Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student's proof scheme, which \"constitutes ascertaining and persuading for that person\" (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove?
Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general.
","abstract_html":"<p>Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student's proof scheme, which "constitutes ascertaining and persuading for that person" (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove?</p> <p>Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general.</p>","abstract_has_math":false,"creators":["Haverhals, Nicolas John"],"institution":"University of Montana","degree_name":"Doctor of Philosophy (PhD)","degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-01T08:00:00Z","date_published":"2011-01-01T08:00:00Z","updated_at":"2026-07-24T03:14:06Z","subjects":["Mathematical proof","Mathematics education","Undergraduate mathematics education"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.umt.edu/etd/923","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Haverhals, Nicolas John"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["University of Montana"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematical proof","Mathematics education","Undergraduate mathematics education"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.umt.edu/etd/923"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student's proof scheme, which \"constitutes ascertaining and persuading for that person\" (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove?
Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general.
"]},{"key":"dc:title","label":"Title","values":["STUDENTS' DEVELOPMENT IN PROOF: A LONGITUDINAL STUDY"]}]}],"canonical_facts":{"dc:creator":["Haverhals, Nicolas John"],"dc:description.abstract":["Despite importance of teaching proof in any undergraduate mathematics program, many students have difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, nine undergraduate students were each interviewed once every two weeks over the course of an academic year. During each interview, the students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it came about. The second framework addressed identifying each student's proof scheme, which \"constitutes ascertaining and persuading for that person\" (Harel & Sowder, 1998). Using these structures as a guide, the question I sought to answer is: What, if any, identifiable paths do students go through while learning to prove?
Unfortunately, the data from this study failed to demonstrate any identifiable path that was common to all participants. In fact, only a single student made clear progress as judged by the criteria laid out at the beginning of this study. Specifically, the way she attempted proofs changed which was reflected in a greater tendency to use a particular proof type as time passed: semantic. Of the other students, six entered the study with a fairly mature view of proof that remained unchanged and thus had little progress to make relative to the frameworks used in the study. These students were also generally successful with the proofs they attempted and were more likely to use semantic proofs. The remaining two students were generally less successful and used semantic proofs rarely. This seems to imply that as students become more comfortable with proof, they become inclined toward the semantic proof type and this coincides with becoming more successful with proof in general.
"],"dc:identifier":["https://scholarworks.umt.edu/etd/923"],"dc:publisher":["University of Montana"],"dc:subject":["Mathematical proof","Mathematics education","Undergraduate mathematics education"],"dc:title":["STUDENTS' DEVELOPMENT IN PROOF: A LONGITUDINAL STUDY"],"dc:type":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:14:06Z"}