© ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions DECONSTRUCTING CALCULATION METHODS, PART 4: DIVISION In the final article of a series of four, Ian Thompson deconstructs the primary national strategy’s approach to written division. The other three articles in the series appeared in MT202, MT204 and MT206. In the introductory section to division on page 16 However, although 84 Ϭ 7 is equivalent to (70 Ϭ 7) of the primary national strategy’s ‘Guidance paper’ ϩ (14 Ϭ 7), it is not equivalent to (84 Ϭ 4) ϩ Calculation (DfES, 2007), we are informed that: (84 Ϭ 3). This suggests opportunities for children ‘These notes show the stages in building up to long to make inappropriate partitionings: they need to division through Years 4 to 6 – first long division remember that only the dividend and not the TU ÷ U, extending to HTU ÷ U, then HTU divisor can be partitioned. ÷ TU, and then short division HTU ÷ U’. The second problem with this algorithm is that This progression seems to me to be somewhat children are expected to be able to make ‘non- ‘logically-challenged’: why long division before canonical partitions’, such as 73 ϭ 60 ϩ 13 or 563 short division? However, the actual five-part ϭ 400 ϩ 150 ϩ 13. Ross (1989) has shown that sequence of stages set out on subsequent pages is children generally find this difficult. Also, if the slightly more logical (although still questionable): wrong partition is made, the result can be mental division using partition; short division of somewhat ‘messy’. TU ÷ U; expanded method for HTU ÷ U; short Ϭ ϭ ϩ Ϭ division of HTU ÷ U; and long division. 81 3 (50 31) 3 ϭ (50 Ϭ 3) ϩ (31 Ϭ 3) ϭ 16Á ϩ 10À Stage 1: Mental division using ϭ 27 partition Anghileri (2001) shows the workings of a child Ϭ Ϭ It is recommended that children record mental calculating 1256 6 by finding, separately, 1000 6, Ϭ Ϭ Ϭ division as follows: 200 6, 50 6 and 6 6. These work out, Ϭ ϭ ϩ Ϭ respectively, to 106 r 2, 21 r 2, 8 r 2 and 1, and are 81 3 (60 21) 3 then added to give 136 r 6. ϭ (60 Ϭ 3) ϩ (21 Ϭ 3) ϭ 20 ϩ 7 ϭ 27 Stage 2: Short division of TU ÷ U I see several conceptual difficulties with this I have written elsewhere (Thompson, 2003) about suggestion, over and above the inappropriate different interpretations of short division – both as formality of the notation, which is, after all, a concept and as an algorithm. This guidance paper recording for a mental calculation. The main states that the short division method is to be problem concerns the distributivity of division. recorded as: Multiplication is both left- and right-distributive 20 ϩ 7 over addition (and subtraction), whereas division is 3)60 ϩ 21 only right-distributive. This means that 7 ϫ 12 can be calculated as (7 ϫ 4) ϩ (7 ϫ 8) or (3 ϫ 12) ϩ This is then to be shortened to: (4 ϫ 12), where either number is partitioned. 27 3)8 21 6 MATHEMATICS TEACHING INCORPORATING MICROMATH 208 / MAY 2008 © ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions We are informed that: succinct notation: ‘The carry digit ‘2’ represents the 2 tens that have 6)196 been exchanged for 20 ones.’ (DfES, 2007: 18). Ϫ180 6 ϫ 30 Interestingly, this is the first time that the 16 concept of ‘exchanging’ has been mentioned in the Ϫ12 6 ϫ 2 entire document, despite the fact that there is a 432 detailed section on subtraction. Written subtraction Answer: 32r4 algorithms are often based on a model that involves The way in which the strategy is presented in base-ten materials and the exchange of ‘flats’ for the document gives the impression that it comes in ‘longs’ (hundreds for tens) and ‘longs’ for ‘ones’. just two forms: you either repeatedly subtract the The subtraction method recommended in this smallest multiple of ten of the divisor or you document, however, is based on the concept of ‘re- subtract the largest. In fact, the strength of the partitioning’ (although the word is never actually chunking algorithm lies in its great potential for used). Given the difficulties that children experi- differentiation: it allows for a range of levels of ence with exchanging in subtraction, even when sophistication in children’s confidence and under- preparatory work has been done with base-ten standing, in that the less confident can remove materials (Hart, 1989), it is difficult to believe that small chunks; the more confident can take away the recommended progression will be any more larger chunks; and the most confident can subtract successful. the maximum-sized chunks. A more detailed The progression from the third calculation to analysis of this strategy can be found in Thompson the fourth calculation also involves a shift from the (2005). quantity value aspect of place value (80 and 1) to the column value aspect (8 tens and 1 one) Stage 4: Short division of discussed in earlier articles in this series HTU ÷ U (Thompson, 2007 and 2008). However, this partic- ular shift is slightly more complicated, in that In addition to the issues raised above, concerning children have to be able to partition 81 into 60 ϩ the introduction of short division in stage 2, the 21, as illustrated in the third calculation, and then following question is offered: Why do we need to switch to interpreting 81 as 6 tens and 21 ones. teach a conceptually difficult strategy for dividing a Given the research mentioned above (Ross, 1989) three-digit number by a single-digit number when that children have great difficulty making ‘non- the chunking method introduced in stage 3 for canonical partitions’ such as 81 ϭ 60 ϩ 21, it solving three-digit by two-digit divisions is much would be informative to ascertain how easy they easier to understand, allows for differentiation and find the idea of 81 being equivalent to 6 tens and is probably more effective? 21 ones. Stage 5: Long division Stage 3: ‘Expanded’ method for The recommended method in this section involves HTU ÷ U estimating to find the maximum amount to subtract The recommended method is the one we have initially, as in stage 3, and then continuing with the come to know as ‘chunking’, where multiples of standard chunking procedure. the divisor are subtracted from the number to be 24)560 divided (the dividend). 20Ϫ480 24 ϫ 20 80 6)196 ϫ Ϫ ϫ 3 72 24 3 60 6 10 8 136 Answer: 23r8 Ϫ60 6 ϫ 10 76 One reason offered for the rather strange Ϫ60 6 ϫ 10 positioning of the 20 and the 3 down the left-hand 16 side is to keep the links with ‘chunking’ – although Ϫ ϫ 12 6 2 I would have thought that the notation down the 432 Answer: 32r4 right-hand side and the calculation procedure itself might do that! A second reason given is that it Having done some preliminary work on the reduces the errors that tend to occur with the development of estimation strategies, children are positioning of the first digit of the quotient. As the expected to progress to the following, more first digit of the quotient is 2, written as 20 on MATHEMATICS TEACHING INCORPORATING MICROMATH 208 / MAY 2008 7 © ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions both the right- and the left-hand side, I have diffi- ‘However, at Key Stage 2 they [teachers] often culty in understanding what this talk of ‘positioning’ overlook the importance of linking pupils’ mental is all about. Also, as the answer is to be written at strategies to the introduction of expanded and the bottom of the procedure, the extra inclusion of compact written methods.’ (p3) the 20 and the 3 down the left-hand side seems to This belief in a natural progression permeates be adding another level of potential confusion. many NNS and PNS publications, QCA documents 23 and HMI reports. However, I think I have shown in 24)560 this series of articles that this progression is not as References Ϫ480 natural as it appears to be, and that more thought Anghileri, J. (2001) 80 needs to be given – and more research carried out Intuitive approaches, Ϫ72 – concerning the seamless links suggested in this mental strategies and 8 standard algorithms, in guidance paper. Answer: 23r8 Anghileri, J. (ed.), Moreover, there is too much emphasis in the Principles and Practices in The document then tries to argue that the Arithmetic Teaching, OU: document on advancing children to compact 91 notation illustrated above is, in effect, the standard methods for all four basic operations as quickly as DfES (2007) Guidance long division method. However, the language and possible. For the vast majority of children it would paper – Calculation thinking associated with the chunking method runs be more useful to focus particularly on algorithms www.standards.dfes.gov.uk something like: /primaryframeworks/ that, unlike compact algorithms, have in-built downloads/PDF/ I need to find out how many 24s there are in 560. I variability that allows for the important principle of calculation_guidance_ know that there are ten of them in 240 and so there differentiation. The following algorithms fit neatly paper.pdf (accessed are twenty of them in 480. If I take this from 560 I into this category: subtraction by complementary January 2008) get 80.