Study Times and Test Scores: What Student's Graphs Show: How Do Children Receive and Represent Information Gathered from Real-Life Contexts? Jonathan Moritz Shows Us How with His Vivid Collection of Work Samples from Students in Schools That Illustrate Varying Levels of Understanding Information
Moritz, Jonathan, Australian Primary Mathematics Classroom
Why should students represent statistical association?
Statistics is becoming increasingly significant in daily life. Many newspaper reports and advertisements make verbal statements that involve causal claims, in the form 'X causes Y'. Often these claims are based on statistical covariation, meaning that two measures vary together, for example, 'as values of X increase, values of Y tend to increase' (or decrease). But not all newspaper reports use graphs to illustrate the statistical data that lie behind the causal claims. Figure 1 shows translation processes among raw numerical data, graphs, verbal statements of statistical association, and causal claims. The arrows indicate the order of data analysis to arrive at conclusions, as often suggested for students in schools. In daily life such as reading the newspaper, however, adults more commonly read a causal claim, and to understand and critically evaluate it, they must imagine what statistical data might lie behind it, that is, think in the reverse direction of the arrows in Figure 1. Hence drawing a graph to illustrate a verbal statement of statistical association is a task that also deserves attention in students' schooling.
Graphing in the curriculum
A National Statement of Mathematics for Australian Schools (Australian Education Council [AEC], 1991) recommended that primary school students graph information and interpret graphs as part of the Chance and Data strand and the Algebra strand. The emphases in the two strands are different, Chance and Data characterised by a set of data values (points), and Algebra often involving idealised generalisations of continuous variables (straight lines). As part of Chance and Data, younger students use pictographs, and by the end of primary school progress to bar graphs and pie graphs. For primary students, commonly data involve one variable across a number of cases, represented in a bar graph, and analysed by averages, whereas for secondary students, bivariate data might be represented in a scatterplot (e.g. in Figure 1) and analysed by correlation. As part of Algebra, primary students should have experiences with functions to 'represent (verbally, graphically, in writing and physically) and interpret relationships between quantities' (p. 193). Possible activities include 'Sketch informal graphs to model familiar events such as variations in hunger through the day' and 'Given a sketch graph (e.g. of the depth of water in the farm water tank), write a story about it' (p. 193).
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Similar recommendations appear in Principles and Standards for School Mathematics in the United States (National Council of Teachers for Mathematics [NCTM], 2000). The standard for Data Analysis and Probability accelerates the suggestions above: in grades 3-5, students 'represent data using tables and graphs such as line plots, bar graphs, and line graphs' (p. 176), and by grades 6-8, 'select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots' (p. 248). The Algebra standard includes expectations that students Pre-K-2 should 'describe qualitative change, such as a student's growing taller' (p. 90), and by grades 3-5, students should 'represent and analyse patterns and functions, using words, tables, and graphs' (p. 158) and 'investigate how a change in one variable relates to a change in a second variable' (p. 158), such as for the growth of a plant, 'describe how the rate of growth varies over time' (p. 163).
Narratives as a way into reading graphs and constructing graphs
Researchers investigating the foundations of algebra have suggested that mathematical narratives may assist students to understand the situation mathematised in a graph. Narratives involving events over time may encourage a verbal language to express generalities of how a quantity varies rather than being limited to individual data points (Nemirovsky, 1996). …