Academic journal article Australian Primary Mathematics Classroom

# Fractions as Division the Forgotten Notion? Doug Clarke Explores a Construct of Fractions That Is Very Useful but Often Neglected

Academic journal article Australian Primary Mathematics Classroom

# Fractions as Division the Forgotten Notion? Doug Clarke Explores a Construct of Fractions That Is Very Useful but Often Neglected

## Article excerpt

About 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 context. Although I have seen it used in a variety of ways, I will describe one way in which I use it with teachers and middle school students:

I place three small chairs out the front of the classroom 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.

[FIGURE 1 OMITTED]

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 middle school students accept readily!

Interestingly, the first couple of people to enter often choose the chairs with either one or two blocks of chocolate. Possibly, they think that there is a trick involved and this is some kind of reverse psychology; or is it that they think, "With fractions, the bigger it is, the smaller it is"? (see Roche, 2005)

When we are down to the last two (Belinda and Sandy, say), I ask them each in turn to pause before entering, and ask the rest of the class, who have been observing, to decide where they think Belinda should go and why. I invite individuals to explain their reasoning, and then ask Belinda to move to where she 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 people might distribute themselves. 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 wherever 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 usually at least one situation for which most people are not able to give an immediate answer about how much each person would get. This might be, for example, two blocks shared between three people (see Figure 2) or three blocks shared between five people.

[FIGURE 2 OMITTED]

My experience is that most middle school teachers and almost all middle school students use the same strategy to share two blocks between three people. They solve the problem by "mentally breaking" each block into three pieces, yielding the correct answer of two-thirds of a block each. Few will explain that two things shared between three must be two-thirds of a block each, without calculating.

I then ask the total group, "How many people knew before they came to class or our session today, without calculating, that two blocks shared between three people must mean two thirds each?" Invariably, less than one-tenth of the group claim to know. …

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