Comparing bundles and associated intentions of expert and novice provers during the process of proving
Open Access
- Author:
- Karunakaran, Shiv
- Graduate Program:
- Curriculum and Instruction
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 18, 2014
- Committee Members:
- Mary Kathleen Heid, Committee Chair/Co-Chair
Glendon Wilbur Blume, Committee Chair/Co-Chair
Rose Mary Zbiek, Committee Member
Yakov B Pesin, Special Member - Keywords:
- Proving
Mathematics Education
Mathematical Knowledge
Undergraduate Mathematics
Graduate Mathematics
Proof Instruction - Abstract:
- The broad purpose of this study, which uses grounded theory methods, is to compare the processes of proving of expert and novice provers during their work on proving mathematical statements. In this study, the process of proving is analyzed using the constructs of resources, actions, bundles, and intentions. An individual’s mathematical knowledge is cast as a network of relations between connected elements of mathematics, such as properties, objects, procedures, definitions, and theorems. These elements of mathematics can be referred to as resources to which the individual has access during the act of doing mathematics. Once individuals have called on these resources, they reason about them or perform certain actions on or with these resources. Some of these actions can be to ask a question, to construct an example, to perform a form of reasoning (e.g., inductive), or to apply a theorem. However, understanding the actions of an individual requires examining the intentions associated with the same actions (Skovsmose, 2005). Bundles are defined as subsections of the proving process that consist of groups of actions and resources that are clustered together by an identifiable intention. Five undergraduate students of mathematics and five advanced graduate students of mathematics were identically asked to validate or refute five mathematical statements. All ten participants were interviewed individually as they attempted the five tasks. The interviews were video– and audio– recorded and then were transcribed for analysis. The data was analyzed using ground theory methods of open coding, axial coding, selective coding and constant comparison. The process of coding and constant comparison resulted in two claims related to bundles involving the generation and use of counterexamples and examples, and a third claim related to the sequencing of bundles. Both the expert and novice provers recognized the implication of finding a counterexample on the validity of the task statement. The expert provers persevered in the search for a counterexample regardless of whether they believed one exists or not. However, the novice provers did not persevere in the search for a counterexample. Both expert and novice provers used constructed examples (i.e., examples that are generated using the conditions present in the mathematical statement) as a resource that satisfy conditions present in the statement to be proved. In addition, expert provers connected the task context to a previously known illustrative example (i.e., examples that are previously known to the prover and may not be completely compatible with the conditions in the task statement). They then used the known properties of these illustrative examples to shed light on the proving process for the task at hand. Novice provers did not call upon illustrative examples. A third claim is presented related to how the expert and novice provers sequenced the bundles. Expert provers may knowingly interrupt the deductive logic in the sequencing of bundles. Novice provers were not willing to allow such interruptions. The findings of the study show great similarity in the proving behavior of the expert provers interviewed, but also great differences when compared to the proving behavior of the novice provers. As such, the findings of this study may offer some characteristics of expert proving behavior.