NUMBER WAR GAMES II: Addition and Subtraction facts

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

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

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

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

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

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

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

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

Before and during the game, the emphasis should be on the use of sight facts, decompositon/recomposition, making ten, and making ten. During first few games, a teacher/parent should participate in the game with the children. During the game she should develop script fot finding the facts. For example, if during the Addition War Game game a child gets two cards: 6 of diamonds and 8 of spades, he needs to find the sum 6 + 8. If the child does not know the answer readily, then the teacher/parent should practice fidning the sum of 8 + 6 by asking questions that help the child practice the necessary pre-requisite skills and develop a script that will help the child to find the answers independently. For example, (a) Look at your cards: (points the cards: 8 and 6). Which is the larger number? “8” (The numbers in quotation marks are child’s answers.) (b) Remember the strategy of adding numbers? “Make ten, first” (c) Good! How do you make 8 as 10? “by adding 2 to 8” (c) Good! You moved the 2 pips to 8 card and it became 10. Where did 2 come from? “from 6” (d) What is left in 6? “4” (e) What do you add now? “10 and 4” (f)) What is 10 plus 4? “14” (g) So, what is 8 + 6? “8 + 6 = 14.” the teacher could also use Cuisenaire rods or the Empty Number Line (ENL), to support the child in developing the script. When children have developed, visualized, and mastered such scripts, they become independent learners.

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

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

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

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

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

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

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

Game Three: Number Addition War

Objective: To master addition facts

Materials:  Same as above

How to Play: 

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

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

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

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

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

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

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

Game Three: Subtraction War

Objective: To master subtraction facts

Materials:  Same as above  

How to Play: 

  1. The whole deck is divided into two equal piles of cards.  
  2. Each child gets a pile of cards.  The cards are kept face down. 
  3. Each person displays two cards face up.  Each one finds the difference of the two cards. The bigger difference wins. For example, one has the three of hearts and a king of hearts (value 10), and then the difference is 7. The other has the seven of diamonds and the seven of hearts, and then the difference is 0.  The first player wins.  The winner collects all cards. 
  4. If both players have the same difference, they declare war.  Each one puts down three cards face down. Then each one turns two cards face up.  The bigger difference of the two displayed cards wins. The winner collects all cards. 
  5. The first person with an empty hand loses. 

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

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

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

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

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

NUMBER WAR GAMES II: Addition and Subtraction facts

Effective Teaching of Mathematics

The framers of CCSS-M have identified Standards of Mathematics Practice (SMP) based on practices and research on teaching and learning that consistently produce mathematics learners with high achievements. To be an effective teacher, one needs to know the content, pedagogy and models to deliver that content, and understand how students learn.

CCSS-M gives us what is important to teach and learn at what level. It describes the content and levels and nature of content mastery. However, teachers decide how to teach and assess. The goal of the first four to five years of students’ mathematics experience is to become proficient and comfortable in number concept, numbersense, and numeracy. After that, students use their understanding and fluency in numeracy skills to learn mathematics—algebraic and geometric models, and their integration.

The Standards for Mathematical Practice (SMP) describe ways in which students should be engaged in increasingly demanding subject matter as they grow in mathematics expertise and content throughout the elementary, middle and high school years.

To support students’ growth in mathematical maturity, designers of curricula, assessments, and professional development should connect mathematical instructional practices to the mathematics content espoused in the CCSS-M. Without connecting content standards with instruction practice standards, we cannot achieve the goals of CCSS-M. Assessment, on the other hand, is to see whether that connection has been made or not.

The Mathematical objects (e.g., numbers of various kinds—natural to complex; geometric entities—shapes, figures, diagrams, functions; operations and procedures of different types—decomposition/ recomposition, manipulations of numbers of different kinds and forms, e.g., long-division; transformations and functions—static and dynamic, congruence and similarity, matrices and determinants, etc. are examples of components and manifestation of content standards.

Mathematics practice standards, on the other hand, describe what actions teachers are to take so that students make the mathematics content—language, concepts, procedures, and skills their own. Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs. The Standards of Mathematics Practice are action steps to make the content of the CCSS-M possible to take hold in the classroom and make students learners of mathematics.

Why SMP?
The National Mathematics Advisory Panel concluded that our students have reasonable, though incomplete, factual and procedural knowledge, but poor conceptual knowledge (e.g., many do not fully understand the base-10 number system, concepts of fractions, how decimals and fractions are related, the differences in arithmetic and algebraic reasoning). This is because of our over emphasis on procedural knowledge throughout children’s mathematics experience.

At the same time, the last 20 years of mathematics education reform indicate that American students, even many college students, neither have automatized fact retrieval nor have achieved fluency with procedures. For example, many students can perform routine procedures (e.g., procedures on fractions) but cannot justify the reasons for the steps involved or provide estimated answers before they execute the procedures.

Our students’ lack of conceptual understanding is a major cause of concern and requires investment after they leave school. For example, many corporations spend large sums of money in training high school graduates in their use of simple arithmetic. The problem has also reached college and university levels. Half of the students at community colleges, and 1 in 5 students at four-year institutions, require remedial courses in writing and mathematics, with community colleges spending more than 2 billion dollars on remediation and four-year colleges $500-million. As one university leader pointed out,

Many of us in higher education have observed an increasing number of students arriving at our doorstep not fully prepared to pursue a college degree. This is our collective problem as a nation. (Chronicle of Higher Education, 2014, June 14)

Framers and supporters of CCSS-M and SMP believe that this latest educational reform, if properly implemented, can alleviate some of these problems and narrow the college and career preparation gap. Their aim is to improve instruction so that students acquire mathematics ideas with conceptual understanding and procedural fluency so they can apply mathematical tools effectively and provide reasons for what they do.

Let us consider an analogy to illustrate this point: You can watch two people swim a length of a pool. They take nearly the same time to swim the same distance, but one of them churns the water more and takes more strokes. When this swimmer gets out s/he is breathing a bit heavier but is in great shape, so it is not too noticeable. The other swimmer took fewer strokes and seemed to glide through the water.

If we assess the performance only by a stopwatch, we will conclude that they are swimmers of the same competence. In actuality, they are not.  As the lengths pile up and the task gets harder, the second swimmer will do much better. No matter what the stopwatch said, s/he is a much better swimmer than the other. And if we ask the first swimmer to swim a long distance over deep water, s/he may very well drown.

Students who only memorize facts, formulas and procedures, without understanding, are like the first swimmer. They churn and work hard, and if they are gifted with an outstanding memory, they can pull it off for a while. These students can – and often do – get by in the early grades when they can rely on their strong counting skills and contextual clues (e.g., concrete materials and pictures) to “find” a fact but they typically hit a wall sometime around 4-5th grade when they have to deal with fractions and decimal numbers with problems involving unfamiliar content and complex mathematics vocabulary words. On the other hand, children who have the rich language, robust conceptual schemas, and clear understanding of and fluency in executing procedures develop into graceful, effortless swimmers in the waters of mathematics.

In cultivating greater conceptual knowledge, effective teachers do not sacrifice procedural or factual knowledge. Procedural or factual knowledge without conceptual knowledge is shallow and unlikely to transfer to new contexts. At the same time, conceptual knowledge without procedural or factual knowledge is ineffectual and inefficient in execution. It needs to be connected to procedures so that students learn that the “how” has a meaningful “why” associated with it. It is more effective to move from conceptual knowledge to procedural knowledge. Increased conceptual knowledge helps students move from competence with facts and procedures to the automaticity needed to be good problem solvers.

When teaching conceptual, procedural and factual knowledge, effective teachers ensure that students gain automaticity. Their students know that automaticity and understanding of procedures and facts is important because it frees their minds to think about concepts and making connections. This requires some memorization and ample practice and the ability to communicate with the support of reason.

What the Research Says
The data from the 13 million students who took the Program for International Student Assessment (PISA) tests shows that the lowest achieving students worldwide are those who use memorization strategies – those who think of mathematics as methods to remember and who approach mathematics by trying to memorize facts and procedural steps. The implication is that we should not encourage rote memorization without understanding (i.e., in dividing fractions, teachers should not continue to use statements such as: “just invert and multiply”) but, instead, we should present appropriate models to create conceptual schemas and arrive at and master the procedure with proper understanding and mastery that is rooted in reasoning.

Because of the emphasis on procedural teaching, the U.S. has more memorizers than most other comparable countries. Perhaps not surprisingly mathematics teachers, driven by narrow and subjective tests (e.g., end of section tests in textbooks that value only the methods considered in the textbook), have valued those students over all others, communicating to other students that they do not belong in the mathematics class. Current research about how we learn and how our brains receive and process information shows that the students who are better memorizers may not have more ability or potential, but, unfortunately, we continue to value the faster memorizers over those who think slowly, deeply and creatively. We need students with mastery or conceptual, procedural, and factual knowledge for our scientific and technological future. Certain things should be memorized to relieve our work memory from mundane fact work to focus on creativity and applications.

Poor conceptual understanding comes at a cost. For example, if a student thinks that an equal sign means “put the answer here in the box,” she will be confused the first time she sees an equation with terms involving variables and multiple operations on both sides of the equal sign. Similarly, when a student first encounters factoring (whether in the case of whole numbers or polynomials), she ought to see its relationship to division and multiplication. But she may not be able to do so unless she has a deep conceptual understanding of multiplicative reasoning—that division and multiplication are inverse operations. She also will be slowed in factoring if she hasn’t memorized the multiplication tables, divisibility rules, prime factorization, and short division.

Factual and procedural knowledge are acquired by practice and stick-to-ness to tasks and practice. But, to be engaged in the task deeply, the task should be meaningful and the student should have understanding of the concepts and skills involved in it.

Of the three varieties of knowledge that students need, conceptual knowledge is difficult to acquire. It is difficult because knowledge is never transferred from one person to another directly nor is it developed without explorations, concrete manipulations, effective reasoning, and questioning. Rather, new concepts must build upon something that students already know when they explore the new concepts and ideas. Examples that are familiar to students and analogous to the current concept are useful to understand the concept.

SMP Principles
The writers of the CCSS-M were careful to balance the development of conceptual understanding, procedural skill and fluency, and application at each grade level. The standards are based on the idea that procedural skill and fluency expectations hinge on conceptual understanding. Fact fluency and procedural fluency help students to develop concepts, make connections, observe patterns, and form relationships between ideas, concepts, skills, and procedures thereby facilitating mathematical thinking. With mathematical thinking, students take interest in mathematics and develop mathematics stamina.

To achieve the different kinds of knowledge, we need to adopt pedagogical principles in every lesson that are informed by the Standards of Mathematics Practice:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Each of these standards adds to a teacher’s ability to develop the different components of knowledge, helping children acquire mathematical ways of thinking, creating interest for mathematics, and recognizing the power of mathematics. We need to incorporate these practices in our lessons if we want to have students who enjoy doing mathematics and achieve higher.

Examples of Standards
In future posts, I want to comprehensively develop how to implement each of these standards. At present, I want to consider a few illustrative examples.

Standard number one, for example, deals with developing understanding and engagement with a problem and creating mathematics stamina in solving it.

Solving a problem almost always depends on what tools (linguistic, conceptual, and procedural) one knows and how to connect those tools with the current problem. As students advance and encounter new problems, new concepts will increasingly depend on old conceptual knowledge. For example, understanding and solving algebraic equations depend on the understanding of the concepts of equality, variable, arithmetic operations, operations on fractions and integers, ability to generalize, etc. In solving problems factual, procedural, and conceptual knowledge all go together. And to stay engaged with the problem requires a student to have mastery of these prerequisite tools.

Familiarity is not the only ingredient necessary for successful problem solving. Students are more likely to understand abstract ideas when they see many diverse examples in the classroom that depict the conceptual components, schemas, and constraints of the ideas. In such instances, they can learn the essential properties in the concept of the problem (e.g., in fractions, the division of the object into equal parts and what is equal there) and which properties are incidental (e.g., in fractions, that the resulting parts need to be whole numbers).

Standard number seven: to look for structure in mathematics concepts and procedures is at the heart of understanding mathematics. Students need to realize that mathematics is the study of patterns—the underlying structures. Students frequently fail to understand the concept if they are not helped to discern patterns—to look for the structure—commonalities among examples and what is different in these examples.

Indeed, when the teacher introduces a concept through an abstract definition alone, e.g., the standard deviation is a measure of the dispersion of a distribution, students miss the conceptual understanding of the concept of standard deviation. The standard deviation is dependent on the spread; therefore, our examples should show the impact of that spread on standard deviation: Two groups of people have the same average height, but one group has many tall and many short people, and thus has a larger distribution and standard deviation, whereas the other group mostly has people’s heights right around the average, and thus has a small standard deviation.

If we introduce students to the formal procedure of finding the standard deviation too quickly, they won’t realize the relationship of spread on standard deviation. To realize the importance of such a component in the concept, we should also consider special cases and non-examples: e.g., to find the standard deviation when every piece of the data is the same, the spread has a particular meaning.

The third standard is the key to developing conceptual understanding in students. No conceptual understanding can be developed without emphasizing reasoning. To emphasize the importance of reasoning— concrete and abstract, examples, non examples, and counter examples play a crucial role.

Let us consider the definition of prime number: A number is called prime if it has exactly two distinct factors, namely, 1 and itself. The definition of prime number has two key features: it has factors and there are exactly two of them. 2, 3, 5, etc. are prime numbers as they have two factors only. It is important to give examples of prime numbers and examples of numbers that are not prime and the reasoning behind the choice. Examples emphasize the components of the concept, and “non examples” help students see the subtleties and nuances of the concept. For example, why is 1 not a prime number? (It has only one factor.) Why is 0 not a prime number? (0 is divisible by any non-zero number; therefore, it has more than two factors.)

The discussion to discern similarities and differences—comparing and contrasting examples, helps students to acquire the language and conceptual knowledge with rigor. With the help of appropriate language and transparent models, conceptual knowledge is converted into procedural knowledge. For this to happen, the conceptual (representational models—concrete and pictorial) should be congruent with abstract, procedural models. And with practice, procedural knowledge is then converted into factual knowledge. This practice should take place in a variety of problems and problem situations that are related to the procedure to arrive at the appropriate level of fluency.

If students fail to gain conceptual understanding, it will become harder to catch up, as new conceptual knowledge depends on the old. Students will also become more likely to simply memorize algorithms and apply them without understanding.

Helping Students Learn Concepts and Procedures
In our schools, much is made of the use of manipulatives to help children understand abstract concepts in mathematics, but many manipulatives and models themselves are abstract (students treat them as a symbol for something else), and not all manipulatives help learning—they sometimes impede it. This is most likely when manipulatives are so visually interesting that they distract from their purpose, when their relationship to the concept to be represented is obscure, or when they are used for rote counting. Manipulatives seem helpful because they are concrete; to be helpful, they should satisfy certain properties.

To illustrate the idea of a fraction, one might divide a cookie in two for the purpose of sharing it with a student. The concreteness of this example is likely less important than its familiarity. In contrast, suppose I cut a hexagon into two pieces and said, “See? Now there are two equal pieces. Each one is half a hexagon.” That example is concrete but less effective because it is unfamiliar; the student has no experience with divided hexagons, and the purpose of sharing is also missing.

Concreteness, in itself, is not a magical property that allows teachers to pour content into students’ minds. It is the familiarity that helps because it allows the teacher to prompt students to think in new ways about things they already know. However, familiarity also may create some misconceptions, half a pizza, half a cookie, half a glass are not precise as key characteristics of fractions may be missed. Students know a fraction when they focus on: (a) What is my whole here? (b) How many parts are there in this whole? (c) Are the parts equal? (d) Do all the parts together make the whole? (e) What is the name of each part? (f) How many of these parts will make the whole? And (g) What is the new name of the whole in the light of these parts? The teacher’s language, questions, and sequence of activities with materials transform the concrete models into representations—pictorial and abstract.

A teacher must move from familiar materials and models to the form that shows all the attributes of the concept and then can lead to abstract representations that are congruent to the abstract procedure. As concepts become more complex, it becomes harder to generate familiar examples from students’ lives to generate mathematics conceptual schemas, and teachers may have to use analogies more often. In such cases, a familiar situation is offered as analogous to the concept under discussion, not as an example of the concept.

An Example of Implementing SMP
In order to focus instruction responding to CCSS-M and applying SMP, teachers need to identify the essential language, concepts, procedure and skills in each major standard. SMP does not advocate a “one size fits all” model or a boxed curriculum. Essential Elements of each lesson informed by SMP include: Integration of language, concepts, and procedures, Multisensory (appropriate and efficient models), decomposition/ recomposition of problem components, Synthetic-Analytic (seeing patterns and analyzing the problem), Structure (logical language categories), Sequential (simple to complex), Cumulative (continually making connections), Repetitive, cognitive (meta), Diagnostic and Prescriptive (design lesson to assure progress and plan next lesson around noted errors that need additional reinforcement).

Let us illustrate this process in one specific mathematics standard. Mathematics standard 4.OA.A.3 (CCSS-M, 4th grade) says:

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

This standard includes only a few sentences, but involves several different terms, concepts, procedures, and skills. To make sense of and understand them, students need to know the meaning of these terms and concepts and execute the procedures.

There is a need to delineate these elements, focus on them, teach them, help students master them, connect them with other concepts and procedures, and then assess all of these elements.

There should be a clear understanding of what and how to represent each concept, procedure and the skill involved in this standard. Every concept and procedure involved in this standard should be transformed into a set of concepts and skills to be learned, mastered, and applied by the students. In the context of CCSS-M, teaching should be to acquire understanding; students should arrive at fluency and should be able to apply concepts and skills contextually.

Step # 1
Language and Concepts

  • Know the meaning of each word and term in order to translate from English to mathematical equations
  • Identify the unknowns and understand the role of these unknowns; know the relationship(s) between knowns and unknowns

Step # 2
Language and Concepts

  • Represent terms and words into appropriate mathematics symbols; translate multi-step word problems into/by equation(s)

Skills and Facts

  • Identify the units and the domain and the range of the variable(s) involved in the problem

Step # 3
Procedures

  • Solve multistep word problems by establishing the sequence of arithmetic operations

Skills and Facts

  • Know and apply the properties of equality; mastery of arithmetic facts; execute procedures for whole numbers efficiently; know the order of operations

Step # 4
Concepts and Procedure

  • Assess the reasonableness of the answer

Skills and Facts

  • Numbersense: Use mental computations such as rounding to estimate the outcome of an operation

Step # 5
Language and Concepts

  • Interpret the answer including the remainder if involved; express the division problems in multiple ways

Skills and Facts

  • Add, subtract, multiply and divide whole numbers fluently with understanding; know the role of numbers in each operation, e.g., know the role of remainder in practical situations

Learning with rigor using SMP means that the students not only understand the concept and procedures but also see that a particular method(s) may have limitations and that the context of the problem defines the applicability and efficiency of the method.

 

Effective Teaching of Mathematics