Fostering remainder understanding in fraction division.

Zembat, Ismail O.

Most students can follow this simple procedure for division of fractions: 'Ours is not to reason why, just invert and multiply.' But how many really understand what division of fractions means--especially fraction division with respect to the meaning of the remainder. Think about the 'bags problem' given in Figure 1, and its solution, using only diagrams.

* What is the remainder (amount of sugar left) in this problem?

* How does the remainder relate to the divisor (amount of sugar per bag)?

* How the divisor, the remainder, and the quotient (total number of bags to be made) all are related?

The purpose of this article is to provide an instructional method as a way of scaffolding and understanding the issues regarding division of fractions for both teachers and elementary students. The method has the potential to inform teachers in their work with students' learning about fraction division. Figure 1. A contextual problem that can be modelled by 2 1/2 / 2/3. Karen has 2 1/2 kilograms of brown sugar and she wants to put them in bags. If each bag can take up 2/3 of a kilogram of brown sugar, what are her options as to how many bags she can use to store the sugar? Solve the problem using diagrams only.

Consider the bags problem from Figure 1, and its mathematical analysis given in Table 1. The overall goal in this problem is to find the number of 2/3 kilogram sugar bags (divisors) to be made from 2 2 kilograms sugar (dividend) at hand. Achieving this overall goal based on drawing a diagram only requires attaining the sub-goals detailed in Table 1. Each given sub-goal and the required working out can be considered as the unique way to be followed by students in solving fraction division problems. The use of diagrams may limit students mental or physical actions in a predictable way for this type of problem.

A crucial understanding required here is the multiplicative coordination (hereafter, referred to as MC), among divisor (2/3 kilogram bags), remainder (3/6 of a kilogram of sugar), and quotient (3 3/4 bags). In other words, understanding fraction division requires focusing on what is being counted, that id, the number of bags, and what the units in the drawings represent (bags, kilograms of sugar), in coordination with the overall goal. That being looking for total number of divisors within dividend.

This kind of an analysis would also be helpful to teachers as a first step in identifying the mathematical goal to be targeted, and the activities to be designed for teaching fraction division. The next step is to decide on an instructional sequence that would help students conceptualise such a MC which is detailed next.

Developing a solid understanding of remainder

An instructional sequence was established as part of a teaching experiment to assist two prospective elementary teachers, Nancy and Wanda (pseudonyms), develop a solid understanding of the MC within fraction division. The given tasks in the applied instruction required prospective teachers (hereafter, referred to as students) to use their available understandings of fractions, whole number division and referents. This sequence included working with diagrams in the following sequential settings: contextual whole number division settings, context-free whole number division settings, contextual fraction division settings, and context-free fraction division settings (refer Figure 2 for samples from each setting/class). Figure 2. Sample problems used in all classes. Sample problem from class #1 A 10-acre farm will be ploughed and sowed with wheat. If every day 3 acres of the farm is being ploughed and sown, what are the farmer's options as to how many days he can spend on this work? Solve the problem using diagrams. Sample problem from class #2 Karen has 6 1/2 kilos of brown sugar and she wants to put them in bags. If each bag can take 3/4 of a kilo of brown sugar, what are her options As to how many bags she can use to store the sugar? Sample problem from class #3 a) Solve the following problem using the given diagram: 9/5 / 2/3 b) What is the left over peice in this problem? c) If you were to find the solution to the same problem using your calculator, what would you expect as an answer? How is it related to the answer you have from your solution?

During the instruction, students were not directly taught. Instead, they were continuously probed to focus on "why". For all the given problems in the sequence, the students initially worked on the problems and solved them independently. This was followed up with a discussion to assist the students to reflect on what they had done. The reasons and the pay-off for following this sequence, as well as a description of students' developing understandings and how each setting contributed to those understandings, will be discussed in the following subsections.

1. Moving from contextual to context-free whole number settings

Since the students already knew whole number division, the instructional sequence started with three whole number division problems (see the sample problem from Class #1 in Figure 3). Once the meanings of quotient-only result and quotient-with-remainder result are negotiated, the students worked on the farm problem through the method of using a diagram. As a result of this work Nancy wrote "3 days and 1 acre left" whereas Wanda's worksheet showed "days." Once they generated their individual solutions, upon my questioning Wanda stated, and Nancy then supported the rationale as:

Wanda: [...] three acres can be done in one day so I marked off three acres for day 1, and then I marked off three acres for day 2, and then three for day 3. I saw there was one acre left out of the three that would fill a whole day. So that was one out of three, so one third.

Once they solved all the problems in this manner, their responses were recorded on the board, a sample of which is illustrated in Figure 3. Then students were asked about the role of those two answers for each problem.

The purpose here was to orient the students' thinking toward the commonality among all given problems, and to have them reflect on the relationship between a fractional part of the dividend and the remainder like 0.5 bags correspond to 1 kg of flour left. Therefore, the question was posed about how each of the answer pairs was related. As a teacher, this contained a desire to have the students reflect on MC. The following dialog illustrates students' approach and reflection:

Nancy: For the first one, you can either fill up foru and a half bags, or you fill four bags and have a kilogram of flour left over but that one-kilogram can fill up half of a bag. So you still have like the same amount on each side.

Teacher: [asking Wanda] How does it apply to the second one?

Wanda: Because you have three days that you can fill up total, like fully with ploughing acres of land. But you'll have one acre left over which you can either just like leave it or it'll take this amount of day [referring to one-third] to do that.

As a result of searching for commonality among those problems and solutions (for example, 0.5 bags refer to 1 kg of flour, 1/3 day refers to 1-acre land, etc.), they understood with the help of the context that the remainder (for example, 1 kg of flour) and the fractional part of the quotient (e.g., 0.5 bags) refer to the same quantity, but might be expressed as a different magnitude (0.5 versus 1) and different units (kg versus bags). It was important for the students to reflect on these two types of answers to understand the MC; however, this was all done in context.

To separate students' thinking from context-dependency, during the same class, I asked the students to determine the remainder for the division problem 1354 / 38 using a scientific calculator only. They found 1354 / 38 = 35.6315 and considered 0.6315 as the remainder itself, which was the evidence for me as a teacher that students' understanding of MC was limited to work with contextual clues and not solid enough to think about fraction division in context-free settings yet. With direct guidance and encouragement, students then generated a contextual problem that could be modeled with 1354 / 38 = 35.6315 and started thinking through that problem. Together, they formulated the word problem: "It's Halloween and we are going egging. We use 38 eggs per house. How many houses can we egg?" Through referring to the referents in this contextual problem they were able to identify that:

Nancy: So it would be 0.6315 times 38 eggs per house.

Teacher: Because?

Nancy: Because 38 eggs for a house but you want to know 0.6315 of that house.

Wanda: What is it of for 38, 0.6315.

Nancy: Which is 38 eggs. Does that make sense? This [pointing to 0.6315] is your part of the house and this [pointing to thirty five] is your whole house. So and you want to find out 0.6315 of 38. So you multiply that by 38 and that'll give how much of that 38 you'll need or will be able to egg that part of the house.

They were now able to pay attention to the involved relationships through referring to referents; eggs, houses, eggs per house. In fact, students' work in context paved the way for understanding the MC in a context-free problem.

The pedagogical issue at this point was to move students from thinking effectively in contextual problems to context-free problems. For this purpose in the follow-up class, students were asked to solve 1379 / 28 as if they were using diagrams only after finding the answer with a scientific calculator. Note that their previous attempt to solve a similar problem, 1354 + 38 = 35.6315, did not involve using diagrams. To solve 1379 / 28 they initially identified their goal as "how many 28s are in 1379" and found 49.25 as an answer using a scientific calculator. Then, when asked about the remainder, they wanted to apply some context to 1379 / 28 and reason within the context. This was however constrained by the directive: "No context." Such an approach provided an opportunity to limit their focus of attention on the involved quantities (1379, 28 and 49.25) in the problem. This constraint also helped students reflect on the MC between those quantities. After Wanda suggested drawing 1379 unit wholes and circling each 28 group until 1379 is completely used up, she said, "28 can go into 1379, 49 times and 0.25 times of another whole time." When probed, Nancy supported this idea by saying:

Nancy: 28 is in the 1379, 49 times, and then 0.25 of another time. So we don't have a whole 28, we have 0.25 of 28. So in order to find what 0.25 of 28 is, we can multiply them by each other.

The students both considered the quotient, 49.25, as having two parts (49 and 0.25) and based on that understanding and the overall goal they realised through coordinating both that both parts need to refer to the divisor, 28, and could be coordinated with the divisor through multiplication. Since they considered each action of making up a full divisor group within the dividend as 'one time,' they now think about the '0.25 times divisor' as "0.25 of another time [referring to group of 28]", which is the partial divisor group. The students anticipated that they needed to multiply 0.25 and the value of divisor to obtain the partial group. In this sense, they carried over their understanding of quantities and referents into the context-free setting.

To check their understanding, they then had to calculate "28 times 0.25" as 7 and were asked them to mentally check whether 7 was the remainder. The method they offered was to "take 28 times 49 [...] and see what you get and then add 7. [...] Because 28 can go into 1379 forty-nine times at least we know but [...] 28 can't go into 1379 a whole other time." However, this was all done in whole number settings and moving to fractional settings was the next step as detailed next.

2. Moving from contextual to context-free fraction settings

The second class continued with the two problems (see the sample question for Class #2 given in Figure 2), to help the students establish the MC within fraction division settings. Similarly this commenced with context to give the students an opportunity to use the contextual clues since fraction division was conceptually more challenging than whole number division. In solving these problems the students were to only use diagrams. The key reasons being that diagrams foster thinking about MC and the involved quantities. For example, in solving the sugar problem Wanda first drew 6 1/2, partitioned each unit whole into four parts and then circled each three-part group (representing 3/4 of a kilogram of sugar) until the last rectangle (as in the left side of Figure 5). She then wrote "8 2/3 bags or 8 bags and 1/2 kilogram of sugar" after considering last unit whole (as in the right hand side of Figure 4).

Each time the students struggled with the remaining parts, they were asked:

1. What are you counting here?

2. What do the units in your drawing represent?

Getting their attention on the referents in coordination with the overall goal within drawn sections in this sense helped the students solve the given contextual problems. For the rest of this second class and the last one, the instructional sequence continued with four more problems with increasing difficulty (see the sample problem for Class #3 given in Figure 2).

For the problem 9/5 / 2/3 they initially shaded in nine-fifths in the given diagram and then horizontally partition each unit whole into three equal sections as illustrated in Figure 5. Each horizontal section (5 small pieces representing 1/15 -unit) is 1/3 of a whole unit and two horizontal sections represent 2/3 of a whole (10/15 of a whole unit). They then circled each 10-unit section twice (by labeling them as 1 and 2) and there were 7 pieces left (labeled as L). Since one "can make 7/10 of a whole package, the result is 2 and 7/10." As seen here the diagram work limited students' actions and helped them think about involved units.

In all these problems, focusing on the remainder as well as its correspondence as a fractional part of the quotient helped the students understand the MC and this understanding was fostered by use of diagrams in context-free settings.

What lessons do we learn from this experience?

Fraction division consists of a network of mathematical relationships (Thompson & Saldanha, 2003). Understanding fraction division is dependent on understanding these relationships (Armstrong & Bezuk, 1995). The instruction illustrated here is promising as a way of scaffolding understanding of division of fractions for teachers and elementary students alike and has the potential to inform teachers in their work with students learning about fraction division.

The instruction was effective in helping students establish MC by starting with contextual (diagram) settings since they foster students' thinking (Sharp & Adams, 2002), and then continuing with work in context-free (diagram) settings as students move from working on whole numbers, as suggested in the literature (For example, Flores & Priewe, 2013), to fractions. Each setting in the instruction fostered a higher level of understanding. In applying the same sequence in regular classrooms it is initially important to encourage students to think about quotient-only result and quotient-with-remainder result as well as the relation between them together.

Throughout the instruction the overall goal was to build understanding of referents and MC. Each step in the instruction fostered the next level of understanding. For example, the use of contextual problems was crucial in helping students deal with the complexity of dividing fractions and it paved the way for understanding MC in context-free problems. Although useful, contextual clues (For example, bags of sugar) carry a lot of the thinking about the referent units for dividend, divisor and remainder, which may be an obstacle to students' development of higher level understanding of fraction division. Context-free settings on the other hand require students to reflect on the involved quantities and their relationship to each other at a higher level than contextual settings do. So what can be done? In order to think about MC, students should have an understanding of the divisor as number of items per group (For example, 3/4 of a kilogram of brown sugar per bag) that connects two quantities, dividend (total number of items, like 6 1/2 kg brown sugar), and quotient (total number of groups, like number of bags). In a context-free setting, one already needs to have such an understanding of the divisor in order to think about MC in the absence of contextual clues (For example, kilogram of sugar, bags). Therefore, teachers should benefit from both contextual and context-free settings since contextual problems can scaffold understanding of fraction division in context-free situations on the one hand and context-free problems help students to think about referents at a higher level on the other hand.

The understanding of the remainder is important but as it relates to all the parts of a division problem. Students will have difficulty in relating the remainder (as part of dividend) to the quotient and divisor unless they understand what a quotient is, more specifically, what a fractional quotient as it relates to dividend and divisor is. The use of diagrams helps students tie all these understandings together. In addition, the diagram work and reflection over it enabled students to divorce their understanding (Steffe & Cobb, 1998) of the MC from context. Asking questions about "what is being counted" and "what the units in the drawings represent" was helpful to students to get their attention to the referents in coordination with the overall goal. Limiting students' actions with diagrams was also helpful for me as a teacher in anticipating the possible mental or physical actions in which they would engage. Using this form of scaffolding would help teachers play the teaching scenario and resolve the complications before entering the class.

Editor's note

Whilst the above article provides examples which uses Imperial units of measurement, it provides innovative and insightful ways which can be directly linked to, The Australian Curriculum.: Mathematics (ACARA, 2014), in terms of how multiplicative coordination supports the development of middle school students' proportional reasoning abilities. It also provides interesting sub-activities in the conversion of units.

Acknowledgements

The data provided here is taken from a dissertation study and re-purposed for this article.

References

Armstrong, B. E., & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp.85-119). Albany, NY: State University of New York Press.

Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2014). Australian Curriculum.: Mathematics. Sydney, NSW. Retrieved 23 April 2017 from http://www.australiancurriculum.edu.au/ mathematics/curriculum/f-10?layout=1

Flores, A., & Priewe, M. D. (2013). Orange you glad I did say "fraction division"? Mathematics Teaching in the Middle School, 19(5), 288-293.

Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.

Steffe, L., & Cobb, P. (1998). Multiplicative and divisional schemes. Focus on Learning Problems in Mathematics, 20(1), 45-61.

Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin & D. Schifter, Research companion to the Principles and Standards for School Mathematics (pp.95-114). Reston, VA: National Council of Teachers of Mathematics.

Ismail O. Zembat

Mevlana University, Turkey

<izembat@gmail.com>

Caption: Figure 4. Wanda's solution method for the question from class #2.

Caption: Figure 5. Representation of the student's solution method for 9/5 divided by 2/3. Table 1. Solution process for bags problem that can be modelled by 2 1/2 / 2/3 Subgoals Contextual Mathematical Diagram work subgoal subgoal required to achieve the subgoal 1 Identify and Identify and Each whole represent the represent the represents one sugar at hand, quantity of the kilogram and 2 1/2 dividend. the shaded kilograms. section represents the dividend, or the sugar at hand. 2 Identify and Identify and Upon horizontal represent 2/3 represent a partitioning kilogram sugar divisor group each unit whole bag. within the (each kilogram dividend. of sugar) into three equal sections, each 4-unit group (shown in blue) will correspond to 2/3 of the whole unit, the divisor. 3 Count the Count the Each 4-unit number of 2/3 number of group (labelled kilogram sugar divisor groups as 1, 2, and 3) bags within 2 within the is a full 1/2 kilograms dividend. divisor group of sugar. and there are three full such groups and three peices the size of 1/6 left. 4 Identify what Identify what Since each 1/6 part of a bag part of the unit (1/6 can be made 3/6 divisor group kilogram sugar) by leftover 6 4(1/6 g unit), corresponds to of a kilogram can be made by 1/4 of the of sugar. leftover (3/6) divisor (fills unit. 1/4 of a bag), the leftover as a totality (3/6) (or 3/6 kilogram sugar) corresponds to 3/4 of the divisor (or 3/4 of a bag). 5 Identify the Identify total * We can make 3 total number number of 3/4 divisor 2/3 of 3 divisors groups (bags) kilogram bags including in total out of including partial the given partial bags. divisors. dividend (2 1/2 kilograms of sugar)--called the quotient-only result. * 3R (3/6), which means that there are three full 3/6 divisors (bags) with 6 of a unit whole (a kilogram of sugar) left (called the quotient-with- remainder result. Figure 3. Representation of the board with the two results of each problem. 4.5 bags 4 bags + 1 kg of flour left 3 1/2 days 3 days + 1 acre left