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, 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 conceptin 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 3? Twenty-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:
- The whole deck is divided into two to four equal piles (depending the number of players).
- Each child gets a pile of cards. The cards are kept face down.
- 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.
- 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.
- The first person with an empty hand loses.
- 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.
- The whole deck is divided into two to 4 equal piles (depending on the players.
- Each child gets a pile of cards. The cards are kept face down.
- 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.
- 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.
- 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
- The whole deck is divided into two to four equal piles (depending on the number of players.
- Each child gets a pile of cards. The cards are kept face down.
- 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.
- 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.
- 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.
See previous posts on Numbersense; Sight Facts and Sight Words; What does it Mean to Master Arithmetic Facts?, etc.
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).