Professor Mahesh Sharma’s second workshop on Proportional Reasoning is on February 15, 2019 at Framingham State University. Here is the link:

Professor Mahesh Sharma’s second workshop on Proportional Reasoning is on February 15, 2019 at Framingham State University. Here is the link:

Summer Slide and Regression

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

List of Games (with identified prerequisite skills)

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

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

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

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

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

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

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

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

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

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

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

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

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

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


[1]Language instigates models,

Models help develop conceptual schemas and instigate thinking,

Thinking instigates understanding,

Understanding produces competent performance,

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

Self-esteem is the motivating factor for all learning.

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

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

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

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

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

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

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

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

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

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

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

[11]Available from the Center.

[12]Available from the Center.

[13]Available from the Center.

[14]Available from the Center.

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

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

Summer Slide and Regression

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

January 25, 2019 

For grade 3 through grade 9 teachers and special educators

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

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

For grade 3 through grade 9 teachers

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

D. Algebra

Arithmetic to Algebra: How to Develop Algebraic Thinking

For grade 4 through grade 9 teachers

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

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

For K through grade 12 teachers

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

Learning Problems in Mathematics (including dyscalculia)

For special education and regular education teachers 

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

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

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

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

Click here to register

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

How To Improve Numbersense – Number Relationships: Counting Part Three

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Screen Shot 2018-07-03 at 3.58.02 PM

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

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

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

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

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

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

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

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

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

Teacher: Great!

Then the teacher moves to another child.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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




How To Improve Numbersense – Number Relationships: Counting Part Three

Role of Homework and Achievement

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

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

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

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

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

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

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

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

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

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

The objective of homework is to:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Role of Homework and Achievement

Mathematics Education Workshop Series with Professor Mahesh Sharma – Spring 2018

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Don’t miss these highly interactive day workshops with Professor Mahesh Sharma!

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

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

In these workshops, we provide strategies, understanding and pedagogy that can help teachers achieve these goals.

Dyscalculia and Other Mathematics Difficulties
Who Should Attend:
For K through grade 11 teachers (regular and special educators)
When: March 30, 2018
Workshop Description:
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics? and (d) how to fill gaps in mathematics learning.
Cost: $49.00 Includes Breakfast, Lunch and Materials

Number Concept, Numbersense, and Numeracy, Part One
Who Should Attend:
For K through grade 2 teachers, special educators and interventionists
When: April 13, 2018
Workshop Description:
Number concept is the foundation of arithmetic. Ninety-percent of students who have difficulty in arithmetic have not conceptualized number concept. In this workshop we help participants how to teach number concept effectively. This includes number decomposition/recomposition, visual clustering, and a new innovative concept called “sight facts.”
Cost: $49.00 Includes Breakfast, Lunch and Materials

Number Concept, Numbersense, and Numeracy, Part Two
Who Should Attend:
For K through grade 3 teachers, special educators and interventionists
When: May 11, 2018
Workshop Description:
According to Common Core State Standards in Mathematics (CCSS-M), by the end of second grade, children should master the concept of Additive Reasoning (the language, concepts and procedures of addition and subtraction). The mastery means (a) understanding, fluency, and applicability. In this workshop, the participants  learn effective, efficient, and elegant ways of achieving this with their children.
Cost: $49.00 Includes Breakfast, Lunch and Materials

How to Teach Fractions Effectively
Who Should Attend:
Grade 3 through grade 9 teachers and special educators
When: May 18, 2018
Workshop Description:
According to Common Core State Standards in mathematics (CCSS-M), by the end of sixth grade, children should master the concept of Proportional Reasoning (the language, concepts and procedures ratio and proportion). The concepts of ratio and proportion are dependent on the mastery of the concept of fractions. The mastery means (a) understanding, fluency, and applicability of fractions and operations on them. In this workshop, the participants will learn effective, efficient, and elegant ways of achieving the concept of fractions and multiplication and division of fractions and help their children achieve that.
Cost: $49.00 Includes Breakfast, Lunch and Materials

Arithmetic to Algebra: How to Develop Algebraic Thinking
Who Should Attend:
Grade 4 through grade 9 teachers
When: May 25, 2018
Workshop Description:
According to CCSS-M, by the end of eighth-grade, students should acquire algebraic thinking. Algebra is a gateway to higher mathematics and STEM fields. Algebra acts as a glass ceiling for many children. From one perspective, algebra is generalized arithmetic. Participants learn how to extend arithmetic concepts to algebraic concepts and procedures effectively and efficiently. Algebraic thinking is unique and abstract and to achieve this, thinking students need to engage in cognitive skills that are uniquely needed for algebraic thinking. In this workshop we look at algebra from both perspectives: (a) Generalizing arithmetic thinking and (b) developing cognitive and mathematical skills to achieve algebraic thinking.
Cost: $49.00 Includes Breakfast, Lunch and Materials

Dyscalculia and Other Mathematics Difficulties
Who Should Attend:
For K through grade 11 teachers (regular and special educators)
When: June 8, 2018
Workshop Description:
In this workshop, participants will learn (a) why learning problems in mathematics (e.g., dyscalculia, etc.) occur, (b) how children learn mathematics, (c) what are effective methods of teaching mathematics? and (d) how to fill gaps in mathematics learning.
Cost: $49.00 Includes Breakfast, Lunch and Materials. Registration, workshop hours,  location, and parking please call: Anne Miller at 508.626.4553

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


FSU | Office of Continuing Education | 508.626.4553

Mathematics Education Workshop Series with Professor Mahesh Sharma – Spring 2018