CAN PICTORIAL REPRESENTATIONS SUPPORT
PROPORTIONAL REASONING? THE CASE OF A ‘MIXING
PAINT’ PROBLEM
Christina Misailidou and Julian Williams
University of Manchester
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.
Item s' D ifficulty
2
With models version
1
2 Cam pers
Printing Press
1 Cam pers
0
-4
-3
-2
-1
0
Books' Price
2 Paint
1
1 Paint
2
-1
Fruits'Price
-2
-3
Class
-4
W ithout m odels version
Table 1: Scatter plot of performance for with-models and without-models items
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:
1. Nothing written on the pictorial part of the question whatsoever.
2. Numbers only written on the pictorial part, mainly the answer to the problem.
3. Simple drawings of cans are drawn, mainly a number of cans denoting the answer
to the problem.
4. 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:
Figure 1
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.
Figure 2
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).
David gave the incorrect answer ‘2’ to the 1 Paint question. He used the ‘constant sum’
strategy (we adopted this term from Mellar, 1987) to answer the problem. He thought that
the sum of Sue’s cans should be equal to the sum of Jenny’s cans: 3+6=9 therefore 7+2=9
and so the answer should be 2.
He did not draw anything on the picture of his test he just wrote the number ‘2’ near the
question mark.
When he was interviewed and was asked to explain his answer he explained:
David:
I counted how many cans it was altogether…Um…six cans of red and
three cans of yellow [he means Sue’s cans] And Jenny uses 7 cans of
yellow. She can only use 2 cans of red paint.
Interviewer: OK. How did you come up with this idea?
David:
Because…Sue can use six reds and three yellows and that equals nine…so
she is only using nine cans of paint. Jenny uses seven, so we got 2 cans
of red, she can only use 2 cans of red.
Then the interviewer tried to investigate whether the pictures influenced his decision.
David commented on the pictures of cans:
David:
I used them. There were these lots of cans and these lots of cans
Interviewer: So you actually prefer the problem with pictures?
David:
Yes.
Interviewer: Did they help you find the solution at the end?
David:
Yes I think so. Because I wasn’t adding the three and the six…I was
looking at the cans altogether and…using the cans instead of the
writing…more…
David stresses that he ‘used’ the pictures and from his interview it seems that he probably
preferred the problem presented in a pictorial rather than a plain form. It is possible that
his concentration on the gestalt of the picture helps to encourage him to adopt the
constant sum strategy: he may be distracted from the logic which is wrapped up in the
verbalisation of the task. But then he may be facilitated in performing the necessary
operations to execute his strategy of adding and subtracting.
Sarah gave the correct answer ‘12’ to the 2 Paint question. Her script can be seen in
Figure 3:
Figure 3
When Sarah was interviewed and was asked to explain her answer she said:
Sarah:
John’s got 5 cans of green paint and then George’s got 20…and it says
they want to use exactly the same colour. So to get from 5 to 20 you have
to times by 4. And then…John’s got 3 cans of yellow paint, how many
would George have? So I timesed by 4 to get the answer because I
timesed by 4 there.
Interviewer: OK. And here what made you decide…on using that method?
Sarah:
We have been doing in class stuff about…how to get from one answer to
another…you have to like…times by 4 or by 5 or whatever…or I just
thought to do it here as well.
Then the interviewer tried to investigate whether Sarah has used the pictures to solve the
problem:
Interviewer: So, I see here that you wrote beside the arrow…and I see that you made
an arrow here as well? Why did you do these?
Sarah:
It helped me.
Interviewer: So…if it wasn’t here, what do you think?
Sarah:
I probably wouldn’t have got it…I don’t know what I would have done!
Sarah, just like David, seems to like the pictorial version of the problem. She doesn’t
actually use the picture to develop the times strategy, which she says comes to her from
what ‘we have been doing in class and stuff’. But she does use it to organise and record
the ‘times’ operations on the appropriate numbers, and her inscription does appear to
confirm her answer.
Pupils like Anna and Chris below belong to ‘Category 1’, which means that there is no
evidence neither in their scripts nor in their interviews that they had used the pictorial
model in any way.
Anna gave the answer ‘10’ to the 1 Paint question by using an additive approach and she
did not write anything to the pictorial part of the problem in her script. Her explanation
for obtaining 10 was:
Anna:
3 add 7 is 10…so the answer is 10 cans.
Interviewer: OK. What helped you think of that answer?
Anna:
I’ve noticed that 6 minus 3 equals 3…so then I’ve used the same amount
of paint
Interviewer: So, it was the numbers that made you think of the answer?
Anna:
Yeah.
When she was asked about the model she appears not to have used it at all
Interviewer: OK. Why do you think the cans of paint are painted here?
Anna:
Probably…to help…cause…you can count them if you want.
Interviewer: Did you use this picture to find your answer?
Anna:
I don’t know…probably not.
Chris gave the correct answer ‘12’ to the 2 Paint problem and just like Anna he did not
write anything to the pictorial part of the problem in his script and in his interview he
admits not to have used the pictures. First he explained how he found 12:
Chris:
The green paint was originally five and it was timesed by four to become
twenty so I thought it would be the same with the three, the three cans of
yellow paint.
And later he comments on the model:
Interviewer: So here, there is a picture with some cans. Was this picture confusing…or
helpful…or it didn’t matter at all having it?
Chris:
I don’t think it would have mattered.
In summary, the qualitative analysis of the test data complemented by individual
interviews provides a different aspect of the use of models than the statistical analysis of
the test data. Judging from their interviews pupils fall in three categories in respect of
how they used the pictorial model and it must be noted that in all three categories belong
pupils that give correct as well as incorrect answers. Rather few pupils belong to
‘Category 1’ though: from the 23 pupils that we have interviewed only 3 explicitly
admitted not to have ‘used’ the model at all. We propose that the analysis shows that the
pictorial models influenced one way or another the selection or use of the pupils’
strategies and the explanations they gave for them.
Presentation of a small group interview
We have already reported results for the case of a group working on the 1Paint item. By
studying this case, we concluded that one of the factors that brought a successful ‘change
of mind’ of two pupils, who had given incorrect answers to this item in the test, was the
pictorial representation of the problem (the same one that was used in the test) that was
presented to them during the discussion (Misailidou and Williams, 2003).
Informed by the results of the above group discussion we designed and then conducted
the group discussion that we present in this paper.
Three 11 year old pupils, Alan, Victor and Soula were selected form the same class to
form a ‘discussion group’ because they had provided three different responses on the 1
Paint item: ‘2’, ‘10’ and ‘14’ respectively. Alan and Victor worked on the ‘non model’
version of the test and Soula on the ‘model’ one. The discussion, with the guidance of
one of the authors (who is denoted below as the ‘interviewer’), lasted approximately 1
hour and was audio taped.
Initially, the pupils were asked to recall their responses by consulting their test scripts.
They were also invited to present an argument for their response to the group.
Alan explained his answer 2, which was the result of the use of the constant sum strategy,
as ‘3 add 6 is 9…so then 7 add 2 equals 9…so then the cans are equal…so the answer is 2
cans of red paint.’
Victor could not repeat, when asked, Alan’s method so at this point the interviewer
distributed the sheet that is shown in Figure 4, as a tool that was supposed to facilitate
discussion. In Figure 4 Soula’s sheet can be seen after the discussion ended. The
drawings on it come not only by Soula but by the other pupils and the interviewer as well.
Figure 4
The interviewer encouraged the pupils to use the sheet as an aid to their explanations and
she tried to use it herself as much as possible during the discussion. Indeed, it is notable
that from now on in the discussion there were continual examples of the way the picture
is used by the children and the interviewer to help explain methods and working. The
references to the picture or elements of the picture are signalled by the use of ‘that’,
‘this’, ‘them’, ‘here’, ‘there’ and ‘she’ throughout the discourse, and of course these were
accompanied by gestures pointing to the elements represented.
Having this visual aid and some time to think about it, Victor was able to repeat Alan’s
method and then he explained his own method, which is the ‘additive strategy’ to obtain
10: ‘If Sue got 3 of yellow and 6 of red…and Jenny got 7…she got 4 more here [points to
the picture]…and so she needs 4 more there [points to the picture]…so it’s 10.’
Finally Soula explained her answer 14: ‘Sue has 3…cans of yellow paint…[she points to
the pictures] she’s got 6 cans of red paint…Jenny’s got 7 cans of yellow paint…and add
another 7 which is 14…cause I thought…if 3 and 3 will be 6…then 7 and 7 is 14.’
Then the interviewer invited the pupils to consider all three answers and explanations and
share with each other their opinions. At one point Alan commented:
Alan: I think Victor’s is the best one…you have to find how many…would
Jenny…how many would she need to make it equal with them [points to Sue’s
cans]…so I don’t think you have to double that to find out… So Victor’s method is
better.
And a little later
Alan:
Yeah…Soula says that 3 and 3 is 6…and 6 there [points to the picture]…so
the answer is going to be 14…so one is going to have 14 cans of paint
and one is only going to have 6…
Interviewer: Yes…
Alan:
So it wouldn’t be right…we have to make it even…
Alan’s use of the phrases ‘make it equal’ and ‘make it even’ brought up a discussion
about the wording of the problem. Some parts of it are presented below:
Interviewer: Maybe we should go back to the problem…all of us…and see what do
they say to us. So, what colours do they need?
Victor:
Red and yellow.
…
Interviewer: Soula? What does the problem say?
Soula:
It says they want to use each exactly the same colour.
Interviewer: So, what do they mean?
Soula:
That they want to have equal cans of paint.
…
Interviewer: What this colour is going to look like when they finish? What colour are
they going to see?
…
Victor:
It might be pink.Soula:
I think that if they want to use exactly the same on each can…they will
have four different walls in each room…and one of them would be
yellow, one would be red, another will be yellow, another will be red.
…
Victor:
They might have mixed the yellow…
Interviewer: If you mix the yellow and red you get…
Victor:
Orange I think.
…
Soula:
I think they are painting separately.
Interviewer: But when they say they want to use each exactly the same colour…and
not colours…then they are going to use one colour…or not? What do you
think Alan? Are they going to paint the walls separately or not?
…
Alan:
They might have the exact same amount.
It is obvious that the wording of this specific problem provides opportunities for
exploring different interpretations: not mixing paint and not using ratio and proportion to
find the answer or mixing paint and using proportional reasoning. Since, in addition to
not using proportionality when needed, the misuse of proportionality in non-proportional
situations is a ‘classical’ misconception as mentioned by De Bock, Van Dooren, Janssens
and Verschaffel (2002) we believe that problems that encourage such explorations as the
above help pupils to clarify the essence of the proportional relationships.
Furthermore, ‘problems’ in real life situations-in contrast to many ‘school textbook’
problems- have often an ambivalent ‘context’ that must be deciphered, just like the
problem used here.
After the clarifications about the context Soula commented
Soula:
It really depends on what shade of orange do you want…because if you
mix them together then it will be a dark orange…
This last comment by Soula gives the opportunity to reject the answer ‘2’
Interviewer: [pointing to Sue’s cans in the picture] If you mix this red and this yellow
what sort of orange do you think it would make?
Alan:
It would be dark because there’s more red than yellow.
Interviewer: OK. [Addressing Alan] Your answer to the problem was 2 reds, wasn’t
it?
Alan:
Yeah.
Interviewer: So if they have 2 red and 7 yellows, what sort of orange would they
make?
Soula:
A lighterAlan:
Yeah…a lighter orange…yeah…cause there’s way more yellow than
red…
Interviewer: So could they use 2 cans?
All of them: No.
Then Victor explains why ‘14’ should also be rejected as an answer:
Victor:
Having here [points to the picture] 14 it would be a dark orange!
Interviewer: And this is what?
Victor:
[On the same time pointing to the picture] That one [he points to the
answer 14] …is one that would make a darker orange than that [he
points to Sue’s cans in the picture]!
Interviewer: Ah…you feel like…if there is 14 here [points to the picture] it’s going to
be darker than the shade that we want. So how many cans do you feel
that we should have here [points to the picture]?
Victor:
10. Because she’s [points to Sue’s cans in the picture] got three more
red…and if she [points to Jenny’s cans in the picture] had 10 she would
have three more.
Interviewer: What do you think Soula?
Soula:
I think that’s OK.
Interviewer: So you feel that we will have the same shade if here [points to the
picture] we have 10. OK? What do you feel Alan?
Alan:
I think that’s OK
At this point of the discussion all the three pupils were convinced that the answer ‘10’
which was the product of the additive method was the correct one. The pupils having
reached an agreement could not see the point of continuing the discussion and so the
interviewer brought up a new element that was proposed by a pupil in a previous group
discussion in order to provoke cognitive conflict. (Misailidou and Williams, 2003a).
Interviewer: Let me ask you something else. [She draws on Soula’s ‘pictures-sheet’
but makes sure that the others can see as well] If we had 1 can here of
yellow paint…let us see what would happen with the two methods.
[Addressing Victor] With your method, how many cans of red paint
would you have?
Victor:
I think…4.
Interviewer: 4. Yes. [She draws 4 cans on the picture sheet] And with that method
how many cans of red paint would we have?
…
Soula:
2.
Interviewer: [she draws 2 cans on the picture sheet] So, this method gives us 4 and
this method gives us 2. So, do you think these two colours are the same?
Victor:
[Points to the two cans of red and 1 can of yellow on the pictures] That
one is lighter.
Interviewer: That one is lighter. But comparing to these [shows Sue’s cans], which of
these two gives us the same shade as this?
Victor:
This [points to the 4 cans of red and 1 can of yellow]
Soula:
This [points to the 4 cans of red and 1 can of yellow as well]
This new element did not provoke cognitive conflict so the interviewer attempted a
‘resource to zero (0)’ strategy
Interviewer: So, if we have 3 cans of red paint [on the same time she draws the cans in
the picture], with your method [she means the additive method] how
many cans of yellow paint should we have?
…
Victor:
None?
Interviewer: It’s none. [Victor draws on the picture something that denotes an empty
can and the interviewer writes in it the word ‘none’] Would that give us
the same colour as here? If we had none? Would that give us the same
colour as here if we had none?
Victor:
It depends which method we are using.
Interviewer: With this method if we had 3 cans of red paint we would have what?
Victor:
1?
…
Interviewer: Not 1. 1 and a half plus 1 and a half how much is it?
Victor: 3 [Soula and the interviewer draw 1 and a half cans on the picture]
…
Interviewer: So, what do you think now?
Victor:
I don’t know.
Interviewer: With this [point to the picture] method we have red. We don’t have
orange.
At this point probably the pupils considered ‘too’ absurd the situation of having ‘none’ as
an answer and red as a colour so they started thinking again the context of the problem:
Soula:
I don’t think they are going to mix them [points to the cans] together…
Alan:
Yeah…
Interviewer: But…as we agreed earlier… if we suppose that they are mixing them?
Victor:
Then mine is not right!
Soula:
Yeah.
Interviewer: Alan?
Alan:
With Victor’s method we can’t get an answer at all!
Unfortunately at this point the bell rang and the session ended. We believe more
discussion was needed in order for the pupils to understand why the additive method does
not work. Nevertheless, a ‘change of mind’ was achieved and the generative element of
the discourse which supported it was the interviewer’s introduction of a ‘conflict
strategy’. It cannot be assumed that the pictorial model ‘generated’ a multiplicative
construction and in fact during the discussion the model afforded explanations of correct
and incorrect strategies. It did appear though to help the children to recognize the relative
‘darkness’ of combinations of yellow and red (like for example, (1, 4) compared to (3, 6))
CONCLUSION
Our aim was to complement what has already been reported on the children’s
proportional reasoning by examining the effect of a pictorial model on pupils’ strategies
while attempting to solve a rather ‘difficult’ type of ratio task: the one which involves
mixing of paint.
We found that pupils –who as a rule were not taught to use pictorial models- were
influenced in their strategies when the pictorial model was presented to them in test
conditions.
We then found when the use of the pictorial model was suggested to pupils in small
group discussions by the interviewer that some children (and the interviewer) could use
the pictorial model to represent their strategy, and to facilitate discussion and argument.
Learning was facilitated by the resolution of conflict, the arguments for which were
successfully backed by the use of the model.
In conclusion, we believe that pictorial models could facilitate pupils' development of
proportional reasoning but that this is mainly through facilitating communication of
strategies.
Hence the role of the teacher or other sources for managing conflict is crucial in ensuring
that the models are used productively by the pupils.
ACKNOWLEDGEMENT
We gratefully acknowledge the financial support of the Economic and Social Research Council (ESRC),
Award Number R42200034284
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