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A bout fifteen years ago, I discovered an interesting activity in some materials that Malcolm Swan from the Shell Centre (University of Nottingham, UK) had developed for the English National Curriculum Council in 1991. The activity, one which has been used by several presenters in professional development workshops in Australia in recent years, involves sharing chocolate in a problem solving DOUG CLARKE context. Although I have seen it used in a variety of ways, I will describe one way in which I use it with explores a construct teachers and middle school students: of fractions that The chocolate block task is very useful but I place three small chairs out the front of the class- often neglected. room as shown in Figure 1. I explain to the group that I am placing one block of chocolate on the first chair, two blocks on the second, and three on the third. I deliberately use chocolate that is not already subdivided into separate pieces, as this would “blur” the concepts which I hope will emerge. I ask ten volunteers to leave the room, spreading the chairs out to give plenty of room. I then invite the ten to return, one at a time, and choose a chair at which to stand, knowing that when everyone has entered the room and made a decision they get to share the chocolate at their chair. I explain to them that the assumption is that “more chocolate is better,” an assumption that most teachers and almost all 4 APMC 11 (3) 2006 Fractions as division: The forgotten notion? selves. It is possible that the last person is faced with three equivalent alternatives if there are, respectively, 5 people standing with the 3 blocks, 3 people standing with the 2 blocks, and 1 person standing with the 1 block. In each case, the person would get half of a block of chocolate wher- ever they choose to go. The calculation involved in sharing the one block is, of course, relatively straightforward, as is the case where the two blocks are shared between two or four people. However, there is Figure 1 usually at least one situation for which most people are not able to middle school students accept readily! give an immediate answer about Interestingly, the first couple of people to enter how much each person would get. often choose the chairs with either one or two blocks This might be, for example, two of chocolate. Possibly, they think that there is a trick blocks shared between three involved and this is some kind of reverse psychology; people (see Figure 2) or three or is it that they think, “With fractions, the bigger it is, blocks shared between five the smaller it is”? (see Roche, 2005) people. When we are down to the last two (Belinda and My experience is that most Sandy, say), I ask them each in turn to pause before middle school teachers and almost entering, and ask the rest of the class, who have been all middle school students use the observing, to decide where they think Belinda should same strategy to share two blocks go and why. I invite individuals to explain their between three people. They solve reasoning, and then ask Belinda to move to where she the problem by “mentally breaking” wishes. I then pose the same question in relation to Sandy’s decision, and after a similar discussion, I ask Sandy to take her place. I then invite the individuals or group at each chair or table to discuss how much chocolate they would finally get, and how they know. The remainder of the class is also asked to discuss how much chocolate participants at each chair would receive. Of course, there are many different ways in which the ten Figure 2 people might distribute them- APMC 11 (3) 2006 5 Fractions as division: The forgotten notion? each block into three pieces, Queensland, who had just participated in this activity yielding the correct answer of two- with me. After my debrief, he asked the indulgence of thirds of a block each. Few will the volunteers for a moment. He asked each group to explain that two things shared lift up their chairs with the chocolate on them. For the between three must be two-thirds chair with three blocks of chocolate and five people, of a block each, without calculating. he showed us in a powerful visual form: the three I then ask the total group, blocks of chocolate on top; the chair which formed “How many people knew before the vinculum of the fraction (the line dividing the they came to class or our session numerator and the denominator); and the five people today, without calculating, that underneath. He had created a stunning and hopefully two blocks shared between three memorable image of three blocks shared between five people must mean two thirds giving three over five or 3/5. It was one of those each?” Invariably, less than one- “magic” moments, and the other teachers present and tenth of the group claim to know. I were all most impressed. This leads me to one of my main intentions in posing the task. I believe that the notion of Key constructs of fractions “fraction as division” or “fraction as quotient” is not a common In what follows, the focus will be on fractions specif- construct in most people’s minds. ically, although the research is often framed in terms If we understand, for example, of rational numbers. [Rational number is the name that one meaning of 2/3 is “2 given to any number that can be expressed as the ratio divided by 3,” then strategies in of two whole numbers. So, for example, 0.6, 4/7 and the above activity become obvious 158% are all rational numbers, while the square root quite quickly, assuming that there of 7 is not, as it can not be expressed in this way.] are not complicated comparisons Fractions are difficult to teach and to learn. There such as comparing two blocks is a substantial research literature on key concepts of shared between three and three fractions and how these develop, and yet this research blocks between five (i.e., has had little impact on state and national curriculum comparing 2/3 and 3/5). documents and even less impact on classroom prac- Ironically, in teaching students tice. Much of the research on fractions is contained to convert 3/7 to a decimal, we within broader research programs in relation to the encourage them to use a calculator rational number system. to divide the 3 by the 7, thus It is not that research has not focused on the notion invoking the construct of fractions of fractions as division, equal sharing and partitioning as division, without possibly (see, e.g., Empson, 2001; Gould, 2005; Siemon, 2003). thinking about why. Similarly, The issue is the lack of impact, to this point, of 17/5 can be represented as the research on practice. mixed number 3 2/5, and students In recent years, the power of the one-to-one assess- tend to “convert” this by dividing ment interview as a professional learning tool for 17 by 5 — the same principle teachers has been increasingly recognised in Australia again. and New Zealand (Bobis, Clarke, Clarke, Gould, As an interesting postscript to Thomas, Wright, Young-Loveridge & Gould, 2005; this activity, I would like to share Clarke, Roche & Mitchell, 2005; McDonough, Clarke & a powerful visual image that was Clarke, 2002). suggested by a teacher in As part of a research project, the team at Australian 6 APMC 11 (3) 2006 Fractions as division: The forgotten notion? Catholic University (Melbourne) has been developing on the number line with whole a range of task-based, one-to-one interview assess- numbers, decimals, etc.”) and ment tasks in fractions and decimals. These tasks focus operator which enlarges or on the “big ideas” which underpin these topics and reduces the size of something common misconceptions, (e.g., Clarke, Roche, (e.g., determining 3/4 of 28 Mitchell & Sukenik, 2006; Mitchell & Clarke, 2004; metres, where the fraction is oper- Mitchell, 2005, Roche, 2005, 2006; Roche & Clarke, ating on the 28) are often absent. 2004). In this research, we have been careful to try to Kilpatrick, Swafford and Findell address important interpretations or constructs of frac- (2001) argue that from a student tions. point of view, rational numbers Thomas Kieren (1976) identified seven different (and therefore fractions) are not interpretations (or constructs) of rational numbers. single entities but have “multiple Different scholars have summarised or re-framed these personalities… the task for over the years (e.g., Post, Cramer, Behr, Lesh & Harel, students is to recognise these 1993). For the purposes of this article, I want to focus distinctions, and at the same time on fractions specifically, and consider five different to construct relations among them interpretations of them (see Figure 3). that generate a coherent concept” (p. 233). They note the relative complexity compared with whole numbers, and that students need to understand that the numerator and denominator are related through multiplication and division, not addition. Students also need to come to grips with a variety of discrete and continuous models, and some are Figure 3 more helpful than others. Within the part-whole interpretation, In introducing students to fractions, I believe that circular models are extremely the part-whole comparison dominates the way in common, when our research and which fractions are presently taught and therefore that of others (see, e.g., Moss & learned. In focusing on part-whole, many teachers Case, 1999; Witherspoon, 2002) provide students with prepared wholes (sometimes have found that this is a poten- discrete models such as counters, sometimes contin- tially unhelpful model, given the uous models such as commercial “pies” or rectangles), difficulty with which students “cut and then ask them to identify and name particular up” the circle, and interpret the parts of the whole. There is usually less emphasis result. Take a moment to consider within this part-whole focus on flexible movement how you would use circles to between whole and part, and between one part and convince a middle school student other part (e.g., if the brown is 4/3, what rod is one? of the relative size of 2/3 and 3/5, If the purple rod is 1/2 which rod represents 3/4?) or fold a circle into thirds. A nice In the same way that quotients (or fraction as divi- classroom activity which gets at sion) may not be given sufficient attention, the notions the notion of fraction as operator of measure (which can be thought of as “fraction as a is the “Estimation of Fractions” number that can be placed in its appropriate position activity from the Mathematics APMC 11 (3) 2006 7 Fractions as division: The forgotten notion? Curriculum and Teaching Program person would get five thirds, which could form one materials, which involves initially and two-thirds blocks. finding 2/5 of the way across the No doubt, the reader may have thought of other blackboard (Lovitt & Clarke, 1988). directions in which this activity might proceed, but even to this point, I would claim that this one lesson has enabled the emergence of a range of important Taking the activity ideas, and broadened hopefully the notion of what fractions are all about for many students. a little further In debriefing the chocolate game, I also encourage students to think Another problem worth posing about their answers to the to middle school students following questions: • (directed to the individuals at A task that forms part of the interview we are devel- the chairs) If, at the end, you oping, originally taken from Lamon (1999), which had the choice to move to a usually brings out a range of interesting responses, is different chair, would you do the following: so? We explain that three pizzas are shared evenly • (directed to the students between seven girls, while one pizza is shared evenly watching the task) Where between three boys. The question is posed: “Who gets would you choose to stand in more pizza: a boy or a girl?” the queue? Is it best to go first Students who understand the notion of fraction as or last? division in a way that they can use it, will quickly • What strategies would you use conclude that each girl gets 3/7 of a pizza and each if you were in the line? boy gets 1/3 of a pizza, and then think about which of I then either act out or discuss these two fractions is larger. If they “benchmark” to the following two situations: one-half, they will usually conclude that 3/7 is larger 1. The case where the chocolate is because it is almost one-half. Not surprisingly, in light out of the packet and clearly of the earlier discussion, few students or teachers use already subdivided. In this this strategy, unless they have recently played the discrete case, the students are chocolate game! now using the fraction as an However, other more interesting methods often operator notion, as they calcu- arise. For example, we are always pleased when a late, say, 1/3 of 24 blocks or student will give one of the following two responses: 1/5 of 20 blocks. • “Well, three boys share one pizza, so I’ll give the 2. The case where there are more first three girls the first pizza, and the next three blocks of chocolate at a chair girls the next pizza, leaving a whole pizza for the than people. In this case, a seventh girl, which must mean [on average] that the context is available to discuss girls get more.” improper fractions, where the • “Seeing as the boys get one-third each, I imagine mixed number equivalent is dividing the girls’ pizzas into thirds, which would either obvious or can be easily give a total of nine thirds. As there are seven girls, determined. For example, five there will be two thirds left over, so [on average] blocks shared between three is the girls get more.” 5/3, but by breaking each block Taking the activity a little further, students can be into three, we can see that each challenged to follow up their answer to “who gets 8 APMC 11 (3) 2006 Fractions as division: The forgotten notion? Kieren, T. (1976). On the mathematical, more?” with a quantification of how much a girl and cognitive and instructional founda- boy get, respectively. Further, how much more does tions of the rational numbers. In R. A. Lesh (Ed.), Number and one get than the other? Measurement: Papers from a Research In an easier variation of this task, we asked 323 Workshop (pp. 101–144). Athens, Grade 6 students to indicate how much each person Georgia: ERIC/SMEAC. Kilpatrick, J., Swafford, J. & Findell, B. would get if three pizzas were shared between five (Eds) (2001). Adding It Up: Helping people (Clarke, Roche, Mitchell & Sukenik, 2006). Children Learn Mathematics. Only 30% of students at the end of the Grade 6 year Washington, DC: National Academy Press. could give a correct answer to this question. 12% did Lamon, S. (1999). Teaching Fractions and so mentally, while 18% used a drawing to reach their Ratios for Understanding: Essential answer. Content Knowledge and Instructional Strategies for Teachers. Mahwah, NJ: The reader is invited to try any of the problems Lawrence Erlbaum. given in this article with individuals, small groups or Lovitt, C. & Clarke, D. M. (1988). The Mathematics Curriculum and the whole class, and explore their potential for Teaching Program Professional assessing and developing student understanding, Development Package: Activity Bank leading to a broader, more connected and applicable Vol. 1. Canberra, Australia: Curriculum Development Centre. notion of fractions. As Kilpatrick, Swafford, and McDonough, A., Clarke, B. A. & Clarke, Findell, (2001) note, “sharing can play the role for D. M. (2002). Understanding assessing rational numbers that counting does for whole and developing young children’s mathematical thinking: The power of numbers” (p. 232). the one-to-one interview for preser- vice teachers in providing insights into appropriate pedagogical prac- tices. International Journal of Acknowledgement Education Research, 37, 107–112. Mitchell, A. & Clarke, D. M. (2004). When is three quarters not three I am grateful to my ACU colleagues Anne Roche and quarters? Listening for conceptual Annie Mitchell for helping me to adapt the tasks understanding in children’s explana- referred to in this article. tions in a fractions interview. In I. Putt, R. Farragher & M. McLean (Eds), Mathematics Education for the Third Millennium: Towards 2010 References (Proceedings of the 27th annual conference of the Mathematics Bobis, J., Clarke, B. A., Clarke, D. M., Gould, P., Thomas, G., Education Research Group of Wright, R., Young-Loveridge, J. & Gould, P. (2005). Australasia, pp. 367–373). Townsville, Supporting teachers in the development of young children’s Queensland: MERGA. mathematical thinking: Three large scale cases. Mathematics Moss, J. & Case, R. (1999). Developing Education Research Journal, 16(3), 27–57. children’s understanding of the Clarke, D. M., Roche, A., Mitchell, A. & Sukenik, M. (2006). In J. rational numbers: A new model and Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds), an experimental curriculum. Journal Proceedings 30th Conference of the International Group of for Research in Mathematics Psychology of Mathematics Education (Vol. 4, pp. 425–432). Education, 30, 122–147. Prague: PME. National Curriculum Council (1991). Clarke, D. M., Mitchell, A. & Roche, A. (2005). Student one-to- Mathematics Programme of Study: one assessment interviews: A powerful tool for teachers. In J. Inset for Key Stages 3 and 4. York, Mousley, L. Bragg & C. Campbell (Eds), Mathematics: UK: Author. Celebrating Achievement (pp. 66–80). Melbourne: Post, T., Cramer, K., Behr, M., Lesh, R. & Mathematical Association of Victoria. Harel, G. (1993). Curriculum implica- Empson, S. (2001). Equal sharing and the roots of fraction equiv- tions of research on the learning, alence. Teaching Children Mathematics, 7(7), 421–425. teaching and assessing of rational Gould, P. (2005). Really broken numbers. Australian Primary number concepts. In T. Carpenter, E. Mathematics Classroom, 10(3), 4–10. Fennema & T. Romberg, Rational Numbers: An Integration of Research, (pp. 327–361). Hillsdale, NJ: Lawrence Erlbaum Associates. APMC 11 (3) 2006 9 Fractions as division: The forgotten notion? Roche, A. (2005). Longer is larger: Or is Siemon, D. (2003). Partitioning: The missing link in building frac- it? Australian Primary Mathematics tion knowledge and confidence. Australian Mathematics Classroom, 10(3), 11–16. Teacher, 59(2), 22–24. Roche, A. & Clarke, D. M. (2004). When Siemon, D., Izard, J., Breed, M. & Virgona, J. (2006). The devel- does successful comparison of deci- opment of a learning assessment framework for multiplicative mals reflect conceptual thinking. In J. Novotna, H. Moraova, M. Kratka & N. understanding? In I. Putt, R. Farragher Stehlikova (Eds), Proceedings of the 30th Conference of the & M. McLean (Eds), Mathematics International Group of Psychology of Mathematics Education Education for the Third Millennium: Vol. 5, (pp. 113-120). Prague: PME. Towards 2010 (Proceedings of the Witherspoon, M. L. (2002). Fractions: In search of meaning. In D. 27th annual conference of the L. Chambers (Ed.), Putting Research into Practice in the Mathematics Education Research Elementary Grades (pp. 133–136). Reston, VA: National Group of Australasia, pp. 486–493). Council of Teachers of Mathematics. Townsville, Queensland: MERGA. Roche, A. & Clarke, D. M. (2006). When successful comparison of decimals Doug Clarke is a Professor of Education and directs the doesn’t tell the full story. In J. Novotna, H. Moraova, M. Kratka & N. Mathematics Teaching and Learning Centre at Australian Stehlikova (Eds), Proceedings of the Catholic University (Melbourne). 30th Conference of the International <[email protected]> Group of Psychology of Mathematics Education Vol. 4, (pp. 425–432). Prague: PME. 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