Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction
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- Clarke, D.M. & Roche, A. Educ Stud Math (2009) 72: 127. doi:10.1007/s10649-009-9198-9
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As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that predicted, with the exception of fractions with the same numerators and different denominators, which proved surprisingly difficult. Students who demonstrated the greatest success were likely to use benchmark (transitive) and residual thinking. It is hypothesised that the methods of these successful students could form the basis of instructional approaches which may yield the kind of connected understanding promoted in various curriculum documents and required for the development of proportional reasoning in later years.
KeywordsFractionsStrategiesAssessment tasksOne-to-one interviewsUnderstandingMisconceptions
1 Background literature on relative size of fractions
It is widely agreed that fractions form an important part of the middle years’ mathematics curriculum (Lamon, 1999; Litwiller & Bright, 2002), underpinning the development of proportional reasoning, and important for later topics in mathematics, including algebra and probability. However, it is clear that it is a topic which many teachers find difficult to understand and teach (Ma, 1999; Post, Cramer, Behr, Lesh, & Harel, 1993) and many students find difficult to learn (Behr, Lesh, Post, & Silver, 1983; Kieren, 1976; Streefland, 1991). Amongst the factors that make rational numbers in general and fractions in particular difficult to understand are their many representations and interpretations (Kilpatrick, Swafford, & Findell, 2001).
Generalisations that have occurred during instruction on whole numbers are frequently misapplied to fractions (Streefland, 1991). Post, Wachsmuth, Lesh, and Behr (1985) note that “children’s understandings about ordering whole numbers often adversely affect their early understandings about ordering fractions. For some children, these misunderstandings persist even after relatively intense instruction based on the use of manipulative aids” (p. 33).
Kilpatrick et al. (2001) commented that “of all the ways which rational numbers can be interpreted and used, the most basic is the simplest—rational numbers are numbers. That fact is so fundamental that it is easily overlooked” (p. 235). A number of researchers have highlighted the importance of students being able to give meaning to the size of a fraction and the many difficulties associated with doing so.
An often-quoted result from the National Assessment of Educational Progress (Carpenter, Kepner, Corbitt, Lindquist, & Reys, 1980) supported the claim of student difficulty in understanding the size of a fraction. When estimating the answer to 12/13 + 7/8, only 24% of 13-year-olds chose the correct answer in a multiple choice set, with the majority choosing 19 or 21. By age 17, only 37% of students were successful on this task. Given that the most common answers were presumably calculated by adding 12 and 7, and 13 and 8 respectively, this appears to be an example of students considering numerators and denominators as separate entities rather than as connected in any way.
Findings suggest that a coordinated use of order and equivalence knowledge, combined with the skill of estimating the size of rational numbers, enabled some children to be successful across all three tasks....An ability to perceive the ordered pair in a fraction symbol as a conceptual unit (rather than two individual numbers) was also found to be an indicator for successful performance. (p. 2)
The connection between the comparison of fractions and development of number sense is clear. Comparing fractions is necessary for obtaining an intuitive feel of the size of fractions. If a fractional number is recognised to be close to 1/3 or 1/2, for example, one has a better feel for its magnitude. This fractional number sense is particularly important when estimating with fractions. (p. 189)
In support of this, Post et al. (1986) noted that “children who do not have a workable concept of rational number size cannot be expected to exhibit satisfactory performance across a set of tasks which varies the context in which the number concept of fraction is involved” (p. 2). Further, Behr, Wachsmuth, Post, and Lesh (1984) ask, “what meaning, for example, do 2/3 × 5/6 or 2/3 + 5/8 have for children who lack a well-internalised concept of the bigness of rational numbers?” (p. 324).
Researchers report frequently that students use strategies, in solving fraction comparison tasks, which they are unlikely to have been specifically taught. The use of residual thinking (Post and Cramer, 2002) and benchmarking (or transitive, Post et al. 1986; or correct reference point comparison, Behr, Post, & Wachsmuth, 1986) is likely to be successful, whilst gap thinking (Pearn and Stephens, 2004) is likely to be less so. The term residual refers to the amount which is required to build up to the whole. So, in comparing 5/6 and 7/8, students may conclude that the first fraction requires 1/6 more to make the whole (the residual), whilst the second requires only 1/8 to make the whole, so 7/8 is larger. The use of benchmarks involves the student comparing two fractions of interest to a third fraction, often 1/2 and sometimes 1. A student using this strategy appropriately would say that 5/8 is larger than 3/7 because the first fraction is greater than one half, whilst the second is less than one half. Post et al. (1986) referred to benchmarking as a transitive strategy where the transitive property is used in relation to an external value, the benchmark fraction.
On the other hand, some students claim that 5/6 and 7/8 are equivalent, because they both require one “bit” to make a whole. In this case, the students are focusing on the gap between 5 and 6 and the gap between 7 and 8, but not considering the actual size of the pieces. This gap thinking is really a form of whole number thinking, where the student is not considering the size of the denominator and therefore the size of the relevant parts (or the ratio of numerator to denominator), but merely the absolute difference between numerator and denominator.
In relation to the residual strategy, Riddle and Rodzwell (2000) noted that students from first to fifth grade could solve a fraction addition problem, 2½ + 3/4, by using a strategy they called “filling up the whole.” “Some of them chose to build onto the 1/2 by taking half out of the 3/4. Others seemed to see that the 3/4 was close to a whole, so they pulled the fourth needed to complete the whole out of the 1/2” (p. 203). For the same task, one teacher noted that 22% of her fifth-graders attempted to solve this problem using strategies like finding a common denominator. None of them, however, were successful. Two-thirds of these fifth-grade students were correct, and all successful students used a version of the filling-up-the-wholes strategy. It will be argued in a later section that exposing these strategies to teachers and students has the potential of enhancing student performance and understanding of fraction size.
The use of language is important also. Post et al. (1986) noted that students’ use of “more” and “greater” (and their counterparts, “less” and “fewer”) can cause confusion, given the critical importance of distinguishing between more pieces and greater sized pieces.
In the Early Numeracy Research Project (Clarke, Sullivan, & McDonough, 2002), a task-based, interactive, one-to-one assessment interview was developed for use with students in the early years of schooling. This interview was used with over 11 000 students, aged 4 to 8, in 70 schools in Victoria (Australia) at the beginning and end of the school year, thus providing high quality data on what students knew and could do in these early grades, across the mathematical domains of whole number, measurement and geometry. There was equal emphasis in the teachers’ record of interview on answers and the strategies which led to these answers. The present study involved further, similar research at Grade 6 level.
The use of student assessment interviews, embedded within an extensive and appropriate professional development programme or a preservice programme, can be a powerful tool for teacher professional learning, enhancing teachers’ knowledge of how mathematics learning develops, knowledge of individual mathematical understanding as well as content knowledge and pedagogical content knowledge (Clarke, Mitchell, & Roche, 2005; McDonough, Clarke, & Clarke, 2002; Schorr, 2001).
Focusing on the rational number constructs of part-whole, measure, division and operator, the “big ideas” of the unit, discrete and continuous models, partitioning, and the relative size of fractions, a range of around 50 assessment tasks was developed, drawing upon tasks which had been reported in the literature, and supplemented with tasks which the research team created. These tasks were piloted with around 30 students in Grades 4–9, refined, and piloted again (see Mitchell and Clarke, 2004).
Following piloting and refinement, a total of 323 Grade 6 students was interviewed on 15 of these tasks. Grade 6 is the final year of primary school in Victoria, and students are typically 11 or 12 years old. The schools and students were chosen to be broadly representative of Victorian students, on variables such as school size, location, proportion of students from non-English speaking backgrounds, and socio-economic status. A team of ten interviewers, all experienced primary teachers, with at least four years’ experience in one-to-one assessment interviews of this kind, participated in a day’s training on the use of the tasks, including viewing sample interviews on video, and discussing the possible strategies students might use for each task. The students had participated in eight similar interviews on different mathematical content in the previous seven years. This is important to note, as these students were quite used to being asked to articulate their reasoning on such problems, although this was the first time rational numbers tasks had formed part of the assessment interview.
The tasks were administered over a 30- to 40-minute period in a quiet place in the students’ own schools, with interviews following a strict script for consistency, and using a standard record sheet to record students’ answers and methods. Each actual response to a question was given a code by the authors, and a trained team of coders took the data from the record sheets and entered it into SPSS. Only the fraction pairs’ data are discussed in this paper.
2.1 The fraction pairs used in this study
No opportunity was given for the students to write or draw anything. Our interest was in mental strategies. There were several reasons for this decision. We wanted to see whether students had confident access to strategies other than common denominators. We also knew that the opportunity to draw pictures would greatly extend the time which the interview would take, with the fraction pairs being only one of 15 rational number tasks. We had deliberately chosen eight pairs which, given access to appropriate strategies, could be solved mentally, without access to common denominators, other than very simple ones involving halves and eighths. Finally, we agreed with Post et al. (1986) that “a crucial point in acquisition of order and equivalence concepts is reached when children’s understanding of fractions becomes detached from concrete embodiments and children are able to deal with fractions as numbers” (p. 47), and we wanted to collect evidence of the extent of this. We do acknowledge of course that access to a pen and paper may have increased the success rate on some of these tasks.
The intention with the order of administration of tasks was that, based on previous piloting, the tasks would get progressively harder. As will be discussed later, this predicted increase in difficulty was not always found to be the case. The reader may note that in every case except the equivalent fractions, the second fraction of the pair was the larger. This was intentional. We noted from piloting that in the course of listening carefully to the large variety of correct and incorrect explanations, the interviewer could sometimes be confused in their own mind as to which of a given pair was larger. Knowing that the second fraction was larger provided some reassurance for the interviewer under pressure.
We are confident that this decision did not reduce the validity of the data, for four reasons. First, each student was shown only one pair at a time, very much reducing the chance of a student noticing any pattern in the location of the correct choices. Second, no student out of 323 commented to the interviewers that they noticed this pattern. Third, only 4.3% of students were able to give correct answers to all 8, and only 5.0% were correct for even 7 of the 8, thus reducing the chance of pattern recognition. Finally, given that students were required to give a clear explanation for their choices each time, anyone who did notice that the second fraction was usually larger would nevertheless have to give a coherent reason why each one was larger.
Residual thinking with equivalence (2/8 > 2/9)
Residual thinking (1/4 > 2/9) with proof
Converts to decimals
Higher or larger numbers
Data analysis involved determining the percentage of students who gave the correct answer, and then for both correct and incorrect choices, the percentage of students using each particular strategy. The list of strategies was expanded during data analysis to incorporate any strategies which were common, from the “Other” category. The results from this analysis are given in the following section.
Percentage of grade 6 students choosing appropriately from fraction pairs with appropriate explanation (n = 323)
Student strategies given for correct choices in each fraction pairs task as percentage of total satisfactory responses for that task (n = 323)
Proportion nominating this strategy
3/8 & 7/8
• Denominator the same and compares numerator
• Benchmarking to ½ and one
1/2 & 5/8
• Benchmarks to one half
• Converting to common denominator
4/7 & 4/5
• Numerator the same and compares denominator
• Converts to common denominator
• Benchmarking to ½ and one
• Residual thinking
2/4 & 4/8
• Equivalent (“the same”)
2/4 & 4/2
• Equates to ½ and 2
• Equates to a half and more than one
• Improper fraction
3/7 & 5/8
• Benchmarks to one half
• Converts to common denominator
• Residual thinking
5/6 & 7/8
• Residual thinking (1/6 > 1/8)
• Converts to common denominator
3/4 & 7/9
• Residual thinking with equivalence (2/8 > 2/9)
• Residual thinking (1/4 > 2/9) with proof
• Converts to decimals
• Converts to common denominator
Sample student strategies for particular fraction pairs
Verbal explanation with strategy summarised in parentheses as appropriate
Satisfactory or unsatisfactory use of strategy
3/8 & 7/8
• “The pieces are the same size — they’re eighths and 7 is more than 3” (Denominator the same and compares numerator)
1/2 & 5/8
• “1/2 is equivalent to 4/8, so 5/8 is 1/8 more” (Benchmark)
4/7 & 4/5
• “4 is closer to 5 than 4 is to 7” (Gap thinking)*
• “They’ve both got four bits but the fifths are bigger” (Numerator the same and compares denominator)
2/4 & 4/8
• “Both equal a half” (Equivalent)
2/4 & 4/2
• This [4/2] is improper, so it must be bigger” (Impropers)
3/7 & 5/8
• “5/8 is more than a half and 3/7 is less than a half” (Benchmark)
5/6 & 7/8
• “Same because each has one left” (Gap thinking)
3/4 & 7/9
• “3/4 of 9 is 6 point something, so I chose 7/9 because it’s more than 3/4” (Converts to decimals)
As is evident from Table 2, benchmarking and residual thinking were used consistently and usually appropriately for a given fraction pair by successful students. Our experience is that unless Australian teachers have conducted interviews of this kind, they are generally unaware of these strategies. We are therefore confident that few students in this study had been taught these strategies prior to the interviews. However, we believe that the percentage of students who used these strategies with success is low (see Table 1). Of course, it needs to be noted that these strategies can be sometimes overused when a more simple explanation could be given. For example, using residual thinking for the first pair would be less efficient than several other strategies.
4 Discussion and implications for further research
In this section, we discuss the relative difficulty of the various fraction pairs, the common strategies used by successful and unsuccessful students in their solutions, and the implication of these results for teacher professional development and classroom instruction. There is an assumption in the discussion that the methods which students claimed to use were those which they actually did use. We cannot be certain of this of course.
4.1 The relative difficulty of the fraction pairs tasks and common strategies
The most straightforward pair (3/8, 7/8) and the most difficult pair (3/4, 7/9) were easily predicted in advance. Having said that, the percentage success on the easiest pair (77.1%), with success being defined as a correct choice coupled with an appropriate explanation, was not high. Given that students were interviewed at the end of their Grade 6 year, after several years of introductory work on fractions, at least one-third of students do not seem to have a basic, part-whole understanding of fractions. The vast majority of successful students (94.8%) noted that the denominator was the same (and hence the size of the parts), and therefore compared the numerators. However, 5.2% benchmarked to 1/2 and 1. Also 38.5% of all incorrect solutions (for which 3/8 is chosen as the larger) gave an explanation to the effect that “smaller numbers mean bigger fractions.”
On the other hand, the most difficult pair (3/4 & 7/9) proved to be very difficult for various groups of primary and junior secondary teachers with whom we have worked in professional development settings. Many teachers have been unable to offer an explanation beyond the use of common denominators, and so the 10.8% success rate for students is probably not surprising. In fact, 54.3% of students who were successful on this task used common denominators and a total of 40% used some form of residual strategy (either 2/8 > 2/9 or 1/4 > 2/9 with some other explanation), whilst 5.7% (2 students, in fact) converted the fractions to decimals in their heads. Some students were able to identify 1/9 as 0.111 and therefore 7/9 as 0.777, noting that 0.777 was greater than 0.75. 35.6% of all incorrect solutions (for which 3/4 is chosen as the larger) were chosen with the explanation that 3/4 has a smaller gap and therefore is the larger. For four of the eight pairs, gap thinking will lead to a successful response. Clearly, success with this strategy may affirm a student’s belief that this is sound thinking. As will be discussed under classroom implications, this faulty thinking needs to be exposed.
As discussed earlier, benchmarking and residual strategies are strategies that appear to be used by students displaying a more conceptual understanding of the size of fractions, yet they do not appear to be in widespread use by teachers in our schools. These strategies would have been most appropriate for the pairs (3/7, 5/8) and (5/6, 7/8) respectively, but the success rates were 20.4% and 14.9%. Of the successful students, 28.8% and 45.8% of students chose to use common denominators for these pairs, respectively. In this way, a procedure was preferred to a strategy based more clearly on number sense. Also, 21.2% of all students used gap thinking for (3/7, 5/8) and 29.4% of all students claimed 5/6 & 7/8 were the same, often using gap thinking as their justification.
Simple equivalences, however, appear to have been reasonably well understood, given the relative success for the pairs (2/4, 4/8) and (1/2, 5/8), although it must be said that 64.4% success for the first pair is not a good result. The observed lack of emphasis on improper fractions in primary grades may account for the difficulty in explaining the relative order of 2/4 and 4/2 (42.7%). Also, the language some students use to label fractions may hinder their understanding. For example, some students were noted to read these as “two out of four” and “four out of two” which is not helpful when considering their respective size. “Two-quarters” and “four-halves,” on the other hand, may help to create an image about the size of the parts that is more likely to lead to a correct solution. It is interesting to note that 25.3% of incorrect explanations were that these two fractions were “the same,” presumably because they contain the same digits. Alternatively, we have noticed students who “flip” improper fractions in an attempt to give them meaning.
As mentioned earlier, the relative difficulty of the pair (4/7, 4/5) was greater than we had anticipated, with only a 37.2% success rate, indicating that it was more difficult than (1/2, 5/8) and (2/4, 4/2). We had expected that more students would have chosen successfully, with an accompanying explanation of the form, “there are four pieces in each, but as sevenths are smaller than fifths, 4/5 will be larger” (see also Post et al. 1985). Mamede, Nunes, & Bryant (2005) offer a possible explanation, in that unlike the comparison of 3/8 and 7/8, “the students have to think of an inverse relation between the denominator and the quantity represented by the fraction” (p. 282). Although explanations were worded in a wide variety of ways, the most common correct response was one in which the student noted the number of parts being the same, but the size of the parts being different (60.0%). It was of some concern that 20.0% felt the need to convert to common denominators. This was a task in which gap thinking was common, with 21.4% of students who chose 4/5 as larger providing inappropriate reasoning (focusing on the difference between 4 and 7 and between 4 and 5). For all students who chose 4/7 as larger, 73.5% of reasons were to do with “larger numbers,” a probable example of “whole number dominance” (Behr et al. 1986, p. 104).
As with a number of the fraction pairs in this article, there are interesting parallels with the seminal work of Noelting (1980), who, as part of a study on the development of proportional reasoning, invited subjects aged 6 to 16 to compare mixtures of varying numbers of glasses of orange juice and water for their “relative orange taste” (p. 220). The closest task to our pair (4/7, 4/5) was his comparison of (1, 2) to (1, 5), where the two figures in each pair are the number of glasses of orange juice and water, respectively. Of course, there was no direct comparison of unit fractions in our study. Noelting identified the stage associated with success with these pairs as being an earlier stage than that represented by his pairs (1, 1) and (2, 2), which could be related to our pair (2/4, 4/8). Not surprisingly, his highest stage, “Higher Formal Operational” (p. 231) related to a comparison of (3, 5) and (5, 8), which has similarities to our most difficult task, the pair (3/4, 7/9).
4.2 Implications for teacher professional development and classroom practice
Mathematics curriculum developers and classroom teachers are routinely faced with the challenge of making the best use of the limited amount of time they have to cover the desired curriculum content. It is unlikely that the level of intensity possible in the Rational Number Project clinical teaching experiment (13 lessons of 3–8 days each, over an 18-week period) will be attainable or desirable for the average classroom. Some careful decisions must therefore be made.
We agree, and believe that the recommendations we offer below, which find harmony with the work of key scholars in the area, will help both teachers and students to build such a firm foundation.
We fail to understand how teachers without a relatively firm foundation could possibly be in a position to present and explain properly, to ask the right question at the right time, and to recognise and encourage high levels of student mathematical thinking when it occurs. (p. 197)
a greater emphasis on the meaning or semantics of rational numbers than on procedures for manipulating them;
a greater emphasis on the proportional nature of rational numbers, highlighting rather than glossing over the difference between rational and whole numbers;
a greater emphasis on children’s natural ways of viewing problems and their spontaneous solution strategies; and
the use of an alternative form of visual representation as a mediator between proportional quantities and their conventional numeric representation (i.e., an alternative to the standard pie chart).
An emphasis on a more careful definition of numerator and denominator for students. The relative difficulty with the pair (4/7, 4/5) and the pair involving an improper fraction highlighted the need for a clear understanding of the meaning of the numerator and denominator and the relationship between them. Such a definition should be inclusive of improper fractions. The definition might take the following form: In the fraction a/b, b is the name or size of the part (e.g., fifths have this name because 5 equal parts can fill a whole) and a is the number of parts of that name or size. The reader can see the way in which such a definition for students lends itself readily to improper fractions, and, if understood, would assist with several of the fraction comparison tasks discussed above. Our experience with students is that they typically identify the denominator as “the number of parts you cut the whole into” and the numerator as “the number of parts you take from the whole or that are shaded” and that this does not generalise well to improper fractions. We also believe that students who use the appropriate language of fractions (halves, thirds, fifths, etc) may be able to conceptualise the size of the fraction better than students who label the fraction 2/3 as “two out of three” or as we sometimes found “two-threes.” If a more careful definition is used, we may achieve the stated aim of Post et al. (1986), namely “a realisation that the relationship between the numerator and denominator defines the meaning of a fraction and not their respective absolute magnitudes when viewed independently” (p. 41).
The explicit sharing, in the context of whole-class discussion, of the strategies of residual thinking and benchmarking (or transitive thinking). These two strategies emerged from the study (and the work of other researchers) as being used by students who were generally successful on these kinds of tasks. These strategies address the difficulty which students have in giving meaning to the size of a fraction. One classroom activity with considerable potential is providing students with a large collection of proper and improper fractions in conventional form on cards, and requiring them to sort them into those which are close to 0, close to 1/2, and close to 1, respectively. A variation of this is to provide fractions which are incomplete (e.g., □/13, 9/□), and ask them to complete the fractions so that they are close to 1/2. Whilst identifying these preferred strategies, it is also important to expose and discuss the inappropriateness of strategies that use some whole number thinking such as gap thinking or “larger numbers mean bigger fractions.”
Greater opportunities for estimation and approximation, which assist to develop a feeling for numbers. It was clear from the interview data that many students lacked any sense of size in relation to the fractions, focusing separately on the numerator and denominator, without often connecting the two in a proportional sense. Having a feeling for numbers of course relates closely to benchmarking. One of our favourite tasks is “Construct a Sum” from the Rational Number Project (Behr, Wachsmuth, & Post, 1985). Students are given cards with the numbers 1, 3, 4, 5, 6, and 7 on them, and a board with □/□ + □/□, and asked to place the number cards inside the boxes so that when you add them, the answer is as close to 1 as possible, but not equal to 1. Interestingly, when used with our 323 Grade 6 students in the interview setting, with no time limit, only 25.4% could create a combination within 0.1 of 1, with a total of 85 different combinations offered. Also, 24.4% of students created at least one improper fraction in their solution, indicating a misunderstanding about the size of improper fractions. This task provides a wonderful basis for discussion about relative size, as well as providing an opportunity to address students’ understanding of improper fractions. We believe this assessment task can readily be adapted for the classroom by altering the solution, e.g., what is the largest/smallest sum you can make? Make a sum that is close to two. How many sums can you make that are close to one and which is the closest possible solution? This classroom activity may lead to fruitful discussions about fraction size and contribute to students’ development of a robust understanding.
In conclusion, we would support strongly the provision of opportunities for teachers to interview at least a small number of their students on the tasks contained within this paper. Fractions are clearly difficult to teach and to learn, but a curriculum emphasis of the kind outlined in this paper and consistent with the work of other key researchers in the field of rational number, is likely to yield positive results. Our study builds on the work of other scholars, offering high quality data on a relatively large number of students at an important stage of their fraction learning (Grade 6), in a one-to-one interview format where students were required to articulate the reasoning behind their decisions. By outlining the percentage of students who articulated different strategies, and noting the use of residual and benchmarking strategies by the more successful students, we contribute to the research base and offer practical, research-based advice for practitioners.
We acknowledge gratefully the contribution of Annie Mitchell in item development and data analysis and the very thorough and insightful comments of the editor and all reviewers.