USING THE DOUBLE NUMBER LINE TO MODEL MULTIPLICATION

Dietmar Küchemann, Jeremy Hodgen and Margaret Brown King’s College London, UK

As part of the research of the ICCAMS project, we have been working with Year 8 students (age 12–13 years) to explore their understanding of multiplicative structures and to develop teaching materials to enhance this understanding. In this paper we discuss some of our work related to the double number line and its use as a model for multiplication.

INTRODUCTION Vergnaud (2009) argues that a concept’s meaning arises from a variety of situations and puts forward the notion of a conceptual field, as a way of binding situations and concepts together. In the case of multiplicative structures, this conceptual field is extremely complex, as can be seen, for example, from the myriad of models put forward by the Rational Number Project (eg Behr et al, 1991), or from the intricate ‘learning trajectories maps’ woven by Confrey et al (2009). Anghileri and Johnson (1992) identify 6 key aspects of multiplication (and division), which they list as:

equal grouping, allocation/rate, number line, array, scale factor, Cartesian product. They argue that “children will need to become familiar with the different situations that embody these aspects” (ibid, p170). Davis (2010) describes how ‘concept study’ can be used to draw out and develop teachers’ mathematical knowledge for teaching. He discusses how a group of teachers worked on the concept of multiplication. One of their activities was to produce a list of ‘realisations of multiplication’, which included these items:

grouping process; repeated addition; times-ing; expanding; scaling; repeated measures; making area; making arrays; proportional increase; splitting; skip counting; transformations; stretching/compressing a number line.

Of course, this list of realisations is in no sense definite and Davis makes the important point that the teachers’ conceptions of multiplication (and how it might be engaged with in the classroom) was continually shifting over time. We would argue that the same will apply to school students, though depending on the extent to which they are allowed to engage with these ideas. In this regard, it is interesting to note the guidance on multiplication offered by the National Numeracy Strategy, which was set up by the UK government to offer advice on the teaching of mathematics. In the Strategy’s Framework document for primary schools (DfEE, 1999), it is suggested that students as early as Year 2 (6–7 year olds) should understand multiplication in terms of repeated addition and rectangular arrays. Similar advice is offered for Year 3 students, and it is suggested that the idea of scaling should also be introduced. Some advice on understanding multiplication is also given for older students, but this is subordinated to advice on performing calculations. Thus the use of models such as

the array is not mentioned again in this document, nor in the Strategy’s subsequent Framework document (DfEE, 2001) for early secondary school (Years 7, 8 and 9). One gets the impression that by the time students reach secondary school, multiplication is somehow meant to be ‘understood’ and no longer needs to be supported by models. So for example, it is stated that Year 7 students should understand that “multiplication is equivalent to and is more efficient than repeated addition” (ibid, Section 4, p82), but there is little indication of what ‘multiplication’ means here, ie in what situations it might be modeled or realised. Unfortunately, the paucity of models in the Framework documents is mirrored in most current secondary school mathematics textbooks in the UK. Extensive work on the didactical use of models has been undertaken in Holland, from the perspective of RME (see eg Van den Heuvel-Panhuizen, 2002). This work makes an interesting distinction between ‘models of’ and ‘models for’, whereby the development of a more formal mathematical understanding is seen as a shift from the construction and use of the former to the latter.

THE DOUBLE NUMBER LINE As part of the work of the ESRC-funded ICCAMS project1 we have been working with Year 8 students (12–13 year olds) to explore their understanding of multiplicative structures, with the aim of developing teaching materials to enhance this understanding. From the work that we have undertaken so far, including exploratory group interviews and short teaching sequences, but also the large-scale use of written tests, it is clear that many secondary school students have a very shaky understanding of multiplication, based in only a very limited way on models that could support and develop their understanding. We think that one interesting and important model for multiplication is provided by the double number line, and it is this model that we focus on in this paper. The double number line also serves as a model of a variety of contexts that should be reasonably accessible to students (eg scales on a map; a one-way stretch; conversions, such as £ to €). An attraction of this model is that it offers a fairly gentle way of departing from an additive approach to multiplication. Repeated addition (perhaps modelled by skips along a number line) provides a salient and reliable (and quite efficient) model for the multiplication of (small) whole numbers (eg 3×7 can be thought of as 3 skips, each of 7 units, along the number line, starting at 0). However, even in situations that involve multiplication by simple rational numbers, eg ×1.5, we have found that many students stick to an additive strategy (in this case, ‘rated addition’). Consider the item in Fig 1 (below), taken from the CSMS Ratio test (Hart, 1981). This was answered correctly by 14 % of a representative sample (N=309) of Year 8 students in 1976, and by a similar proportion (12%) of a representative sample (N=754) of Year 8 students in 2008. When we have interviewed students on this item, we have found that many students use an inappropriate ‘addition strategy’ (Karplus & Peterson, 1970) arguing along the lines of ‘8+4 = 12, so RS = 9+4 = 13’.

Those who do solve the task successfully tend to use an argument of this sort: ‘8 + half of 8 = 12, so RS = 9 + half of 9 = 9 + 4.5 = 13.5’. Students tend not to go for the

more direct approach of scaling by ×1.5, along the lines of ‘12 = 8×1.5, so RS = 9×1.5 = 13.5’. It is worth pointing out that this scaling approach is actually more appropriate here, since rated addition does not fit the geometric situation: one can’t really add a curved line of length 8 units to a smaller version of the line of length 4 units, and thereby make a larger version of the line of length 12 units! It is perhaps for this reason that the task is so difficult.

Fig 1: Curly Ks item (Hart, 1981)

Thus, in a study using items derived from Hart (1981) but with the numbers more closely matched, Küchemann (1989) found that while a Ks items was answered correctly by only 25% of students (N=153), a recipe item, where rated addition makes perfect sense, was answered correctly by 64% of students from a comparable sample (N=154). The double number line provide a neat way of representing (or indeed embodying) multiplicative relations, such as ×1.5. Consider the pair of lines A and B in Fig 2a, where 0 and 8 on line A are lined-up with 0 and 12 on line B.

0 12

8

B

A0

0 1 2 3 4 5 8 12

8

B

A

117 106 9

0 2 4 6 71 3 5

Fig 2a Fig 2b

We know that 12 is 8×1.5, and we can thus make the number lines represent the mapping ×1.5 (or x → 1.5x, or y = 1.5x, etc) by drawing linear scales on each line (Fig 2b): any number on line A is now mapped onto a number positioned directly below it on line B that is 1.5 times its value. The great strength of this representation is that it is not just mapping 8 onto 12, but showing the mapping ×1.5 regardless of any particular pair of numbers that we may be considering. Thus the operation ×1.5 is brought to the fore. Of course, this may not be perceived in this way by all students. The diagram can, for example, be read as showing a move along a number line from 0 to 12, in 8 skips of 1.5 units (with line A representing the number of skips and line B the distance skipped); in other words the diagram can also be interpreted as showing repeated addition. Scales on a map are well known examples of double number lines. One kind shows distances on the map (measured in cm, say) and corresponding distances on the object depicted by the map (measured in km, say). The ruler in Fig 3a is of this sort. Another kind shows how distances represented on a map can be read in different units (eg feet and metres). Fig 3b shows a scale of this sort used by Google Maps.

Fig 3a: A ruler for scaling feet to mm Fig 3b: A scale used by Google Maps

A double number line can also be used to represent an enlargement, or more specifically, a one-way-stretch. An example is shown in Fig 4b, which shows the result of stretching scale A in Fig 4a by a factor ×1.5. (Note the similarity to Fig 2b.)

0 1 2 3 4 5 8 12 B

A

117 106

1 2 3 4 5 876

9

0

0 1 2 3 4 5 8 12 B

A

117 106 9

1 2 3 4 5 8760

Fig 4a: Two identical scales, A and B Fig 4b: A ×1.5 stretch applied to scale A

RELATED WORK Our approach to the didactical use of models is perhaps similar to the Dutch approach, as embodied by RME, though we would not be as strict about not handing ready-made models to students. Van den Heuvel-Panhuizen (2002), in describing RME, gives a nice illustration (ibid, Fig 11, p12) of how the number line can be used to support students’ learning. She adds that

The number line begins in first grade as (A) a beaded necklace on which the students can practice (sic) all kind of counting activities. In higher grades, this chain of beads successively becomes (B) an empty number line for supporting additions and subtractions, (C) a double number line for supporting problems on ratios, and finally (D) a fraction/percentage bar for supporting working with fractions and percentages. (p12)

The fraction/percentage bar and, to a lesser extent, the double number line, feature strongly in materials developed by RME, as can be seen in the TAL-project materials in the Netherlands (eg, Van Galen et al, 2008) and in the Mathematics in Context materials in the USA (eg, Keijzer et al, 2006). The use of the double number line is not yet widespread in the UK, although the ‘fraction wall’, which has similarities to the fraction bar, has been around for a long time (eg, Watt et al, 1967, p103). However, since the introduction of the National Numeracy Strategy (DfEE, 1999) the single number line has been used extensively in UK primary schools, which may partly explain the dramatic rise in facility of the CSMS Decimals item (Brown, 1981) shown in Fig 5. When we gave this to a representative sample (N=294) of Year 8 students in English schools in 1977, it had a facility of 37%, but with a similar sample (N=767) in 2008/2009 this had risen to 78%2.

Fig 5: CSMS Decimals item 6e

TASKS AND FINDINGS Our data on Year 8 students’ use of the double number line are still exploratory. However, we have sufficient data to give a sense of some of the affordances of the double number line and of some of the difficulties that students encounter. Put another way, our current data suggest that the development of classroom activities involving the double number line is worth pursuing for us, but that learning to construct and use the model may be far from trivial for students. We report informal data from three sets of tasks involving students from two classes. The first task asks students to use a double number line to represent equivalent fractions or to evaluate a percentage. The second task involves a one-way-stretch, while the third asks students to convert information on a map from metres to feet. Fractions and percentage tasks The task in Fig 6a gives some indication of whether students can, in certain circumstances, appreciate the need for a linear scale (although it clearly tests much more than this). The task in Fig 6b can be solved in a variety of ways. One approach

Fig 6a

Fig 6b is to see the double number line as representing a mapping, whose value (leaving aside the units, or in this case the %) is given by ×100÷40, ie ×2.5. We gave these tasks to an above-average attaining Year 8 class as a homework. The tasks were given cold, ie without any kind of introductory work. For the task in Fig 6a, most students made effective use of the given scale (of 12ths) to mark off the required 4ths (as in Fig 7a). Some partitioned the line into 4 parts but seemed to ignore the given scale (Fig 7b), while a few students tried to use the given scale but did so erroneously (Fig 7c).

Fig 7a Fig 7b Fig 7c

Fig 8 shows responses to the percentage task shown in Fig 6b. Fig 8a shows a successful response to the item, but it seems likely that the student achieved this by making use of their (impressive) knowledge of fractions, decimals and percentages, rather than making use of the structure embodied by the double number line. On the

other hand, the students giving the responses in Figs 8b and 8c do seem to have made use of the double number line and interpreted it successfully as representing the mapping ×2.5 or, in the case of Fig 8c, seeing this in terms of the more grounded, rated addition approach of ‘double and add a half’.

Fig 8a Fig 8b Fig 8c

Elastic strip task Part of the elastic strip task is shown in Fig 9. We tried the task as a starter activity with several Year 8 classes and also interviewed small groups of students on the task. In one Year 8 class of roughly average attainment we displayed the task on the whiteboard and also acted it out using a long elastic strip. The demonstration was quite dramatic and intriguing but the task still proved very demanding for this group. After some small-group discussion, three possible solutions emerged for the new distance of the red mark from the left hand edge, namely 16cm, 12cm and 9cm. The class teacher wrote these on the board and asked for a vote. The three responses received 12, 4 and 9 votes respectively. The 16cm response comes from using the addition strategy (either ‘the end has moved 30cm–20cm=10cm, so the 6cm mark will move 10cm’, or, less often, ‘the 6cm mark is, and will remain, 14cm from the right hand side’). We were aware of only 3 students who had themselves come up

with the correct value of 9cm, though there might of course have been others. Two had used rated addition (‘The stretched strip is half as long again; 6cm + half of 6cm = 9cm’) while one student had used an argument based on the idea that if the strip were 10cm long the red mark would be 3cm from the end (this is quite a sophisticated argument since it would not be possible to enact this in practice).

Fig 9: The elastic strip task

Fig 10 shows a drawing that arose during an interview with 4 students (A, B, C and Z) from the previously mentioned above-average Year 8 class. The drawing of the 20cm strip and the 30cm stretched version was done by the interviewer. The vertical marks labelled A, Z, B, C, on the 30cm strip were drawn by the students. We started by considering the image of a line 5cm from the left hand end of the 20cm strip after the strip had been stretched to 30cm. Reading from the left, the first set of vertical

lines (labelled A, Z, B and C) on the 30cm strip were students’ estimates of the position of the image of the 5cm line. [The second set of lines labelled C, B, Z and A

Fig 10: Interview responses to the elastic strip task

were their initial estimates for the image of the central dot drawn 10cm along the 20cm strip.] Student C justified his mark in terms of the addition strategy (ie the right hand end of the strip had moved 10cm, so the 5cm mark would move 10cm). Other students thought it would move a bit less than this (and hence drew their images to the left of C’s image). However, initially their estimates were purely qualitative, though eventually, after considering the image of the midpoint, they adopted a successful rated-addition approach (the 5cm mark moves half this distance, ie 2.5cm, so it ends up 7.5cm from the left hand end). A map task: Westgate Close Our third task was based on a map of a short private road, Westgate Close. One version of the task is shown in Fig 11. As with the elastic strip task, we used this as a starter with some Year 8 classes. We would argue that it is more obvious in this task, than with the elastic strip, that the scales are linear (it would be very odd if for part of the road, a given number of feet matched a certain number of metres but that somewhere else on the road the same given number of feet matched a quite different number of metres!). Nonetheless, the addition strategy was still common here, with students calculating that the distance of El’s house along Westgate Close was 107 ft

from Roman Road (50–15=35, 35+72=107, or 72–15=57, 50+57=107). However, the task also provoked an interesting variety of correct, or partially correct strategies. For example, several students in a low attaining Year 8 class, estimated the distance by marking-off 50 ft lengths. Often this lead to quite good Fig 11: Westgate Close

estimates (Fig 12a), of between 200ft and 250ft (the actual answer is 240ft). Though these students simply ignored the information about the distances in metres, their estimates might stand them in good stead once they do try to calculate, especially if they use an inappropriate strategy.

Fig 12a Fig 12b

As with Elastic Strip, the task proved difficult for the average-attaining Year 8 class. One student (Fig 12b) used what might be called a ‘function’ approach to arrive at the multiplier 3.3 (50 ≈ 15×3.3), and used this to calculate 72×3.3 (but as can be seen, he made arithmetic and transcription errors to arrive at a widely-off answer of 91.87ft). Another student used an effective rated-addition strategy to move along the road in steps of 15m/50ft, to arrive at 75m=250ft; however, having thus overshot the 72m distance by 3m, she then subtracted 3 from 250 to arrive at 247ft.

In the above-average attaining Year 8 class, one pair of students used a ‘scalar’ approach; they determined that 72 is 4.8 times 15, and used this multiplier to calculate the distance of El’s house in feet: 50ft × 4.8 = 240ft. This pair also made use of a ratio table (Fig 13) which may well have helped them structure their work.

Fig 13: Ratio table

However, not all students in this class were immune from using the addition strategy, and one pair used a hybrid scalar/addition strategy to come up with an answer of 212ft, based on the observation that 72 can be expressed as 15×4 + 12 (leading to 50×4 + 12 = 212).

DISCUSSION The findings reported above suggest that some students can make productive use of the double number line as a model of contexts that involve multiplication-as-scaling, even when they may have had little or no prior experience of using it in this way. But equally, there are students who do not readily see the multiplicative structure of the contexts (or of the model), which suggests that developing this insight is not a trivial matter. Nonetheless, we would argue that is worth giving students experience of the model (both as a model of multiplication-as-scaling contexts, and as a model for multiplication-as-scaling concepts), since scaling is an important aspect of multiplication, and since it is important to be able to discern whether a context involves scaling. Also, the model is well suited to showing the contradictions inherent in the addition strategy when applied to scaling contexts (for example, in the

case of a ×1.5 stretch applied to a 20cm elastic strip fixed at one end, if students use the addition strategy to argue that points on the strip will move 10cm further from the fixed end, it is fairly easy to provoke a contradiction by asking what happens to the mid-point of the strip, or to a point very near the fixed end). The double number line also links nicely to other powerful representations and we should help students develop these, in particular links with ratio tables, and with mapping diagrams and Cartesian graphs (the latter are also based on number lines, but in the case of Cartesian graphs the lines are orthogonal rather than parallel). At the same time the double number line may not always be the most appropriate model for representing and/or solving multiplication tasks, and we would argue that students, including those at secondary school, need extensive and ongoing experience of other models of multiplication, in particular arrays and the area model.

NOTES 1. ICCAMS (Increasing Student Competence and Confidence in Algebra and Multiplicative Structures) is a 4-year research project funded by the Economic and Social Research Council as part of a wider initiative aimed at identifying ways to participation in Science, Technology, Engineering and Mathematics (STEM) disciplines. Phase 1 of the project consists of a large-scale survey of 11-14 years olds’ understandings of algebra and multiplicative reasoning in England. This is followed in Phase 2 by a collaborative research study with two teacher-researchers in each of four secondary schools. The aim is to examine how formative assessment can be used to improve attainment and attitudes, and finally how the work can be disseminated on a larger scale.

2. In 2008/2009, we re-administered three of the CSMS tests that had been developed in the 1970s - Algebra, Decimals, and Ratio. In general performance on the tests was very similar to the 1970s; the most notable exceptions were some Decimals items that related to measurement, where performance improved in 2008/2009.

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