Two-Column Proofs Should be Three-Column Proofs

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:

  1. a is true.
  2. a –> b is true because of a postulate, definition, or some other proof we have already done.
  3. Therefore, b must be true.

However, a very common line that might appear in my students’ two-column proofs is:

  1. Statement:  m<a + m<b + m<c = 180.
  2. 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:

  1. Observation:  a, b, and c are the interior angles of a triangle (i.e. a is true)
  2. Postulate:  The sum of interior angles of a triangle is 180 degrees (i.e. a –> b is true)
  3. Conclusion:  m<a + m<b + m<c = 180 (i.e. b must be true).
Two-column proofs have it all wrong.  Not only do they not represent valid deductive thought, but they are formulaic.  The fact that my students like them shows how much they prefer to rely on formulaic thinking rather than their own thinking.
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One Response to Two-Column Proofs Should be Three-Column Proofs

  1. Hilary says:

    You should post this on the SWN, Matt. There’s actually a conversation about proofs already going.

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