# NUMBER WAR GAMES III: Multiplication and Division Facts

Teaching Multiplication and Division Facts

In my more than fifty-five years of teaching mathematics from number concept to Kindergarteners to pure and applied mathematics to graduate students (in mathematics, engineering, technology, and liberal arts), and preparing and training teachers for elementary grades to college/university, I am strongly convinced that no student should leave the fourth grade without mastering multiplicative reasoning—its language, conceptual schemas/ models, multiplication and division facts, and its procedures—including the standard algorithms.

A. Concept, Role and Place of Multiplication in the Mathematics Curriculum: 1. After number concept[1], additive reasoning, and place value, the next important developmental concept in mathematics is multiplicative reasoning. Multiplicative reasoning is an example of quantitative thinking that recognizes and uses repetition of groups to understand the underlying pattern and structure of our number system. Multiplicative reasoning is the key concept[2]in the mathematics curriculum and instruction in grades 3-4. Multiplication and division are generalizations and abstractions of addition and subtraction, respectively, and contribute to the understanding of place value, and, in turn, its understanding is aided by mastering place value. It helps students to see further relationships between different types and categories of numbers and it helps in the understanding the number itself.

2. Whereas, in the context of addition and subtraction, we could express and understand numbers in terms of comparions of smaller, greater and equal, with multiplication and division, numbers can be expressed in terms of each other and we begin to see the underlying structures and patterns in the number system. Multiplicative reasoning provides the basis of measurment systems and their interrelationships (converting from larger unit to smaller unit (you multiply by the conversion factor and vice-versa. It is the foundation of understanding the concepts in number theory and representations and properties of numbers (even and odd numbers; prime and composite numbers, laws of exponents, etc.), proportional reasoning (fractions, decimals, percent, ratio, and proportion) and their applications.

3. The move from additive to multiplicative thinking and reasoning is not always smooth. Many children by sheer counting can achieve a great deal of accuracy and fluency in learning addition and subtraction facts, and at least for some multiplicaion facts.  However, it is not possible to acquire full conceptual understanding (the models of multiplication and division), accuracy (how to derive them efficiently, effectively, and elgantly), fluency (answering correctly, contextually, in prescribed and acceptable time period), and mastery of multiplicative reasoning by just counting.

B. Definition: Qualitatively and cognitively, for children, multiplicative reasoning is a key milestone in their mathematical development.  It is a higher order abstraction: addition and subtraction are abstractions of number concept and number concept is an abstraction of coutning. Addition and subtraction are one-dimensional cocnepts and are represented on a number line. Multiplication and division, as abstractions of addiotn and subtraction, start out as one-dimensional (as repeated addition and groups of), but they become two-dimensional concepts/ operations (i.e., as an array and area of a rectangle representations).  Lack of complete understanding and mastery of multiplicative reasoning can be a real and persistent barrier to mathematical progress for students in the middle years of elementary school and later. Compared with the relatively short time needed to develop additive thinking (from Kindergarten through second grade), the introduction, exploration, and application of ideas involved in multiplication may take longer. Understanding of multiplicative reasoning (i.e, the four models–repeated addition, groups of, an array, and area of a rectangle) is truly a higher order thinking as the basis of higher mathematics.

1. The main objective of the mathematics curriculum and instruction, particularly in quantitative domain, for K through grade 4, is to master numeracy. Numeracy means: A child’s ability and facility in executing, standard and non-standard, arithmetic procedures (addition, subtraction, multiplication, and division), correctly, consistently and fluently with understanding in order to apply them problem solving in mathematics, other disciplines, and real-life situations. To achieve this: children by the end of fourth grade, should master multiplicative reasoning.  They should master multiplication concept, facts, and procedure by the end of third grade and by the end of fourth grade, they should master concept of division, division facts, and division procedure. Mastering multiplicative reasoning means mastering multiplication and division and understanding that multiplication and division are inverse operations. They should be able to convert a multiplication problem into a division problem and vice-versa.

2. The reasons for difficuties in mastering multiplication and multiplication tables: The first real hurdle many children encounter in their school experience is mastering multiplication tables with fluency. Even many adults will say: “I never was able to memorize my tables.  I still have difficulty recalling my multiplication facts.” It is a worldwide phenomenon. Everyone agrees that chidren should master multiplication tables, but there is disagreement in opinions about what it means to master multiplication tables and how to achieve this mastery. Mathematics educators, teachers, and parents have formed opposing camps about it. One group believes in achieving understanding of the concept and believe that fluency will be reached with usage, whereas the other group believes in memorizing the tables and insist that conceptual understanding will come with use. Both of these extreme approaches are inadequate for mastering mutiplication tables for all children. Both work for some children, but not for all.

At the time of evaluation for a student’s learning difficulties/disabilities/ problems, when I ask him/her, ‘Which multiplication tables do you know well?’ Inevitably, the reply is ‘The 2’s, 5’s and 10’s.’  Some of them would add on the tables of 1’s, 0’s and 11’s to their repertoire.  If I follow this up by ever so gently asking the answer for 6 × 2, then the response is: “I do not know the table of 6.” On further probing, I get the answer.  Most frequently, the student finds the answer by counting on fingers 1-2, 3-4, 5-6, 7-8, 9-10, 11-12.  6 × 2 is 12. Some will say: 6 and then 7, 8, 9, 10, 11, and 12. 6 × 2 is 12. All along, the student has been keeping track of this counting on his/her fingers. Another way the answer is obtained by reciting the sequence: 2, 4, 6, 8, and 12.  Here also the record of this counting is kept on his/her fingers. Both of these behaviors are indicative of lack of mastery of multiplication facts. They are also indications of the child having inefficient strategies for arriving at multiplication facts. Skip counting forward on a number line or counting on fingers is not an efficient answer to masering multiplication facts.

On the other hand, repeated adddition and array model are limited to whole number multiplication. And, groups of model is helpful in conceptualizing the concept of multiplication of fractions and decimals. Children also acquire the misconception that “multiplication makes more” when they are exposed to only repeated addition and the array model. In such a situation, I say to them: “you are right.  But what happens when you have to find the product of two fractions ½ × ⅓? What do you repeat how many times? The answer, invariably is: “You cannot. You multiply numerator times numerator divided by denominator times denominator.” Or, “what do I repeat when I want to find 1.2 × 1.3?” At this time, most teachers will give me the procedure of multiplying decimals. “Multiply 12 and 13 and then count the number of digits after the decimal.” If I pursue this further by asking: “How do we find the product (a + 3) (a + 2)?’ I begin to loose many in my audience. If, a person has complete understanding of the concept of multiplication, they can easily extend the concept of multiplication from whole numbers to fractions, decimals, and algebraic expressions. Only, the models “groups of” and the “area of a rectangle” models help us conceptualize the multiplication of fractions, decimals, integers, and algebraic expressions. And, only the area of a rectangle model helps us to derive the standard procedure for: multiplication of fractions/decimals, binomilas, distributive property of multiplication of arithmetic and algebraic expressions.

As one can see from this exchange, according to most teachers, the model or definition for conceptualizing multiplication changes from grade to grade from person to person. Rather than understanding the general principle/concept of multiplication, students try to solve problems by specific or ideosyncratic methods. Later, they find it difficult to conceptualize schemas/models/procedures for different examples of multiplication problems (with different types of numbers) and they give up. For example, they have difficulty reconciling the multiplication of fractions and decimals with their intial schema for multiplication (repeated addition or array andd even groups of, in some situations). We beleive, they should be exposed to and should be thoroughly familiar to the four models of multiplication before we introduce them to procedures. They should practice mastering multiplication tables when they have learned and applied these four models of multiplication. Then, they can accomodate different situations of multiplication into their schema of multiplication and create generalized schema for multiplication.  The most generalized model for multiolicaiton is the area of rectangle.

Some of the difficulties children have in learning the concept of multiplication are the result of the lack of understnading of these different schemas and the emphasis on sequential counting in teaching multiplication in most classrooms.  Students are not able to organize them in their heads, see the connections between them, and the importance of learning these models. They also think that different number types (whole numbers, fractions, decimals, integers, rational/irrational, algebraic expressions, etc.) have different definitions of multiplications. They do not see that the definitions and models should be generalizable.

3. Another reason for the difficulty is the teaching of multiplication: Children learn the tables and multiplication procedure in mathematics curriculum as mere procedures--a collection of sequential steps, sometimes the facts are derived just with the help of mnemonic devices, songs, and rote memorization as ‘a job to be done.’ This means: give a cursory definition of the term (e.g., multiplication is easy way of doing addition), give the procedure (e.g., this is how you do/find it), practice the procedure (do these problems now), and then apply the procedure (let us do some word problems on multiplication). It is a little exposure and then practice of the narrowly understood procedure.  It is not mastery with rigor.

4. Mastering a concept means, the student has the language, the conceptual schema(s) (effective and efficient strategies), accuracy and fluency in skills and procedure, and can apply it to other mathematics concepts and problem solving. The procedure of mastering multiplication tables should be based on solid understanding of the language and the concept. Students and the teacher should arrive at strategies and procedures by exploring and using the language, the conceptual schemas, and efficient and effective models. And then from several of these procedures should arrive at those that are efficient and generalizable (the standard algorithms). Students should develop, with the teacher, the criteria for efficient and effective conceptual schemas for deriving facts and procedures for multi-digit multiplication.  The teacher should also help develop an efficient script for students to follow the steps needed to executeprocedure. Once children have arrived at an efficient procedure or procedures, they should practice it to achieve fluency and automatization. The fluency should be achieved by applying it in diverse situations. It means, ultimately, they have understanding, fluency, and applicability. Children learn tables successfully when teachers give them efficient strategies, enough practice in doing so and make it important to do so. They understand and are able to apply them according to how well they are taught.

From the outset, we want to emphasize that it is important for children to learn (understand, have efficient strategies for arriving at the facts, accuracy, fluency, and then automatization) their multiplication tables. Eventually, by deriving the facts using efficient strategies and applying them to problems, they will be able to recall multiplication facts rapidly (8 times 3Twenty-four!), and then use this knowledge to give answers to division questions (24 ÷ 3? Eight!); use these multiplication and division facts to do long multiplications and divisions; and use them appropriately in solving problems. When the concept of multiplication is understood, then one should introduce division concept and help them see that multiplication and division are inverse operations. Cyisenaire rods are the best material for making this relationship clear. (See How to Teach Multiplication and Division, Sharma 2018).

C. Transition from Addition to Multiplication: Pre-requisite Skills for Multiplication and Multiplication Tables: 1. Counting by 1, 2, 10, and 5. The instructional practice of having students count groups—skip counting—is an essential transition between additive and multiplicative reasoning. This counting should be limited to counting by 1, 2, 10, 5, and possibly 9. All other groups, when being added should be done by decomposition (adding 6 to 36 should be accomplished by asking: What is the next 10s? “40” How do I get there? “add 4” Where did the 4 come from? “from 6” What is left in 6? “2” What is 40 + 2? “42” So, what is 36 + 6? “42” Encouraging to count after 36 to add 6 does not amke the child acquire a robust numbersense. Just like visual clustering or representation of number as a group is a generalization and abstraction of discrete counting, skip counting, emphasizes the structure and efficiency that grouping gives to counting and, therefore, to addition. For example, counting by fives (using the fingers on hands as a starting model, then moving to TenFrame, Visual cluster cards representing 5, and then the 5-rod (yellow) of the Cuisenaire rods is the right progression for learning to count by 5. or twos (using eyes, or stacks of cubes, Visual Cluster Card representing 2, then the 2-rod (red) of the Cuisenaire rods) is very productive. Similarly, counting tens rods (in base 10 blocks or the 10-rod in Cuisenaire rods, however, using the Cuisenaire rods is better) as: 10, 20, 30, 40, and so on, emphasizes the concept of repeated addition and grouping. However, if these counting sequences are learned by discrete counting (Unifix cubes, fingers, number line, etc.) or without models to support the grouping and repeated counting activity then the order and the outcome will be learned without the concept and significant meaning about multiplicaiton or division.

2. Additive Reasoning pre-requisite Skills for learning and masrering Multiplication Tables: (a) 45 sight facts of adddition, (b) Making ten, (c) Making Teens’ numbers, (d) What is the next tens, (e) Adding multiples of Tens to a two-digit number (e.g., 27 + 30 = ? 59 + 50 = ? 40 + 10 =?), (e) Commutative property of addition, (f) Counting forward and backward by 1, 5, 10, and 2 from any number.

3. The Order of Teaching Multiplication Tables: Derivation of multiplication facts/tables is easier when the four models: repeated addition, groups of, an array, and the area of a rectangle; commutative and associative properties of multiplication; and distributive property of multiplication over addition and subtraction: a(b + c) = ab + ac and a(b – c) = ab – ac have been mastered. Multiplication tables should be mastered only after the groups of and area of a rectangle is clearly understood. If we use Cuisenaire rods for modeling multiplication, particularly for showing it as area of a rectangle, then the repeated addition and groups are already embedded in it and children can see the commutative, associative, and distributive properties also. Using these propeties, the teacher should derive multiplication tables up to 10 (i.e., 10 × 10 = 100 facts), in the following order (I cannot oveemphasize this order).

(i) Commutative property of multiplication: This reduces the work of deriving 100 facts to only 55, an easier task.

(ii) Table of 1 (19 facts), (iii) Table of 10 (17 new facts), (iv) Table of 5 (15 new facts), (v) Table of 2 (13 new facts).

(vi) table of 9 [11 new facts] The table of 9 has several clear patterns hidden in it. Children need to see them. For example, (a) the sum of the digits in the table of 9, from the facts we already know (from tables of 1, 10, 5, and 2) is always 9: 9 × 1 = 9 = 09, 0 + 9 = 9; 9 × 2 = 18, 1 + 8 = 9; 9 × 5 = 45, 4 + 5 = 9; 9 × 10 = 90; 9 + 0 = 9; (b) the tens’ digit in the table of 9 is 1 less than the number being multiplied with 9, 9 × 1 = 9 = 09, 1 – 1= 0; 9 × 2 = 18, 2 – 1 = 1; 9 × 5 = 45, 5 – 1 = 4; 9 × 10 = 90, 10 – 1 = 9. Let us, therefore, apply these two patterns to derive 9 × 7 = ? We use the two patterns: here in the ten’s place will be 7 – 1 = 6, and, then to make the sum of the two digits as 9, we know that 6 + 3 = 9, thus, 9 × 7 = 63, and by commutative property of multiplication, we have 9 × 7 = 7 × 9 = 63. This process helps children to easily memorize the table of 9. We can also derive the fact 9 × 7 in several other ways: (a) by using the distributuve property of mulitplication over subtraction: we already know that , 10 × 7 = 70 ; 9 × 7 = (10 – 1) × 7 = 10 × 7 – 1 × 7 = 70 – 7 = 63; (b) using distributive property of multiplication over addition, 9 × 7 = 9 × (5 + 2) = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 20 – 2 = 65 – 2 = 63, Or, 9 × 7 = 9 × 5 + 9 × 2 = 45 + 18 = 45 + 10 + 8 = 55 + 5 + 3 = 60 + 3 = 63.]

(vii) Table of 4 (9 new facts). Since 4 is double of 2, the entries ib the table of 4 are double of table of the corresponding entries in the table of 2. For example, 4 × 7 = 2(2 × 7) = 2 × 14 = 2 × 10 + 2 × 4 = 20 + 8 = 28; Or, 4 × 7 = 4 (5 + 2) = 4 × 5 + 4 × 2 = 20 + 8 = 28. Or, 4 × 7 = (2 + 2)7 = 2 × 7 + 2 × 7 = 14 + 14 = 28.

(viii) Remaining facts: The total number of multiplication facts derived so far: 19 + 17 + 15 + 13 + 11 + 9 = 84. The remaining 16 facts are: 3 × 3; 3 × 6, 6 × 3; 3 × 7, 7 × 3; 3 × 8, 8 × 3; 6 × 6; 6 × 7, 7 × 6; 6 × 8, 8 × 6; 7 × 7; 7 × 8, 8 × 7; and 8 × 8. And, because of the commutative proeprty of multiplication, the number is reduced to 10. These 10 facts can be mastered by children in a week. These remaining facts should be derived by decompositon/ recomposition. For example, let us consider: 8 × 6 = ?. Teacher: Do you know the answer? Student: No! Teacher: Which is the bigger number? Student: 8. Teacher: Good! Do you know 8 × 5? Student: Yes! Teacher: Good! Break 6 into 5 and 1. What is 8 × 5? Student: 40! Teacher: What is 8 × 1? Student: 8! Teacher: What is 40 + 8? Student: 48! Teacher: Now, what is 8 × 6? Student: 40 + 8 = 48. 8 × 6 = 48. First, If necessary, students form this fact as area of a 8 by 6 rectangle concretely with the help of Cuisenaire rods. 6 brown rods forming a 8 × 6 (vertical side = 8 and horizontal side = 6) rectangle and then breaking it into two rectangles (8 × 5 and 8 × 1). Students, now, derive these, by seeing the graphic organizer: 8 × 6 = 8 (5 + 1) = 8(5) + 8(1) = 40 + 8 = 48. Then, they should repeat it by visualizing it. All of this work should be done orally creating the script as described above. There are several ways the result can be derived by decomposition/recomposition.

5. Improving Times Table Fluency: The Institute for Effective Education (IEE) in the UK has published a new report on improving times table fluency, as a result of study of 876 children in 34 Year 4 (grade 3 in the U.S.) classes. All groups had similar pre-test scores and similar groups of children–same distribution of children with similar abilities. Each class used a different balance of conceptual nad procedural activities during times tables lessons. Conceptual activiities were games that focused on the conncetions and patterns in table facts, while procedural activiities were games in which students practiced multilication facts. All grous had same pre- and post tests. The report concluded that times tables may be best taught by using a balanced approach–teaching both the concepts behind them and practicing them in a range of ways with low-stakes testing.

In the light of many similar studies, concept-based instruction involving efficient and effective methods that can be generalized and uses pattern-based continuos materials (Cuisenaire rods, Visual Cluster cards, etc.) that help in developing the script are better. Once children know the tables of 1, 2, 10, and 5 and can derive the other facts by using effective scripts, they should paractice the tables with games. We have found the following games using Visual Cluster cards to be very effective.

Game Four: Mastering Multiplication Facts

Materials:  A deck of Visual Cluster Cards (Playing cards without numbers) without face cards or with face cards. Each face card is, intially, given a fixed value (Jack = 2, Queen = 5, and King =10), later they are given values as: Jack = 11, Queen = 12, and King =15).

How to Play

1. The whole deck is divided into two to four equal piles (depending the number of players).
2. Each child gets a pile of cards.  The cards are kept face down.
3. Each person displays two cards face up.  Each one finds the product of the numbers on the two cards. The bigger product wins. For example, one has the three of hearts and a king of hearts (value 10), the product is 30. The other has the seven of diamonds and the seven of hearts, the product is 49.  The second player wins.  The winner collects all cards.
4. If both players have the same product, they declare war.  Each one puts down three cards face down. Then each one turns two cards face up.  The bigger product of the two displayed cards wins. The winner collects all cards.
5. The first person with an empty hand loses.
6. Initially, the teacher or the parent should be a player in these games. Their role is not only to observe the progress, mediate the disputes, keeping pace of the game and encouragement, but also to help them in deriving the fact when it is known to a child. For example, if the child gets the cards: 8 of diamond and 7 of spade. Teacher asks: What is the multiplication problem here? “8 × 7” The teacher asks: Do you know the answer? “No” Which is the bigger number? “8” Can you break the 7 into two numbers (point ot the clusters of 5 and 2 on the 7-card)? “5 and 2” If the 7-card was 5-card, then the problem would be 8 × 5. If the 7-card was 2-card, then the problem would be 8 × 2. Now, 7-card has 5 and 2, so the problem is: Is 8 × (5 + 2). Is 8 × 7 is same as 8 × (5 + 2). “Yes!” So, 8 × 7 = Is 8 × 7 = 8 × (5 + 2) and is made up of two problems: 8 × 5 and 8 × 2. What is 8 × 5? “40” What is 8 × 2? “16” Now, What is 8 × 5 and 8 × 2 together? “40 + 16” What is 40 + 16? “56” Good! What is, then, 8 × 7? “56.” All this should be done orally.

In one game, children will derive, use, and compare more than five hundred multiplication facts.  Within a few weeks, they can master multiplication facts. Once a while, as a starting step, I may allow children to use the calculator to check their answers as long as they give the product before they find it by using the calculator.

Game Five: Division War

Objective: To master division facts

Materials:  Same as above

How to Play: Mostly, same as above.

1. The whole deck is divided into two to 4 equal piles (depending on the players.
2. Each child gets a pile of cards.  The cards are kept face down.
3. Each person displays two cards face up.  Each one finds the quotient of the numbers on the two cards. The bigger quotient wins. For example, one has the three of hearts and a king of hearts (value 10). When 10 is divided by 3, the quotient then is 3 and 1/3. The other has the seven of diamonds and the seven of hearts, the quotient is 1.  The first player wins.  The winner collects all cards.
4. If both players have the same quotient, they declare war.  Each one puts down three cards face down. Then each one turns two cards face up.  The bigger quotient on the two displayed cards wins. The winner collects all cards.
5. The first person with an empty hand loses.

In one game, children will use more than five hundred division facts.  Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

Game Six: Multiplication/Division War

Objectives: To master multiplication and division facts

Materials:  Same as above

How to Play: Almost same as the other games

1. The whole deck is divided into two to four equal piles (depending on the number of players.
2. Each child gets a pile of cards.  The cards are kept face down.
3. Each person displays three cards face up.  Each one selects two cards from the three, multiplies them, and divides the product by the third number (finds the quotient of the numbers). The bigger quotient wins. For example, one has the three of hearts, the seven of diamonds, and a king of hearts (value 10). To make the quotient a big number, the player multiplies 10 and 7, gets 70, and divides 70 by 3. The quotient is 23 1/3. The other player has the seven of diamonds, the seven of hearts, and the five of diamonds.  He/she decides to multiply 7 and 7, gets 49, divides 49 by 5, and gets a quotient of 9 4/5. The first player wins.  The winner collects all cards.
4. If both players have the same quotient, they declare war.  Each one puts down three cards face down. Then each one turns three cards face up.  The bigger quotient on the three displayed cards wins. The winner collects all cards.
5. The first person with an empty hand loses.

In one game, children will use more than five hundred multiplication and division facts. They also try several choices in each display as they want to maximize the outcome.  This teaches them problem solving and flexibility of thought. Within a few weeks, they can master simple division facts. I allow children to use the calculator to check their answers as long as they give the quotient before they find it by using the calculator.

[1]See previous posts on NumbersenseSight Facts and Sight WordsWhat does it Mean to Master Arithmetic Facts?, etc.

[2]See previous posts on Non-Negotiable Skills at the Elementary Level.  For a fuller treatment on the topic see: How to Teach Multiplicative Reasoning by Sharma (2019).

# NUMBER WAR GAMES II: Addition and Subtraction facts

In the quantitative domain, the focus of three-years of mathematics instruction, from Kindergarten to second grade, is that by the end of second grade, children master the concept of additive reasoning. Additive Reasoning means mastery of: (a) concepts of addition and subtraction, using multiple conceptual and instructional models, settings, diverse vocabulary and phrases that translate into addition and or subtraction concepts and operations, (b) related procedures (in standard and non-standard forms, using place-value and decomposition/ recomposition at one-digit and multi-digit levels), (c) relevant applications to solving problems in learning other mathematics concepts (e.g., multiplication), solving problems in other subject areas (e.g., time line), and relevant real-life problems (e.g., money, time, measurment), and, (d) the understanding that adddition and subtraction are inverse operations (e.g., given an addition equation, one can express it in subtraction form and a subtraction equation into an addition form and can use this knowledge in solving problem in various situations).

To achieve this goal of quantitative domain, at the end of Kindergarten, a child should have mastered: (a) Counting forward and backward by 1, 2, and 10 starting from any number up to at least 100; (b) Number vocabulary (lexical entries for number) of at least up to 100; (c) Number concept: visual clustering (generalizing subitizing)–recognizing, by observation (without counting), a cluster of objects up to 10, numberness–integrating the size of a visual cluster, its orthographic (shape of the number–“5”), and audatory (saying: f-i-v-e) representations of a number, the skill of decomposition/ recomposition: visualy and mentally breaking a cluster of obejects into two sub-clusters and, then, a number into two smaller numbers and joining two clusters into one larger number (e.g., a cluster of 7 objects is made up of a cluster of 5 and a cluster of 2, therefore, 7 = 5 + 2 and 5 + 2 = 7; (d) the 45 sight facts (using decomposition/ recomposition, by sight, one finds that a number, up to 10, is the combination of two numbers (e.g., sight facts of 5 are: 1 + 4, 2 + 3, 3 + 2, and 4 + 1)); (e) Commutative and Associative properties of addition (e.g., on a Visual Cluster card of 9, one can see that 4 + 5 = 5 + 4 and (3 + 2) + 4 = 3 + (2 + 4); (f) of Making ten (what two numbers make 10); (g) Knowing teens’ numbers (combination of 10 and a number, i.e., 10 + 5 = 15, 10 + 7 = 17, 15 = 5 + 10); (h) Concept and role of zero in forming larger numbers (10, 20, 30, etc.) and adding to and subtracted from a number; and, (i) Place-value of 2-digit numbers: what two digits make a number? and what two numbers make a number? (e.g., digits 1 and 5 make 15 and numbers 10 and 5 make 15).

Mastery of number concept is the foundation of arithmetic. The ten numbers/digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are the alphabets of the quantitative language and numeracy. Sight facts are the sight words of this language. Decomposition/recomposition is the arithmetic analog of phonemic awareness. By the help of numberness, decomposition/ recomposition, sight facts, making 10, and the knowledge of teens’ numbers, one acquires the “number-attack skills” — mastery of arithmetic facts, beyond the 45 sight facts. For example, 8 + 6 = 8 + 2 + 4 (decomposition of 6 into 2 + 2) = 10 + 4 (knowledge of the sight facts of 10, making 10, and recomposition) = 14 (knowledge of making teens’ number). The child further extends to problems such as: 68 + 6 = 60 + 8 + 6 = 60 + 8 + 2 + 4 = 60 + 10 + 4 = 70 + 4 = 74, 68 + 6 = 68 + 2 + 4 = 70 + 4 = 74.

Similarly, in the quantitative domain, at the end of second grade, a child should have mastered: (a) Concepts of addition and subtraction and extending decomposition/recomposition to numbers greater than 10 (as mentioned in the previous paragraph); (b) Addition facts (sums of two single-digit numbers up to 10 by the end of first-grade and corresponding subtraction facts (by the end of second grade); (c) Place-value of three-digit numbers, both in canonical (e.g., 59 = 50 + 9) and non-canonical forms (e.g., 59 = 50 + 9 = 40 + 19 = 30 + 29, by the end of first grade) and four-digit numbers (both in canonical and non-canonical forms, by the end of second grade); (d) Addition procedures (standard and non-standard using place-value and decompistion/recomposition at one- and two-digit level), by the end of first grade and subtraction procedures (standard and non-standard using place-value and decompistion/ recomposition at one- and two-digit level), by the end of second grade.

In the quantitative domain, the focus of three-years of mathematics instruction, from Kindergarten to second grade, is that by the end of second grade, children have mastered the concept of Additive Reasoning. Aquiring additive reasoning means mastery of: (a) the concepts of addition and subtraction, with multiple conceptual and instructional models, settings, and diverse vocabulary and phrases that translate into addition and or subtraction (b) related procedures (standard and non-standard forms, using place-value and decomposition/recomposition at one-digit and multi-digit levels), (c) Application of additive reasoning to solve problems (learning other mathematics concepts, solving problems in other subject areas, and relevant real-life problems); (d) Understanding that adddition and subtraction are inverse operaions; (e) Understanding that the operation of Addition is commutative and associative, but the operation of Subtraction is not.

The concept of mastery of mathematics concept/skill/procedure means: (a) the child possesses the appropriate numerical language (vocabulary and phrases, syntax, and ability to translate from native language to mathematics language and from mathematics language to native language) for understanding and applying; (b) appropriate strategies (effective, efficient, and elegant) for deriving an arithmetic fact, skill, or procedure accurately in standard and non-standard forms; (c) appropriate level of proficiency and fluency in producing an arithmetic fact (e.g., 2 seconds or less for an oral arithmetic fact, 3 seconds for writing a fact); (d) appropriate level of numeracy: can execute an arithmetic procedure correctly, accurately, fluently, in non-standard and standard forms (algorithm) with understanding; (e) can estimate the answer/outcome to an addition and subtraction problem in acceptable range (without counting/writing or applying a procedure), and, (f) can apply the skill, concept, and/or procedure in–learning a new mathematics concept/skill/procedure, solving a problem in another subject/discipline/area, or a real-life problem.

A strategy is appropriate if it effective, efficient, and elegant if yields result with less effort and energy. It uses the principles of decomposition and/or recomposition, place-value, or peoperty of numbers/operations. It is transparent. It does not tax the working memory and processing ability too much, i.e., it is accessible, but moderately challenging. It is applicable not just to a specific or particular problem, but is generalizable, can be extrapolated and abstracted into a principle/concept/proecedure. The learner experienced being in the “zone of proximal development.” It results in a definite expereince in metacognition for the learner. Strategies based on counting experiences (e.g., addition and subtraction facts derived by coutning forward or backward, making change by counting, finding perimeter by counting units, etc.). A strategy can be at concrete, pictorial, visualization, or abstract level. However, if it is only at the concrete or pictorial level, it should be advanced to the abstract mathematical level also.

Learning and mastering arithmetic facts is dependent on three kinds of pre-requisite skills: (a) Mathematical: number concept (numberness, 45 sight facts, making ten, knowing teens numbers, properties of operation, and the most important skill decomposition/ recomposition), (b) Executive function skills: working memory, inhibition control, organization, and flexibility of thought, and (c) cogntiive skills: ability to follow sequential directions, discerning and extending patterns, spatial orientaiton/space organizaiton, visualization, estimation, deductive and inductive reasoning. Since these skill categories have operlaps, it is important that instructional activities embed as many of these skills as possible. Integration of the use of concrete instructional models, playing games, and interactive activities is the most pedagogically sound approach to mathematics instruction, whether it is regular (intial), intervention, or remedial instruction. Moreover, these skills, when practiced in isolation do not have lasting effect, learners do not see relationships between concepts, and do not last long. As a part of regular instruction, intervention, and then in reinforcement activities, to get maximum benefit, I plan lessons that follow the principle of six levels of knowing: intuitive, concrete, pictorial, abstrct, applications, and communications. I take a child from intuitive to communicaitons. In addition, I have found that students, at all grade level, from pre-Kindergarten to Algebra, find the Number War Games exteremly engaging and productive. They incorporate many of the principles included earleir.

All of these questions, with the help of visual cluster cards (Cuisenaire and Empty Number Line), should be answered and practiced orally. This process develops many of these pre-requisite skills individually and then helps integrate them. For example, working with the patterns on the Visual Cluster cards and then visualization of the cards aids in the development of the working memory. The organized sequential script helps them focus, organize and develop deductive reasoning. The reorganizing the pattern on the first card into sub-patterns and then integrating them with the patterns of the second card helps with the acqusition of decomposition and recomposition skills. The game setting: playing the game involves practicing these skills again and again and soldiifes these skills. For example, in playing the Number Addition War involves making, hearning, and practiicing more than 500 addition facts. Neurologically, questions instigate neural firing and making connections, that in turn invites oligodendrocytes–(oligo) to instigate the production of myelin–creating covering around the nerve fibers, that in turn controls and improves the impulse, and the impulse speed is skill. Each time a child practicies the script, the nerve fibres get stronger and wrapping wider and wider making the pathway of the nerve impulses into a major “highway.” The integration of (a) practicing the script, (b) visualizing the action guided by the script, (c) acceleration of the neural firing (better myelination), (d) and reducing the refractory time (the wait required between one signal and the other) makes learning optimal. The increased speed abd decreased refractory time No child will practices the number examples in a formal setting as he practices in one game. With the Number Addition War Game, children master their addition facts in a very short time. And that too with great deal of pleasure.

To make the learning robust and making children super-confident, we should practice finding the answers, even to one simple fact, in multiple ways. In the script developed and used above, the practice strengthened certain nerual pathways and it opened certain “files” (e.g., sight facts, making ten, and making teen’s number files) in the long-term memory (the practice was being performed in the working memory and it was transferrdd to long-term memory), but the retrieval is easier and more useful, when the infromation is transferrd to long-term memory in more than one way (different instructional materials, stategies, models, scripts, order, modality of learning, levels, occasions, times, groupings, and settings). For example, the fact 8 + 6 can be derived using counting objects (e.g., objects, fingers, on number line, etc.), Ten Frames, Rek-n-Rek, Visual Cluster cards, Cuisenaire rods, Invicta Balance, decompositin/ recomposition, Empty Number line, orally, visualization, and abstractly (notice the order–from less efficient to more efficient, from concrete to abstract, from lower level to higher level, from less understnading to more understanding, etc.). To provide the flexibility of thought, let us consider the following. In the following discussion, child’s answers to a fact problem are dispalyed in quotations.

Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14.” What about 6 + 8? “14.” How did you know that quickly? “Because 8 + 6 = 6 + 8.” What property is that? “Turn-around-fact.” What is another name for that property? “Commutative Property of Addition.” Is there any way you can find 6 + 8? “I do not know.”

Display two Visual Cluster cards: 6 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “6 + 6.” Good! What is 6 + 6? “16.” How did you find the answer? “6 + 4 is 10 and then 2 more is 12. So, 6 + 6 is 12.” What about 6 + 8? “14.” How did you know that quickly? What peoperty is that? Doubles property.” Good!

Display two Visual Cluster cards: 8 of dimonds and 8 of clubs. Do you know what addition problem can you make form these numbers? “8 + 8.” Good! What is 8 + 8? “16.” Can you find 8 + 6 using the fact that 8 + 8 = 16? “I do not know.” Is 8 + 6 is less than 8 + 8 or more than 8 + 8? “It is less.” If, the child begins to count. The teacher/parent should intervene. Look at the second 8-card. If you cover the 2 from the card, what do you see on the card. “a 6.” What addition problem do you have now? “8 + 6.” Can you figure out the answer for 8 + 6? “Yes, it is 14.” How do you know? “I know 8 + 6 = 14.” So, 8 + 6 is how much les than 8 + 8? “2 and 8 + 6 = 8 + 8 – 2.” Good!

Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14. Or, 6 + 6 + 2 = 14. Or, 8 + 8 – 2 = 14.” Do you know any other way? “I do not think so!” What if you took the one pip from the 8-card an put it on the 6-card, what problem would you have? “7 + 7.” What is 7 + 7? “14.” How do you know? Doubles property. Great! Can you apply making 10 strategy to this problem? “Yes! 7 + 3 is 10 and 10 + 4 = 14.” Great! Now, you know several ways of finding 8 + 6 or 6 + 8. How far apart are 8 and 6? “2 apart.” What number is between 6 and 8? “7.” So, 6 + 8 is same as 7 + 7. When two numbers are 2 apart, their sum is double of the middle property.

Practicing multiple strategies for finding the answer improves a child’s cognitive potential. They begin to see more realtionships, patterns, and concepts. They do not get helpless when they do not have the answer. They take action. This is an anti-dote to math anxiety.

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 two cards face up.  Each one finds the sum of the number represented by these cards. The bigger sum wins. For example, one has the three of hearts (value 3) and a 10 or a king of hearts (value 10). The sum of the numbers is 13. The other person has the seven of diamonds (7) and the seven of hearts (7). The sum is 14. The person with the sum of fourteen wins. The winner collects all the displayed cards and puts them underneath his/her pile.
4. The face card and the wild card can be assigned any number value up to ten.
5. If both players have the same sum, there is war. For example, one has the five of hearts (value 5) and the seven of clubs (value 7), and then the sum is 12. The other person has the six of spades (value 6) and the six of clubs (value 6). They declare war.
6. Each one puts three cards face down. Then each one displays another two cards face up.  The bigger sum of the last two cards wins.
7. The winner collects all the cards and places them underneath his/her pile.
8. The first person with an empty hand loses.

This game is appropriate for children who have not mastered/automatized addition facts.

Initially, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters on the cards to recognize and find the sums.  In one game, children will encounter more than five hundred sums. Within a few weeks, they can master all the addition facts. Initially, if the child does not know his sight facts, the game can be played with dominos or with a deck of Visual Cluster cards with numbers only up to five. Then, include other cards.

I sometimes allow children to use the calculator to check their sums.  The only condition I place on calculator use is that they have to give the sum before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts. After they have learned the 10 ×10 arithmetic facts (sums up to 20), you can assign values to the face cards:  Jack = 11, Queen = 12, and King = 13. The joker has a value assigned by the player. Its value can be changed from hand to hand.  The joker is introduced with a variable value so that children can get the concept of variable very early on.

Variation 1:  After a while, you might make a change in the rules of the game.

Each child displays three cards, discards a card of choice, and finds the sum of the remaining two cards.

Variation 2: Each child displays three or four cards, finds the sum of the three or four cards, and the bigger sum wins.

Game Three: Subtraction War

Objective: To master subtraction 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 two cards face up.  Each one finds the difference of the two cards. The bigger difference wins. For example, one has the three of hearts and a king of hearts (value 10), and then the difference is 7. The other has the seven of diamonds and the seven of hearts, and then the difference is 0.  The first player wins.  The winner collects all cards.
4. If both players have the same difference, they declare war.  Each one puts down three cards face down. Then each one turns two cards face up.  The bigger difference of the two displayed cards wins. The winner collects all cards.
5. The first person with an empty hand loses.

As in addition, children can initially count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the difference. In one game, children will use more than five hundred subtraction facts.  Within a few weeks, they can master subtraction facts. Initially, the game can be played with dominos.

I allow children to use the calculator to check their answers. As mentioned above, the only condition I place on calculator use is to give the difference before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts.  This game is appropriate for children of all ages to reinforce subtraction facts.

Variation 1:  After a while, you might make a change in the rules of the game.  Each child displays three cards, discards a card of choice, and finds the difference using the remaining two cards.

Variation 2: Each child displays three cards, finds the sum of any two cards, and subtracts the value of the third card.  The bigger outcome of addition and difference wins.

Variation 3 :Each child displays three or four cards, an objective number is decided and finds the result by adding or subtracting of any combination of cards gets the declared number as the result. The bigger outcome of addition and difference wins.  No number can be used more than once.

# 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 (maheshsharma@me.com).

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.

[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.

# PROCESSING SPEED AND WORKING MEMORY: MATHEMATICS ACHIEVMENT

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 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.

# TEACHING MATHEMATICS AS A SECOND LANGUAGE

MATHEMATICS AS A SECOND LANGUAGE

A ONE-DAY WORKSHOP

BY

MAHESH C. SHARMA

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.

1. MATHEMATICS IDEAS HAVE THREE MAIN COMPONENTS (in this order):
• Linguistic component,
• Conceptual component, and
• Procedural component

LINGUISTIC COMPONENTS

• VOCABULARY
• SYNTAX
• TWO-WAY TRANSLATION

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, 34, 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.

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.

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.

“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.

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?

Question:  Have you ever seen this word before?

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.

 Mathematics Literacy 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 numbers 2.     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 operations 3.     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) strategies 4.     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 others 5.    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.

http://www.mathematicsforall.org

## 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

### DVDs

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):

CENTER FOR TEACHING/LEARNING OF MATHEMATICS

754 Old Connecticut Path, Framingham, MA 01701

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

info@mathematicsforall.org

http://www.mathematicsforall.org

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

Mahesh@mathematicsforall.org

Center for Teaching/Learning of Mathematics

754 Old Connecticut Path

Framingham, MA 01701

info@mathematicsforall.org

www.mathematicsforall.org

Mathematics Blog :www.mathlanguage.wordpress.com

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# LEARNING PROBLEMS IN MATHEMATICS: DYSCALCULIA

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.