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81 votes
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Why are induction proofs so challenging for students?

The following list comes from a combination of reading various research articles and my own experience helping students in my Maths Learning Centre for the last seven years. Some reasons why ...
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55 votes

How to explain Monty Hall problem when they just don't get it

For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me. Rather than talk about probabilities as fractions, I explain it this way: "If you ...
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40 votes
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Why do inequalities flip signs?

I'm slightly concerned that Is there a mathematical reason (like a proof) of why this happens? is a purely mathematical question, but since you write "we just warn students" I will assume that ...
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36 votes

Why are induction proofs so challenging for students?

In my experience, the biggest issue is that students don't have a clear grasp of quantifiers, so they don't see the distinction between "for all n P(n)" and "consider an n such that P(n)". This leads ...
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33 votes

How can I explain why we need proofs to someone who has no experience in mathematical thinking?

More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot ...
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32 votes

I want a "true" proof by contradiction of an implication P => Q

As you've noticed, there are (at least) three potential ways of proving an implication $p \Rightarrow q$: Assume $p$, and conclude $q$. Assume $\neg q$, and conclude $\neg p$. Assume both $p$ and $\...
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31 votes

Good, simple examples of induction?

The problem with induction proofs is that too often the problem is given by "Prove that..." After a few examples and explanations of induction, if the students know elementary calculus, the following ...
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31 votes
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Impressive examples where a "proof by picture" goes wrong

This one can be presented to students at any level, really, although the way to explain "repeat to infinity" will certainly change for your audience. It can be used to teach them that weird things ...
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30 votes

Why are induction proofs so challenging for students?

There is a fair amount of research on students' understanding of (and difficulties with) proof by induction. Some good places to start: Palla, M., Potari, D., and Spyrou, Panagiotis. (2012) ...
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29 votes
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How to explain Monty Hall problem when they just don't get it

Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is ...
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28 votes

What am I supposed to be learning with long proofs of the main theorems in class?

Hmm apparently I will be the dissenter here. I think that long proofs taught in lectures are very much a good thing. This is particularly true for hard proofs. I will try and split the reasons why I ...
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27 votes
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Explaining why volume of cone is a third of cylinder

This is an experiment which can lead you to guess that the volume of a cone is approximately $\frac{1}{3}$ the volume of a cylinder with the same base and height. It is not a proof in any sense of ...
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26 votes

Good, simple examples of induction?

Two more examples. Proving that a $2^n \times 2^n$ chessboard with a single square missing can be covered using L-shaped (made out of three squares) pieces. Proving that a convex $n$-gon can be ...
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25 votes

Why are induction proofs so challenging for students?

For CS students specifically, there is another approach that would work better than the usual way induction is taught, namely by teaching structural induction, which goes like this: If you want to ...
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24 votes

What do mathematicians call a proof?

From Bill Thurston: When I started as a graduate student at Berkeley, I had trouble imagining how I could “prove” a new and interesting mathematical theorem. I didn’t really understand what a “proof” ...
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24 votes

Is Euclid dead? or Should Euclidean geometry be taught to high school students?

I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, ...
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23 votes
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Good examples of proof by contradiction?

Here are some good examples of proof by contradiction: Euclid's proof of the infinitude of the primes. (Edit: There are some issues with this example, both historical and pedagogical. See Mike F.'s ...
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23 votes

Impressive examples where a "proof by picture" goes wrong

The Curry Paradox is a classic. This animation resolves it:
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23 votes
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What am I supposed to be learning with long proofs of the main theorems in class?

I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes ...
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23 votes

When do college students learn rigorous proofs?

From day one. In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus ...
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22 votes

Good, simple examples of induction?

Historical remark. Though it has already been mentioned in an answer, I can't resist posting a bit more about the following wonderful example of a proof by induction. I quote directly from the ...
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22 votes

Why are induction proofs so challenging for students?

Lots of good answers here (I've upvoted many). I'm won't try to add to the discussion about why induction is hard, but I can suggest some approaches that have helped some of my students. Many have ...
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21 votes

Is there a good age/level to start learning mathematical proofs?

Not to be annoying, but what is a proof? Here's my best take: a proof is an explanation that could convince all conceivable skeptics. Proof, then, is absolutely crucial at all levels of math ...
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21 votes

Good, simple examples of induction?

Proving DeMorgan's Laws for $n$ sets. I like this example because it requires the $n=2$ case in the induction step. It's common to have students prove that $\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\...
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21 votes
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Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?

The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century. You ask about its effectiveness; unfortunately,...
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21 votes
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Inability to work with an arbitrary mathematical object

I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?" When you say "Suppose $A⊆R$ is nonempty ...
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  • 19.1k
21 votes

Why do some linear algebra courses focus on matrices rather than linear maps?

Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of ...
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  • 1,873
20 votes

How to explain Monty Hall problem when they just don't get it

There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals ...
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20 votes

Why are induction proofs so challenging for students?

I think the main problem students have with induction proofs is that the ordinary direct proof works by reducing a statement with unknown truth value to one that is known as true. The bulk of an ...
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20 votes
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Should students be given partial scores when they gave an incomplete proof by contradiction?

There's no abstract reason that an imperfect proof by contradiction should categorically fail to get credit. A proof should generally get partial credit based on how much knowledge of the relevant ...
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