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Supporting students' additive thinking: The use of equal additions for subtraction

Jennifer Young-Loveridge and Judith Mills
Abstract: 

How can teachers support students' additive thinking? This article focuses on the study of a lesson designed to teach the equal additions strategy for subtraction, in which many teachers, despite having a strong commitment to promoting conceptual understanding, struggled with various aspects of the material and resorted to teaching procedurally. The authors conclude that teachers need to have a deeper, more connected understanding of addition and subtraction in order to develop their pedagogical content knowledge in mathematics.

Journal issue: 

Supporting students’ additive thinking

The use of equal additions for subtraction

JENNIFER YOUNG-LOVERIDGE and JUDITH MILLS

Key points

Additive thinking involves using knowledge of the way numbers can be split into parts and recombined to solve problems more easily. In this study, teachers were observed using the equal additions strategy to help students solve subtraction problems.

The teaching of addition and subtraction observed was far more problematic than anticipated. This appeared to be due to teachers’ lack of deep understanding of addition and subtraction processes, and to the fact that NDP Book 5 did not contain enough information to clarify the underlying rationale behind each lesson.

Teachers need to make a point of understanding the relationships among the strategies students can use to solve addition and subtraction problems in order to develop the deeper and more connected understanding that is needed in order to teach mathematics more effectively to their students.

It is also important for teachers to understand that the lessons in the NDP books are illustrative, not prescriptive.

This paper looks at how teachers can support students’ additive thinking. It reports on a study of a lesson designed to teach the equal additions strategy for subtraction, contained in Numeracy Development Project (NDP) Book 5. Nine teachers were observed teaching mathematics. Despite having a strong commitment to promoting conceptual understanding, many teachers struggled with various aspects of the material and resorted to teaching procedurally. We concluded that teachers need to have a deeper, more connected understanding of addition and subtraction in order to develop their pedagogical content knowledge in mathematics.

Introduction

It is now more than a decade since the Numeracy Development Project (NDP) began. This initiative was designed to raise students’ achievement in mathematics and build teachers’ professional capacity to teach mathematics (Ministry of Education, 2001). Professional development programmes, led by a team of numeracy facilitators, introduced teachers to these key components and provided them with feedback and support in their own classrooms as part of the initiative’s implementation.

A key aspect of NDP is the Number Framework, which describes a series of increasingly sophisticated mental strategies used by students to solve problems. At the lowest level are the counting strategies (stages 0 to 4). Stage 5 marks the transition to the use of “part−whole” strategies to solve problems: addition and subtraction; multiplication and division; and proportion and ratio. Part−whole thinking is a major goal for students and involves learning how to split numbers into parts (“partitioning”) in order to solve problems more easily. For example, 9 plus 5 can be solved by splitting the 5 into 1 and 4, combining the 9 and 1 to make 10, then adding on the 4 to get 14.

This part−whole thinking depends on knowing how the parts make up the whole number. For example, 10 can be thought of as made up of 5 and 5, 6 and 4, 7 and 3 and so on. This is sometimes referred to as “additive composition”, which refers to the way numbers can be split into parts and recombined to solve problems more easily. It is important to remember that the so-called “additive domain” includes both addition and subtraction. Figure 1 shows a range of different strategies that can be used to solve addition and subtraction problems.

The release of The New Zealand Curriculum (Ministry of Education, 2007) and Mathematics Standards for Years 1−8 (Ministry of Education, 2009) has embedded the Number Framework within the system of expected outcomes for all New Zealand students (see Table 1). Thus, by the end of Year 4, most students should be able to split and recombine small numbers (stage 5), and by the end of Year 6 they should be able to choose from a range of mental strategies to solve multidigit addition and subtraction problems (stage 6). This means it is really important for teachers to have a deep understanding of the mathematics they are teaching, how their students’ mathematical ideas are likely to develop and how they can help their students to learn mathematics effectively. In other words, teachers need adequate pedagogical content knowledge in mathematics.

Table 1 shows that by the end of the year’s NDP professional development, fewer than half (38 percent) of the Year 6 students were at stage 6 or higher on the additive domain (Young-Loveridge, 2010). This is well short of the majority (75 to 80 percent) expected at this level. The proportion of Year 5 students at stage 6 was considerably lower (25 percent). The present study looked at what happened in classrooms as teachers used the lesson in NDP Book 5 (Ministry of Education, 2008, p. 38) on using the equal additions strategy to extend students’ repertoire of part−whole strategies for solving subtraction problems (for further details, see Young-Loveridge & Mills, 2010).

Method

The participants were nine teachers of Years 5/6 students from four schools (deciles 2−10). The teachers had a range of classroom experience (two to 25 years) and work with NDP (two to seven years). One teacher from each school had previously worked with the researchers, and that teacher invited the other teacher/s working at the same year level to be involved in the research. Teachers worked with a group of their students judged by them to be ready for instruction in advanced additive thinking (stage 6). The study focused on the equal additions lesson (see Ministry of Education, 2008, pp. 38−39). Each teacher wore an audio-recorder with lapel microphone, and both researchers observed the lesson and made notes.

FIGURE 1 PART-WHOLE (PARTITIONING) STRATEGIES FOR SOLVING ADDITION AND SUBTRACTION PROBLEMS

TABLE 1 THE PERCENTAGES OF STUDENTS IN YEARS 1–8 AT OR ABOVE A PARTICULAR STAGE ON THE NUMBER FRAMEWORK AND THE CORRESPONDING CURRICULUM EXPECTATIONS

In the equal additions lesson, the first scenario presented in Book 5 is:

Problem: Debbie has $445 in her bank account, and her younger sister Christine has $398. How much more money does Debbie have?

Make piles of $445 and $398. Now suppose that Grandma gives Christine $2 to give her a ‘tidy’ amount of money. To be fair, Grandma gives Debbie $2 also. Discuss why 445 – 398 has the same answer as 447 – 400 and then record 445 – 398 = 47 on the board or modelling book.

Book 5 then presents further examples of word problems and equations for use in the lesson.

Results

Examples of effective teaching

We observed that most teachers drew students’ attention to the learning intention right at the beginning of the lesson. Cara taught this particular lesson very effectively, and we observed several key features in her teaching that were not found to the same extent in the others’ teaching.

Personalising the problem for the students

Cara changed the names of children in the problem to those of students in her group and got them to act out the scenario, with different people responsible for each denomination of “play” money ($1, $10, $100). With subsequent problems, the names of other students in the group were used, and responsibility for managing the money was rotated, so that all students became involved in helping to solve the problems.

Fairness

Cara emphasised the idea of fairness as a justification for giving the same amount of money to the two children. She talked about Grandma giving Christine $2 to round up the money in her bank account to $400 (a tidy number), but to be fair to Christine’s sister, Debbie, Grandma also gave Debbie $2, taking the money in her bank account up to $447.

Proving that the difference remained constant

Cara checked that the children understood that the difference between the larger quantity and the smaller quantity after equal additions was the same as before:



Cara: $47 [referring to the difference between 447 and 400]. Easy to see, isn’t it? Would that difference between these two numbers [pointing to 445 and 398] have been 47?

Some children disagreed, expressing doubt that the magnitude of the difference was unaffected by the equal additions process. When Cara asked about the difference between the original two numbers (445 and 398), at least one child thought that the difference was now larger by two (49 instead of 47).

Cara then got the children to check the difference by building the smaller quantity up until it equalled the larger quantity, keeping track of how much money they added. The student with $398 was given $2 by the bank, and that amount was recorded. The resulting 10 $1 notes were traded for one $10, which was added to the pile of nine $10 notes. The resulting 10 $10 notes were then traded for one $100 note, which was added to the pile of three $100 notes to make $400. The students were then asked how much money was needed to make $445. All agreed that $45 more was needed to build up from $400 to $445. Cara then commented:

Let me get this straight. The difference between this amount here [398 and 445] and this amount here was $47. Nana gave you both $2 more, and the difference between those two amounts [400 and 447] is …

One child responded, “It would be exactly the same.”

Later one child raised the possibility of turning the larger quantity into a “tidy number” rather than the smaller quantity. Cara supported the child in investigating whether it mattered which number was turned into a tidy number. Eventually the child recognised that only when the smaller quantity was a tidy number did the problem become much easier to solve (see Figure 2).

FIGURE 2 COMPARING THE PROCESS FOR “TIDYING” THE LARGER AMOUNT (ABOVE) AND THE SMALLER AMOUNT (BELOW)

Not all teachers realised that making the smaller quantity tidy was the most efficient use of the equal additions strategy. Dot commented:

They’re both right so it doesn’t matter which number we turn into a tidy number.

However, adding 5 to both numbers (rounding 445 to 450 and 398 to 403) meant that subtracting 403 from 450 involved regrouping, which is harder than subtracting 400 from 447.

Barriers to effective teaching

Consistency of language and mathematical structure

Although it was not clear whether Cara understood about different problem structures, she was very careful in her use of mathematical language and special terminology with the children. Other teachers confused compare with separate problems.

For example, Dot took the first problem (a compare or difference problem) and turned it into a separate problem by saying to her students:

I had $445, right? [D4] asked me if she could have a loan of $398 and … I said sure. How much money did I have left over? Right, I want you to think about the tidy numbers ...

D1 was concerned that if 2 was added to one number, it needed to be subtracted later. Dot explained:

[D1], what I think you’ve been confused with is if we did it to one of these numbers, if we added 2 to one number, then yes, we do have to take it away, but we did it to both numbers. If we just added 2 to 398 and left 445 the same, then yes, we would have to take that 2 away, but because we do the same treatment to both numbers, the gap remains the same.

Several students were obviously confused, so Dot ended the lesson saying they had run out of time for further explanations. When she was asked after the lesson what she would do later to follow up on the lesson, she did not mention clarifying the confusions that emerged during the equal additions lesson.

To be fair to the teachers, there is no mention in Book 5 of different problem structures for addition and subtraction, or that the equal additions strategy is about comparing two quantities rather than operating on (i.e., subtracting from) just one quantity.

Ben began by asking the students, “What sort of problem are we looking at?” Student B1 suggested that the problem should be 398 +  = 445, an addition structure with a missing addend (i.e., join change unknown). Ben would not accept this addition structure because the learning intention specifically referred to subtraction. Another child suggested, “445 take away 398”, and Ben affirmed that response.

FIGURE 3 ADDITION AND SUBTRACTION WORD PROBLEM TYPES ADAPTED FROM THE WORK OF CARPENTER ET AL. (1999, P. 12) AND FUSON (1992, P. 246)

Ben’s unwillingness to accept the missing addend structure as a possible representation for the problem may have reflected a rather inflexible understanding of addition and subtraction, and as a result he missed a nice opportunity to exploit the inverse relationship between addition and subtraction that leads to the reversibility strategy shown in Figure 1. Alternatively, he may not have wanted to take time away from his main focus of developing the students’ understanding of equal additions. The incident does highlight the need for teachers to understand the relationships between the strategies presented in Figure 1. If teachers simply follow the lessons in the NDP books, the links between different strategies may be overlooked.

Ben then asked whether adding 2 to both numbers would change the answer. Some students thought it would increase the answer by 4, but others believed it would be the same. Ben asked those students to explain why. There was some discussion but it did not produce what Ben wanted, so he explained:

Okay, with subtraction, we’re really looking at the difference between these two numbers, so the difference between 445 and 398 is 47. So we’re just looking at difference, so the numbers here, you can change the numbers either way and it’s not going to affect the outcome. Does that make sense?

At least one child agreed, but another was concerned about what happens if different amounts are added to different numbers. Ben responded by saying that it was a good question, and then gave them another problem, with a separate structure. This led to more discussion, but when Ben pressed the students to check their understanding, it was not clear whether they in fact understood what had happened.

The importance of recording the process clearly

Teachers differed in the clarity of their recording. Ann recorded the equal additions process vertically so that the adjusted equation was shown horizontally below the original equation (in the same way as the equations in Figure 2).

In the equal additions lesson, the adjustments were so small (less than nine) that place value was not an issue. However, sometimes equal additions involves adding a quantity to more than just the “Ones” in a number (e.g., 445 – 287 = ). We noticed the importance of recording each rounding step separately (e.g., rounding up the “ones” digit to the nearest ten [7 + 3 = 10], then rounding the “tens” to the nearest hundred [290 + 10 = 300], and summing the two amounts to find the total difference [3 + 10 = 13]) (see Figure 4). It is important that these two amounts are recorded separately. We noticed some teachers simply inserting subsequent digits to the left of the one before (e.g., 3 became 13). This made it difficult to go back and work out where the “13” came from. That is important for equal additions because the “13” needs to be added to the minuend as well as the subtrahend in order to keep the difference between the two amounts constant.

A common problem for many teachers was to allow the steps in recording addition or subtraction to run on, so that the equation was no longer true and did not balance. Teachers (and students) often see the equals sign as signalling that the answer follows, rather than as a statement about the equivalence of both sides of the equation. For example, in the problem:

If John has $360 and Troy has $298, how much more money does John have than Troy?

students tended to first record 298 + 2 = 300. Then, instead of starting a new line to add 60 to 300 (300 + 60 = 360), they recorded the second step straight after the first, violating the requirement of equality within the equation. They recorded the process as:

298 + 2 = 300 + 60 = 360

Hence, they failed to recognise that 298 + 2 does not equal 360.

FIGURE 4 STEPS "TENS" SHOWING THE RECORDING PROCESSES FOR EQUAL ADDITIONS IN TWO STEPS (FOR “ONES” THEN FOR “TENS”) USING FULL PLACE-VALUE RECORDING VERSUS ABBREVIATED RECORDING SHOWING INSERTING THE DIGIT “1” INTO THE “TENS” PLACE TO THE LEFT OF THE DIGIT “3”

Ensuring a maximum of nine for any
place-value denomination

Care is needed in working with the materials. We noticed that not all of the teachers adhered to the rule of no more than nine of any single denomination of money. For example, Ed arranged for the student with eight $1 notes to be given two $1 notes, making 10 altogether, but did not ask the student with the 10 $1 notes to exchange them at the bank for one $10 note before giving them to the “tens person” (holding nine $10 notes). The “tens person” then had a mixture of denominations—potentially a very confusing situation for students trying to understand place value.

Ensure students always refer to total value (not face value)

Another important place-value issue was the need to ensure that students describing addition or subtraction refer to the total value of the digits they are operating with—not their face value. For example, 900 – 298, adjusted to 902 – 300, should be described as “300 from 900” rather than “3 from 9” (then add two zeros before adding on the two from 902).

Discussion

The teaching of addition and subtraction we observed was far more problematic than we had anticipated. This may have been because teachers’ understanding of addition and subtraction processes was not deep, and because Book 5 did not contain enough information to help them understand the underlying rationale behind each lesson. There was no indication that the problem structure for problems presented in the equal additions lesson was subtraction as difference (the compare structure) rather than take away (the separate structure). Teachers seemed unaware of different problem structures for addition and subtraction, categorised according to the types of action or relationships described (Carpenter, Fennema, Franke, Levi, & Empson, 1999).

Problem structure

The equal additions strategy provides students with an elegant solution strategy, suitable for solving compare difference unknown problems. It is also valuable for solving separate result unknown problems (e.g., 9 – 6 = ), and any other problem type with a part unknown (e.g., join change unknown). However, it was interesting to note that very few students chose to use the equal additions strategy in the post-lesson assessment. This may have been because their teachers did not appreciate that the lesson was about understanding subtraction as difference (rather than take away). Compare (i.e., difference) problems involve the static comparison of two quantities, whereas separate (i.e., “take away”) involves the action of removing part of one amount. Teachers in this study instead focused their attention on making tidy numbers. Hence neither teachers nor students appreciated the suitability of equal additions for solving subtraction with numbers close to a whole decade or century. Unfortunately, students simply learned a rule for adding the same amount to both numbers and rounding one quantity to a tidy number, making the problem easier to solve. This is quite contrary to the goal of NDP, which is to teach mathematics conceptually rather than procedurally.

In the post-lesson assessment, most students continued to use bridging through ten/hundred or reversibility to solve the problems. In fact, students tended to use their favourite strategy for all problems, regardless of its suitability. This raises questions about whether students are being empowered with a flexible range of strategies, or whether the traditional written algorithm has simply been replaced by other algorithms applied in an equally mindless way (Kamii, with Livingston, 1994; Pesek & Kirshner, 2000; Skemp, 2006).

It might have been valuable to let students use their preferred strategies initially, and then compare these to equal additions, discussing the simplicity and efficiency of each strategy. We would like to have seen far more discussion among students about their solution strategies (see Whitenack & Yackel, 2002). Hunter (2010) argues that students need to be encouraged to question each other to check their understanding and to convince each other about their chosen strategy. Teachers tended to control their lessons quite tightly, giving little power to the students, although this may have been because they wanted to ensure the lesson went the way they had planned in front of the researchers. We would also like to have seen students being given more opportunities to record their solution strategies in the group recording book instead of teachers doing it for them.

Teachers tended to stick very closely to the lesson description in the book, beginning with the difference between two three-digit numbers. None began with single-digit quantities to get across the initial idea of difference by comparing two quantities. None used materials such as Unifix blocks to model two towers of different heights (e.g., 5 versus 7) to show how the difference in height between the two towers would be unaffected by adding two blocks to each (7 versus 9). Only one teacher (Gail) used a number line to show how the distance between two numbers on a number line is constant provided the same quantity is added to both numbers. A cardboard strip and clip-on pegs could be used to record the initial difference between two numbers, then used again to check that the difference has not changed after the same quantity was added to both numbers. A piece of string would work just as well.

Although Book 5 is reasonably substantial, it contains few lessons to deepen students’ understanding of part−whole strategies and to give them a range of strategies for solving addition and subtraction problems (to support advanced additive part−whole thinking). This is a particular problem for teachers who treat the NDP resource books as prescriptive rather than illustrative. We have seen some teachers use the equal additions strategy for teaching addition even though it is not appropriate to do so; for example, when adding 18 + 25, they added 2 on to both the 18 and the 25, arriving at a final answer of 47 instead of 43. It appears that some teachers do not understand the relationships among the strategies in Figure 1, which give rise to balancing in the case of addition (17 + 15 = 20 + 12 = 32), and equal additions in the case of subtraction (445 – 398 = (445 + 2) – (398 + 2) = 447 – 300). Equal additions works for subtraction because the focus is on difference, but equal additions does not work for addition: balancing is the corresponding strategy for addition.

Notable omissions from Book 5 are the theoretical underpinnings and problem structures for addition and subtraction problems (see Carpenter et al., 1999, p. 7). These are important for teachers to help them develop a deep and connected understanding of the mathematics required for teaching with a conceptual emphasis, and to enhance their pedagogical content knowledge in mathematics (Ball, Thames, & Phelps, 2008).

Our finding that teachers need to understand the relationships among the strategies presented in Figure 1 is consistent with the work of several writers who have emphasised the importance to students of appreciating the structure of mathematics (Mulligan & Mitchelmore, 2009; Young-Loveridge & Mills, 2009). Writers who have focused on the use of structured materials argue that they make it possible for students to create mental images of the problems and use these images to help them solve problems before they develop a full understanding of the abstract properties of numbers (e.g., Pape & Tchoshanov, 2001; Wheatley, 1991).

The NDP professional development programme for teachers working at the Years 5/6 level lasted little more than one year. That may have been enough time for teachers to get their heads around the components of the NDP approach (Number Framework, assessment tools, teaching model and published resources), but it is questionable whether teachers were able to develop a deep and connected understanding of the complexity of addition and subtraction, and to shift from a calculational to a conceptual orientation to mathematics (Philipp, 2007).

It is generally assumed that learning to add and subtract is a straightforward process. This overlooks the enormous complexity of these operations. There is little appreciation of the challenge for teachers in understanding, at a deep conceptual level, what addition and subtraction with whole numbers is about. As Bahr and de Garcia (2010) note, “elementary mathematics is anything but elementary”.

Conclusions

Several key recommendations emerged from this research:

Teachers need to have a deeper, more connected understanding of addition and subtraction in order to develop their pedagogical content knowledge in mathematics. Figure 1 provides a succinct summary of a range of possible strategies that students can use to solve addition and subtraction problems. By making a point of understanding the relationships among these strategies, teachers can develop the deeper and more connected understanding that is needed in order to teach mathematics more effectively to their students.

Teachers need to understand the different types of problem structure for addition and subtraction. In particular, they need to understand that subtraction can be about comparing two quantities to find the difference between them. The compare structure for subtraction is about difference—it is not about take away or separating a quantity into parts. Equal additions is probably easier to understand initially within the context of compare problems. Only later, once students understand the idea of difference between two quantities, should teachers extend the use of equal additions to separate problems that involve taking a part away from a single quantity.

Teachers need to understand that the lessons in the NDP books are illustrative, not prescriptive. It would be useful for teachers to undertake something like the Lesson Study process that occurs in Japan, where teachers meet to refine and improve particular lessons and share their work with others (e.g., Post & Varoz, 2008; Takahashi & Yoshida, 2004).

It is important for teachers to ensure that all parts of an equation are equal. To ensure this, the steps involved in solving a problem need to be recorded separately, on a new line for each step.

Students should be encouraged to discuss the usefulness of particular strategies to solve particular problems, focusing on making connections between different strategies and considering the efficiency of particular strategies. They also need to question their peers and justify the strategies they have used, as part of good mathematical communication.

The emphasis needs to be on building conceptual understanding rather than training students to use particular procedures. Teachers should be alert to the dangers of simply replacing one algorithm with another.

The equal additions strategy should be introduced using small (single-digit) quantities before moving to larger numbers. The process needs to be modelled initially with materials such as Unifix (linkable) blocks to demonstrate how adding the same number to two quantities does not change the difference between them.

The number line should also be used to show the constancy of the difference between two numbers when the same quantity is added to each number. A cardboard strip with clip-on pegs could be used to record the initial difference between two numbers, then used again to check that the difference has not changed after the same quantity is added to both numbers.

Students should be encouraged to investigate the efficiency of tidying the smaller number (subtrahend) versus tidying the larger number (minuend) in equal additions.

When teachers use “play” money to model addition and subtraction, they need to emphasise that there must be a maximum of nine notes of any one denomination, and whenever there are 10 notes they should be exchanged for one note of the next denomination (10 $1 notes for one $10, 10 $10 notes for one $100 note, etc.).

Students should be encouraged to refer to a digit in a multidigit numeral as its total value rather than its face value (e.g., in 398, the 9 should be referred to as 90, and the 3 as 300).

The so-called “modelling book” could be renamed the “group workbook” to avoid implying that it is a place where the teacher provides good examples or “models” to students. Students should be encouraged to take responsibility for the group workbook to record their strategies and the discussions they have about the efficiency of those various strategies.

Teachers need to understand how number strategies apply differently to addition and subtraction (e.g., equal additions is used only with subtraction).

Acknowledgements

Sincere thanks to the students, their teachers and schools for participating so willingly in this research. The views presented in this paper do not necessarily represent the views of the New Zealand Ministry of Education. This paper is a modified version of a paper by Young-Loveridge and Mills published in the 2009 Compendium.

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JENNY YOUNG-LOVERIDGE is an Associate Professor at the University of Waikato. She teaches mathematics education, research methods, and human development. She has directed several large research contracts, developed Checkout/Rapua (a shopping game to assess numeracy as part of the School Entry Assessment kit sent to all primary schools), and been one of the researchers involved in the evaluation process for the Numeracy Development Projects. Jenny’s research focuses on the development of students’ mathematical thinking and ways this can be enhanced.

Email: jenny.yl@waikato.ac.nz

JUDITH MILLS is a numeracy adviser and lecturer in mathematics education at the University of Waikato. She has an interest in the pedagogical practice of classroom teachers in relation to the teaching of numeracy. Her doctoral research examines teachers’ content knowledge in mathematics together with their beliefs about teaching mathematics and the role these play in ongoing professional learning and development designed to raise students’ mathematics achievement.

Email: judith@waikato.ac.nz