Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

Abstract

This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

Fifth Through Eighth Grade Students’ Difficulties in Constructing Bar Graphs: Data Organization, Data Aggregation, and Integration of a Second Variable

“…Accordingly, research in the field of mathematics education has determined that graphing includes the processes of axes selection and labeling, scaling, unit identification, and plotting in Cartesian space (Leinhardt, et al, 1990;Parmar & Signer, 2005;Rangecroft, 1991). As Leinhardt, et al (1990) reported, the most difficult step in graph construction is establishing the axes and scales with the appropriate labels (i.e., in Kosslyn's terms, establishing the relationship between the graph's components). We agree that the creation and labeling of appropriate axes and scales are important parts of the process of bar graph construction, but Kosslyn (2006) states that it is essential to understand the interrelated connections between single constituents (framework, specifier, and labels) before putting them together to choose the most suitable one for each particular set of data.…”

“…Thus, graphing requires not only an understanding of each of the components but also a style of organization that facilitates our seeing the relationships among them (Kosslyn, 2006). Accordingly, research in the field of mathematics education has determined that graphing includes the processes of axes selection and labeling, scaling, unit identification, and plotting in Cartesian space (Leinhardt, et al, 1990;Parmar & Signer, 2005;Rangecroft, 1991). As Leinhardt, et al (1990) reported, the most difficult step in graph construction is establishing the axes and scales with the appropriate labels (i.e., in Kosslyn's terms, establishing the relationship between the graph's components).…”

“…Analyses of students' graphing techniques have been common in the science and mathematics education literature of the last two decades (Friel, Curcio, & Bright, 2001;Janvier, 1987;Leinhardt, Zaslavsky, & Stein, 1990;Postigo & Pozo, 2004;Shah & Hoeffner, 2002;Watson & Fitzallen, 2010). However, research has investigated children's construction of graphs less frequently (diSessa, 2004;Nemirovsky, Tierney, & Wright, 1998;Watson & Fitzallen, 2010), and research devoted to bar graphs is even less common (Äberg-Bengtsson, 2006;Lehrer & Schauble, 2000Lehrer & Romberg, 1996;Watson & Moritz, 2000Wu & Krajcik, 2006).…”

“…Again, the analyses of Cartesian graphs by Kosslyn (2006) and Leinhardt, et al (1990) inspired the fourth criterion (labeling the second variable with the legend) and the seventh and eighth criteria for axes (x and y) labeling. Finally, the fifth (coherence between what the participants represented and the labels that they used) and the sixth criteria (the scaling of the measures on the axes) were based on the need to measure the relationship between the components described above (see the detailed coding rubrics for the bar graph in Appendix B).…”

“…We integrated Kosslyn's (2006) proposal that there are three main components of graphing (framework, specifier and labels) with the definition of bar graphs by Friel, et al (2001) and the analysis of Cartesian graphs by Leinhardt, et al (1990). The first criterion, format, corresponds to Kosslyn's framework concept (whether students represented height and frequencies on the y-and x-axes, respectively).…”