We are currently coding and analyzing data from year two.  The following are descriptions of preliminary results from first year data.

For more information contact Sharon McCrone or Tami Martin.

Preliminary Findings Related to Students' Beliefs:

1) Some students rely on context to determine truth value of conditional statements and their variations.
2) Many students, especially those in non-honors sections, believe that several examples are sufficient to justify a general statement.
3) Many students believe that general proofs are only valid for the accompanying diagram, even if the proof does not depend on specific features of the diagram.
4) Students in the non-honors sections never developed a sense that proof was valuable or necessary.
5) We have developed a framework for characterizing student beliefs about proof, which is an expansion and modification of the six principles of proof understanding propsed by Dreyfus and Hadas (1987).

Preliminary Findings Related to Students' Proof Construction Ability:

1) Students from all four classes performed most poorly on open ended local deductions. Students in all classes also had difficulty writing open-ended proofs. (This echos the findings of Senk (1983) and others.)
2) Students in non-honors classes had more difficulty with the non-analytic proof item than the analytic proofs. Conversely, students in honors classes had more difficulty with the analytic proof items than the non-analytic proof.
3) Overall, the honors students peformed better in the Proof Construction Assessment than the non-honors students.
4) From interviews, we found that students had different criteria for evaluating proofs. In three of the four classes, students primarily focused on factors such as: number of steps and format. In two of the classes (honors classes), focus students also mentioned creativity. elegance, and logical reasoning as additional criteria for evaluating proof.

Preliminary Findings Related to Classroom Microculture and Teachers' Pedagogical Choices:

1) Teachers in honors-level classes included more discovery investigations in their classroom activities than the teachers of the non-honors classes.
2) Teachers in honors-level classes also involved students in critiquing each other's proofs. Their classes used proofs as a basis for building and justifying relationships in a geometric system. Student proof critiques were rare in non-honors level classes. In fact, proofs were treated as exercises in the sense that if the teacher modeled one example, the students were expected to write several similar proofs modeled after the given example.
3) One of the honors-level teachers stressed the difference between "elegant" and "brute force"
proofs.
4) Teachers of the honors-level classes also encouraged students to look back on complete proofs to determine if there were other valid methods for proving the same relationship or if steps could be eliminated to create more concise proofs. They also modeled this process during class. In one of the non-honors classes, the teacher generally led students to one particular method for a proof.