# Three approaches to one-place addition and subtraction: Counting strategies, memorized facts, and thinking tools. Liping Ma

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2 Counting counters and memorizing facts : Two approaches often used in the US Counting counters The counting counters approach focuses on helping students to get the answer for addition and subtraction by counting counters. Besides students own fingers, in early elementary US classrooms one can see various kinds of counters: chips of different shapes and colors, blocks of different sizes and materials, counters shaped like animals or other things that students like. Also, the dots on the number line posted on the classroom wall are often used as counters. In the counting counters approach, the main task of instruction is to guide students to improve their counting strategies. As we know, computing by counting is the method children use before they go to school, it s their own calculation method, and the counters most often used are their fingers. As research points out, the more experience children get with computation, the more they improve their strategies. i A concise summary of the strategies that children use is given in Math Matters (Chapin & Johnson, 2006, pp ). The process of optimizing addition strategies goes from counting all, to counting on from first, to counting from larger. For example, to calculate =? by counting all: show 2 fingers and show 5 fingers, then count all the fingers, beginning the count with 1. A more advanced strategy is counting on from first. Instead of counting from 1, a child calculating =? begins the counting sequence at 2 and continues on for 5 counts. In this way, students save time and effort by not counting the first addend. An even more advanced strategy is counting on from larger: a child computing =? begins the counting sequence at 5 and continues on for 2 counts In this way, students save even more time. Because we only have ten fingers, the strategy of counting all can only be used for addition within 10. But the other two strategies can be used to calculate all 1-place additions. There are also counting strategies for subtraction: counting down from, counting down to, counting up from given (Chapin & Johnson, 2006, p. 64). Draft, June 21,

3 Figure 1. Student using counters, student using fingers, counters used in school, grade 2 textbook. Counting strategies can be used to solve all 1-place additions and subtractions. However, the ability to use these strategies does not ensure that students develop the ability to mentally calculate 1-place additions and subtractions. The more proficient students become with counting strategies, the more they likely they are to develop the habit of relying on these methods. This habit may last for a long time. Thus, these methods may hinder the growth of the ability to calculate mentally. And, unless elementary mathematics education decides to give up the goal of teaching students the algorithms for the four operations, the ability to calculate mentally is necessary. Imagine that when students are doing multi-place addition and subtraction or multiplication and division, they still rely on counting fingers to accomplish each step of addition and subtraction. The whole process will become fragmented and lengthy, losing focus. Memorizing facts The goal of memorizing facts is to develop students capability to calculate mentally. The content that students are intended to learn includes 200 number facts. As we know, the ten digits of Arabic numbers can be paired in a hundred ways. The hundred pairs with their sums (0 + 0 = 0, = 1,,,,, = 18) are called the hundred addition facts. Corresponding to the hundred addition facts are the hundred subtraction facts (0 0 = 0, 1 0 = 1,..., 0 = 9 9). These two hundred facts, which cover all possible situations, are called number facts. If one can memorize all these facts, then one is able to do mental calculations with 1- Draft, June 21,

4 place numbers. In this approach, the instructional method is to ask students to memorize the two hundred facts. The number facts are shown to students with counters, then students are drilled. Drills are done in various ways. Mad Minute is an instructional practice often seen in US classrooms. A display shows 30 or 40 problems and students are asked to complete as many problems as they can in one minute. Each problem is written with the numbers in columns similar to the form used in the algorithms for multi-place computations. Each sheet of drills may have a particular characteristic: all addition problems, all subtraction problems, or a mixture of the two. The numbers in the problems may lie within a particular range. For example, the left side of Figure 2 shows addition and subtraction problems within 10. There are also more interesting drill sheets (shown on upper right of Figure 2). A newer, more efficient variation of this method is to group related facts together, for example, grouping addition facts with the sum 7, by forming a 7-train for students to memorize (shown on lower right of Figure 2). Figure 2. As a way to develop students ability to do 1-place calculations mentally, the rationale for the memorizing facts approach is straightforward. However, its two elements, memorizing and considering 1-place calculation as 200 facts, have some defects with respect to activating students intellects. First, although memorizing is an important intellectual function, children are considered to be especially good at memorizing. However, if students only memorize, then other types of Draft, June 21,

9 Fifth, to help those students who haven t mastered addition and subtraction with sums of 4 or 5 to master this skill. Because of differences in family background or everyday environment, some students may not yet be comfortable in calculating these sums. In summary, the task of teaching the first stage Calculations with numbers between 1 and 5 is to take stock of the knowledge of quantities that students already have. Through this stocktaking, students develop a clearer understanding of this knowledge, at the same time establishing a conceptual foundation for their future mathematical learning. At this stage, there is no extrapolation. However, the learning tasks of this stage draw on students prior knowledge in order to build a foundation for extrapolation in later stages. Stage 2: Learn how to extrapolate with thinking tools The second stage is addition and subtraction with sum or minuend between 6 and 9. In contrast with the numbers from 1 to 5, the numbers 6, 7, 8, 9 are much less familiar to students. Thus, mental calculation with these numbers starts to feel harder to handle for students, although this is not uniform. Facing the calculations in this stage, many students unconsciously use their fingers to help. At this moment, instruction comes to a fork in the road: letting students acquire the method of counting on fingers or leaving them to use a new method: using mathematical thinking tools to extrapolate from known to unknown. In other words, what aid should students use: fingers as counters or mathematical thinking tools? Extrapolation chooses the latter. The five teaching tasks accomplished at the previous stage have prepared students to learn extrapolation at this stage. The difficulty at this stage is that students are not familiar with the four numbers, 6, 7, 8, 9. To dissipate this difficulty, the prior stage prepared students in two ways. First, based on students prior knowledge that 2 is composed by 1 and 1 and 3 is composed by 2 and 1, by revealing that 4 is composed by 3 and 1, 4 is composed 2 and 2, and 5 is composed by 3 and 2, the concept of a big number is composed of smaller numbers was established. Second, from the tasks they did at the previous stage, students became more familiar with the numbers from 1 to 5. Thus, although students are not familiar with 6, 7, 8, 9, they have established the concept of a large number is composed of smaller numbers, and they are very familiar with the smaller numbers that compose 6, 7, 8, 9. They have a foundation for understanding these larger numbers, lessening the initial difficulty. For example, a student might feel that 6 is unfamiliar and hard to comprehend. However, when he or she sees that 6 is composed of 5 and 1, of 4 and 2, and of 3 and 3, because the numbers from 1 to 5 are already familiar, 6 becomes less unfamiliar. The student is one step closer to comprehension of 6. Also, students already learned that because 3 is composed of 2 and 1, therefore = 3. Thus, when they see 6 is composed of 5 and 1, 6 is composed of 4 and 2, 6 is composed of 3 and 3, without counting, they can extrapolate that = 6, = 6, = 6. Extrapolation based on the composition of numbers relies on the comprehension of previous numbers. At this stage, each new larger number is treated separately, in increasing order: sums and minuends of 6, then of 7, 8, and 9. Draft, June 21,

12 On the one hand, the numbers involved are large, so large that they are beyond what students can comprehend. On the other hand, the notation involved is positional notation, involving the abstract concept that the place of a numeral changes its value. Indeed, the difficulty of this stage is an intellectual height that most first graders are not able to scale in one attempt. However, at the previous three stages, instruction led students to use extrapolation to climb, step by step, dissipating the difficulties of these stages. At this stage, along with the knowledge gained at the first three stages, extrapolation requires two more thinking tools: understanding of the meaning of the numerals from 11 to 20, based on positional notation, and the associative law. With these tools, students can solve the problems of this stage, using their minds, without fingers or other counters. For example, based on the knowledge that students already have, they can tell that the answer for =? will be larger than 10. But, what is the exact number? By understanding the associative law and by knowing the composition of 9 and 2, they can separate a 1, which with 9 composes a ten, from 2; make a ten, with 1 remaining. Based on positional notation, one ten and one one is written as 11. Therefore, = 11. From this, by using the commutative law and subtraction as the inverse operation of addition, students can extrapolate = 11, 11 2 = 9, 11 9 = 2. Using the same idea, they can extrapolate solutions for all the problems of this stage. Moreover, once they use the associative law to find a solution, students can also use the compensation law to extrapolate further additions and subtractions. For example, from = 11, they can extrapolate = 12, = 13, = 14, = 17, and so on. Each of these equations, with extrapolation, generates another group of equations. The thinking tools and computational skills that students have already gained prepare them to use several ways to extrapolate. When the learning of this stage is accomplished, first graders capacity for mental calculation with 1-place numbers has been achieved. This capacity will contribute to their learning to use algorithms for the four operations with multi-place numbers. Of course, future calculations will reinforce this capacity. Draft, June 21,

13 Overview of the thinking tools The extrapolation approach just described can help students develop the capacity to calculate mentally with 1-place numbers, without counters, as used in China and some other countries. However, this cannot be done instantly, it requires sustained effort from students and teachers for about 20 weeks. vii Figure 3 shows how the various thinking tools are introduced over four stages. The dotted lines separate the four stages. 1-place addition and subtraction Computations: 1 5 Stage 1 Sum or minuend 6 9 Stage 2 Sum or minuend 10 Stage 3 Sum or minuend Stage 4 Associative law of addition The meaning of the numerals Composition of 10 and positional notation Compensation law of addition Subtraction as the inverse operation of addition Commutative law of addition Names of the quantities in an equation Addition and subtraction equation and symbols Composing a number Figure 3. Thinking tools introduced during the four stages In Figure 3, we see that in the process of developing the capacity of mental computation the extrapolation approach leads students to climb nine steps. The instructional content of these steps is thinking tools needed for extrapolation or preparation for the introduction of a thinking tool. Except at the first stage, the thinking tools introduced at a given stage are used at that stage for the learning tasks of that stage and all later stages. Moreover, these thinking tools are useful for not only 1-place addition and subtraction, but will be used throughout elementary mathematics. Draft, June 21,

15 Figure 4. (Moro et al., 1980/1992, p. 9) The title of the lesson is +,, = the symbols needed to connect numbers to make addition and subtraction equations. The section for introducing the new concept is composed of three pairs of pictures, each pair accompanied by an equation. The pictures show objects that are familiar to students. The addition and subtraction equations = 2, = 3, 2 1 = 1 look very simple. However, they are expressed in the standard notation used throughout the world, which is the result of centuries of notational evolution. viii By using the notation and form shown in these equations, students can express their own mathematical knowledge in a way that allows them to communicate with the rest of the world. What students learn in this lesson is only the format for mathematical expressions. There seems to be no new conceptual content. However, students still make significant intellectual progress. To go from the everyday two pieces of cake and one piece of cake make three pieces of cake to the mathematical equation = 3 involves two intellectual leaps. One is from concrete numbers to abstract number. The other is from everyday language to the mathematical language used by the rest of the world. Another meaningful arrangement in the lesson that students might not notice: when they first encounter addition and subtraction equations, the two operations occur in the same lesson. In the previous lessons on addition and subtraction, these concepts occurred independently. In this lesson, a connection between the two operations is unobtrusively represented. It is a preparation for later revealing the common quantitative relationship that underlies addition and subtraction. Draft, June 21,

17 The title of the lesson is Interchanging Addends. The section that introduces the new concept depicts an everyday scene that students might encounter, accompanied by two equations = 3 and = 3. In this section, students don t need to calculate. Their learning task is to observe and discuss the new concept, guided by the teacher. The exercise section includes two groups of problems related to the concept just introduced. The first group of problems leads students to notice that the phenomenon illustrated by the top picture also applies to other numbers: when addends are interchanged, the sum does not change. From the pictures, students can clearly see why there is such a pattern. The second group of exercises are calculations. Students are supposed to extrapolate the solution for each lower equation from the one above it, based on the pattern that they just noticed. This is also a chance for students to check the validity of the pattern they observed. Eventually, after all these intellectual activities, the lesson presents the statement of the commutative law of addition. By learning the commutative law, students acquire a thinking tool that can be used for extrapolation. This tool will be used mainly to extrapolate the sum of a small number and a large number from the sum of the large number and the small number. On later pages of the textbook (pp. 52, 53, 54), many groups of exercises involve finding the sum of a small number and a large number, allowing students to use the commutative law to extrapolate the sum, and appreciate the convenience of this strategy. In the textbook, the thinking tools for calculation at later stages are usually introduced by pictorial problems involving numbers from previous stages, with which students are already familiar. For example, although the lesson above occurs at the second stage, the section that introduces the new concept uses numbers from the first stage. In the exercise section, students can use the new concept to solve problems with numbers at the second stage. This approach has several advantages: 1. the smaller numbers reduce the cognitive load of calculation, allowing students to focus on the new concept; 2. effective use of the knowledge that students already have to support the acquisition of new knowledge; 3. utilizing knowledge that students already have allows them to see new features of that knowledge that deepen their understanding; 4. the new thinking tools students just learned are immediately used for calculations with larger numbers, helping students to solve new kinds of problems. Students can appreciate the significance of these tools and also get a chance to use them. Draft, June 21,

18 There is one more interesting aspect that deserves mention. Although the task of the lesson is to introduce the commutative law of addition from its title Interchanging Addends to the end, in the statement A sum does not change if the addends are interchanged the term commutative law of addition never appears. We can see that the lesson is designed to be intellectually honest but also considerate of learners by not containing anything superfluous. These principles, to efficiently utilize the knowledge that students already have, and to be intellectually honest and at the same time considerate of learners, occur in every lesson that introduces a new concept. Subtraction as the inverse operation of addition Subtraction as the inverse operation of addition is another thinking tool used at the stage of addition and subtraction with sums and minuends between 6 and 9. This concept is introduced in the lesson entitled How to Find an Unknown Addend. Figure 7. (Moro et al., 1980/1992, p. 55) The quantities used in the section that introduces the new concept are the numbers from the first stage. The three pictures are like three cartoon panels. At the top are five jars in a cupboard whose doors are open: three at left and two on the right. This picture illustrates the equation = 5. In the next picture, the lefthand cupboard door is closed, hiding the three jars. Only two of the five jars can be seen. This picture illustrates the equation 5 3 = 2. The bottom picture shows the left door open and the right door closed, hiding two jars and illustrating 5 2 = 3. Draft, June 21,

21 Associative law and single-place addition with sums from 11 to 18 The associative law is very important for helping students to understand addition with composing and subtraction with decomposing. It is also an important thinking tool for extrapolation at this stage. The textbook uses four lessons to introduce the associative law for addition and its application to subtraction: Adding a Number to a Sum (p. 106), Subtracting a Number from a Sum (p. 113), Adding a Sum to a Number (p. 125), Subtracting a Sum from a Number (p. 142). xi This arrangement takes care of different ways in which the associative law is used, beginning with topics that are easy for students and progressing to more difficult ones. The structure of these four lessons is similar: a section that introduces the new concept, composed of three lines of cartoon, each with three panels accompanied by the equations they illustrate. The cartoon topic for addition is birds on a tree and the topic for subtraction is fish in a tank. The exercises of the lesson are groups of calculations that ask students to solve each problem in three different ways. Because adding a sum to a number is key to helping students understand the rationale for addition with composing, the lesson on Adding a Sum to a Number will be used as an example. Figure 9. (Moro et al., 1980/1992, p. 125) Draft, June 21,

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