# Tag Info

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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 ...

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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 ...

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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, ...

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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).

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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 connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. ...

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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 ...

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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 $\neg q$, and derive a contradiction. If I understand you right, you're asking for a proof which is of the third kind. Moreover, you're asking for a proof which ...

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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 $x\cdot e^x$?...

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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 ...

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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 ...

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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 ...

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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.

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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 ...

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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 theorems ...

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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 ...

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

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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 (which is a part of the high school curriculum here) and half an introduction to proofs. And the very first homework assignments in analysis and linear algebra ...

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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, polyominoes of $... 22 Here are some good examples of proof by contradiction: Euclid's proof of the infinitude of the primes. (Edit: There are some issues with this example, both historical and pedagogical. See Mike F.'s answer and the ensuing discussion.) The famous proof that$\sqrt{2}$is irrational. (I don't particularly like this one---there are better ways of proving ... 22 I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes unnecessarily complicated proofs are skipped, but not often enough to my taste. But there is a point to proving these "big theorems". The proofs contain new ... 21 Proving DeMorgan's Laws for$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$and ... 21 Lots of good answers here (I've upvoted many). I'm won't try to add to the discussion about why induction is hard, but I can suggest some approaches that have helped some of my students. Many have seen induction as an algebra exercise - for example, summing the first$n$squares. Those examples are tedious and work badly. Students have trouble with the ... 21 Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of matrices, linear maps and systems of linear equations. It is a fascinating result that when you formalize these three views the mathematical objects are ... 20 Not to be annoying, but what is a proof? Here's my best take: a proof is an explanation that could convince all conceivable skeptics. Proof, then, is absolutely crucial at all levels of math instruction, because explanation is crucial to learning. I figure that you're imagining something different, a moment when you would start asking kids to offer the ... 20 The two-column proof form has been the dominant mode of presentation for proofs in secondary geometry in the United States for most of the past century. You ask about its effectiveness; unfortunately, I think that question is ill-posed, because the goal state isn't clearly defined (effective at what?) and so there's no way to measure whatever it is you want ... 20 There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't ... 20 There's no abstract reason that an imperfect proof by contradiction should categorically fail to get credit. A proof should generally get partial credit based on how much knowledge of the relevant material it demonstrates. An incomplete proof by contradiction could correctly get the main idea but omit some of the technical material needed to make the ... 19 If$\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 ...

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Perhaps it's not the explanation that's the problem. I suggest you have them explain to you their understanding of the problem. Listening to their justification might reveal why your explanation is not gaining traction. Even in the cases where people are saying they now agree with you, you don't necessarily know that they understand the problem. It's ...

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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 induction proof however is reducing a statement with unknown truth value to a statement with unknown truth value. And not just that, it's even the same statement, ...

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