http://articles.latimes.com/1998/feb/06/local/me-16064

If only more professional mathematicians would publicly refute Simon’s rather hyperbolic and biased piece.

]]>As the possessor of a math degree who has also worked on research, I can say my experience with high school two-column proof not only was useless but actively harmful. It’s not anything like writing or thinking through a real proof.

Note also two column proof is a US-only thing.

]]>Thus, I mention the de Villiers text because it’s an excellent examination of how proof can serve many other purposes. If I have to demonstrate that these other purposes fit your definition of cognitive dissonance in order to have my point /suggestion matter, I’ll pass. I think it’s not hard to do so, but my sense is that you’re more worried about what I take it struck you as a criticism rather than an attempt to ensure that we don’t reinforce a too-narrow notion of proof. Sometimes, the larger context requires that we treat our particular concerns with due caution.

Note that I don’t disagree with the idea that the “different” prompt you offered is the one most in keeping with the spirit of mathematical exploration, discovery, etc., and one that has a somewhat better chance of leading students to more than mere going through the motions in order to please the teacher. Giving the end of the story away before anyone has had a chance to even think about possibilities usually seems like a bad idea to me, in and out of mathematics classrooms. In that sense, doubt is a very useful tool. But I really don’t want to feed the conservative/reactionary monster that claims that verification of (almost always already-known facts and theorems) is the sole reason for doing proofs.

]]>Name my priors: I think introducing proofs with numbers and/or counting is easier for kids to understand, though I can’t prove that.

Thumbs up. I’m a big fan of finding the sum and product of three consecutive whole numbers. Some truly surprising properties (to me, and several other math teachers) arise.

**MPG**:

I think restricting matters to “doubt” severely under-estimates the number and nature of the headaches

I acknowledged the breadth of purposes the proof act satisfies. But this headache metaphor is explicitly tied to Piagetian ideas of cognitive dissonance — order arising unexpectedly from disorder or disorder arising unexpectedly from order. While “proof as verification” falls along these lines, you’d have to clarify how the other approaches create this dissonance, if you’re posing them as headaches.

**Scott Farrar**:

I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool A losing sight of the very reason for its development. So, we lay a hook by presenting concept B first.

Big. Or as Daniel Willingham writes:

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question.

Shoot, it’s worth putting these two comments side by side in their own post.

**Mr Ruppell**

We almost always do an always-sometimes-never to motivate a particular proof. Mine are usually teacher-generated (here’s a list of 5 statements about rhombi – tell me if they are always, sometimes, or never true). Then we prove the always and the never.

Nice. Useful structure.

**Joey Kelly**:

I think the headache for “simplifying sums of rational expressions with unlike denominators” could be a graph. If I type that equation into desmos, I notice a vertical asymptote at x=6, which makes sense. But I also notice that there is not a vertical asymptote at x=-1, and I thought there would be. Plus there is an x-intercept at x=4, and I have no idea why. This graph creates a need for explanation, and simplifying sums of rational expressions with unlike denominators provides the explanation/aspirin.

Every headache so far has generated from an easier, more concrete act than the one we’d like to teach. Cognitive dissonance is easier to generate if students first feel confidence in some belief or skill. So I’m wondering if a student who doesn’t yet know how to simplify rational expressions would feel sufficiently confident in *graphing* those expressions to make that the basis for dissonance.

[…] Scott Farrar, on my last post on motivating for proof: […]

]]>1. Proof as Explanation

2. Proof as Discovery

3. Proof as Verification (this seems to come closest to the “doubt” headache)

4. Proof as Challenge

5. Proof as Systemization

Let’s not restrict mathematics or the role of proof therein to dealing with doubt.

]]>http://mctownsley.blogspot.com/2010/04/proofs-triangle-congruence-and-reality.html

]]>I can’t resist linking to my (brief) thoughts on CPCTC — can we please remove this from the school math lexicon?

https://iheartgeo.wordpress.com/2012/12/20/stop-saying-cpctc/

]]>I would like to make this a student-driven set of conjectures in the future…

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