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

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