NUMBER WAR GAMES III: Multiplication and Division Facts

Teaching Multiplication and Division Facts

When I reached middle school, the headmaster welcomed us and gave a little “talk” on what was expected of us in the middle school. He talked about forming the habit of reading everyday for pleasure and for school, importance of doing homework every night, selecting a sport that we could enjoy even after we left school, keeping a good notebook for classwork, writing everyday something of interest, become proud of our school, and then he said:  “Those of you who have mastered your multiplication and division facts, you will be finishing eighth grade with a rigorous algebra course and then finish high school with a strong calculus course.” After laughter subsided, we realized the importance of the statement our elementary mathematics teacher–Sister Perpetua used to make as she was making sure that we had mastered our multiplication tables by the end of third grade and division facts by the end of fourth grade. We had heard about headmaster’s welcoming speech from her and the students who had gone before us.  As headmaster and a demanding math teacher, he was very popular and respected by teachers, parents, and most students.  He would repeat the ideas many times after that. It was more than sixty-five years ago, but his words are still fresh in my mind.

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

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

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

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

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

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

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

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

Similarly, during my workshops for teachers, when I ask them to define “multiplication.” Most people will define multiplication as “repeated addition,” which is something that most of them know about multiplication from their school experience. Then I ask, according to your definition, what do you think the child would do to find 3 × 4?  What does that mean to the child? The answer is almost immediate.  “It means that child thinks 3 groups of four.  He would count 4 three times.” As we can see, the person is mixing the two models of multiplication: “repeated addition” (3 repeated 4 times: 3 + 3 + 3 + 3 and “groups of” (4 + 4 + 4). Their definition and the action for getting the answer do not match. There is incongruence between their conceptual schema for the concept of multiplication and the procedure for developing a fact. Many children when deriving multiplication facts have the same confusion. To derive 6 × 8, A child would say (if he knows the table of 5–a very good sign): “I know 8 × 5 is 40 and then I add 6 so the answer for 6 × 8 = 46. The reason for wrong answer is this confusion in mixing the definitions. Children should understand different definitions of multiplication. The concept and problems resulting in multiplication emerge in several forms; repeated addition, groups of, an array, and area of a rectangle.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Game Four: Mastering Multiplication Facts

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

How to Play

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

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

Game Five: Division War

Objective: To master division facts

Materials:  Same as above

How to Play: Mostly, same as above.

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

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

Game Six: Multiplication/Division War

Objectives: To master multiplication and division facts

Materials:  Same as above

How to Play: Almost same as the other games

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

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

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

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

NUMBER WAR GAMES III: Multiplication and Division Facts

Attend to Precision: The Foundation of Mathematical Thinking

The sixth of the Standard of Mathematics Practice (SMP) in Common Core State Standards (CCSS-M) is: Attend to Precision. The key word in this standard is the verb “attend.” The primary focus is attention to precision of communication of mathematics—in thinking, in speech, in written symbols, in usage of reasoning, in applying it in problem solving, and in specifying the nature and units of quantities in numerical answers and in graphs and diagrams. With experience, the concepts should become more precise, and the vocabulary with which students name the concepts, accordingly, should carry more precise meanings.

The word “precision” calls to mind accuracy and correctness—accuracy of thought, speech and action. While accuracy in calculation is a part, clarity in communication is the main intent of this standard. The habit of striving for clarity, simplicity, and precision in both speech and writing is of great value in any discipline and field of study. In casual communication, we use context and people’s reasonable expectations to derive and clarify meanings so that we don’t burden our communication with too many details that the reader/listener can surmise anyway. But in mathematics (thinking, communicating, and writing), we base each new idea/concept logically on earlier ones; to do so “safely,” we must not leave room for ambiguity and misconceptions.

Students can start work with mathematics ideas without a precise definition. With experience, the concepts should become more precise, and the vocabulary with which we name the concepts can, accordingly, should carry more precise meanings. But we should strive for clarity and precision constantly. Striving for precision is also a way to refine understanding. By forcing an insight into precise language (natural language or mathematical symbols), we come to understand it better and then communicate it effectively. For example, new learners often trip over the order relationships of negative numbers until they find a way to reconcile their new learning (–12 is less than –6) with prior knowledge: 12 is bigger than 6, and –12 is twice –6, both of which pull for a intuitive feeling that –12 is the “bigger” number. Having ways to express the two kinds of “bigness” and the sign defining the direction helps distinguish them. Learners could acquire technical vocabulary, like magnitude or absolute value, or could just refer to the greater distance from 0, but being precise about what is “bigger” about –12 helps clarify thinking about what is not bigger. With such a vocabulary, one can express the relationship between the two numbers more precisely.

The standard applies equally to teachers and students and by extension to textbooks, modes and purpose of assessments, and expectations of performance. To achieve this, teachers need to be attentive to precision in their teaching and insist on its presence in students’ work. They should demonstrate, demand and expect precision in all aspects of students’ interactions relating to mathematics with them and with other students. Teachers must attend to what students pay attention to and demonstrate precision in their work, during the learning process and problem solving. This is not possible unless teachers also attend to the same standards of precision in their teaching.

Teachers, while developing students’ capacity to “attend to precision,” should focus on clarity and accuracy of process and outcomes of mathematics learning and in problem solving from the beginning of schooling and each academic year. For example, teachers can engage their students in a “mathematics language talk” to describe their mathematics activity. The emphasis on precision can begin in Kindergarten where they talk about number and number relationships and continues all the way to high school where they furnish mathematics reasoning for their selection and use of formulas and results.

Attention to precision is an overarching way of thinking mathematically and is essential to teaching, learning, and communicating in all areas of mathematical content across the grades.

For the development of precision, teachers should probe students to defend whether their requirements for a definition are adequate as an application to the problem in question, or whether there are some flaws in their group’s thinking that they need to modify, refine and correct. Just like in the writing process, one goes through the editing process, students should come to realize that in mathematics also one requires editing of expressions to make them appealing, understandable and precise.

However, communication is hard; precise and clear communication takes years to develop and often eludes even highly educated adults. With elementary school children, it is generally less reasonable to expect them to “state the meaning of the symbols they choose” in any formal way than to expect them to demonstrate their understanding of appropriate terms through unambiguous and correct use.

The expectations according to the standard are that mathematically proficient students

  • communicate their understanding precisely to others using proper mathematical terms and language: “A whole number is called prime when it has exactly two factors, namely 1 and itself” rather than “A number is called prime if it can be divided by 1 and itself.
  • use clear and precise definitions in discussion with others and in their own reasoning: e.g. “A rectangle is a four straight-sided closed figure with right angles only” rather than “A four-sided figure with two long sides and two short sides.”
  • state the meaning of the symbols they choose, use the comparison signs ( =, >, etc.) consistently and appropriately, for example, the names of > and < are not greater and smaller than respectively, but depend on how we read them: x > 7 is read as: x is greater than 7 or 7 is less than x; 2x + 7 = -5 + 3x is bidirectional (2x + 7 => -5 + 3x and 2x + 7 <= -5 + 3x).
  • are careful about the meaning of the units (e.g., “measure of an angle is the amount of rotation from the initial side to the terminal side” rather than “measure of an angle is the area inside the angle or the distance from one side to the other”), identifying and specifying the appropriate units of measure in computations, and clearly labeling diagrams (e.g., identify axes to clarify the correspondence with quantities and variables in the problem, vertices in a geometrical figure are upper case letters and lengths are lower case letters, and the side opposite to the <A in ΔABC is denoted by “a”, etc.).
  • calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context (e.g., the answer for the problem: “Calculate the area A of a circle with radius 2 cm” is A = 4π sq cm not A = 12.56 sq cm; if x2 = 16, then x = ± 4, not x = 4, whereas √16 = 4, etc.).
  • know and state the conditions under which a particular expression, formula, or procedure works or does not work.

Beginning with the elementary grades, this means that students learn and give carefully formulated explanations to each other and to the teacher (at Kindergarten level it may mean that the child explains her answer for 8 + 1 = 9 as “I know adding by 1 means it is the next number. I know 9 is next number after 8” or can show it concretely as “Look here is the 8-rod add the 1-rod and I get the 9-rod.” By the time they reach high school, they have learned to examine claims—their own and others’ in mathematical conversations, make explicit use of definitions, formulas, and results, and proper and adequate reasoning. At the high school level the explanations are rooted in any or more of these:

  • demonstrating it concretely,
  • showing by creating and extending a pattern,
  • application of analogous situation, or
  • logical reasoning—proving it using either deductive or inductive reasoning or using an already proved result.

What Does the “Attention to Precision” Look Like?
Effective mathematics teachers who use precision and efficiency in their teaching and encourage precision in their classrooms produce mathematically proficient students. Mathematically proficient students understand the role of precision in mathematics discourse and learning. They understand that mathematics is a precise, efficient, and universal language and activity. Precision in mathematics refers to:


  • Appropriate vocabulary (proper terms, expressions, definitions), syntax (proper use of order of words), and accurate translation from words to mathematical symbols and from mathematical symbols to words.
  • Knowledge of the difference between a pattern, definition, proof, example, counter example, non-example, lemma, analogy, etc. at the appropriate grade level.
  • Reading and knowing the meaning of instructions: compute or calculate (4 × 5, √16, etc., not solve), simplify (an expression, not solve), evaluate (find the value, not solve), prove (logically, not an example), solve (an equation, problem, etc.),
  • Know the difference between actions such as: sketch, draw, construct, display, etc.
  • Precise language (clear definitions, appropriate mathematical vocabulary, specified units of measure, etc.).

Teacher instruction about vocabulary must be clear and correct and must help children to understand the role of vocabulary in clear communication: sometimes formal terms and words distinguish meanings that common vocabulary does not, and in those cases, they aid precision; but there are also times when formal terms/words camouflage the meaning. Therefore, while teachers and curriculum should never be sloppy in communication, we should choose our level of precision appropriately. The goal of precision in communication is clarity of communication and achieving understanding.

A teacher can use familiar vocabulary to help specify which object(s) are being discussed—which number or symbol, which feature of a geometric object—using specific attributes, if necessary, to clarify meaning. Actions such as teaching writing numerals to Kindergarten by “song and dance” is a good starting point, but ultimately the teacher should use the proper directional symbols, e.g.,

  • To write number “4” the teacher first should point out the difference between the written four (4) and printed four (4). Then she needs to show the direction of writing (start from the top come down and then go to the right and then pick up the pencil and start at the same level to the right of the first starting point and come down crossing the line).
  • When discussing a diagram, pointing at a rectangle from far away and saying, “No, no, that line, the long one, there,” is less clear than saying “The vertical line on the right side of the rectangle.”
  • Compare “If you add three numbers and you get even, then all the numbers are even or one of them is even” with “If you add exactly three whole numbers and the sum is even, then either all three of the numbers must be even or exactly one of them must be even.”
  • Compare giving an instruction or reading a problem as “when multiply 3 over 4 by 2 over 3, we multiply the two top numbers over multiply two bottom numbers” to “find the product of or multiply three-fourth by two-third, the product of numerators is divided by the product of denominators.”

Elementary school children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all of their vocabulary is acquired from use in context. Children build their own “working definitions” based on their initial experiences. With experience and guidance, the concepts should become more precise, and the vocabulary with which children name the concepts will carry more precise meanings. Formal definitions generally come last. Children’s use of language varies with development but typically does not adhere to “clear definition” as much as to holistic images. If the teacher and curriculum serve as the “native speakers” of clear Mathematics, young students, who are the best language learners around, can learn the language from them.

Accuracy (know the difference between exact, estimate, approximation and their appropriateness in context) and appropriate level of precision in use of numbers (level and degree of estimation, significant digits, significant powers, units of measurement), correct classification and location of number on the number line (e.g., to locate ⅞, one divides the unit segment into halves and then each half into fourths, and then each fourth into eighths and then locates ⅞ rather than arbitrarily divide the unit segment into eight parts), correct relationships between numbers (e.g., √(140) is between 11 = √(121) and 12 = √(144), because, we have 121 < 140 < 144, therefore, √(121) < √(140) < √(144), but √(140) much closer to 12 as 140 is much closer to 144 than 121), selection of appropriate range and window on graphing calculator, tool selection (when to use what tools–paper-pencil, concrete models, diagrams, abstract, or calculator), and appropriate meaning of numbers in the outcome of operations (what role do the quotient and remainder play in the outcome from the long division algorithm, etc.). Precise numbers (calculate accurately and efficiently; given a context, round to an appropriate degree of precision)

Teachers should use written symbols correctly. In particular, the equal sign (=) is used only between complete expressions and signals the equality of those two expressions. To explain one way to add 42 + 36, we sometimes see it written (incorrectly) this way: 40 + 30 = 70 + 2 = 72 + 6 = 78. This is a correct sequence of calculator buttons for this process but not a correct written mathematics expression: 40 + 30 is not equal to 70 + 4; only the last equals sign is correctly used. We need the = sign to have a single, specific meaning. Also, the equal sign should not be misused to mean “corresponds to”: writing “4 boys = 8 legs” is incorrect.

Appropriate choice of concepts and models in the problem solving approach: choice of strategy in addition/subtraction (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 10 + 4 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, or 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14 rather than “counting up” 6 from 8 or 8 from 6), appropriate multiplication/division model (the only models of multiplication work for fraction multiplication are “groups of” or “area of a rectangle” not “repeated addition” and the “array” models), which exponential rule, which rule of factoring, which rule for differentiation, what parent function to relate to, what formula to use, etc.

Reasoning, Symbols, and Writing Mathematics
Appropriate and efficient use of definitions, reasons, methods of proof, and order of reasoning in solving problems and explanations. For example, children should know the reasons for using the “order of operations” or that the solutions of equations have domains and range. Precise usage of symbols and writing:

  • Choose correct symbols and operators to represent a problem (knowns and unknowns; constants and variables),
  • State the meaning of the symbols and operations chosen appropriate to the grade level (multiplication: 4×5, 45, 4(5), (4)5,(4)(5), a(b), (a)b, (a)(b), ab),
  • Label axes, shapes, figures, diagrams, to clarify the correspondence with quantities in a problem, location of numbers,
  • Show enough appropriate steps to communicate how the answer was derived,
  • Organize the work so that a reader can follow the steps (know how to use paper in an organized and systematic form—left to right, top to bottom),
  • Clearly explain, in writing, how to solve a specific problem,
  • Use clear definitions in discussion with others and in reasoning
  • Specify units of measure and dimensions,
  • Calculate accurately and efficiently.

At the elementary level, even the simplest of things such as: the proper way of forming numbers and mathematical symbols, writing the problems solving steps in a sequence: ([3(4 + 8) – (4 ÷ 2)] = [3(12) – (2)] =[36 − 2]= 34 rather than 4 + 8 = 12 × 3 = 36 −2 = 34). Similarly, clarity in reading numbers and mathematical symbols needs to be  emphasized from the beginning (e.g., ¾ is read as “3 parts out of 4 equal parts” rather than “3 out of 4,” “3 divided by 4” rather than “3 over 4.”

It is difficult to change inappropriate and incorrect habits later on. For example, when elementary grade teachers do not emphasize the importance of aligning multi-digit numbers in their appropriate place values, this creates problems for children later. New symbols and operations are introduced at each grade level, so it is important for the teacher to introduce them correctly and then expect precision in their execution.

Similarly, when middle and high school students are not instructed to write fractions properly, it creates problems. The following high school lesson illustrates the point. The problem on the board was:


To solve the equation, in order to eliminate fractions in the equation, the student suggested we multiply the whole equation by the common denominator of all the fractions in the equation (a correct and efficient method). When I asked for the common denominator, the student said: 9x because the denominators are 3x, 9 and 3. The error is purely because of lack of precision in writing fractions in the equations.

Precision often means including units when specifying numerical quantities. But not always. The purpose of precision is never to create work, only to create clarity. Sometimes a number is clear by itself, other times a unit is needed, sometimes a whole sentence is required: the situation determines the need. For the same reason, label graphs and diagrams sufficiently to make their meaning and the meanings of their parts clear.

Exposure and consistent questions from the teacher such as the following help students to be accurate, precise and efficient:

  • Is this the right way of writing the expression (number, symbol, etc.)?
  • Does the diagram you have drawn show the elements asked for or given in the problem?
  • Is this the right unit for the quantities/numbers given in the problem?
  • What mathematical terms apply in this situation?
  • Is the term you used the right one in this situation?
  • How do you know your solution is reasonable and accurate?
  • Explain how you might show that your solution answers the problem?
  • How are you showing the meaning of the quantities given in the problem (e.g., problem says: “the length of the rectangle is 3 more than twice the width)? Does your rectangle demonstrate the right dimensions? Your rectangle looks like a square.
  • What symbols or mathematical notations are important in this problem?
  • What mathematical language, definitions, known results, properties, can you use to explain ….?
  • Can you read this number (symbol, expression, formula, etc.) more efficiently?
  • Is ___ reading (saying, writing, drawing, etc.) correctly? If not, can you state it correctly and more efficiently?
  • How could you test your solution to see if it answers the problem?
  • Of all the solutions and strategies presented in the classroom, which ones are exact/correct?
  • Which one of the strategies is efficient (can achieve the goal more effectively)?
  • What would be a more efficient strategy?
  • Which one is the most elegant (can be generalized and applied to more complex problems) strategy? Etc.

The number and quality of questions in a classroom bring the attention of students to appropriate and precise conversation. In a fourth grade geometry lesson, I had the following exchange with the students: 

Sharma: Look at this rectangle (I was holding one of the 10 by 10 by 1 rectangular solids in my hand) while touching the 10 by 10 face, I asked: What are the dimensions of this rectangle?

A student raised his hand and said: “That is not a rectangle. It is a square.”

I said: “yes, it is a square. Can you also call it a rectangle? Is it also rectangle?”

“No!” He declared emphatically.

I asked the class: “How many of you believe that it is not a rectangle?” Almost every hand went up.

When I asked them what the definition of a rectangle was, almost all of them said: “A rectangle has two long sides and two shorter sides.” I drew a quadrilateral with 2 long sides and 2 short sides that did not like a rectangle.

Another student said: “The sides are parallel.” I drew a parallelogram.

The student said: “No! That is not what I mean. Let me show you what I mean.” He drew a correct rectangle.

One student said: “A rectangle has four right angles and 2 longer sides and 2 shorter sides. Like this.” He drew a correct rectangle.

We had a nice discussion and came to the conclusion that a rectangle is: A straight-sided closed figure with four right angles. I also emphasized the meaning of the word “rectangle.”  It is made up of two words “recta” and “angle.” The word “recta” means right.  Therefore, a rectangle has only right angles. With this discussion and the precise definition, they were able to accept and see the face of the object I was showing as a rectangle.

This episode, in one form or the other, is repeated in many classes, from urban to rural classrooms, in many elementary schools. The same misconception is present even in many classrooms in many middle and high schools students. This is an example of lack of precision in teaching and, therefore, lack of precision in student understanding and expression.

There are many examples of such misconceptions. For instance, children often misunderstand the meaning of the equal sign. The equal sign means is “the same as,” “equal in value” “equal in some specified characteristic—length, area, quantity, volume, or weight,” but most primary students believe the equal sign tells you that the answer is coming up to the right of the equal sign. When children only see examples of number sentences with an operation to the left side of the equal sign and the answer on the right, this misconception is formed and generalized. Teachers should, therefore, emphasize the true meaning of the equal sign. From the very beginning—Kindergarten children should be shown that the equal sign “=” is a two-way implication. For example, Kindergarteners should be shown and know the simple facts as: 2 + 8 = 10, 8 + 2 = 10 & 10 = 2 + 8, 8 + 2 = 10 and first graders need to see equations written in multiple ways, for example 5 + 7 = 12, 7 + 5 = 12, 12 = 5 + 7, 12 = 7 + 5, and 5 + 7 = 2 + 10, 5 + 7 = ☐+10, ☐ + 2 = 9 + ☐. Although most above average and many average children are able to realize this level of understanding of the concept of equal or equal sign, there are many average and children with learning disabilities who have difficulty in reaching that level of understanding. This level of precision in understanding can be achieved by using Cuisenaire rods, the Invicta math balance for teaching arithmetic facts, and proper and appropriate language usage and questioning by teachers.

If students are taught using imprecise language, they will necessarily learn imprecise language and concepts, because language is the basis of mathematics learning. Later, they will not only resist when asked to use precise language in mathematics, but they will also have difficulty applying the concepts. A sequence of ideas begins to take place in students’ mind when we ask questions and emphasize language.
Questions instigate language.
Language instigates models.
Models instigate thinking.
Thinking instigates understanding.
Understanding produces conceptual schemas.
Conceptual schemas produce competent performance.
Competent performance produces long lasting self-esteem.
Self-esteem produces willingness to inquire and learn.

With proper language and conceptual models a great deal can be achieved. It is not too late to instill precision even at the high school level; however, if it is not emphasized at the elementary and middle school levels, it is much more difficult to do so. This does not mean we give up; it only means we redouble our effort and find better ways of doing it, such as using concrete models, patterns, and analogies when we are introducing new mathematics concepts and procedures.

As students progress into the higher grades, their ability to attend to precision will expand to be more explicit and complex if we constantly use proper language and symbols.

As students develop mathematical language, they learn to use algebraic notation to express what they already know and to translate among words, symbols, and diagrams. Possibly the most profound idea is giving names to objects. When we give numbers names, not just values, then we can talk about general cases and not just specific ones.

Correct use of mathematical terms, symbols, and conventions can always achieve mathematical precision but can also produce speech and writing that is opaque, especially to learners, often to teachers, and sometimes even to mathematicians. Good mathematical thinking, therefore, requires being correct, but with the right simplicity of language and lack of ambiguity to maintain both correctness and clarity for the intended audience. If we are particular about this in the first few grades, it becomes much easier to attend to precision in later grades.

Attend to Precision: The Foundation of Mathematical Thinking

Model with Mathematics: Real World to Mathematics and Back

No problem can withstand the assault of sustained thinking. Voltaire

The Standards for Mathematical Practice (SMP in CCSS-M) describe mathematically productive ways of thinking that support both learning and applications by modeling mathematics in the classroom. Providing these experiences has to be an intentional decision on the part of the teacher. In other words, students learn mathematics concepts and procedures using models, on one hand, and use mathematics to model real problems, on the other. This activity is the basis of scientific and many social science innovations. Students need to experience this aspect of mathematics in the classroom from the very beginning.

Models and practical applications of mathematics have three distinct roles in mathematics learning. The first purpose of modeling and applications of mathematics is to motivate students to learn, engage, and see the relevance of mathematics. Here students learn mathematics by using concrete and representational models. This requires choosing the right manipulative, instrument, model or pedagogical tool to learn a mathematics concept, procedure or mathematical language.

In the second case, students apply mathematics—when they have learned a concept, skill, or procedure, to solving real life problems. The second case brings the appropriate mathematical knowledge and methods to match the demands of the problem. The third aspect deals with generating new mathematics or a model to solve a problem where one direct mathematics idea is not available. Throughout history, this twin process of modeling to learn new mathematics and solving novel problems by developing/ discovering models has solved real problems and generated new mathematics ideas—concepts, procedures. This is the interplay of pure and applied mathematics.

The facility of modeling mathematics is an example of the mathematical way of thinking and demonstration of competence in mathematics.

The modeling process, as application, spans all grade levels and applies mathematics that students know up to that grade level to solve “real” and “meaningful” problems. A simple example of modeling is the application of fractions to solve problems relating to rates of increase and decrease in various situations.

Deep Mathematical Understanding and Flexibility
A great divide often exists between students’ conceptual understanding, their procedural skills, and their ability to apply what they know. An even larger divide is that students may have conceptual and procedural knowledge but they have difficulty in applying mathematics ideas and realizing the power and relevance of mathematics.

The belief that applying mathematics is complex and complicated for many students and is separate from learning the concept and skills often leads many teachers to stop short of this most important step of teaching problem solving as part of each lesson. However, application of mathematics should not be a separate activity. While students who learn mathematics in a traditional fashion perform well on customary, standardized assessments, they tend to do poorly on tasks that require them to apply the math concepts to real problems. Students who learn mathematics through a modeling lens are better able to perform on both traditional and non-routine assessments.

Students too often view what happens in the math classroom as removed from and irrelevant to the real world. When a task can tap into a student’s innate sense of wonder about the world around him or her, that student becomes engaged in the problem-solving process. But when we can pique the interest of students through problems that have a basis in reality, we encourage them to question, investigate, and problem solve. Modeling bridges this gap and allows students to understand that to resolve many of the situations around them involve and require mathematics. When students engage in rich modeling tasks, they develop powerful conceptual tools that increase their depth of understanding of mathematical concepts and improve their abilities and interest in mathematics.

The concept of mathematical modeling, as a mathematics practice, has an important place in implementing the Common Core State Standards for Mathematics (CCSS-M). This practice emphasizes a student’s ability to realize the power of mathematics by applying mathematical tools to solve problems. Mathematical modeling demonstrates the power of mathematics for learners. Throughout their schools, students should use mathematical models to represent and understand mathematical relationships.

Levels of Knowing and Modeling
At each stage of mathematics learning (intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication) and in mastering its components (linguistic, conceptual, and procedural) problem solving plays an important role. At the intuitive and concrete levels, a real life problem not only acts as a “hook” for students to see the role of mathematics as an important set of tools but also gets them interested in that concept.

At abstract/symbolic and the applications levels, applying the concepts, procedures, and skills shows how those elements are used and integrated, so students learn the strength and limitation of a particular mathematical tool.

When students have acquired a set of concepts and procedures and face a real life problem, they try to model the problem in mathematical form and solve it. This takes several forms: word problems, problem solving, and modeling. Because these are not isolated activities, problem solving, modeling, and application must be embedded throughout students’ learning of mathematics.  To make sense of developments in the natural, physical, and even social sciences and to solve the related problems involves looking for and developing mathematical models.

By incorporating mathematical modeling in their classrooms, teachers can motivate more students to enter STEM fields and to solve real life problems in social sciences and humanities. Integrating computers and calculation tools with mathematics methods, many of the social science problems are amenable to mathematical modeling.

Problem Solving: Model for Introduction to Mathematics Concept
Real life examples can introduce mathematics concepts and bring the real world into the mathematics classroom.  A real world scenario motivates students to see mathematics as relevant to their lives and increases the desire to learn that mathematics idea. In this situation, a teacher moves students, explicitly, from real-world scenarios to the mathematics in those scenarios.

For example, an elementary school teacher might pose a scenario of candy boxes with an equal number of candies in each box and represents it as repeated addition and then relates and extends “the repeated addition of a number,” “groups of objects,” or the tile pattern in the yard to see the “area of a rectangle” into the concept of multiplication.

An upper elementary grade teacher poses a scenario of candy boxes with a number of candies with different flavors in each box to help students identify ratios and proportions of flavors and ingredients.

A middle school teacher might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted.

A high school teacher shows several kinds of receptors (parabolic dishes) and poses a set of questions to instigate a discussion why parabolic receptors are optimal shapes to receive the sound, radio, and micro-waves. This discussion instigates the study of parabolas, in particular, and quadratic equations in general. Similarly, the discussion of waves of different kinds might instigate a discussion of Sinusoidal curves in an algebra, trigonometry or pre-calculus class.

A statistics teacher brings in a big bag of MMs to the class and asks:  “Without counting all of the MMs, how do we determine the number of MMs of different colors, as close as possible to their distribution in the bag?” Students might say: “It is easier to count them, why go through all that?” The teacher responds by posing the problem: “Yes, you can count the MMs, but how do we determine the population of fish in the pond or the number of particular species of animal in the wild as we cannot directly count them?” In this process, she shows the power of sampling method in real life and therefore the reasons to learn it.

The role of these problems is to motivate students to learn mathematics and show the power of learning the tools of mathematics. To achieve this goal, the problems have to be of sufficient interest and diversity. They should show the relevance to the topic being studied. The mathematics in them should be transparent.  Finally, they should be accessible to children.

Problem Solving: Applications of Mathematics
The first application of a mathematics concept, procedure or a skill is in the form of word problems. Word problems, while more demanding than pure computation problems, are typically presented in the context of a specific mathematics content area or skill, and are solved with a particular method or algorithm; therefore, students do not apply much mathematical reasoning. Word problems can serve as one example of problem solving; however, typical word problems in mathematics classrooms are “concocted.”  Often, they have no resemblance to realty, so to call them as applications is stretching the meaning of the word “problem solving.” However, if teachers routinely mix different types of problems, involving several mathematics concepts, they can help solicit mathematical reasoning.

Problem solving, on the other hand, is when students need to decide what mathematics, concept, skill, or procedure is involved for solving the problem. Problem solving is more advanced than word problems because it requires students to (a) translate native language to mathematics expressions, relationships (equations, inequalities, formulas, etc.), (b) interpret what mathematics skills, concepts, and procedures are needed to solve the problem, (c) make assumptions and approximations to simplify a complicated situation, and realize that these may need revision later, (d) determine how to find the answer, and (e) to make sense of the solutions in terms of the conditions of the problem and the solution sought.

Problem solving takes place when students have acquired a certain set of mathematics concepts, procedures and skills and the teacher presents problems that they can solve by using these newly learned skills. The role of the teacher is to identify and present these problems to students. These are focused application problems where the math content and the skills needed to solve problems have a close match between the problem and skill set, but they are not recipe oriented.

There are three kinds of applications of mathematics: (a) intra-mathematical, (b) interdisciplinary, and (c) extra curricular.

In intra-mathematical applications, a student learns a new mathematical skill, concept, and procedure and can apply this to solve problems in other parts of mathematics. To be successful in this context, teachers must be cognizant of the connections that can be made in different parts of the mathematics curriculum at that grade level and even higher grades. The role of modeling in mathematics, in this context, is making connections between different branches of mathematics and discovering new relationships about mathematics concepts. Students who engage in modeling in the math classroom have increased mathematical autonomy and flexibility in the ways they use mathematics.

For example, in early grades, students may learn the property of commutative property by using Cuisenaire rods:
2 + 5 = 5 + 2
models with math 1

In this case, the model is used to learn a mathematics concept. On the other hand, they may use Cuisenaire rods to solve an addition problem at first grade (I spent $9 on Monday and $7 on Tuesday.  How many dollars did I spend?) by constructing and then writing an addition equation to describe the situation.
model with math 2

In the middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. On the other hand, a student might use paper folding to see the division of fractions or visual cluster cards to learn the operation on integers.

In inter-disciplinary applications, a student learns a new concept, mathematical skill, or a procedure and can apply it to another discipline.  For example, a student learns the concept and operations of fractions and now can apply this knowledge in the “shop class.”  A student learned the concept of transformations in geometry and now can create a collage by using tessellations in art class.  The student just learned how to solve linear equations, so now she can use this skill to solve a problem in chemistry class. A school-based project integrating learning from several disciplines is a good example of this type of application. To achieve this objective of inter-disciplinary applications, teachers should be aware of the interconnections of the mathematical concepts and the use of mathematics in other disciplines of students’ curriculum.

In extra-curricular applications, a student learns a mathematics concept, procedure, and a skill and applies these to problems in everyday situations outside of the curriculum.  Here the teacher finds problems from the real world to connect with mathematics skills.

The goal of mathematical modeling, at this stage, is for students to pose their own questions about the world and to use mathematics to answer those questions. Quite naturally, most students want to know there is some utility in what they’re learning, that a lesson is not just an isolated lesson with no future use. In each section, in each module, they should be able to see what they are learning as relevant to their own lives and their own careers.

Discovering Mathematics: Modeling as Content Category
Throughout history, individuals have generated mathematics knowledge to solve practical problems. On the other hand, some mathematicians focus on mathematics for the sake of mathematics. Many others are interested in mathematics for its power, its tools, its approach to problem solving and modeling problems. The mathematical tools available at any given time are the means for innovation, inventions, determining the standards of living at that time. For example, in the twentieth century, most science, engineering and technology problems were tackled by the tools of calculus, but with the advent of calculators and computers, it is possible to extrapolate the data and find solutions using discrete methods. In such problems, there is need to integrate the mathematical tools that are based on continuous models (functions, calculus, etc.) and discrete models (finite difference methods, probability, statistics, etc.). In this scenario, students use their mathematics skills to discover new mathematical tools and skills.

Modeling as conceptual content category means using mathematics models to generate and learn new mathematics concepts. It is more than just using a concrete material, pictorial representation to learn a mathematics concept. For example, the study of transformations (both rigid and dynamic) to geometric and algebraic objects gives rise to the study of geometrical concepts, understanding of curves, functions, and conic sections. Similarly, in statistics and probability, we create, model, or simulate an idea to study it.

Modeling with Mathematics
When students themselves find or encounter real world problems and want to solve them, they are modeling mathematics at the highest level. One distinct difference between typical problem solving and mathematical modeling is that modeling frequently involves interpretation or analysis of an essentially nonmathematical scenario. This content conceptual category reflects a modeling cycle involving a series of operations.  Students must:

  1. identify a problem,
  2. study the scenario that gave rise to the problem to determine what the important factors or variables are, interpret these mathematically, identify variables in the situation and selecting those that represent essential features,
  3. observe the nature of the data—looking for regularities (e.g., if the data is increasing at a constant rate, it may be modeled by a linear system; if the change is constant at the second level of iteration, it can be modeled by a quadratic function; if each entry in the data is a constant multiple of the previous entry, it is modeled by a geometric/exponential function, etc.),
  4. develop and formulate a tentative mathematical model by selecting arithmetical, geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
  5. use the model to analyze the problem situation mathematically, draw conclusions, and assess them for reasonableness of the solution,
  6. analyze and perform operations on these relationships obtained by the modeling process to draw conclusions,
  7. test the solution to determine whether it makes sense in the context of the problem situation,
  8. interpret the results of the mathematics in terms of the original situation, and
  9. if the solution makes sense and they have a mathematical model for this type of problem, validate the conclusions by comparing them with the situation, then either improve the model or express the model formally in mathematical terms – if it is acceptable, and report the conclusions and the reasoning behind them.

However, if the problem is not adequately solved, the learning from this trial is incorporated to improve the model.

The iterative process and interpretation of the solution are hallmarks of the modeling process. A vital part of modeling is interpreting the solution and comparing it to reality.

By high school, students might use pictorial, numerical, algebraic, geometrical, trigonometric, functional, probabilistic, statistical, and computational methods to solve real life—social and physical science problems.  Choices of functions, diagrams, assumptions, range of constraints, and approximations are present throughout the modeling cycle.

Why Modeling?
There are common misconceptions about mathematical modeling. Many teachers view mathematical modeling as a process of showing the students how to approach or solve a problem. The first two types of applications described above are not true examples of mathematical modeling. They are uses of models to learn mathematics. The key feature of mathematical modeling, as defined by the NCTM and the CCSSM, is that students seek or encounter a problem, to solve. The teacher is the facilitator and guide in the process, but the modeling is done primarily by the students under her guidance: students select a problem, select the mathematics, integrate the skills and concepts, and then explain what they have done.

Mathematical modeling is not just a type of word problem or problem solving—it is mathematics being practiced; it is applications of mathematical ways of thinking. Modeling represents a shift from learning math to doing math. Modeling can be differentiated from word problems as it does not usually call for the use of one method or algorithm in order to solve the problem. Standard word problems or even problem solving in school mathematics curricula do not model realistic problem situations for problem solving, whereas modeling presents students with realistic problem-solving experiences requiring strategizing, using prior knowledge, and testing and revising solutions in real contexts.

The inclusion of modeling in the math classroom increases student engagement, depth of understanding, and provides opportunities for investigation, contribution, and success for all learners. Students involved in problem solving and inquiry-based activities such as modeling develop a positive disposition toward mathematics.

There is a place in the classroom for each type of problem, but it is possible to take a typical word problem and adapt it in such a way that it increases the depth of knowledge required to solve it. Open-ended modeling problems allow students to use mathematical tools and prior knowledge including measurement, proportions, map reading, scale drawings, and geography to make decisions and justify those decisions in a real-life problem. Students may use multiple methods to come to a conclusion and multiple representations to demonstrate their understanding. They must explain their solutions and use writing in mathematics to explain their reasoning.

Mathematical modeling is an effective practice for all students even for those who have a history of poor performance on traditional mathematics tasks. By incorporating modeling tasks into the classroom, we recognize all students as important contributors to the decision- making and investigation of the problem at hand. Because a good modeling task will be based in real-life experience, all students have the ability to make contributions based on their prior knowledge. Such tasks also emphasize and require a broader range of mathematical abilities than algorithmic exercises, and therefore allow a broader range of students to emerge as being capable. Students who may have a history of poor performance in math when their abilities and understanding are assessed solely on narrowly defined tasks and assessments can demonstrate significant ability and potential when given the opportunity to problem solve in a real-life modeling context. It is therefore important for teachers to emphasize to students that any (mathematically valid) solution for which they can make a strong argument is “correct.” This flexibility in thinking, and departure from the idea of only one correct solution, encourages students.

When students understand that they each have a unique contribution and valid voice in problem solving, they are more likely to become involved. Even students who have struggled in the past will contribute and share their thinking, and will be less likely to rely on the work of more successful students. Because modeling tasks can utilize a broad range of mathematical abilities, a broader range of students can emerge as capable mathematical learners.

Not all modeling problems have several feasible solutions, but the process always presents opportunities for different approaches and diversity of thinking, resulting in greater chances for success for students with diverse backgrounds and experiences. As these students gain confidence in their ability to contribute to the problem solving, they begin to develop a sense of mathematical autonomy.

Mathematical Empowerment
Our job as teachers is to present the tools, show students when and how they are used, and then provide a context in which they can choose the appropriate tools for a given problem. When we provide a larger toolkit, students begin to approach problems in a variety of (perfectly valid) ways and gain confidence in their abilities. The more opportunities are given to students to make mathematical decisions, the more they are encouraged to use their tools to explore and reason about mathematical problems. They grow increasingly confident in offering their ideas and methods and are better able to take the initiative when presented with a novel task.

Students should also see the teacher as a problem solver. When a teacher participates in problem solving as a senior learner and persists in solving a problem with students and explore different ways of solving the problem, students learn from that cognitively and affectively. When they see the teacher excited about learning, interested in students’ ideas and thinking, and willing to explore new ways of teaching and learning, even unmotivated learners convert into students who are excited about learning and proud to consider themselves mathematical learners. When teachers are open to finding different ways of solving a problem, they are more likely to create mathematically confident students.

Success and empowerment are the key factors to student motivation. The student who has tasted success is more persistent in solving problems and demonstrates meta-cognition in learning. Successful learners are curious and desire greater challenge. As they build confidence in their own thinking, their intrinsic motivation increases. They are viewed, and view themselves, as developing math experts. They have the confidence to make mathematical decisions, to approach a problem from one direction and, if necessary, change direction and try another. They will use the mathematical tools in their toolkit and decide when and how they might be helpful in a given problem. Therefore, a vital part of our work as math teachers is to encourage our students to become originators of ideas rather than merely recipients of content.

The introduction of modeling into the math classroom across all grade levels will increase student understanding, interest, and appreciation for the power of mathematics tools. Whether engaged in shorter, more focused modeling problems or more extensive, multi-day projects, students gain confidence in their proficiency as mathematical learners and make connections between mathematics concepts and real-life applications. All students, regardless of background or history with mathematics, have the opportunity to contribute and learn through experiences with mathematical modeling.

Characteristics of Effective Problem Solvers
First, to be a problem solver, one has to be mathematically proficient. Mathematically proficient problem solvers can and do apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Students who regularly experience modeling with mathematics as a problem solving tool acquire unique characteristics and think of themselves as mathematicians.

They do not shy away from selecting complex problems and they:

  • simplify a complex problem and identify important quantities to look at relationships and they can represent this problem mathematically,
  • ask what mathematics do I know to describe this situation either with an equation or a diagram and interpret the results of a mathematical situation,
  • look for the mathematics learned to apply to another problem and try to solve the problem by changing the parameters of in the problem.

They regularly ask:

  • What model (quantitative, geometrical, algebraic, statistical, probabilistic, or mixed) could be constructed to represent the problem?
  • What are the ways to represent the information in the problem (e.g., create a diagram, graph, table, equations, etc.)?
  • What tools and approaches are appropriate to the problem at hand?
  • How to select and decide which argument makes sense and is reasonable in the context of the problem?
  • How to justify the appropriateness of the solution, explain why it makes sense, and how to convince the group of the reasonableness of the solution?
  • How to make sure that the results make sense?
  • How to improve/revise the model?
  • How do I incorporate the comments and concerns of others in the approach?
  • What is the best way of presenting the solution to others?
  • What further extensions, generalizations, investigations might be interesting or necessary?

In order to support such learners, the classroom must be a place that encourages choice and provides positive feedback regarding competence.  Teachers in such classrooms:

  • Assure all students that they are capable and competent, and their ideas are worth sharing with others and encourage student collaboration,
  • Presents problems that encourage student initiative and provide the opportunity for a variety of approaches and representation,
  • Make available appropriate manipulatives and instructional materials for exploration,
  • Practice and integrates the three roles: didactic, Socratic, and coaching,
  • Spend less time talking and more time listening to student questions and reasoning,
  • Ask more questions, give measured and focused feedback without curtailing creativity and initiatives, seek suggestions for improving solutions, encourage alternative solution approaches.

In such classrooms, students are more likely to dive into a problem and less likely to ask a question like “What do we do now?” As students succeed, they will be more likely to identify a problem, discuss methods for approaching the problem, and begin investigating and discussing different possible methods of solving the problem. When students are motivated from within—when they are excited about participating in their own learning for learning’s sake rather than because of pressure or external rewards—they become empowered learners.

Example 1  In the early grades, students have concrete, pictorial and arithmetic models available. This might be as simple as writing an addition equation to describe a situation. For example:
My team scored 91 points on Monday and 37 points on Tuesday.
How many more points did they make on Monday than Tuesday?
What is the difference in points on the two days?

This problem could also ask:
How many fewer points did my team score on Tuesday than Monday?
How many more points should my team have made on Tuesday to have the same score as on Monday?

A second grade class worked out the problem as:
One child said: ‘I subtracted 1 from 91 to make it 90 and then subtracted 30 from 90 to get 60 and then subtracted the remaining 6 from 60 to get 54. I know 6 and 4 are pairs to make 10. I took 37 away in all. My answer is 54.’

David was all excited. He said: ‘I have a better method. I subtracted 40 from 91 to get 51 and then added 3 to 51 to make sure that I actually subtracted 37. I subtracted and also added.’

Another child said: ‘I first added 50 to 37 and then 3 and then 1 to get the same answer 54.’ He said: ‘I know 7 and 3 are pairs to make 10.’ He showed the work on the empty number line as follows:

Another child said: ‘I added 50 to 37 to get to 87, but then I just added 4 directly to 87 to 91 as I know that 7 plus 4 is 11, so 87 plus 4 is 91.

Another child said: ‘I did it a little differently. I added 3 to 37 to make it 40. And then I added 1 to 40 to make it 41. And then I added 50 to 41 to get 91. I also got 54.’

Another said: ‘I added 4 to make 41 and then added 50 to get 91.’

Another said: ‘I added 3 to get 40 and then 50 to get 90 and then added 1 to get 91. And the answer is the same 54.’

Then students wrote their equations:  91 − 37 = 54 or 37 + 54 = 91.  They concluded that the team made 54 more points on Monday than Tuesday.

All of these examples show that the students are able to apply their understanding of (a) number concept (as demonstrated in the decomposition/ recomposition of numbers), (b) addition and subtraction facts (as seen in number sense), and (c) place value. The discussion and the recording of different number relationships demonstrate all the standards of mathematics practice.

Example 2  In the middle grades, students have pictorial, arithmetical, algebraic and geometrical methods available. For example, a student might apply proportional reasoning to plan a school event or analyze a problem in the community:
Our school survey revealed that 7 out of 8 students have access to iPads for their homework. If there are 128 students in our grade, I wonder how many students have access to iPads. Is that more than the number of students with iPads in the eighth grade where, according to our English teacher, 91 students have access to iPads?

Students in a seventh grade class demonstrated modeling by constructing the following table and then corresponding equation.

Some students in the class approached the problem as:
# of Students with iPads: ___ ___ ___ ___ ___ ___ ___ (7 equal line segments)
Total # of Students:  ___ ___ ___ ___ ___ ___ ___ ___ (8 equal line segments)
# of Students with iPads in my grade: ___ ___ ___ ___ ___ ___ ___
# of Students in my Grade: ___ ___ ___ ___ ___ ___ ___ ___ = 128
The goal now is to find: “What does one line segment represent?”
Now, 8 (___) = 128,
# of students represented by ___ = 128 ÷ 8 = 16
# of Students in my grade with iPads = 7 × 16 = 112.
Thus, the number of students with iPads in my class = 112. Then they extended it to the 8th grade to compare the corresponding numbers.

Example 3 Observing social and political world events, a group of researchers wanted to know how things go viral on the Internet. In other words, the team wanted to understand how political and social movements, ideas, or products could catch on or fail to do so.

Since two phenomena—disease and social movements use the same word “spread,” the team borrowed from mathematical models used in epidemiology. In biology and medicine, scientists have studied the spread of viruses, disease, and epidemics by using mathematical models. This suggests a possible model for the spread of social and political phenomena.

In medicine, there are susceptibles, infected, and impacted (dead, cured, quarantined); similarly, the researchers reasoned that they could construct a new model to examine the spread of ideas. The team showed that while an individual’s resistance to the spread of a “contagion” might be high, when bombarded by that contagion from many directions, such as happens through Facebook or Twitter, transmission occurs, i.e. you view the activity or participate in it as well. That synergy leads to explosive transmission and we say that something has gone “viral.” This is not only a wonderful example of the use of mathematical modeling to explain a real-world phenomenon but also an example of the generalizability of mathematics and mathematical models. The same mathematics and the same types of mathematical models that can be used to study, for example, the spread of Ebola can be used to study the spread of ideas.

Example 4 Modeling with mathematics means that the students not only understand the concept and procedures but also see that particular method(s) may have limitations and that the context of the problem calls for the applicability and efficiency of model and the method. For example, after students have learned to recognize and use the linear system, they explore real life situations and learn that some situations are modeled by linear systems only under certain conditions. For students to become proficient in modeling, they should have experiences that relate to modeling. Here is an example of a problem (from a textbook commonly used in high schools) a teacher presented to his students to understand modeling:

When a cake is first removed from the oven, its temperature is 370°F. After 3 hours, its temperature is approximately 70° F, the temperature of the kitchen.

  • Does this situation represent a linear system? Why do you think so? If so, represent it as a linear system. If not, why not?
  • Use the information above to write two ordered pairs (x, y), where x represents the time (in hours) since the cake was removed from the oven and y represents the temperature (in degrees Fahrenheit) of the cake at that time.
  • If it is a linear system, write the linear relationship between x and y, in any of the following forms, with general values (two point form; a point and slope form; slope-intercept form; standard form)
  • Find the slope of the line through the two points identified in step 2.
  • Write the linear equation in slope-intercept form or point-slope form.
  • Use the equation from step 5 to estimate the temperature of the cake after 1 hour, after 2-hours, and after 4 hours.

The problem, as given in the book, was straightforward; however, the questions above are reformulated to make sure that the students not only understand the problem but also have a deeper understanding and make connections between different concepts and relate the problem to a realistic situation. After this, the teacher made this problem even more rigorous by asking a series of further questions:

  • Why do you think the information given to you in the problem satisfies the conditions of a linear relationship?
  • Under what conditions can this be modeled by a linear relationship?
  • You know from geometry that two points determine a line, is that condition satisfied here?
  • What does a linear relationship look like in general?
  • What minimum conditions do you need to be able to find the linear relationship in this situation?
  • What is unknown in the linear function you just gave?
  • What is unknown in this equation?
  • When you look at your ordered pairs, will the slope be positive or negative?
  • What will be the orientation of the line?
  • How will you find the slope of this line?
  • What is the formula for slope?
  • Can you find the slope geometrically?
  • You said: “The formula for slope is .”
  • What do m, (y2−y1) and (x2−x1) represent in the formula?
  • What do y2, y1, x2, x1 represent?
  • Will the formula…give the same slope for your line?
  • Why do you think so?
  • Can you prove that the two formulas represent the same slope?
  • Please draw a rough sketch of the line.
  • Based on this sketch, what can you predict about the temperature in the future?

Then the teacher asked his students to solve the problem. Students calculated the slope by considering two points (two ordered pairs): (0, 370) and (3, 70). As the teacher was walking around in the room looking at their work, he asked students:

  • What does -100 mean here?
  • What will be the temperature in five hours? 10 hours?

At this point there was a great deal of discussion amongst students and they began to question whether it was really a linear model. Students came to the conclusion that it was a linear model only till the temperature of the cake reached room temperature, and after that it was not a linear model. The teacher introduced several examples of non-linear and mixed models. Students even brought the idea of a step function.

This is an example of teaching with rigor, making connections, and how mathematics is used to model real-world prolems. The teacher focused only on one problem during the lesson, but students understood the concept at a deeper level rather than solving several problems just applying a procedure.

The requirements of rigor—understanding, fluency, and ability to apply, are parallel to our expectations in reading. A child is a good reader when he or she (a) has acquired fluency in reading (displays speed in decoding, chunking, blending of sounds using efficient strategies indicating phonemic awareness, and word attack), (b) shows comprehension (understands the context, intent, and nuances of meaning in the material read), and (c) is able to use it in real life with confidence (pragmatics—able to read a diversity of materials from different genres and reads for interest and purpose). Mastery in any of these elements alone is not enough because reading is the integration of these skills.

Similarly, rigor in mathematics means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, real-life problems. Finally, it is demonstrated in their ability to communicate this understanding. To achieve the same level of mastery as in reading, mathematics educators need to balance these elements in expectations, instruction, and assessments.

The writers of the CCSS-M were careful to balance conceptual understanding, procedural skill and fluency, and application at each grade level.


Model with Mathematics: Real World to Mathematics and Back

Reason Quantitatively and Abstractly: Specific vs. General

Common Core State Standards-Mathematics (CCSS-M) define what students should understand and be able to do in their study of mathematics. But asking a student to understand and do something also means asking a teacher to first help the student to learn it and then assess whether the student has understood it. So how do teachers gauge mathematical understanding? One way is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity and to the context of the problem and concept, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding results from the practice of these justifications and, in the process, procedural skills are strengthened, particularly when mathematical tasks experienced by students are of sufficient richness.

Reason Abstractly and Quantitatively
Mathematics learning is the continuous movement between the particular and universal. Resolving the tension in mathematics between understanding at an abstract, context-free level and providing some kind of context for the problem at hand is at the heart of teaching and learning of mathematics. For children, mathematics begins with specific and concrete tasks, and they ultimately reach the most important and high-level thought process in mathematics—the abstraction process. It means to know the abstract and general, on the one hand, and the particular and specific, on the other. Taking the child from understanding a concept at the specific, concrete level to generalizing and extrapolating it to the abstract, symbolic level is the mark of a good teacher.

Abstraction is to capture essential properties common to a set of objects, problems, or processes while hiding irrelevant distinctions and uniqueness among them. Abstraction gives the power to deal with a class of problems that are diverse and complex. For example, children encounter specific shapes, figures, and diagrams in geometry in different contexts. At the same time, all geometrical shapes are abstractions, that is representations of concrete objects from multiple settings and contexts, e.g., a circle drawn on a paper represents a family of circular objects. Similarly, students encounter different kinds of numbers and diverse relationships between them. On the other hand, definitions, theorems, and standard procedures are abstractions, that are general cases derived from specific contexts and relationships—properties such as: associative and commutative property of addition, distributive property of multiplication over addition/subtraction, long division procedure, prime factorization, divisibility rules, solving equations. Abstract thinking enables a learner to bend computation to the needs of the problem.

A mathematically proficient student makes sense of quantities and their relationships in a given problem situation, looks for principle(s) applicable to that problem, and takes the problem situation to a general situation. The specific case is dependent on the context, but generalization happens only when we decontextualize the relationship(s). For example, the expression 12 ÷ 3 in a specific case represents: if 12 children are divided into 3 teams of equal number of students, how many are in each team? However, it is an abstraction of several situations, the numbers 12 and 3 can represent a variety of objects—concrete and abstract and from several settings and forms:

    1. How many groups of 3 are there in 12? (repeated subtraction)
    2. If we divide 12 into 3 equal parts/shares/sets/groups, what is the size of each part? (groups of/partitioning model)
    3. If we organize 12 chairs in 3 rows with equal number, how many will be in each row? (array model)
    4. If we organize 12 unit square tiles into a rectangle with a vertical height of 3 units, what will be the size of the horizontal side? (area model).

Thus, the expression 12 ÷ 3 no longer represents a contextual, concrete problem; it has been decontextualized; it is context free. To abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own is the key to true mathematical thinking.

Decontextualizing, thus, means abstracting, going from specific situations to general and representing them abstractly, symbolically and then to manipulate these symbols without necessarily attending to their referents and contexts. However, once the solution is found, it needs to be interpreted from the context of the original problem.

Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

For effective learning of mathematics and solving problems, students need two complementary abilities—understanding quantitative and abstract relationships—how to contextualize and decontextualize. Many students, even when they may show skills at each of these levels separately, show gaps in reasoning at these two levels simultaneously or making connections between them. Proficient students reason at both levels—to reason quantitatively and abstractly, to understand the context of the problem and then to decontextualize it.

For most students, to understand a problem and apply mathematical reasoning, the context of the problem matters. However, the ultimate goal is a context-independent understanding of problem solving. Everyday examples, models, context, analogies, and metaphors are critical in linking the problem to students’ prior knowledge and to illustrate different aspects of the subject matter and facilitate students’ transition from specific to general and vice versa. Mathematically proficient students make sense of quantities and their relationships in problem situations. At the same time, they are able to generalize and abstract from these specific situations.

As an example of transition from specific to abstract, consider this problem:

Children collected 45 bottle caps each school day for a week. How many bottle caps did they collect?

Initially, children see this as a series of additions (45 + 45 + 45 + 45 + 45)—a context specific approach to the problem, but then they abstract it into a multiplication concept connecting with the schemas of multiplication as repeated addition or (5 ×45) or “5 groups of 45”—a one-dimensional concept. When several such problems are handled successfully, they begin to see the general situations that are translated to a × b, where a and b are numbers representing a variety of settings and later the multiplication is extended to the array and area model—two-dimensional models and application to a diversity of numbers (multi-digit, fractions, decimals, algebraic expressions) and mathematical entities, such as: functions, determinants, matrices, etc.

Similarly, at the high school level, students know the role of numbers in a situation represented by algebraic relationships. For example, in the linear equation p = 25n + 45, they understand that p describes the cost in $ of n items where the cost of manufacturing per item is $25, and $45 represents the start up costs. This is the context—this is a specific case. Representing these in a table and developing a pattern helps students to reach the general case.

Case/State # of items p = Total Cost in $
Start 0 45
1st 1 25×1+ 45
2nd 2 25×2+ 45
3rd 3 25×3+ 45
100th 100 25×100+ 45
nth n p=25×n+ 45=25n+ 45

Decontextualizing here means that if the cost per item or the start up costs are changed, then we will have different numbers in place of 25 and 45; we will have a new equation. In the most general case, the equation will be p = an +b, where p is the cost of n items, a is the cost of manufacturing one item and b is the start up costs. This is a complex idea and many students have difficulty arriving at this point. Only with a great deal of scaffolded questioning and examples can a teacher achieve this with all students.

Contextualizing is also the movement from general to specific or seeing the role of context on quantities and probing into the referents for the symbols and numbers in the problem. It is to take an abstract symbol or an equation and to look for its context—its special case. In the manufacturing equation, it means that if we want to find the cost of manufacturing 1 item, we will change n to 1 and if we want to know how many items we can manufacture for $245, we will change p to 245. Here we are going from general to specific. And we understand the specific case that even if no item has been manufactured, there is a cost of $45.00 incurred. Or, when the variables in the equation are changed, the student still understands the roles of the variables.

For example, in a right triangle ABC, with the right angle at vertex C, when the 2 legs and hypotenuse are given, in several settings, one observes and then derives: the sum of the squares of the legs is equal to the square of the hypotenuse. Then, generalizes this result into, form specific right triangle to any right triangle, a2 + b2 = c2, the decontextualized form as Pythagoras Theorem. Further, one applies this universal result into specific contexts (special cases) in solving problems. Every middle and high school student understands and masters the specific and general result about right triangles. However, when the name of the triangle is changed to ABC with the right angle at vertex B (e.g., a2 + c2 = b2), or with the triangle PQS, with the right angle at Q, (p2 + r2 = q2), they have difficulty relating to the Pythagorean result. In other words, for them the result is contextual to a particular right triangle. Thus, mathematics learning is closing the loop:

capt 1
In meaningful problem solving, the decontextualizing and contextualizing processes are intertwined. The process starts when students first read the problem and understand the context of the quantities. They

  • understand and convert what they have read into mathematical equivalents—numbers, symbols, operators (contextualize),
  • use knowledge of arithmetic, algebra, geometry, calculus, etc., to write expressions, equations/inequalities, functions, systems (de-contextualize),
  • compute, evaluate, solve equation(s) and systems, simplify expressions, etc., to generate answers to the questions posed in the problem (context to general and back to context),
  • refer the solution/answer back to the original context of the problem, interpret and understand the meaning of the answer to realize a solution (contextualize and decontextualize), and
  • extend the solution approach to other similar problems to generalize the approach (contextualize and decontextualize).

Decontextualizing and contextualizing also mean thinking about a problem at multiple levels—going beneath the surface and making connections. It goes beyond the ability to merely find the value of the unknown (say, x) in the equation. It is also to find the meaning about the solution and the uniqueness and efficiency of the solution process. For example,

Find the distance between a submarine, 250 ft below the surface, and a satellite tracer orbiting 23,000 ft directly above the submarine at a particular time.

The following steps describe the contextualizing to decontextualizing process that provide entry to the solution process.

  •  As a start, student represents this information on a vertical line (contextualizing) locating the zero as the sea level and the locations of these two objects as points on the vertical line with relative positions and distances (submarine = −250, satellite = +23,000 (de-contextualization);
  • The student tries to remember how to find the distance between two points (e.g., y1 and y2) on a number line (in this case, y-axis) as distance = |y2−y1| (decontextualizing); and
  • Relate the formula to the objects = |23000 −250| (contextualizing).
  • Finally, they simplify the expression and respond to the question in the problem and express the result contextually: The distance between the satellite and the submarine is 23,250 ft.

Let us take a similar problem and use another approach for solving it and make connections to make generalizations to prior knowledge.

The temperature in the morning was 450F and in the evening it went down to -120F. How much colder was in the evening? How much warmer was in the morning? What was the difference in temperature in the morning and evening? The temperature from morning to evening went down by how many degrees?

In a seventh grade classroom, when students initially saw the problem, quite a few of them answered it quickly as 330F. These students did not contextualize it. Others wrote: 45 – 12 = 330F. These students started with quantities without contextualizing the problem.

However, if they had represented the problem (contextualized), they would have been able to solve this problem, answer all the questions raised in the problem, and even others of the same type (decontextualized).

blog 22 capt 2

By the help of this diagram, they compute the distance between the points to 45 –(-12) = 45 + 12 = 57 and infer that it is 570F cooler in the evening. Therefore, it is 570 F warmer in the morning than evening. And, the difference between the temperature in the morning and evening is 570F.

This problem can also be solved by starting from 450F and getting to -120F by moving left rather than right adding a level of generalization (decontextualize).

Quantitative reasoning is important in its own right; however, the goal is to learn, apply, generalize, and reason with numbers and use them to make meaningful inferences, create conjectures to arrive at generalizations.

For successful execution of the solution process with understanding, quantitative reasoning should be comprehensive—contextualized, decontextualized, and contextualized; it must go beyond mere computational proficiency.

Comprehensive quantitative reasoning entails the habits of creating a coherent representation of the problem; considering and understanding the units involved; attending to the meaning of quantities and efficiently computing with them; and knowing and flexibly using different properties of operations and objects. Thinking quantitatively and abstractly also means that students know the proper use of mathematical symbols, terms and expressions.

Comprehensive reasoning—to think abstractly and quantitatively separately and then together, develops when teachers employ a range of questions to help students focus on understanding quantities (e.g., type and nature of numbers), language (vocabulary, syntax, sentence structure, and translation), concepts and the associated schemas, and operations involved in the problem. We need to help students focus on the specific as well as the general and abstract, particular and the universal. It means:

1. Making sense of quantities in the problem (units, size, meaning, and context) and their relationships:

  • What do the numbers/quantities in the problem represent?
  • What is the relationship between these quantities?
  • How is _____ related to ______?
  • What is the significance of units associated with these quantities?
  • Are all the units of measurement uniform?
  • What are the relationships _____ units and _____ units?

2. Creating multiple representations of quantities and relationships in the problem (concrete, iconic and pictorial representations, symbolic expressions—equations, inequalities, diagrams, etc.).

These representations should be appropriate to the grade level (for example, thinking of division “as groups of” and performing it by sequential counting is appropriate at the third grade level, but it is not appropriate at the sixth or seventh grade levels. At that time, we should be thinking of the area model of division).

The teacher should provide a range of representations of mathematical ideas and problem situations and encourage varied solution paths.

  • What are some of the ways to represent the quantities and their   relationships?
  • Is there another form that the numbers can be represented by (table, chart, graph, bars, model, etc.)?
  • What is an equation(s) or expression(s) that matches the pattern, diagram, number line, chart, table, graph, …?
  • What formula(s) might apply in this situation? Why?

As an illustration let us consider the problem: 91− 59.

At the concrete level the solution can be derived by using BaseTen blocks or Cuisenaire rods. But Cuisenaire rods are more efficient as there is no counting involved. Then we can use Empty Number Line in multiple ways (ENL) to find the difference. The ENL helps develop numbersense and mental arithmetic. Once students have facility with ENL, they should explore this problem using compatible numbers and decomposition/re-composition. For example,

Screen Shot 2016-03-30 at 11.56.58 AM

All of these problems are equivalent and develop a deeper understanding of numbersense, quantitative reasoning and mental arithmetic.

3. Forming and manipulating equations (attending to the meaning of the quantities, not just computing them):

  • Is it the most efficient relationship or equation representing the quantities in the problem?
  • Which property or rule can make this equation simpler?
  • What property of the equation (equality, procedure, number, operation, etc.) did you apply in solving the equation?
  • Could you use another operation or property to solve this task? Why or why not?

4. Making sense of the given problem and applying that understanding to consider if the answer makes sense.

  • How does this solution relate to the problem?
  • Can you relate the solution of the problem to a real life situation?
  • What does this answer mean? For example, what does the slope of this line mean in the context of the problem?
  • Can this solution approach be generalized to other number systems, operations, ……, ……?

Levels of Knowing Mathematics
For any concept or procedure to be mastered by a child, it has to go through several levels of knowing: intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication.

Intuitive level of knowing means the student is trying to connect the new concept with the schemas of prior knowledge—language, concepts, skills, and procedures. It is like relating subtraction to addition, division to multiplication, laws of exponents with base 10 to other bases, or laws of exponents in the case of whole numbers to integers, rational, or real numbers. In the process, previous schemas get transformed—extended, amalgamated, reorganized, even destroyed and replaced by new schemas. This is how a person enters into the new mathematics concept, learning, or problem.

Concrete level of knowing means the student represents the concept, procedure, problem through concrete models based on the intuitive level understanding. The concrete model should be efficient and transparent in representing the concept or problem. Of all the efficient models, one should look for elegant models. A model is efficient and elegant when it takes the student to representation level easily.

Pictorial and representational level of knowing means seeing the concept using pictures (iconic or representational), diagrams, or graphic organizers. There is a difference between an iconic representation and pictorial representation. For example, representing a problem with pictures of Cuisenaire rods or Base Ten blocks is iconic, whereas Empty Number Lines or Bar diagrams are pictorial. Iconic representation is the true copy of the concrete model and keeps the learner longer on a concrete and contextual level. As a result, many children do not become proficient in abstract or de-contextualization. On the other hand, an efficient pictorial representation leads the student to generalization and abstract representation of the concept. Efficient and elegant models facilitate such decontextualization.

When a concept is learned at the abstract level, it is easier for a student to apply it to general problems (applications level of knowing) and the exposure from intuitive to concrete to pictorial to abstract helps the student to become fluent in communicating understanding and mastery (communications level of knowing).

Let us consider an example of writing an addition equation to describe a situation (first grade level) that illustrates the transition from contextualizing to decontextualizing:

The team scored 33 and 25 points in two games, respectively. How many points in all did the team score in the two games?  

First step, using Cuisenaire rods, 33 can be represented by 3 tens (3 orange rods) and 3 ones (1 light green rod) and 25 can be represented by 2 tens (2 orange rods) and 5 ones (one yellow rod), then the sum is 5 tens (5 orange rods) and the 3-rod and the 5-rod gives the 8-rod (brown) equals 58 (concrete).

Second step: the sum can be represented by an empty number line (pictorial level). Several ENLs can be created for this computation. Finally, the total score in two games can be expressed as a sum of 33 and 25. Total = 33 + 25. (abstract)

blog 22 capt 3

Only after students understand the concept should a teacher move to abstract.

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After the understanding is gained from this decomposition/recomposition, we should move to the standard addition procedure.

capt 3
After this, one can use the procedure to solve problems or extended to multi-digit additions with regrouping.

Let us consider another example to examine how to go from specific to general.

The length of a rectangle is 3 more than two times the width. The perimeter is 78 in. What is the width of the rectangle?

Solution One: (Contextualizing: Quantitative reasoning)
Each expression from the problem is translated into mathematical expressions:

Screen Shot 2016-03-31 at 6.21.13 AM

Solution Two: (De-contextualizing: Generalizing)
We express the length in terms of the width: length in inches = 2x + 3, where x = width in inches.

capt 2

To have proficiency in mathematics, to decontextualize and to represent abstractly, students need to learn to use symbols correctly. This begins with number concept and the fundamental concepts such as equality. Many students misunderstand the concepts of equation and equality. Their misconceptions originate from not knowing the concept of “=” in its proper form.

It is difficult to understand the concept of and working with equations, without understanding the concept of equality. Understanding and using the concept of equality is a good example of going from a particular situation to a general situation. Though the concept of equality is so germane to mathematics, most children have difficulty in answering problems such as (a question that has appeared on several national standardized tests):

What should be placed in the place of in the equation?  9 + 5 = + 7.

Many students from second to eighth grade would place 14 in place of . These students have no idea what the symbol “=” means. For them it is an operation and is used when two numbers are added. They see it as one-way implication (blog 22 capt 4). They do not have the idea that the two expressions on either side of the equal symbol need to be compared to see if they are equal. They need to see it as a two way implication (blog 22 capt 5 equivalent to =).

When students, in the early grades, have not experimented with materials such as a mathematics balance or Cuisenaire rods to see the equivalence of two expressions, they have difficulty understanding the concept of equality or equation.

blog 22 capt 6

The diagram suggests that 9 + 5 = 7 + 2 + 5; therefore, there should be 7 in the box. The use of concrete models is a good starting point for proper understanding of these fundamental concepts. Practice without conceptual understanding does not lead to generalizations and abstractions.

A group of teachers was asked how they or their students would respond to 4 = 6? Almost everyone replied: “Well, we just know it is not true.” When asked how they would prove their statements, one of them said: “If you compare 6 items and 4 items by one-to-one correspondence, you find that six has two more items, so 6 does not equal 4.” This shows that they have the reasoning for the concept of inequality.

When they were asked: “How they or their students would explain 2 + 3 = 5,” one of them answered: “My students would get 3 things and then 2 things and put them together and you would know you have five things.” That is finding a total of 2 objects and 3 objects. That is not a proof for equality. That is right, but that is not the question.

To prove the equation “2 + 3 = 5” concretely, we put on one side of a balance two Unifix cubes and three more with the two already in the rocker balance. Now we place 5 Unifix (of the same size and weight) cubes on the other side of the balance, and the balance balances. Now we see that 2 and 3 are 5. Now if we take 5 cubes on one side, once again, we find that 5 is not equal to zero. However, if we put 3 cubes and then 2 more, we find that the two sides balance. We have shown that 2 + 3 equals 5 and 5 equals 2 + 3. It shows it as a two-way implication.

Similarly, if we take the red Cuisenaire rod (representing 2, if the white represents 1) and place the light green rod (representing 3) next to the red rod making a train, we find that the yellow rod (representing 5) is equal in length to the two rods. Now we can read (in color): red + light green = yellow and yellow = red + light green. Therefore (in numbers), 2 + 3 = 5 and 5 = 2 + 3. In both cases, we have shown that the equation is true using concrete materials. We can do the same in later grades using abstract formal arguments using the properties of numbers and axioms.

When a group of middle and high school students were asked: “What is the definition of an equation?” Answers varied:
“When two sides are equal.”
“When we have an equal sign in it.”
“When we are solving something.”
“When there is variable in it.”
Although there was a lot of discussion, none of them could clearly define an equation.

We have an equation when two mathematics statements/expressions are equated. Examples: (a) 2 + 3 and 4 + 1 are two mathematical expressions. When they are equated we have an equation: 2 + 3 = 4 + 1.   (3x + 5) + 9 and x2 + 2(3x + 7) are two mathematical expressions, when we equate them, we get an equation: (3x + 5) + 9 = x2 + 2(3x + 7).

In the early grades, we need to ask students to use quantities and units as descriptions whenever possible. We should inundate them with questions that ask how many, how many more, how many less, what is the total, why can you do this, what is the reason, what do you infer from this, what conclusion can be drawn from this, can we form a conjecture from this, can you give another example for this procedure, concept or word, etc. The role of examples, counter examples, non-examples, specific cases of a definitions, and theorems are effective means of relating to the specific and general. Unless students regularly connect different concepts, procedures, and language, they will have difficulty in focusing on the specific and general and the quantitative and abstract.


Reason Quantitatively and Abstractly: Specific vs. General

Make Sense of Problems and Persevere in Solving Them: Engagement with Mathematics

On encountering a new problem that they cannot solve, many students immediately give up. It doesn’t have to be this way.

There is a difference between students who welcome and remain engaged in the problem and those who give up easily. The difference is not due to innate factors, but it is mostly the outcome of teaching. With effective teaching all children can acquire attitudes and strategies to become proficient in problem solving—understand the problem, approach the solution process, and stay engaged in the problem using different perspectives.

According to the framers of the Common Core State Standards-Mathematics (CCSS-M) and Standards of Mathematics Practices (SMP), helping students to understand a problem, initiate a solution process, remain with the problem by exploring it from multiple perspectives are important characteristics of teaching. This helps students acquire the ability to enter the solution process and develop mathematical stamina.

Making Sense of Problems
Making sense of the problem means understanding the language, the concept, and the conditions and parameters involved in the problem. Students identify the objectives of the problem. They may not engage in the problem and remain engaged in the problem if they do not understand the problem. To initiate a solution process and to pursue it, students should associate appropriate schemas and procedures with the language, symbols, and concepts involved in the problem.

Mathematically proficient students read a problem carefully, understand the meaning and context of the problem, and explain to themselves the role of particular numbers, expressions, and actions in the problem. They analyze the givens, study the constraints on the quantities in the problem, understand, identify, or determine the unknowns in the problem. They understand intra- and interrelationships amongst knowns and unknowns. They understand the nature of these relationships. They seek entry points to the solution process keeping focus on the goal/s of the problem.

Mathematically proficient students analyze the problem, consider analogous situations, try special cases, and simpler forms of the problem (changing numbers, e.g., changing fractions into whole numbers, relaxing constraints in the problem, or reducing the number of variables) to gain insight into the problem and solution process.

They classify and organize the information into tables, charts, or groups. They search for regularity, patterns, or trends. They make conjectures about these patterns. They observe and explain correspondences between variables (knowns and unknowns) by forming equations, verbal descriptions, inequalities or diagrams of important features, relationships, and representations. Through these conjectures about the form and meaning of the data, they plan solution pathways and enter the solution process, rather than simply jumping into a solution attempt by choosing a formula or procedure.

Students who have acquired a concept, skill, or procedure using diverse language and a multiplicity of strategies have flexibility of thought to explore multiple ways of entering the problem.

Building Mathematics Stamina: Perseverance
Perseverance means having the self-discipline to continue a task in spite of difficulties and dead ends. It is a function of skills and attitudes. Albert Einstein said, “It’s not that I’m so smart, it’s just that I stay with problems longer.” Perseverance is a necessary ingredient for student achievement. One of the reasons students do not persevere in solving problems is lack of flexibility of thought. When they exhaust their ability and options to think about the problem, they do not have stamina for solving problems. Students develop perseverance when they are taught with rigor.

The requirements of rigor—understanding, fluency, and ability to apply, means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, mathematics and real-life problems. Finally, it is demonstrated in the ability to communicate this understanding. To achieve a level of mastery/rigor among students, mathematics educators need to balance expectations, instruction, and assessments.

We can help students continue thinking about a problem by modeling the many different questions they can ask about a difficult problem.

Asking Questions
Effective teachers use a variety of language, questions, and methods to derive a concept or procedure. For example, let us consider a problem:

A science book has 251 pages and a mathematics book has 197 pages.     

Teacher: What question can we ask so we have a subtraction problem from this information? Students formulate questions. If they do not, she articulates several questions:

  • How many more/extra pages are in the science book than the math book?
  • How many less/fewer pages are in the math book than the science book?
  • What is the difference in the number of pages in the science and math books?
  • How many pages should be added to the math book so that it will have the same number of pages as the science book?
  • How many fewer pages should be in the science book to have the same number of pages in the math book?
  • How many pages are left in the science book if we took away as many pages as the math book?

Through this process of generating questions, over a period of time, students develop flexibility of thought about additive reasoning, in general, and subtraction, in particular. These children, in future, will find several ways to enter the solution process of any subtraction problem involving numbers other than whole (e.g., fractions, decimals, integers, algebraic expressions, etc.). Effective questions build student stamina for problem solving. This should be a regular process in a mathematics class.

Using Effective Concrete Materials
To build stamina, younger students should be exposed to a multiplicity of concrete objects (e.g., Visual Cluster cards, TenFrames, Cuisenaire rods, fraction strips, Base-Ten blocks, Unifix cubes, pattern blocks, Invicta Balance, etc.) and diagrams and pictures (number line, Venn diagram, empty number line, bar model, graphic organizers, tape diagrams, tables, charts, graph paper, etc.) to understand strategies based on decomposition/ recomposition of numbers and facts and solve problems. Counting materials and strategies based on them build neither the flexibility nor the stamina for problem solving.

Middle and high school students may, depending on the context of the problem, transform numbers (fractions, decimals, and percents, algebraic expressions) using concrete models and the properties of numbers, operations (associative, commutative, or distributive properties to simplify numbers and expressions), change the viewing window on their graphing calculator to get the information they need (e.g., to observe the behavior of a polynomial, trigonometric, or rational function near the origin or at a specific point; compare it with the “parent function,” etc.), or use Algebra tiles, Geoboard, geogebra, Invicta Balance, etc., to arrive at relationships and equations involving variables.

Monitoring Progress and Evaluating Success
Mathematically proficient students monitor and evaluate their progress and change course if necessary. They check their answers to problems using a different method, and they continually ask themselves: “Does this make sense?” “Does this answer the questions in the problem?” (e.g., analyze partial and final answers). They can explain their solution approach and try to understand others’ approaches to solving problems, and they identify correspondences between different solution approaches. All of these activities, habits, and attitudes help them to be engaged in the problem resulting in perseverance. Students develop and improve perseverance when they realize that mathematics is thinking and making mistakes. It is also a process, not just finding the answer. It happens when we ask:

  • Step One: What is it that we are trying to find out here? This is the question we ask in the real world. And this is the most important part of doing mathematics. People, including our students, need practice and opportunities in asking the right questions. This should be a group activity as group work as a strategy is critical to good mathematics work and student engagement. Group work generates better understanding of problems and then multiple entry points. It is also critical in countering inequities in mathematics achievement by different groups of students in the classroom.
  • Step Two: Next is to take that problem and turn it from a real world problem into a mathematics problem—express it as a relationship between the elements (variables and quantities) that define or have created the problem. This translation from real word situation expressed in the native language to mathematics language is an important step in doing mathematics.
  • Step Three: Once we have defined a relationship (an expression, an equation/inequality, or a system of equations/inequalities, etc.), we manipulate these relationship(s) and that involves formal mathematics—this is the computation step. Through computation, we transform the relationships into an answer in a mathematical form. This is an important step, but for developing interest in mathematics, we should not begin with this step.
  • Step Four: When we have dealt with the computation part of mathematics, we need to then turn it back to the real world. We ask the question: Did it answer the question? And we also verify it—a crucial step.
  • Step Five: To create interest and involvement, we need to now engage students in collective reflections by sharing different strategies and their relative efficiencies and elegance.

Completing this loop keeps our students grounded in the reality and power of mathematics. The majority of students will repeat these steps in their real life. And a small percent of students will have the satisfaction of repeating the steps in the context of mathematics and sciences only.

Teacher Attitudes
Teachers need certain attitudes, skills, and habits of mind for developing children into effective problem solvers with stamina. They need to practice the following:

  • Believe in each child’s ability to improve and achieve higher in mathematics.
  • Expect and help them to finish what they start and when they are stuck, providing scaffolding with enabling questions to continue in the task.
  • Avoid accepting excuses for unfinished work.
  • Give positive feedback when a child puts forth extra effort or takes initiative.
  • Help students realize that everyone makes mistakes, but what is important is to keep trying.
  • Demonstrate and motivate them to try new things.
  • Encourage children to take responsibility for their work and make constructive choices.

Students become mathematically proficient and persevere in solving problems when teachers model these skills and choose meaningful problems to solve. They create conditions for students’ engagement in problems; that in turn develops perseverance. Students are engaged when problems are contextual, moderately challenging yet accessible, have multiple entry points, and are amenable to various solution approaches (intuitive, concrete, pictorial, abstract, on the one hand, and arithmetical, geometrical, and algebraical, on the other). It develops a variety of tools.

For example, using the Empty Number Line (ENL) approach to solving addition and subtraction problems rather than jumping into applying the standard procedure has many more entry points to the solution and can be solved using multiple ENLs (e.g., the problem: the difference 231 – 197 can be arrived at by at least different ENLs with a deeper understanding of numbersense (number concept, arithmetic facts, and place value) and problems solving. Arriving at the answer this way will keep them engaged.

blog 21 capt 1

Similarly, the Bar Model (BM) is an effective problem solving tool involving fractions, decimals, percents, and deriving algebraic equations easily. The area model of multiplication and division is effective for whole numbers, fractions, decimals, and algebraic numbers and for deriving properties of operations (e.g., commutative, associative, and distributive properties of multiplication and subtraction, etc.).

Tools are not enough, however, unless teachers scaffold student work. Questioning, based on formative assessment, is the key to the scaffolding process. Scaffolding is a function of a teacher’s ability in

  • doing task analysis—know and establish the trajectory of the development of a concept, skill or procedure, and help students to know the goal of the task,
  • being aware of the student’s capabilities, as well as their limits;
  • doing continuous formative assessments of students’ assets—cognitive and content (conceptual and skill sets),
  • asking enabling questions to move students toward the goal, and gradually fade and remove the support structures, and
  • knowing models and approaches best suited for connecting concepts with students.

The response to “good” questions develops conceptual understanding, stick-to-it-ness, and helps them refine the tools—make them effective, efficient, and elegant. The better a teacher gets at asking “why” questions, the better her students are at understanding concepts, staying on, applying tools, and solving problems.

Effective questioning is more than giving students a solution approach, steps for solving a problem, or identifying the typology of the problem. Effective questions invite students to enter the solution process and stay with it. They may include:

  • What question(s) are you trying to answer in the problem? What are you trying to find? Can you state that in your own words?
  • What information do you have that can help you answer the question in the problem? Do you have enough information to answer the questions raised in the problem?
  • Do you know any relationships among the information you have and what you do not have?
  • Can you write this information using mathematical symbols?
  • Can you write a fact, equation, inequality, formula or a relationship between symbols in the problem?
  • How would you show the information in the problem in a different way?
  • What other information do you need to answer the question?
  • Where might you get that information?
  • What other questions do you need to answer before you can answer the question in the problem?
  • Have you solved another problem like this before?
  • How is this problem like that problem? What is different about this problem?
  • Could you solve the problem if the numbers were simpler?

When students have solved the problem, the teacher reengages them by asking:

  • Do you have the answer to the problem/question?
  • Have you answered the question raised in the problem?
  • Which question in the problem does this answer?
  • What does your answer mean?
  • Does this answer make sense?
  • Have you expressed the answer in the appropriate units of measurement or order of magnitude?
  • What did you learn from this problem?
  • Is there any information in the problem that was not necessary for answering the problem?
  • Can we relax the conditions of the problem and still answer the problem?
  • Can you write another problem similar to the given problem?
  • Can you formulate a more difficult problem?

To demonstrate some of the questions, let us consider a problem:

In a village, 20% of voting age people did not vote during the last election. If only 4,280 people voted, what was the voting age population of the village?

Teacher: What are we looking for?
Students: The total voting age population of the village.
T: What information do we have?
S: The number of people voted? 4,280
S: The percentage of people did not vote? 20%
T: What else do we have? What information can we derive from the given information?
S: The percentage of people who did vote: 80%
S: The percentage of voting age population: 100%
T: What are we trying to find?
S: The voting age population of the village.
T: Can you represent the information by diagram, table, equation, or relationship? Make a start and try to solve it. We will discuss all of the methods used by the class. I will visit all of you and keep an eye on your progress. You can ask me questions when you need help.

At the end she asks children to share all of their methods and their relative merits are discussed. The approaches are shown here.

Method One: Visual Representation Method (line segment, Bar Model or Pie Chart)
The following bar represents the total voting age population.

# of people of voting age = ______________________________ =100% = ?

# of people who voted     = _______________________    = 80% = 4,280

# of people did not vote = ______                                  = 20% =

As we do not know the total population, we represent it by a “?” mark, which is made up of those who voted (longer line or a bar) and those who did not vote (shorter line or shorter bar) (see the bar model below).
Because of 80% and 20% distribution, the line/bar is divided in two sections: the larger section is 4 equal parts and the smaller section is 1 part. The number 4,280 is equal to 4 equal parts and the missing part is one part. Therefore, one part is equal to 4,280 ÷ 4 = 1,070. Then the total number of people of voting age is 5 parts (4 parts + 1 part): 1,070 × 5 = 5,350.

Method Two: Applications of Fractions
The fraction of people who did not vote = 20% (= ⅕) of total number of people of voting age. The fraction of people who voted = 80% (= ⅘) of total number of people of voting age = ⅘ of total = 4280 (the 4 parts out of the 5 equal parts). So 1 part is 4,280 ÷ 4 = 1,070. Therefore, the total = 5 parts =1,070 × 5 = 5,350.

Method Three: Ratio and Proportion Method
blog 21 capt 2
Here part = number of people voted, whole = number of people of voting age, percent of people voted is 80% as percent of people did not vote is 20%. We can compare the number of people who voted in two forms: 80 percent vs actual number (4,380) and similarly compare the total # of people of voting age as 100 percent vs. actual number that we do not know and we consider as “?”. We have
blog 21 capt 3

4×? = 5(4280) (multiply both sides by 5 and ?; or cross-multiply);

? = 5(4280) ÷ 4 (isolate the “?,” divide both sides by 4),

? = 5(1070)

? = 5,350 (# of people of voting age).

Or, the total number of people of voting age = Number of people who voted + number of people who did not vote = 4280 + 1070 = 5350.

Method Four: Algebraic Method
Let us assume the number of people of voting age is x.
The number of people who did not vote is 20% of x. The number of people who did vote is 80% of x.

Thus,       80% of x = 4,280
                  .80 × x = 4,280 or of x = 4,280 or x = 4,280÷
                  x = 4280 ÷ .8 = 5,350.
Therefore, the total number of people of voting age = 5,350.

Method Five: Shortcut
To solve the problem, many teachers will just give the formula:blog 21 capt 4
They will say to solve this problem is easy:
If 80% of a number is 4,280, then what is that number?
First, underline is and of in the problem. Then, the number just before is is the number to be placed in place of is and the number in place of of is to be placed in place of of, in the formula. Therefore, we have blog 21 capt 5. Then, they will ask students to solve next ten to twenty problems on a sheet of paper.

This is purely a procedural method and does not emphasize much mathematics. The consequence is that students are unable to apply it if the problem is slightly different or the numbers are placed in a different form or different language. In this method, there is no involvement with language or concepts of mathematics. There are no connections made with other procedures or concepts. Students get the impression that mathematics is just a collection of procedures, and if they can recall the formula but can’t apply it, they give up.

Shortcut methods do not develop perseverance. Perseverance is reached when teachers apply methods that have mathematics and thinking behind them rather than methods that appear like tricks. Students who are familiar with the above four methods will be able to see where this formula comes from and then use it effectively.

Exposure to multiple approaches helps students understand concepts and acquire “stamina” for problem solving. As in any exercise, the stamina is a function of optimal (conceptually efficient) methods, regular and intentional practice, guided reinforcement (coaching and well-designed exercises and homework) and discussions of mathematics processes.

When teachers encourage students to share with the class, their

  • understanding of the problem—language and concepts involved in the problem,
  • entry points to the problem,
  • approaches and strategies to and nature of the solution, and
  • the mathematics concepts and procedures involved

students work hard and their mathematics stamina is strengthened. The crucial point is that students need to understand, know and experience that mathematics is not equal to computation. That is what develops perseverance.

Well-crafted mathematics classroom tasks, exercises, and assignments (including homework) hold the potential to make learning and teaching of mathematics focused and relevant and making all students achieve. In planning lessons, effective teachers make decisions about context, mathematics language, content, and rigor. Since homework is generally for reinforcement and practice, they assign homework that achieves those goals and needs to ensure that large chunks of class time are devoted to “why” and “how” questions to develop and reinforce mathematics concepts.

If mathematics is taught using deep learning—emphasis on concepts, language, and multiple models, instead of a performance subject—applying just “closed end” standard procedures, students will see it as important knowledge. Mathematics will become a collection of powerful tools that empower them to think quantitatively to solve problems in their work and lives.

We need to give all students the opportunity to taste real mathematics. Once students have acquired and mastered numeracy and algebraic skills with understanding, fluency and have the ability to apply, then we should use more efficient methods for computations. For example, computers and calculators can do a better job than any human as long as we know what we are doing and when such tools should be used. When relevant and efficient, we ought to use calculators and computers to do computation and engage students to spend more effort on conceptualizing and solving problems.



Make Sense of Problems and Persevere in Solving Them: Engagement with Mathematics

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

  • 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

CCSS-M: Arithmetic and Algebraic Thinking

When we arrived as freshmen at my high school, our headmaster—a very popular, caring, and tough mathematics teacher (yes, he still taught), greeted us warmly and during his welcoming speech remarked: “Those of you who are fluent in fractions will end up in calculus and you know that fractions are dependent on multiplication. And those who do not have the mastery of multiplication tables and fractions will not enter into fields such as science, technology, engineering, mathematics, physics, and even economics. I do not want mathematics to be a gatekeeper for your aspirations. You should have all the skills that give you freedom of choice of options.” Most of my friends laughed at the remark.

After fifty years of teaching, I am convinced, more than ever, about the validity of that statement. All students should have the option to pursue any field, and a lack of proper preparation in mathematics should not close doors too early. Mathematics has become an entry to exciting and rewarding fields. Today, it is not just the STEM fields that require higher mathematics; even in the social sciences success and competence are dependent on skills in mathematics.

Throughout the twentieth century, most problems in natural and physical sciences, engineering, technology, and even social sciences could be modeled by functions – mostly by continuous and differentiable functions, but sometimes other functions such as: piece-wise, step, etc. Due to the advent of computers and related new technologies, today we can model many of the problems with only a few data points. Therefore, discrete mathematics (e.g., probability, statistics, linear programming, numerical analysis, etc.) and computer science play an important role in modeling problems in social sciences, physical and natural sciences. Similarly, the role of compu-graphics and graphing utilities in gaming systems and simulations is important in problem solving, therefore, in mathematics. This requires quantitatively and qualitatively a different kind of preparation in mathematics.

A different preparation in mathematics means that students, from the beginning of middle and high school, need to be made aware of the cumulative nature of mathematics: e.g., that mastery of multiplicative reasoning facilitates the understanding and the mastery of proportional reasoning (e.g., fractions/decimals/percent, rate, unit change, scale factor, and slope (ultimately, differential coefficient—rate of infinitesimal change in one variable with respect to the corresponding change in the other variable). The understanding and mastery of fractions are essential for success in algebra. And success in calculus is greatly dependent on facility in algebraic concepts, manipulations, and operations.

In high school, from ninth to 11th grade, students should formalize and extend the relationships between and integration of logical, arithmetic, algebraic, functional, geometric, probabilistic, linguistic, and statistical reasoning, their models and applications. Students should experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. With this aim in mind, the framers of CCSS-M recommend three courses in high school: Algebra I, Geometry, and Algebra II.

The fundamental purpose of CCSS-M Algebra I is to formalize and extend the mathematics that students learned in the middle grades. CCSS-M’s Algebra I is built on the middle grades standards and is a more ambitious version of algebra traditionally taught in grades eight or nine. The topics deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. Solving algebraic problems procedurally, without the appropriate development of algebraic thinking, does not take students far in mathematics and the rigor of Algebra I escapes them. Such procedural work does not prepare them for higher mathematics and meaningful problem solving.

It is evident from children’s classroom work and performance on mathematics tests and examinations that most of their errors in solving problems in algebra I, such as algebraic equations, functions, and quadratic equations, are related to lack of mastery of facts and errors in operations on integers, fractions and lack of algebraic thinking. Because of these errors, even on easy word problems, the overall success rate of students using algebraic methods to solve problems is low and reflects a great deal of variation among students. In solving algebraic problems, a large number of students use few algebraic models, instead relying on arithmetic reasoning or guess and check, minimally useful but ineffective methods. The table below contrasts arithmetic and algebraic thinking.

Arithmetic Thinking Algebraic Thinking
Work from knowns to unknowns Work on unknowns to knowns
Thinking in natural language Thinking in symbolic terms and mathematics language
Unknowns transient Unknowns defined by the conditions of the problems and fixed for the particular problem


Equation as a formula to produce answers Equations and inequalities as descriptions of the relationships, parameters, and situations
Chains of successive calculations Chains of logically linked equalities or inequalities to transform them into simpler forms


Solution found to a specific problem Method and solution found to a category and classes of problems

The transition from arithmetic to algebra takes time and experience. It is the focus of middle school mathematics. However, till the rigor of CCSS-M holds in our schools, it should also get teachers’ attention during the high school years. It is important that the intervention work during the high school years focus on this aspect of transition from arithmetic to algebra.

During middle school and early part of high school, students need to see and understand the difference between arithmetic and algebraic reasoning. They need to see that for some simple problems, arithmetic methods are adequate although they do not lead to generalizations. Unless students experience several problems that are amenable to arithmetic and algebraic methods and see the inadequacy of arithmetic methods, they will have difficulty appreciating the importance of algebraic thinking. Let us consider a simple problem:

Two players, David and Mark, scored a total of 37 points in a game. David scored 5 points more than Mark. How many points did each score?

This problem can be solved by several methods.

Solution 1: Guess and check

14 + 23 = 37 but the difference is not 5

15 + 22 = 37 but the difference is not 5

16 + 21 = 37 the difference is 5

21 + 16 = 37 is also an answer as the difference is 5.

Since David scored 5 points more than Mark. David: 21 points; Mark: 16 points.

Solution 2: Arithmetic reasoning

If they both score equally, then I divide 37 by 2 = 18.5. But they did not score equally. One scored 5 more than the other. To have such a score, one is higher than 18.5 and the other is lower than 18.5 with a difference between the two scores of 5 points. Since 18.5 is the average, one is 18.5 + 2.5 = 21 and the other is 18.5 − 2.5 =16. David is 21 and Mark is 16.

Solution 3: Logical arithmetic reasoning

David’s score 5 points more than Mark’s score. I find 37 − 5 = 32. Since, 32 is the average of the two, now. I divide 32 equally. (37 − 5) ÷ 2 = 16.

David’s score = (37 – 5)/2 + 5 = 32/2 + 5 = 16 + 5

Mark’s score = (37 – 5)/2 = 32/2 = 16

Solution 4: Geometrical/pictorial

Mark’s score: ____________ (the length of the line segment represents Mark’s score)

David’s score: ____________ _____ (the line segment of the same length plus 5 points)

+ 5

Mark’s score + David’s score = 37

____________ ____________ _____ = 37 (the total = two line segments of same length + 5)

____________ ____________ = 37 – 5

____________ ____________ = 32

____________ = 32 ÷ 2 = 16

Mark’s score: 16 points

David’s score: 16 + 5 points = 21 points.

(In many countries in place of this line segment, representing an unknown, a bar is used.)

Solution 5: Algebraic approach to solution

Let us say Mark scored x points. Then David scored x + 5 points. Together they scored 37 points. So, x + x + 5 = 37.

x + (x + 5) = 37 <-> 2x + 5 = 37 <-> 2x = 32  <->   x = 16.

Mark scored 16 points; David scored 21 points.

The five approaches to solutions used by students from middle to high school are progressively algebraic in reasoning beginning with guess and check to arithmetic reasoning to more generalized algebraic thinking. These solution approaches indicate some of the ways in which arithmetic leads to algebra. For developing algebraic reasoning and the flexibility of thought and problem solving facility, students should be helped to realize the strengths, weaknesses, and limitations of each approach.

A guess and check solution is quite easy, accessible even to upper elementary school children, especially as the answer does not involve non-integral numbers and the numbers involved and the conditions of the problem are quite simple. It, thus, has limitations.

The next two solutions, the second using arithmetic reasoning and third using logical arithmetic reasoning are also simple. These methods are helpful in visualizing the problem and provide entry into the problem solving process, but they also have limitations if the numbers are not easy and if the parameters of the problem are complex. The fourth method begins to introduce the concept of unknown and the relationships between unknowns and knowns. It becomes the basis of the symbolic representation. The spatial representation (whether line segment representation or the bar method), even when the numbers in the problem are fractions, decimals, or percents, provides easy access to algebra early and effectively. Most high performing countries on mathematics use spatial representation of problems before symbolic representations.

The fifth method involves algebraic thinking and can be suggested by the spatial reasoning method, the arithmetic reasoning, and the translation from natural language to symbolic, mathematical language. Effective teachers always begin with discussing some kind of visual or spatial representation of the problem and then using logical reasoning lead to an algebraic representation.

In the first category of approaches—three methods based on arithmetic reasoning—students’ solution approaches began with the ‘knowns’ and moved to ‘unknowns.’ On the other hand, in the second category of approaches—the last two methods—the spatial reasoning and formal algebraic method, one begins from unknowns and establishes relationships between unknowns and knowns in the form of an equation. Then one applies a procedure, based on logical reasoning and properties of numbers, operations, and equality, to solve the equation.

There are fundamental differences in these two categories of solutions approaches. The three methods based on arithmetic reasoning are less applicable as they cannot be generalized. In the case of more complicated problems, the ‘guess and check’ is less straightforward and difficult to generalize, particularly when the answer is not an integer. Generalizing logical arithmetic reasoning method of Solution 2 is also hard, and even generalizing the method of Solution 3 is challenging for many students. As a result, the power of algebra in the form of integration of quantitative and spatial reasoning is needed. Although many students are unable to change or extend their approach, it can be accomplished if we begin with

  • translation from natural language to symbolic language
  • construct diagrams (Bar Graph, tables, charts, or Empty Number line), and
  • then translate into algebraic equation (integrate spatial, quantitative, and symbolic representations).

To truly understand algebraic problem solving, students should be taken through the first four methods before leading to algebraic methods. And they should arrive at the realization that when the problem is a little more difficult and involves non-integer numbers and solutions, the first four non-algebraic methods become much more difficult.

Solutions 2 and 3 of the problem above illustrate how arithmetic solutions to a problem work from ‘knowns’ (actual numbers) to unknowns (the answer) by a process of calculation. For example, from the known numbers 37 and 5, one finds the new quantity 32 (in Solution 2, the number of points to be shared equally after the extra 5 have been awarded to David), and then from this new known, one finds 16 (the points that they both score). The unknowns are transient: first the aim is to find one, by finding the first unknown, then one finds the next one, and so on in a chain of successive computations.

In contrast, the algebraic method is quite different. Instead of immediately progressing towards a solution, one understands and describes the situation in natural language: ‘One of them scores 5 more points than the other’ and then translates it into spatial and/or symbolic algebraic language. So if one scores say: x points (unknown) then, the other scores x + 5 (unknown). Then a relationship is found between unknowns and the known in the form of equation. So we have: x + (x + 5) = 37. Now from unknowns one moves to knowns by working towards a solution by calculation using the properties of numbers, operations, and equality; one then works on the whole equality, producing more (equivalent) equations until eventually the solution appears (here the transitive property of equality is hidden: first equation => second equivalent equation => etc.; the solution to the last equation, therefore, should be the solution to the first equation).

Statement                     Reasoning

x + (x + 5) = 37       Translating natural language into symbolic and relational language

(x + x) + 5 = 37       (Applying associative property of addition)

2x + 5 = 37            (Combining like terms—definition of addition)

2x + 5 –5 = 37 –5 (Addition/subtraction property of equality)

2x + 0 = 32             (Property of additive inverse and subtraction operation)

2x = 32                  (Property of additive identity), and finally, the equation:

x = 16.                 (Division property of the equality)

Solution: Mark’s score (x points) = 16 points; David’s score (x + 5 points) = 16 + 5 = 21 points.

The kind of thinking where one begins with identifying unknowns, establishes relationships between unknowns (an equation or expression), and finally moves to the known (solution process using rules, relationships, theorems, definitions, properties, etc.) is not natural. It is contrary to intuitive and arithmetical thinking.

In arithmetic, we begin with knowns and lead to unknown(s) using the logic of natural language. The translation from natural language and arithmetical thinking to algebraic language and relational thinking is a major transition for students. It requires experience and effective teaching—the development of mathematical (spatial and algebraic) language—vocabulary, syntax, and two-way translation from natural language to math and vice-versa), forming conceptual schemas (wherever possible concrete and visual and relevant models must be used), and arriving at efficient procedures.

Certain arithmetic and pre-algebraic skills need to be mastered in order to be successful in algebra. I find that along with the basic arithmetic facts and fractions[1], a few key mathematics skills are also needed for a student to be successful in algebra. The most important of these is how to operate on integers. Many of students’ problems in algebra can be traced to their limited numbersense and misconceptions about integers and lack or mastery on their operations. For example, most students conceptualize subtraction of whole numbers as ‘take away’ and multiplying and dividing integers only as repeated addition and repeated subtraction respectively. These schemas are sufficient for conceptualizing and operating on whole numbers, but they are limiting and inadequate for operating on fractions, decimals, integers, rational numbers, and algebraic expressions. Therefore, a strong understanding of numeracy (number, number relations, number operations, patterns in numbers, and properties of numbers) and proportional reasoning is essential for learning algebra, but there are also marked transformations (mathematical and cognitive) that have to occur in students’ thinking to become comfortable with problem solving using algebra rather than arithmetic. With this change, a student can access the power of tools from algebra and other higher mathematics. These tools are the language of STEM fields and social sciences such as: economics, business, psychology, geography, etc.

When students experience and engage in discussions about the efficiency and efficacy of different methods early on, they begin to think flexibly and acquire the ability to augment, extend, and adapt their methods of problem solving. When all (or almost all) of the students have solved the problem or made attempts in solving the problem, the teacher should solicit (orally or written form) all the approaches (successful or not successful). Place them on the board or show them under the document camera or a picture. She should then need to engage in the discussion that help students to see each other’s approach and thinking behind it. Then discuss the approaches using the criteria: whether the approach

  • furthers our thinking and understanding of the problem (that is the first criteria for the acceptance of a solution approach/method),
  • provides ‘exact/correct’ solution,
  • gives the correct solution efficiently (out of all the correct/exact solution approaches, we need to ask which one is more ‘efficient’. In other words, which method gives us the solution easier and with less consumption of resources and time?), and
  • is elegant

In the final stage of discussion, the whole class should focus on which method is ‘elegant’. A method is called elegant when it can be generalized, abstracted and works for many situations (class of problems rather than individual or specific problem)—a method that also leads to the standard procedure.

Arithmetic is generalizing concrete experiences to concepts and procedures. Algebra is generalizing arithmetic to relationships between numbers and concepts and then developing the concept of relationships between mathematical entities (e.g., numbers, etc.) and extending them to functions appropriate to model problems. For example, in the elementary and middle school, the formula for the area of a circle was intuitively understood or just accepted, whereas, in high school starting with a concrete model and by the use of the concept of limits, it is derived into A = πr2. The idea of deriving the formulas using fundamental principles, logic, and methods such as limits (a sum of infinite terms) is what differentiates earlier mathematics and the high school mathematics courses. Such thinking plays an important role in almost all topics of mathematics, for example, showing that a regular polygon approaches a circle when the number of sides approaches infinity or the price of a car decreases toward 0 as the number of years approaches infinity. Similarly, a diagram of a cone sectioned into cylindrical slabs gives a reasonable estimate for volume of the entire cone. The volume could be determined if these slabs were very thin, their volumes calculated and then summed. This leads to the basic idea behind integration—an important concept in calculus.

The key concepts: quantitative and spatial reasoning reach fruition by high school. For example, the concept of spatial reasoning that began in easier grades reaches its formal form in high school.

Spatial reasoning is observing objects and simple relationships between them—from spatial organization to formal concepts in geometry, trigonometry, and visualization and representation of transformations of geometrical objects. With the help of formal logic, it develops into rigorous treatment of understanding formal geometry: deriving formal definitions and formulas; making connections and inferences; proving and justifying claims using formal logic and language; describing relationships; integrating quantitative and spatial ideas to model problems into systems of equations and figures.

Quantitative reasoning ranges from conceptualizing number relationships to algebraic principles–generalizing, abstracting, extrapolatingalgebra is generalized arithmetic; reversibility of thought; pattern analysis—recognizing, extending, creating and applying patterns; propositional reasoning; analogies; moving from knowns to unknowns; from facts and procedures to relationships; expanding the set of integers to include irrationals, imaginary numbers. Once algebraic reasoning is achieved, with the integration of the spatial tools of geometry, all algebraic, geometric, trigonometric, probabilistic and even calculus tools are within the reach of a student. Of course, this is subject to the availability of effective and efficient methods teaching, support, and resources.

The focus in high school is to prepare students to see the role of mathematics in the world of natural, physical, and social sciences and to prepare them for higher education and work. At the end of high school, they should be able to apply the tools of arithmetic, algebra, geometry, trigonometry, and probability to diverse situations, such as (a) intra-mathematical (higher concepts in mathematics, e.g., calculus), (b) interdisciplinary (e.g., STEM fields, social sciences—economics, psychology, etc.), (c) extra-curricular (real life applications).

To be prepared to solve problems through mathematical modeling, at the end of high school, students should be sensitive to the presence of numerous patterns in the relationships between a variety of variables from diverse situations:

  • Direct and inverse variation—as one variable increases, another also increases (or decreases) at a similar rate.
  • Accelerated variation—as one variable increases uniformly, a second increases at an increasing rate.
  • Converging variation—as one variable increases without limit, another approaches some limiting value.
  • Cyclic variation—as one variable increases uniformly, the other increases and decreases in some repeating cycle.
  • Stepped variation—as one variable increases, another changes in jumps.

Nature of the Mastery of Curricular Elements
To make sure that students have learned the material we teach, we need to pay attention to the mastery of individual curricular elements:

  • Mastery of mathematics language: Possesses adequate vocabulary, the syntax, and can translate math expressions and equations into English and vice versa; can explain his or her thinking using mathematics language, symbols, and processes
  • Constructs, develops, and understands concepts: can demonstrate the related multiple mathematical models of a concept (e.g., for systems of equations, they know the rationale and reasons for different—graphical, substitution, elimination, matrix, and determinant—methods of   solving them)
  • Develops and executes procedures: Can execute standard procedures (including the development of them) accurately, consistently, fluently, efficiently with understanding (e.g., factoring trinomials, polynomial division, or synthetic division)
  • Automatizes skills: Produces the result in acceptable time, fluently, and consistently (e.g., can give facts about integers or laws of exponents orally in 2 seconds or less and written 3 seconds or less)

Problem solving and communication: Integrates language, concepts, procedures and skills in problem posing, solving and interpreting and communicating of results and the solution process.
As proposed by the framers of the CCSS-M, the central idea of a beginning algebra course is to become fluent in using and interpreting symbols so as to generalize the concepts from arithmetic and to see algebra as generalized arithmetic and to explore and study relationships and functions and their multiple representations. This means:

  • Understanding, mastering, and applying the operations on and properties of real and complex number systems and their applications;
  • Justifying operations on real numbers with rigor (e.g., the set of real number is complete—between any two real numbers there exists a real number);
  • Understanding the arithmetic of algebraic expressions—extending the understanding and mastery of arithmetic operations (e.g., extending long division process to division of a polynomial by a monomial and binomial) and understanding the geometric and algebraic behaviors of polynomials and rational fractions;
  • Extending the concept and operations of factorization and prime factorization to factorization of polynomials, particularly with integer coefficients;
  • Understanding—defining, mastering operations on functions—addition, subtraction, multiplication, division, composition (including trigonometric functions), and inverse of functions (including exponential and logarithmic); understanding their behavior and applications; modeling applications using functions; impact of transformations (rigid and dynamic) on functions;
  • Understanding and mastering the representation of and operations on information (data) linguistically (expressing ideas in words), arithmetically (number relationships including determinants and matrices), algebraically (expressions, systems of equations and inequalities), geometrically (pictorially, tabular, curves, figures), functionally (operations and compositions), discrete methods (determinants, matrices, flow charts), and probabilistically;
  • Intra- and interrelationships between algebra and geometry (understanding the relationship between two and three dimensional objects—generating 3-dimensiional objects from 2-dimensional objects and vice-versa), coordinate geometry—an equation represents a geometrical entity and most geometrical objects can be expressed as a system of algebraical equations; polar vs. Cartesian, etc.); e.g., a circle—a collection of points equidistant (r units) from a given point (h, k) drawn on a paper (geometrical representation) can be expressed by the Cartesian equation: (x – h)2 + (y –k)2 = r2 (derived using the distance formula—algebraic); can be converted into polar equation by sing the transformation: x = h + r cos (t) and y = k + r sin (t).
  • Understanding transformations (rigid and dynamic) and their representations (functional, matrices, etc.) and their impact on geometrical, discrete, and, algebraic systems (transformation, congruence, similarity, composition); studying systems through transformations;
  • Applications of tools—arithmetic, algebraic, geometrical, probabilistic, trigonometric, functional and technological, in learning concepts, acquiring skills, and solving problems.

Students should realize that the idea behind learning properties of whole families of relations is typical of all mathematics: recognition of structure and similarities in apparently different situations allows applications of successful reasoning methods to new problems.




[1] For fractions see How to Teach Fractions Effectively by Mahesh Sharma, 2008.

CCSS-M: Arithmetic and Algebraic Thinking

CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight

The goal of elementary school (K through 6) arithmetic is for students to master additive reasoning—the inverse relationship of addition and subtraction (achieved K through 2), multiplicative reasoning—the inverse relationship of multiplication and division (achieved 3 through 4), and proportional reasoning—a relationship between two quantities (achieved 5 through 6).

With additive reasoning we can compare only two quantities by their sizes (smaller than or greater than, or how much smaller or greater) whereas multiplicative reasoning helps students to compare (beyond relative sizes) two quantities in terms of each other, e.g., one quantity can be expressed in terms of the other, such as it is twice as much or is going to be three times larger than now. The transition from additive reasoning (a one dimensional linear concept) to multiplicative reasoning (a two-dimensional concept—represented by an array or the area of a rectangle) is cognitively an important milestone.

Multiplicative reasoning provides the ability to understand and apply proportional reasoning—ability to make comparisons and define relationships between objects and quantities in multiplicative form rather than additive form (5 through 6). It provides the ability to predict the state of the two quantities in a future state, if the relationship persists. Having multiplicative reasoning and extending it to proportional reasoning is a great achievement on the part of children and is the beginning of algebraic thinking in the true sense. The understanding and competence in proportional reasoning facilitate students’ transition from arithmetic to algebra.

Many topics in upper elementary and middle school mathematics and science curricula require proportional reasoning, including problem solving with fractions, decimals, percentages, scale, transformations of objects/shapes, probability, trigonometry, conversions of measurements, geometry of shapes, density, molarity, speed, acceleration, force, mapping, scale drawing, etc.

At the end of elementary school, children should have strong numeracy skills (number concept, numbersense, and the four operations of addition, subtraction, multiplication, and division on whole numbers), on one hand, and proportional reasoning, on the other.

The success in the concepts to be learnt during the upper elementary and middle school mathematics is dependent on a few important concepts: mastery of ten counting numbers (particularly decomposition/composition and sight facts); numbersense (arithmetic facts, particularly multiplication facts and place value); numeracy; divisibility rules; prime factorization; short-division; and concept and operations on fractions.

Middle School Years
The core of elementary and middle school mathematics features understanding of and operations of addition, subtraction, multiplication, and division on whole numbers, fractions, integers, rational numbers, and real numbers. With this understanding and competence, children can understand and master other related content easily and effectively.

The goal of mathematics education during the middle school years is to arrive at generalizations, extrapolations, and abstractions from arithmetic problems and express them through algebraic symbolism and then manipulate these abstractions/symbols. At the elementary school level, the focus and competence are on solving problems arising from specific contexts whereas, in algebra, the algebraic models are applied to solving classes and systems of problems rather than specific problems in particular contexts.

This generalization process begins with students realizing that algebra, initially, is generalized arithmetic and then extends to arithmetic of algebraic expressions, relations and functions.

In the middle school years, the objective is for students to make the transition from arithmetic to algebra. In other words, during the middle school years, the focus is on developing algebraic thinking and applying it. However, it should be remembered that algebra is not a collection of procedures or manipulation of symbols; it is making connections.

School mathematics and algebra have always had the goal of training students to manipulate numerical and algebraic symbols. The purpose of this manipulation is to solve problems not only by arithmetical models but also through algebraic models and systems of equations, inequalities, and representations—both algebraical and geometrical.

In CCSS-M, the framers insist that algebra is not to be introduced just as a collection of isolated procedures, and they assume that students have a reasonably well developed understanding of arithmetic principles and procedures (especially fluency in the execution of the four operations on whole numbers and integers, and the understanding that every arithmetic operation has its inverse operation, etc.).

The framers of CCSS-M advocate that algebraic concepts and operations should be introduced by developing arithmetic patterns and then converting them to the algebraic generalizations to ensure that students recognize why algebra is an important area of mathematics to learn. If the students do not see this early on, it is much harder to get them interested later. It means students clearly understand the concept of rational numbers and, in the seventh grade, extend the arithmetic operations on integers and fractions to operations on the rational number system.

In eighth grade, students should extend their understanding of the rational number system to the real number system and understand and master the arithmetic operations on the real number system. They should understand the nature of the “real number” line. They should see the applications of algebraic models in a variety of quantitative and spatial situations using real numbers.

They should generalize quantitative reasoning to algebraic reasoning—understand how to operate on algebraic expressions, equations, and inequalities; express relationships through functions, which provides the ability and tools to understand relationships between systems; know operations on functions; expand the intuitive and concrete spatial relationships (such as transformations) to formal geometrical truths and relationships; integrate quantitative and spatial reasoning and relationships in problem situations; and solve problems through arithmetic, algebraic, geometrical, statistical, and probabilistic models. More specifically, the content in these grades are:

Seventh Grade
The focus of seventh grade is to extend the number system to include rational numbers. Students should understand and master the concepts of integers and rational numbers, their relationship with other numbers and mastery of operations on them. It means:
(a) understanding the difference between numeral (representation of quantity, e.g., I took three steps is represented as the numeral 3) and number (a directed numeral, e.g., I took three steps forward is represented as +3 and I took three steps backward is represented by –3) and operations on them;

(b) understanding the definition—a number that can be written as the ratio Capture blog 6.02, where a and b are integers, b ≠ 0, and a and b are relatively prime—with 1 as the greatest common factor; representing them as decimals—a rational number can be represented by (i) terminating decimal (.459), or (ii) repeating, non-terminating decimal Capture blog 6.03; and using rational numbers in multiple contexts—quantitative and spatial reasoning;
(c) understanding and mastery of operations on rational numbers, for example, it means that: operations (such as multiplication) are extended from fractions to rational numbers by requiring that they continue to satisfy their properties, particularly the distributive property, leading to products such as (−1)(−1) = +1;
(d) understanding that the rules for operations on signed numbers are well-founded; for example, by emphasizing the properties of operations, students are able to extend the situations such as: (−m)(−n) = mn for any integer m and n to the product of rational numbers. They see them as a general case rather than a special one—indicating entry into algebraic thinking in the formal sense;
(e) interpreting the operations (e.g., products of) of rational numbers by describing real world contexts.

The second focus of seventh grade mathematics is acquiring facility in working with and applying rational numbers in multiple contexts such as extending the concept of proportional reasoning and mastering it. It should include concepts such as: proportion, unit price, scale factor (stretching and shrinking), slope, conversions, etc.
(a) from extending the understanding of and applying proportional reasoning and relationships to learning new mathematical concepts and solving real world problems
(b) deeper understanding of and operations on rational numbers involving expressions, linear equations and inequalities;
(c) solving problems involving similarity, scale drawings, rate of change, slope, and informal geometric constructions;
(d) working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume, first treating them intuitively and concretely—conducting experiments such as three cones with the same base and height can fill the cylinder with the same height and base, showing that the volume of the cone is one-third the volume of the cylinder—and similarly considering generalized situations of particular situations (e.g., the surface area of a prism is equal to the product of the perimeter of the base and height plus the sum of the area of the two bases; the volume of the prism is equal to the product of the area of the base and the height) of simple and compound shapes, figures, and diagrams; and
(e) drawing inferences about populations based on samples.

Eighth Grade
In eighth grade, the focus is on understanding and operating on real numbers—each real number has a unique place on the real number line (e.g., a real number can be located on the number line, and each point on the number line represents a real number). The set of real numbers is the union of the sets of rational and irrational numbers and a real number can be represented as (a) a terminating decimal, (b) repeated non-terminating decimal, or, (c) non-repeating, non terminating decimal (e.g., .o1oo1oo1ooo1oooo1….; or .12123123412345123456…, etc.). The mastery of numeracy skills and operations on integers and rational numbers is good preparation for understanding and operating on real numbers.

Students should be fluent in using symbols so as to generalize arithmetical procedures to algebraic operations, abstracting from quantitative and spatial situations, and developing algebraical and geometrical reasoning to understand the algebra of functions, linear equations and inequalities. In particular:
(a) Formulating, reasoning, and performing operations on algebraic expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations and inequalities using multiple methods and knowing the appropriateness, efficiency, and limitations of a particular method;
(b) Grasping the concept of a function and using functions to describe quantitative, spatial (geometric), and probabilistic relationships;
(c) Analyzing two- and three-dimensional space, shapes, figures, and diagrams using distance, angle, similarity, and congruence; relating 2- and 3-dimensional shapes; understanding and applying quantitative and spatial relationships and results such as the Pythagorean theorem; the trigonometric ratios in a right triangle; making conjectures based on observed patterns in quantitative and spatial relationships and then supplying reasons to prove or disprove those conjectures.

In eighth grade, students should use their understanding and mastery of operations on the real numbers to solve arithmetic, algebraic, geometric, and probabilistic problems.

They should understand patterns in quantities and graphs relating two variables. The conceptual heart of the matter is understanding relations among several quantities whose values change. Eighth grade mathematics must also include

  • thinking about variables as measurable quantities that change as the situations in which they occur change;
  • understanding that variables are not usually significant by themselves but only in relation to other variables and context.

And finally, apart from considering algebra as generalized arithmetic, students should see algebra as a system with a basis in the concept of relations and functions and understand that the most useful algebraic idea for thinking about relations is the concept of function. This helps students to relate one set of representations, ideas, and properties to another and their relationships (linguistic expression, iconic, tabular, graphical/spatial, equation and systems of equations, quantitative/abstract). For example, the typical relation among two or more varying quantities may look like:

  • As time passes, the depth of water in a tidal pool increases and decreases in a periodic pattern.
  • As bank savings rates increase, the interest earned on a fixed monthly deposit also increases, but when the interest earned is compounded, the new amount increases exponentially.
  • In a sequence of squares having sides 1, 2, 3, 4, 5, …,n, …, the areas of those squares are 1, 4, 9, 16, 25, …,n2, …and the perimeters are 4, 8, 12, 16, 20, …., 4n,….
  • For any rectangle of base b and height h, the perimeter p is 2b + 2h.        

Students should know the difference between a conjecture, definition, and a theorem. They should know the difference between proof, example, and counter example; between direct and indirect proof; between justifying and providing a counter example, etc.

With strong numeracy skills and access to these tools, our students—the future mathematicians, can search for patterns in much the same way that scientists explore results from experiments by systematically manipulating variables. The experimental data of mathematics—calculations are made using appropriate algorithms and tools, and then data and calculations are displayed graphically to reveal patterns, regularities and variations. This data can be sorted and analyzed; and then patterns are observed and inferences are made. Further calculations are made to prove or refute these inferences. They should understand and appreciate that ultimate standard for verification remains a formal proof by reasoning from axiomatic foundations.

Further, students should be familiar with and able to use, when necessary and appropriate, computational capabilities of machines—both existing and envisioned. These tools suggest some exciting curricular possibilities. Calculators and computers have a profound effect on students’ understanding of the nature of mathematics. Thus, calculators and computers can be efficient means to generate understanding and interest in algorithms, in particular, and mathematics concepts in general. In this way, the role of tools (calculators, computers, sketchpad, geogebra, software apps, etc.) could be to enhance mathematical thinking rather than detracting from understanding and mastery of arithmetical and algebraic algorithms or just mindlessly learning or applying procedures.

Finally, CCCSS-M has a provision for meeting the needs of talented students in middle school. Students with high aptitude and ability in mathematics are offered pre-algebra in the seventh grade as an accelerated course that provides a transition from arithmetic to algebra and a challenging algebra course (see CCSS-M Algebra One course) in eighth grade. CCSS-M recognizes that these students are our future as they are going to invent more mathematics to provide the language for science, technology, and engineering fields. The framers understand all students should be challenged to realize their potential, but these students in particular should be provided the challenging mathematics they are capable of handling.




CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight

CCSS-M: Non-Negotiable Skills in Elementary Grades

Primary Grades: Kindergarten through Second Grade
During the first three years—from Kindergarten through second grade, the goal of CCSS-M is that children understand, master and apply additive reasoning. This means children acquire true understanding and mastery of counting numbers (number concept), addition and subtraction concepts, facts, procedures and see addition and subtraction as inverse relationships—given a subtraction problem, they can solve it by addition and vice-versa. They should also understand the concept of place value (as a pattern of representing numbers in the base-ten system.

At the same time, children are able to name, recognize, and represent (draw) the commonly found objects in their environment and are able to perform simple operations such as finding their perimeters.

CCSS-M recommends three years to achieve this fundamental goal, which is the foundation of all future mathematics. Moreover, mastery should reflect understanding, fluency, and applicability of this important concept of additive reasoning.

The focus of mathematics teaching and learning for children in Kindergarten is to acquire the number concept and relationships between numbers. Number concept[1] means:

  • Representing, comparing, and recognizing whole numbers in different forms and modes: discrete (sets of random objects—counting objects), continuous (visual/spatial—comparing visually the length and area of objects, such as Cuisenaire rods to determine larger, smaller, etc.), pictures (like marks on a paper, clusters, number line, etc.), abstract (forming and recognizing numbers);
  • Fluency in number relationships up to ten—mastery with understanding (decomposition/recomposition of a number, e.g., the number 7 can be seen as made of 7 and 0 (0 and 7), 6 and 1 (1 and 6), 5 and 2 (2 and 5), 4 and 3 (3 and 4);
  • Fluency of 45 sight facts[2] (can recognize the facts by sight just like children can recognize certain words by sight);
  • Expressing numbers (two-digit) through place value representation (e.g., 56 = 50 + 6 = 40 + 16 = 30 + 26 = 20 + 36 = 10 + 46 = etc.);
  • Fluently able to count numbers and understanding role of number words (difference between a quantitative and non-quantitative words, difference between cardinal and ordinal numbers); and
  • Recognizing, identifying, and naming commonly found objects and shapes in their environment and spatial relationships and corresponding vocabulary.

In order to achieve the appropriate level of competence in these concepts, a great deal of teaching and learning time (almost 70% of allotted instructional time) in Kindergarten should be devoted to the development of number concept and its mastery. In addition, at the end of Kindergarten, children should be able to represent the first 30 numbers on the number line and fluently count forward and backward by 1, 2, and 10 from any number up to 100 and beyond.

First Grade
The focus of first grade work is to build on the number concept. Children learn number relationships and place value, specifically

  • Understanding addition and subtraction concepts and learning   efficient strategies (using decomposition and recomposition of numbers and sight facts) for and fluency in (10 by10) addition facts and constructing and arriving at subtraction facts within 20 (without counting);
  • Developing an understanding of whole number relationships and place value, including groupings in tens and ones to understand two and three digit numbers fluently (e.g., 346 = 300 + 40 + 6 = 300 + 46 = 340 + 6 = 306 + 40 = etc.);
  • Developing an understanding of linear measurement and measuring lengths as iterating length units (moving from egocentric measurement to using “go between” units, e.g., the room is 35 foot lengths to it is 40 book lengths); and
  • Reasoning about, attributes of, and composing and decomposing geometric shapes commonly found in the child’s environment.

A great deal of time, in first grade, should be devoted to understanding the concept of addition, achieving mastery of addition facts[3] (adding up to 20) and understanding the concept of subtraction.

Fluency (automatization with understanding) is achieved by using efficient strategies based on decomposition/recomposition of number and sight facts (e.g., 9 + 7 is achieved by seeing that by decomposing 7 as 1 and 6 and then adding 1 to 9 transforms the sum as 10 + 6, or since 10 + 7 = 17, thus, 9 + 7 is 16; 9 + 7 = (6 + 3) + 7 = 6 + (3 + 7) = 6 + 10 = 16, or, 9 + 7 = (2 + 7) + 7 = 2 + (7 + 7) = 2 + 14 = 16, or, 9 + 7 = 9 + (9 – 2) = (9 + 9) – 2 = 18 – 2 = 16, etc. Deriving the arithmetic facts using multiple strategies provides children flexibility of thought. Similarly, in place value, they should know not only that 124 = 100 + 20 + 4 (expanded form with canonical decomposition) but also that 124 = 100 + 24 = 12 tens + 4 = 120 + 4 = 114 + 10 = 110 + 14 = etc. (expansion with non-canonical decompositions).

In first grade and then in later grades, children realize that concepts such as finding the perimeter, working with money and time are applications of numbersense (number concept, number relationships, and place value). If students have “good” numbersense, teachers can teach and students can easily learn these topics. Without numbersense, these topics are difficult for many children to learn. Teaching them in isolation, even with concrete instructional materials, limits a child’s learning and a teacher’s instructional time.

Second Grade
The focus of second grade is to master subtraction and integrate addition and subtraction into additive reasoning. This means:

  • Mastery of (fluency with strategic understanding—using decomposition and recomposition of numbers and sight facts) addition and subtraction facts;
  • Properties of numbers: even and odd;
  • Fluency in executing standard and alternative procedures of addition and subtraction with understanding that these procedures are based on place-value;
  • Mastering place value beyond hundreds; expressing the number as:

5,467 = 5,000 + 400 + 60 + 7 (expanded form–canonical decomposition)

= 5400 + 67 = 5000 + 467 = etc. (expanded from–non-canonical form);

  • Using standard units of measure (understanding that the smaller the unit of measurement the larger the iterated quantity and vice-versa); and
  • Describing and analyzing shapes.

In the second grade, a great deal of time should be devoted to achieving mastery of addition and subtraction facts (for numbers up to 20).

Fluency in addition and subtraction should be achieved using strategies and understanding (decomposition/recomposition of number and sight facts) and the addition strategies should be used and extended to deriving the subtraction facts. For example, students should able to think the subtraction problem 16 – 9 = (6 + 10) – 9 = 6 + (10 – 9) = 6 + 1= 7, or, to find 16 – 9, I know 9 is 6 + 3, so I take 6 from 6 and 3 from 10, so the answer is 7, or 10 + 6 is 16, 9 is 1 less than 10 so 9 + 7 is 16, so the answer is 7.

By the end of second grade, students should have mastered additive reasoning and its applications. Additive reasoning means: students understand the inverse relationship of these two operations and given a subtraction problem, it can be treated as an addition problem and addition problem as a subtraction problem.

[1] For details of how to teacher number concept, please see the blog on Number concept and Numbersense.

[2] Each counting number up to 10 has sight facts associated to it, e.g., 7 = 1 + 6 = 2 + 5 = 3 + 4 = 4 + 3 = 5 + 2 = 6 + 1. The total number sight facts, thus is 45. For the list see: Visual Cluster Cards and Number Concept by Sharma, 2016.

[3] For strategies of teaching addition facts see: How To Teach Arithmetic Facts Easily and Effectively by Sharma, 2008.

CCSS-M: Non-Negotiable Skills in Elementary Grades

CCSS-M Focus: Non-Negotiable Skills

Most of mathematics and mathematics activities at the school level are goal oriented, and every vocabulary term, concept, procedure, or skill should have a purpose and place in the curriculum and in the overall development of mathematical thinking. According to CCSS-M, there should be focus on particular aspects of mathematics at each grade. Focus requires that in order to learn mathematics meaningfully, we significantly narrow the scope of content, both in a lesson and at the grade level, particularly in the formative earlier grades in order to build a strong foundation for future mathematics learning.

Focus: The Importance of Non-Negotiable Skills
The focus of arithmetic teaching in the K—5 standards is to build an important life skill and to develop mathematical ways of thinking—to observe patterns, make conjectures, generalize, and abstract—and to have mastery of numeracy skills with understanding and processes that make higher mathematics accessible to all children. Mastery of the content at the elementary level (numeracy skills) should prepare them to be able to apply these skills and concepts to learn mathematics and solve problems not only in other parts of mathematics and other disciplines but also in real-world situations.

In essence, such a focus means mastering certain non-negotiable skills at each grade level.

Non-negotiable skills form the basis for other concepts, procedures and skills at that grade level and build the foundation for future mathematics learning. For example, in CCSS-M, learning the concept of “decomposition and recomposition” is identified at the Kindergarten level, but it should be used extensively in deriving the addition and subtraction arithmetic facts at the first and second grade levels. Similarly, multiplication is introduced in the second grade as repeated addition, equal groups of objects, and arrays. By the end of third grade, however, children should have mastered the concept as repeated addition, groups of, arrays, and the area of a rectangle so that they understand the distributive property of multiplication over addition and subtraction better and have automatized multiplication facts. At the same time, the multiplication as the area of a rectangle should be developed from repeated addition, groups of, and arrays. Such an understanding of multiplication prepares student to multiply fractions and decimals in fifth grade using the area model of multiplication and arriving at the standard algorithm of multiplying fractions. Then they can apply multiplication of fractions to master the standard operations of addition and subtraction on fractions with efficiency and understanding.

Using the same principles, they should automatize division facts by the end of fourth grade and division of fractions by sixth grade along with the mastery of operations on integers and operations of rational numbers in the seventh grade. Similarly, concepts of fractions, e.g., adding and subtracting fractions with same denominators are introduced in the third and fourth grades, and by fifth grade, children should have mastered the operations on fractions with deeper understanding. This is possible because the mastery of multiplication and division facts and the additive and multiplicative reasoning was developed as focus areas in the third and fourth grades.

In the early grades, the standards rightly concentrate on the mastery of arithmetic skills (number concept, numbersense, and numeracy) along with spatial orientation/space organization, and measurements relating quantitative and spatial reasoning. As we expect all children to read fluently with comprehension by the end of the third grade, we should expect all children to have mastery of numeracy by the end of fourth grade so that they can learn higher mathematics easily, effectively, and efficiently in later grades.

Numeracy means a child can execute the four whole number operations correctly, consistently, fluently, in standard/efficient forms with understanding. They may begin with idiosyncratic methods and explorations of number operations, but ultimately they should be fluent in standard algorithms that are the distillation of centuries of efforts by mathematicians from many cultures.

The objective of mathematics throughout the middle and high school grades is to deepen and broaden their mathematics understanding and competence and to prepare students for college and meaningful careers. As a result of many surveys, it is clear that postsecondary instructors value greater mastery of prerequisites skills for higher mathematics over shallow exposure to a wide array of topics with doubtful relevance to postsecondary work. At present, the shallow exposure in our curriculum to mathematics concepts is true for all students, including exceptional students at institutions with high expectations. For instance, according to the director of undergraduate studies in mathematics at Johns Hopkins University, high performing students “have the grades and the test scores to be there” but lack “a deep understanding of why the techniques they’ve been taught work, the actual underlying mathematical relationships. They walk into to my classroom in September and don’t have the study habits or proper foundation to do the work.”

Just like number concept, numbersense, and numeracy are expected to be mastered during the early elementary school and are used to learn future mathematics. Similarly, appropriate algebraic concepts and skills (generalization of arithmetic facts, concepts, and arithmetic operations and procedures on algebraic expressions, the idea of functions and operations on functions, and modeling problems by algebraic equations and solving them) are acquired and used during the middle and high school years. However, aspects of these concepts are acquired throughout the curriculum and not reserved only for the middle grades. They are introduced and expanded appropriately throughout the grades, but they are expected to be mastered by the end of eighth grade.

Mastery of non-negotiable skills at each grade helps maintain continuity of learning from grade to grade and provides for a seamless transition from arithmetic to algebra in the middle grades and preparation for higher mathematics in later grades. The mastery of the eighth grade algebraic and geometric concepts helps students to learn and master high school algebra and geometry comprehensively and provides access to higher mathematics such as calculus and discrete mathematics. Students leave high school with the ability of applying arithmetic, geometrical, algebraical, probabilistic, discrete and continuous models to problems in college and at work.

Mastery of non-negotiable skills also ensures that the next grade level teacher can begin the teaching of mathematics concepts at the grade level rather than endlessly review work from previous grades. Because of lack of mastery of non-negotiable skills (focus concepts and procedures), every grade has such a diversity of skill and mastery levels that teachers end up teaching several groups at different grade levels, thereby devoting limited time on grade level tasks. The reinforcement and practice of skills, going deeper, making connections, and providing special care for needy children (on both ends of the spectrum—from slow learners to gifted and talented) should be done in smaller groups or at the individual level after the main grade level concept has been presented to the whole class or taking meaningful but brief digressions to review previous concepts and skills and providing extensions.

Mastery of non-negotiable skills avoids the over-reliance on review of concepts and procedures in the beginning of every grade, causing many high achieving children to lose interest in mathematics. It sends students the message that what is learned can be forgotten by the end of each grade because it will be reviewed in the next grade anyway. The over emphasis on pre-tests and review also indicates that the claim the previous grade teachers made that “they taught the material and children learned it” is not true and sends teachers the message that their colleagues cannot be trusted. In any case, it indicates that the material was not learned in the first place.

In the process of developing these non-negotiable skills, teachers should keep in mind that concepts and procedures are not taught in isolation but need to be interwoven with linguistic, quantitative, algebraic, and geometric reasoning. Children should see the fundamental nature of that concept or operation. They should see the trajectory of the development of the concept—how does it begin and how does it transform over the years? Teachers should make these connections transparent. For example, taxonomy of actions involving quantitative reasoning and numeracy skills shows that operations on numbers can be matched with operations on objects that numbers describe:

  • Addition models begin as putting “things” together, combining and joining collections and then become shifting and translating forward (a transformation),
  • Subtraction models begin as take away from a collection, comparisons of collections and quantities and then become shifting and translating backwards (a transformation), or recovering an addend,
  • Multiplication models begin as repeated addition, grouping, organizing and area of a rectangle and then become size change (stretching and shrinking) (a transformation), or use of a rate factor, and,
  • Division models begin as repeated subtraction, grouping, organizing and then become ratio, rate, rate division, size change (a transformation), or recovering a factor.

Similarly, when students have learned operations on fractions, finding the perimeters, areas, and volumes should involve figures and shapes with measurements in terms of fractions and decimals. Formulas or memorizing interesting facts is fine in context but only after students have understood the concept. Memorizing formulas and facts has a very important place, just as learning vocabulary does— however, the presence of rich schemas for concepts together with automatized essential skills helps students to create richer, more nuanced work, and the possibility of applying it to problem solving with ease and flexibility.

CCSS-M Focus: Non-Negotiable Skills