https://mrchasemath.wordpress.com/2017/01/11/proving-identities-whats-your-philosophy/

]]>I feel like most of the apparent disagreement about this post comes from people talking about different things.

You, John, and many others have offered thoughtful criticisms about this work as a *proof*. In particular, your post about this kind of problem is spot on.

I, on the other hand, am evaluating this work as a response to an exam item. My point is not that this is a good proof (it isn’t), but that it is not invalid for the reason officially cited, and thus, it is as valid as the other responses that were awarded full credit (none of which would meet any commenter’s criteria of a good proof).

]]>Thanks for the interesting post and for sharing your thoughts. I’ve read some of the comments here and I find them very interesting and thought provoking. However, I have to admit that I disagree with you, and I agree with John Chase quite strongly. In fact, I work very hard to break students of writing exactly this type of argument. If they had put “if and only if” between each line I’d feel better, but even still it isn’t great. The xâ‰ 2 bit makes it messy. The first two expressions are equal for all xâ‰ 2, but the last two are equal for all x. I’d much rather see a string of equalities that starts with the LHS and ends with the RHS.

I wrote a (now that I reread it seven years later, somewhat obnoxious) blog post about this in 2009 https://divisbyzero.com/2009/12/27/showing-two-expressions-are-equal-stop-the-madness/. (Interestingly, John Chase’s father Gene Chase commented on the post!!) In fact, your example is a better example of what I don’t like than the example that I gave in my blog post.

As a final comment, I also don’t like the question. I don’t like the “where” construction. It isn’t terrible, but I try to discourage students from using it. It is too easy for them too confuse themselves with how it fits in with the quantifiers. I’d much rather see “if xâ‰ 2, then…” or “for all xâ‰ 2…”

Dave

]]>Thus, starting with the equation is *absolutely* the same as “starting with what you want to prove”. Experienced mathematicians sometimes do this with both equalities and inequalities so that we can get an idea of how a rigorous proof will go, but it certainly does not qualify as a rigorous proof.

So, I agree with the test, both with the way it’s phrased and with the assessment of the incomplete answer. But I despair that this basic idea is not being taught in US algebra courses.

]]>“For the most part” meaning it doesn’t constitute proof. Watch:

4=5

0*4=0*5

0=0

QED.

This method of proving A=B, where you start with A=B and do operatioms to both sides and end with C=C is not valid. You are assuming what you are trying to prove, as others have told you.

To prove A=B, you need to start with A=A, do some identity operations on the right hand side, and then end up with A=B.

]]>In one, a student writes a string of equivalent expressions with no equal sign; does that student understand that the application of transitivity requires equality? In another, a student replaces 1 with (x^3 + 8) / (x^3 +8); does that student understand this is only true as long as x <> 2 ?

The truth is, none of those responses provide any more justification than the response under scrutiny here. I’m not sure why we should hold this work to such a high standard and not the others.

]]>The original response discussed up above is a little shakier to me. It is true that ak = bk => a = b for k not 0 and by this fact the reasoning in the response works, but I’m not sure ak = bk => a = b counts as common knowledge that can go without stating. Does the student understand that their reasoning relies on that fact? Do they understand this is true for multiplication but not for instance squaring? Have they considered a 0 denominator? I have enough doubts I’d probably go with 1 point myself but if someone else at the grading table felt strongly it should get 2 I wouldn’t fight it. ]]>

I’m not sure how I would score it, however. I might let it slide and just add a written comment. Depends on the in-class conversation we’ve already had about this issue.

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