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59

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 students find induction difficult: Many students don't know what proof is. Many students don't realise it's actually about statements. Many students don't ...


50

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 picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching." (stop here ...


41

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 this question is purposefully asked here on Math Educators StackExchange. As to a proof: Given $a>b$, subtract $a$ from both sides: $0 > b-a$. Next, ...


30

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 happen with limits and we can't just pass things through to the other side. It's also a good way to jumpstart a discussion of definitions: what's a proper way to ...


30

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 to common errors like using P(n) to prove P(n), or to thinking the method is circular because we assume P(n) to prove P(n).


28

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) Secondary school students' understanding of mathematical induction: Structural characteristics and the process of proof construction. International Journal of ...


27

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 think so into a couple of points. Hard proofs are no only hard to create but also hard to learn on your own. Have you tried learning a complex proof on your ...


25

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 precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of ...


24

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 sequence might prove interesting: Find the first ten derivatives of $x\cdot e^x$. What seems to be the formula for the $n$th derivative of $x\cdot e^x$?...


24

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” was. By going to seminars, reading papers, and talking to other graduate students, I gradually began to catch on. Within any field, there are certain ...


23

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 divided into $n-2$ triangles.


22

The Curry Paradox is a classic. This animation resolves it:


21

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, than Euclidean Geometry is a great way to implement this: Visuality Euclidean Geometry deals with objects that can be easily visualized. It can be properly ...


21

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 prove that a collection $S$ of finite structures (such as binary trees) satisfy a property $P$, then all you have to do is to show that for any arbitrary ...


21

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 unnecessarily complicated proofs are skipped, but not often enough to my taste. But there is a point to proving these "big theorems". The proofs contain new ...


19

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 answer and the ensuing discussion.) The famous proof that $\sqrt{2}$ is irrational. (I don't particularly like this one---there are better ways of ...


19

If $\gcd(a,b)=1$, there exists a multiplicative inverse for $a$ modulo $b$. (Otherwise, look at the $b-1$ multiples of $a$, namely $a,2a,3a,\dots,(b-1)a$. They must fall into congruence classes that aren't 0 or 1, but there are only $b-2$ of those.) $R(3,3)\leq 6$, and other Ramsey-style arguments Give any domino tiling of a $6\times 6$ checkerboard, there ...


19

I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove. Furthermore, seeing several techniques ...


18

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 original printing of Polyominoes (1965) by Solomon Golomb: T R O M I N O E S It is impossible to cover an $8 \times 8$ board entirely with trominoes, ...


18

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 $\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$. A great follow up is to assume you have a sequence $\langle a_k\rangle$ that satisfies $\sum_{k=1}^n a_k^3 = \left(\sum_{k=1}^n a_k\right)^2$ and ...


18

This collection of essays: Reuben Hersh: Experiencing Mathematics: What do we do, when we do mathematics?, Amer. Math. Soc., 2014 contains, among other topics, also lots of excellent discussion of the problems you mention. I highly recommend it, some of the articles are even available online if you google for them.


18

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 with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't ...


17

One example that I recently came across was to prove that the function $$ f(x)=\begin{cases} e^{-1/x},&x>0\\ 0,&x\leq0 \end{cases} $$ is smooth (i.e. $f\in\mathcal{C}^\infty$). Here a link to a proof online.


17

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, I think that question is ill-posed, because the goal state isn't clearly defined (effective at what?) and so there's no way to measure whatever it is you want ...


17

Perhaps not pointing out that the obvious steps are obvious but that the insights are insights. I believe students don't feel bad for not seeing the "magic steps" by themselves, so pointing out that those are hard is not a problem. The opposite is what you mention: they would feel bad for not seeing the obvious. Hence, only treat the difficult as difficult.


17

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 material it demonstrates. An incomplete proof by contradiction could correctly get the main idea but omit some of the technical material needed to make the ...


16

Think of a maze, like the ones you can find in kids' books that you trace through with a pencil. This may sound like a silly question, but what's the definition of a maze? If you ask your students this question, I suspect you'll get some puzzled reactions. I mean, you know a maze when you see one, right? But that very intuition supports the inclusion of "...


16

For a very basic example, how about proving that 59549121058965346178 can be expressed as a product of primes? It is much easier to prove the stronger result that every positive integer can be expressed as a product of primes.


16

Be careful here. You're teetering close to the edge of checking whether the student has guessed your solution to the problem rather than whether the student has solved the problem. This is easy to do, especially in cases like this where a problem is constructed with the specific purpose of exercising a particular technique (eg, your solution). This ...


16

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 seen induction as an algebra exercise - for example, summing the first $n$ squares. Those examples are tedious and work badly. Students have trouble with the ...


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