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I taught the proof inclusion-exclusion principle to CS students yesterday. While the proof is not too long, it does involves quite a bit notations. I could feel that most students lost interests a few minutes into the proof.

When we teach a proof that takes longer than 5 minutes, how can we help students to follow? Or maybe we should just ask students to read proofs by themselves?

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    $\begingroup$ It might help to know what your learning objectives are for this proof. Are your students expected to produce similar proofs? $\endgroup$
    – TomKern
    Sep 6 at 23:44
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    $\begingroup$ Instead of proving it, can you have them program something that uses this principle? $\endgroup$
    – Sue VanHattum
    Sep 7 at 14:31
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    $\begingroup$ I only want them to know how to apply the principle to solve combinatorics problems. --- Given also your comment in @Alan's nice answer, maybe just carefully show the $n = 2, 3, 4$ cases (separately) and clearly indicate how and why the pattern continues. I haven't looked at a formal proof of this in a long time, but my guess is that the formal proof you gave is mostly about how to correctly express what is otherwise intuitively clear, so unless one of your goals is for them to learn this type of "math programming skill", perhaps skip a formal proof (assuming you have that teaching option). $\endgroup$ Sep 7 at 14:53
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    $\begingroup$ If they're getting board with the notations, you may be either writing or talking through simple definitions as a pace slower than they can comprehend. Maybe use a little shorthand or colloquialisms, and breeze through the simpler steps. Formal proofs are often tediously precise and not the way people naturally talk or think. Do make sure the formal proof and proper notations are available, but maybe make it assigned reading students can get through at their own pace. $\endgroup$
    – yesennes
    Sep 7 at 15:16
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    $\begingroup$ Tell them up front that you will be making a mistake somewhere in the proof and challenge them to point it out?? $\endgroup$
    – Flydog57
    Sep 7 at 15:54
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You want the students to work, not be an audience, because being an audience is highly ineffective. (Some of the best students do know how to work even when they are treated as an audience.)

Can the proof be broken into smaller pieces, some of which the students could do? Are there exercises that are the same as parts of the proof? Is the entire proof something that they could manage if working on it as a class, perhaps after you have given a high level idea of the proof?

Just showing or covering material does not lead to that much learning, so you might just as well not do it and instead spend that time on some activity that makes the students think. (Some exceptions might exist; some clever tricks, for example, might well be illustrated by a short and to the point proof, but hopefully with activity, too.)

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I was taught to follow the CRA model: Concrete, Representational, Abstract. That's the order you should present things.

So for instance, you could start the concept off with actual physical objects that have multiple qualities (Say, stickers on them, shape, color, etc.), and show that if you count all the qualities you overcount by everything that's in 2 categories, so you subtract those but now you have undercounted by anything that was in 3, etc.

From that you can take it to the representational on the board, with a Venn Diagram of 3 sets, pointing out the overcount,undercount/corrections.

Only at the end do you go the abstract with the terminology and the general case.

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  • $\begingroup$ Well, I did this. But it was the last abstract part (the actual proof) students get lost. $\endgroup$
    – ablmf
    Sep 7 at 6:03
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I always had fun following maths proofs, both in school and uni, but that was mostly through self-motivation/interest. I wager that most of my co-students more or less slept through the same lessons or were only kept awake because at that time (pre-internet/pre-digital) we were forced to write down whatever the teacher/prof wrote on the chalkboard.

So you're not alone and it does not seem to be a trivial endeavour.

As you wrote in a comment, you do not need your students to repeat the proofs, or come up with their own; you just want them to be more engaged in their time they sit in your classroom.

So I would suggest for you to do exactly that: engage them. Don't just present the proof, but at every single step, ask them what they think comes next. Don't just grab the one pupil that you know "groks" that stuff, but try getting a discussion going, or ask those more at risk of nodding off...

You might or might not get through the whole proof using this method, but at least you will engage them and force/encourage them to think about the individual steps. You'd need to tailor your proof so that they have a running chance to come up with good ideas of course; maybe give a few more hints when it comes to solve the crux of the problem.

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    $\begingroup$ I tried similar to what you wrote in my last class. It seemed students are engaging more. But I guess I will only know much they learned after the exam. $\endgroup$
    – ablmf
    Sep 7 at 23:47
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Several tricks:

  1. Surprise: present your theorem as surprising (i.e. nonobvious). How would I go about showing this is actually true? Then develop the line of attack to show them that this can actually be done; ideally, but not necessarily with a trick. I got more than one "wow/awesome" using this method. This is actually a good sign, because this means that students made the transition from treating math as accounting to seeing that it's actually an exciting endeavour.

  2. Intuition: this is exactly the opposite of 1., but sometimes more suitable. Explain what the theorem says and show how this is actually intuitive. Now show how to cast intuition into formal language.

  3. Processual: sometimes neither 1 nor 2 are possible. Show them the technical steps. Repeat and rinse. This needs practice. This is very local and again complementary to 1 and 2, but as students see and take similar steps again and again, they gain familiarity and thus lose fear of approaching proofs themselves, by having the "muscle memory" already entrained. This is essentially implementing von Neumann's dictum: "Young man, in mathematics you don't understand things. You just get used to them."

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