LOOKING FORWARD TO Making Math Real: Geometry
Success with Deductive Proof: Synthesizing the Perceptual Big Picture
Although the ability to perceptually synthesize the complete whole picture from partially given information is needed in all geometry, it is most particularly required for success with deductive proof. With deductive proof, some partial information is provided from which the student must prove a specified relationship such as congruence or similarity. To successfully structure a deductive proof, students must be able to generate the complete whole picture of the proof from the partially given information prior to picking up their pencil. In the correct incremental teaching of geometry, it is necessary to help the student synthesize the complete big picture from the partially given information, then mentally hold that synthesized perceptual picture as the basis from which to structure each step of the deductive proof. This means the student has already “proven” the relationship prior to writing anything down. It is in this manner students can successfully manage the challenges of structuring and organizing deductive proofs. The students’ ability to synthesize the big picture from a partial picture requires significant developmental practice and experience, and is therefore not indicated until well into the high school year, and not immediately at the beginning of the year as it is typically presented in textbooks. Presenting deductive proof prior to students’ development of synthesizing the perceptual big picture can make it unnecessarily difficult for students to be successful with deductive proof, which is most unfortunate since the developmental outcome of successful proof is directly related to significant increases in executive processes such as organization, structure, sustaining focus, sequential processing, cause and effect, inhibiting distractors, and mental flexibility.