Understanding and Constructing Proofs: A Design Experiment Kerry McKee Annie Selden New Mexico State University Department of Mathematical Sciences
[email protected] [email protected] April 4, 2009 Southwestern Section of the MAA
Joint work with John Selden Funded by Educational Advancement Foundation
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Outline of our talk • Description of the design experiment • Distinctions we have been making on types of proofs • Our perspective on affect • Our view of behavioral schemas or habits of mind • Several specific examples of student work • Conclusion 3
The Design Experiment • A one-semester, 3-credit, special topics course for prospective and beginning math graduate students. • Meets 2 times/week for 75 min. • Purpose is to teach proof construction
• Significant modification of the R. L. Moore Method (Jones, 1977: Mahavier, 1999) 4
• Notes with statements of theorems, definitions, requests for examples, but no proofs and only minimal explanations. • No lectures. • Students work outside of class and present their proofs at the blackboard. • We read and check each proof, “thinking aloud” so students can see what we are checking. 5
• We offer, sometimes extensive, criticism and advice • Halfway through the course, we have students validate each other’s proofs • Course has practical value because professors assess students’ understanding by asking them to prove theorems
• We are now in our 4th funded iteration of the course. 6
• There are two versions of the course. Either or both can be taken for credit. • One version covers some basic ideas of sets, functions, real analysis, and semigroups.
• The other covers sets, functions, some real analysis, and topology.
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• Our course is consistent with a constructivist point of view in that we try to maximize students’ proof constructing experiences.
• It is also somewhat Vygotskian in that we represent to the students how the math community writes proofs. 8
• Everything is video recorded: both the classes and our planning sessions. • Field notes are taken. • This information is analyzed in planning sessions between class meetings in order to influence students’ learning trajectories.
• Later, we often reanalyze the data for research purposes. 9
Our perspective on affect Traditionally, many math ed researchers have considered affect as consisting of beliefs, attitudes, and emotions, which are of increasing intensity and decreasing stability with emotions being the most intense and changing the most rapidly. (McLeod, 1992) 10
To these three, DeBellis & Goldin (1999) have added a fourth, namely, values, e.g., mathematical integrity and mathematical intimacy. We now add a fifth, feelings, especially nonemotional cognitive feelings, such as the feeling that an argument is correct or that one is “on the right track.” 11
Difficulty of obtaining information on feelings Only occasionally does one get first person reports of feelings. During “think aloud” problem solving, reports are likely to be about the problem solver’s immediate actions, except in the case of very intense feelings.
Since it is rare that one obtains direct information about feelings, one often has to infer feelings from behavior. 12
Distinctions we make in regard to proofs We distinguish two aspects or parts of a proof: • the formal-rhetorical part and • the problem-centered part. 13
The formal-rhetorical part of a proof is the part that one can write based only on logic, definitions, and sometimes theorems, without recourse to conceptual understanding, intuition, or genuine problem solving. Students sometimes call this the “set-up.” We call the remainder of the proof the problem-centered part and it does require conceptual understanding and genuine problem solving. (Selden & Selden, 2009) 14
Example of the “set up” Theorem. The identify function on the reals is continuous. Proof: Let f be the identity function. Let a be a real number. Suppose e>0. Let d = ___ . Note d>0. Let x be a real number. Suppose |x – a|
