Assessing Understanding of the Concept of Function: A Study Comparing Prospective Secondary Mathematics Teachers' Responses to Multiple-choice and Constructed-response Items

Open Access
Feeley, Susan Jane
Graduate Program:
Curriculum and Instruction
Doctor of Philosophy
Document Type:
Date of Defense:
June 27, 2013
Committee Members:
  • Glendon Wilbur Blume, Dissertation Advisor
  • Mary Kathleen Heid, Committee Chair
  • Nigel David Higson, Committee Member
  • Rose Mary Zbiek, Special Member
  • Understanding
  • Function
  • Secondary Mathematics Teachers
  • Assessment
  • Multiple-Choice Items
  • Constructed-Response Items
  • Sierpinska
  • Indicators of Understanding
  • Certification
The purpose of this study was to determine whether multiple-choice and constructed-response items assessed prospective secondary mathematics teachers’ understanding of the concept of function. The conceptual framework for the study was the Dreyfus and Eisenberg (1982) Function Block. The theoretical framework was Sierpinska’s (1992, 1994) Indicators of Mathematical Understanding: identification, discrimination, generalization, and synthesis. Fourteen prospective secondary mathematics teachers were recruited, nine from a land-grant university and five from a private college. Three sets of items identified as assessing the concept of function were administered during the study. Two sets were taken from items released by the Educational Testing Service PRAXIS SeriesTM, 21 multiple-choice items and 5 constructed-response items. A third set, the researcher-developed constructed-response items (RD CR), was based on a framework developed by the investigator. Four experts in mathematics and mathematics education reviewed the framework as appropriate for creating items and recommended the inclusion of six RD CR items for the study. Participants met with the researcher for four sessions to respond to the three sets of items in a problem-solving interview setting. They shared their process of solution aloud as they solved each item. Audio and video recordings were made of each session and then transcribed for purposes of analysis. Both quantitative and qualitative methodologies were used to analyze data: quantitative (correctness of response) and qualitative (foci of items, correctness of mathematical reasoning, mathematical reasoning involving the function concept, extended Function Block, strategies, and Sierpinska’s Indicators of Mathematical Understanding). The investigator’s findings affirm the potential of the constructed-response items, particularly the RD CR items, to assess a subject’s understanding of the concept of function as characterized by Sierpinska’s Indicators of Mathematical Understanding, particularly generalization and synthesis. The investigator questions the potential of the multiple-choice items to assess accurately a subject’s understanding of the concept of function because of the implementation of multiple-choice test-taking strategies (matching, elimination, substitution, etc.) and the use of mathematical reasoning not involving the concept of function.