The Concept of Function: The Effect of the Form of Representation
The aim of this study is to identify the effect of the form of representation when students in early grades deal with the concept of function. In mathematics teaching, three forms of representation of functional dependencies have prevailed: the table of values, the graph, and symbolic algebra. Although Borba & Confrey (1996) question the dominance of symbolic algebra over other representations, school teaching of the concept of function is still dominated by symbolic algebra. Therefore, the concept of function is rarely taught before middle school or even high school (Blanton & Kaput, 2011).
Teaching functional thinking in early grades entails using forms of representation that are adapted to the mathematical ability and cognitive development of the students. Alongside the table of values and the graph, the bar chart could be a helpful form of representation when dealing with functions. In spite of the arguments for implementing the teaching of functional thinking in early grades, there is a lack of research on the suitable forms of representation.
In mathematics education, two aspects of the concept of function are considered to be essential (Confrey & Smith, 1994; Thompson, 1994). In the correspondence view, also called the assignment view or input-output view, every value of the domain corresponds to exactly one value of the range. In the covariational view, the co-variation of the dependent variable induced by a variation of the independent variable is scrutinized.
In our opinion, the covariational view can be further divided into a quantitative and a qualitative analysis of covariation. In the quantitative analysis of covariation, change is analyzed in numbers, i.e., by calculating rates of change or extrapolating the function in a numerical way. In a qualitative analysis of covariation, the students have to explore the covariation of the function value in a qualitative manner.
Students should be able to easily solve correspondence tasks (CSP) with a table of values and a bar chart as we use a specific bar chart in which the function values are printed above the bars. However, the graph requires more mental effort to identify the correspondence, because the students have to read values off the axes. Therefore, we conjecture that the performance for tasks addressing the correspondence view is higher if a table (CSP(T)) or a bar chart (CSP(B)) is used rather than a graph (CSP(G)): CSP(T) > CSP(G) and CSP(B) = CSP(T).
Concerning the quantitative analysis of covariation (QNC), we put forward similar hypotheses. First, the students have to read off the values, keep them in working memory, and finally execute calculations. Therefore we conjecture: QNC(T) > QNC(G) and QNC(B) = QNC(T).
With regard to the qualitative analysis of covariation (QLC), there are two contradicting argumentations. First, it can be assumed that the visual impression of a graph provides additional help. As the bar chart also provides visual information, its performance is supposed to be similar to the graph. This results in the hypotheses: QLC(G) > QLC(T) and QLC(B) = QLC(G).
Second, an alternative argumentation leads to a contrary hypothesis. The graph provides a quick intuitive answer to a qualitative question on the basis of visual appearance. But, as known from previous research, visual impressions can lead to incorrect conclusions. The graph-as-picture interpretation (Kerslake, 1981) or the Müller-Lyer illusion (Müller-Lyer, 1889) are two of many examples of cognitive errors when processing visual perception. The same reasoning applies to working with bar charts whereas a table of values prevents the students from solving the task in a quick and fuzzy way based on visual impressions. This argumentation leads to the hypotheses: QLC(T) > QLC(G) and QLC(B) = QLC(G).
A paper-and-pencil test was administered in 15 classes of grade 6 and grade 7 (N = 377). The students had not been taught in the concept of function before. The test consisted of three testlets. The first testlet (bathtub-testlet) specified a water-filling process of a bathtub by a piecewise linear function. The second testlet (winterday-testlet) described the non-linear development of the air temperature within a 24 hour period of a winter day and the third testlet (Peter’s-growth-testlet) dealt with the growth function of a boy called Peter. Every testlet contained 12 items, of which four items each added up to the variable for the correspondence view (CSP), the variable of the quantitative covariational view (QNC), and the variable of the qualitative covariational view (QLC). For testing, three booklets were constructed so that each testlet could be presented with a table of values (table-group), a bar chart (bar-chart-group), and a graph (graph-group). Thus the form of representation established a factor with three levels in a between-subjects design.
With the additional condition that the booklet versions were as evenly distributed as possible within a class, students were randomly assigned to a booklet version. We used an alpha level of .05 for all statistical tests.
Concerning the correspondence view, there was a relatively high performance level in all forms of representation. In every testlet and every group there was a solution rate of more than 90%. The high performance rates in the correspondence view tasks stress that the students of grade 6 and 7 understood the three discussed forms of representation in principle. Our hypotheses CSP(T) > CSP(G) and CSP(B) = CSP(T) were corroborated in the winterday-testlet and the Peter’s-growth-testlet as determined by a Welch ANOVA and a contrast analysis.
With regard to the quantitative analy¬¬sis of covariation, the winterday-testlet and the Peter’s-growth-testlet confirm our hypotheses. The graph-group performed statistically significantly worse than the bar-chart- and the table-group. A noteworthy clue is that the more difficult the testlet is (indicated by the average solution rate), the bigger the differences between the forms of representation become.
Concerning the qualitative analysis of functions, the data indicate that the visual information of the graph is of almost no help if the students have not been previously instructed on how to interpret it. In the winterday-testlet and the Peter’s-growth-testlet the graph-group’s performance was significantly lower than the table-group’s and the bar-chart-group’s performance. Therefore the hypothesis QLC(G) > QLC(T) can be rejected and the almost contrary hypothesis QLC(T) > QLC(G) can be confirmed.
In the bathtub-testlet, there could not be identified any differences in the mean between the three forms of representations for any of the three variables. That the tasks were the easiest of the test may account for the nonsignificant results.
To sum up, statistically significant performance differences concerning the forms of representations were proven. In correspondence tasks as well as in quantitative and qualitative covariation tasks, the performance of the graph-group was worse than the table-group and the bar-chart-group in two of three testlets.
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