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

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

What Does it Mean to Master Arithmetic Facts?

Question: Everyone is focusing on children knowing mathematics ideas. What does it mean to know arithmetic facts? Is mastery different than knowing? Is it important to memorize arithmetic facts?

Answer: In the context of mathematics curriculum, the term mastery is associated with several words: know, understand, master, being fluent, proficient, etc. Although it has been clearly defined in the Common Core State Standard in Mathematics (CCSS-M) documents, in our schools, the definition of mastery of mathematics curricular components still varies with individuals, individual school systems, and textbook series.

Students need to satisfy certain characteristics to have acquired mastery of mathematics curricular elements. Mastery of different curricular elements is complex; it is multi-dimensional.

Components of Mastery
A student has mastered an arithmetic fact when three conditions are satisfied: The child

  • demonstrates the understanding of the concept—can provide an efficient strategy (based on decomposition/recomposition and select the most efficient strategy out of several strategies), without counting, for arriving at the arithmetic fact;
  • demonstrates fluency/automatization (able to produce a fact orally in less than two seconds and in writing in less than three seconds), and
  • can flexibly apply the knowledge of the fact in different situations in solving problems without counting and using concrete or pictorial models.

Learning with rigor (understanding, fluency, and applicability) is at the center of the Common Core State Standards for mathematics (CCSSM). Teaching with rigor produces flexible and long lasting mastery.

Understanding, here, means a student is able to derive a fact or procedure with effective, efficient strategies and able to explain his approach, for example, an addition fact is derived by using decomposition/recomposition of numbers, not just by counting up. This also includes flexible use of these strategies. It is widely acknowledged that practice, drill and memorization are essential if students are to become mathematically fluent. However, practice without understanding is of very little value as the facts and skills are not integrated and applications under such limited level of mastery become difficult.

The concept of “fluency” refers to knowing key mathematical facts and methods and recalling them efficiently. Fluency is automatized mastery of a fact, skill, or procedure.

Understanding and fluency facilitate the application of facts and procedures in problem solving. Applications may be learning other mathematics concepts using those facts. It may be problem solving in other disciplines or extra curricular situations.

Without this level of automatization, children become dependent on concrete or pictorial materials used in instruction such as: counting objects—number line, hash-marks, fingers, TenFrames, discrete objects, etc. Strategies based on counting materials do not help children to achieve either the understanding and fluency or the ability to apply such knowledge.

The elements of rigor (mastery) are inseparable. The absence of any one of them is problematic. For example, when children do not have understanding of the strategy and have not automatized facts, they are not able to apply their knowledge to newer situations effectively. They digress from the main problem to generate the facts needed to be used in solving a problem. Their working memory space is filled mostly in constructing facts. Then it is not available to pay attention to the instruction, to observe patterns, focus on concepts, nuances, relationships, and other subtleties involved in the concepts, procedures, and applications.

When children possess number combination mastery, their achievement increases at a steady rate in arithmetic, whereas children with low mastery in arithmetic facts make little to almost no progress in later grades. When students, without mastery of arithmetic facts, are provided classroom instruction in procedures, although they make good strides in terms of facility with these algorithms and procedures and even solving simple word problems, deficits in the retrieval of basic combinations remain. These deficits inhibit their ability to understand and participate fully in the mathematical discourse and to grasp the more complex multi-step and algebraic concepts later. Failure to instantly retrieve a basic combination, such as 8 + 7, often makes discussions of the mathematical concepts involved in algebraic equations more challenging.

Before we can have effective mathematics teaching and children can achieve higher in mathematics, everyone involved needs to have a well-defined and commonly-agreed upon definition of knowing/mastery for a concept, procedure, or skill—with clear markers for mastery. Then and only then can it be taught well, retained by students, and assessed and monitored effectively.

Essentials for Higher Achievement
To assure the conditions for higher achievement, we first need to
(a) identify non-negotiable concepts, skills, and procedures to be achieved by each child at each grade level,
(b) develop common definitions and criteria for “knowing” concepts, “mastering” skills, and achieving “proficiency” in executing procedures, across the school system, and
(c) identify and discuss the most “effective,” “efficient,” and “elegant” ways of teaching the key developmental mile-stone concepts and procedures at each grade levels. These include: number concept, arithmetic facts, place value, fractions, integers, and algebraic thinking.

When we have identified these, training should be provided for all teachers (all classroom teachers and interventionists of various kinds) and administrators by using content embedded pedagogy. The training should also include how to observe children’s work and learn from the error analysis about their level of mastery.

Observing Children’s Work
In a second grade classroom I observed a teacher assessing children’s mastery of addition and subtraction facts. The purpose of the test was to assess understanding, fluency, and applicability of addition and subtraction. The test had 25 problems on addition and subtraction.

One of the problems on the test was 17 − 9 = ☐. I identified five children from the class to observe work on the problem. Problems were written both in horizontal and vertical forms:



Here are my observations.

Child One
One child read the problem and solved it in about 20 seconds by sequentially counting on his fingers: 10, 11, 12, 13, 14, 15, 16, and 17. Then he recounted the fingers used: 1, 2, 3, 4, 5, 6, 7, 8 to find the answer and wrote the correct answer 8.

Child Two
The second child answered the problem in about 50 seconds. She drew seventeen tally marks in front of 17 and nine tally marks in front of 9. She crossed one tally mark in front of 17 from the top and one from the 9. After all the nine tally marks were exhausted, she counted the remaining tally marks in front of 17 and correctly wrote the answer as 8.

Child Three
The third child answered the problem in about 36 seconds. The child counted: 17, 16, 15, 14, 13, 12, 11, and 10 on his fingers and counted the fingers used. And then he wrote the correct answer in the right place.

Child Four
The fourth child used the number line pasted on his desk. The child located numbers 9 and 17 on the number line and then counted the numbers from 9 to 17 and wrote down the correct answer in about 27 seconds.

Child Five
The fifth child read the problem and thought for a moment and wrote down the answer (8) in 2 seconds in the correct place.

The teacher collected the papers of all the children from the class. During the debriefing, I asked her to check the problem: 17 − 9 = ☐ on these five children’s papers. She did. She put a check mark (✓) in front of the problem on their papers.

I asked her whether these five children “knew” the answer to the problem 17 − 9 = ☐ and did she have enough information to judge the responses to satisfy the criteria for knowing?

She said: “Of course. They have the correct answer on their papers.” I asked other participants to also look at the response of these five children. They agreed with the teacher.

It is true that the children had the correct answers, but they knew the fact 17 − 9 = 8 at different levels of knowing.

The first and the third child knew the fact: 17 − 9 = 8 at the concrete level and used counting as a strategy. The second and the fourth child knew the fact at the pictorial level and also used counting strategy. The second child used a simple one-to-one correspondence counting. Unlike the others, the fifth child answered the problem without any overt strategy and very quickly.

In the case above, only the fifth child, who answered the problem in 2 seconds (within the expected time for response), had automatized this subtraction fact. However, we do not know whether she just memorized the fact in a rote manner or with some strategies. Therefore, her example does not satisfy the definition of knowing.

The four children arrived at the correct answer but used inefficient strategies. They also have not achieved fluency in arriving at the fact. Very few children arrive at fluency using inefficient strategies such as counting.

The best method of automatization of arithmetic facts is to practice the facts first orally. When children have shown the mastery orally using efficient strategies, then the children should practice the facts in writing. A fact is mastered orally when the child can answer the fact in 2 seconds or less. A fact is mastered in written form when the child can answer a fact in 3 seconds or less and able to furnish a strategy when asked.

Assessment of arithmetic facts should be oral first with immediate feedback about the strategy and its efficiency. When children can answer facts orally in the prescribed time, the teacher should ask for the strategy used. Then there should be intentional effort to make children flexible by providing alternate strategies. There should be a class discussion on which strategy is most exact and of all the exact strategies which ones are efficient and then which ones are elegant.

Without focusing on children using efficient strategies and correcting papers only for correctness, we send a message to the children that they can keep using inefficient strategies. When children become fluent in using inefficient strategies, they may become fluent counters.

Several studies found that a significant area of difference between students with number combination mastery and those without was the sophistication of their strategies. The poor combination mastery group in second and third grades continued to use fingers to count when solving problems. In contrast, their peers increasingly used verbal counting or decomposing/recomposing numbers without fingers, which led much more easily to the types of mental manipulations that constitute mathematical proficiency.

In the development of counting knowledge in young children, one can observe that children use an array of strategies when solving simple computational problems. For example, when figuring out the answer to 6 + 8, a child using an unsophisticated, inefficient strategy would depend on concrete objects by picking out first 6 and then 8 objects and then counting how many objects there are all together.

A slightly mature but still inefficient counting strategy is to begin at 6 and “count up” 8. Still more mature would be to begin with the larger addend, 8, and count up 6, an approach that requires less counting. However, all of these are based on counting, and many of these children will not reach fluency in addition and subtraction. Effective teachers promote efficient use of number relationships and ultimately help children transcend counting. They help children acquire efficient number combination strategies based on decomposition/recomposition (e.g., sight facts, making ten, teens numbers, doubles, near doubles, the missing double, etc.).

For example, using decomposition/recomposition, a child might say that 6 + 8 = 6 + 4 + 4 = 10 + 4 = 14 (making ten and teens numbers), or 6 + 8 = 4 + 2 + 8 = 4 + 10 = 14 (making ten and teens numbers). Some children might give this as: 6 + 8 = 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 6 + 8 = 6 + 6 + 2 = 12 + 2 = 14, or 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14 (decomposition/recomposition and doubles). When children can furnish more than one strategy (other than counting) in arriving at a fact, they show flexibility of thought. Children with flexibility of thought are able to apply their fluency in novel situations and go higher in mathematics.

Some children will simply have this combination stored in memory and remember that 8 + 6 is 14. Ultimately, that is what we want children to reach—a level of automatization. But this should be reached with understanding and with the use of efficient strategies. Memorization of a fact with repeated use remains an isolated fact. Its application also remains isolated. However, it does have some advantages if that fact mastery is used to gain other fact mastery by using decomposition/recomposition. For example, when a child can retrieve some basic combinations (sight facts of a number, say 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10), then he or she can use this information to quickly solve other problems (e.g., 6 + 5) by using decomposition (e.g., 6 + 4 + 1 = 11).

The ability to derive, store, and easily retrieving information in memory helps students to build both procedural and conceptual knowledge of abstract mathematical principles, such as commutative and the associative laws of addition and mental fluency and numbersense (e.g., 9 + 7 = 9 + 1 & can be extended to 59 + 7 = 59 + 1 + 6; 149 + 7 = 149 + 1 + 6). Immature finger or object counting creates few situations for learning these principles. Research also suggests that maturity and efficiency of strategies are valid predictors of students’ ability to profit from later mathematics instruction.

In order to achieve fluency, teachers, parents, and students should understand what fluency looks like. To achieve fluency there is a definite progression. For example, in the case of addition, it involves:

  • mastering number concept—subitizing and visual clusters, decomposition/recomposition, 45 sight facts (what are sight facts, see previous blog on sight facts);
  • mastering addition strategies—commutative property of addition, N + 1, making ten, teens numbers, N + 9, double numbers, near doubles, missing double, N + 2, near tens (9 and 11), and the remaining four facts (8 + 4, 4 + 8, 8 + 5, 5 + 8) using decomposition/recomposition;
  • “working out” using efficient strategies with understanding to efficiently generating an answer (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 8 + 6 = 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14);
  • practicing “rapid recall” and finally,
  • the ultimate goal of “instant recall.”

The time taken and the expectations of efficient recall vary per topic. In mastering addition facts, the role of sight facts, decomposition/ recomposition, making ten and teens numbers is crucial. For example, a child may be able to recall the sight facts of 9 and then practice adding other numbers to it using decomposition/recomposition (9 + 2 = 9 + 1 + 1; 9 + 3 = 9 + 1 + 2, etc., or 9 + 7 = 10 + 7 – 1, etc.). Students need to keep working on decomposition/recomposition strategies till they are able to generate the facts and then keep on practicing automatization.

When a child can read fluently, we are not able to detect the strategies used for acquiring the reading skill. Fluent reading means the child has transcended the strategies and skills used for arriving at that fluency.  Similarly, when we ask a child to find the sum of numbers 8 and 7 and the child counts 9-10-11-12-13-14-15 and says: 8 + 7 = 15, we should not be content with that. Instead, we need to give the child better strategies and work with the child until he or she arrives at fluency. Sequential counting is decoding of numbers; it is not mastery of a fact. By giving the feedback—“good job” to such a strategy, we are sending the wrong message that decoding of numbers is adequate. Knowing addition as “counting up” and subtraction as “counting down” are not strategies.

Mistakes are an integral part of learning, but it is equally important for teachers and children to be aware of the need for accuracy. When a child gives a wrong answer (for example, 8 + 6 = 15), the teacher needs to redirect the child:

  • 8 + 7 = 15, our problem is 8 + 6, can you use that fact to find 8 + 6?
  • what other strategy would you use for finding the answer? How would you make 8 as 10? Etc.
  • can you use another strategy? Etc.

Through these activities—by constantly bringing their attention to the appropriate strategy, the child should derive the fact. Once the child derives the correct answer using an efficient strategy, direct the child to another equally efficient strategy.

After that, observational assessments should be used to ensure that all children are being accurate. Accuracy should be achieved first at oral level with immediate feedback when an incorrect answer or strategy is produced. This involves listening to children’s verbal responses, targeting specific children with differentiated questions and checking responses on whiteboards. Then, children should also be given responsibility for self-assessing their own work for accuracy.

Automatization and fluency at appropriate level (speed) is achieved by practice. Practice sessions for achieving speed should be brief, paced, and create a buzz of excitement in which children’s recalling and using their knowledge efficiently gives the feeling of achievement (immediate feedback).

It is important, though, to recognize that fluency is not solely about memorizing and recalling facts; it also means being able to work flexibly and choose the most appropriate method for the problem at hand. Children do need drilling in the basics, but this can be delivered in open-ended, rich and engaging ways. The key is to balance the three components outlined above. As is often the case in teaching, getting the balance right is crucial. Fluency of facts is essential, and if we teach in a style and order that suits the development of this fluency, we do not risk sacrificing creativity and contextual richness in mathematics tasks.

A short modeling demonstration by the teacher should be followed by game-like activities lasting between three and four minutes. The fast pace of these activities, combined with the emphasis on aiming to beat the child’s own personal best, makes the session exciting and engaging for the child. Each child should be working on identified facts. For example, The Addition Fact Ladders[1] and Fact War Games[2] are good for achieving accuracy and speed.

Thus, for automatization, first the child must have the understanding, and then accuracy, followed by speed. For that, a great deal of practice is essential.

Difficulty of the tasks (facts to be mastered) should be gradually increased while practicing and testing. The first practice and test should not be difficult, and oral practice with immediate practice is required. Teachers should point out not only the strategy but also the name of the strategy. For example, when a child is practicing a fact, say 8 + 6, the child says: I take 2 from 6 and give it to 8 and then I add 10 and 4. The answer is 14. The teacher should ask: Which strategy did you use? The child should say: “Making ten and then teens’ facts.” The strategies can be shared with parents, and the Addition Ladders can be given to them so they can practice them with their children at home.

In my experience with thousands of children, I have found that children overwhelmingly want more challenge in mathematics; they want mastery, but they also want efficient methods. What is important is giving every child a challenge that is personal to them and is attainable with the right amount of effort and practice. Children like to set their own mathematics targets in consultation with and with scaffolding from their teacher or tutor. These targets should be specific and achievable within a period of a few weeks. This motivates students to work hard and keep pushing themselves towards new goals.

During the “Tool Building Time” of daily lessons, the teacher should identify an ‘arithmetic fact of the day’ or adding a particular number to other 10 numbers (using the Addition Ladder). She should ask that fact orally and in multiple forms from all children. This process should be repeated until the fact(s) has been mastered orally. Then give them 20 facts on a sheet of paper (randomly organized and including the ten facts just practiced interspersed with previously mastered facts). The goal is one minute. At the end of one minute, children put a marker and continue till finished. The teacher reads the answers and children correct their papers. Children, in turn, give a correct strategy for a wrong answer.

One of the important elements in helping a student to memorize arithmetic facts is the questioning process and the type of questions the teacher asks. It works in the following way: Once the teacher has chosen an arithmetic fact, then she asks children the various forms of that fact.

The nature of questions to be asked should be at the level and competence of the individual student. For example, if you ask a student the question: what is 8 + 7? and the next student has difficulty with most addition facts, then the teacher should just ask: “what did the previous student give as the answer for the questions: 8 + 7? “What question did I ask and what was his/her answer?” “You know 7 + 7 is 14, then what is 7 + 8?” “What is 8 + 7?”

If the student is able to repeat it, then ask: if 8 + 7 = 15, then what is 7 + 8 = ? A child who already knows 8 + 7, then the teacher, in the next turn, should ask the child: 18 + 7 = ?, or 48 + 7 = ?, etc. She can then ask that student questions like this, out of turn also. This will help him to participate in the lesson and increase his confidence and deepen his understanding. This is an example of differentiation.

To develop flexibility in arithmetic facts, I suggest the following types of questions to be asked for each arithmetic fact.

7 + 4 = ______; 7 + ____ = 11;        ____ + 4 = 11; ____ + ____ = 11; ____ = 7 + 4; 11 = 7 + ____; 11 = ____ + 4; 11 = ____ + ____

Once a student has mastered the family of facts related to a given arithmetic fact orally, then the teacher could give an Addition Fact Ladder to a pair of students and ask them to practice it orally by quizzing and helping each other.

The speed is also achieved by using the Visual Cluster cards. The teacher displays one card after another and asks questions using different strategies (see the list of Addition Strategies below):

  • What is this card?
  • What is one more than this number?
  • What number will make it 10?
  • What is the double of this number?
  • What is one more than the double of the number? Etc.

This is repeated for the whole deck of Visual Cluster Cards. The same questions are repeated for making ten, doubles, double and one more, two more, making 9, making 11, etc.. After the practice has been done for each strategy individually, the teacher displays two cards and asks:

  • What are these two numbers?
  • What is the sum of these two numbers? If the student gives the correct answer, then ask:
  • What strategy did you use?
  • Can you use another strategy?
  • Any other strategy? Etc. If the child doe not give the correct answer, the teacher helps him to arrive at the appropriate strategy to be used.

When To Automatize Arithmetic Facts?
To achieve mastery of numeracy by the end of fourth grade, mastering of fractions in fifth and sixth grade, and integers by the end of sixth grade, children should have mastered 45 sight facts by the end of Kindergarten; 100 addition facts by the end of first grade, 100 subtraction facts by the end of second grade, 100 multiplication facts by the end of third grade, 100 subtraction facts by the end of fourth grade. This should be with understanding, fluency, and applicability.

[1] Available electronically free from the Center for Teaching/Learning of Mathematics

[2] See Games and Their Uses by Sharma (2008).

What Does it Mean to Master Arithmetic Facts?