Sample Key Stage 3 curriculum framework
Multiplication as scaling
In this section of the Guidance document, the authors reproduce several pages, including the diagram in Figure 2 again, of a section about the ‘key idea’, Understand the mathematical structures that underpin the multiplication of fractions from a Key Stage 3 document about the ‘core concept 2.1’, Arithmetic procedures from ‘theme 2’, Operating on numberB. One might have thought the diagram would belong to Theme 3, Multiplicative Reasoning, but that’s the trouble with mathematical ideas - they don’t fit neatly in any one place!
The suggestions made in these extracts from the Arithmetic procedures document are also rather dogmatic and often unclear. Nonetheless, the extracts initiate a useful discussion about the ideas that one might want students to engage with, or as the document puts it, that ‘students need to understand’. For example, it is suggested that students should be encouraged to think of a calculation like 3 × ¾ in different ways, depending on whether 3 or 3/4 is seen as the operator (Figure 3).
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| Figure 3: Two interpretations of a fractions calculation (p B24 or A92) |
This is very useful in that it helps us to see that students can make sense of mathematics in different ways. However, it seems incomplete, since we can also interpret multiplying by ¾, or ‘¾, of’ as partitioning. This, I would argue, is a particular form of repeated addition, or grouping. However, according to the extract, this is emphatically not the case:
if the fraction is the multiplier, students are forced to re-think multiplication as scaling (as in ‘one third of two’). p B24, A92,
Forced? To be sure, scaling offers a more powerful way of interpreting multiplication by a fraction. However we are not compelled to use it here, since repeated addition, in the form of partitioning, works perfectly well and is more familiar. This is followed by another emphatic statement:
This is important because when both numbers are fractions, multiplication cannot be thought of as ‘groups of’.
Cannot? This is all rather strange, especially as the extract then goes on to consider multiplication of fractions in terms of the area model. This is quite legitimate, but the treatment is unclear. Scaling, which a moment ago was seen as vital, is no longer mentioned, but nor is repeated addition or the Cartesian product, all of which provide valid ways of interpreting the area model. We don’t know which the authors have in mind. Finally, the diagram from Figure 2 reappears, which can be interpreted perfectly sensibly without recourse to scaling or the area model by using partitioning.
The point surely is, that these ideas feed into one another. They are not formed in a rigid order. Familiarity with the area model is useful for interpreting the diagram from Figure 2. But the diagram, and knowledge of partitioning, can also help students to develop a better understanding of the area model.
Similarly, we could start with the multiplication-as-partitioning diagram and use it to enhance students’ understanding of scaling. For example, we could relate the diagram for ⅓ × ⅕ in Figure 2 to one or other diagram shown in Figure 4.
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| Figure 4: Forming a part of a whole, and a part of a part, by scaling |
The boldness of the claims about the need to think in terms of scaling when multiplying by a fraction contrasts with the timidity with which scaling is described. The diagram in Figure 5 (page B25, A93) is designed to show that 4 × ⅓ (grouping) and ⅓ × 4 (scaling) are ‘the same’. The first bar effectively shows the idea of ‘lots of’ or ‘groups of’ ⅓, but it is less clear that the third bar shows 4 being ‘reduced to one third’. It is more easily read in terms of partitioning.
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| Figure 5: multiplication as grouping and scaling (p B25 or A93) |
When a geometric shape is scaled, a crucial outcome is that every feature of the object is scaled. So the diagram would be clearer if we replaced the bottom two bars in Figure 5 with, say, the first or second pair of bars in Figure 6.
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| Figure 6: Multiplication as scaling |
Evidence about scaling
Evidence suggests that the concept of scaling is far from trivial. Consider the item in Figure 7, from the CSMS Ratio test. Here scaling comes into its own, since one cannot enlarge the Curly K by some form of addition. When it was given to representative samples of Year 9 students in 1976 and 2008/9 (for the CSMS and ICCAMS projects) it was answered successfully by only 20% and 13% of students respectively. The issue here is not just about scaling the numbers: students first need to recognise that scaling applies in this situation. The most popular response was purely additive, with 40% of the students in 1976 giving the response 13, rather than 13½.
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| Figure 7: A scaling item from the CSMS Ratio test |
Measurement-as-scaling
This evidence suggests that the advice in the Arithmetic procedures extract that students ‘need to deepen their understanding of multiplication to include scaling’ (page B24, A92) is something of an understatement, as even more so is the notion that this can somehow be ‘mastered’ in Year 7. In turn it is intriguing to find that a Year 4 NCETM guidance documentC (for ‘segment 3.6’) gives more or less the same advice: ‘Children will learn that multiplying a whole number by a fraction can be thought of in two different ways: repeated addition and scaling’ (page C1). The Year 4 document has a similar discussion to the one in the Key Stage 3 Arithmetic procedures document about the two ways of interpreting 3 × ¾ (see above), but this time the scaling example involves the calculation ⅔ × 60, where the result is a whole number. This is illustrated with a diagram (see Figure 8) that is rather better than in the Key Stage 3 extract, though it is not made clear what interpretation of multiplication is offered by the partitioning diagram that accompanies it! It turns out that it is not until we are more than halfway through the 50 page document that there is a switch from partitioning to scaling for interpreting a simple sentence like ‘½ of 10 = 5’, after first replacing it by ‘½ × 10 = 5’:
At this point, introduce the term ‘scaling’ and explain that we can think of this as scaling the number down. Ten has been scaled-down by ½ to make 5. (page C29)
Incidentally, notice the danger here of using such a simple example. What would happen with ‘⅓ × 12 = 4’, say? Has 12 been ‘scaled-down’ by ⅓, or to ⅓? |
| Figure 8: Multiplication as scaling (p C2) |
Another Year 4 documentD (for ‘segment 2.17’) looks at scaling in the context of measurement. Here continuous quantities, such as lengths, are compared. For example, students are asked to compare the height of a 12cm sunflower with its height when it has grown to be 10 times as tall. This allows a nice distinction to be made between repeated addition and scaling: the 120cm mature sunflower is not the same as ten 12cm seedlings, although of course we can still think of its height in that way. The examples in the document, thoughtful though they are, would have been more salient and ‘purposeful’ if they had involved stretching, but this idea is touched on only once.
Interestingly the introduction to this measurement-as-scaling document (page D1) claims that scaling has already appeared in two Year 2 documents, one involving doubling and halving of discrete quantities (‘segment 2.5’) and one involving multiplying and dividing by 10 and 100 (‘segment 2.6’). Using a simple operation like doubling, and applying it to discrete quantities is not a very compelling way of introducing scaling, especially when the document goes on to say that the actual term ‘scaling’ is not introduced to students until the Year 6 ‘segment 2.27’ document - though we know that this is not entirely correct. This documentE, Scale factors, ratio and proportional reasoning, introduces the notion of ‘scale factor’ by transforming a unit square (page E27). This means it is not entirely clear whether the resulting figure is due to scaling the unit square or to duplicating it several times. So here is another example where keeping things simple, or taking ‘small steps’, obscures what is going on.
I doubt whether any of the examples in these documents would convince me of the need to switch from the cosy world of repeated addition to the potentially more demanding world of scaling. Nonetheless, the notion that students should meet scaling in Year 4, or even Year 2, but that they will still need to grapple with it in Year 7 (and beyond!) is, in principle, fine by me. But how does this fit the rhetoric of ‘deep, embedded and sustainable understanding’ (page A12) and the ‘vital’ importance of prerequisite knowledge? What is the worth of this fanciful language, and where does it come from?!
The Arithmetic procedures document makes (indirect) reference to the earlier Year 4 document and to a Year 6 documentF that addresses multiplication of a fraction by a fraction. Scaling is mentioned there too, but again we start with examples that are so simple it is difficult to see what is going on. Thus ‘½ of ⅓’ is construed as this:
If one of the thirds is made half as long, what fraction of the whole will one of the new equal parts be? (p F5)
Leaving aside that if we scale one of the thirds there is only one rather than several ‘new parts’, when I think of halving or making something ‘half as long’, my first thought is of cutting something in two, not shrinking it to half its size.
The accompanying diagram is shown in Figure 9, but is the blue 1/3 segment being scaled here, or partitioned?
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| Figure 9: ½ of ⅓ as scaling? (p F6) |
The Arithmetic procedures document also provides a list of 15 bullet points under the heading ‘Prior learning’, thought the only one directly relevant to multiplication of fractions is this (Figure 10):
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| Figure 10: Prior learning outcome, Upper KS2 (p B3) |
In the Guidance document, many of the Arithmetic procedures pages are reproduced verbatim, but strangely it has its own list of ‘prior learning’ under the slightly revised heading ‘Arithmetic procedures including fractions’. This provides 11 bullet points, only one of which is the same as those in the Arithmetic procedures list from 2 years earlier. It also doesn’t refer to the aforementioned Year 4 and Year 6 documents, but instead mentions some ‘Key Stage 2 ready-to-progress criteria’. None of these items seem directly relevant to multiplication of fractions, which leaves one wondering what to make of this stirring sentence from the Guidance document:
Being clear about the important prerequisite knowledge from Key Stage 2 and allowing time to consolidate this and then build new Key Stage 3 ideas on these firm foundations is vital. (page A16)
Summary and conclusion: see the next post
