Fractions: A Weeping Sore in Mathematics Education Nerida Ellerton and M. A. (Ken) Clements Faculty of Education Deakin University Geelong Abstract Data is cited from recent studies in Australia, Papua New Guinea and Malaysia showing that most primary school children do not link what they learn about fractions in mathematics classrooms with situations involving fractional quantities in their personal worlds. For example, many children who correctly answered pencil-and-paper fraction questions such as 5/11 x 792 = q could not pour out one-third of a glass of water, and of those who could, only a small proportion had any idea of what fraction of the original full glass of water remained. A theoretical model aimed at achieving links between children's personal worlds and the verbal and symbolic language of fractions is proposed. Paper presented at the 1992 Joint Conference "Educational Research: Discipline and Diversity" of the Australian Association for Research in Education and the New Zealand Association for Research in Education held at Deakin University, Geelong, November 22-26, 1992. FRACTIONS: A WEEPING SORE IN MATHEMATICS EDUCATION Nerida F. Ellerton and M. A. (Ken) Clements Deakin University (Geelong) The Need to Establish Links in Cognitive Structure In two complementary papers, Clements and Lean (1988, 1990) reported data from a study in which they investigated the "discrete" and "continuous" fraction concepts of 59 students in Grades 4, 5, and 6 in three Papua New Guinea Community Schools. These data derived from three different kinds of tasks, namely "sharing" tasks, tasks involving formal fraction language, and "symbol" manipulation tasks. (The distinctions between these types of tasks are suggested by the examples given in Figure 1.) All tasks were directly or indirectly concerned with the fractions 1/2, 1/4, and 1/3, and while all 59 students were confident and accurate when performing the "sharing" tasks, they were much less successful on the corresponding tasks that involved formal verbal fraction language and symbol manipulation. Clements and Lean (1988, p. 220) commented that the analysis of their data indicated that the reality-based sharing concepts which all of the interviewees possessed were rarely linked, in the interviewees' minds, "with the formal language of fractions or with the symbolic language of fractions; also formal language of fractions was not linked with symbol manipulation." The need for these links is suggested by Figure 1. The absurdity of the situation described by Clements and Lean was emphasised by the fact that, although most of the Grade 6 students in their sample were able to give correct answers to pencil-and-paper questions such as 5/11 x 792 = o, many of the same students did not respond correctly when asked by an interviewer to pick up "one-third of" a set of 12 marbles which were arranged in a 3 x 4 array in front of them. Similarly, given a full glass of water and some identical but empty glasses, many of the students did not know how to pour one-third of the full glass of water into another glass, and of those who could, only a small proportion had any idea of what fraction of the original full glass of water remained. Figure 1. Establishing links in cognitive structure (from Clements and Lean, 1988, p. 222). SOME SOUTHEAST ASIAN FRACTION DATA Recently, the same tasks as those used by Clements and Lean in their Papua New Guinea (PNG) study were given to 21 Grade 4 students attending an international school in Malaysia. This Malaysian study was carried out by Leong and Ferrer as an extension of their earlier fraction studies in Malaysia (Leong & Ferrer, 1990; Ferrer, 1991), and although the data for the extension study are as yet unpublished, it is clear that the pattern of the results was strikingly similar to those obtained in the PNG study. This similarity can be seen in the following summary analyses of data from two of the tasks. The "Moving Around an Equilateral Triangle" Task As an example of the type of fraction data obtained by Leong and Ferrer in Malaysia, consider the responses given by the international school Grade 4 students to a task which required them to indicate half-way, a quarter of the way, and one-third of the way around the boundary of an equilateral triangle. Interviewees were given an A4 piece of paper on which three equilateral triangles, each with vertices labelled A, B, C, were drawn (see Figure 2). They were told that Salan wanted to move around the sides of the triangle, starting at A and going through B, then C, before arriving back at A. While these instructions were being given the interviewer demonstrated the meaning of what was being said by pointing to a triangle. Then the interviewee was asked to indicate on the triangle where Salan would be after he had moved 1/2 (then 1/4, then 1/3) of the way around the triangle. Figure 2. Moving around an equilateral triangle The percentages of correct responses by the 21 Grade 4 students at the international school in Malaysia on this task for the fractions 1/2, 1/4, and 1/3 were 29%, 19% and 5% respectively. While these percentages were similar to those reported by Clements and Lean (1988, p. 219) for their PNG sample (corresponding percentages for the 59 PNG students were 14%, 5%, and 7%, respectively), Leong and Ferrer (1992) report that even when correct responses were given by the students in Malaysia, in almost all cases they were obviously guesses. The cognitive structures of the interviewees clearly did not enable links between movement around the boundary of a triangle and fractions to be made. One might have reasonably expected that on this task a fairly high percentage of correct responses would have been given for the "one-third" task, but this expectation was not realised in either Papua New Guinea or in Malaysia. The "Fraction of a Glass Filled With Water" Task For this task interviewees were shown five identical clear plastic "glasses," one of which was full with water and the other four empty. They were handed the glass full of water and told: "This is full of water. By pouring, and using any of these other glasses, show me exactly a half [or a third, or a quarter] of a glass of water." The percentages of Grade 4 international school interviewees in Malaysia giving correct responses for this task corresponding to the fractions 1/2, 1/4 and 1/3 were 76%, 67% and 54% respectively. Although these percentages are significantly higher than the percentages for Clements and Lean's PNG sample (which were 39%, 17% and 14% respectively), it is still a matter for concern that so many students experienced difficulty in applying their fraction concepts to such a relatively simple situation. Interviews suggested that interviewees who experience difficulty could not reconcile the displays of cups and water with their concepts of fractions. For these interviewees relating cups of water with fractions was a novel, and confusing idea. Many students simply guessed an answer, pouring a little bit of water from one glass to another, and then looking to the interviewer, hoping that some form of assistance might be forthcoming. If asked to state the fraction of the water they had poured into a "glass" they said "one-quarter," or "one-half," or, it seemed, whatever first came into their head. Interestingly, in both the PNG and Malaysian samples only a small proportion of those who correctly poured out one-quarter (or one-third) of a glass of water when asked to do so knew that the fraction remaining in the previously full glass was three-quarters (or two-thirds). Yet many of these same students could respond correctly to the pencil-and-paper question: "What is the value of 1 - 1/3 ?" FRACTION DATA FROM LARGE-SCALE AUSTRALASIAN STUDIES Data from pencil-and-paper tests containing items on fractions can be of limited value unless they are complemented by qualitative data (for example, from interview transcripts) which throw light not only on the fraction concepts that children have, but also on how these relate to different aspects of their cognitive structures. Another form of data, only rarely reported, is whether children can express fraction ideas spontaneously, that is to say, without responding to someone else's direct question involving fractions. (Clements and Del Campo (1987, p. 109) refer to this as "expressive mode" data.) Examples of these three forms of data, gathered in studies carried out in Australia and New Zealand, are now given. Traditional "Percentage Correct/Percentage Incorrect" Analyses of Pencil- and-Paper Data Large-scale studies based on "percentage correct/percentage incorrect" analyses for pencil-and-paper fraction items invariably produce results which suggest that, despite the large amount of time devoted to teaching fractions, the related concepts are poorly understood. For example, in a representative sample of 6247 14-year-old children from all Australian states, 37% and 30% respectively gave incorrect answers to the following questions (Bourke & Lewis, 1976): Question 4: Which fraction is closest in size to 3/16? [37% incorrect] Question 28: What is the yearly interest on $900 at 10%? In Ellerton's (1989) study of over 10 000 secondary school students in Australia and New Zealand, it was found that 21%, 71% and 78% of Grade 10 students (average age about 15 years) gave incorrect answers to the following four fraction questions: Question 16: If 3/4 = 21/o, what is the value of o ? [21% incorrect] Question 25: If x = 6 and y = 21, what is the value of 2/x + y/7 ? [71% incorrect] Question 35: If 1 Ö 4 = o x 2, what is the value of o ? [78% incorrect] Interview Data Fraction studies which have reported qualitative data (for example, interview transcripts), have invariably showed that (a) few students have a relational understanding of fractions, and (b) the responses most children give to pencil-and-paper questions such as those above are usually based on mechanical, rote-learnt procedures. For instance, Ellerton (1989, p. 377) reported the following transcript from an interview with Nina, a 12-year- old Grade 7 girl who had answered Question 25 (" If x = 6 and y = 21, what is the value of 2/x + y/7 ?") incorrectly. Nina: x is six and y is twenty-one. So, I can't remember how you add fractions. Do you find a common denominator when you're in fractions? Interviewer: You may not need to if you look at them carefully. Nina: That equals - no, yes. That equals one-third. And three goes, seven into that ... uhm ... three times, and goes into that once. And that's a third, so it would equal one and two-thirds. Interviewer: You've got one-third and three over one. Nina: Three over one equals ... [pauses] Interviewer: Tell me something about three over one. Nina: Well, it's - oh, what do you call it? It's an improper fraction I think. Is that what you call it? And so you have to change it to a proper fraction. So you go one into three won't go. How did I get that? I'm not sure how I got that. I really couldn't tell you. Even students who gave correct responses usually had little understanding of the methods they used. This is illustrated in the following transcript of an interview with Marion, a 12-year-old girl (from Ellerton, 1989, pp. 374-375): Marion: [Crosses out x and writes 6, crosses out y and writes 21.] You've got to find a common denominator. Twelve, eighteen. Oh, you've got to go on to forty-two, I guess. Well, six sevens are forty-two, so seven twos are fourteen. Six twenty-ones are six, one six is six, six twos are twelve, fourteen and one hundred and twenty-six is four and six is nought carry the one, four, one, one hundred and forty over forty-two. (Writes out long division.) Twice, three times. (Subtracts 126). That leaves me with Interviewer: You've been trying to bring one hundred and forty over forty- two back to a whole number. You've got three, and you've got fourteen left over. What are you going to do with the fourteen that's left over? Marion: Ah, yes. Three and fourteen forty-twos. Yeah. And the fourteen forty twos then goes into ... seven goes into forty-two, so seven goes into fourteen goes twice. Six - that goes in again. One-third. That makes it three and one-third. "Make Up a Problem" Data Marriott's (1978) study showed how students whose teachers emphasised fraction algorithms came to think of fractions purely in terms of symbol manipulation. The end result was that, in the minds of the students, fractions became abstractions divorced from any reality outside school mathematics. Ellerton (1986, 1988, 1989), who investigated children's perceptions of the nature of mathematics by asking them to make up and then solve a difficult mathematics problem, found that many children were pre- occupied with fraction algorithms. The following "problem" and "solution" (from Ellerton, 1986, p. 40) were proffered by John, a Grade 6 student: Interviewer: What did you do? John: Three and five-eighths, times eight-tenths divide two and nine- twelfths. Well, first we change the three and five eighths into an improper fraction, check twenty-nine over eight, and then the eight-tenths was alright. Then I changed, then I divided ... then I changed the two and-nine-twelfths into thirty-three over twelve. Then I did it again; I did twenty-nine over eight times eight over ten, and then instead of having to divide, I timesed, I multiplied, I swapped them really around so that the twelve would be on the top, twelve over thirty-three. Then I cancelled the eight into the eight, twenty-nine times one is twenty-nine times twelve is three hundred and forty-eight. One times ten is ten and ten times three is thirty, ten times thirty-three is three hundred and thirty. So that'd be an improper fraction, so I have to change it to a mixed number equals one and eighteen over three hundred and thirty. Interviewer: Just out of interest, which would you have preferred - if I'd given you the choice at the beginning - if I'd said to you write a problem that would be difficult for you to do, as against writing a problem that would be difficult for a friend? Would you have still done the same problem? John: No. Interviewer: You wouldn't have? You would have done something quite different, would you? John: Yes. Sometimes I don't get ... how it goes, say there's a dollar a kilo and there's three ... uhm ... forty-nine kilos, like that, something like that. I'm not sure how they go, but sometimes I don't get them. Interviewer: So if I had a problem . . . going to a shop and buying three packets of something at a dollar forty-nine. John: No, it's the ... you have to double it and add on, like one-and- a-half kilos is so many dollars - say you have three dollars forty-five - sometimes I get them. Interviewer: Like if it's $3.45 a kilo, and you've got to do one-and-a- half kilos? John: Yeah. Data from the Application of a Projective "Write a Letter" Technique In another study, Ellerton (1988) used a projective strategy to investigate the conscious and unconscious reactions of Grade 6 children to the teaching and learning of mathematics. She devised the following task: Imagine that you have a friend who has been ill and has missed about 3 weeks of school. Your friend has sent a message to you, asking you to explain what the class has been doing in mathematics so that he or she can do some extra work at home to catch up. Write a letter to your friend describing what mathematics you've been doing in class. In the letter, don't forget to give examples of some of the mathematics questions that your friend would need to be able to answer so that he or she could catch up. Make sure you explain how to answer these questions. Stephanie, in writing to Kimberly, said that they had been doing a lot of mental and written work in mathematics. Her letter, which was concerned with fractions, is shown in Figure 3. SOME CONCLUDING COMMENTS The data in this paper reflect the fact that many children are unable to "do" fractions; furthermore, it appears to be the case that their lack of understanding is related to their perception of fractions as being solely concerned with conundrums that require the application of fixed rules and terminology that have no meaning for them outside the classroom. Interview data reported in this paper point to an overwhelming belief among students that fractions are all about "getting lowest common denominators," "improper fractions" and remembering and applying rules. The interview with John, for example, suggests that even if students do attempt to apply fraction concepts to some practical situation, they find it difficult to do so because they have had very little opportunity to articulate such relationships. We do not believe that the answer to the challenge of improving the teaching and learning of fractions lies in developing ever more complex models of children's fraction behaviour (and that is why we have not cited papers and books describing detailed theories that purport to account for children's development of rational number concepts). More positively, we do accept the views expressed by Clements and Del Campo (1990), Hiebert and Wearne (1988), Lean, Mousley and Rice (1991), and Mack (1990), that teachers need to recognise that children's fraction concepts are developed through multiple embodiments, and that the concepts should be developed prior to instruction which aims at automatising the application of rote procedures to fraction symbols. Figure 3. Stephanie's letter to Kimberly. The major theoretical challenge, then, so far as the teaching and learning of fractions is concerned, is to help teachers find ways and means to assist students to build on informal knowledge so that they can give meaning to fraction symbols and procedures. Specifically, helping students to develop appropriate mental links between fraction related real-life situations, formal fraction language, and written fraction symbols should be the first step in any agenda for action aimed at improving children's understanding of fractions. Although some work has already been done towards this end (see, for example, Clements & Del Campo, 1989; Mack, 1990), much more is needed if, by the year 2000, schools are to break free from the historical straight-jacket of rote symbol manipulation of fractions. REFERENCES Bourke, S. F., & Lewis, R. (1976). Australian studies in school performance: Literacy and numeracy in Australian schools item report (Vol. 2). 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