Ahmed Benkhalti

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs. In order to understand where students are “coming from” and to help them learn to construct proofs. We have analyzed students’ examination papers from several such courses. We have identified process, rather than mathematical content, difficulties such as not unpacking the conclusion, and not using definitions correctly.

We present the results of an analysis of undergraduate students’ examination papers from an IBL transition-to-proof course. Students’ papers were considered from the point of view of their actions (mental, as well as physical), instead of their possible misconceptions. In doing so, we identified process, rather than mathematical content, difficulties, and this has resulted in the detection of both beneficial actions and detrimental actions that students often take.

Thus far, we have identified the following categories of difficulties: omitting beneficial actions; taking detrimental actions; inadequate proof framework (e.g., not unpacking the conclusion); mathematical syntax errors; wrong or improperly used definitions; misuse of logic; insufficient warrant; assumption of all or part of the conclusion; extraneous statements; assumption of the negation of a previously established fact; difficulties with proof by contradiction; inappropriately mimicking a prior proof; mathematical syntax errors, failure to use cases when appropriate; incorrect deduction; assertion of an untrue result; and computational errors. Examples of some of these difficulties will be presented

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. We understand that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof framework. Generating a first-level proof framework is often easy, provided the theorem is stated in the standard “If …, then …” form. However, formulating a second-level proof framework requires knowing how to use the relevant mathematical definitions, that is, being able to put them in an operable form. For example, the definition of the inverse image of a set D under a function f:X→Y is usually given as f -1 (D) = {x∊X| f(x)∊D}. However, in constructing a proof, one needs to be able to use this in an operable way: If a∊ f -1 (D), then one can say f(a)∊D, and conversely, if f(a)∊D, then a∊ f -1(D). This may seem obvious, but it not obvious for some beginning students.

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our understanding that many community colleges may want to begin teaching such courses. We have students start by writing a proof framework which is based on the logical structure of the theorem statement and associated definitions. Often there is both a first-level and a second-level proof framework. Generating a first-level proof framework is often easy, provided the theorem is stated in the standard “If …, then …” form. However, formulating a second-level proof framework requires knowing how to use the relevant mathematical definitions, that is, being able to put them in an operable form. For example, the definition of the inverse image of a set D under a function f:X→Y is usually given as f -1 (D) = {x∊X| f(x)∊D}. However, in constructing a proof, one needs to be able to use this in an operable way: If a∊ f -1 (D), then one can say f(a)∊D, and conversely, if f(a)∊D, then a∊ f -1(D). This may seem obvious for us, but it not obvious for some beginning students.

This PowerPoint details a case study that documents the progression of one non-traditional individual’s proof-writing through a semester. We analyzed the videotapes of this individual’s one-on-one sessions working through our course notes for an inquiry-based transition-to-proof course. Our theoretical perspective informed our work with this individual and included the view that proof construction is a sequence of (mental, as well as physical) actions. It also included the use of proof frameworks as a means of initiating a written proof. This individual’s early reluctance to use proof frameworks, after an initial introduction to them, was documented, as well as her later acceptance of, and proficiency with, them. By the end of the first semester, she had developed considerable facility with both the formal-rhetorical and problem-centered parts of proofs and a sense of self-efficacy.

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to create proofs in order to prepare them for proof-based courses in their junior and senior years. It is our understanding that now many community colleges do not offer such courses, but may want to begin doing so. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on proof writing. This is the technique of writing proof frameworks, based on the logical structure of the statement of the theorem and associated definitions. Often there is both a first-level and a second-level of a proof framework. We discuss how we came to the idea of proof frameworks and demonstrate the writing of several proof frameworks.

We document Alice's progression with proof-writing over two semesters. We analyzed videotapes of her one-on-one sessions working through the course notes for our inquiry-based transition-to-proof course. Our theoretical perspective informed our work and includes the view that proof construction is a sequence of mental and physical, actions. It also includes the use of proof frameworks as a means of getting started. Alice's early reluctance to use proof frameworks, after an initial introduction to them, is documented, as well as her subsequent acceptance of and proficiency with them by the end of the real analysis section of the course notes, along with a sense of self-efficacy. However, during the second semester, upon first encountering semigroups, with which she had no prior experience, her proof writing deteriorated, as she coped with understanding the new concepts. But later, she began using proof frameworks again and regained a sense of self-efficacy.