CAN PICTORIAL REPRESENTATIONS SUPPORT PROPORTIONAL REASONING? THE CASE OF A ‘MIXING PAINT’ PROBLEM
Christina Misailidou and Julian Williams
University of Manchester
Paper presented at the British Educational Research Association Annual Conference, Heriot-Watt University, Edinburgh, 11-13 September 2003
Abstract
In this paper we report on the use of ‘pictorial representations’ in ratio problems about ‘mixing paint’. Two such problems (1Paint, 2Paint) were included in a diagnostic test for ratio and proportion which was constructed in two versions: one with models thought to facilitate proportional reasoning and one without. For our study sample (N=212) of 11 to 13 years old pupils, the statistical analysis of the test data showed that for both of the 'paint' items, the version with the picture was much easier than the one without it. By interviewing selected pupils that took the test, we found that several pupils answered the ‘paint’ items correctly by using the pictorial information, others did not make use of it and yet others were confused by it. We present extracts from the individual interviews that support these findings. Then we present a ‘discussion group’ dialogue consisting of three pupils who provided different responses to one of the ‘paint’ items when they took the test. These children were found to ‘change their mind’ as a result of cognitive conflict, which was facilitated by the pictorial representation. We suggest that models can facilitate pupils’ development of their strategies, by facilitating communication about them.
Keywords: Ratio, Proportional Reasoning, Models.
Introduction and background
In our study we focus on the area of ratio and proportion, a topic in which research in mathematics education reveals that secondary school pupils experience difficulties.
In previous papers (e.g. Misailidou and Williams, 2003b; 2003c) we described how we aim to contribute to teaching by developing a test that can help teachers diagnose their pupils’ use of the additive strategy in a variety of tasks, some easier than others, and some more suggestive of additive strategies than others. These tasks were short written test items, but with different numerical demand and different contexts known to provoke various errors.
We constructed two versions of many items, one with ‘models’ thought to be of service to children’s thinking and one without: the aim was to compare the difficulty of the parallel items and explore the potential of ‘models’ to induce conflicts or changes of strategy. In this paper we will evaluate the use of one particular model by children from our sample.
Our ideas for alternative presentations of items came from the literature, and was finalised only after discussions with primary and secondary school teachers on effective ways of teaching ratio problems. Lamon (1994) suggests that ‘the presentation of the situation in the form of pictures’ facilitates pupils’ proportional reasoning. Middleton and van den Heuvel-Panhuizen (1995) mention that the presence of tables helps children to solve ratio and proportion problems and Streefland (1984) reported the use of the number line as a helpful tool in the classroom teaching of ratio. Finally, some teachers that we have interviewed suggested the use of models (like cubes) that would help children reason proportionally. We decided to investigate whether models could be implemented in test items, and if so to what effect. We thought that if our test data revealed a significant difference in children’s responses to items with or without models then we could use these productively in conflict discussions with children (as in Ryan and Williams, 2002). We drew on the literature to find models suitable for use in test conditions, showed an initial list of models to teachers to comment on their suitability and hypothetical effectiveness and finally we came up with four categories of models: pictorial representations (two types), tables, double number lines and diagrams. A limitation of our approach due to the scale of our study is that we were not able to exhaust all the possible alternative presentations of the items (e.g. all the possible pictorial representations). In any case our aim was not to research the comparative effectiveness of all the models available but to identify some tools for facilitating conflict discussion and classroom teaching in the context of ratio.
Initially, we tested an item bank that consisted of 38 items in total. 24 such items were adopted with slight modifications from previous research studies and others have been created based on findings of that research. For 13 of these items we created an additional, alternative version, taking advantage of the ‘models’ mentioned above. (Misailidou and Williams, 2003c)
The statistical (Rasch) analysis of some preliminary test results of our study showed that the presence of models affected the difficulty of both easy and difficult items (Misailidou and Williams 2002a) and the statistical analysis of all our test results showed that the pictorial representations and the use of the double number lines were more successful than the rest of our models in supporting ratio reasoning (Misailidou and Williams, 2003b; in press). On the other hand, the qualitative analysis of the test data complemented by individual interviews showed that even the successful models helped as well as hindered individual pupils in their investigations. They definitely though influenced the strategies that pupils used to solve the problems and the explanations they gave for them (Misailidou and Williams, in press).
A Rasch analysis of the data combined with interview data allowed us to select the most interesting diagnostically items and test how they would behave as a test.
The resulting diagnostic test was given to a new sample of 212 pupils aged 10 to 13 in two schools in the north west of England in two linked forms: a ‘with-models’ version (13 items) and a ‘without-models’ version (13 items). Both of the test versions can be seen in Misailidou and Williams, 2003c. In each class, half of the pupils were given the without-models version and half the with-models one and the data were subjected again to a Rasch analysis. This analysis allowed us, amongst other things, to examine the difficulty of the same item in its with-model and without-model form. This was possible since the two test forms were linked through 5 common items.
Table 1 presents the difficulty estimates for the two versions of our items (with and without-models) from the scale which measures ‘ratio reasoning ability’.
The graph shows that for most items, the parallel forms were equally difficult, that is, there was no significant difference in difficulty of the with-models and without-models versions of items. There were a few exceptions, and the most extreme were the Paint items. The difference in difficulty between the two forms of the 1Paint item was approximately 1.5 logits and the same difference for the 2Paint item was more than a half logit (0.64) which is significant enough for children of this age.
Is in this paper we focus on the effect that a pictorial representation has on a ‘mixing paint’ task. Our research question is:
How do (untutored) children sometimes use (pictorial) models to support their proportional reasoning?
In order to answer it, we complement the results from the test data with analyses of selected individual interview data. Based on the analyses of these data we planned and conducted group discussions and we present here the results of the analysis for one of them.
Presentation of the paint tasks and the models
The Paint items that were used in our test have been adapted from Tourniaire’s (1986) study. One of the original items that Tourniaire (1986) used was
Sue and Jenny want to paint together.
They need the same colour.
Sue uses 1 can of yellow paint and 3 cans of blue paint.
Jenny uses 5 cans of yellow paint.
How much blue does she need?
By adapting this we came up with the item that we call 1Paint and which is presented below:
Sue and Jenny want to paint together.
They want to use each exactly the same colour.
Sue uses 3 cans of yellow paint and 6 cans of red paint.
Jenny uses 7 cans of yellow paint.
How much red paint does Jenny need?
Generally, two kinds of ratios can be compared in a given proportion. The ratio of quantities of the same nature, which denotes the ‘scalar relationship’ in the proportion, and the ratio of quantities of different natures, which denotes the ‘functional relationship’.
In item 1Paint the ‘easier’ ratio is the functional one (3:6). This ratio, which involves one quantity that evenly divides another, is called an ‘integer ratio’
The 2Paint item has exactly the same wording as 1Paint but different numerical structure: the functional ratio is (3:5) and the scalar, which is the ‘easier’ in this item, is (5:20)
Tourniaire (1986) comments that:
The presence of a mixture does appear to increase the difficulty of problems. It may be because mixture situations are less familiar to the subjects, because continuous quantities are involved, or because mixtures are more difficult to conceptualise. (p. 408)
Hence, we decided to enrich the Paint items with a pictorial representation, the idea for which came from the textbook ‘SMP 11-16, Ratio 1’ (1983):
Sue and Jenny want to paint together.
They want to use each exactly the same colour.
Sue uses 3 cans of yellow paint and 6 cans of red paint.
3 cans of yellow paint 6 cans of red paint
Jenny uses 7 cans of yellow paint.
7 cans of yellow paint How many cans of red paint?
How much red paint does Jenny need?
We hypothesized that by providing the pictures of the cans we introduce a discrete quantisation in the continuous paint context and with the additional diagram we attract the pupils to the ‘easier’ integer ratio. The latter may well account for the difference in difficulty of the two modes of presentation of the item: perhaps this model is in itself of little help to the children, as was found with many of the other models. This was a key question for the interviews in the next stage.
methodology
In addition to these test analyses, we drew on semi-structured clinical interviews and semi-structured small group interviews with selected pupils about the test items.
By analysing and coding the textual responses to the paint models version of the tests, we found the following categories:
- Nothing written on the pictorial part of the question whatsoever.
- Numbers only written on the pictorial part, mainly the answer to the problem.
- Simple drawings of cans are drawn, mainly a number of cans denoting the answer to the problem.
- A construction drawing is presented which illustrates a method/strategy
From each of these categories we selected randomly pupils for individual interviews and the final available sample was 23 pupils. The purpose of the individual interviews was to allow us to validate the items, to confirm our interpretations about the strategies that were used to solve them and to gain a deeper understanding of the way the pupils made use of the model that was provided to them. We decided that a semi-structured type of individual interviews would better serve our purpose and so we constructed a simple interview guide, which consisted of three main points of inquiry rather than specific questions:
1. What was the method that the pupil used to obtain their answer.
2. Whether the pupil had used the picture
3. How the pupil had used the picture.
Thus, it was possible for the pupils to influence the interview agenda and provide possibly more and better quality information. We analysed the individual interview data by searching through them for emerging themes or patterns (Taylor and Bogdan, 1984) rather than imposing on the data an existing categorisation.
The purpose of the group interviews was again to validate the test items and confirm our interpretations about the pupils’ strategies but most importantly we wanted to investigate whether conflict groups can learn to reason proportionally through discussion and with the help of appropriate tools.
Again the interview guide did not consist of specific questions but rather it was a general structure of each interview:
Part 1: The interviewer asks each pupil to present her or his answer and method for obtaining the answer. If possible the presentations starts with the most primitive answer/method, moves on to more sophisticated methods and ends with the correct answer and method.
After each pupil’s presentation another pupil is asked to repeat what s/he had just heard so that the interviewer makes sure that everyone understands everyone’s method
Part 2: The pupils are prompted to discuss/compare/discard/defend answers and methods. During this discussion when the interviewer senses that the pupils need help with their explanations, distributes the tool (a sheet of paper with pictures) that is supposed to facilitate their thinking.
Part 3: The pupils are asked to write down what was their original answer, if they had changed their minds during their discussion and why.
This structure was meant to be only indicative and the pupils were allowed to influence the interview agenda more than the case of the individual interviews.
results
Effect of the models-Individual interview data
Based on some preliminary qualitative analyses of pupils’ work in the scripts and their individual interviews, we reported (Misailidou and Williams, 2002b) that the addition of pictures sometimes helped and sometimes hindered the pupils depending on the item they were dealing with.
Here, we examine more systematically these interviews and scripts. The analysis of the individual interviews provided us with three hypothetical categories of the way that the pupils ‘interacted’ with the model that was provided to them in the test:
Category 1: No evidence of use of the pictorial model
Category 2: Use of the pictorial model to ‘keep track’ of the numbers and/or operations (tally marks etc)
Category 3: Use of the pictorial model to represent or help develop a strategy
Thus, we assign pupils like Keith and Jane below, to ‘Category 3’, which means that they used the model in a way that essentially affected their answer.
Keith, for example, was helped by the pictures as shows his test script in Figure 1 to keep track of the numbers and so to find the correct answer ‘14’ for the item 1Paint:
When he was interviewed and asked to explain his answer, Keith said that:
Keith: What I worked out is…one equals two red, so I worked it out here as well (shows the pictures) so I thought one can of yellow paint is two red so it’s seven…so we should have fourteen…so that’s what I put down.
Interviewer: OK and I see here that you drew some cans. Why did you do that?
Keith: So I don’t get mixed up. I drew one there (shows the yellow paint circle) and then I drew two there. So I can work it out. Because I put seven there (shows the yellow paint circle again) and then I worked it out there (shows the red paint circle)
Keith suggests it was easier for him to figure out the ratio 1:2 by drawing cans of paint, then work on the cans, find the answer and then report with numbers his actions on the cans: a classic use of a modelling tool to keep track of what is going on!
Jane, on the other hand did not use the cans in the same way.
Jane’s script is shown in Figure 2. She gave the incorrect answer ‘18’ to the problem 2Paint which is the result of using the “constant difference” or “additive” strategy. This is a frequently used error strategy where “…the relationship within the ratios is computed by subtracting one term from another, and then the difference is applied to the second ratio.” (Tourniaire & Pulos 1985, p.186)In this particular problem, the answer 18 can be obtained either by thinking that 5+15=20 so 3+15=18 or by thinking that 3+2=5 and so 20-2=18.
When Jane was interviewed she explained her work as:
Jane: There is 3 and there is 5 there…so you take 2 off there…so you take
2 off 20…and 18 is your answer.
Interviewer: I can see that you’ve crossed out 2 cans…
Jane: If you take 2 cans off here (she shows the circle with the 5 cans) you get the same as there (she shows the circle with the 3 cans)…so there was 20 cans…I took off 2…and then I found out that it was 18 cans left.
Later in the interview she was asked whether she has used the model:
Interviewer: In this problem they are not only giving us the numbers here, they are
giving us pictures as well. Did you use them in any way?
Jane: Yeah…I tried to check in the picture how many to take off to get the right answer.
Jane was not ‘helped’ by the model to find the correct answer 12. Instead she used the additive strategy and found the answer 18, and then confirmed this by ‘taking away’ in the model. We do not believe that the model provoked her additive approach but it might have acted as a visual confirmation for her approach. In neither case (Jane or Keith) did the model appear to be instrumental in helping the child to identify a strategy, but in both cases they were able to use the picture to communicate more clearly what they were doing.
We assign Pupils like David and Sarah to ‘Category 2’ which means that although we cannot claim that their strategy was essentially influenced by the model, they noticed it, and maybe they used it to ‘keep track’ of their operations. In addition, the model served to externalise their strategy on their inscriptions, and hence may have supported them, and helped them in confirming their adopted strategies (whether correct or not).