Chik y Watson_Estudio de Caso

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Journal of Mathematical Behavior

21 (2002) 371400

Collaborative influences on emergent statisticalthinking a case study

Helen L. Chicka, , Jane M. Watson b

a Department of Science and Mathematics Education, University of Melbourne, Melbourne 3010, Vic., Australia

b University of Tasmania, Hobart, Tasmania, Australia

Abstract

The purpose of this case study is to examine how collaboration affects the emergent statistical thinking of a

group of three Grade 6 boys. Results of previous studies of students in Grades 3, 6, and 9 suggested that (a) when

finding and justifying associations in data sets students working in groups may produce higher level outcomes

than those working individually, and (b) there are numerous factors that influence the success or otherwise of

collaborative activity. The current study, based on detailed analysis of video tape and transcripts of a group working

collaboratively on a data handling task, documents various factors that affect collaboration and how these contribute

to the attainment of desirable cognitive outcomes in terms of the task set. These outcomes are classified by emergent

statistical themes and insight is gained into how nave statistical thinking begins to develop during the collaborativeprocess. Implications for educators and researchers are considered.

2002 Elsevier Science Inc. All rights reserved.

Keywords: Emergent statistical thinking; Collaboration; Elementary students; Data handling; Cognitive change

Collaboration while completing mathematical tasks has been the subject of numerous studies anddebates by mathematics educators. Many authors (e.g., Davidson, 1985; Lindquist, 1989; Phelps &

Damon, 1989; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990) have found evidence that it can be

a successful approach to mathematical learning. They highlight the benefits that come from providingstudents with opportunities for verbalization, allowing students to resolve conflicts in understanding, and

giving students increased responsibility. Collaboration allows a diversity of problem solving techniquesand has great scope for accommodating students needs and interests. In contrast, Stacey (1992) gives

some examples of collaborative problem solving episodes that show that there can be a tendency amonggroups to choose ideas and approaches that are easily accessible, but not necessarily appropriate or correct,thus showing that a collaborative environment need not lead to successful conceptual development.

Corresponding author. Tel.: +61-3-8344-8324; fax: +61-3-8344-8739.

E-mail address: [email protected] (H.L. Chick).

0732-3123/02/$ see front matter 2002 Elsevier Science Inc. All rights reserved.PII: S0732-3123(02)00135-9


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372 H.L. Chick, J.M. Watson / Journal of Mathematical Behavior 21 (2002) 371400

The main difficulty associated with trying to draw conclusions about the value of collaboration is that

there are many factors involved. Although acknowledging that cooperative learning can increase studentlearning, Slavin (1989/1990) calls for the identification of the specific conditions required to achieve it.

Various authors (e.g., Good, Mulryan, & McCaslin, 1992; Johnson & Johnson, 1985; Noddings, 1989)propose a number of important variables, such as the nature and purpose of the task, the learning strategiesemployed, the occurrence of academic disagreement among group members, the time spent on task, the

support and feedback of peers in the group, the ability levels of group members, the composition of thegroup, the role of the teacher in preparing students for small group activities, classroom management, theage of the students, and the role of explanations given and received within the group, but acknowledge

the difficulty in isolating these factors for study. One report (Good et al., 1992, p. 190) emphasized thatthe way that students experience group work is so varied that it is virtually impossible to predict the

outcomes of a collaborative event.Furthermore, although it has been suggested that certain statistics-oriented activities be conducted in

a collaborative setting (see, e.g., Russell & Corwin, 1989), there have been only a few studies of thebenefits to the learner when encountering statistical and probabilistic ideas in that setting. There arereports of some successes but the data are by no means complete or conclusive (Carvalho & Csar, 2000;Lajoie & Lavigne, 1994; Lajoie, Jacobs, & Lavigne, 1995). Lehrer and Romberg (1996) examined the

choices children make about how to transform observed phenomena into data and construct suitablerepresentations from which they can make meaningful statistical inferences, but group work per se wasnot considered by those researchers.

There are thus two questions of interest for the current study. One concerns the effectiveness of col-laborative activity in the statistics part of the mathematics curriculum. This is particularly relevant at the

stage of development where students are moving from what Shaughnessy (1992) termed nave-statisticalconceptions to emergent statistical ones. As part of an earlier study that investigated how higher order

thinking in statistics can be recognized and evaluated, Watson, Collis, Callingham, and Moritz (1995)speculated that students working in groups while interpreting data seemed to produce higher level out-comes than individuals. For students in the middle school years, this study provides the opportunity toconsider in detail the types of emergent statistical insights that arise in a collaborative setting.

The second question involves the determination of factors that influence collaborative outcomes. Thishas already been addressed in some detail by the present authors (Watson & Chick, 2001a) in a precursor

to the current study. Three levels of cognitive outcome were observed there during sessions with eightgroups of three students. Lifting was the term used when collaboration was seen to raise the cognitivelevels of one or more group members. The term falling was used when the outcome of collaboration

was a reduction in the level of cognitive functioning. Finally, the term hovering was used to describe

the outcome of a collaborative event that neither lifted the group, nor caused it to fall. It is importantto note that lifting and falling are associated with a change in cognitive functioning as a result of the

collaboration for one or more of the group members. Sometimes this is seen in terms of relative changeor in sensible decision-making, for example, which may not lead directly to optimal performance but is

beneficial at least in the short-term.Further to the observation of collaborative outcomes, Watson and Chick (2001a) documented 17 factors

in the group setting that appeared to influence directly the observed outcomes. These are summarized

in Table 1 under three general headings. The first nine are cognitive factors, associated with aspects ofcognition and metacognition that can influence problem-solving outcomes. The next five are social orinterpersonal factors that can have an effect on collaborative outcomes. Finally, there are three external


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H.L. Chick, J.M. Watson / Journal of Mathematical Behavior 21 (2002) 371400 373

Table 1

Factors associated with collaborative outcomes

Cognitive factors

Cognitive ability: The intellectual capability of an individual or the group; the influence of this factor depends on thelevel of ability and leadership skills, and can lead to lifting or falling

Previous experience: Understanding from another setting that is applied to the current task; outcome depends on

nature of understanding and persuasiveness of person who brings it

Cognitive disagreement: Cognitive dissonance that, if not resolved, may result in hovering or falling; outcome

depends on nature of arguments and sometimes who puts them

Doubt: Re-evaluation of current understanding based on new evidence from the task or group; outcome depends on

confidence of student and quality of arguments presented

Misunderstanding: Ideas that are misheard, not clearly expressed by speaker, or misinterpreted by hearer; outcome is

often negative if not resolved quickly

Tenacity of ideas: An obsession for an idea, brought up repeatedly until acknowledged by the group; outcome

depends on quality of the idea and whether the group takes it on

The big picture: Ability to keep in mind the overall goal of the group; outcome may be negative if group is distractedfrom overall objective or positive if goal can be kept in sight

Picking the easiest ideas: Making unsophisticated choices of strategy or technique when better ones are available;

outcome is usually negative

Organizational collaboration: Collaboration that is on-task but not directly cognitive. It is usually concerned with the

mechanics of assigning tasks and is deemed to be metacognitive as it requires an overview of the task and

knowledge of the strengths of group members

Social or interpersonal factors

Leadership factors: Ability or otherwise of a group member to lead the group and convey ideas; outcomes depend on

both the leadership skills and the worth of the ideas espoused

Social disagreement: Social dissonance that may affect further interactions within the group

Egocentrism: Being locked in ones own world-view and unable to listen to others or refusing to be influenced by

others opinions; outcome usually negative (or non-positive)Social collaboration: Off-task collaboration in which the group still functions as a group but is not engaged in the

task, or collaboration based on social conventions rather than cognition; the outcome is often negative distraction

Other social factors: Pre-existing friendships, gender balance, location of interviewer, etc.; outcome depends on factor

External factors

Task factors: Use of materials, conceptions of expectations of the task, open/closed nature of task, etc.; outcome

depends on factor

Outsider: Influential person from outside the group; such a person may be a teacher or interviewer or a fellow student;

outcome depends on nature of intrusion from this person

Environment: External physical factors such as classroom noise, physical positioning in group, outside interruptions,

etc.; outcome depends on factor

factors that may contribute to outcomes. The first two sets of factors represent influences of individualsin the group in their attempts to construct ways of handling the data presented and of the group in

devising mathematical or statistical practices appropriate for the task. These factors are consistent withthe emergent perspective suggested by Cobb and Yackel (1996), combining aspects of a socio-culturalperspective and a constructivist perspective. In the study in which these factors were described (Watson &

Chick, 2001a), these outcomes and influences were observed in eight collaborative groups, each involvingthree students, from Grade 3 (two groups), Grade 6 (four groups), and Grade 9 (two groups). Althoughthe groups worked on tasks from the chance and data part of the curriculum, the prime focus of that study

was on collaboration and the factors that influenced it, not on the mathematical outcomes of the tasks.


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In that study the groups varied in the degree of collaboration and the levels of outcomes, with both often

disappointing. One of the groups observed, however, appeared to engage in more collaborative behaviorthan the others, and also attempted to deal with quite sophisticated statistical ideas for their age. This

group is the subject of the current study, verifying that successful outcomes can occur in a group settingand that difficult statistical ideas can be grappled with by relatively inexperienced students.

1. Research aims

In contrast to most previous research on collaboration, this study considered (i) a task that was notwell-defined; (ii) a task from the probability and statistics section of the mathematics curriculum; and

(iii) the interactions that appeared to cause cognitive change to take place, recorded as they happened bymeans of a fully transcribed video recording. This case study investigated issues raised in the literature (cf.

Watson & Chick, 2001a) by observing a group of three students working in a collaborative environment.The fully transcribed video recording provided a nearly complete picture of exactly what took placein interactions to produce cognitive change. This was essential, as the open-ended task produced some

dynamic interactions.There were two closely interrelated aims of the research. The first was to corroborate the existence of

the factors and collaborative phenomena identified by Watson and Chick (2001a) that appeared to have a

direct influence on cognitive outcomes. Of particular interest were the changes that took place in the levelof cognition of the group: lifting, falling, and hovering. This will go further to address Slavins(1989/1990) concern to document specific conditions contributing to collaborative outcomes. The second

aim within this collaborative framework was to identify the emergent statistical ideas and outcomes ofnave but motivated students and how these were influenced by the collaborative phenomena. For statistics

educators this may inform decision-making in terms of teaching methodology and the curriculum.

2. Methodology

2.1. Procedure

This case study considers the collaborative group work of three Grade 6 boys, Sam, Mark, and Luke

(names have been changed). Their group was one of eight groups of students from Grades 3, 6, and 9who worked in group settings on two protocols from the chance and data section of the mathematics

curriculum (see Watson & Chick, 2001a; this group was Group 6 in that study). The boys workedtogether on the protocol described below for two sessions separated by 2 days; the first was 40 min longand the second 75 min. The boys were selected by their teacher, who was asked for three above-averagestudents who would work well together on a cooperative task. The boys had previously participated in

collaborative classroom activities, but not necessarily with each other. The sessions took place in a quietroom, removed from the classroom, in the presence of a researcher, and both sessions were video-taped.

Complete transcripts were prepared for both sessions. The transcripts were annotated by the transcriber,the researcher who had attended the sessions, and the two authors.

The researcher who worked with the boys was a classroom teacher who was working with the second

author as a research fellow at the time the sessions took place. Her role was to explain the task, facilitate


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the boys work on it, and answer their questions. The sessions were not intended to provide instruction nor

were they in the style of a traditional interview. Most of the time the boys worked without intervention,but with the researcher in the room. This implies that the setting for this collaborative work might be

regarded as ideal, in that the presence of the researcher ensured that the group remained on task for theentire period. Her presence alone seemed enough to achieve this as she did not have to remind the studentsto focus on the task. On one occasion when she left the room, however, the boys immediately stopped

work and shared some sweets among themselves, seemingly oblivious to the video camera. This indicatesthat the students did not necessarily limit their off-task behavior for fear of having such misbehaviorrecorded; thus the use of a video camera probably did not distort or inhibit the students natural conduct.

Since the students were focused for virtually the entire activity, time-on-task is not a variable of interestin this study. It should also be pointed out that this group required no special encouragement to work

collaboratively nor was there provision of explicit instruction about the social behavior that might beexpected to facilitate successful collaboration.

The desire to mimic a classroom environment, but one where high quality data collection could occur,determined the conditions as described above. The interrelationship of the collaborative setting, the taskset, and the cognitive and social skills brought by the boys, was complex and fits within the frameworkdescribed by Yin (1994) as appropriate for employing a case study.

The beginning propositions for the case study were based on the 17 factors identified by Watson andChick (2001a) and associated with cognitive outcomes reflecting lifting, hovering, or falling. The datacollected on the unit of study the group consisted of the complete identification of the factors as well

as the statistical concepts employed and developed as part of the collaborative activity. Although a bodyof statistical concepts existed that could be expected to be used by a statistician (e.g., Moore, 1991), these

were not delineated a priori because the students had had no formal exposure to them. The identificationof the emergent statistical concepts used by the students took place as part of the data analysis, as did

the associated judgment of whether the outcomes improved, diminished, or remained unchanged. Thisdesign is consistent with that of Yin (1994), where for this study the findings are interpreted in the lightof the relationships among collaborative factors and cognitive outcomes, the expectations for meaningfulstatistical inquiry, and the implications for the classroom.

2.2. Data cards protocol

The students were supplied with a set of 16 cards, each containing information such as age, weight,and food consumption for an individual. An example of one of the cards is shown in Fig. 1 and the entiredata set is produced in Appendix A. The data provided were designed to describe 16 children ranging

in age from 8 to 18 years, with plausible favorite activities, and weights that were within the normalrange as published by the National Health and Medical Research Council (1957). At the commencementof the session the students were given time to examine the data cards and were invited to produce their

own cards with information in the same categories about themselves, thus producing a total of 19 data

Fig. 1. Example card from the concrete materials for data cards interview items.


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cards for analysis. The students were then asked to find any interesting connections among the variables

represented on the cards and justify these for the researcher. Although the sample size arising from theset of data cards would not be considered appropriate for a statistician to carry out an analysis, it was

considered adequate for students of this age to discover relationships using the concrete methods availableto them. This sample size is consistent with those used by other statistics educators with school studentssomewhat older than those in this study (e.g., Estepa, Batanero, & Sanchez, 1999; Ross & Cousins, 1993).

A larger sample size would have required tools needing additional explanation (e.g., a spreadsheet) orwould have involved greater computational complexity. The resultant level of abstraction may also haveinterfered with students ability to demonstrate understanding. The desire was to observe the students

data handling skills in a context they had not met before with a manageable data set.The seven variables (including gender) for the data cards set were chosen for several reasons. Because

the original study in which the cards were used involved students in Grades 39, it was necessary to havevariables that young children could easily interpret. It was also important to have categorical as well as

measurement variables, both to assist younger children if they decided to use counting techniques and toallow for a variety of methods of comparison for older students. From a statistical education point of view,it is of interest to observe which variables the students of different ages focus on. Eye color, although ofno interest to a statistician, was found by Watson et al. (1995) to be of interest to many Grade 3 and 6

students. Gender was intended as an implicit variable that could be determined by consideration of thenames, and many groups in earlier studies compared boys and girls on the weight and fast food variables.

Students were provided with materials to produce graphical representations of the data: pens, rulers,

paper, glue, scissors, blocks, and counters. To give the students an opportunity to demonstrate theirknowledge or understanding of a variety graphical forms, they were invited to contribute their own

ideas. Near the beginning of the second session, students were also shown prepared examples of datarepresentations such as tables, pie charts, line graphs, scattergrams, and column graphs, and were asked

if they had used any of these before. They were then given the opportunity to use one or more of themto relate variables from the data. Further description of the protocol is available in Watson et al. (1995),with examples of responses largely from individual students in Lidster, Chick, and Watson (1997), andfrom groups in an earlier study in Watson and Callingham (1997).

2.3. Transcription of the sessions

Both sessions were video-taped, then fully transcribed and annotated in an attempt to provide both anaccurate account of what was said and to illustrate the interactions and activity that were taking place.

On some occasions, the annotator (occasionally the transcriber or researcher at the sessions, but more

usually one of the authors) has speculated about the exact nature of the observed behavior; the fact thatsuch conjecturing may be inaccurate is indicated by question marks or other comments when it occurs.In the excerpts from the transcripts the annotators comments are within square brackets; a [ . . . ] in thetranscript means that material has been edited from it; and an ellipsis with no enclosing brackets means

a conversational pause. The researcher present at the sessions is identified by R.

2.4. Synopsis of the interview sessions

The following summary is provided to assist with comprehension of the tasks attempted and their place

in the time-frame of the sessions. In the first session the boys initially spent time examining the cards and


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creating data cards bearing information about themselves, before deciding, after some discussion, to graph

average weight against age. After completing this task the researcher suggested that they examine someother aspect, which led to a discussion about why graphs are good ways to summarize information. The

group then decided to examine the relationship between fast food and weight, a task made complicatedby the fact that they wanted to incorporate some notion of fat, despite the lack of data for it, andby the decision to represent the details with towers of Unifix blocks. The struggle to reconcile various

difficulties continued in the second session, until the researcher helped them to focus on simpler ideas.This was achieved when she presented various forms of graphical representation (such as bar charts, piecharts, and scattergrams) that the boys interpreted, before they decided to produce their own bar charts

to show, separately, favorite activity and dominant eye color. Having completed these they produced agraph showing the relationship between average fast food consumption and age, and then redid a graph

of average weight for age, a task previously done in the first session. The session closed with a discussionon working collaboratively led by the researcher.

2.5. Analysis

The transcripts and the video recording were examined in order to determine the cognitive changes thattook place leading to better, worse, or unchanged performance and the factors that influenced them.This was done during the analysis for the larger study (Watson & Chick, 2001a) by means of a type of

clustering process (Miles & Huberman, 1994). Transcripts were successively analyzed and similar typesof collaborative events were grouped together and given succinct descriptors. The nature of the outcome,

whether positive, negative, or neutral, was also recorded. Validation of the decisions regarding theseclassifications occurred through a triangulation process (Maxwell, 1996), involving discussions amongthe authors, the researcher at the sessions, and the transcriber. The complete transcript for this group of

boys was then analyzed in detail to identify collaborative factors and outcomes associated with identifiableemergent statistical concepts. Extracts were selected and clustered to provide evidence of these.

3. Results

The open-ended task chosen for this study was selected to allow for a range of outcomes and to providean opportunity for students to display high order statistical reasoning. As in the study of other Grade

6 students by Watson et al. (1995), no formal training in analyzing complex data sets or in providingevidence to support hypotheses had been provided for the students. It was hence of interest to observe

the level of outcome associated with nave attempts to analyze the information on the data cards. Theachievement observed for the group in this study equated with the highest level observed by groups inWatson et al. This level was characterized by consolidation and application of concepts in a mannerthat integrated understanding of the relationships between the different aspects of elements involved in

the task (p. 250). Although it is acknowledged that the boys in this group did not always perform atthis high level, the opportunity to observe the groups behavior over a relatively long period of time

provides evidence of the variation in performance even for those who do achieve high level outcomes.Such an analysis was not possible in the Watson et al. study because only final outcomes were analyzed.Hence false starts or dead ends, such as seen when the boys struggle with percent in Vignette 6, were

not documented in the earlier work. Although statisticians might criticize the analysis carried out by this


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Table 2

Emergent statistical thinking in the Vignettes

Statistical concept Brief description

Association (Vignette 1) Consideration of two or more variables and possible causeeffect relationships

Average (Vignette 2) Idea of average as: typical, middle, a summary value. Ability to calculate with meaning

Graphing (Vignette 3) Use of graphing skills to tell a story

Evidence (Vignette 4) Realization of the need to prove suppositions, i.e., to find and support implications

from the data

Part/whole (Vignette 5) Recognition of part/whole relationships in data sets

Mathematical tools (Vignette 6) Trying available mathematical tools to tackle a problem, e.g., percent, totals

Limitations (Vignette 7) Recognition that ideas and conclusions are limited by the information at hand

group, it represents a high level of performance for students of this age. Table 2 contains a summary of theemergent statistical thinking that occurred during the groups collaboration and the results are presentedin seven Vignettes that each illustrate a different statistical theme. The order of these themes was chosen

to reflect the chronology of the sessions as far as was possible. A final Vignette contains the studentsbeliefs about working in groups.

The results of the analysis of collaboration are summarized in Table 3, which indicates the relationshipof the factors identified as influencing the outcomes of collaboration and the type of cognitive changeobserved (lifting, hovering, or falling). The statistical concepts associated with the outcomes are noted

Table 3

Vignettes illustrating collaborative outcomes and influencing factors

Influential factors Outcomes

Lifting Hovering Falling

Cognitive factors

Cognitive ability 1, 2, 3, 4 1, 5, 7

Previous experience 1, 2, 3 1, 3, 6, 7 6

Cognitive disagreement 3, 4, 5, 6 1, 4, 7 6, 7

Doubt 3, 4, 6 2, 3, 5, 6, 7

Misunderstanding 3

Tenacity of ideas 1, 3, 4 3, 5, 6 6, 7

The big picture 2, 3, 4, 5 1, 2, 3, 6, 7

Picking the easiest ideas 3 3 6

Organizational collaboration 2, 3 1, 3, 6

Social or interpersonal factorsLeadership factors 1, 2, 3 1, 6

Egocentrism 6

Social collaboration 2

Other social factors 3, 4, 8 5

External factors

Task factors 3, 4 2, 3, 5, 7 6

Outsider 1, 3, 4 6

Note: Vignette 1 concerns association, 2 = average, 3 = graphing, 4 = evidence, 5 = part/whole, 6 = mathematical tools,

7 = limitations, 8 = beliefs. Social disagreementand environmental factors were not observed for this group of students.


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in the table in the cells whose characteristics they display. Most of the influencing factors described do

not exist in isolation, and, indeed, interact extensively; for example, doubt can lead to conflict that maythen produce lifting or falling. Furthermore, the group and the individuals in it are not always consistent

in their levels of functioning, as exemplified for average when Luke first teaches Sam how to calculateaverages, but then later encounters difficulties himself.

3.1. Vignette 1: association [cognitive ability, previous experience, cognitive disagreement, tenacity of

ideas, the big picture, organizational collaboration, leadership factors, outsider]

The following extract is from early in the first session, immediately after the researcher has introduced

the boys to the problem, shown them the materials, and invited them to work together. It gives an exampleof the usefulness of the task in providing a context with which the boys can immediately engage, exploringideas of association.

Mark: Why dont we check out the favorite activities first and check out the weight and everything[after]. [Leadership factors and an attempt at organizational collaboration. Outcome hovers for

a while.]

Luke: Like if the favorite activity is something lazy like TV.

Mark: Yes, theyd have a big weight. [Probably previous experience, which will lift them.]

Luke: And then check out what the weight is . . . .

Sam: Shall we make a graph? [This query has no impact initially.]

Luke: Some people can be heavy but its really muscle. So it depends on favorite activity. [More

previous experience.]

Sam: Most of the people who weigh a lot like watching TV 74, 60 . . . . [An early sense of the big

picture that remains undeveloped, implying hovering.]Luke: Or it could be his height because Zack is about 50.

Mark: Cos the weight and swimming . . . 15 . . . hed have strong arms . . . and the netball . . .

Luke: Hes got two fast foods a week that might add on . . . . [Cognitive ability from all the boysshowing recognition of multiple variables and causeeffect relationships.]

Sam: How can we say that in a graph? [Tenacity for graph idea.]

Luke: We could put ages and put . . . .

Sam: . . . the weight for their ages . . . .

The students are getting a feel for the data and the task. Sams sense of the big picture is such that hebelieves that drawing a graph is a relevant way of summarizing the data they have, and he is able to

guide the group to consider it. This is an early indication that Sam may be a charismatic intellectualfor the group, that is, someone with cognitive ability and good leadership skills who can direct the groupproductively.

Later, after the boys have completed their work on weight and age, the researcher prompts furtherconsideration.

R: What sort of things were you hoping to show when you were looking at fast food and weight?

[The students had only suggested this; they had not studied it.]

Sam: The ones that have a lot of fast food are . . . theyd have a higher weight than people who dont.

[Previous experience and cognitive ability.] [ . . . ]


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R: So if someone came through the door you could say, Look here, this is what we have found out

about this group of people. [Mark starts to sort the cards into age order.]

Luke: Say the average of. . . shall we focus on the fast food map [his term for graph?], the weight or

what?Sam: Well . . . weightandfastfoodIthinktheyreinthesameone . . . somehow connected . . . . [Nascent

idea of correlation.]

Luke: Mmm?

Sam: Well, the more fast food you eat the fatter you get so . . . . [Correlation idea clarified.]

Mark: Well, if youve got a fast metabolism it may not make the difference. [ Cognitive disagreementbased on previous experience.]

Sam: Yes, but if the person was 74, the average we got was 70 so . . . . [Cognitive ability.]

It is possible that Sam may be thinking about people being above average and below average, and

suspecting that this is an indication of fatness. Unfortunately Sam does not get a chance to pursue thisconnection because Luke proposes an idea that moves the group in another direction.

At one point the group, while struggling to consider several variables at once, actually realizes that it

is falling. Sam, perhaps in an attempt to regain a view of the big picture, suggests that at least they donot have to worry about eye color and Mark agrees. The researcher the outsider finally intervenesin an attempt to help the group refocus on fewer variables at once.

Sam: Weve done weight and age already so . . . weight and activity?

Luke: Or we could do activity and food.

R: You could do activity and fast food. . . that would be another interesting one wouldnt it?

Luke: We could do weight/activity, weight/food because activity and food both goes with weight and

we could do two charts like that. [Again the ability to consider complex relationships is shown.]

A confounding difficulty is that by this stage the boys are dealing with data that they have groupedaccording to age, and the weights they are working with are average weights. This may well make it moredifficult for them to achieve a useful representation and keep the big picture in sight. The researchers

intervention lifts them to focus on a narrower, more straightforward task.

3.2. Vignette 2: average [cognitive ability, previous experience, doubt, the big picture, organizational

collaboration, leadership factors, social collaboration, task factors]

There are several extracts from the sessions illustrating the boys previous experience with average.

The following is from early in the interview when the boys are examining the cards; at this stage noaverages of any kind have been calculated or even mentioned.

Luke: With the guy who watches TV and has 12 fast foods a week, I mean 12 is way over average.

Luke uses the expression way over average quite confidently. There is also a hint of the same idea inthe following aside from later in the interview.

Mark: Richards 65 [kg] . . . and hes only 10 [years old].

Sam: Richard? [It would seem that Richard is a classmate who is a bit large.]

Luke: Yeah, but hes above average. [All dissolve into fits of giggles.]


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Lukes original remark may well have influenced their decision to start looking at average weights for

age, thus lifting them.The boys are quite comfortable with the mechanics and the concept of average, showing their cognitive

ability. At one point later in the second session they are sidetracked discussing a recent mathematicstest, which also illustrates social collaboration and hovering because no change in cognitive functioningoccurs.

Mark: What did you get in that maths test?

Sam: Which one?

Mark: The one that we did with the worksheet . . . .

Sam: 10.

Luke: 11.

Mark: I got 12 . . . James got 13!

Sam: I know.

Luke: 10, 11, 12, 13 . . . Im the average!Mark: [With dry good humor] Thank you Luke!

Although Lukes last statement above is not strictly correct, he has appropriate understanding, and it ishis previous experience and knowledge of average that lifts the group in the following extract. In order to

work out average, Luke reminds Sam of the process involved (interestingly, Luke does not just take overthe calculations), and then, after collaboratively deciding on the organization of the graph, it is Mark whoguides and supervises Luke in its production. They want to work out age-based weight averages.

Sam: We could work out the average of what their weight is for their age. [ . . . ] Ill just write down allthe ages. [Sam directs proceedings here, exhibiting organizational collaboration and leadershipfactors. He makes a list of all the age groups and they then fill in all the weights by the ages.

Mark organizes the pile of cards and Luke calls out the values.]Sam: [Uncertain] We add them up then? And . . . you work out . . . .

Luke: The average . . . .

Sam: Yeah, the average . . . but theres more people for some of them. [Sam shows he is aware of whatan average is supposed to take into account, because he has more people in some age groups.There is cognitive ability here, even if he does not know or remember how to calculate the mean.]

Luke: You know how you add those [points to the list of 8-year-olds, of which there are two] . . . youonly divide by two [continues down the list], divide by two, divide by two, divide by one, [reaches

the four 11-year-olds] then you divide by four . . . so it would be different . . . averages. [Lukes

cognitive ability and previous experience with average computation lifts the group.]

Sam: So if you take the first one, which is 54 [the sum of the weights of the two 8-year-olds]. . .

divideit by two . . . 27. The average for eight is 27. . . . So, Mark, do you want to take a piece of graphpaper and put the ages along the bottom and put up the side, the weights up the side. [Sams

organizing skills lead to positive cognitive outcomes for the group at this point. Mark and Luke

work on their assigned tasks, while Sam continues the calculations with a calculator.] [ . . . ]29.5 . . . does that seem right? Yup, 29.5 is OK for age nine. [Self-corrected doubt.] So you get

a pencil and put them on the graph with the first one 27 and 29.5 is the average for the people.[Organizational collaboration.]

Luke: Oh, so this chart will show the averages of weight, will it? [Doubthere as he tries to get a sense

ofthe big picture. Mark is doing most of the preparation for graphing and so, with Sam doing the


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calculations, Luke is a little unoccupied at present. This task factormay contribute to hoveringfor Luke.]

Although Sam needs Lukes advice on the actual procedure for calculating the averages it is Samwho decides that it is averages that they want to have on their graph, perhaps influenced by the earliercomment of Lukes. Thus, one member of the group has helped to liftthe group by proposing a task thatwill represent the data in which they are interested, whereas another has liftedthe group by providing the

mechanics of a particular subtask. It is interesting to note that whereas Luke helps Sam here, it is Samwho helps Luke in a similar situation described in the next Vignette. In this case, organizational abilityhas helped the group, too, as Sam provides directions that will enable the group to achieve its objectives.

Sams dominant role provides more evidence that he is the charismatic intellectual. Towards the end ofthe excerpt, Luke does seem to have some doubtabout what they are doing; perhaps he has temporarilylost sight of the big picture, or, indeed, has not quite seen it at all yet. Shortly after, as he realizes the

point, he is liftedto a higher level of functioning.

3.3. Vignette 3: graphing [cognitive ability, previous experience, cognitive disagreement, doubt,

misunderstanding, tenacity of ideas, the big picture, picking the easiest ideas, organizational

collaboration, leadership factors, other social factors, task factors, outsider]

There are several issues associated with graphing that arise for the boys during the sessions. Theserange from decisions on which type of graph to use, to assisting each other represent the informationcorrectly and to scale.

In the following short extract lifting or falling is not at issue but the group is hovering as groundworkis being laid for later experiences. As they continue to work on the graph of average weight against age,

Luke and Mark bring in some ideas for some alternative representations. The mention of pie charts islikely to be based on previous experience.

Luke: Wecoulddoapiegraph.[Previous experience.Continuestoworkonchart,withMarkoverseeinghis work.]

Sam: Should we put . . . look this ones 50 so its obviously below . . . . [Sam continues to work out

averages on his calculator. The others continue to add the weights to the chart: organizational

collaboration.]

Luke: 61?! Its gone down. [He seems to have noticed an increasing trend of weight against age, and

this decrease surprises him. He has some sense ofthe big picture.]

Mark: How can we work out the graph with blocks?

Sam: How about the average fast food?Mark: Yeah, we could do that . . . .

Luke: Or pie charts? [Tenacity.]

Mark: Or we could use the blocks . . . . [More tenacity.]

Marks tenacity for using the blocks may provide an impetus for their use when the boys later try to deal

with the issue of fat (Vignette 6), but the evidence is not clear. The tenacity for the idea of pie graphsreappears 2 days later, although it is Mark and Sam who propose it then and Luke who suggests that theydo not use it. At other points, Luke also suggests they use a logic graph and a tree graph, again using

his previous experience. Both times these ideas lapse.


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Prior to the comments elicited below the researcher has shown the boys a series of graphs such as

a pie chart, bar graph, and scatter graph, intended to illustrate different kinds of data representation.The researcher then gives them the chance to consider the data again using some of the new ideas. The

extract contains three illustrations ofpicking the easiest ideas. First, there is the choice to graph eye color,which only had three categories; second, there is the rather flippant remark by Sam about avoiding graphsaltogether; and, third, there is the choice of a bar graph as easier to construct than a pie graph. This last

suggestion is likely based on previous experience. These and the other seven factors noted throughoutthe extract produce a rapid alternation of lifting and hovering.

Luke: Maybe if we had this sort of graph with ages [along the top] to kilograms [down].

Mark: I reckon what we could do is with the sport they do we could like . . . we could have a total of

how many people play football . . . .

Sam: But what does that mean? . . . . [Doubt.]

Mark: Ball games. . .

whats most popular. . .

. [Cognitive ability.]Sam: Yeah . . . suppose so . . . we could do what the most popular eye color is.

Ss: Oh, yeah! [Here they pick the easiest idea, although perhaps only because they have not con-

sidered eye color yet.]

Luke: We havent really thought about eye color.

Sam: Do you have to make the most . . . .

Mark: Yeah, do eye color . . . .

Sam: . . . like the question you asked like what was the most . . . [to researcher].

Luke: How we can tell them . . . [. . . ]

Sam: Why cant we just show them the cards? [Doubt and tenacity from earlier when he made thesame suggestion (see Vignette 5), and perhaps a desire to pick the easiest idea.]

R: Well if you can imagine you were producing a newspaper or what have you, you dont print allthe information thats on . . . .

Sam: Its got to be in a graph . . . .

R: Yes, an efficient way of showing out what youve found out. . . it summarizes the information.[Conversation continues about graphing. Note the outsiders influence. Other social factors mayalso affect what happens, such as a desire to satisfy the researcher.]

Sam: OK, Mark, do you want to do their favorite activity over all? Luke and I will do eye coloring overall . . . andthenwellsetitoutinagraph.[Leadership factors and organizational collaboration.]

[. . . ]

Sam: So Ill call out the eye color first. [Reads out list of students and the others write the details.]

Mark: What kind. . .

pie? [Previous experience.]Sam: Pies pretty good.

Luke: Three sorts of colors, so its a bit hard . . . .

Sam: [Looking at Marks lists of favorite activities.] These seem pretty close . . . the favorite seems

to be . . . .

Mark: TV!

Luke: The one you can sort of read very well is you know . . . bar graph because its easy to read. Inpie graph you have to guess your percentage and we dont really have the equipment to do itproperly. [Picking the easiest ideas based on previous experience.]

Sam: So we do a bar graph. Do we have squared paper?


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The most obvious aspect of their collaboration illustrating the phenomenon of picking the easiest ideasis their choice of doing a graph of eye color, which reduces the task to a univariate problem that requiresonly simple tallying. The group has, of course, already considered much more complicated situations

and, in fact, it appears that eye color and favorite activities are really only considered because they haverun out of other ideas. At the end of the extract above the boys reject the possibility of doing a pie graph,

because it is too hard for them given their resources. This might be regarded as a task factor. In a sensethe resulting choice of a bar graph is picking the easier option, too, although they present cogent reasonsfor that choice.

The boys ability to collaborate without social conflict in correcting errors is shown on several occasions.The following occurs when they are plotting averages for each age group, with Sam and Mark payingattention to what Luke is graphing.

Luke: Right, Ill just mark it with this and then color it in different colors. So if the average for 8 is

27. Average for 9, 29.5. [Luke enters this into chart.]Mark: Wrong one . . . thats about the 27 . . . [Cognitive disagreement.]

Luke: Oh yes, I see now. [Adjusts the graph with Marks help.]

Sam: [Working on averages] See it goes up . . . [The big picture and cognitive ability: Sam seems tohave expected the graph to show this.]

The following extract provides another good example of collaborative behavior, as the boys discuss theway that they are going to present their graphs. Sam appears to liftLuke to choose a good scale, although

it should be recognized that perhaps Luke has perfectly good ideas about the scale anyway this is, infact, the crux of the misunderstanding . It is not clear, however, whether Luke is content with the decisionbecause he genuinely believes that Sam has made a good choice of scale, or because he is just happy to

let Sam decide. Nevertheless, Luke can see the reason behind Marks proposed choice of colors for thebars in the eye color bar chart.

Sam: [Takespaperandbegins.]OK,thisisforeyecolor . . . Whatwasthe favorite eye color? . . . Brown. . . next one, green and . . . What do we make zero? . . . . So should it go up one or should it go uphalf sort of. . . cos there are only five six seven . . . ? [Organizational collaboration necessitated

by the task factorof the size of the grid on the supplied graph paper.]

Luke: Yup . . . I think it should go up 2s . . . . [Has Luke misunderstoodwhat Sam meant?]

Sam: Then youre going to have a tiny graph though. [Perhaps, in fact, Sam misunderstands Luke: itis possible that Luke meant that each unit should be represented by two squares on the graph

paper. Sam makes a decision and there is no further disagreement.] Go up ones. . .

whats thehighest number we had? . . . . Seven. What was your second highest number, Mark? [Sensiblegraph-making decisions rely on this sense of the big picture.]

Mark: They all tied, four of them. [Mark is working on favorite activities.]

Sam: What were the lowest . . . swimming and computers?

Mark: One . . . Yup. Shall we color it in different colors? You can do brown, green and blue. [Color

choice based on a sense ofthe big picture.]

Luke: Im going to do brown, red and green . . . .

Mark: No, blue, because theyre blue [eye colors]. [Cognitive disagreement.]

Luke: Oh, yeah! [Comprehension dawns and Luke is lifted.]


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Similar issues of computation and graphing were considered later in the sessions and the boys show

successful cooperation to ensure the graph and calculations are done correctly. During the next exchangeSam and Mark watch Luke to make sure he does the right thing the previous experience of Sam and

Mark with graphing and computational mathematics seem to be stronger, so they have more completecognition of graphing processes. Mark exhibits tenacity with his name idea, although at its second mentionhe is not urging its use it seems that he just wants the group to acknowledge his observation. They

are working on a line graph that will show average weight (y axis) against age; this is initially the task ofSam and Luke, while Mark works on a graph of average of fast foods against age. For most of the timethe impression is that oflifting.

Luke: Oh, so were doing a line graph and joining up the dots? [Sam goes on writing the ages acrossthe bottom of the graph up to 18.] Well have to get the information . . . oh, youve got it.

Sam: [Working up the vertical axis, which is for weights, counting] 14, 15, 16 . . . . Oh no! Well have

to go up to 74! [Task factors as the graph paper again influences what they do.]

Luke: Thats a bit . . . you should have done it in . . . .Sam: Shall we do it in 2s? [Organizational collaboration, perhaps based on previous experience.]

Luke: 5s.

Mark: I reckon 5s.

Luke: Definitely 5s.

Sam: 5s it is. [. . . ]

Mark: Their names dont mean anything do they? [Previous experience.]

Sam: Average of names! [Chuckles.] No. Can you just get the next one Luke? [Organizational col-

laboration. Works out average on calculator.] 29.5. Next one is 29. [Luke enters it on the graphincorrectly and Sam intervenes (Cognitive disagreement) and corrects it.] [. . . ]

Luke: Maybe I should do the adding. [Implying that Sam should do the entering on the graph. Theychange jobs. Again, organizational collaboration is evident as appropriate task delegation leads

to better outcomes, because tasks have been delegated according to perceived strengths andweaknesses.] What about tens?

Sam: Theres only one . . . and its 29. [. . . ]

Mark: OK [Mark has finished his graph of average fast food consumption against age, and showsit to the others.] Write what mine is. Average for fast foods. [Writes in the title, a graphicalconvention based on previous experience. Meanwhile Luke is calculating.] [. . . ]

Luke: What! [Surprised at the result of his calculation.] Thats 19 kg. [Doubtarising from a sense of

the big picture.]

Sam: Yeah . . . dont think so Luke.

Luke: Ive made a boo-boo.Sam: Did you do how many two? You sure you havent done how many four? Have you cleared

it? [This time Sams cognitive ability and leadership liftLuke with the average calculations.]

Luke: [Luke recalculates.] 38. I must have done how many fours and got confused. Now thats 55and thats the end.

Sam: Theyre getting fatter! [Shows his sense ofthe big picture.] [. . . ]

Mark: We cant really do anything with the age and the name! [Note Marks tenacity with names.]

The previous experience the boys have had with graphing stands them in good stead as they appreciate

the need for scale, labeling, and giving their graph a title. From this point the boys are tidying up and


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checking their work. As they do so, they initially think that there are some cards that they have overlooked.

They start to redo their calculations until a comparison with their previous analysis helps them to realizethat the results have already been included. This again illustrates collaborative lifting with all three boys

contributing as they ascertain that the details ofthe big picture are complete.

3.4. Vignette 4: evidence [cognitive ability, cognitive disagreement, doubt, tenacity of ideas, the big

picture, other social factors, task factors, outsider]

Having completed the initial graph of average weight against age, the boys are undecided about what todo next and there is some indication ofcognitive disagreement. The following extract is the first instance

where the idea of proof is mentioned. It appears that although Sam does not possess the vocabulary todescribe it, he is describing the dilemma of drawing conclusions from the data. His comment later about

a single data card and a whole would seem to support this contention. Overall, the impression is one

oflifting.

Luke: Shall we make a whole lot of graphs that show fast food, weight and umm . . . ?

Sam: What does it prove though? Its only facts. [Doubt.]

Luke: Yeah, thats what were meant to do, I think. . . . Whats the main question? [His question is

directed at the researcher who has moved to sit away from the group. Her position away fromthem may be a social factor that allows them to interact with greater ease. The question helpsthe group clarify the objectives of the task; that the task requires them to do this is a task factor.]

The group interaction and questioning, and even the doubtthat Sam voices about what, exactly, is required,helps them to keep the big picture in sight. The placement of the researcher possibly has an influence on

the extent of collaboration.Later Sam suggests an association of fast food and weight.

R: OK, so it would be interesting if you put forward that thats what you think it is, and have a look

at the information and. . . .

Sam: We have to prove it. [Tenacity of ideas.]

R: Could you show it in a graph or a table? [Outsider: the researchers comment almost certainlyinfluences what they do subsequently.] [. . . ]

Sam: Somehow we have to prove our point, cos we all think that these people who eat a lot of fastfood are fat and these people arent because they dont. [Tenacity of ideas for proof, togetherwith cognitive ability and a sense ofthe big picture.]

Sams observation here is very astute. They all seem to believe there is a relationship, but Sam is awareofthe big picture and the need to prove our point. This is further evidence of Sams cognitive ability.

3.5. Vignette 5: part/whole relationship [cognitive ability, cognitive disagreement, doubt, tenacity of

ideas, the big picture, other social factors, task factors]

On a number of occasions the boys show understanding of the part to whole proportionality ideasassociated with indicating relative frequencies. When recording favorite activities, for example, Luke

comments that we should show how many this is out of. . . 19. Late in the interview the boys look


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at a pie graph shown by the researcher utilizing data from the original 16 data cards. They endeavor to

imagine what the pie graph of their own data would have looked like if they had done it.

Luke: [Reading the values on the pie graph.] 38%, 31. . .

31. . .

and theyre meant to be the same asours but, no, brown is . . . [he notices that the supplied graph is different from what they would

have had] . . . oh, yeah, but we put ourselves on . . . . [The big picture.]

Ss: Yep.

Mark: So what would be one persons percentage? [. . . ]

Luke: But I dont get . . . blue and green tied, but if we didnt have ours on, blue and green theyd alltie wouldnt they?

Mark: Hey, but what is one persons percentage? How do you work that one out? [ Tenacity of ideas.]

Sam: What?! [Doubt, mingled with laughter as Sam and Luke look at Mark in amazement.]

Luke: You mean how many people fit into 31%?

Mark: Yes.

Luke: Ah! Well that would be 5 according to that graph. So that would be 5, that would be 5 and thatwould be . . . .

Sam: 6, and with us added on that would win.

Luke: Yes, because there are 16 people in it.

Sam: Plus us . . . oh, were not in the group. [. . . ]

Mark: This all adds up to 100%! [The others agree in a friendly Yes, Mark, we understand that! tone:

Cognitive ability from all of them, together with other social factors based on their friendship.]

They then attempt to grapple with how to modify the researchers pie graph to reflect their own data, butkeeping track of both the changes in frequencies andthe two different totals involved proves too difficultfor them. This task factor, along with the other factors, ultimately contributes to a hovering outcome.

On another occasion the boys are discussing how to represent the information on the cards.

R: What could you show somebody about that group of people?

Luke: [Insistently] So if we did the charts it would be showing them . . . .

Mark: The weights.

Luke: It would be giving them the information.

Sam: But wouldnt this be giving them the information [holding up a single card; this causes apause] . . . or do you want a whole . . . ? [Expresses some cognitive disagreement.]

This is the idea that Sam presents again, perhaps tenaciously, when later involved in graphing (reported

in Vignette 3: Graphing).

3.6. Vignette 6: mathematical tools [previous experience, cognitive disagreement, doubt, tenacity of

ideas, the big picture, picking the easiest ideas, organizational collaboration, leadership factors,

egocentrism, task factors, outsider]

The longest and most complex interaction in which the boys engage deals with the issue of fat, which

is the name they use for their perception that some of the people described on the cards are overweight.As noted earlier, they are aware that there are possible connections among fast food consumption, weight,and favorite activity. Not surprisingly, however, they find it difficult to extract the various relationships

from the data. This very task factor is what makes the data cards protocol an open-ended task that is


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appropriate for evaluating higher order functioning. In exploring the relationships there are points where

it is clear that their previous experience in mathematics lets them down. This is particularly evident withrespect to their concept of percent. Various collaborative phenomena are revealed as they struggle to

resolve these issues. Although in the short term there are instances of lifting and hovering, overall theimpression cognitively is one offalling. This extract provides an example of a case where not picking aneasy idea has led to falling: the group is unable to satisfactorily resolve all the complex relationships.

After some discussion of association and the need for evidence to support their claims, the boys decideto work with the small colored blocks provided, which lock together easily to form towers. Luke uses amathematical term that sounds good to him but that is not entirely appropriate: like say . . . we work out

the percentage of how much fat and the percentage of how much strength and we work it out in blocksso its like a percentage. After some confusion over whether they are representing information for the

groups of different ages or for individuals, they settle on individual data to obtain a recognizable output.

Sam: Weve done weight and age so we might as well do weight and food. [Organizational collabora-

tion and leadership factors again.] What well do . . . just say were doing Mark, then well have40 thats what he weighs [makes a tower of eight red blocks; each red block represents 5 kg]

and two fat pink [places two pink blocks, which represent Marks two fast food meals per week,beside the tower of red ones]. About 2% of his 40 is fast food.

This time it is Sam who tenaciously wants to keep using the blocks, and he also wants to maintain the useof different colored blocks for various aspects, in particular, the use of pink blocks for fast food, which

they regard as being indicative of fat. Here the absolute number 2, representing the actual number offast food meals is (a) being compared with blocks which represent 5 kg of weight and (b) used by Sam asa value for percent of fast food fat in the weight (at least, this seems to be how Sam sees it). Mark, as seen

in the excerpt below, at least makes an attempt to reconcile problem (a), but does so erroneously; however,he convinces them, probably because there is so much to keep track of that they are all overwhelmed. The

group does not identify problem (b), suggesting that the understanding of percent that they are bringingto this problem as previous experience is not fully formed.

Mark: No, thats 10 [Each block represents 5 of something. Cognitive disagreement.]

Sam: Sorry, 10 out of his 40 must be fast food or things like that.

Mark: 30 natural fat!

Sam: Dont forget there are other fats. [Tenacity as he had acknowledged this earlier when talkingabout 8-year-olds. It is based on previous experience and shows his understanding of the bigpicture.]

The boys produce more columns, consistently using a base five system to represent weight with differentcolored blocks (seven red and one yellow is 36 kg) and using a pink block for each fast food meal(multiplied by 5 and treated as a percentage). Hence Sam, who is 36 kg and has three fast food meals per

week, is deemed to be 15% fat.In the next exchanges doubtis expressed several times. Luke expresses doubt about Sams claim, which

indicates that within this group even the person who may be considered the charismatic intellectual can bequestioned. It may also be that the boys feel such a sense of mutual equality that they are not necessarilyaware of Sams role. Although this expression of doubt has the potential to be good for the groups

outcomes, none of the three can suggest a successful change of course.


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Luke: Its amazing how small . . . it doesnt seem right. [Doubt.]

Mark: But what if the person had like that guy with 12 [fast food meals]? That would be . . . .

Sam: His fat would be nearly all of his body!

Mark: Well do that 18-year-old guy first. [They are referring to an 18-year-old in the data cards whoweighs 74 kg and eats 12 fast food meals per week.]

Luke: [Referring to the towers showing Sams data with eight blocks for weight and three blocksfor fat.] If you say thats 100%, thats a third of his body. [He points to the small pinkcolumn with three blocks against the last red/yellow column with eight. Note that this is now a

different interpretation of comparing the columns, and is correct at least as far as a proportionfor comparing column heights is concerned.]

Mark: There he is [the 18-year-old]. Seventy-four weight and each of them is 5. [He gets 16 reds/4

yellows which they will realize is incorrect shortly and 12 pinks and compares thecolumns.]

Sam: Oh boy!Mark: 60%.

Sam: I dont think its 60%. [Cognitive disagreement.]

Mark: [Counts, 5 per pink block] 60%.

Sam: The most takeaway [fast food] you can haveis 21 [per week].So out of his whole body . . . 60% . . . .

Luke: 60% of his weight is fast food!

Sam: And the rest of its like . . . some of this could be other food and the rest of its like body weight.[Previous experience.]

Luke: Struth!

Mark: Hey, how can it be that because that would be like 20 . . . isnt it 74? Weve got 84. [Doubtbasedon a sense ofthe big picture. Their basic mathematics skills are okay.]

Sam: [Removes two red.] So now its even worse! [The columns are more alike in height, and yetnone of them acknowledge the idea of percent as a proportion: as far as they are concerned thepercentage fat is still 60% because the fast food column is 12 blocks high.]

Luke: Lets just call him Mr Fat! [Laughter.]

In the latter half of the above excerpt there is additional evidence that the boys understanding of percent

brought from previous experience is incomplete. At one point Luke gives a reasonable interpretation ofproportion, when comparing Sams own data of fast food blocks against weight blocks (ignoring the fact

that there is an additional complication because of the different values of blocks). Later, however, whenthey are making adjustments to the columns representing the heavy 18-year-old, Sam acknowledges that

the proportion of fast food to weight is even worse after the correction. There is, however, no recognitionof any implications for their interpretation of percent: their rule for obtaining a number for percentageappears to have nothing to do with proportion comparing fast food to weight.

The diversions and dead-ends pursued by the boys in this extract are similar to what occurs for practicing

statisticians when they are problem solving; it is justthatstatisticians are debatingabout more sophisticatedtools. Given the stage of development of statistical ideas in these boys, their attempts to deal with

comparisons of variables using percent is not totally unreasonable if they had the appropriate data. Itmight be considered equivalent to statisticians trying various sophisticated statistical techniques to seewhich one better tells the story of their data. The problem for the boys is their incomplete understanding

of the technique they wish to employ. This extract illustrates the major incidence of falling experienced


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by the boys; in fact they do not resolve it in a statistically satisfactory way but move on with the assistance

of the researcher.The other mathematical tool that they suggest using during their sessions is the total. This arises

several times as Mark suggests considering the data per year, instead of per week, hence multiplyingby 52. Towards the end of the interview Mark has a turn at being tenacious, and, as the following excerptshows, nearly brings Luke into sharing his obsession. The boys have been looking at and discussing

another groups graph of fast food and weight. Almost out of the blue, Mark reintroduces his idea ofconsidering total fast food consumption for a year (it had, in fact, occurred to him an hour earlier). Againthe outcomes for the group are hovering.

Mark: But what would be 52 times 12? [Tenacity of ideas.]

Sam: What?!

Luke: [Simultaneously, with incredulity] What?!

Sam: What are you talking about now?

Mark: 520 plus . . . .

Sam: What are you doing? . . . Ah! Calculator! [They work it out] . . . Why are you working this out,Mark? [Doubt.] 624!

Mark: 624 fast foods he had per year.

Ss: Oh really! Who wants to know that!

Mark: Yeah, well . . . .

Luke: It might be good actually . . . . [Luke nearly falls, influenced by Marks suggestion.]

Mark: But if we added them up for a year we could show how much fat that would put on. [ Tenacityfor fat again.]

The idea of fat that influenced so much discussion in the first session is also still being held by Mark.They return to the task and consider another two graphs, before the researcher the outsider givesMark another chance to bring up his idea about annual results. Here his tenacity is verging on egocentrism:

he may well be insisting on the idea just for effect.

R: If we had time is there anything else youd like to graph?

Mark: How much they would eat in the year, thats what Id like to do.

Luke: You could take that male and female thing a bit more.

Mark: Oh yeah. Male and female per year.

Luke: Males weight and females weight.

Mark: You could do that for a year. [Tenaciously determined to get it in!]

It is interesting that these boys do not consider the possibility of a gender split earlier, as was done byother groups studied (e.g., Watson & Callingham, 1997; Watson & Chick, 2001a).

3.7. Vignette 7: limitations [cognitive ability, previous experience, cognitive disagreement, doubt,

tenacity of ideas, the big picture, task factor]

The issue of using blocks dominates the groups activity at the time of this extract. What to representwith them, however, is difficult to decide. Luke is the one most tenaciously focused on the fat idea and

the use of blocks. As can be seen in what follows, Sam and Mark have some reservations that recognize


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limitations in the data, but the idea seems to linger and their obsession with it pulls their performance

down.

Luke: Well why dont we. . .

like. . .

we could use red colors to show that theyre strong and say wethink half of its strong and half of its fat. Pink could represent fat. [Grabs some pink and redblocks: tenacity again.]

Sam: But anybody could be strong. [Doubtfor the idea, revealing further cognitive ability.]

Luke: Yeah, but what we think. . . .

Sam: Cos if they watch TV they could still be strong. [Previous experience.]

Luke: Yeah, I know . . . but if someone liked football but eats a lot of fast food that could be the fat and

the strength in the weight. [Has two pink cubes with three red cubes on top in a column.]

Sam: Or this could be the good in the food and that could be the bad in the food [points to each of thetwo colors in turn].

Luke: Buthowdoweknowwhatotherfoodstheyeat?[Cognitive disagreement, in which the limitationsof the data are acknowledged.] [. . . ]

Luke is keen to do something, perhaps because considering fat was his idea, and although Sam stillhas some doubtbecause he knows fat comes from sources other than fast food, he proposes an approach

that will try to take it into account. His approach represents individuals whereas Luke is still thinking ofage groups. Mark cannot follow them and asks for clarification that Luke attempts to provide.

Luke: Well, say . . . this is all the fat, this is all the strength, we work out how much fat there is in the11-year-olds, like say . . . we work out the percentage of how much fat and the percentage ofhow much strength and we work it out in blocks so its like a percentage. [More tenacity for the

fat idea; the introduction of percentage is probably based on previous experience.]

Luke uses a mathematical term here that sounds good to him but that is not entirely appropriate. It soundsconvincing enough to Mark, however, although Sam still has reservations about the whole plan.

Mark: That sounds pretty good.

Sam: You cant do strength cos youcant work out how much strength they have. [Doubtand cognitiveability: the limitations of the data are again recognized.]

Luke: This is like what we think, and what were showing . . . we dont know how much fat theyvegot either . . . so, you cant really say . . . .

Sam: Why not instead of calling that fat [points to the pink blocks] we call it fast food because we know

how much fast food they have? [Cognitive disagreementas a reflection ofcognitive ability.]Sam highlights that there is a problem with this vague fat concept, which is similar to the earlierreservation of Luke about what foods people eat. Sam clearly indicates that he is aware of the need to usethe data that they have, an important early statistical recognition.

Two days later when the session resumes the boys continue to wrestle with the problem. Interestingly,it is Luke who now acknowledges the difficulty of relating fast food consumption directly to weight.

Luke: But havent you got to know the weight you get from the fast food to umm . . . . [Both Luke andSam had mentioned this problem 2 days earlier. Tenacity and the big picture.]

Sam: Yeah, we need it . . . .


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Luke: Because thats just saying that one fast food equals one kilogram and thats not true is it?[Cognitive disagreement. The units have changed since the previous session: one fast food is nolonger turned into five but is kept as one.]

Sam: No, but thats the percent of your body that has fast food in it per weight.Luke: We dont really know that do we? [Tenacity.] [. . . ]

Sam: But we still wouldnt know what the weight would be.

Finally, at one point the researcher asks the following question.

R: What would someone think if they saw that?

Luke: Whats the average weight for that age.

Sam: Out of the people weve got . . . . [Sam acknowledges the fact that the data form a small sample.Although not included here, he expresses the idea on other occasions (see Watson & Chick,

2001a, Group 6): tenacity.]

Sam is aware of the data set as but a small, perhaps non-representative, sample of a larger population. It

is not clear, however, that this idea was not already known to Sam, nor does it appear to lift the others;hence it results in an indeterminate hovering outcome at the end.

3.8. Vignette 8: beliefs [other social factors]

At the end of the second session the researcher asked the students about how they felt about working in

a group. Their responses are interesting, and indicate that students are capable of determining the benefits,effectiveness, and constraints of working as a collaborative team.

R: One other question I wanted to ask: do you prefer to work on your own or do you prefer to work

in a group?

Ss: Group! [unanimous]

Mark: Cos you get it done quicker . . . like Im doing the coloring in and hes doing the calculator andhes drawing up another graph [pointing to the others in turn].

Sam: As long as youve got people . . . .

Mark: . . . who co-operate . . . .

Sam: Yup . . . because if youve got people who just sit around . . . .

Mark: . . . watching . . . .

Sam:. . .

or if someone tells you to do it all the time.R: So its quicker, and as long as you get on with the other people . . . . Is there any other reason

you like group work?

Sam: You get shared ideas and you get to listen to other peoples ideas and you probably come up

with a solution which would be better . . . or better than what you were thinking . . . not all thetime but some of the time you would . . . .

R: Is that what you were thinking?

Ss: Yup!

R: Youll agree with him because youre co-operating!

Ss: Yeah. [Laughs all around!]


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It appears that these boys, due to their natural friendship (a social factor) and probably due to previouswork together in the classroom, had acquired collaborative skills and recognized their benefits, whichmay have been a significant influence on the overall lifting experiences of the group.

4. Discussion and conclusions

The implications of the observations from this case study will be discussed from several perspectives,including what is learned about collaboration and its potential effect on cognitive outcomes especiallyin relation to emergent statistical concepts, the implications of these observations for educators, and

suggestions for future research in this area.

4.1. Collaboration and its effect on cognitive outcomes

There is ample evidence from the Vignettes to demonstrate that lifting that is, an improvementin the level of cognitive functioning can occur for individuals within the group and for the group

as a whole as a result of collaboration. In contrast, it is also apparent that the level of cognitive func-tioning may fall or hover. Whether it is lifting, falling, or hovering that occurs depends on a numberof factors that influence the nature of the collaboration and what is achieved through that collabora-

tion. Thus, for example, there are situations in which a students tenaciousness in bringing up a par-ticular idea can be beneficial, as in Sams persistence with the idea of doing a graph. Alternatively, it

can be detrimental, as demonstrated by Luke who clings to the idea of trying to consider fat, evenafter he acknowledges that they do not have the data to do so. Resolving conflicts can also producevaried outcomes, depending on the correctness of the arguments and the force with which they are

presented.In terms of mathematical ability and the cognitive factors identified in Table 1, the boys in this group

appear able and often attempt to bring previously acquired skills to this new context. Although for the

concept of average they are quite successful, for the concept of percentage they are less successfuland the outcomes show the potential for falling when there is not a member of the group with the previousexperience to prevent it. Their communication skills, however, are such that they successfully teach each

other techniques as required, as when Luke teaches Sam how to calculate average. They also often checkeach others work, even when they are struggling with percent. It should be noted that in all the examples

of lifting illustrated here, the group lifts because one of the group members knew an appropriate strategythat could be explained to the group.

An interesting special case of the factor of previous experience prevalent in this group is the use ofjargon, where some individuals use terms that sound impressively technical. This may occur for a numberof reasons: (a) the individual knows what the term means and that it will be useful; (b) the individualknows what the term means, but is proposing it as a straw man, unsure if the idea is worthwhile; (c)

the individual is unsure of the terms meaning but thinks it may be relevant to the situation; or (d) theindividual is unsure of the terms meaning, and it is said in order to impress or to create an illusion

that the speaker is knowledgeable about such things. In this study, Luke uses a number of terms that aremathematical but seem to be introduced in part because they sound good. Although he seems to hopethat they will be helpful, it is not immediately clear that he always knows how appropriate they are, so

he probably fits into the third category above.


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Turning to social or interpersonal factors, this case study took place with individuals who were prepared

to be cooperative with the researcher and her goals, as well as with each other. The camaraderie exhibitedby the boys in the group suggests they were close friends. They shared ideas effectively, tasks were

assigned and undertaken readily, leadership roles were shared naturally, difficult issues were addressedwithout social disagreement, and ideas were treated with respect, even those regarded as incorrect. It ishypothesized that the positive relationship between the boys and their willingness to engage in the task

are conducive to positive outcomes; it seems reasonable to expect that poor rapport and/or a negativeattitude to the task would reduce the likelihood of improved cognition. The observations of the studywere thus made in the most favorable circumstances for collaboration, and hence the presence of both

helpful and hindering collaboration is significant. It is difficult to imagine a scenario involving an extendedopen-ended task where both outcomes of collaboration will not occur.

Finally, two of the external factors task factors and an outsider influenced the collaborativeprocess. Although the researcher had only 6% of the turns at talking in the extracts shown (and less than

11% overall), there were occasions when her comments appeared to cause significant changes in approach.In particular, she encouraged the boys to consider fewer variables and sanctioned the use of graphs. Inboth these cases her recommendations lifted the group, and enabled the boys to achieve recognizableoutcomes more efficiently than they would have managed on their own.

Although no comparison was carried out with other types of open-ended mathematical tasks, it appearsthat this task provided a motivational context in which these boys could collaborate successfully. That itwas slightly above their current level of experience and expertise provided challenges that were sometimes

met (leading to lifting) and sometimes not (leading to hovering and falling).In generalizing the findings of this study, the importance of four groups of factors are highlighted in

Fig. 2. These were identified individually or in clusters in Table 1. First there are the two sets of factors thatare dependent on the members of the collaborative group. These individuals bring a variety of cognitive

and social attributes to the collaborative event, which vary from individual to individual and hence from

Fig. 2. Key factors that influence cognitive outcomes during a collaborative event.


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group to group. Some of these factors may be more influential than others cognitive ability is an

obvious example but there is no guarantee that these factors will come into play in a predictable way.Secondly, there seem to be two particularly important factors that are outside the control of the group: the

task itself and the influence of an outsider. In most school-based events, external influences govern thechoice of task, but any task has attributes that may make it more or less conducive to use in a collaborativesituation. In the present case, the data cards task was sufficiently open-ended to stimulate productive

statistical collaboration. The other factor the outsider, who was the interviewer in this study but who ismore usually the teacher in classroom situations plays a significant role in regulating the course of thecollaboration. In the present study, if the researcher had not intervened, particularlyin the fat episode (see

Vignette 6), the outcomes would have been less satisfactory. This highlights the importance of appropriateintervention by teachers, not necessarily to instruct but to steer the group towards fruitful avenues.

What is striking in this study is that it seems to be impossible to predict the outcomes from the interplayof these factors. The point made by Good et al. (1992) about the difficulty of predicting collaborative

outcomes is borne out clearly: lifting, hovering, and falling may all occur as a result of collaboration,and almost any of the factors may lead to any of these cognitive outcomes (see Table 3). The frameworkprovided in Fig. 2, however, may provide a useful starting point for future research.

4.2. Mathematical and statistical cognition

As far as statistical cognition is concerned, it is appropriate at this point to summarize this groupsoverall performance in the context set by the data cards protocol. The use of this task supports the views

ofPhelps and Damon (1989) about the benefits of using collaborative activities requiring higher orderreasoning skills. From the starting point of their previous experience these three boys achieved high levelcognitive outcomes for students in Grade 6 (cf. the study ofWatson et al., 1995). This may not be surprising

in the light of their ease of collaboration, apparent ability levels, and the time devoted to the task.In particular, as highlighted by Table 2 and the Vignettes, there were a number of statistical concepts of

which the boys were cognizant even if they had limited strategies for dealing with them. The concept of

average appeared well-understoodthroughout the extracts. Although the boys strategy of computing aver-ageweightsforagewouldnothavesatisfiedastatisticianduetothesmallnumberineachagegroup,itover-came the difficulty of repeated ages encounteredby some groups in the study ofWatson and Chick (2001a).

One Grade 9 group in that study, for example, opted for an awkward doubling up scheme when plottingbars for two weights associated with one age. The ideas of causeeffect relationships and correlation were

evident, with the boys previous experiences suggesting some of the relationships among variables. Theyalso had reasonable expectations for what some of the two-variable relationships would look like when

depicted graphically (e.g., weight against age). The limitations of the data, both because of the variablesinvolved and the size of the set as a sample, were recognized by the boys. Their graphing skills were, ingeneral, sufficient for their needs, and graphing conventions were employed without any prompting.

That the techniques used and outcomes achieved may not be considered the best by practicing

statisticians does not detract from the boys higher order reasoning. These are emerging skills, not yetrefined ones.

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Chik y Watson_Estudio de Caso