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Research in Mathematics Education

ISSN: 1479-4802 (Print) 1754-0178 (Online) Journal homepage: http://www.tandfonline.com/loi/rrme20

EXPLORING THE COMPLEXITY OF THEINTERPRETATION OF MEDIA GRAPHS

Carlos Monteiro & Janet Ainley

To cite this article: Carlos Monteiro & Janet Ainley (2004) EXPLORING THE COMPLEXITY OF THEINTERPRETATION OF MEDIA GRAPHS, Research in Mathematics Education, 6:1, 115-128, DOI:10.1080/14794800008520133

To link to this article: http://dx.doi.org/10.1080/14794800008520133

Published online: 21 Apr 2008.

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EXPLORING THE COMPLEXITY OF THE INTERPRETATION OF MEDIA GRAPHS

Carlos Monteiro and Janet Ainley Institute of Education, University of Warwick

Studies which have investigated the interpretation of graphs give evidence that this is not a straightforward action. However these studies fail to approach ‘informal’ and ‘personal’ elements of the interpretation of graphs developed by readers. For example, more recent research discusses the importance of previous knowledge and experiences in certain contexts of interpretation. In this paper, we discuss the notion of Critical Sense in graphing as an ability which readers can develop to mobilise and balance several aspects involved in the interpretation of media graphs, such us: mathematical knowledge, personal opinion personal experience and affective exhibition. Particularly, we discuss a study that explores Critical Sense amongstudent teachers. The analyses will help us to think about teaching and learning graphing in ways that will support the development of Critical Sense.INTRODUCTION The increasingly widespread use of graphs in the media [1] for communication purposes assumes that graphs are transparent in communicating their meanings (Ainley, 2000). The interpretation of graphs is viewed as an extraction of the information displayed. In this paper, we argue that this assumption is false, and that the interpretation of graphs is a complex activity in which people establishrelationships within the data and infer information based on prior knowledge and experience. A graph is not an isolated or neutral construct. We initially discuss studies that point to interpretation as an activity that is more than restricted data decoding (Curcio, 1987), including two perspectives on the reading of misleading elements on graphs which are present in the media (McKnight, 1990;Watson, 1997). However, we argue that this work overlooks some aspects of the interpretation of media graphs, aspects that make interpretation a complex activity. Inthe following sections, we argue that when readers are interpreting media graphs, they evoke previous knowledge and experience that is linked to the emergence of meanings related to the data being interpreted. In particular, we develop the notion of Critical Sense in graphing [2] as an ability of readers to mobilise and balance elements of previous experiences and knowledge during the interpretation of media graphs. We analyse extracts from interviews with student teachers in which they display Critical Sense as they interpret media graphs. These analyses help us to think about the teaching and learning of graphing, including the construction and interpretation of statistical data, in ways that will support the development of CriticalSense in graphing.

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INTERPRETATION OF GRAPHS IS NOT ONLY “READING THE DATA” The interpretation of graphs was conceptualised for a long time as a straightforward action. For example, according to Kerslake (1981), the main purpose of a graph mustbe to illuminate numerical data in a visual form. Consequently, failures and mistakes of interpretation could be explained as misunderstandings or ignorance of the correct way to read a graph. However, this traditional perspective on the interpretation of graphs has gradually been revised. An important contribution to understanding of the process of the interpretation of graphs was made by Curcio (1987), who emphasised that graphs might be viewed as a type of text. According to Curcio, the effect of prior knowledge about structural components of graphs (topic, mathematical content and graphical form) influences the reader’s ability to comprehend mathematical relationships expressed in graphs. Curcio classified three main different types of graph reading:

Reading the data: ‘lifting’ information to answer explicit questions for which the obvious answer is right there in the graph;

Reading between the data: interpolating and finding relationships in the data presented in a graph;

Reading beyond the data: extrapolating, predicting, or inferring from the representationto answer implicit questions (p. 384).

Friel, Bright and Curcio (1997) argue that the process of interpretation involves extrapolating from the data displayed on the graph, which suggests that students build on what they already know. Thus, readers use their background knowledge and experience when processing information, whether they are reading prose, tables, or graphs. Even though Curcio’s perspective has contributed to understanding nuances of the interpretation of graphs, this approach only investigates the kinds of graphstraditionally used in schools. These graphs have limited purpose, in terms of analysing or communicating information that relates to problems of interest to the students who interpret them. Interpretation of such graphs is connected to a limitedrange of experience. In addition, Curcio’s approach only highlights technical aspects of graphs. For example, the three different types of reading can be developed in reading a graph which is technically accurate but which might present unrealistic and incoherent data. Therefore, we might read beyond the data (extrapolating, predicting, or inferring from the representation) without being prompted to question the main idea presented in the graph. Curcio did not investigate how students evaluate and criticise the information displayed. McKnight, Kallman and Fischer (1990) argue that little investigation has been done to understand the nature of the ability to think critically in the presence of argumentswith essential quantitative elements, such as in the interpretation of graphs. McKnight (1990) investigated the interpretation of graphs that would be encountered either in relatively popular media (e.g. Scientific American) or in academic texts and

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monographs. Some graphs were related to propositions that were patently false (e.g. “storks bring babies”) or to others that would seem more likely to be true (e.g. “population will increase faster in developing countries than in developed countries”). Seven participants from academic backgrounds (professors and graduatestudents) participated in the study, answering multiple-choice and open-ended questions, which were related to a five level taxonomy of information processing tasks:

• Observation of facts in the graph;

• Observation of relationships in the graph(s) as graphs;

• Interpretation of relationships in the graph(s) in the ‘real-world’ context;

• Evaluation of the value of the graphical data as evidence for the truth of the related proposition;

• Assessment of the basis on which each subject made his/her evaluation of the evidential value of the data (pp. 174-183).

The results indicated that only the tasks related to the observation of facts in the graphs seemed to be unproblematic (McKnight, 1990, p. 183). A preliminary catalogue of processing errors that occurred emphasises the importance of the reader becoming distracted by extraneous knowledge (McKnight, Kallman, and Fisher, 1990):

Translation from the ‘clean’ world of abstract mathematics to the ‘messy’ world of everyday reality – in which all of our knowledge has links to other knowledge as well as links to personal beliefs and emotional reactions – introduces yet another complexity. Sometimes that other knowledge – or what one thinks is other relevant, linked knowledge – or those beliefs and affective reactions interrupt the more cognitive, information processing tasks of interpreting the graph (p. 14).

McKnight and her colleagues recognised that the reactions and feelings expressed by readers when they are interpreting media graphs are relevant aspects which need further investigation. However, it seems that this perspective emphasises these ‘everyday’ non-mathematical or non-statistical components as causing interference in the interpretive process. Watson (1997) states that statistical thinking needs to be assessed as it occurs in social settings outside the classroom. She suggests that “unusual” media graphs might provide good examples to motivate students in school (p. 107). Watson therefore proposes a three-tiered hierarchical model for assessing statistical literacy based onstudies using authentic extracts from the media:

Tier 1: A basic understanding of statistical terminology;

Tier 2: An understanding of statistical language and concepts when they are embedded inthe context of wider social discussion, recognising, interpreting, and using these in applied contexts;

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Tier 3: Being able to question unrealistic claims made by the media or others. A questioning attitude that can apply more sophisticated concepts to contradict claims made without proper statistical foundation (pp. 108-111).

Watson (2000) argues that the assessment of these skills goes hand-in-hand with teaching strategies in which students can be aware of misleading data, which they must constantly question. The hierarchical tiers proposed by Watson (1997) toclassify interpretations highlight the statistical and mathematical knowledge involved in interpreting graphs. Her categorisation does not take into account the reader’s informal knowledge. Watson’s approach also seems to assume that when higher statistical thinking has been acquired it can then be applied in all interpretation of graphs. Therefore, the context of interpretation is of minor importance to the interpretative process. McKnight’s and Watson’s studies are innovative in investigating the interpretation of media graphs which most people read in everyday situations. However, we argue that they attribute an excessive importance to the graph itself. In particular, theydeliberately used ‘misleading’ media graphs in their experiments and surveys as stimuli which provide the opportunity for a specific type of interpretation. However, misleading aspects are not always visible and even accurate graphs can be misinterpreted in a specific context of interpretation. APPROACHING INFORMAL AND PERSONAL ASPECTS OF THE PROCESS OF INTERPRETATION OF MEDIA GRAPHS We argue that the interpretation of graphs is a dynamic process in which people interact with the data displayed. When people are engaged in interpretation they evoke previous knowledge related to the actual facts or emotional experience of their lives. This knowledge and experience influences the reader’s interpretation of the data displayed. Mathematical and statistical knowledge can be mobilised, as well as everyday experiences. The use of the term mobilisation emphasises that we see someone engaged in interpreting a graph not as ‘transferring’ knowledge and experiences from previous situations, but as triggering those elements for use in the current interpretation. In addition, the use of this knowledge and experience is not simply a question of direct application, since mobilisation happens concomitantly with the emergence of the different meanings. For example, Evans (2000) observes that readers can make relationships between the data displayed and ‘painful’ or ‘pleasant’ previous experiences. Gal (2002) suggests that people can engage in different process of interpretation of graphs depending on the context in which the person is involved. Gal cites two main kinds of contexts in which the interpretation of graphs might be developed: ‘enquiry’ and ‘reading’. In enquiry contexts (as suggested by Wild and Pfannkuch, 1999) people act as ‘data producers’ and usually have to interpret their own data and report their findings (e.g. like researchers or statisticians). Reading contexts emerge in everyday situations, in which people see and interpret graphs (e.g. watching TV,

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reading newspapers, looking at advertisements while shopping or visiting internet sites). Even though differentiated, each context is not homogenously defined because people can develop different types of participation. For example, people engaged in a reading context can be actors, speakers, writers, readers, listeners, or viewers, in either passive or active roles. Gal (2002) also argues that the same person might be a reader and/or a producer, depending on their engagement in a particular context. We suggest that a central component of the interpretation process is Critical Sense, which is an ability to deal with three dynamic components: the mobilisation of previous knowledge and experiences; the emergence of meanings; and the balance of these elements involved in a context of interpretation for particular graphs. The term Critical Sense is not only referring to the action of ‘criticising’ the data. The reader needs to take a critical approach to the whole process, including him/herself. Therefore, Critical Sense also comprises the sensitivity of readers to critique their own ideas, beliefs, feelings, conceptions, and conjectures about the data being interpreted. The term ‘sense’ is used to emphasis this broader dimension of being critical. Our perspective on ‘critical’ is close to the conceptualisation of Freire (1972). He developed an original concept of critical consciousness (from the Brazilian Portuguese ‘consciência crítica’) which is an aspect that each individual should develop to perceive social, political, and economic contradictions, and take action in a conscious and creative manner against the oppressive elements of reality. Several studies have extended the early notion of critical consciousness in the field of mathematics education (e.g. Frankenstein, 1989; Skovsmose, 1994), and specifically for interpretation of graphs (e.g. Moreira, 2002). However, our notion of CriticalSense in graphing is close to the later Freirean perspective, which highlights other human dimensions to critical consciousness such as affective elements (Freire, 1997,1994). Our use of Critical Sense is also based on socio-cultural approaches in psychology (e.g. Valsiner, 2000; Vygotsky, 1978), which argue that cognition and affectivity are related in an inclusive separation (Da Rocha Falcão, Araújo, Andrade, Hazin, Nascimento, Lessa,, 2003). Therefore, Critical Sense in graphing is an ability to involve the reader as a whole in the interpretation of graphs. EXPLORING THE NOTION OF CRITICAL SENSEIn this section, we present part of a study involving primary-school student-teachers. The participants were second-year students from an undergraduate education course, following specialisms in mathematics, science and English, and post-graduate education (PGCE) students who had degrees in a range of different areas. In this paper, we focus on data from interviews in which we asked questions about media graphs. Specifically, we discuss the interpretations of one graph reprinted from an annual report called Quality of life in Warwickshire (Warwickshire County Council, 2001), that includes economic, social, and environmental indicators (see figure 1, below).

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Figure 1: graph reprinted from Quality of life in Warwickshire, 2001, p. 93.

The surrounding text mentions that Central Government in March 2000 expected a 40 per cent reduction in the number of people killed or seriously injured in road accidents. Neither the written text nor the graph mentions an intended figure (about 379 casualties) for the end of the target period (2010). This omission does not allow an accurate comparison between the actual figures (2000) and the planned results for 2010. The graph also does not refer to the data sources on which it is based (Warwickshire County Council for the actual numbers of deaths and serious injuries, and national government for the projected targets).This graph was chosen as a research task for three main reasons. Firstly, we anticipated that the topic was related to the interests of the students, most of whom were living and studying inWarwickshire. Secondly, the graph also seems to present accessible levels of complexity in mathematical relationships and concepts: the graph presents absolute and rational numbers, and percentages. Thirdly, we tried to choose a media graph which was free from technical errors or misleading elements, even though we also recognized that the way in which the target has been represented is not very clear. The graph was presented to participants in the study in isolation from the surrounding report, however, because we intended to focus on the interpretation of the graph itself. The interview was the second occasion in which the 13 volunteers had read this graph. It was previously given as an item in a survey previously completed by 118 students prior to a data-handling section of their course. All students involved in thesurvey were asked if they would take part in further interviews and these interviews were developed with those who volunteered. The questionnaire task related to the graph consisted of three open items: If you could talk to the person that produced thisgraph, are there any questions you would like to ask? If the information from these two graphs were combined, what would the graph look like? Do you think that these targets are realistic? At the beginning of the interview participants, were asked whether they remembered the survey which they had completed. All of them answered that they vaguely remembered that the survey items were related to graphs. However, they immediately

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recognised the graph when it was presented to them during the interview. The interviews were audio- and video-recorded which provided more accurate data for analyses, such as aspects of body language and intonation. We structured the main questions of the interview based on the typology of Curcio (1987), in order to have some measure of the participants’ technical ability in reading the graph:

Reading the data questions: What is the total of number of deaths and serious injury per year? What is the lowest actual death and serious injury rate?

Reading between the data questions: Between 1994-1995, and 1997-1998, there was a decline in the number of deaths and serious injuries. Which period represents the greatest decline? Which years represent the highest and lowest number of deaths and serious injuries?

Reading beyond the data questions: What is your prediction for death rate and serious injury in 2001? If the targets for 2000-2010 were met, what do you think the pattern might be for 2010-2020?

However, the way in which we utilised the typology was different from Curcio’s original study. For example, Curcio proposed conventional school graphs in multiple choice tasks which restricted the range of responses which could be given. Thus, students might not have the opportunity to display aspects of their knowledge and experiences which might play an important role in their reading of the graph. The formulation of these types of questions in the context of an interview, however, provided an opportunity for the participants to think about their own interpretation of the graphs. STUDENT TEACHERS INTERPRETING MEDIA GRAPHSAlthough we present data relating to only one media graph, we emphasise that these findings emerged from the whole process [3]. Generally, the questions involving “reading the data” and “reading between the data” demanded direct answers which the participants could easily identify on the media graph. These questions provided an opportunity to carry out an initial exploration of the graph, although the participants did not generally express their evaluation of the data displayed. Therefore, we can infer that it is possible that the students could start the process of mobilisation of previous knowledge and experience, as well as the emergence of meanings of data displayed by unspoken interpretations. On the other hand, the responses to the “read beyond the data” questions demanded that the participants verbalised their thoughts and opinions. From the analyses of these more explicit interactions with the data, we could infer their utilisation of Critical Sense. Therefore, in this section, we illustrate four aspects that came out of our analysis:

1. Mathematical knowledge: the participant interprets the media graph based on her mathematical knowledge of the quantitative relationships displayed.

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2. Personal opinion: the participant expresses her opinion about the data displayed which might be supporting or not the main tendency suggested. Generally this opinion is based on information from the media.

3. Personal experience: the participant exemplifies or explains the data based on previous situations she has actually experienced.

4. Affective exhibition: the participant exhibits her feelings about her interpretation of data displayed on the media graph.

Julia In the interview with Julia, a 19-year-old second-year student taking English, we can observe that she started her comments emphasising her opinion about the apparent trend displayed. It seems that, even when explicitly considering the data displayed on the graph, she was combining reading the data with her opinion about the realistic context to which the data might be related [4]:

R “What is your prediction for death rate and serious injury in 2001?” J It could be anything, couldn’t it? It’s not going to go down. They put…the

target is going down…so just because they stuck a pin in it and draw a line down it doesn’t mean going down, does it?…[Observing]…It is probably going to be middling about the same, maybe…615 something…just been fluctuating…It’s gone down a bit there, like compare to those it seems going down a level a little bit…something like that…between 600 and 615 (…) [I am observing] the patterns…I don’t think it is going necessarily going down.

Julia demonstrated her knowledge about the mathematical meaning of the straightline on the graphical representation. However, she made a distinction between what the media graph was representing, and what she believed about that representation. She developed the same type of approach when answering the following question:

R “If the targets from 2000 to 2010 were met, what do you think the pattern might be from 2010 to 2020?”

J If the…like it really did that, the line going down…than I suppose…if that works and than they want to…just keeping going down for the next 10 years. It will be 200 or something maybe. If it really works, which it probably wouldn’t? And than just the line kept going down like that…that will be the pattern. It will be 200. It is very nice, isn’t it? It’s quite reassuring.

From these transcripts alone, we cannot get a sense of the emotional aspects which were involved in her interpretation. However, analysis of the video data as a whole revealed that the intonation of her comments explicitly expressed sarcasm which displayed her distrust of what the data suggested. Even considering the structure of the media graph, Julia did not believe in the “tendency” represented. In these exchanges from the interview with Julia, we can identify three elements which comprise her use of Critical Sense: she expresses her opinion (personal

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opinion), she ‘manages’ her sceptical feelings about the data (affective exhibition), and displays her ‘technical’ interpretation of media graph (mathematical knowledge) to give a ‘sensible’ answer [5]. Hillary Affective exhibition of the reader’s interaction with the data is not always expressed. Evans (2000) suggests that latent content which constitutes these interpretations is not always observable, but might be inferred from the analysis of the participant’s reactions during the interview (p. 182). An example comes from the interview with Hillary, a 35-year-old PGCE student with a degree in music. She developed an interpretation which expressed her feelings about the data displayed on the road accidents graph:

R “What’s your prediction for death rate and serious injury in 2001?” H 2001? Right. Hum…Yeah…I would say…eh…It would be…I mean I know

it is going up…I know it is going up a little bit there. I think it would be down again about…says 600. At moment is going up at…Yeah…I think it will reduce it…I am not really going by…the graph, the flow of the graph…I am just going by a gut feeling more than anything. You’d like to think that it’s coming down.

In this part of the interview, the student begins by looking closely at the media graph, noticing the upward trend over the last two years, but then responds in terms of her feelings about the issue of traffic accidents. The interviewer then encourages her to try to specify a prediction:

R So do you think it would be some…If you guess some number, some rate? H Yeah, again…I am not…it’s very hard to say because…I’m thinking that

it’s…I am just thinking of…basically the media coverage on this type of thing…And…especially around Christmas time around…there is always afocus to control the number of accidents on the road, and I think this country…Well, I know this is Warwickshire, but I think this…the government does do…does make an effort…and obviously there are reductions. So I am basing my information on that, not just what the graph is telling me. But obviously going from last…Going from year 2000. And…yeah…hum…600. I don’t think there will a dramatic decline. But yeah...if I would say figure, say 600.

When asked for a ‘figure’ she gives reasons for the limits of her answer, in terms of her knowledge about attempts to improve road safety. She tries to get a balance between the information displayed, her ‘feelings’ and her knowledge about the social context to which the ‘figure’ might be related. At the end of her interpretation, she gives a reasonable conclusion based on the different aspects that were involved in her reading. In the following exchange, she seems to want to believe in the trend, but it also seems that she did not find a strong argument to base her answer on:

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R “If the target for 2000-2010”…there is a target there…“What do you think the pattern would be from 2010 for 2020?”

H 20…All right…hum…I think provided that technology doesn’t take over people’s well being…Than…I think the pattern should decline. But there are so many other things that might influence that pattern, like population rates…and…It is difficult to say…it is really difficult…it is hard question that…But I think…I think it would be a decline. I think there always be a decline, because it is such important issue…And then…There obviously… italways has been history of some kind of decline. But obviously things come along the way that interrupt the flow…obviously here [pointing to 1997figure on the graph] there is…more deaths on the roads. There was obviouslyreason…Well, I don’t know. It is hard to say whether its death and injuries. (…) But obviously that was addressed, because there was a big drop there [1997-98]. So, I think there always a kind of picture of a decline, or an attempt for a decline. With something as serious you know…as this issue.

Hillary moves between looking at the patterns shown on the media graph (mathematical knowledge), considering the context in which the road accident occurs (personal opinion), and expressing her desire to see safer roads with lower levels of accidents (affective exhibition). She seems to be reluctant to face up to the complexity of the question. However, when she was encouraged to try to specify a prediction, Hillary managed to “guess” an answer that seems to be based on the media graph, but also considers aspects such as the “hope” that was implicitly present on the interpretation:

R If could say a rate as well? H Rate?…Do you want me to say what I think that death and injury rate might

be…? Right. So if it’s starting at 500 which its obviously that’s what they’re hoping…I don’t think its actually going to hit the bottom. I think there is always be deaths and injuries on the road. I don’t think you ever avoid that happening, but it might be…For instance, a target…a realistic target might stretch from 500 to…say 300…Yeah, it seems a realistic target.

R Because you don’t think it will be zero. H No, I don’t think that will never happen. No, I don’t think that…I think there

is always be accidents and deaths on the road. Yeah. I think that would be very hard to control…to…[avoid completely]…Completely, yeah. I thinkthere is always be some statistics about this, because obviously there will be bad drivers.

The indication of emotional components which contributed to Hillary’s interpretation was emphasised when she verbalised an important aspect which had been omitted before: she had actually been involved in an accident (personal experience). The following exchange shows this specific part of the interview when the interviewer

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invited her to reanalyse the answers produced on the questionnaire some months before:

R You produced these questions at that time. H All right, that is interesting. R Do you comment this? H [Laughs]R …Just if you want to…H It’s just interesting to see…That might…hum…I came up with similar things

with this graph compare to other one. And that has more to do with the fact that I feel quite stronger about this…And than, I think I can perhaps relate to this more…you know, as a person.

R Why do you? H I have been involved in an accident myself… R Hah…I’m sorry.H I think…But…Oh no…It wasn’t a particular serious accident. But…I can

perhaps relate to the statistics more…I think. I can actually see what it’s telling me.

This moment of the interview was an opportunity in which she compared both situations in which she had read the media graph. It was a moment in which she could make explicit a factor which might be meaningful for her interpretation. We can infer that Hillary’s motivations and wishes played a prominent role in her interpretation. The fact that she cares about the road accidents and that she was actually involved in one herself, was an essential part of the meaning of the media graph for Hillary. She was trying to see what she wished to see, even though criticising and recognising the limits of her interpretation. Hillary’s interpretation seems to be characterized by a ‘conflict’ between the different elements involved: mathematical knowledge, personal opinion, personal experience, and affective exhibition related to the data interpreted. However, her interpretation is an indication of her use of Critical Sensebecause she mobilised her previous experience and knowledge, provoked the emergence of new and different meanings, but managed them to answer the question proposed. CONCLUSIONS We have argued that, while previous approaches, such as those described in the earlier parts of this paper, consider a range of perspectives on graphing that arerelevant to the school context, they fail to consider a range of factors which we observed during the interviews with student teachers interpreting media graphs: mathematical knowledge, personal opinion, personal experience, and affective exhibition. Particularly, we suggest the notion of Critical Sense in graphing as an important aspect which mobilised previous knowledge and experiences, provoked the

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emergence of particular meanings and managed the influence of these elements in interpretation. We propose that Critical Sense is an important notion which marks the complexity of the process of interpretation of media graphs. Our analysis of interview data indicates that the way in which we asked about predictions from the data helped the students in building an interpretation that involved an interaction with the data. It seemed that, as the participants workedthrough the interview, the media graph became more transparent for them. They seemed to be aware that technical knowledge about the interpretation was not enough to answer the questions. The participants built interpretations which combined different kinds of previous and emergent knowledge, experiences and feelings, which all played an important role in their reading of the media graph. Therefore, we do not see these elements as a form of ‘interference’ that can disturb the interpretative process (e.g. McKnight, 1990). Our analysis also indicates that it is quite difficult to fit the student-teachers’ responses into hierarchical classifications which, for example, would make critical thinking the highest tier of sophisticated statistical literacy. In part, this is because we argue that the Critical Sense is an ability which is dependent on contexts of interpretation.The analyses seem to us to provoke further discussion about the nature of interpretation of media graphs, and about the pedagogical implications of the notion of Critical Sense. For example, we argue that bringing media graphs into the classroom as a pedagogical approach to teach graphing will not, in itself, embrace the complexity of the interpretative process which can be established by readers of media graphs. On the other hand, the analyses which come from one-to-one interviews cannot be generalised to conventional pedagogical contexts in which many other factors can be involved. However, we suggest that the teaching of graphing should be based on opportunities to learn how to be aware of, to experience and to balance the diversity of elements involved in the interpretation of media graphs, in other words, how to develop Critical Sense in graphing. NOTES 1. In this paper we use the term media graphs referring to statistical graphs which are published in newspapers, magazines, periodicals and other publications that provide news and information for the public.

2. The process of interpretation of graphs does not only happen during the situation in which people read data displayed in a ‘ready made graph’. During the construction of graphs it is necessary to establish interpretations of data. For example, the choice of the type of graph to present the data is in itself an interpretation. Therefore, the term graphing refers to both construction and reading processes which are related to the interpretation of graphs.

3. There are several factors involved in students’ participation which need to be taken into account. For example, the fact that the students volunteered for interviews during a very busy term time, that they established good rapport with the interviewer, and that they did not seem to feel that they were

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being assessed, are aspects which may be relevant to understanding their engagement in the interpretative situation.

4. The extracts which start with R are associated with the researcher’s speech. The others start withthe initial letter of participant’s names (e.g. J = Julia). The excerpts of researcher’s questions which are in inverted commas are related to parts which were read from the interview script. The words or sentences in square brackets were added to clarify the participants’ actions as well as to complete some sentences. Ellipsis … in the transcripts means that the sentence was not started or finished properly, or that the participant gave a pause when was formulating a sentence. Ellipsis in brackets(…) means that a small part of the participant’s speech has been omitted because it repeated what she said immediately before.

5. Julia’s response was ‘sensible’ for her because she could express her opinion, consider her feelings and gave an ‘approximate technical’ answer which satisfied the question asked. Her answer is also sensible because even though the graph did not explicitly display figures for 2010, she could provide an approximation very close to the hypothetical answer which, according to the government target, would be a reduction of 40 per cent for 2010 (379 casualties) and about 227 casualties on 2020.

ACKNOWLEDGEMENTSWe wish to thank: members of the audience at the BSRLM Day Conference held in BirminghamUniversity on 15th November 2003; our reviewers for very helpful comments in early versions of this paper; CNPq – Conselho Nacional do Desenvolvimento Científico e Tecnológico (Brazil) for financial support for our study.

REFERENCES Ainley, J.: 2000, ‘Constructing purposeful mathematical activity in primary classrooms’, in C. Tikly

and A. Wolf (eds.), The Maths We Need Now: Demands, Deficits and Remedies, University of London, London, 138-153.

Curcio, F.: 1987, ‘Comprehension of mathematical relationships expressed in graphs’, Journal forResearch in Mathematics Education 18 (5), 382-393.

Da Rocha Falcão, J.T., Araújo, C.R., Andrade, F., Hazin, I., Nascimento, J.C., Lessa, M.M.L.:2003, ‘Affective aspects on mathematics conceptualisation: from dichotomies to an integrated approach’, in N. Pateman, B. Dougherty, and J. Zilliox (eds.), Proceedings of the 2003 JointMeeting of PME and PMENA, University of Hawaii, Honolulu, HI, vol. 2, 269-276.

Evans, J.: 2000, Adult’s Mathematical Thinking and Emotions: A Study of Numerate Practices, Routledge, London.

Frankenstein, M.: 1989, Relearning Mathematics: A Different Third R – Radical Math(s), FreeAssociation Books, London.

Freire, P.: 1972, Pedagogy of the Oppressed, Penguin, London.

Freire, P.: 1994, Pedagogy of Hope, Continuum, London.

Freire, P.: 1997, Pedagogy of the Heart, Continuum, London.

Dow

nloa

ded

by [

T&

F In

tern

al U

sers

], [

Mr

Lea

h St

anle

y] a

t 08:

12 1

2 Ju

ly 2

016

Research in Mathematics Education Volume 6

128

Friel, S., Bright, G. and Curcio, F.: 1997, ‘Understanding students’ understanding of graphs’,Mathematics Teaching in the Middle School 3 (3), 224-227.

Gal, I.: 2002, ‘Adult statistical literacy: meanings, components, responsibilities’, International Statistical Review 70 (1), 1-25.

Kerslake, D.: 1981, ‘Graphs’, in K. Hart (ed.), Children's Understanding of Mathematics: 11-16, John Murray, Oxford, 120-136.

McKnight, C.: 1990, ‘Critical evaluation of quantitative arguments’, in G. Kulm (ed.), AssessingHigher-order Thinking in Mathematics, American Association for the Advancement of Science, Washington, DC., 169-185.

McKnight, C., Kallman, C., and Fisher, M.: 1990, ‘Task analyses of critical evaluations ofquantitative arguments: first steps in critical interpretation of graphically presented data’, Proceedings of Annual Meeting of American Educational Research Association, Boston, MA., 2-22.

Moreira, L.: 2002, ‘Mathematics education and critical consciousness’, in A. Cockburn and E. Nardi (eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, University of East Anglia, Norwich, UK, vol. 3, 369-376.

Skovsmose, O.: 1994, Towards a Philosophy of Critical Mathematics Education, Kluwer Academic Publishers, Dordrecht.

Valsiner, J.: 2000, Culture and Human Development, Sage, London.

Vygotsky, L.: 1978, Mind in Society: The Development of Higher Psychological Processes, Harvard University Press, Cambridge, MA.

Watson, J.: 1997, ‘Assessing statistical literacy through the use of media surveys’, in I. Gal and J. Garfield (eds.), The Assessment Challenge in Statistics Education, IOS and Press International Statistical Institute, Amsterdam, 107-121.

Watson, J.: 2000, ‘Statistics in context’, The Mathematics Teacher 93 (1), 54-58.

Warwickshire County Council: 2001, Quality of Life in Warwickshire, Warwickshire County Council, Warwick

Wild, C., and Pfannkuch, M.: 1999, ‘Statistical thinking in empirical enquiry’, International Statistical Review 67 (3), 223-65.

Dow

nloa

ded

by [

T&

F In

tern

al U

sers

], [

Mr

Lea

h St

anle

y] a

t 08:

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2 Ju

ly 2

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