The Relationship Between Meta-Representational Competency Skills and Problem Solving Outcomes by College Students
Jones, Jennifer L.
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The study of meta-representational competency (MRC) is of emerging interest in the field of mathematics education due to the foundational role that problem solving abilities play in the study of mathematics. While previous research in this area has focused on the invention MRC skills young children exhibit during problem solving tasks, little is known about the invention MRC skills exhibited by adult learners, or about other types of MRC skills. Furthermore, to date there has been little research related to the impact of meta-representational competency skill usage on problem solving outcomes. This study fill the void in research by investigating both the invention and functioning MRC skills exhibited by college aged students, as well as an analysis of which specific MRC components are significant predictors of academic achievement. One hundred and one college students from two institutions completed a series of problem solving tasks designed to investigate the MRC that students exhibit and its impact on problem solving outcomes. These tasks consisted of two extended word problems from the New York State Common Core Curriculum that required a visual representation to solve. Students' solutions to these word problems, as well as elements of students' visual representations, were coded using the qualitative technique of analytic induction proposed by Lincoln and Guba (1985). In all, the participants in this study exhibited five categories of invention MRC skills. They were (1) representation of motion/direction, (2) representation of key situational features, (3) representation of inferred details, (4) inclusion of textual explanations and (5) inclusion of mathematics expressions/ equations. These categories were created based on evidence of specific problem solving actions exhibited by students during the problem-solving tasks. A taxonomy regarding the functioning MRC skills exhibited by participants was also created. The results of this study suggest a five part functioning MRC taxonomy that describes student-generated responses to the utility of their visual representation on problem solving efforts. They included the ability to (1) visualize key components, (2) determine the relationship between time and distance, (3) aid in comprehension of the problem or procedure, (4) visually display the answer, (5) prevent errors in problem solving and (6) not use a visual to problem solve. Results of a series of regression analyses suggest that specific invention and functioning meta-representational competency skills can be used to predict students' problem-solving achievement. Students' use of numerical labels to differentiate and describe key relationships was a significant positive predictor of problem solving success. Students' inclusion of spatial details from the text into the visual representation to demonstrate key relationships between variables was also a significant positive predictor of problem solving success. Students' use of mathematical equations and expressions has mixed results. While evidence was significant, some elements were positive predictors of academic achievement while others were negative predictors. Finally, a students' inability to articulate why a visual representation was helpful, or their failure to use a visual representation while problem solving was a significant negative predictor of problem solving achievement. This study suggests that meta-representational competency skills vary among college students. Though all the participants from the study were enrolled in a college program and were from a fairly uniform geographic region, the nature and sophistication of invention and functioning MRC skills they used was diverse. Through regression analyses, it was determined that a number of these skills are positive predictors of academic achievement. These results suggest that further enhancement of these skills may be an important instructional avenue for mathematics educators to consider in curricular design and instructional development.