The development of statistical reasoning in primary school students
While formal statistical practices are not generally accessible to students in the primary years of schooling, the principles underpinning statistical thinking and reasoning—such as posing questions, collecting data, comparing groups, and representing and inferring from data—are relevant in primary mathematics (Watson et al., 2018). Recent Australian studies by English (2012, 2013, 2018), Fielding-Wells (2014, 2018 a,b), Kinnear (2013, 2018), Makar (2014, 2016, 2018), Mulligan (2015) and Watson (2018) have focused on primary school students’ capacities to engage in data modelling and on statistical reasoning more broadly. An early years’ approach to the teaching of statistics involves including students’ personal experiences, encourages self-collected data sets, and emphasises the reasoning process rather than outcomes or conclusions (Doerr, et al., 2017). How young students’ develop and apply the modelling and refinement process is not clearly understood however, especially when working with an abstract or complex data set. This thesis aimed to gain a more coherent understanding of the developmental aspects of Grade 1 through 4 students’ statistical reasoning and metarepresentational competence with explicit emphasis upon predictive reasoning.
Three interconnected design studies on model-based reasoning and predictive reasoning were conducted with 46 Australian students drawn from one cohort of a single, independent, metropolitan primary school. In the first design study, nine high-ability Grade 1 students created a word-based model for categorisation of self-portraits drawn by students in other grades, and assessed the model using three reasoning tasks. Of interest were the features of the modelling process observed in Grade 1 students, and how students’ used test data collected from the model to inform judgements regarding its efficacy and limitations. The second design study focused on predictive reasoning. How Grade 2 students used the variability of the temperature table to inform their predictions, how they justified predictions and their use of probabilistic language was the focus. Ten high-ability Grade 2 students, including seven students retained from the previous study, predicted maximum monthly temperatures from a temperature table then plotted their predictions against background temperature readings using TinkerPlotsTM.
For both design studies, student predictions, representations and explanations were coded using three levels of statistical reasoning: idiosyncratic, transitional and quantitative (Leavy, 2008). Seven of the Grade 1 students were observed using data-based reasoning when justifying and revising their decisions. Six of the Grade 2 students made predictions similar to other monthly values in the data table, increasing to nine students after plotting the predictions with TinkerPlotsTM. All ten students used probabilistic language when describing the data set, including terms such as outliers, clusters and range.
Following this pilot work, the main study employed 46 students from Grade 3, and 44 of the same students from Grade 4 in a longitudinal teaching experiment. Students predicted maximum monthly temperatures for the current year using a data table containing past maximum temperatures, represented the data table using informal freehand inscriptions or graphing and described their predictive strategies in verbal and written form. Data were collected at the beginning of Grade 3 and the beginning and end of Grade 4 using the same tasks. Data were coded using a data lenses framework (Konold et al., 2015) in Grade 3 and a framework for analysis of structural features (Awareness of Mathematical Pattern and Structure [AMPS]) (Mulligan & Mitchelmore, 2009) in Grades 3 and 4. Most Grade 4 students (87%) made predictions within the historical range, relative to half in Grade 3 (54%). Representations included co-ordinate graphing including column, line and dot plots and were more sophisticated in Grade 4, with 57% demonstrating data transnumeration, while in Grade 3 they were predominately idiosyncratic or copies of the data table. Grade 4 students were more likely (79%) than Grade 3 (51%) to use and describe predictions based on extraction, clustering, aggregation, noticing seasonal trends and range, identifying causal and random variation, and observing measures of central tendency. Large individual differences emerged: three developmental pathways are illustrated through case studies of high, average, and low ability students. This range suggests that pathways for predictive reasoning are somewhat flexible or idiosyncratic.
The design studies in this thesis demonstrated the advanced potential of some young students to reason statistically: Grade 1 students developed a viable word-based model using a complex data set, and Grade 2 students employed TinkerPlotsTM to critique their data predictions. Levels of statistical reasoning in these students was higher than previously reported in studies of students in first and second grade such those by Makar (2016) and Lehrer and Schauble (2000b), as demonstrated through their use of data when justifying their reasoning.
The longitudinal study on student predictive reasoning and meta-representational competence contributes to a more in-depth or fine grained analysis of the possible developmental sequence of these capacities across Grades 3 and 4. Primary school students used contextual cues and data content when they make predictions, and appear to make realistic predictions from data tables prior to being able to describe viable prediction strategies, or to select data for representational purposes. However, other skills appear to develop unevenly— some students developing meta-representational competence and formal graphing prior to reasoning about their strategies, while other students developing reasoning strategies prior to meta-representational competence. Intermediate stages of transnumeration of data tables to formal graphs were described, providing a comprehensive longitudinal set of student representations from a single data set. The studies contribute to a growing body of research that investigates the predictive and data-modelling capacities of young students, and makes a distinct contribution by reporting on the use of TinkerPlotsTM as a visualisation tool with second graders. The research supports the inclusion and extension of curriculum reform highlighting data-driven learning, and the development of statistical concepts that are integral to statistical literacy and mathematics learning. Research implications include arguments for more explicit outcomes in the Statistics and Probability strand of the mathematics curriculum on informal statistical inference and data exploration in the early years. This needs to be accompanied by newly developed professional development programs, resources and support for teachers’ acquisition of pedagogical content knowledge in statistical reasoning, and for primary school students to have extended opportunities for informal data representation prior to the introduction of formal graphing instruction.
