Children’s representation and organisation of data

Mathematics Education Research Joinhal 2003, Vol. 15, No. 1, 42 58

Children's Representation and Organisation of Data Steven Nisbet

Griffith University

Graham Jones, Carol Thornton, Cynthia Langrall and Edward Mooney Illinois State University

This study investigated how children organised and represented data and also examined relationships between their organisation and representation of data. Two protocols, one involving categorical data and the other involving numerical data, were used to intmwiew 15 students, 3 from each of Grades 1 through 5. Although there were differences between Grade 1 students and the rest, the study suggested that numerical data was significantly harder for children to organise and represent than categorical data. Children beyond Grade 1 could make connections between organising and representing data for categorical data but their connections for numerical data were more tenuous. The process of reorganising numerical data into frequencies was not intuitive for the children in this study but they showed greater readiness in recognising and interpreting data that had already been reorganised as a frequency representation. Given this latter result, a pedagogical approach that asks students to make links between raw data and a frequency representation of it may prepare students to create and construct their own frequency representations.

Recent reforms in m a t h e m a t i c s educa t ion ( A u s t r a l i a n Educa t ion Counci l , 1994; N a t i o n a l Council o f Teache r s o f M a t h e m a t i c s , 2000) h a v e ca l l ed for more ex tens ive a nd ea r l i e r in t roduc t ion to d a t a h a n d l i n g in t h e e l e m e n t a r y school. T h i s b roade r p e r s p e c t i v e h a s h i g h l i g h t e d t h e need for f u r t h e r r e sea rch on t h e l ea rn ing and t e a c h i n g of d a t a h a n d l i n g a t t he e l e m e n t a r y school level an a r e a t h a t is under represen ted in t h e m a t h e m a t i c s educa t ion r e sea rch l i t e r a t u r e ( S h a u g h n e s s y , Gar f i e ld , & Greer, 1996). A l t h o u g h t h e r e is ev idence t h a t r e sea rch on s tuden t s ' s t a t i s t i c a l t h i n k i n g and learning is beginning to emerge (Br igh t & Friel , 1998; Curcio, 1987; Friel , Curcio & Br igh t , 2001; Greer, 2000; L e h r e r & Romberg, 1996; Mokros & Russell , 1995; Moritz, 2002; W a t s o n & Moritz, 2000), t h i s r e sea rch h a s l a r g e l y focused on s tudents ' s t a t i s t i c a l t h i n k i n g beyond e a r l y e l e m e n t a r y . In fact , even t h o u g h g r a p h i n g h a s been p a r t o f t he e l e m e n t a r y school curriculum for t h e las t t wo decades , t h e r e is a lmos t no r e sea rch on h o w young c h i l d r e n organise and represent r a w d a t a .

The presen t s tudy , a l b e i t exp lo r a to ry , a t t e m p t e d to address t h i s gap by examin ing h o w ch i ld r en organise d a t a w h e n t h e y are in tile process of represen t ing d a t a . More spec i f i c a l l y , t h i s s t u d y asked t h e fo l lowing questions: (a) H o w is c h i l d r e n ' s s t a t i s t i c a l t h i n k i n g in o rgan i s ing and represen t ing d a t a r e l a t e d to g rade l eve l ? (b) W h a t is t h e effect o f t h e type o f d a t a (ca tegor ica l versus numerical ) on c h i l d r e n ' s s t a t i s t i c a l t h i n k i n g in o rgan i s ing and represen t ing d a t a ? (c) Is c h i l d r e n ' s s t a t i s t i c a l t h i n k i n g in represen t ing d a t a r e l a t e d to t h e i r t h i n k i n g in o rgan i s ing d a t a ? Is t h i s r e l a t i o n s h i p inf luenced by w h e t h e r t h e d a t a is c a t ego r i ca l or numer ica l? (d) Is s tudents ' pe r fo rmance in recognising organ isa t ions o f d a t a r e l a t e d to t h e i r p e r f o r m a n c e in creat ing organ isa t ions of d a t a ?

Children's Representation and Organisation of Data 43

Theoretical Considerations This study is based on a Statist ical Thinking Framework (Framework) (Jones et

al., 2000) tha t characterises children's statist ical thinking according to four cognitive levels and across four key constructs: describing data, organising and reducing data, representing data, and analysing and interpreting data. The present study is concerned wi th two of the constructs (a) representing data, and (b) organising and reducing data. Represen ring data involves the construction of visual representations of data, for example, bar graphs, t a l ly graphs, tables and line plots. Of particular importance in this study were representations t h a t incorporated different organisations or groupings of data. Ovganising arid reducing data incorporates mental actions such as grouping, ordering and summarising da ta (Moore, 1997). We were especially interested in children's organisations of da ta tha t were generated whi ls t they were representing the given data. That is, we were interested in children's preferred organisations of data tha t arose as part of their actions in representing the data. In addition we also examined children's reactions to a representation and organisation of data generated by an imaginary child.

The Jones et al. (2000) Framework provides expectations of the levels and kinds of thinking children engage in when representing and organising sets of data. For representing data, children exhibiting Level 1 thinking (Idiosyncratic) produce idiosyncratic or incomplete displays of the data set. (The authors of the Framework regard both idiosyncratic and incomplete responses as baseline indicators of children's thinking in relation to data representation.) Children at Level 2 (Transitional) produce displays tha t represent tile data but do not a t tempt to reorganise it. Children at Level 3 (Quantitative) produce displays tha t show some attempt to reorganise or regroup the data. Children at Level 4 (Analytical) produce multiple val id displays, some of which reorganise the data. Wi th respect to organising arid reducing data, the Framework reveals tha t children exhibiting Level 1 (Idiosyncratic), t ha t is baseline thinking, give idiosyncratic groupings or do not group or order the data at all. Children at Level 2 (Transitional) give groupings or ordering tha t are not consistent, or group data into classes using criteria tha t they cannot explain. Children at Level 3 (Quantitative) order or group da ta into classes, and can explain the basis for their groupings. Children at Level 4 (Analytical) order or group the data into classes in more than one way, explain the basis for their different groupings or orderings of the data.

Inval ida t ing this framework, Jones et al. (2000) noted limitations in the way tha t they had assessed young children's organisation of data. In particular, t h e y were concerned tha t their protocol question dealing wi th the organisation of the Beanie Baby data had focused the children's thinking in a particular way (by animal categories) rather than allowing the children to create they own organisations of the data. They advocated further research into the cognitive processes used by children when they were confronted wi th tasks tha t provided opportunities for them to represent and organise data in multiple ways. This study is a response to their recommendation.

Two other studies (Lehrer & Schauble, 2000; Nisbet, 1998) also provided a stimulus for this research. Using a task involving categorical data similar to the

44 Nisbet, Jones, Thornton, Langrall & Mooney

transport protocol presented in this study, Nisbet examined the representations generated by 114 teacher education students. These students created 11 different types of representations ranging from lists (ungrouped and grouped) to various types of tables, pictographs, line plots and bar graphs. Their representations also revealed the complete range of thinking levels identified by the Jones et al. (2000) Framework. Lehrer and Schauble worked wi th elementa]Tf school children in Grades 1, 2, 4 and 5. They examined how these children developed and justified models to categorise and differentiate (by grade level) drawings made by children in the same grade levels as themselves. The Grades 1 and 2 children were reluctant to use attributes to classify the drawings by grade level and tended to justify the i r categorisations based on idiosyncratic perceptions of a drawer's grade levels or age. By way of contrast, the Grades 4 and 5 children developed category systems t h a t used dimensional attributes associated wi th the drawing characteristics shown for arms, fingers, necks, feet, and other body parts. The results suggest tha t as grade level increases children demonstrate increasing sophistication in their strategies for organising data.

Method

Participants Students in Grades 1 through 5 from a Midwestern U.S. elementary school were

the population for the study. The school population was representative of a broad spectrum of socioeconomic and cultural backgrounds. The sample for the study comprised 15 children, three being selected from each of the five grades. At each grade level children were purposefully sampled (Miles & Huberman, 1994), based on their previous mathematics achievement one h igh (upper quartile), one middle (middle quartiles), and one low (lower quartile) in order to increase the representativeness of the sample However, the study did not at tempt to make comparisons among children categorised as high, middle or low on mathematics achievement.

Children in these grade levels had undertaken instruction in data exploration. This instruction focussed largely on learning to construct particular kinds of graphs such as bar graphs, circle graphs, and line graphs, using data tha t was provided by the teacher or the textbook and had ah 'eady been organised. However, the children had li t t le experience in dealing wi th raw data, and even less experience in creating their own representations and organisations of data.

Interview Protocols The first author interviewed all children in the sample using two researcher

designed Statist ical Representation Protocols. These were presented in two separate sessions of approximately 20 minutes each, which were separated by a week approximately. Both protocol sessions were audio taped, and the children were provided wi th paper and felt pens to draw their pictures or graphs.

For each child, Protocol 1 was administered in the first session, and Protocol 2 in the second session. We maintained this order so tha t any learning effects would

Children's Representation and ©rganisation of Data 45

be consistent across the sample. Protocol 1 (Figure 1), involved da t a on how a class of 10 students in a rural school t r ave l l ed to school. The da t a was categorical w i t h each student l is ted by name and mode of transport . Each ch i ld in the s tudy read the story and associated d a t a once, fol lowed by a second reading w i t h t he researcher. In the case where a chi ld was not able to read the information, t he researcher read it the first time, and the chi ld and researcher read it a second t ime together . Following the reading of the story and data , the researcher probed t he chi ld w i t h simple questions like, " H o w does Brendan come to school?" to ensure t h a t the ch i ld was f ami l i a r w i t h the story and the data . The researcher then focused on the major task t h a t involved t he chi ld in drawing a picture or g r a p h t h a t showed how the students in the story got to school. After giving the ch i ld time to construct a picture or graph, the researcher asked the follow up questions presented in Figure 1.

Story: Some children were talking about how they came to school. This is what they said.

Alice comes by bus. Brendan comes by car. Cathy rides a bike. Denis comes by bus. Elouise walks to school. Francis comes by bus. Gaff comes by car. Herby walks. Ilsa comes by car.

Jack comes by bus. Task: I'd like you to draw a picture or graph that shows how the children in this class get to school. Follow up questions: Can you tell me what you have drawn here? Why did you draw it that way? If someone came into the room and saw your graph, what would they learn from it? Is there anything else they would learn? What title could you write at the top?

Figure 1. How some children came to school.

Protocol 2 (Figure 2) involved the number of pet f ish belonging to a group of 10 students. The f ish d a t a were numerical w i t h each student l is ted by name and number of fish. The procedures for reading the story and associated da t a were exact ly the same as in Protocol 1. As before, following the reading of the story and data , the researcher probed each ch i ld w i t h simple questions such as, "How many f ish does Bruce have?" Then the researcher focused on the major task t h a t involved the chi ld in drawing a picture or g raph t h a t showed the number of pet f ish these students had . After giving the chi ld time to complete her picture or graph, the researcher asked the follow up questions presented in Figure 2. The first set of questions re la ted to the ch i ld ' s graph. After those questions h a d been asked, the researcher showed a g r aph of the f ish d a t a (Figure 3) d rawn by an i m a g i n a r y chi ld Mary and asked the second set of follow up questions. By


Story: Ten children from the Pet Fish Club met at school one day. They were talking about how many fish they had. Amy had 4 fish. Bruce had 2 fish. Cary had 5 fish. Don had 4 fish. Ember had 7 fish.

Francio had 3 fish. Gary had 0 fish. (They all died.) Hugo had 2 fish. Izra had 9 fish.

Janita had 4 fish. Task: I 'd like you to draw a picture or graph that shows something about the numbers of pet fish these children have. Follow up questions (Set I): Can you tell me what you have drawn here? Why did you draw it that way? If someone came into the room and saw your graph, what would they learn from it? Is there anything else they would learn? What title could you write at the top? Follow up questions (Set 2): Another" person called Mary has drawn this graph. Can you read what it says? What do you think Mary was doing when she drew this graph? What do these crosses mean?

Figure 2. Pe t F i sh C lub .

X X X

X X X X X X X

0 1 2 3 4 5 6 7 8 9

Number of fish

Figure 3. M a w ' s g r a p h .

i nc lud ing MaTT's g r a p h ( w i t h i t s i n h e r e n t o r g a n i s a t i o n o f t h e d a t a ) w e p r o v i d e d an o p p o r t u n i t y for c h i l d r e n to c o m p a r e t h e i r o r g a n i s a t i o n a n d r e p r e s e n t a t i o n o f t h e d a t a w i t h an a l t e r n a t i v e t h a t m a y h a v e been d i f f e r e n t f rom t h e i r s . In t h e case o f c h i l d r e n w h o h a d not a b l e to r e o r g a n i s e t h e r a w d a t a , i t e n a b l e d us to d e t e r m i n e i f t h e y could recognise a n d i n t e r p r e t a r e o r g a n i s a t i o n o f t h e d a t a . In t h e case o f c h i l d r e n w h o w e r e a b l e to r e o r g a n i s e t h e d a t a , i t e n a b l e d us to d e t e r m i n e w h e t h e r t h e y r ecogn i sed c o m m o n a l i t i e s a n d d i f fe rences b e t w e e n t h e i r r e p r e s e n t a t i o n a n d t h a t of M a r y .

Data Sources and Analysis The sources o f d a t a for t h i s s t u d y c o m p r i s e d t r a n s c r i p t s of t h e a u d i o t a p e d

i n t e r v i e w sess ions , a n d s t u d e n t s ' a r t i f a c t s in t h e form of p i c tu re s or g r a p h s .


Adopt ing a double coding procedure (Miles & Huberman, 1994), the first two authors used these sources to prepare summaries of each student 's responses to Protocols 1 and 2. More speci f ica l ly , t h e y coded the chi ldren 's representa t ions according to the following character is t ics : level of th inking on the Framework (Levels 1 to 4), type of representa t ion (bar graph, table, line plot, t a l l y p l o 0 , manner of organising the d a t a (categories, groupings), and use of conventions ( labeling axes, and scale). Following a s imi la r procedure to Jones et al. (2000), t h e two researchers independent ly assessed th inking levels for each of the 15 students on the two constructs and the two protocols, t h a t is, 60 assessments of levels by each researcher . The leve l of agreement between researchers for the independent p h a s e was 90% (54 out of 60 assessments). Following the independent assessments, t h e researchers then c la r i f ied points of difference until consensus was reached .

During the coding and levels ana lys is the two researchers produced " w i t h i n case d isplays" to synthesise the s ta t i s t i ca l th inking of each chi ld on each construct ( representat ion & organisation) and each protocol (categorical versus numerical) . Based on these i nd iv idua l w i th in case d isp lays , the researchers generated a clustered mat r ix (Miles & Huberman, 1994, p. 127) t h a t showed key s t a t i s t i ca l pa t terns for the sample of chi ldren by grade, th inking level , and protocol. These pat terns were used to report and in te rpre t the results of the study.

The Wilcoxon Signed Ranks Test (Siegel & Caste l lan, 1988) was used to tes t differences between the students ' th inking levels when representing categor ical versus numerical da ta . The same test was also used to assess differences between the chi ldren 's th inking levels in re la t ion to the organisaHon of categorical versus numerical da ta . Spearman rank correlat ions coefficients were ca lcula ted to find the degree of consistency between students' levels of th inking when representing categorical and numerical d a t a versus t h e i r corresponding th inking levels for organising these two types of data . A Spearman rank correlat ion was also ca lcula ted to assess the degree of consistency between students ' ab i l i t y to a c tu a l l y organise da t a and t he i r ab i l i t y to recognise ano the r organisat ion of the same da ta . In each case non parametric tests were used because levels of th inking were assumed to be ordinal r a t h e r t han in te rva l or ra t io data . Moreover, distributions of levels of th inking were expected to dev ia t e from normality.

Results The presenta t ion of the results is l inked to the four research questions posed in

the introduction. Even though organising d a t a presumably precedes representing d a t a as a mental action, we h a v e discussed representa t ion prior to organisat ion because the basic source of d a t a was chi ldren 's representat ions, and organisa t iona l th inking was inferred from these representat ions.

Question 1" How is Children's Statistical Thinking in Representing and Organising Data Related to Grade Level?

Representing data. Table 1 gives the chi ldren 's levels of th inking, for t h r ee chi ldren in each grade, for the construct representing data . The levels of th ink ing were based on t h e i r representat ions of both categorical and numerical da ta .

48

Table 1 Children

Nisbet, Jones, Thornton, Langrall & Mooney

's Levels of Thinking by Grade and Type of Data (categorical v numerical)

Grade 1 Grade 2 Grade 3 Grade 4 Grade 5

C a N b C N C N C N C N

1 1 1 2 4 3 3 2 3 2

1 1 3 3 3 2 3 2 2 2

1 1 3 1 1 1 3 2 3 3

Note. n 15, with 3 children at each grade level, aC stands for categorical data; bN stands for numerical data.

Ca tegor ica l d a t a were assoc ia ted w i t h the protocol on ch i ld ren ' s modes of t ranspor t (see Figure 1) w h i l e numerical d a t a were assoc ia ted w i t h the protocol on numbers of f i sh (see Figure 2).

W i t h respect to s t a t i s t i c a l th ink ing in represent ing da ta , Table 1 r evea l s a s h a r p e r difference between Grade 1 th ink ing leve ls and th ink ing leve ls beyond Grade 1 t h a n it does be tween th ink ing leve ls for Grades 2 to 5. A l t h o u g h the re a re differences between students ' s t a t i s t i c a l th ink ing in r e la t ion to ca tegor ica l and numerical da t a , the d icho tomy be tween Grade 1 chi ldren and the rest of t h e chi ldren holds for both types of da ta . Moreover the s i m i l a r i t i e s in leve ls of th ink ing across Grades 2 to 5 are consistent for both types of d a t a .

As shown in Table 1, a l l 3 chi ldren in Grade 1 exh ib i t ed Level 1 th ink ing in represent ing both ca tegor ica l and numer ica l da t a . By w a y of contrast, 11 out of t h e 12 chi ldren beyond Grade 1 exh ib i t ed a t l eas t Level 2 th ink ing in one or more protocols. For ca tegor ica l da t a , 2 (out of 12) ch i ldren exh ib i t ed Level 1 th inking , 1 ch i ld exh ib i t ed Level 2 th inking , 8 chi ldren exh ib i t ed Level 3, and 1 ch i ld exh ib i t ed Level 4 th inking . For numerical da ta , the corresponding numbers were 2 chi ldren a t Level 1, 7 chi ldren a t Level 2, and 3 chi ldren a t Level 3.

Al l 3 Grade 1 chi ldren produced Level 1 d i sp lays , t h a t is, t h e i r representa t ions of the d a t a were incomplete or id iosyncrat ic . For the ca tegor ica l da ta , t h e y could iden t i fy the four t ranspor t categories (bicycle, bus, car, and w a l k ) and were able to represent a l l or some of these in t h e i r drawings. However , t h e i r representa t ions showed no evidence t h a t t h e y made links between chi ldren and t ranspor t categories. K y l i e ' s represen ta t ion was typ ica l . She s i m p l y d rew a bus, a car, and a ch i ld wa lk ing . Jason wen t a l i t t l e fur ther . He a t t e m p t e d to represent a r e l a t i o n s h i p be tween modes of t ranspor t and number of chi ldren. However , h i s a t t e m p t was e s sen t i a l ly id iosyncrat ic , ut i l is ing "his own d a t a set" r a t h e r t h a n t h e given d a t a set. For example , w h e n asked " W h y did you d raw 3 people [in t h e bus]?" Jason responded, "One 's a mum and 2 are kids." There was no l ink w i t h t h e four chi ldren in the ac tua l d a t a w h o came by bus. W h e n dea l ing w i t h numerical da ta , the Grade 1 chi ldren produced l a r g e l y id iosyncra t ic representat ions . T h e y usually e l i m i n a t e d the l ink w i t h the people w h o owned the f i sh and drew pictures of f i sh or groups of f i sh t h a t did not r e l a t e to the given da ta . For example , Bruce d rew 14 f i sh t h a t bore no re la t ion to the context of 10 chi ldren and 40 f i sh .

The students beyond Grade 1 w h o exh ib i t ed Level 1 responses produced


incomplete representations rather than idiosyncratic ones for both categorical and numerical data. Their incomplete representations were similar to the incomplete representations of the Grade 1 students. Only one student beyond Grade 1, Joshua (Grade 5), exhibited Level 2 thinking for categorical data but 7 students exhibited Level 2 thinking for numerical data. Joshua did not reorganise the categorical da ta but maintained the essence of the data by showing the name of each child in the sample together wi th an icon tha t represented the mode of transport used by t h a t child. Joshua's reluctance to create a display tha t reorganised the categorical da ta was not typical of students in Grades 2 to 5. However, this reluctance to reorganise was more typical for numerical data. For example, the 7 children exhibiting Level 2 thinking for numerical data generally produced val id graphs tha t maintained the original organisation of the data (child by number of fish). Four of the children drew pictographs and 3 drew bar graphs.

Of the 8 students beyond Grade 1 who exhibited Level 3 thinking in representing categorical data, 6 drew bar graphs, 1 drew a t a l ly graph and 1 drew a pictograph. They showed a clear link between mode of transport and the number of children who used tha t mode. As such they reorganised the data and accommodated the data reduction tha t resulted from the children's names being omitted. Wi th respect to numerical data, 1 student in each of Grades 2, 3 and 5 exhibited Level 3 thinking. Two of them produced a t a l ly table (number of fish by frequency) whi le the thi rd (Aldo) generated a line pictograph tha t was based on frequencies and used people icons to represent the number of people (frequency). Figure 4 shows Aldo's representation of the data.

Figure 4. Aldo's representation.

Organising data. Table 2 gives the students' levels of thinking, by grade, for organising data. The levels of thinking for organising refer to both categorical and numerical data.

The changes in levels of thinking, by grade, for organisation of data, mirrored those for representation of data. Once again the organisational thinking of the three Grade 1 students was assessed at Level 1 and was noticeably lower than the children in the other grades for categorical data but less so for numerical data. The overall differences among Grades 2 to 5 children were re la t ively small, even though the levels of thinking for numerical data were substantially lower than those for categorical data.


Table 2 Students' Levels of Thinking, by Grade, with Respect to Organisation of Both Categorical and Numerical Data

Grade 1 Grade 2 Grade 3 Grade 4 Grade 5

C a N b C N C N C N C N

1 1 1 2 3 3 3 1 3 1

1 1 3 3 3 1 3 1 1 1

1 1 3 1 1 1 3 1 3 3

Note. n 15, wi th 3 chi ldren at each grade level, aC s t ands for categorical data; bN s t ands for numer ica l data.

The organisations of d a t a by the Grade 1 students ref lec ted Level 1 th ink ing for both categorical and numerical data , and were e i t he r incomplete or idiosyncrat ic . T w e n t y five percent of students beyond Grade 1 also exh ib i t ed Level 1 th inking on the categorical data . Like the Grade 1 students, t h e y genera l ly l i s t ed the t ransport categories w i thou t links to the number of chi ldren. Joshua was t h e exception in t h a t he reproduced the raw d a t a w i th o u t reorganising it. W i t h respect to numericaldata, the s i tuat ion for tile 9 (out of 12) chi ldren in Grades 2 to 5 who exh ib i t ed Level 1 th inking was quite different . W h e r e a s a l l Grade 1 students exh ib i t ed id iosyncrat ic th inking w i t h respect to numerical d a t a organisation, 7 out of these 9 chi ldren produced a v a l i d representat ion. T h e i r representat ions were t y p i c a l l y p ic tographs or tables showing the students' names and the number of f i sh essent ia l ly as t h e y were presented in the raw data . These students saw no need to reorganise the da ta . In t h e i r minds it could be represented per fec t ly in its raw form.

Nine students in Grades 2 to 5 exh ib i t ed Level 3 th inking w i t h respect to t h e categorical data . T h e y correctly organised the d a t a by t ransport categories and were able to justify t h e i r organisation. Alex (Grade 4) was t yp i ca l of th is group when he remarked, "Tha t just tells me how many walk , how many ride a bike .... " Unl ike the Level 1 th inkers , chi ldren exhibi t ing Level 3 th inking did not seem to h a v e any anxie ty about losing the chi ldren 's names through reorganisat ion of t h e categorical da ta . The s i tuat ion was d i f ferent for numerical d a t a in t h a t only 3 students beyond Grade 1 per formed a reorganisat ion of the data . Ti l l i (Grade 2), S a l l y (Grade 3), and Aldo (Grade 5) reorganised the d a t a by numerical groups (those who h a d 0 fish, those who h a d 1 fish, etc.) and used frequencies to represent the number in each numerical group. For example, Aldo drew a p ic tograph w i t h the number of f i sh shown v e r t i c a l l y and the frequencies (represented by st ick people) shown h o r i z o n t a l l y (see Figure 4). Moreover, a l l of these students were able to expla in t he i r reorganisat ion of the data . For example, Aldo exp la ined h is organisat ion as follows: "I 've got numbers from 0 through 10:1 person h a d zero f ish , 0 people h a d one f i sh . . . . and 1 person h a d nine f ish." S a l l y and Ti l l i used a s imi l a r reorganisat ion to Aldo but represented it using t a l l y graphs . However , T i l l i did not show al l of the numbers of f i sh on her t a l l y graph. In fact, she omit ted numbers l ike 1 and 5 (fish) where the frequencies (of students) were zero. Hence, a l t h o u g h the re were differences in t he i r representat ions, the key feature t h a t d is t inguishes


these th ree students ' th inking is t h e i r readiness to carry out and explain a frequency reorganisat ion of numerical da ta .

Question 2: What is the Effect of Data Type (Categorical Versus Numerical) on Children's Statistical Thinking in Representing and Organising Da ta ?

A Wilcoxon Signed Ranks Test (Siegel & Caste l lan, 1988) was used to test for differences between students ' representations of categorical and numerical data . In implement ing th is m a t c h e d pa i r test we compared each ch i ld ' s level of th ink ing in representing categorical d a t a w i t h t h e i r level of th inking in representing numerical data . The Wilcoxon Test showed t h a t the re was a s ignif icant difference between the two sets of representat ions in favour of the categorical d a t a representat ions (Mdn [categorical] 3, Mdn [numerical] 2, p .05).

Looking at th is s i tuat ion more closely (see Table l) we note t h a t t h e differences in represen ta t iona l th inking for tile two d a t a sets occurred for s tudents beyond Grade 1. The Grade 1 students ' were a l l at Level 1 for both categorical and numerical da ta . For the students beyond Grade l, Table 1 reveals t h a t 9 out of 12 were at Level 3 or above when representing categorical d a t a compared w i t h 3 out of 12 for representing numerical da ta . As discussed ear l ier , th is difference in th ink ing leve l resulted from the fact t h a t Level 3 th inking in representing d a t a required students to show evidence of being able to reorganise the d a t a w h i l e Level 2 did not. The fact t h a t the re were 50% more reorganisat ions of categorical d a t a t h a n numerical d a t a by these chi ldren h i g h l i g h t s the key difference in the ch i ld ren ' s represen ta t iona l th inking w i t h categorical d a t a versus t h a t w i t h numerical da ta .

A Wilcoxon Signed Ranks Test (Siegel & Caste l lan, 1988) was also used to tes t differences between chi ldren 's organisations of categorical and numerical da ta . Chi ldren ' s levels of th inking for each type of d a t a organisat ion were again used as the comparison measure. The Wilcoxon Test showed t h a t the re was a s ignif icant difference between the two types of organisa t ional th inking in favour of t h e categorical d a t a (Mdn [categorical] 3, Mdn [numerical] 1,p 0.03).

The result is a s l i gh t l y more pronounced version of the differences between categorical representat ions and numerical representat ions. Once again the k ey difference was t h a t the major i ty of chi ldren (9 out of 12) beyond Grade 1 were able to reorganise categorical d a t a (Level 3 thinking) but only 3 out of 12 were able to reorganise numerical da ta .

Question 3: Is Children's Statistical Thinking in Representing Data Related to their Thinking in Organising Data? Is this Relationship Influenced by Whether the Data is Categorical or Numerical?

For categorical data , the re was a s ignif icant Spearman rank correla t ion between chi ldren 's th inking levels on representa t ion and t h e i r th inking levels on organising da t a (r~ 0.95, p < 0.005). There was also a s ignif icant but smal le r correlat ion between chi ldren 's th inking levels on representa t ion and organising numerical d a t a (r~ 0.87, p < 0.005). W h i l e we cannot assume a causal or even d i rec t ional r e l a t i onsh ip between organising and representing data , t h e


correlat ions do indicate a strong association between the cognitive ac t iv i t i es t h a t chi ldren under take in organising and representing da ta .

In responding to the first research question we h a v e a l r e a d y p rov ided qua l i t a t i ve evidence t h a t he lps expla in the difference between the strengths of the correlat ions for categorical d a t a and numerical data . W i t h categorical da ta , organisat ion appears to be a necessary and sufficient condition for representa t ion of data . T h a t is, the only students who could not represent d a t a in a v iab le way were the ones who could not organise the d a t a into t ransport categories. Moreover, a l l of the students who could organise the d a t a by t ransport categories could also represent it in a v a l i d way .

W i t h i n numerical da t a the s i tuat ion was d i f ferent because 7 out of the 15 students were able to provide a v a l i d representa t ion but did not reorganise t h e data; t h a t is, did not build an organisat ion of the d a t a t h a t was s t ruc tura l ly d i f ferent from the raw data . In fact, our q u a l i t a t i v e evidence suggests t h a t these students did not see a need to reorganise numerical data . For example, note how eas i ly Leslie (Grade 5) supports main ta in ing the organisat ion of the raw da ta , "Oh, I know a p ic tograph. I worked out how many f ish t h e y had , l ike 7, so I d rew 7 fish. I just looked at how many f ish each one had , so I d rew t h a t many fish." The fact t h a t these students chose not to reorganise migh t indicate t h a t t h e y were reluctant to lose the da t a labels (chi ldren 's names) or to engage more genera l ly in d a t a reduction. A l though such a position may h a v e l imi ted t h e i r predisposi t ion to generate mul t ip le representat ions of data , it was not an impediment to representing numerical da t a in its raw form.

W h i l e the size of the sample in th is s tudy necessitates caution in drawing conclusions about the influence of d a t a type on s ta t i s t i ca l th inking, the s tudy provides some evidence t h a t chi ldren 's th inking in organisat ion and representa t ion are in grea ter synchronisat ion when the d a t a are categorical t h an when t h e y are numerical. In an ove ra l l sense the re is some evidence in th is s tudy to suggest t h a t chi ldren 's s t a t i s t i ca l th inking in both organising and representing d a t a need to be in ha rmony for them to demonstrate rea l f l ex ib i l i t y in representing da ta .

Question 4: Is Students' Performance in Recognising Organisations of Data Related to their Performance in Creating Data Organisations?

In order to assess students ' performance in recognising organisations of da t a , Protocol 2 (see Figure 2) contained a series of questions on MaTT's h y p o t h e t i c a l representa t ion of the Pet Fish Club d a t a (see Figure 3). MaTT's representa t ion was a line plot w h i c h organised the d a t a by number of f i sh and frequency. Each ch i ld ' s response to MaTT's organisat ion was assigned a level of th inking by adap t ing t h e descriptors of the Framework t h a t r e l a t ed to organising/grouping da ta . In essence, the descriptors were changed so as to ref lect recognition of grouping r a t h e r t h a n creation of grouping. For example, the Level 3 descriptor, "groups or orders d a t a into classes and can explain the basis for grouping" was a d a p t e d to "recognises a grouping or ordering of d a t a into classes and can explain the basis for the grouping." Subsequently the students ' levels of th inking for recognition of Mary's organisat ion were corre la ted against t he i r levels of th inking for ac tua l ly organising the Pet Fish Club data . There was a s ignif icant correlat ion between recognition and


creation with , '~ 0.69 a n d p 0.005. This represents a p p r o x i m a t e l y 48% of s h a r e d va r i ance between the two menta l actions.

In ana lys ing ch i ld ren ' s i n d i v i d u a l da ta , we observed t h a t a l l 3 of the s tudents w h o reorganised the Pet F i sh d a t a (Till i , Sa l ly , and Aldo) also recognised and exp la ined the organisa t ion in Mary ' s l ine plot . T h a t is, these chi ldren e x h i b i t e d Level 3 th ink ing w h e t h e r creating or recognising d a t a reorganisat ion. Moreover , t h e y did th i s w i t h consummate ease connecting Mary ' s represen ta t ion to t h e i r own. For example , T i l l i (Grade 2) said, "It 's the same in a d i f fe ren t way , i t ' s got X's, I 've got t a l l y marks ." H a v i n g noted the connection w i t h t h e i r own represen ta t ion t h e y a l l wen t on to expla in the meaning of the crosses as i f t h e y were t a l l y marks. T i l l i ' s response was typ ica l , "The crosses mean t h a t t h e y h a v e f i sh t h a t m a n y people h a v e fish. Two people h a v e 2 f i s h . . . . . "

A more interest ing result is the fact t h a t 3 chi ldren (Marcia, Lesley, and Joshua) gave v a l i d and complete exp lana t ions (Level 3 th inking) of M a r y ' s organisa t ion of the Pet F i sh d a t a even though t h e y h a d not previous ly reorganised the d a t a in constructing t h e i r own representa t ion. These students also h a n d l e d t h e exp lana t ion w i t h ease and spon tane i ty suggesting t h a t t h e y were able to understand th i s k ind of d a t a reorganisa t ion even though t h e y h a d n ' t chosen to use it in t h e i r own representat ion. Marc ia exempl i f i ed the kind of response given by these 3 chi ldren, "The number o f f i sh . Oh, 1 person h a d zero fish, 2 people h a d two fish, 1 h a d three , 3 h a d four f i s h ..... Each of the crosses is a person."

Three o ther chi ldren (Anton, Adr ian , and Meg) gave p a r t i a l exp lana t ions to Mary ' s organisa t ion of the Pet F i sh d a t a even though t h e y h a d not shown any incl inat ion to reorganise the d a t a in t h e i r own representa t ion. The i r th ink ing was less spontaneous t h a n the prev ious ly ment ioned 6 chi ldren and exh ib i t ed Level 2 th ink ing r a t h e r t h a n Level 3. However , as the following t y p i c a l excerpt of Anton (A) shows, t h e y were able to make accommodat ions in response to the r e sea rche r ' s (R) questions.

R: Can you read what it (MarT's graph) says? A: Yes. Actually it's not a vet T good graph, because it doesn't tell w h o has 3 and who

has 1. [Italics added.] R: What do you think these crosses mean? [Points to 3 crosses above the 4 fish mark] A: 3 fishes. Urn, which one? R: What if I picked that cross there? [R points to 1 of the crosses above the 4 fish

mark.] A: Ah! It means one 4. Which way is it going? R: What does this say along here? [R points to the horizontal axis.] A: Number of fish. R: OK. So what does this cross mean? [R points to the cross above 9 fish.] A: Oh! 9. Izra had 9 fish. [Anton looks back at the raw data.] R: I wonder what that cross next to the zero means? A: Ga W.

At first, Anton was confused because he couldn' t find the ch i ld ren ' s names on Mary ' s g raph . He considered the crosses to be f i sh possibly because t h a t was more compa t ib le w i t h the format of the raw da ta . Even tua l ly , Anton was able to recognise t h a t each cross represented a person but only a f t e r he h a d a c t u a l l y l inked up the single crosses for 9 f i sh and 0 f i sh w i t h the chi ldren in the raw da ta . Anton wen t on to expla in the organisa t ion in Mary ' s representa t ion , "Two chi ldren h a v e 2 f ish, 1 h a s 3 f i s h . . . . " but his th ink ing was very f rag i le and he constantly re fe r red to the raw d a t a .


Accordingly, w h i l e the re is a s ignif icant association between chi ldren ' s ab i l i t y to recognise and create d a t a reorganisat ion, some of the unexplained var iance (52%) may result from the fact t h a t chi ldren 's th inking in the recognition mode is stronger t han t h e i r th inking in the creation mode. Our d a t a r evea l t h a t 6 students exh ib i t ed h i g h e r l eve l th inking in tile recognition tasks than the creat ion task, and no students per formed at a lower level on the recognition task. Hence, the re is evidence, a lbe i t l imi ted , t h a t the mental actions required to create a reorganisat ion lag behind those needed to recognise one.

Discussion This is an exp lora to ry s tudy t h a t examined how young chi ldren organised d a t a

w h i l e t h e y were in the process of representing da ta . More speci f ica l ly , the s tudy used the cognitive th ink ing levels of the Jones et al. (2000) Framework to gain a picture of how 15 chi ldren, 3 in each of Grades 1 to 5, represented and organised both categorical (school t ransport) and numerical (pet fish) data . The s tudy also examined chi ldren 's responses to a representa t ion and organisat ion of the pet f i sh numerical da t a t h a t was generated by an imag ina ry chi ld .

The results of the s tudy showed t h a t chi ldren in Grade 1 were more id iosyncrat ic and incomplete in t h e i r th inking w i t h respect to organising and representing da t a than t he i r counterparts in Grades 2 to 5. These younger ch i ld ren did not make connections between category of t ransport and number of riders (nor number of f i sh and frequency), whe reas chi ldren beyond Grade 1 made these connections, at l eas t for the categorical data . The finding t h a t Grade 1 ch i ld ren were unable to organise the categorical t ransport d a t a surprised us, given t h a t Jones et al. (2000) concluded t h a t chi ldren in Grade 1 could reorganise Beanie Baby d a t a into categories according to an imal name. The fact t h a t the chi ldren in Jones et al . could organise the Beanie Baby d a t a may h a v e been due to tile t ransparent nature of the Beanie Baby categories and the chi ldren 's f a m i l i a r i t y w i t h these toys. It may also h a v e been due to the fac t t h a t the chi ldren could move concrete versions of the Beanie Babies but could not man ipu la te the t ransport categories or the f ish . W h a t e v e r the reason, the result again points to the importance of mode of presenta t ion and context in da t a explora t ion espec ia l ly w i t h young ch i ld ren (Curcio & Folkson, 1996; Friel , Curcio & Bright , 2001; Greer, 2000, Konold & Higgins, in press). C lea r ly fu r the r research is needed on mode of presentation and context in da t a explora t ion w i t h a view to ident i fy ing and classifying tasks and contexts t h a t are sui table for young chi ldren. Young chi ldren in kindergarten and in the p r i m a r y grades regula r ly engage in sorting ac t iv i t i es w i t h blocks and o the r manipu la t ives . However , the re is obviously a need for research to build a more pe rvas ive understanding of chi ldren 's sorting schemata not only in the context of m a n i p u l a t i v e blocks but also in the context of d a t a explora t ion .

The ab i l i t y to make connections between di f ferent aspects of the d a t a (e.g., between transport category and number of riders) enabled students beyond Grade 1 to produce more normative (in a m a t h e m a t i c a l sense) organisations and representat ions of the data . Interest ingly, and in sha rp contrast to ear l ie r findings (Jones et al., 2000; Lehrer & Schauble, 2000), t h e r e were minimal g r a d e level differences for students in Grades 2 through 5 w i t h respect to the processes t h e y


used in organising and representing da ta . The lack of g r a d e level differences beyond Grade 1 may wel l be a mani fes ta t ion of the l imi ted sample size in th i s study; however , the re are recent studies in tile f ie ld of p robab i l i t y and s ta t is t ics where ne i the r age nor grade h a v e not been s ignif icant indicators (e.g. Batanero & Serrano, 1999; Fischbein & Schnarch, 1997). W h a t may be of more importance in th is s tudy is the fact t ha t , as a group, the students in Grades 2 to 5 showed a preference for categorical organisations of d a t a and tile use of p ic tographs , bar graphs , and t a l l y graphs when representing data . Ta l l y graphs were also h e l p f u l in reorganising numerical d a t a by frequencies and 2 of tile 3 students who were able to produce such a reorganisat ion used a t a l l y graph. The t h i r d student, Aldo, used a line p ic tograph (see Figure 4), w h i c h was conceptually equiva len t to a t a l l y g raph .

Another p o t e n t i a l l y impor tan t finding of th is s tudy is t h a t numerical d a t a appears to be more d i f f icul t for students to organise and represent t h an categor ical data . The difference between the two kinds of d a t a did not arise w i t h the Grade 1 chi ldren because none of them produced connected organisations of e i t he r t h e categorical or the numerical data . However , the difference in complexi ty between numerical d a t a and categorical d a t a was ap p a ren t w i t h respect to the students beyond Grade 1. Only 3 out of 12 made the reorganising link between number of f i sh and frequency for the numerical data , whereas 9 out of 12 made the l ink between modes of t ransport and numbers of students for the categorical data . Our corre la t ion analys is also suggests t h a t students' th inking in organisat ion and representa t ion is in grea ter synchronisat ion for categorical d a t a than for numerical data . The d i f f i cu l ty these e l ementa ry chi ldren experienced in reorganising the numerical d a t a according to frequencies is consistent w i t h Br igh t and Friel 's (1998) f inding for middle school students.

Br igh t and Friel (1998) found t h a t reorganising ungrouped d a t a into grouped d a t a was not an in tu i t ive sorting process for midd le school students. The students beyond Grade 1 in our s tudy ref lected th is same di f f icul ty . May be it is too much of a mental leap to expect e l emen ta ry and middle school students to create an organisat ion incorporating frequencies. A l t e r n a t i v e l y these students may s imp ly see no need to reorganise the da t a by frequencies when the raw organisat ion and i ts representa t ion by chi ldren 's names and number of f i sh is per fec t ly meaningful and comfortable. The l a t t e r a l t e rna t i ve has even more credence when it is noted t h a t students in th is s tudy expressed reluctance about breaking the link between t h e chi ldren 's names and the number of f i sh t h e y owned (see Figure 2). In te res t ingly breaking the l ink between chi ldren 's names and t h e i r mode of t ransport did not seem to produce the same resistance for the categorical data . Is it easier to keep t rack of d a t a reduction (loss of students ' names) in deal ing w i t h categories t h an in deal ing w i t h numbers7 Do frequency representat ions of numerical d a t a t h a t h a v e numbers on both axes, t h a t is, "double numeric" labels, produce accommodations t h a t are beyond e l emen ta ry students ' conceptions7 We s imply do not know t h e answer to th is d i lemma. Fur ther research on organizing and representing d a t a should incorporate teaching experiments or microworlds t h a t produce perturbat ions capable of s t imulat ing chi ldren to create frequency distributions. Such perturbat ions would need to chal lenge t h e i r reluctance to reduce data . In add i t i on such research migh t try to explain the dispositions, meta cognit ive processes, and


conceptions tha t lead some elementary students but not others to reorganise numerical data into frequencies.

Although the creat ion of frequency representations was rare and seemingly non intuitive, students beyond Grade 1 revealed stronger sense making when t h e y were asked to analyse and interpret Mary's hypothe t ica l line plot (Figure 3) t h a t was organised by number of fish and frequency. Fif ty percent of the students beyond Grade 1, including all 3 who were able to create their own frequency representation, interpreted Mary's line plot fluently and spontaneously. The ones who had created a frequency representation simply linked Mary's representation to their own. The others discerned wha t was meant by the crosses (people) in Mary's line plot and then built an explanation tha t showed they understood reorganisation by frequencies even though they had not ini t ia ted such a reorganisation when asked to represent the original data. In addition to these 6 students, there were 3 other students who interpreted Mary's line plot after some probing. Interestingly, all 3 could only make the connection between the crosses (people) and the number of f ish after they had assigned names to the crosses. In other words they appeared to have to mental ly rebuild the original raw data before they could make sense of the frequencies and their connection to the number of fish. The evidence associated wi th Mary's line plot suggests an implication for instruction. Given the students' abi l i ty to interpret a frequency representation constructed from raw data, teachers might use recognition tasks like Mary's hypothet ica l line plot to build an understanding of frequency representations and to forge in students a stronger sense of the usefulness of frequency in reorganising data. The idea of presenting both the raw data and its frequency representation and having the students make links between them seems especially powerful. Such an approach might well resonate wi th students like those in this study who could only make sense of the frequency representation when they assigned original data names to the crosses in Mary's line plot. Consistent wi th this pedagogical approach, Perry et al. (1999) argued that , before learning how to construct an unfamiliar visual display, students should begin by examining and analysing the characteristics of the visual d isplay and its connections to the original data .

The size of the sample limits the generalisabil i ty of the conclusions evidenced in this study. Further research might explore organisation and representation of data wi th larger groups of elementary students and in instructional settings t h a t trace students' thinking over an extended period. Notwithstanding the l imitat ion associated wi th sample size, this study has built on the statist ical thinking framework of Jones et al. (2000) and has generated new insights into our understanding of elementary students' cognitive functioning when they are faced wi th tasks tha t provided them wi th the opportunity to organise categorical and numerical data while they were in the process of representing such data.

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Authors Steven Nisbet, Faculty of Education, Griffith University, Brisbane, <[email protected]>.

QLD 4111. Email:

Graham A. Jones, Department of Mathematics, Illinois State University, Normal, IL 61790 4520, USA.

Carol A. Thornton, Department of Mathematics, Illinois State University, Normal, IL 61790 4520, USA.


Cynthia W. Langrall, Department of Mathematics, Illinois State University, Normal, IL 61790 4520, USA.

Edward S. Mooney, Department of Mathematics, Illinois State University, Normal, IL 61790 4520, USA.

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Children’s representation and organisation of data