Tackling the Division Algorithm -Shreya Khemani and Jayasree Subramanian

Introduction

!  Three week study conducted in tackling the division algorithm with Class 6 students of the Raipur Govt. Girls Middle School, Hoshangabad District, Madhya Pradesh.

Motivation

!  The need for a knowledge of division in the teaching and learning of fractions arose during our longitudinal study conducted by Eklavya

!  A string of lively discussions on activemath concerning the division algorithm, with contrasting views presented by various educators

What is it that we are tackling? : The justification for an alternate approach !  We believe, (as much research on teaching division suggests)

that the standard long division algorithm, based fundamentally on the place value system, can be counter-intuitive to children’s existing whole number sense

!  What's more, introducing division solely through algorithmic procedures can reduce the process of division to a series of mechanical, unthinking steps which makes it difficult for a student to relate the taught procedure to the meanings that can be identified with division

!  In particular, it doesn't relate in a natural way to the student's existing intuition of sharing and distributing

Cues from the Classroom

!  Students had already been introduced to the division algorithm in their regular classes

!  Conducted a preliminary 'test' to gauge students' grasp of the division algorithm consisting of five problems:

609 ÷ 3 360 ÷ 6 399 ÷ 7 512 ÷ 4 348 ÷ 4

Examples of students performance:

Tabulated results from Group A, B & C

! In an even earlier class on the number board, we'd asked some students to solve a few simple division problems. ! Some of the informal techniques they used seemed to point in a natural way to the alternate method that we later introduced

Historical and contemporary considerations: A look at the literature ! Much of the teaching of arithmetic in the early part of the century was characterised by the belief in drill and practice that favoured computational skill over conceptual understanding

! Thorndike, one of the biggest proponents of this approach to math education, believed that arithmetic is a 'set of rather

specialised habits of behaviour toward certain sorts of quantities

and relations' and that learning it was like 'learning to typewrite'

! However, more recently, several studies point to the fact that teaching division concepts through the algorithm makes it difficult to grasp the mathematical concept of division, and results in a host of confusions, since the algorithm doesn't lend itself to children's informal, intuitive notions of division.

! Windsor, Will and Booker, George.(2005) point to the fact that a historical analysis of the division algorithm might provide insights for improved teaching methods

! They attribute the 'natural origin' of the 'ineffective teaching strategies' to the understanding provided for division concepts and procedures by the Egyptian, Chinese and Hindu-Arabic number systems, arguing that memorisation of rules and processes in various historical settings arose out of practical circumstances

! They assert that the use of 'inappropriate language' (such as the 'bringing down' of a digit, and 'putting a number into another') confuses students and hinders the understanding of the division concept;

! They feel that the language of division should be established through support of materials that provide ample 'division experiences' before working with the algorithm

! Leung, I., Pang, W., and Wong, R. (2006) contest that the 'guess-and-match type mental process' of maximising the quotient confuses the concept of division and is not consistent with their life experiences associated with grouping and sharing. Initial results of their studies suggest that children taught by a new method, very similar to ours, perform better on a test than those who learn it through the traditional method.

! Anghileri, J., Beishuizen, M. , Van Putten, K. (2002) conducted a comparison study between Dutch and English children to explore their written calculation methods for division, and found that Dutch students, who had been taught division through a 'careful progression' of informal strategies (similar to the method adopted by us) to a more structured, efficient procedure performed significantly better than the English children who have been mostly taught the steps of the standard algorithm

Method ! After the preliminary test, we began with an activity of sharing

! Using matchsticks bundled into groups of 10, we asked students to pick a number less than 100 at random, and gave them a 'divisor' to work with, asking them to pull out an equivalent number of empty matchbox trays

! The student was then asked to equally distribute these matchsticks-exchanging a bundle of ten for ten loose matchsticks if necessary-into the empty trays. As the student begins this process, she was required to spell out all the steps she is taking so that another student could simultaneously record these steps on the blackboard in a representation resembling that of the algorithm

! The number of matchsticks found in each of the trays at the end of the process represents the quotient, while those that remain undistributed-if any-represent the remainder

The representation looked typically like this:

Drawing on the process of sharing and distribution and the idea that multiplication is distributive over addition, this method allows for a decomposition of the quotient into parts that are chosen by the students themselves based on their comfort with multiplication facts.

So instead of having to optimise the quotient right away, the student can choose 'chunks' or what we call 'partial quotients' that will eventually add up to be the quotient

As in the example above, 98 = 4 x 24 = 4 x (10+10+4).

We will refer to this method as the partial quotients method, and highlight the advantages and difficulties we faced in using it in the classroom as we discuss the results of our study.

Ability to distribute vs ability to divide

The link between the two methods And the partial failure of

the note model Problems of hybridisation

A reduction in errors

Meaning attributed to division

Results

Meaning attributed to the process of division

One of the striking differences in students' responses to the two methods is in the apparent meaning they attribute to each of the processes. We take up and dissect the following example of one students work to support our claim

Multiple meanings

!  Distribution in several different ways, the possibility of choice in partitioning

Linking the idea of share in different contexts: A recognition in the parallels between division as a sharing activity and fractions explained by the share model.

Ability to distribute vs ability to divide: A note on formal representation !  Necessary to distinguish between the ability to distribute n objects equally among m places and to be able to divide the number n by the number m !  Demands each of them places on a student is different. Eg the knowledge of subtraction ! Several of the students who were unable to use the algorithm correctly, were in fact able to distribute matchsticks among trays with much ease and deftness. ! Writing out their steps in a formal representation introduced by us seemed rather unnecessary since they had just demonstrated the whole act of division using the objects in front of them. ! Unlike several private schools and perhaps government schools in urban areas, the students we work with are not in the habit of making meticulous records of their work in the classroom.

The physical act of distribution

Solving a problem using the PQ method

Solving a problem using the long division algorithm

1. Does not involve formal representation

involves formal representation involves formal representation

2. Relates in a natural way to students intuitive notion and real life experience of division as an act of distribution

Does not relate in a natural way to students intuitive notion and real life experience of division as an act of distribution

3. Is counter-intuitive to what they've been taught about place value, forcing them to look at a digit/digits at a time for the whole number dividend

4. Does not demand the knowledge of subtraction, and thus reduces the scope of subtraction errors

Demands the knowledge of subtraction. But while there is scope for subtraction errors, it is also perhaps easier for students to take the whole number approach for subtraction, since the numbers directly relates to the quantity being distributed.

Demands the knowledge of subtraction and there is a scope for a whole host of subtraction errors. It is perhaps more difficult to use a whole number approach while subtracting, say, 18 from 23 of the 236 as in the above example, since this method of representation is alien to what they've been taught so far.

5. Grants students the freedom to choose chunks for distribution

Grants students the freedom to choose chunks for distribution, and thus draw upon their existing multiplication facts thereby reducing errors that arise from the need to maximise the quotient

Does not allow this choice. Demands that they find at each step a maximum multiple of the divisor so that it is less than or equal to the dividend/virtual dividend

6. Less efficient that the algorithm The most efficient method of division

The special case of Group D !  Students of category D are at a level much below any of the

other groups; cannot recognise number symbols or pick number quantities greater than 30

!  However, they were able to distribute matchsticks into trays rather efficiently (as the table below indicates)

A reduction in errors and the success of Groups B and C We classify the most common types of errors associated with the division algorithm as follows:

1. Place value error

2. Errors arising out of the problem of maximising the quotient

What happens to these errors in the partial quotients method?

Some data-despite its limitations-on the errors made in each method

The link between the two methods, and the partial failure of the note model

!  We hoped to use the partial quotients method to arrive at the algorithm through suitable process of maximisation of quotient

!  Some students from Group A recognised a link and began maximising quotients

! Some used the results of one to solve the other

! Some even explained quite clearly to the class the link between the two methods

!  However, most students were not able to do this

! They recognised early on that the larger the chunks they distribute (as they maximised the partial quotient) the faster they would solve the problem

!  When working with bundles of matchsticks, this attempt could correspond somewhat more naturally to the logic of the algorithm because matchsticks were bundled into 10s and 100s

! However, without the material, there is no guarantee of choosing multiples of powers of 10, thus the idea of maximisation associated with the algorithm is replaced with that of distributing the maximum possible

! Unless forced into working with base 10 in choosing chunks, the partial quotients method doesn't naturally lend itself to arriving at the algorithm

! We thus introduced the note model : A or the long division algorithm based on currency notes of 100, 10 and 1

! Tried to emphasise that if we were distributing money among children, then the partial quotients method dealt with the quantity of money to be distributed, while in the algorithm, we looked at the number of

notes to be distributed.

! Several educators have claimed to have used this model successfully

! Our situation: We needed 317 rotis for a wedding at home, and we decided to cater them from 3 different people - Kamala, CharanLal and Imran. We used the Jodogyan fraction kit to make denominations of 100, 10 and 1 on different sized fraction pieces. We then pasted 3 pieces of 100, one 10 and seven 1s on the board (the fraction pieces allows us to do this by using water). In solving the problem using the partial quotients method, this 317 represented the number of rotis we needed. However, in using the long division algorithm, we supposed that each roti costs Rs. 1. And interpreted the problem as paying the three caterers

Shortcomings of the model

!  Not 'realistic' enough: What happens to notes of 50 and 20?

!  To counter this argument, we employed the use of material

!  Yet another problem, that of the choice of notes

! Several students picked 3 hundreds and 17 ones instead of 3 hundreds, one ten and 7 ones

!  And lastly, why must I write down if I can distribute

! We created worksheets with kullaks/ boxes beside each division problem, expecting students to forcibly fill in the number of 100-notes, 10-notes and 1-coins

! While some students caught on to what had to be done, others used the algorithm without relating it to the note model, and several just used the partial quotients method

! So we allowed for a choice of method and found that a vast majority chose the partial quotients method

Problems of Hybridisation

! Recurrent confusions that arose when methods were simultaneously addressed

Concluding Remarks !  The division algorithm is counter intuitive to children's

whole number sense and challenges what they've been taught about place value

!  It doesn't relate in a natural way to their intuitive notions of sharing and distribution

!  Student's informal techniques suggest that introduction to division using the partial quotients method seems natural

!  We believe that beginning with the partial quotients method would erase the problems of hybridisation and allow for a meaningful introduction to the division algorithm at a later stage

References !  Anghileri, J., M. Beishuizen and K. van Putten. (2002). 'From informal strategies to structured

procedures: mind the gap!' Educational Studies in Mathematics 49: 149-170.

!  Leung, I., Wong, R., & Pang, W. (2006). 'Departing from the Traditional Long Division Algorithm: An Experimental Study' . Paper presented at the 29th Annual Conference of Mathematics Education Research Group of Australasia. Canberra, Australia (http://www.merga.net.au/documents/RP382006.pdf)

!  Slesnick, T. (1982). 'Algorithmic skill vs conceptual understanding'. Educational Studies in Mathematics, 13, 143-154

!  Windsor, Will and Booker, George.(2005) 'An Historical Analysis of the Division Concept and Algorithm Can Provide Insights for Improved Teaching of Division' [online]. In: Bartlett, Brendan (Editor); Bryer, Fiona (Editor); Roebuck, Dick (Editor). Stimulating the 'Action' as Participants in Participatory Research: Volume 3

!  Kamii, C., & Dominick, A. (1997). 'To teach or not to teach algorithms.' Journal of Mathematical Behavior, 16(1), 51-61

!  English, L. 'Cognitive psychology and mathematics education.' Reprint of English, L. (1995). Cognitive psychology and mathematics education. In L.D. English & G.S. Halford, Mathematics Education: Models and Processes (pp. 1-20)

!  Klein, D., Milgram, J. (2000) 'The Role of Long Division in the K-12 Curriculum'