24 SES 02 B, Representations in Mathematics Education
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).
Blanton, M. L., & Kaput, J. J. (2011). Functional Thinking as a Route Into Algebra in the Elementary Grades. In J. Cai & E. Knuth (Eds.), Early Algebraization. A Global Dialogue from Multiple Perspectives (pp. 5–23). Heidelberg: Springer. Borba, M. C. & Confrey, J. (1996). A Student's Construction of Transformations of Functions in a Multiple Representational Environment. Educational Studies in Mathematics, 31 (3), 319–337. Confrey, J. & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26 (2-3), 135–164. Kerslake, D. (1981). Graphs. In K. M. Hart (Ed.), Children's understanding of mathematics. 11-16 (pp. 120–136). London: John Murray. Müller, F. C. (1889). Optische Urteilstäuschungen. Archiv für Physiologie Suppl., 263–270. Thompson, P. W. (1994). Students, Functions, and the Undergraduate Curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in Collegiate Mathematics Education I (pp. 21–44). Providence, RI: American Mathematical Society.
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