CBL ACTIVITIES AND STUDENTSí CONCEPTUAL KNOWLEDGE OF FUNCTIONS
Brenda B. Cates
Mount Olive College
634 Henderson Street
Mount Olive, NC 28365
Fundamental to the study of mathematics, the function concept has been identified as the single most important concept from kindergarten to graduate school (Dubinsky & Harel, 1992). Eisenberg and Dreyfus propose that having a sense for functions is one of the most important facets of mathematical thinking in that it allows students to gain insights into the relationships among variables in problem solving situations (1994). Reform documents such as Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus (American Mathematical Association of Two-Year Colleges [AMATYC], 1995) also advocate an increased emphasis on the function concept.
Despite widespread agreement that the undergraduate mathematics curriculum should be centered on the function concept, the complex process of developing a conceptual understanding of functions continues to be difficult for students to master. Students are often unsuccessful at establishing the correct connections between the various functional representations (Rich, 1990). The qualitative interpretation of functional graphs is especially problematic. Sfardís research (1989) offers one possible explanation for the difficulties students encounter in terms of developing a sound understanding of the function concept. Sfardís findings suggest that students acquire a procedural or "process" conception of function first, enabling them to understand the function concept on a rudimentary level, but have difficulty making the transition or "reifying" to a structural or "object" conception of function. Students who can reify functions perceive functions as abstract notions whose representations may be algebraic, graphical, or numeric. The knowledge of these different representation systems and of the linkages between them can provide students with a rich conceptual understanding of functions (Kaput, 1989).
There is a growing body of research indicating that the use of graphing calculators significantly affects studentsí understanding of the function concept (Hollar & Norwood, 1999). Evidence has also emerged from the science education community suggesting that the use of microcomputer-based laboratory (MBL) activities improves studentsí ability to interpret and use graphs (Thornton & Sokoloff, 1990). This study, which merged MBL and graphing calculator technology through the use of calculator-based laboratory (CBL) activities, investigated the following research questions: (a) Are CBL students able to interpret functions in their different representations better than students in a TI-83 only college algebra curriculum? (b) Are CBL students able to translate among the various representations of functions better than students in a TI-83 only college algebra curriculum? (c) Are CBL students able to interpret functions from a structural perspective better than students in a TI-83 only college algebra curriculum? (d) What effect does the use of CBL activities have on the ability of college algebra students to model real world phenomena with functions? (e) Are CBL students able to overcome graph-as-picture misconceptions better than students in a TI-83 only college algebra curriculum? (f) Are CBL students able to overcome slope vs. height misconceptions better than students in a TI-83 only college algebra curriculum?
The participants in this study were 56 students enrolled in four intact classes of a college algebra course at a small liberal arts college. This study utilized a quasi-experimental non-equivalent control group design in which two experienced mathematics instructors each taught one experimental and one control college algebra class. For each instructor involved in the study, one college algebra class was randomly assigned to the experimental group and one college algebra class was randomly assigned to the control group.
Over a period of six weeks, students in the experimental group (n = 29) took part in ten CBL activities designed to actively engage students in the learning process and to promote conceptual understanding of the functions used to model the observed physical phenomena. Students in the control group (n = 27) participated in non-CBL graphing calculator activities that paralleled the CBL activities with respect to goals and objectives. The CBL lab activities were a combination of researcher-constructed activities as well as activities adapted from previously published CBL materials. Students in both groups were required to have access to graphing calculators and no restrictions were imposed concerning their use. The experimental and control groups were taught the same algebraic topics and used the same textbook.
Both quantitative and qualitative methodologies were used to investigate the research questions. A survey was administered at the beginning of the semester to obtain demographic information (age, gender, high school mathematics background, experience working with groups, experience with graphing calculators, and comfort level with graphing technology). SAT-Math and SAT-Verbal scores were used as indicators of studentsí mathematical and verbal ability. In order to assess studentsí function-related knowledge and graphical interpretation skills, a modified version of a diagnostic instrument developed by OíCallaghan (1998) was administered at the beginning and at the end of the semester. The Function Test consists of questions designed to assess the following aspects of a conceptual knowledge of functions: (a) modeling real world phenomena with functions, (b) interpreting functions in terms of a real world situation, (c) translating between different representations of functions, and (d) reifying functions.
Post-treatment data were also collected via the College Algebra Questionnaire and student interviews. The College Algebra Questionnaire contained items designed to identify or highlight differences in functional understanding between the experimental and control groups. Three students from each class were interviewed to investigate the nature and extent of differences in studentsí understanding of the function concept and their misconceptions associated with interpreting graphs of functions. The interview data were subsequently used to support, clarify, and extend the quantitative and qualitative findings.
A comparison of SAT-Math and SAT-Verbal scores for the two groups, using ANOVA, revealed no significant differences between the experimental and control group, at the a = .05 level, with respect to mathematical or verbal ability. A comparison of demographic variables, using Chi-Square tests, likewise revealed no significant differences, at the a = .05 level, between the experimental and control group with respect to these characteristics. To analyze the Function Test pretest data, a MANOVA was performed on the four components (modeling, interpreting, translating, and reifying) and an ANOVA was performed on the total score. The MANOVA and ANOVA results revealed no statistically significant differences, at the a = .05 level, between the experimental and control groups with respect to their conceptual knowledge of functions.
Three items from the Function Test, subsumed under the modeling component, were used to assess studentsí potential graph-as-picture misconceptions. Student scores on these items were compared using ANOVA. No significant group differences, at the a = .05 level, were found with respect to studentsí potential graph-as-picture misconceptions. Three additional items, subsumed under the interpretation component of the Function Test, were used to assess studentsí potential slope vs. height misconceptions. Student scores on these items were also compared using ANOVA. Results from the ANOVA revealed that the experimental groupís mean slope vs. height score was significantly higher, at the a = .05 level, than the control groupís mean slope vs. height score.
To analyze the Function Test posttest data, the same statistical procedures were used as those used to compare the experimental and control groupís Function Test pretest scores. The MANOVA and ANOVA results revealed significant differences, at the a = .05 level, in favor of the experimental group, with respect to their conceptual understanding of functions. The experimental groupís mean scores were significantly higher than the control groupís mean scores on all four components of the Function Test, indicating that the use of CBL activities to investigate functions in an active learning environment has the potential to significantly increase studentsí understanding of the function concept.
To investigate whether students in the CBL group would be better able to overcome graph-as-picture misconceptions than students in a graphing calculator only curriculum, an ANOVA was performed on the graph-as-picture pretest to posttest gain scores for the two groups. Results from the ANOVA indicated that the experimental group had a significantly higher mean gain score, at the a = .001 level, than the control group. This finding led to the conclusion that the CBL activities were particularly effective at improving the experimental groupís ability to overcome graph-as-picture misconceptions.
To investigate whether students in the CBL group would be better able to overcome slope vs. height misconceptions than students in a graphing calculator only curriculum, an ANCOVA was performed on the slope vs. height pretest to posttest gain scores for the two groups, with pretest slope vs. height scores as the covariate. Results from the ANCOVA indicated that there was no significant difference, at the a = .05 level, between the experimental and control group adjusted means, indicating that, although students in the CBL group made improvements with respect to overcoming slope vs. height misconceptions, the CBL students were not able to overcome slope vs. height misconceptions significantly better than the students in a TI-83 only college algebra curriculum.
Qualitative data obtained from the College Algebra Questionnaire and the student interviews were analyzed using the constant comparative technique. An analysis of student responses from these instruments led to the following conclusions: (1) the experimental and control subjects were characterized by different profiles of functional understanding with the experimental subjects demonstrating a more intuitive understanding of functional relationships than the control subjects, and (2) the experimental subjects were more adept at qualitatively interpreting distance-time graphs than the control subjects, and viewed graphs of functions more globally.
The findings of this study indicate that CBL activities may aid students in constructing appropriate webs of related concepts and promote studentsí structural understanding of the function concept and ability to qualitatively interpret graphs. As students use CBL activities to explore and investigate functional behavior in a realistic, non-contrived problem solving environment, the potential exists for powerful intuitive understanding to take place. However, the effectiveness of the CBL activities depends on the extent to which they are employed and the nature of the activities. The findings of this study suggest that in order to promote conceptual change, CBL activities should emphasize the following: (a) the development of a structural, rather than a procedural, conception of functions, (b) the qualitative interpretation of graphs, and (c) student communication. The evidence collected in this study further suggests that CBL activities will be more effective at engendering conceptual change if the activities are purposefully designed to build on studentsí prior knowledge and encourage students to challenge their beliefs about functions and graphs.
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