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A few weeks ago, I started tutoring a student in Discrete Mathematics (a subject I took a year ago).

I have previously tutored both pre-calculus and calculus, but never a proof based class. I have been approaching the job by helping the student work through the homework and helping her figure out content that she couldn't follow during class.

I am afraid that I might actually be doing the student more harm than good, as Discrete Mathematics is a proof based course (the first of this kind for her), and thus I am afraid that she will begin relying on me to get intuition for problem solving and writing proofs in general.

I was thinking about transitioning to a structure where she proves everything that they proved in class again by herself (with help where needed). The idea is inspired by this blog post I found last night. Does anyone with experience tutoring a proof based math course have any advice for me?

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    $\begingroup$ You could try to help her by working with her on strategies about approaching, understanding and writing proofs. I would just like to add that while that strategy can yield great benefits, there are a few warning signs to be aware of. Cal Newport was very experienced with working in depth (his book titled Deep Work is a great read, by the way) by the time he was in college. The student you are tutoring probably does not share a similar skillset. A proof based course generates a greater toll on cognitive capacities than algorithmic courses, and that may be part of the problem. $\endgroup$
    – Mark Fantini
    Oct 29, 2017 at 23:34
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    $\begingroup$ Rather than "she proves everything that they proved in class again by herself," which strikes me as possibly boring, you might explore this nice collection: Good, simple examples of induction?. $\endgroup$
    – Joseph O'Rourke
    Oct 30, 2017 at 0:54

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I have found the best thing to do in tutoring a proof based class is to constantly ask the questions "why is that true?" or "what is the definition of that?". These questions help them build relations between definitions and highlight the connections to proofs as filling in the details relating things.

Yes, on occasion you might have to do an example of a proof technique. I remember induction being a challenge myself until I saw it in action a few times. But if they are working a problem themself, start with asking "what is the definition of the beginning object?" and "are there any theorems or propositions we can use relating to it?". Ask "what is our goal?" and "what do you think we need to show to get there?". Correct them on their mathematical grammar and clarify that they write (not just say!) their implications correctly.

Your goal as tutor is to demonstrate good proof technique and to further train them on knowing the definitions and statements of things that they need to know. Questions serve as training their guidance to when to use their statements. Yeah, they'll probably struggle a bit, but you need to help build their confidence and logical thinking skills.

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