YOUNG CHILDREN’S STATISTICAL THINKING: A TEACHING EXPERIMENT

Arsalan Wares	Graham A. Jones	Cynthia W. Langrall	Carol A. Thornton
Illinois State University	Illinois State University	Illinois State University	Illinois State University
wares@ilstu.edu	jones@ilstu.edu	langrall@ilstu.edu	thornton@ilstu.edu

This study designed and evaluated a teaching experiment in data exploration with two grade 1 classes one of which collected the data used in instruction. The teaching experiment was informed by a cognitive framework that described elementary students’ statistical thinking. Following the teaching experiment, the children showed significant gains on some but not all of the statistical thinking processes associated with the framework. While the two classes (collection and noncollection groups) changed in different ways, the evidence did not support stronger overall growth for the collection group. Case-study analysis revealed that: experience with the data context reduced children’s idiosyncratic descriptions, data values of zero were problematic for these children; children possess intuitive knowledge of center and spread; and making meaningful predictions from data was difficult for these children.

In response to the critical role that data play in our technological society, there have been ongoing calls for reform in statistical education beginning in the primary grades (NCTM, 2000). Notwithstanding these recommendations there has been relatively little research on primary children's statistical thinking and even less research on the efficacy of instructional programs in data exploration (Shaughnessy, Garfield, & Greer, 1996). Moreover, the studies that have been undertaken have not developed and used the kind of cognitive models that researchers like Fennema et al. (1996) deem necessary to guide the design and implementation of instruction.

This study addressed the above-mentioned void in the research literature by developing and evaluating a teaching experiment on data handling with young children. More specifically, the study sought to: (a) use a cognitive framework that describes students’ statistical thinking to design and implement a teaching experiment with two grade 1 classes, and (b) to evaluate the effect of the teaching experiment on children’s learning.

Theoretical Perspectives

The conceptualization of this study drew on two theoretical perspectives. First, it was grounded in teaching experiment theory (Cobb, 1999). Second, the teaching experiment was informed by a cognitive Framework (Jones et al., 2000) (Figure 1) that describes students’ statistical thinking. In making the link between these two perspectives the Statistical Thinking Framework served as the research base for designing a hypothetical learning trajectory (Simon,1995) for students as they engaged in the teaching experiment. We also used the Framework to interpret classroom events and

Process/ Level	Level 1 Idiosyncratic	Level 2 Transitional	Level 3 Quantitative	Level 4 Analytical
Describing Data Displays (D)	•gives a description that is unfocused and includes idiosyncratic/irrelevant information; has no awareness of graphing conventions [e.g., title, axis labels] of the display •does not recognize when two displays represent the same data OR indicates some recognition but uses idiosyncratic/ irrelevant reasoning •considers irrelevant/subjective features when evaluating the effectiveness of two different displays of the same data	•gives a description that is hesitant and incomplete, but demonstrates some awareness of graphing conventions •recognizes when two different displays represent the same data, but uses a justification based purely on conventions •focuses only on one aspect when evaluating the effectiveness of two different displays of the same data	•gives a confident and complete description and demonstrates awareness of graphing conventions •recognizes when two different displays represent the same data by establishing partial correspondences between data elements in the displays •focuses on more than one aspect when evaluating the effectiveness of two different displays of the same data	•recognizes when two different displays represent the same data by establishing precise numerical correspondences between data elements in the displays •provides a coherent and comprehensive explanation when evaluating the pros and cons of two different displays of the same data
Organizing and Reducing Data (O)	•does not group or order the data or gives an idiosyncratic/ irrelevant grouping •does not recognize when information is lost in reduction process •is not able to describe data in terms of representativeness or "typicality" •cannot describe data in terms of spread; gives idiosyncratic / irrelevant responses	•gives a grouping or ordering that is not consistent OR groups data into classes using criteria they cannot explain •recognizes when data reduction occurs, but gives a vague/irrelevant explanation •gives hesitant and incomplete descriptions of data in terms of "typicality" •invents a measure--usually invalid--in an effort to make sense of spread	•groups or orders data into classes and can explain the basis for grouping •recognizes when data reduction occurs and explains reasons for the reduction •gives valid measures of "typicality" that begin to approximate one of the centers (mode, median, mean); reasoning is incomplete •uses an invented measure or description which is valid, but the explanation is incomplete	•groups or orders data into classes in more than one way and can explain the bases for these different groupings •recognizes that data reduction can occur in different ways and gives complete explanations for the different reductions •gives valid measures of typicality that reflect one or more of the centers; reasoning is essentially complete •uses the range or invented measure that has the same meaning as range
Representing Data (R)	•constructs an idiosyncratic or invalid display when asked to complete a partially constructed graph associated with a given data set •produces an idiosyncratic or invalid display that does not represent or reorganize the data set	•constructs a display that is valid in some aspects when asked to complete a partially constructed graph associated with a given data set •produces a display that is partially valid, but does not attempt to reorganize the data	•constructs a display that is valid when asked to complete a partially constructed graph associated with a given data set; may have difficulty with ideas like scale or zero categories •produces a valid display that shows some attempt to reorganize the data	•constructs a valid display when asked to complete a partially constructed graph associated with a given data set; works effectively with scale, zero categories, •produces multiple valid displays, some of which reorganize the data
Analyzing and Interpreting Data (A)	•makes no response or an invalid/irrelevant response to the question, "What does the display not say about the data?" •makes no response or gives an invalid/incomplete response when asked to "read between the data" •makes no response or gives an invalid/ incomplete response when asked to "read beyond the data"	•makes a relevant but incomplete response to the question, "What does the display not say about the data?" •gives a valid response to some aspects of "reading between the data" but is imprecise when asked to make comparisons •gives a vague or inconsistent response when asked to "read beyond the data"	•makes multiple relevant responses to the question, "What does the display not say about the data?" •gives multiple valid responses when asked to "read between the data" and can make some global comparisons •tries to use the data and make sense of the situation when asked to "read beyond the data;" reasoning is incomplete	•makes a comprehensive contextual response to the question, "What does the display not say about the data?" •gives multiple valid responses when asked to "read between the data" and can make coherent and comprehensive comparisons •gives a response that is valid, complete, and consistent when asked to "read beyond the data"

Figure 1: Statistical Thinking Framework

examine changes in children’s statistical thinking.

The Statistical Thinking Framework incorporates four key data handling processes adapted from Shaughnessy et al.(1996): describing, organizing and reducing, representing, analyzing and interpreting data. Describing data involves the explicit reading of data contained in visual displays. Organizing and reducing data involves ordering, grouping, and summarizing data using measures of center and spread. Representing data incorporates the construction of visual displays and analyzing and interpreting data includes what Curcio and Artzt (1997) refer to as “reading between the data” and “reading beyond the data” (p. 124). The Framework descriptors for each statistical process built on previous research (e.g. Beaton et al., 1996; Bright & Friel, 1998; Zawojewski & Heckman, 1997; Curcio & Artzt, 1997). For each of these processes, four levels of thinking were hypothesized and validated (Jones et al., in press). Level 1 is associated with idiosyncratic thinking, Level 2 is transitional between idiosyncratic and quantitative thinking, Level 3 involves the use of informal quantitative thinking, and Level 4 incorporates analytical and numerical thinking. These levels of thinking are consistent with neo-Piagetian theories that postulate the existence of thinking levels that recycle during developmental stages (e.g., Biggs & Collis, 1991).

Method

Children from 2 intact grade 1 classes in a midwest school participated in the teaching experiment on data exploration. One class collected the data on number of missing teeth from its own members (Collection Group, n=20) and the other class also used this data (NonCollection Group, n= 18). We hypothesized that this collection activity might work in favor of the Collection Group by building a better contextual base for data exploration. In addition to the overall analysis involving all children, 3 children from each class were purposefully sampled as target for more detailed case-study analysis. For each class,1 student was chosen from each of the upper and lower quarters and 1 from the middle 50% on mathematics achievement.

The intervention phase of the teaching experiment for these grade 1 children comprised four 40-minute sessions spread over a 3-week period. Sessions opened with a whole-class exploration posed by the third author. Ten teacher education undergraduate mentors facilitated children’s solving of data exploration problems that were based on the missing-teeth data and linked to the Framework. To ensure that the inquiry orientation of the intervention phase was met, mentors participated in weekly seminars that explored strategies for fostering children’s statistical thinking.

Data were gathered from three sources: (a) researcher-designed interview assessment protocols administered in the weeks preceding and following the intervention, (b) mentor evaluations from each instructional session, and (c) researcher field notes on the 6 target students. The assessment protocol (Jones et al., in press) based on the Framework comprised 37 items (7 on describing, 13 on organizing and reducing, 2 on representing data, and 15 on analyzing and interpreting data) in three contexts: How Many Friends Came to Visit? Beanie Babies and The Beanbag Game. A double-coding procedure (Miles & Huberman, 1994) was used to establish pre and postintervention statistical thinking levels for all students in both classes. In this procedure, the first two authors independently coded, according to the levels of the framework, all questions on each child’s protocol. The modal response level for each statistical thinking process was used to determine children’s dominant thinking levels. The authors reached agreement on 82% of children’s dominant thinking levels. A Wilcoxon Signed Ranks Test (Siegel & Castellan, 1988) was used to compare pre and postintervention statistical levels for each of the two classes. Both “within” and “cross-case displays”(Miles & Huberman, 1994) were used to guide the analysis of qualitative data from the 6 target students. Mentor evaluations and researcher field notes were coded and synthesized to discern learning patterns exhibited by these students during the intervention.

Results and Conclusions

The effect of the teaching experiment: Quantitative analysis. The Wilcoxon Signed Ranks Test

(Siegel & Castellan, 1988) revealed differences between the pre and postintervention thinking levels of the grade 1 children that were significant for some statistical thinking processes and not for others. For describing data, only the NonCollection Group showed a significant difference (Collection Group, p< .08; NonCollection Group, p< .01); for organizing and reducing data, both Groups showed significant differences (Collection Group, p< .04; NonCollection Group, p< .01); for representing data, neither group showed significant differences (Collection Group, p<.17; NonCollection Group, p< .42); and for analyzing and interpreting data only the Collection Group showed a significant difference (Collection Group, p< .01; NonCollection Group, p< .65). While the statistical thinking of the children in the two groups changed in slightly different ways, the evidence does not support a stronger overall growth in favor of the Collection Group.When the two classes were combined the differences between children's pre and postintervention thinking levels were significant for all statistical processes except representing data. For the three significant statistical thinking processes, the most salient feature of the data was that the number of children exhibiting Level 3 increased following the intervention and this was accompanied by a corresponding decrease in the number exhibiting Level 1 thinking.

The effect of the teaching experiment: Case-study analysis. A number of learning patterns and trends were discerned by examining the relationship between target students' thinking during instruction and their thinking at the pre and postintervention assessments. These patterns are described and interpreted for each of the four statistical thinking processes. With regard to describing data, students brought some prior knowledge to the classroom. For example, they recognized how categorical data such as "days of the week" were shown on a scale, how to find values on a line plot using counting, and how to read data in a table. Target students had more difficulty reading bar graphs than line plots and they made only cosmetic comparisons between a line plot and a bar graph of the same data. However, during instruction they gave less idiosyncratic responses when reading visual displays and became more facile in comparing different displays and organizations of the missing teeth data. With regard to organizing and reducing data, students’ informal knowledge prior to instruction was limited. However, during instruction and in the postassessment most target students demonstrated informal notions of mode or middle (median) in dealing with center. A smaller number showed some idea of clustering in discussing spread. Although most children were able to organize the familiar Beanie Baby data, they were unable to construct different organizations of the missing teeth data. Nevertheless, when the children were shown two different organizations of the missing teeth data they were able to recognize that they represented the same data. With regard to representing data, students were more capable at completing unfinished graphs based on a given set of data than they were at representing a given graph in a different way. Their limited skills in reorganizing data also constrained their representations of data. Interestingly more than 50% of the target students were not able to represent “zero” on a display and zero data values also caused problems when children analyzed and interpreted data. With regard to analyzing and interpreting data, our students, like those of Curcio and Artzt (1997), demonstrated more normative thinking in tasks that involved reading between the data than in tasks that involved reading beyond the data. However, even though students read zero values they ignored them when analyzing and interpreting data. The instructional sequence helped target students to gain better facility in dealing with zero values and in reading beyond the data particularly as they became very familiar with the missing teeth data. By the end of the intervention, more students in both classes were able to make and justify meaningful predictions of how many friends would visit Sam in the next month based on the data for the previous week.

Given the prior knowledge and growth that children showed on the four statistical processes, there is evidence that they can accommodate a broader approach to data exploration. However, if instruction is to reach its full potential in the elementary grades, further research is needed to build learning trajectories that link different levels of children's statistical thinking.

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