management, understanding of math content better, and effective pedagogy that might have been missing in their teacher training programs. On the other hand, teachers not fully prepared to teach math are a major factor in the achievement gap—poor student performance, and summer slide. Schools with large numbers of low-income students tend to have the least qualified teachers when they should have the most qualified.

Professional development that is content-embedded, clinically demonstrated, and related to understanding the developmental trajectory of each concept being taught at a particular grade level is the key to improving the mathematics proficiency of the teachers. Understanding the trajectory of a concept means: where, how and in what form the concept was introduced in the curriculum, what is each teacher’s role in the development of the concept at different grade levels, how and in what form children are going to encounter this concept in the future grades. That means each teacher should know the trajectory of the content for n ±3 grades.

4. Focused Practice and Math Achievement
Research shows that reading just six to eight books during the summer may keep a struggling reader from regressing. Similarly, we have found that just learning and mastering one key developmental strategy (e.g., decomposition/ recomposition of numbers,making ten, double number strategy, empty number line, distributive property of multiplication,addition and multiplication facts using decomposition/recomposition, the role of pattern and cycle in place-value, divisibility rules, short-division, prime-factorization, etc.) and related ten problems a day at the grade levelcan not only check the slide, but can prepare students better for the next grade. During the academic year, the same approach is an antidote to summer slide, reduces the achievement gap and prepares the student for the next grade.

5. Role of Pre-Requisite Skills in Mathematics Learning
Learning disabilities of students compromise the development and acquisition of the prerequisite skills for mathematics learning. They are non-mathematical in nature but affect mathematics learning as their presence in a child’s skill-set makes it possible to acquire mathematics concepts. These skills act as anchoring skills. For example, following sequential directions an essential skill for standard procedures), pattern recognition for understanding concepts, spatial orientation/space organization for number relationships and geometry, visualization for transferring information from working memory to long-term memory, estimation for numbersense, deductive and inductive reasoning for understanding and developing mathematical way of thinking.  These pre-requisite skills are best learned through games and toys and use of concrete materials.

Therefore, in all interventions, during the summer as well, emphasis should be on efficient models that involve Concreteand Pictorialrepresentation activities, Visualizationof models and patterns, and then Abstract representation(CPVA)[6]. Concrete models should be appropriate for the concept and procedure (counting materials are not appropriate for understanding and constructing conceptual schemas and deriving procedures (e.g., using Cuisenaire rods using area model best derives multiplication facts and procedures). Choice of conceptual models and selection of concrete and pictorial representations should be such that they facilitate visualization, abstraction, and extrapolation.

The most important characteristic of CPVA is the congruence between concrete, pictorial, visualization, and abstract. For example, iconic representation of physical objects (even Cuisenaire rods) is not pictorial. For pictorial representation, one should use either Empty Number Line (ENL), Barmodel, rectangles for multiplication and division, orTransparent diagrams.

Appropriate concrete and pictorial materials and toys and games not only are necessary for learning mathematics content but also help in developing prerequisite skills for mathematics learning. Only efficient, effective and elegant materials provide students a preparation for grade level mastery and preparation for future grades. When a child has not mastered the previous grade’s skills and developmental milestones, during interventions (whether during the summer or during the academic year), the child should practice these non-negotiable skills and their relationships with the new skills.

6. Mastery of Non-Negotiable Skills and Achievement
When children leave the grade with the expected mastery of non-negotiable skills at that grade, they are better prepared for the future grade. Non-negotiable skills are the focus elements (language, concepts, and procedures) of the curriculum at that grade level. When a student has mastered the non-negotiable skills at the grade level, they can easily learn and master all the other concepts of the curriculum at that grade level. Such students are better prepared for next grades. For example, children leaving Kindergartenshould have mastered: 45 sight facts[7](two numbers that make a number up to ten(e.g., 10 is made up of 1 and 9; 2 and 8; 3 and 7; 4 and 6; and 5 and 5), teens’ numbers (e.g., 16 = 10 + 6, 17 = 10 + 7, etc.), know numbers up to at least 100, and recognition of 12 commonly found geometric figures/shapes in the environment. Children who have not mastered this family of addition facts up to 10 have difficulty mastering other addition facts (a non-negotiable skill at first grade.

Children leaving first grade, should have mastered 100 addition facts (using strategies based on decomposition/recomposition of number) and 3-digit place value (with canonical and non-canonical decomposition of numbers); second grade, mastery of 100 addition and 100 subtraction facts (using strategies based on mastery of addition facts and decomposition/ recomposition of number), place value into 1000s, describing 12 commonly found geometrical figures/shapes.

By the end of second grade, children should have mastered additive reasoning (addition and subtraction concepts and that addition and subtraction are inverse relationships) and third gradeshould master multiplication concept, multiplication tables (10 by 10), procedures, and place value up to millions.

If students lack the mastery in math non-negotiable skills in elementary and then in middle school, they are less likely to be prepared for the more advanced math courses required for graduating from high school and preparation for college and careers They will also face hurdles in most jobs.

What is important to emphasize as “mastery”? Up to second grade, one can answer all of the questions on a test by just counting and without retaining the outcome of this counting.  These students might have done fine on the exit test from Kindergarten through 3rdgrade by using the counting strategies, but they will have difficult time where counting does not work well (multiplication, division, fractions, proportional reasoning, algebraic thinking, etc. ). When there is true mastery, the amount of regression is minimal.

What Can Parents Do?
Research shows that parental involvement in a child’s education and in school has a powerful influence on their academic performance.It could include: reading aloud, discussing the numbers/quantities children encounter in their environment, helping children to master arithmetic facts, creating physical and emotional learning conditions so they can study, checking homework, attending school meetings and events, setting expectations, relating current behavior and skills with future accomplishments, setting academic and personal goals, and discussing school activities at home. Research shows that when students understand their personal learning goals and receive timely and meaningful feedback as they progress, there is a positive impact on student learning.

Mathematics is everywhere around us. There can be many opportunities for families to build positive memories around mathematics as part of the daily conversations about mathematics. This helps students see the relevance and importance of mathematics in their lives.

The basis of mathematics is: Quantitative reasoning—observing, creating, extending, and using patterns in quantity/numbers—number concept, numbersense, numeracy; Spatial reasoning—patterns in space, shapes and their relationships; and Logical Reasoning—deductive and inductive reasoning.  Developing mathematical way of thinkingis to help children integrate these reasonings. Talking to children about numbers, quantities, shapes, number relationships, and involving them in making quantitative and spatial decisions is one of the ways to foster their numbersense and spatial sense.

Games and Their Uses in Learning Mathematics
Prerequisite skills for mathematics learning are best acquired through games and toys. To get children interested in games and toys, adults should introduce children to their own favorite games. Playing such games is like sharing a favorite book. I remember, in the summer vacations from school, during our visits to my grandfather’s village in India, we designed games, made toys, and enjoyed those games and toys for several hours every day. Invariably, villagers would stop by and offer their suggestions in designing games. Then, during the play, they would offer strategies for winning the game, new ways of playing old games. They introduced us to their favorite games. Our elders ramped up the game experience by asking other family members to explain their reasoning and strategies while playing. Those memories are still so fresh in my mind.

Games invite us to solve problems—learning rules of the games, following instructions, understanding and meeting the goals of the game. Observing and evaluating others’ strategies helps improvising and improving one’s own strategies. By engaging in logical and spatial reasoning and productively struggling in the game, children learn to lose and win gracefully. Games help prepare a player to visualize quantitative and spatial information, communicate ideas, and plan ahead—essential skills necessary for learning mathematics and solving problems. Such experiences will make them better mathematics learners and lower summer slide.

Playing games and toys that use dice, dominos, and visual cluster cards teaches numbersense and spatial sense.

Games involving playing cards (particularly Visual Cluster Cards), dominoes, or dice bring together the essential number skills. Many card and board games reinforce number concept and numbersense, but most importantly they develop logical reasoning and the communication of ideas.

Benefits of Board Games and Toys in Learning Mathematics
Because of their intrinsic entertainment value, board games provide educators and parents with an effective tool for engaging students. Games facilitate a welcoming learning atmosphere because students think they’re just having fun.

The benefits of board games are not limited to mathematics. They can build vocabulary, spelling, and logical reasoning skills. Here are few examples.

  • Memory[8]: to learn basic terminology and hold information in the mind’s eye (e.g., short-term memory receives more information because games and toys are multi-sensory); visualization improves working memory; and making connections and applying information strengthens long-term memory.  For example, the game Simonimproves sequencing, visual and auditory memories, etc.;
  • Inductive thinking(going from specific examples to generalrules):the game Battleshipsvery quickly transfers the rules from the board game to locatingpoints on the coordinate plane;
  • Deductive thinking(applyinggeneral rules to specific problems and situations): the game Master Mindimproves deductive reasoning;
  • Spatial orientation/space organization(learning relational words, such as: close to me, to my left, above me, below the table, under the plate, etc.): the game Connect Fouror Cubichelp children learn spatial relationships; and
  • Task Analysis: In board games, we break down a given/larger problem into smaller, manageable, solvable moves/tasks that help in problem solving.

Games and toys teach childrenskills that help them learn, retain, and master formal concepts, skills, and procedures in mathematics.

Characteristics of “Good” Games and Toys
Many commercial and homemade games and toys and apps help children prepare for learning. However, to develop necessary skills successfully, games and toys should have certain characteristics:

  • Games should be based on strategies,not on luck. In other words, becoming proficient in a game means proficiency in the strategies of the game.  A child’s encounter with the game or toy should help him/her discover something more about the game, i.e., a new strategy or getting better at an old strategy, a new perspective, or a new relationship between moves. For example, the board game Mankalah(it has different names in different continents) is “easy to learn, but a life time to master.” Such games are interesting to novice and expert alike and help children improve their cognition, inquisitiveness, perseverance, visualization, and executive functions (working memory, inhibition, organization andflexibility of thought)[9].
  • In general, a game should last on an average of ten to fifteen minutes so that children can see the end of the game in a fairly short period of time. This helps them understand the relationship between a strategy and its impact on the game and its outcome. This teaches children the foundation of deductive thinkingor the relationship between cause and effect. When a child has more interest and maturity and is able to handle delayed gratification, complex strategy games such as Chess,Go, and multi-step/concept games are meaningful.
  • Each game should help develop at least one prerequisite mathematics skill. For example, the commercially available game Master Mindis an excellent means for developing pattern recognition, visual memory, visualization, and deductive thinking. The Number Master Mindgame, on the other hand, is excellent for developing numbersense. The advanced version, Super Master Mind, makes it very challenging.

Following is a list of games and toys I have used extensively with children and adults to develop prerequisite skills for mathematics concepts and thinking skills. Most of these games and toys are commercial. It is not an exhaustive list and changes constantly.  When I find a new game or a toy I play with it, examine it for its usage, use it with children, assess its impact on children, and identify the corresponding prerequisite skills it develops for mathematics learning. Sometimes, I modify it and when it satisfies the conditions, I include it in my list.[10]

For example, the toy Invicta Balance  (Math Balance), originally was designed by mathematician Zolton Dienes to teach children number concept and the concept of equality. I have modified it not only to derive addition, multiplication, and division facts but also to teach rules and procedures of solving equations with one variable effectively. Cuisenaire rods(designed by Belgian educator Cuisenaire), Montessori colored rods(Italian educator and physician Montessori), and Base-Ten blocks(Zolton Dienes)  were originally developed for teaching number concept and whole number operations. I have modified them to teach all standard arithmetic operations, teaching time, money, and measurement, operations on fractions, decimals, percents, and algebraic operations, and solving linear and quadratic equations.

List of Games (with identified prerequisite skills)

  • Battleships (spatial orientation, visualization, visual memory)
  • Black-Box(logical deduction)
  • Blink(pattern recognition, visual memory, classification, inductive reasoning)
  • BritishSquares(spatial orientation, pattern recognition)
  • CardGames(visual clustering, pattern recognition, number concept—visual clustering, decomposition/recomposition of number, number facts) (see Number War Games)
  • Checkers (sequencing, patterns, spatial orientation/space organization)
  • Chinese Checkers (patterns, spatial orientation/space organization)
  • Concentration (visualization, pattern recognition, visual memory)
  • Cribbage (number relationships, patterns, visual clusters)
  • Cross Number Puzzles (number concepts, number facts)
  • Dominos (visual clusters, pattern recognition, number concept and facts, decomposition/recomposition, number) (Number War Games)
  • FourSight(spatial orientation, pattern recognition, logical deduction)
  • Go Muko(pattern recognition, spatial organization)
  • Go Make___(number concept, number facts, decomposition/ recomposition)
  • Hex(pattern recognition)
  • InOneEarandOuttheOther[11](number relationships, number facts, additive reasoning)
  • Kalah, Mankalah,or Chhonka(sequencing, counting, estimation, visual clustering, deductive reasoning)
  • Krypto(number sense, basic arithmetical facts, flexibility of thought)
  • Math Bingo Games(number facts)
  • Guess My Number (Numbersense, deductive reasoning)
  • MasterMind(sequencing, logical deduction, pattern recognition)
  • Number Master Mind(number concept, place value, numbersense)
  • NumberSafari[12](numbersense, equations)
  • Number War Games[13](visual clustering, arithmetic facts, mathematics concepts, deductive reasoning, fluency of facts)
  • Othello (pattern recognition, spatial orientation, visual clustering, focus on more than one aspect, variable or concept at a time)
  • Parcheesi (sequencing, patterns, number relationships)
  • PinballWizard[14](number facts, a paper/pencil game)
  • Pyraos (spatial orientation/space organization)
  • Quarto (spatial orientation/space organization, patterns, classification)
  • Qubic (pattern recognition, spatial orientation, visualization, geometrical patterns)
  • Reckon(number facts, estimation, basic operations)
  • ScoreFouror ConnectFour or3-D Connect Four(pattern recognition, spatial orientation, visual clustering, geometrical patterns)
  • Shut the Box andDouble Shut the Box (sequencing, number concept, and number facts—making Ten)
  • Simonor Mini Wizard(sequencing, following multi-step directions, visual and auditory memory)
  • Snakes and Ladders (sequencing, following multi-step directions, visualization, number facts)
  • Stratego (spatial orientation, logical deduction, graphing)

Selection of a game or toy to play with should reflect the prerequisite skills the child needs. Once children begin to get interested in a game/toy, they are inclined to play with other games.

Number War Games[15]
A category of games that I designed and started using with childrenalmost 40 years ago arebased on the popular Game of War. They are played using Visual Cluster CardsTM.These games are a versatile set of tools for teaching mathematics from number conceptualization to introductory algebra.

Visual Cluster Cards are numberless cards designed with specific patterns of objects (icons) on them. The cluster of icons on the card represents the numeral to be used in the game with children up to age 11. After that, the cards can be used for operations on integers. Then, the cluster on the card represents the numeral and the color of the cluster gives the sign of the numeral to make it into a number.  For example, the five of clubs or spade represents +5 (based on the idea “in the black”) and five of diamonds or hearts represents -5 (based on “in the red”).

Number War Games are played essentially the same way as the popular American Game of War and are easy to learn.

Children love to play these games. I have successfully used them for initial, regular, and remedial instruction. And, later on, I use them  for assessment. The games are also very good for reinforcement of facts. These games are ideal for formative assessment. They are particularly suited for learning number, arithmetic facts, comparison of fractions, and understanding and operations on integers.

Once children master arithmetic facts (addition, subtraction, multiplication, and division) with these cards, using decomposition/ recomposition, one could extend the games to fractions, integers, and algebra wars. In the Algebra War game, one with bigger value for P = 2x + 3y, wins, where x is the value of the red card and yis the value of the black card.

The algebraic expression for P changes (P = x2+ y2, P = 2x/3y, P = |x| − 3|y|, etc.)with each game (See Number War Games[16]for detailed instructions).

Furthermore, games and play provide opportunities for discussions of strategies, outcomes, and feedback to improve thinking and strategies. Conversations invite children to communicate concepts while sharpening their thinking skills such as their ability to make inferences, to support their arguments with reasons, and to make analogies—skills essential to learning and applying mathematical skills.

Where discussions are encouraged, children begin to ask questions. They learn to evaluate answers, draw conclusions, and follow up with more questions. They begin to differentiate between convergent (a question that calls for a yes, no or a short answer) and divergent (a question that calls for an answer with explanation) types, which strengthens their facility of reasoning. Learning and using reasoning is the core of mathematics learning.

Without discussions, children may become procedurally oriented. Children who hear talk about quantity—counting and use of numbers at home, begin school with more extensive mathematical knowledge—more number words, comparative words, and sizes of numbers, relating numbers, and combining and breaking numbers apart—knowledge that predicts future achievement in mathematics.

Similarly, discussions about the spatial aspects of their world have an impact on their understanding about the spatial properties of the physical world—how big or small or round, sharp objects, angles, or sides are—relationships between geometric objects. Both quantitative and spatial discussions give children’s problem-solving abilities that create an advantage in future mathematics.

Mathematical objects (numbers, concepts, operations, symbols, etc.) seem abstract and unreal, but when a child begins to enjoy mathematics they become real, almost concrete objects. Doing real mathematics is like playing a game; it is thinking about and acting upon mathematical objects and discovering multiplicity of relationships among them. Mathematics uses and develops the same mental abilities that we use to think about physical space, other people, or games and toys. To engage children in mathematics and excite them about mathematics learning, they need to see mathematics as a collection of interesting games and a means of communication.  This communication is enhanced when there is an intentional effort to talk about mathematics to children.

Summer slide is the result of what happens during the whole year.  The antidote for this condition is to provide quality mathematics instruction during the academic year. I urge administrators, teachers, and parents to provide the best possible mathematics education to all children throughout the year so that when they come back after the summer, we do not devote time on endless review.


[1]Language instigates models,

Models help develop conceptual schemas and instigate thinking,

Thinking instigates understanding,

Understanding produces competent performance,

Competent performance results in long-lasting self-esteem, and

Self-esteem is the motivating factor for all learning.

[2]For example, multiplication is not just counting up, it is ‘repeated addition’, ‘groups of ,’ ‘an array,’, and ‘area of a rectangle.  It is an abstraction of addition, just like addition is abstraction of counting. These four models of multiplication give rise to the corresponding four models of division.

[3]An effective and efficient strategy becomes elegantif and when it can be generalized, extrapolated, and abstracted. Elegant strategies result into conjectures.  These conjecturesmany times result into theorems, procedures, and important mathematics relationships.

Such strategies give rise to the understanding of the patterns and regularities that underlie the deep mathematics structures.

These strategies result in developing and understanding properties of numbers, operations, and procedures. They are the basis of long-standing standard arithmetic procedures, algebraic systems, and geometric relationships.

[4]All arithmetic procedures involve a series of sequential steps: long-division, adding fractions with different denominators, solving simultaneous linear equations, etc. Students with poor visualization are poor in mental arithmetic and multi-step problem solving.

[5]Questions instigate language, language instigates models, and models….

[6]For more comprehensive treatment see Levels of Knowing in Mathematics Learning(Sharma, 199–)

[7]See the post on Sight Words and Sight Factson this blog.

[8]See several posts on Working Memory and Mathematics Achievement on this Blog.

[9]See several posts on Executive Functions and Mathematics Achievementon this blog.

[10]I am always looking for new games and toys. If you come across a new game and want to discuss a game or a toy, please contact me at the Center (

[11]Available from the Center.

[12]Available from the Center.

[13]Available from the Center.

[14]Available from the Center.

[15]The Descriptive Booklet (Games and Their Uses) available from the Center.

[16]Available from Center for Teaching/Learning of Mathematics

Summer Slide and Regression

Framingham State University: Mathematics Education Workshops with Mahesh Sharma – 2018-2019

Several national professional groups, the National Mathematics Advisory Panel and the Institute for Educational Sciences in particular, have concluded that all students can learn mathematics and most can succeed through Algebra 2. However, the abstractness and complexity of algebraic concepts and missing precursor skills and understandings–number conceptualization, arithmetic facts, place value, fractions, and integers–may be overwhelming to many students and teachers.

Being proficient at arithmetic is certainly a great asset when we reach algebra; however, how we achieve that proficiency can also matter a great deal. The criteria for mastery, Common Core State Standards in Mathematics (CCSSM), set for arithmetic for early elementary grades are specific: students should have (a) understanding (efficient and effective strategies), (b) fluency, and (c) applicability and will ensure that students form strong, secure, and developmentally appropriate foundations for the algebra that students learn later. The development of those foundations is assured if we implement the Standards of Mathematics Practices (SMP) along with the CCSSM content standards.

In these workshops, we provide strategies; understanding and pedagogy that can help teachers achieve these goals.  All workshops are held on the Framingham State University campus from 8:30am to 3:00pm. Cost is $49.00 per workshop and includes breakfast, lunch, and materials.

PDP’s are available through the Massachusetts Department of Elementary and Secondary Education for participants who complete a minimum of two workshops together with a two-page reflection paper on cognitive development.

A. Creating A Dyscalculia Friendly Classroom
Learning Problems in Mathematics (including math anxiety)

For special education, regular education teachers, interventionists, and administrators

September 28, 2018
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning. The major aim is to deliver mathematics instruction that prevents learning problems in mathematics from debilitating a student’s learning processes in mathematics.

B. Number Concept, Numbersense, and Numeracy Series
Additive Reasoning (Part I): How to Teach Number Concept Effectively

For K through grade second grade teachers, special educators and interventionists

October 26, 2018
Number concept is the foundation of arithmetic. Ninety-percent of students who have difficulty in arithmetic have not conceptualized number concept. In this workshop we help participants learn how to teach number concept effectively. This includes number decomposition/recomposition, visual clustering, and a new innovative concept called “sight facts.”

Additive Reasoning (Part II): How to Teach Addition and Subtraction Effectively

For K through grade third grade teachers, special educators and interventionists

November 30, 2018
According to Common Core State Standards in Mathematics (CCSS-M), by the end of second grade, children should master the concept of Additive Reasoning (the language, concepts and procedures of addition and subtraction). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students.

Multiplicative Reasoning (Part III): How to Teach Multiplication and Division Effectively

For K through four second grade teachers, special educators and interventionists

December 14, 2018
According to CCSS-M, by the end of fourth grade, children should master the concept of Multiplicative Reasoning (the language, concepts and procedures of multiplication and division). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving this with their students.

C. Proportional Reasoning Series
How to Teach Fractions Effectively (Part I): Concept and Multiplication and Division

January 25, 2019 

For grade 3 through grade 9 teachers and special educators

According to CCSS-M, by the end of sixth grade, children should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and multiplication and division of fractions and help their students achieve that.

How to Teach Fractions Effectively (Part II): Concept and Addition and Subtraction

For grade 3 through grade 9 teachers

February 15, 2019
According to CCSS-M, by the end of sixth grade, students should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and operations on fractions-from simple fractions to decimals, rational fractions and help their students achieve that.

D. Algebra

Arithmetic to Algebra: How to Develop Algebraic Thinking

For grade 4 through grade 9 teachers

March 15, 2019
According to CCSS-M, by the end of eighth-grade, students should acquire algebraic thinking. Algebra is a gateway to higher mathematics and STEM fields. Algebra acts as a glass ceiling for many children. From one perspective, algebra is generalized arithmetic. Participants will learn how to extend arithmetic concepts to algebraic concepts and procedures effectively and efficiently. On the other perspective, algebraic thinking is unique and abstract and to achieve this thinking students need to engage in cognitive skills that are uniquely needed for algebraic thinking. In this workshop we look at algebra from both perspectives: (a) Generalizing arithmetic thinking and (b) developing cognitive and mathematical skills to achieve algebraic thinking.

E. General Topics
Mathematics as a Second Language: Role of Language in Conceptualization and in Problem Solving

For K through grade 12 teachers

April 12, 2019
Mathematics is a bona-fide second language for most students. For some, it is a third or fourth language. It has its own vocabulary, syntax and rules of translation from native language to math and from math to native language. Some children have difficulty in mathematics because of language difficulties. Most children have difficulty with word problems. In this workshop, the participants will learn how to teach effectively and efficiently this language and help students become proficient in problem solving, particularly, word problems.

Learning Problems in Mathematics (including dyscalculia)

For special education and regular education teachers 

May 17, 2019
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics, and (d) how to fill gaps in mathematics learning.

Standards of Mathematics Practice: Implementing Common Core State Standards in Mathematics

For K through grade 11 teachers (regular and special educators)

June 7, 2019
CCSS-M advocates curriculum standards in mathematics from K through Algebra II. However, to achieve these standards, teachers need to change their mind-sets about nature of mathematics content; every mathematics idea has its linguistic, conceptual and procedural components. Most importantly, these standards cannot be achieved without change in pedagogy-language used, questions asked and models used by teachers to understand and teach mathematics ideas. Therefore, framers of CCSS-M have suggested eight Standards of Mathematics Practice (SMP). In this workshop, we take examples from K through high school to demonstrate these instructional standards with specific examples from CCSS-M content standards.

Click here to register

Framingham State University: Mathematics Education Workshops with Mahesh Sharma – 2018-2019

How To Improve Numbersense – Number Relationships: Counting Part Three

We want children to have a ‘feel’ for numbers—the ability to work flexibly in solving number problems. That is called numbersense. Numbersense is the mastery of number concept, number relationships, and place value and their integration. Mastery means (a) understanding, (b) effective and efficient strategies, (c) fluency, and (d) applicability. Numbersense leads to the mastery of numeracy.

Mastery of numeracy should be an essential outcome of the elementary school (grades K through 4) mathematics curriculum. It is the facility in executing the four whole number operations, including standard algorithms, correctly, consistently, and fluently with understanding.

Poor numbersense in children is due to inefficient strategies such as relying on sequential and rote counting of objects (e.g., blocks, chips, fingers, or marks on a number line). Learning facts and procedures through rote memorization without understanding does not help children in making connections between numbers, arithmetic facts, concepts and procedures. When they encounter new concepts or need to apply mathematics ideas to problems, they find it difficult. And, they give up easily. As a result, many are termed “slow learners.” Often, our pedagogy turns them into slow learners.

Able children are shown and practice efficient, effective, and elegant strategies. Less able or children with special needs simply are not shown the same techniques. With inefficient and less effective strategies, children end up spending enormous amounts of time deriving even the simple facts. This makes the tasks laborious and they either do not succeed or lose interest and lag behind.

Definitions of arithmetic operations such as: addition is counting up, subtraction is counting down, multiplication is only skip counting forward, and division is skip counting backward, do not lead to efficient strategies. For example, less successful children see subtraction as an isolated concept without connecting it with addition. They do not capitalize on learned addition facts. A similar situation happens with division. They end up spending more time on acquiring mastery of subtraction and division facts with limited results. These children have difficulty becoming flexible and fluent in arithmetic facts. Mastery of arithmetic facts is an essential element of numbersense. When addition and subtraction are shown as inverse concepts/operations, the mastery in one reinforces the other. Similarly, after initial introduction of multiplication, children should be taught that multiplication and division are inverse operations.

Mastering arithmetic facts using efficient and effective strategies and models frees children’s working memory. Then, they can engage in learning and mastering higher order thinking skills and applications, easily and effectively. Higher order thinking is dependent on flexible numbersense and the mathematical way of thinking.

The mathematical way of thinking is the ability to: observe patterns in quantity and space, visualize relationships, make conjectures, predict results, and then communicate observed connections and their possible extensions using mathematics language and symbols. Mastering arithmetic facts is a necessary, though not sufficient, condition for higher mathematics. Competence in numbersense translates into effective mental math—the hallmark of mathematical thinking.

Number Relationships
Children with numbersense make connections, generalizations, abstractions, and extrapolations of number patterns they observe and processes they have mastered. They link new information to the existing knowledge and develop insights about number and their relationships.

Understanding Number: Spatial and Quantitative Relationships
The fundamental relationships between numbers at elementary level are expressed in two forms:

Spatial: The spatial aspect of number is determining the relationships between numbers by their locations and proximity with each other. The child knows a number when she can locate and place the number on an empty number line in relation to other numbers (to the right of, left of, how far from, or how close to a given number). Being able to point to a number and its place on a number line is not enough to understand number relationships.

Spatial aspect also relates to positional aspect of number: e.g., the second from the start, third person in the row, tenth’s book in the row, etc. The numbers in this form are called ordinal numbers. Children learn ordinal numbers before they learn the quantitative aspect of number.

Quantitative: The quantitative aspect of number is the value of the number. How big? How small? More than? Less than? It is the understanding that a number represents the magnitude of a collection. It is knowing that number is the property of the collection, not just the result of counting. And that the last number used in the count, from any direction, indicates the size of that collection. This value has a unique place on the number line. This is the cardinal aspect of the number (the magnitude, numberness).

Numberness is to know: Is the number bigger than another number? What number is half-way between 10 and 20? Can you place ‘three numbers’ between 45 and 55? What digit is in the tens’ place? What is the value of the digit in the tens’ place? What is 10 more than 67? What digit in the given number has the highest value? What is 8 + 6? What should we add to 9 to get to 17? What is the difference between 17 and 9?

Any question about number relates to both aspects of the number, but questions such as the following mainly relate to the spatial relationships between numbers:


On the whole number line above, what number comes after 17? Place 22 on the number line. What number comes before 45? Is the number 29 closer to 20 or 30? To answer these questions, the child refers to the spatial idea: Where is that number located in relation to other number(s)?

The following questions are related to quantitative relationships between numbers: Without referring to or drawing a number line: Give a number between 23 and 29. What number is 10 more than 54? What is 8 + 6? Give a number between and ½? What number is 3 less than 23? What number is more than 3? What is the next tens’ number after 53? (Tens number are: 10, 20, 30, 40, 50, 60, etc.)

If a child can answer these questions only by the help of a number line, then that is not indicative of mastery of the number. Applying only spatial, sequential counting to derive arithmetic facts disadvantages children.

Many children use only the spatial aspects of number in deriving and understanding number relationships (facts). When they have to answer questions such as: What is 5 more than 7 or 2 groups of 7 they get the answer by counting on a number line, objects, or on fingers. Truly understanding number relationships and acquiring efficient strategies for mastering arithmetic facts, one needs to integrate spatial and quantitative aspects of number. To learn efficient strategies for mastering numbers facts one needs: (a) sight facts (including making ten), (b) what two numbers make a teen’s number (e.g., 16 = 10 + 6, and (c) what is the next tens after 43, etc.) Instructional models such as Visual Cluster Cards, Cuisenaire rods, and Empty Number Line help in this integration and acquiring effective, efficient strategies.

The concept of place value is an example of this integration. To know the whole number 235 well, one has to focus on the spatial aspects of the digits (1’s, 10’s, and 100’s places; although the numbers increase to the right, the place values of digits in a multi-digit number increase to the left) and the values of these individual digits contribute to the understanding of the value of the whole number itself. For example, both the Standard (5,694) and the Expanded Forms (5000 + 600 + 90 + 4) and later on, Place-Value form (5×1,000 + 6×100 + 9×10 + 4×1), and Exponential form (5×103 + 6×102 + 9×101 + 100) of the number take advantage of understanding and mastery of quantitative and spatial aspects of number. The same concept is then extended to factions and decimal numbers.

Making Numbers and the Number Line Friendly
Daily Counting Using Number Line
To develop number relationships, forming a visual image of a number line is important. This means: (i) mentally locating numbers on the number line, (ii) recognizing the patterns and structure of the number system, (iii) extending those patterns (e.g., 3 comes after 2, so 23 comes after 22, 73 comes after 72, 173, comes after 172), and, (iv) applying these patterns to solve quantitative problems. This competence is the beginning of developing a robust numbersense.

Games and Toys
Children develop number relationships through routine counting while interacting with their environment as part of normal growth and development. Playing with games, toys and remembering number rhymes and stories bring out counting and number relationships. Board games, using dice, dominos, and playing cards are opportunities for learning number relationships. However, informal and infrequent play may be slow or inefficient for the development of number relationships. Formal exposure to appropriate, diverse activities and effective strategies assures efficient development of numbersense in children. For example, Number War Games[1] using Visual Cluster CardsTM (VCC), dominos, and dice are excellent examples of such activities.

Formal Counting
Children’s practice of meaningful, strategic counting is an important preparation for developing efficient calculation strategies. Counting is a complex process. It involves several sub-skills and takes considerable time to become fluent and competent. Unitary counting (sequential counting from 1) is children’s first exposure to the structure of number line, but it becomes progressively complex with age and grade.[2] For example, it should progressively include counting by 2s, 5s, 10s, 100s, by a unit fraction, proper fraction, mixed fraction, decimal, etc., starting with any number and moving forwards and backwards. Such progressively complex counting strengthens numbersense.

As children become competent in counting, they begin to visualize the number line—number patterns, locations of numbers and their relationships. Crossing the decade/century and realizing the counting patterns is an important achievement for children in understanding the structure of our Base Ten number system. Children’s observed number patterns on real and visualized number line help them develop strategies that give them power to develop and understand efficiency of arithmetic operations. For example, a child observes that 42, 52, 62, 72, and 82 is a sequence of numbers increasing by 10 and they occur 2 after respective decades (tens). She, later, uses it to solve a real problem: what is the change when she has spent 52 cents from a dollar? She discovers that the change could be calculated by counting up by 10 from 52 till she reaches 92 and then 3 more to 95 and 5 more to the dollar (e.g., 4 dimes + 3 cents + 1 nickel = 40 + 3 + 5 = 48 cents). And a little later, she realizes, 52 and 5 tens is 102, that is 2 more than the dollar, so change is 48 cents. Or, 50 + 5 tens = 100, but we should have started at 52, so it 50 – 2 = 48 cents. Similarly, at a later date, to find the product 16×3, a child thinks: 16 is 10 + 6. 10×3=30, 6×3=18, 30+10=40, 30+18=48, so 16×3=48.

What To Do During Counting?
Counting should be a whole class activity, first oral and then in writing. Counting should begin with a number line (with numbers marked and displayed from 0 to a number beyond 100). A portion of the number line, preferably from 0 to 35 should be at children’s eye level and rest on the wall.

During the counting activity, the teacher should emphasize when a decade is complete. She should help children see that something important is happening when they reach a new decade—a new group of tens. Similarly, she should point out what is happening immediately after and before tens numbers (e.g., multiples of 10). She should emphasize what is before and after the new decade (the new ten). Knowing what is before and after that decade is a difficult concept for many children. For example, she should point out that when the count reaches a new tens, e.g., 30, 40, 50, …, the cycle of the count of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 is complete and then repeats again and again.

In counting whole numbers on the number line, children should be able to realize the cyclic pattern of the base-ten system:

…29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, …etc.

This cyclic pattern is the key to understanding number relationships. Initially, counting should be done on a number line in linear form (as seen and discussed above) so children develop the idea that numbers are continuously increasing to the right and decreasing to the left. In later grades they will extend it and understand the idea of positive infinity (+∞) and negative infinity (−∞).

When children see the progression of number in a 10×10 grid form, they see cyclic patterns much more clearly. This understanding of number relationships and structure leads children to arithmetic operations.

1,   2,   3,   4,   5,   6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, …

Most children develop this structure independently with little help from others. However, for others, it is important to formally develop it.

Mid-line Crossing Problem
It is important to begin counting on the horizontal grid (shown above). Some children, because of their mid-line crossing problem (MLCP) may not discern the pattern easily as they do not see the horizontal numbers on a number line as “equidistant.” For example, many children with MLCP, see the equidistant numbers displayed in the first row (below) as in rows two or three where the numbers are not equidistant. In row two, they are jumbled up in the two ends and in the third row, they are jumbled up in the middle.

1,   2,   3,   4,   5,   6,   7,   8,   9,   10    (row one)
1, 2, 3,   4,     5,     6,           8,9,10    (row two)
1,   2,   3,   4, 5, 6, 7,      8,  9,   10     (row three)

When the numbers are organized vertically, it is easier for them to see the patterns.

Counting Using Number Grid
Number Grids are horizontally (figure one) and vertically organized (below) One). The Horizontal Grid is a 10×10 grid, with entry of 1 in top left most cell. Each row ends with a multiple of 10. The Vertical Grid is a 10×10 grid with top left most cell with entry of 1. Each column ends with a multiple of 10. The procedure for counting using the grids is the same as the number line. Counting on grids can be done horizontally and vertically.

Screen Shot 2018-07-03 at 3.49.37 PM

Locating Numbers on an Open/Empty Number Line
Kindergarten and First Grade
On one side of the room hangs a clothes line (low enough so children can reach it and high enough so it does not interfere in their movement). On clothes pins write numbers in dark ink from 1 to through 100. The multiples of ten numbers are written in red. Similarly, the numbers with 5 in the one’s place are written in green.

The numbers: 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 are known as Bench Mark Numbers—important in our Base-Ten system. With practice, children see all numbers in relation to bench-mark numbers. Bench-mark numbers serve as the markers for estimation, location of numbers, and approximating the outcome of arithmetic and algebraic operations.

Place the numbers in two buckets. Numbers 1 through 30 in one bucket and numbers 1 through 100 in the second.

In later grades, children encounter other bench mark numbers, such as: ½ (.5, 50%, etc.), square numbers, certain products of numbers (e.g., multiplication tables), π, standard trig functions (e.g., trigonometric function values of 30°, 45°, 60°, etc.) and important parent functions in algebra.

The teacher points to the clothesline and asks each child, in turn, to pick a number from the bucket and place it on the clothes line in its place. Children take their turn placing their numbers. A child can move or adjust the place of the numbers already placed on the line in order to locate his/her number. Teacher should ask children the reason for the placement of their numbers, adjusting the numbers on the number line, and the relationship of their number with other numbers, particularly with the bench-mark numbers.

In the beginning of the year, the teacher should use numbers from 1 to 30. After about 2 months, she should use numbers up to 100 and beyond. After half-year, the teacher should give children an empty number line (ENL) on a sheet of paper where two end numbers are written, and she dictates numbers and children locate the number’s place and write the numbers on the ENL.

Grades Two and Three
The teacher should give children a sheet of paper (2”×11”) each side having an Empty Number Line (ENL) drawn on it with two end numbers. The end numbers change every week.

Screen Shot 2018-07-03 at 3.52.26 PM

Ÿ She dictates 10 random numbers between the two end numbers and children locate the number’s place and write the numbers on the ENL as the numbers are dictated. After children have located the numbers they compare their ENL with their partners and come to agreement on the locations of these numbers. The corrected locations are placed on the ENL on the other side of the paper. This activity should be part of a daily math lesson.

Grades Four through Six
The same activity as in the grades 2 through 3, but the choice of numbers changes. The numbers can be whole numbers, fractions (unit fractions, proper fractions, mixed fractions), decimals, and percents.

Grades Seven through Nine
The same activity as in grades 4 through 6, is repeated but the choice of numbers changes. The numbers can be real numbers (whole numbers; fractions—decimal numbers, percents; integers; rational numbers and irrational numbers).

Activity Two
Every day, before children arrive, the teacher places cards with random numbers written on them (numbers appropriate to grade level). Each child picks a card, and when children line up, each child follows the order by the number on his/her card. Children keep their cards ready; before any classroom activity, the teacher calls on them by specific criteria: The person with the card between 1.5 and 1.6 will answer the next question. The choice of numbers changes every day.

Daily Oral Counting
Daily counting is a warm-up activity for grades K through 8. It could be part of the calendar activity in grades K through 2. The choice of number to count with should be related to the main mathematics concept taught in the classroom that day. For example, when children, in the third grade, have been introduced to fractions, it is a good idea to count by unit fractions. Similarly, when children are adding and subtracting fractions with same denominators, counting should be backward and forward by a proper fraction. It is one of the tools for helping children to have a deeper understanding of number.

Each child should have a Math Notebook where all of his/her mathematics work is recorded. It is the sequential record of classroom mathematics writing: language, concepts, operations, definitions (examples and counter examples), conjectures, proofs, formulas, calculations, constructions, drawings, sketches (geometrical shapes, figures, diagrams, etc.), summary and reflections on class mathematics work. The written work should be done in pencil. Only when the teacher wants a short answer, an example, immediate feedback to a definition, concept, or a procedure, children can use individual white boards. In math notebooks, one can have several examples in succession, so children see emerging patterns. Individual white board work does not leave a history of their work and they may not be able to observe patterns. We, as math teachers, should remind ourselves and our children that: “Mathematics is the study of patterns. It has deep structures.” For this reason, we should help children to observe these patterns in their work and recognize the structures that emerge from these patterns.

Procedure and Language for Counting
Here are the points to consider during the daily counting process (at least five minutes). Each teacher should adapt these to suit her students’ and her instructional needs.

  • The teacher should announce the counting number and start number (later children can select the starting number and the counting number). These numbers should change each day.
  • During counting, when children give their numbers, the teacher should repeat each number clearly enunciating each word. This is particularly important at the Kindergarten through second grade.
  • The teacher should record the numbers from the count on the board creating columns and rows. Children record the numbers on their graph papers in the same way, in columns and rows. The starting number should be placed in the uppermost left corner of the paper (in the first full column of the paper). Leave one column between the columns for comments. As the columns of numbers emerge, the number of entries in each column must be same. For example, begin with 4 numbers in each column. Each day, change the number of rows up to about 10. Having the same number of entries in each column will produce patterns both horizontally (in rows) and vertically (in columns). It makes counting a rich activity. It also provides opportunities for differentiation. “High flyers” can be asked to give numbers horizontally and others vertically.
  • During the counting, the teacher should ask specific children to come forward and record the number on the board. The child writes the number on the board in the appropriate place. The teacher takes this opportunity to model the writing of multi-digit numbers: Are they of the same size? Are they at the same level? Are the digits equidistance? Are they aligned with each other?
  • Counting activity should include counting both forward and backward (not necessarily on the same day).
  • Each child should have the opportunity of responding a few times during the counting.
  • The choice of a number for counting begins from the easier one in the beginning of the year to bigger and more difficult numbers as the year progresses. For example, one should begin counting by 1 forward and backward in the beginning of the Kindergarten and counting by 10 toward the end of the year.[3]
  • Counting by 2 can be assisted by using the Number line, Hundred’s chart, using the Cuisenaire rods’ staircase, the standard number grid, or Vertical Number Grid.[4]
  • Counting by 10 can begin concretely, using the Cuisenaire rods and then without them. For example, begin counting by a number, say 7, ask a child to pick up the 7-rod (black). Write the number on the board. The child gives the rod to the child to his right and that child adds the 10-rod and calls out the number (17). The teacher writes the number on the board, starting the first row or column (below is the example of the first row). The process is continued for several more times. And then the teacher encourages children to extend the pattern without the rods. She keeps it on till children can give the next few numbers without the help of rods. Next day the counting by ten can begin with picking another rod. Toward the end of this forward counting begin counting backward from the last number.

Screen Shot 2018-07-03 at 4.06.51 PM

  • When the teacher begins any counting, she asks who has the next number, and then the next one, till several numbers in the count are generated. This should be done by volunteers first and then by randomly selecting children or the ones who need support. One should take advantage of high flyers’ knowledge of numbers as a starter. Never give the number easily. Try to derive the number with the help of children using decomposition-recomposition process. Someone will come forward. I have never been disappointed in any class, in any school. Some child in every school, in every class comes up with the next number and then others pick up the theme and the pattern and the learning process and counting begins. When a particular child is stuck on getting a number, give him clues: start with the facts he already knows. For example, if the child (Kindergarten level) does not know what comes after 54, go back to the child who gave 51, and continue, most times the child will come up with the number. If he still does not come up with the number, ask him: what comes after 4? If he answers correctly. Ask him: what comes after 14? Etc.
  • It is important that the teacher openly acknowledges the child who gives the correct number by children clapping twice in unison. Never leave a child without success. Help each child to taste success, even if it is just what comes after 7 or before 7.
  • Once a pattern begins to emerge and children understand the task and the count, ask them to write the next five numbers and place them in the proper places—in the correct columns so that they can observe the emerging pattern in numbers. As children write the numbers, the teacher walks around the room asking each student to give an example. Some children will readily observe the emerging patterns, both vertically and horizontally. Avoid having children give the pattern too soon. Instead, devote enough time discussing the numbers so most of the children see the patterns. It may take several days.
  • Do not disclose the pattern, let children arrive at the pattern. Ask teams of two children to discuss the number relationships and the process. Let them arrive at different patterns. Only when most children are able to give the correct entry, then ask a child (not a high flier) to articulate the pattern.

This counting will result in a Dynamic Vertical Grid. In the beginning of the year, the counting will take longer; however, as it becomes a routine, it will take less and less time and more will be accomplished. Soon the counting process becomes an important means for making formative assessment of children’s numbersense.

  • During the counting, the teacher should ask a great deal of questions about the numbers—place value, digits, values of digits, location of the number, one before, one after, 10 more, 10 less, what is the next tens, what is the next whole number, etc. These questions instigate mathematics language, concepts, and mathematical ways of thinking.
  • The teacher can make up impromptu mathematics problems: What is the difference between 2 consecutive numbers (in the same row, in the same column, both in the same row but few cells apart, both in the same column but a few rows apart, etc.).

Counting Example Grade One/Two:
Following is an example of daily count (for first grade during the middle of the year and in the early part of the year in higher grades):

Teacher: Let us count by 5 forward starting with 49. We will write the numbers in the first column. You do the same on your graph paper.

The first column and the beginning of the second column is derived together.

Screen Shot 2018-07-03 at 3.58.02 PM

Now ask children to write the next five numbers in the count. They should begin from the cell marked by “*”. As they are writing their numbers, the teacher should go to the child who is struggling and help generate one or two numbers. For example, she asks:

Teacher: What are we adding to 99?
Child: 5.

Teacher: What is 1 more than 99?
Child: 100.

Teacher: Good! We have added 1 to 99. Where did we get 1 from?
Child: Did it come from 5?

Teacher: Yes! That is good. Now, what is left from 5 to be added after adding 1?
Child: 4.

Teacher: 4 is added to what number?
Child: 100.

Teacher: Very good! What is 100 + 4?
Child: 104.

Teacher: Good! So, 99 + 5 is what number?
Child: 104.

Teacher: Now continue. What is the next number?
Child: That is easy. 104 + 5. I know 4 plus 5 is 9. So, 104 + 5. That is 109.

Teacher: Great!

Then the teacher moves to another child.

If a child has finished writing 5 numbers, she checks his work. If it is correct, she asks him/her to check other children’s work as they finish the task. If two children have finished writing the numbers, you ask them to compare their entries with each other and make corrections. If they have any disagreements, they should come up with consensus by supporting their arguments. When many more children have written all the five numbers, ask them to compare the answers in pairs. Keep on making pairs to correct each other’s work and checkers to check other children’s work. The teacher, with the children’s help, writes the five numbers on the board. The teacher should begin with the child who was struggling and then move on to others.

Screen Shot 2018-07-03 at 4.00.10 PM

Children who are able to complete the task earlier, are asked to write 7, 8 or more entries. The numbers in red are the entries provided by children.

Then the teacher asks children to work in pairs to identify the numbers in the place indicated by “___”.

When children (in teams) have found the number in the indicated place, she asks them to supply the numbers. She records them on the board in a separate place than the grid. She discusses their answers and asks for their methods of finding these numbers. Children defend their answers. Exact answers are identified. Then, the most efficient methods for finding the exact answers are identified. Entry in the place is made. If time permits, she creates more places with the red line (___).

The Modified Vertical Grids are effective in helping children improve the numbersense and to assess if children have acquired the structure of the number system. In vertical grids some of the entries are left blank to perform formative assessment. Children are asked to give the missing numbers by counting horizontally and vertically.

Counting Example at Grade Three/Four Level:
Numbersense is a constantly evolving skill for a child. One of the processes is to relate the new numbers being introduced to the numbers the child already knows. In the third grade a new set of numbers (fractions) are introduced in earnest. Therefore, it is important to begin to relate fractions with whole numbers and with each other.

In third grade, counting using whole numbers (1, 2, 5, 10, 100, and 1000), starting from any number should continue. However, towards the end of the year, when the concept of fractions is being introduced to children, the teacher should introduce counting by a unit fraction. A unit fraction is a fraction with numerator as 1. Following is an example of counting by a unit fraction (e.g., ⅕) starting from 4. All other elements of counting procedure remain the same as before.

Screen Shot 2018-07-03 at 4.02.44 PM

Counting Example at Grade Four/Five Level:
During grades four through six, our focus is on understanding and operating on fractions (and all the other related concepts). In the fourth through sixth grades, we need to relate fractions to each other and whole numbers, decimals, and percents. Although in fourth grade counting using whole numbers (1, 2, 5, 10, 100, 1000, and unit fractions) starting from any number continues, children should begin counting by proper fractions. In the fifth grade, they should add counting by mixed fractions. Following is an example of counting by a mixed fraction (e.g., 1⅖) starting from 4. All other elements of procedure remain the same. In these grades, they can also count by decimals.

Screen Shot 2018-07-03 at 4.04.11 PM

The objective of the daily counting tools (Number line, Open/Empty Number Line, Horizontal and Vertical Number Grids, Modified Number Grids, Hundreds Chart, Modified Hundreds Chart, Dynamic Number Charts, etc.) is to develop and improve numbersense. This objective can be achieved if children are given enough practice in counting using these tools and if they achieve the bench-marks in this activity at their grade level[5].

Each classroom should have clear display and use of these tools. However, they should not be overused for deriving addition, subtraction, multiplication, and division facts and operations. Their overuse makes children dependent on counting as the only strategy for developing arithmetic facts.

[1] See Games and Their Uses in Mathematics Learning by Sharma (2008).

[2] See the goals of counting at each grade level in Part One of this series on Numbersense.

[3] For the numbers to be used at each grade level see previous posts on Numbersense on this blog.

[4] See the Numbersense Part 1 in this series of posts.

[5] See first post in this series on Numbersense.




How To Improve Numbersense – Number Relationships: Counting Part Three

Role of Homework and Achievement

The role and amount of homework to be assigned is the most controversial topic of discussion among educators: teachers, parents, administrators, psychologists, and researchers. Even politicians get into the fray.

Researchers have been trying to figure out just how important homework is to student achievement. The Organization for Economic Cooperation and Development (OECD) looked at homework hours around the world and found that there was not much of a connection between how much homework students of a particular country do and how well their students score on tests (OECD, 2009).  However, in 2012, OECD researchers drilled down deeper into homework patterns, and they have found that homework does play an important role in student achievement within each country.

They found that homework hours vary by socioeconomic status. Higher income 15-year-olds, for example, tend to do more homework than lower income 15-year-olds in almost all of the 38 countries studied by the OECD. Furthermore, the students who do more homework also tend to get higher test scores.

An important conclusion of the study is that homework reinforces the achievement gap between the rich and the poor. For example, in the United States, students from independent schools do more homework than students from Christian/parochial and other religious schools.  And students from suburban public schools do significantly more homework than those in urban public schools except the urban public examination schools. It is not just that poor children are more likely to skip their homework, or do not have a quiet place at home to complete it. It is also the case that schools serving poor children often do not assign as much homework as schools for the rich, especially private schools.  Other findings from this study are also instructive. For example,

  • While most 15-year-old students spend part of their after-school time doing homework, the amount of time they spend on it shrank between 2003 and 2012.
  • Socio-economically advantaged students and students who attend socio-economically advantaged schools tend to spend more time doing homework.
  • While the amount of homework assigned is associated with mathematics performance among students and schools, other factors (teacher competence in subject matter and classroom management; higher expectations from students, parents, and teacher; amount of classroom time allotted to content, etc.) are more important in determining the mathematics performance and achievement of school systems as a whole.
  • Homework patterns among 15 year-olds, revealed that the children in western countries get much less homework than children in eastern countries. For example, students in United States and UK are assigned an average of five hours of homework a week compared to nearly fourteen hours in Shanghai, China, and nearly ten hours weekly in Russia and Singapore.

There are many other studies about the role and utility of homework with conclusions ranging from assigning no homework to students should be assigned substantial daily homework. However, most such studies are survey types that describe the state of homework and opinions about homework. There are some correlational studies where students (or their parents) are asked about the amount of homework they do and the status of their mathematics achievement. These studies are also suspect as the responses are purely subjective. The problem with such studies is that the quality and nature of homework vary and is self-reporting. These are not causal studies.

Most research indicates that it is not necessary to assign huge quantities of homework, but it is important that assignments are well thought-out—systematic and regular, with the aim of instilling work habits and promoting autonomous, self-regulated learning.

Researchers emphasize that homework should not exclusively aim for repetition or revision of content, as this type of task is associated with less effort and lower results.  Research has consistently found that students who work on their own on their homework, without help, performed better—score higher than those who ask for frequent or constant help.  Most studies show that self-regulated learning is aligned to academic performance and success. Self-regulation, organization, and perseverance are important components of the complex of executive functions.

When it comes to homework, how is more important than how much.

The Purpose of Homework
Meaningful homework is a means to reinforce classroom learning in the home.  Homework transfers learning from formal, socially guided learning to individualized responsibility and accountability.  The impact of supervised and independent practice using effective classroom instructional techniques and well-organized homework is well known to teachers. Teachers know that both provide students with opportunities to deepen their understanding and skills relative to content that was initially presented and practiced in the classroom. Most teachers and parents know them as important factors in student achievement; however, what and how to make them real and useful is a problem. The objective of teachers’ assignments should always aim to have impact. Effective teachers, therefore, plan activities in such a way that they have the most impact. For assignments to have impact, students need to practice (a) choosing strategies and (b) have retention.

The objective of homework is to:

  • Communicate to students that meaningful learning can continue outside the classroom;
  • Help children to develop study habits and foster positive attitudes toward school;
  • Reinforce and consolidate what has been learned in the classroom;
  • Helping students recall previously learned material;
  • Prepare, plan, and anticipate learning in the next class;
  • Extend learning by making students responsible for their own learning.
  • Practice to achieve fluency by initiative, preparation, reinforcement, preparation, and discipline of independent learning.

For these reasons, daily homework assignments should not be busy work but should always be well thought out, meaningful, and purposeful.  To achieve the stated goals of homework, it should have three components: cumulative items, current practice exercises, and challenge tasks. The integration of these tasks adds a key element of learning—reflection on one’s learning. Reflecting on one’s learning aids in the development of metacognition—a major ingredient for growth and achievement.

Principles Guiding Homework
Research shows that homework produces beneficial results for students in grades as early as second.  I remember, my daughter wanted to do her Kindergarten homework too when her older brother was doing his homework. A routine was set. The earlier these routines are set, the earlier the formation of life time habits.

There are three parties to homework: teacher, parents, and the student.  When homework compliance does not take place, we need to work on all the components and find ways of removing any hurdle. Teachers design the homework; parents support the homework completion, and students complete it, alone or with someone’s guiding support. The following principles can guide teachers and parents:

  • The school, with teachers, should establish and communicate a homework policy during the first week of school. The policy should be uniform across grade and subject levels. Students and parents need to understand the purpose of the homework, the amount to be assigned, the positive consequences for completing the homework; description and examples of acceptable types of parental involvement should be provided.
  • The amount of homework assigned should vary from grade to grade. Even elementary students should be assigned homework even if they do not complete it perfectly.
  • Research indicates, to a certain limit, homework compliance and mathematics achievement are related. The curve relating the time spent on homework and mathematics achievement is almost an inverted “wide” parabola. For about every thirty minutes of additional homework a high school student does per night, his or her overall grade point average (GPA) increases approximately half a point. In other words, if a student with a GPA of 2.00 increases the amount of homework he or she does by 30 minutes per night, his or her GPA will rise to 2.5. On the other hand, oppressive amounts of homework begin to reduce its benefits. Homework is like exercise, difficult to start and keep up, but the more we do it, the better we get at it and, within limits, we can do more.
  • Parents should keep their involvement in homework at a reasonable level. At the same time, parent involvement in the classroom should be welcomed.  Parents should be informed about the amount and nature of homework, and they should be encouraged to have moderate involvement helping their children. Parents should organize time, space, and activities related to homework.  Parents should be careful, however, not to solve content problems for students; they can give hints, or explain the method, but not give a method, which the student does not understand. Giving “tricks” to solve problems is not useful in the long run. There are no tricks in mathematics only strategies. An efficient strategy for others looks like a trick because they may not have the reason why it works.
  • Not all homework is the same. That is, homework can be assigned for different purposes, and depending on the purpose, the form of homework and the feedback provided to students will differ.
  • All assigned homework should be commented on and responded to because the benefit of homework depends on teacher feedback.Homework with the teacher’s written comments has an even greater positive effect on students. It provides a formative assessment, information how the student is doing. This also offers information for parents about standards, pedagogy, and methods of assessment. When homework is assigned but not commented upon, it has limited positive effect on achievement.  When homework is commented on and graded, the effect is magnified. In addition to teacher corrected homework, homework can be self-corrected by the student with the teacher providing the answers.  The homework can be peer corrected. Some homework is corrected publicly under the teacher’s guidance. Still, at least once a week, the homework is commented upon by the teacher. These comments should address common problems – lack of concept, misconceptions, poor language, inefficient procedures, poor organization, and misunderstanding of standards – as well as the efficient and elegant methods and concepts used by students.
  • Homework is practice. Students should practice at least 30 minutes a day on their academics just as they would an instrument or a sport. If one plays multiple instruments or multiple sports, does one give only 30 minutes of practice for both? Of course not! The same goes for reading and math, science and social studies. Research shows about 1 to 1 hour per day (7.5 hours a week) of homework, on consistent basis, can achieve the goals of homework.
  • Parents should be active participants in their child’s academic career. However, that does not mean doing the homework for their child because it would be counterproductive. They can make sure to remind their child to do the homework and that it gets completed. They can give suggestions when necessary and review completed homework. Homework is a child’s academic practice. He/she needs rewards and consequences and a great deal of encouragement.
  • Administrators and teachers should do everything to impress upon parents to make sure that they, in turn, make learning a priority for their children and practice every day. However, schools should not make children’s achievement solely dependent on this variable. They should make sure that all children get enough practice in the school itself. The lessons should be planned and delivered in such a way that there is enough practice in the classroom so that children feel confident in tackling the homework themselves.

A teacher should always ask: “Does the completion of homework have any impact on her instruction? Does it inform her instruction? Does it contribute to the teaching and learning of the new material? Does she learn something about the child and/or her teaching from it?”  If the answer is affirmative to any of these questions, the homework is worth assigning. Can the goals of the curriculum and her instruction be achieved through some other means? If so, then there is no need to assign homework.  However, if there is no homework, we have to find more time for instruction in the day or reduce the allotted time for regular instruction to be redirected to practice, reinforcement, and reflection.  Both situations are costly.  Therefore, we should always look for ways to improve homework compliance.

Composition of Homework
Most teachers assign homework at the end of each section in the book: “OK, now do problems 1-25 on page ____.” Does this work for students? Not based on what I have seen and heard in my 56 years as a mathematics educator. To help students develop competence and confidence in math, teachers should be concerned with the quality of problems they assign in the classroom and for homework rather than the quantity.

Most mathematics assignments (homework as well as practice activities) consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use that formula, which means that students know the strategy before they read the problem. Most times, they do not even read the instructions before solving a problem. Problem sets made up of only one kind of problem deny students the chance to practice choosing a strategy—that means thinking about the problems (reflection). When faced with a mix of types of problems on an exam, such students find themselves unprepared. These classroom or homework problem sets are called blocked assignments. The grouping of problems by strategies is common in a majority of practice problems in most mathematics textbooks.

The framers of CCSSM (2010), recommend thatstudents must learn to choose an appropriate strategy when they encounter a problem. Blocked assignments deny such opportunities. For example, if a lesson on the Pythagorean theorem is followed by a group of problems requiring the Pythagorean theorem, students apply it before reading each problem. If all the problems for practice are direct application of the Pythagorean formula (a2+ b2 = c2, where a and b are the legs of a right triangle and c is the hypotenuse), then this direct “blocked” practice is a practice in algebraic manipulation, not a practice in understanding and applying an important geometrical result about right triangles and its role in higher mathematics.

An alternative approach to practice is when different kinds of problems—varying concepts, procedures, and language, appear in an interleaved order (mixed and uncategorized) problems. Such problem sets require students to choose the strategy on the basis of the problem itself. Such problem sets are also referred to as distributed or spiraled practice. For example, consider the problem:

A bug flies 6 meters east and then flies 14 meters north.

Her starting point is at point A and her final destination is represented by a point B, represent her flight by a diagram on the coordinate plane. How far, in terms of tenth of a meter, is the bug from where it started? Give reasons for your choice of solution approach. How much distance did it travel? Why are these two distances different? (No calculators)

This problem is ultimately solved by using the Pythagorean theorem. The distance travelled by the bug is different than the distance between points A and B—the hypotenuse of the right triangle, they drew, with sides 6m and 14m. To find the length of the hypotenuse, they used the formula:

Screen Shot 2018-06-21 at 1.48.51 PM

The bug flew 20 m to reach point B and the distance between A and B is about 15.3m.

In this problem, students first draw a diagram. The diagram suggests a strategy. Then, they choose a strategy (Pythagorean theorem) to apply and then they execute the strategy.

The choice of strategy means that a student is observing a pattern (mathematics is the study of patterns), recalling a theorem or formula suggested by the situation (learning is the residue of experiences and recall shows its presence), or noting the presence of certain conditions or language that suggest a concept, or a procedure (integration of learning). The choice of a strategy is dependent on understanding language, concept, and procedures involved in the problem situation.

Learning to choose an appropriate strategy is difficult, partly because the superficial features of a problem do not always point to an obvious strategy. For example, the word problem about the bug does not explicitly refer to the Pythagorean theorem, or even to a triangle, right triangle, or hypotenuse. This kind of assignment is called interleaved practice where a majority of the problems (practice, homework, assessment tasks, etc.) are from previous lessons, current work, mixed problems (new concept mixed with previous concepts and procedures) so that no two consecutive problems require the same strategy. Students must choose an appropriate strategy, not just execute it, just as they would be required to choose a strategy for a problem during a cumulative examination or high-stakes test. Whereas blocked practice provides a crutch that might be optimal when students first encounter a new skill, only interleaved practice allows students to practice what they are expected to know.

(a) Cumulative Homework
One-third of the homework assignment must be cumulative in nature. It should include representative problems from previous concepts and procedures. Whatever has been covered in the classroom during the year should find its representation in daily practice and assigned homework. In other words, what was covered in the months of September or October should continue to appear in the month of March or April. Such an assignment makes connections and achieves fluency. Consider, for example, the connections between multiplication and fractions, fractions and ratios, and equivalent fractions and proportions. Or, the relationship between algebra and arithmetic. There is such a close relationship between algebra and arithmetic that algebra is often referred to as “generalized arithmetic.” Using the distributive property in multi-digit multiplication procedure, combining like terms, applying the laws of exponents, and other rules and procedures are the same for algebraic expressions as they are for arithmetic expressions (e.g., long division for whole numbers and division of a polynomial by a binomial; short division for whole numbers and synthetic division for polynomials).

This part of the homework plays an integrative role in learning the material in the curriculum and provides opportunities for reflection. This part is to improve fluency and smoother recall of learned material. Familiarity and success on these problems emphasizes and meets the need for structure and success of the R-Complex and the limbic system. Another objective is to consolidate learning and connect concepts, procedures, and language. The topics, skills, and procedures mastered must be revisited on a regular basis. The memory traces of the learned skills must be retouched regularly because knowledge atrophies over time if not maintained.

(b) Practice Problems
Another one-third of the homework must be a true copy of the work done in the classroom that day. The objective of this component of the homework or practice problems is to consolidate the material learned in the classroom and continues the learning outside the classroom. It also helps to remain current in the material. If the teacher has covered the odd problems in the section of the book, then she can assign the even problems for homework.  When homework is assigned for the purpose of practice, it should be structured around content. Students should have a high degree of familiarity with the material assigned.  Homework relating to topics that have not been clearly understood and a level of competence has not been achieved should not be assigned. Practicing a skill with which a student is not comfortable is not only inefficient but might also serve to habituate errors and misconceptions, and high probability of non-compliance.

True mastery requires practice. But again, quality often matters more than quantity when it comes to practice. If students believe they can’t solve a particular problem, what is the point of assigning them 20 more similar problems? And if they can solve a problem in their sleep, why should they do it again and again?

The objective of this segment is to develop procedural fluency. Both fact fluency and procedural fluency can be developed in the class and through homework. Procedural fluency builds conceptual understanding, strategic reasoning, and problem solving. It involves applying procedures not only accurately but also efficiently and flexibly and recognizing when one strategy or procedure is more appropriate than another. To help students develop procedural fluency, teachers must therefore assign problems that are conducive to discovering and discussing multiple solution strategies. And once again, this doesn’t require elaborate problems. Sometimes it’s just a matter of recognizing the learning potential within straightforward problems. Here’s a simple problem that generates rich discussion and helps students develop procedural fluency and number sense related to fractions: Find five fractions between 2/5 and 4/6. Describe your approach and reasoning for it.(This problem can be assigned as you’re practicing the ordering of fractions. It makes students think about all the ways of approaching it: common denominator, common numerator, converting to decimals, and comparing with a benchmark fraction such as 1/2).

You do not have toavoid using or assigning problems from textbooks. Make thoughtful, intentional choices that help students learn and like mathematics and feel good about themselves in the process.

(c) Challenging Problems
The problems in the last one-third of the homework should be (a) moderately challenging or (b) one or two-word problems from a previous topic. These problems are not mandatory but for those who want to solve these problems. Students can trade one problem from this set with two in other parts. These problems should add some nuance or subtlety to the problems of the type done in the classroom, or application of the concepts and procedures discussed in class during previous topics. Or, this component may introduce a related concept or procedure.

This component helps to prepare students for new content or to have them elaborate on content that has been introduced. Through these problems, even when students have demonstrated mastery of a skill, students can gain a deeper understanding of the math involved. For this to happen, teachers must assign the right problems and be prepared to scaffold students’ understanding. Here is one such problem that stretches students including those who have mastered or memorized the laws of exponents:  Which is greater: 240 + 240or 250 ? Can you prove your assertion?  Assign problems students are likely to mess up, and then help them learn from their mistakes so that they don’t make the same mistakes again. Mistakes make us learn more. Do not prevent students’ mistakes, prepare them for learning through them. Discuss student mistakes, misconceptions, and lack of understanding them in class. Help them to find mistakes and their causes. When they do make a mistake, give a counter example and create cognitive dissonance in their minds.

The problems, in this section, become the starting point for the next day’s lesson. In that sense, they are a kind of preview of the next lesson. For example, a teacher might assign homework to have students begin thinking about the concept of division prior to systematically studying it in class.  Similarly, after division of whole numbers has been studied in class, the teacher might assign homework that asks students to elaborate on what they have learned and how this will extend division of whole numbers by simple fractions.

In both situations, it is not necessary for students to have an in-depth understanding of the content. The objective of these problems is to further the learning. It doesn’t matter if students do not solve any of the problems from this part of the homework as it will become the introduction to the next lesson. This part of the homework is to challenge the student and should be of a moderate level of novelty. It might invite participation from other members of the family. These problems are assigned so that students who need a challenge get it. These problems satisfy the needs of the neocortex.

Every quiz, every test, and examination are set from the problems (or a very close replica) assigned in the homework throughout the year.

Homework with these components is an example of interleave practice homework. The students take a while to warm to this new type of homework because it has been so long since they have actually seen how to do a particular problem. But once they get used to it, students like the new homework.  When they are reviewing the old concept or procedures, there is an aha moment — “oh I remember that.” This increases confidence and compliance.

This interleaving effect is observed even though the different kinds of problems are superficially dissimilar from each other. Interleaving of instruction and homework improves mathematics learning not only by improving discrimination between different kinds of problems but also by strengthening the association between each kind of problem and its corresponding strategy.

Interleaved practice has these two critical features: Problems of different kinds are intermixed (which requires students to choose a strategy), and problems of the same kind are distributed, spaced, across assignments (which usually improves retention). Spacing and choosing strategies improves learning of mathematics and performance on delayed tests of learning.

The interleaving of different kinds of mathematics problems improves students’ ability to distinguish between different kinds of problems. Students cannot learn to pair a particular kind of problem with an appropriate strategy unless they can first distinguish that kind of problem from other kinds and interleaved assignments provide practice to learn this discrimination. In other words, solving a mathematics problem requires students not only to discriminate between different kinds of problems but also to associate each kind of problem with an appropriate strategy, and interleaving improves both skills. Aside from improved discrimination, interleaving strengthens the association between a particular kind of problem and its corresponding strategy. The abilities to discriminate and associate strategies are critical skills for doing well on cumulative examinations, such as standardized tests, SAT, achievement tests of different kinds. Since most of these examinations are cumulative and different kinds of problems are organized in it, students need to have mastered the critical skill of discriminating between the different types of problems.

Research on error analysis showsthat the majority of test errors take place when students have practiced using the blocked assignments but much fewer when they have practiced interleaved conditions. The errors occur because students, in blocked practices, are accustomed to choosing strategies corresponding to the assignments; they have learned identifying strategies in isolation, so when they encounter the problems in combination they mix them up. Through practice in interleaving situations, fewer errors are possible because interleaving improves students’ ability to discriminate one kind of problem from another and discriminate one kind of strategy from another.

Blocked assignments often allow students to ignore the features of a problem that indicate which strategy is appropriate, which precludes the learning of the association between the problem and the strategy.Blocked problems also lack subtleties and nuances. Blocked practice allows students to focus only on the execution of the strategy, without having to associate the problem with its strategy.

Helping students develop the discipline of completing homework is key to becoming independent and lifelong learners. Ifa student is not able to complete the homework on the first try, the teacher should ask the student to complete it after the material has been covered in the class. Although the teacher may collect work to record students’ progress, it should not detract from the responsibility given to the student for successful completion of all problems.

CCSS (2010)
OECD (2009)
OECD (2012)

Role of Homework and Achievement