Two-column proofs, written as statement/reason, do not represent valid deductive thought. They should be thrown out of the math curriculum.
A valid deduction is supposed to follow these lines:
- a is true.
- a –> b is true because of a postulate, definition, or some other proof we have already done.
- Therefore, b must be true.
However, a very common line that might appear in my students’ two-column proofs is:
- Statement: m<a + m<b + m<c = 180.
- Reason: The sum of interior angles of a triangle is 180 degrees.
Two-column proofs have taken a three-part process and turned it into a two-part process. Essentially, the “Reason” is supposed to be the postulate, definition or conditional we have already proven, i.e. the a –> b. The “Statement” is supposed to be the conclusion we are allowed to draw, i.e. the b. What the hell happened to a being true in the first place? Without a being true in the first place, we can’t make a valid deduction.
Part of me wants my kids to write three column proofs:
- Observation: a, b, and c are the interior angles of a triangle (i.e. a is true)
- Postulate: The sum of interior angles of a triangle is 180 degrees (i.e. a –> b is true)
- Conclusion: m<a + m<b + m<c = 180 (i.e. b must be true).