Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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87 votes
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Why are induction proofs so challenging for students?

This forum already has many good, simple examples of induction proofs, a great resource. As I am soon to teach induction for the $n^\textrm{th}$ time—this time to some perhaps under-prepared ...
Joseph O'Rourke's user avatar
47 votes
24 answers
20k views

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

Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job. ...
Tutor's user avatar
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44 votes
28 answers
12k views

Good, simple examples of induction?

Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
vonbrand's user avatar
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41 votes
6 answers
4k views

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

It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important....
user avatar
37 votes
4 answers
3k views

Taxonomy of bad proofs

I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ...
Patrick Lutz's user avatar
35 votes
7 answers
5k views

Uninsulting way to say "this will eventually be easy"

When presenting a proof, there are usually a lot of parts which look like "obvious", "routine" manipulation to me, and between zero and two genuinely insightful steps. I want to point out the ...
David E Speyer's user avatar
33 votes
5 answers
2k views

Inability to work with an arbitrary mathematical object

This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to ...
Brendan W. Sullivan's user avatar
32 votes
13 answers
7k views

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

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
Yiorgos S. Smyrlis's user avatar
30 votes
8 answers
4k views

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

I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math. What is a good point to start learning proofs? what are ...
Keith Nicholas's user avatar
29 votes
12 answers
2k views

Why do students like proof by contradiction?

Every-so-often I come across proofs of the form Assume $X$ is false. Prove $X$ is true (without using that it is false). This contradicts that $X$ is false. Hence $X$ is true. I've seen students ...
Jessica B's user avatar
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29 votes
7 answers
2k views

Teaching logic with a proof assistant

I am thinking about teaching a university-level "introduction to proofs" class (mainly for math and CS majors) making use of a computer proof assistant like Coq. I feel like there is a lot of ...
Mike Shulman's user avatar
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28 votes
11 answers
4k views

Impressive examples where a "proof by picture" goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...
Markus Klein's user avatar
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27 votes
6 answers
5k views

Students confusing "object types" in introductory proofs class

In my intro to proofs (and discrete mathematics) class, I see a common mistake where students make nonsensical statements because, for lack of a better term, they confuse the types of the mathematical ...
Daniel's user avatar
  • 381
25 votes
23 answers
5k views

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

I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). ...
user15257's user avatar
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25 votes
7 answers
4k views

Why do we care about multiple proofs of the same theorem?

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
Amanda's user avatar
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25 votes
1 answer
764 views

Is there a Piagetian age at which proofs can be comprehended?

I am wondering if there is literature on the developmental age (pre-adolescent?, adolescent?) at which the notion of a "proof" can be understood? I am less interested in mathematical proofs and more ...
Joseph O'Rourke's user avatar
23 votes
8 answers
2k views

What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?

One useful trick in mathematics is to prove something stronger instead of the question asked. This works well in induction proofs (because strengthening the claim also strengthens the induction basis)...
user11235's user avatar
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23 votes
9 answers
3k views

"A computer program IS a proof": Introducing rigor via programming

This provocative essay Igor Rivin. "Some Thoughts on the Teaching of Mathematics—Ten Years Later." Notices of the AMS, Jun/Jul 2014. (PDF download link). suggests that a discussion of Igor'...
Joseph O'Rourke's user avatar
23 votes
0 answers
578 views

What are some good simple examples that getting the right result is not enough? [closed]

When I want to point out to my students that getting the right result is not enough, I like to show them the example: $$\frac{16}{64} = \frac{1\hskip-.1cm- \hskip-.4cm{6}}{-\hskip-.2cm{6}\,4} = \...
user11235's user avatar
  • 1,280
22 votes
19 answers
8k views

Math Proofs - why are they important and how are they useful?

My 13yr old has leapt forward in math during the pandemic. He's taking discrete math right now but is running into a bit of a wall with proofs. I have a feeling he needs to find reasons why they can ...
Agent Zebra's user avatar
22 votes
3 answers
768 views

How can I discourage proof by patchwork?

I have a student who is working in their spare time on proving or disproving a conjecture of the form $$\exists x.\forall y.\phi(x,y).$$ Right now their strategy is to construct an $x$ and then show ...
forritari's user avatar
  • 323
21 votes
6 answers
7k views

Examples and applications of the pigeonhole principle

The Pigeonhole Principle (or Dirichlet's box principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble ...
András Bátkai's user avatar
21 votes
4 answers
436 views

Multiple Solutions Methods vs. Encouraging a Particular Approach

It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic. One ...
dtldarek's user avatar
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21 votes
2 answers
4k views

Example "bad proofs"?

As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ...
vonbrand's user avatar
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20 votes
4 answers
903 views

What do mathematicians call a proof?

In mathematics education, "proof" is widely used for many kinds of argumentation. For instance, one example could be called a proof, if it is paradigmatic in the following sense: The argument can ...
Anschewski's user avatar
  • 4,801
20 votes
4 answers
6k views

Explaining why volume of cone is a third of cylinder

I came across this video explaining to kids why the volume of a cone is a third of the cylinder of same cross-sectional radius and height. Essentially the explainer presents pre-created cylindrical ...
Bravo's user avatar
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20 votes
6 answers
12k views

Good examples of proof by contradiction?

In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
vonbrand's user avatar
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20 votes
3 answers
531 views

Critiquing Proof Style During Class

I would like to spend a day with my students analyzing mathematical writing. One way I might accomplish this is to offer multiple proofs (some good, some poor) of the same simple statement and ask ...
Austin Mohr's user avatar
19 votes
5 answers
2k views

Should my 8th graders see a proof of the Pythagorean Theorem?

I've been teaching the Pythagorean Theorem in my 8th grade class, and I noticed something odd. In the book I'm using, the sequence goes something like this: Motivate the idea of distances on a grid ...
Wmol's user avatar
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19 votes
8 answers
4k views

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

I hope the essence of the question is clear from the title. There are obvious advantages to making the linear map the central notion of a linear algebra course: the notion can be illustrated with ...
Kostya_I's user avatar
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19 votes
3 answers
1k views

"Proof" meaning in maths and society

When we ask students to prove a particular result in a math class, students often reply with examples. For example, if I state: if a number is even its square will be even, and ask the students to ...
user avatar
19 votes
3 answers
2k views

Is it natural for self-learners to forget most proofs of the theorems they learn?

When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it? I ask this question as I'm a self learner ...
FNH's user avatar
  • 291
19 votes
3 answers
847 views

Constructive refutation of student misconception

Although @Gareth Shepherd recently posted Addressing fundamental math errors close to the issue, I experienced my problem of misunderstanding in class, where two good K10 students were asked to ...
Morten Engelsmann's user avatar
19 votes
0 answers
248 views

Research on the use of outlined / structured proofs in instruction

Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs? For the ...
Willie Wong's user avatar
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18 votes
7 answers
3k views

Do any middle-school texts indicate that irrationality requires proof?

I believe that most middle-school math curricula have at least a brief section about irrational numbers, in which students are taught (among other things) that $\sqrt{2}$ is irrational and $\pi$ is ...
Timothy Chow's user avatar
18 votes
5 answers
1k views

Rigorous proofs vs. illustrative examples

No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as ...
user35603's user avatar
  • 281
18 votes
1 answer
388 views

How to assign grades to proofs: what do(es) the literature/experts suggest?

I am teaching an introductory course on proofs in mathematics in a mid-size American public university, and trying to develop some kind of consistent grading meta-scheme for grading proofs. I am ...
David Steinberg's user avatar
17 votes
10 answers
2k views

How to teach Mathematical Induction mathematically?

I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I ...
Sufyan Naeem's user avatar
17 votes
3 answers
897 views

How to use false theorems or proofs?

I would like students to be critical and not believe that every proof they see is correct. Lecturers make mistakes and students should not think: "That must be a valid argument/proof/syntax because it ...
Joonas Ilmavirta's user avatar
17 votes
4 answers
654 views

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

Here is the problem that I asked undergraduate students of an introductory number theory course to prove: Prove that if $N$ is a nonsquare natural number, then $\sqrt N $ is irrational. Many ...
Amir Asghari's user avatar
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17 votes
7 answers
763 views

Unique steps leading to a non-unique answer

When asked to show a math problem has a unique solution, students sometimes think that if an algorithm leading to a solution has unambiguous instructions at each step (no need to make choices at any ...
KCd's user avatar
  • 3,456
17 votes
1 answer
1k views

Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?

I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol ...
Linear's user avatar
  • 1,006
17 votes
2 answers
331 views

Teaching the use of pictures in proofs

Today I realized that many of my students in an upper-division undergraduate (projective) geometry class don't really understand how to use pictures in proofs. Some draw no pictures at all, making ...
Mike Shulman's user avatar
  • 6,560
16 votes
3 answers
588 views

Teaching strong induction instead of induction

After teaching induction and then strong induction (i.e. the version where you assume $\forall k<n, P(k)$ and prove $P(n)$), one of my students asked why we ever use ordinary induction, since ...
Mike Shulman's user avatar
  • 6,560
15 votes
6 answers
5k views

I want a "true" proof by contradiction of an implication $P \Rightarrow Q$

When teaching proofs by contradiction of an implication $P \Rightarrow Q$, one starts by assuming both $P$ and (not $Q$), and then reaches a contradiction. The problem is, most elementary proofs of ...
Juan Tolosa's user avatar
15 votes
7 answers
695 views

Should theorems be proved to students who are not majoring in mathematics?

My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not ...
Zuriel's user avatar
  • 4,265
15 votes
3 answers
377 views

How to teach students the value of concrete counterexamples?

I teach exercise sessions for a Linear Algebra course for 1st semester students in Europe. Students have to prepare some exercises at home. In class, I call on students to present their solutions. ...
Haudie's user avatar
  • 151
15 votes
1 answer
2k views

Proving theorems on one's own: how long should one persist?

I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found ...
Adrian's user avatar
  • 253
15 votes
1 answer
339 views

Order of Topics in Introductory Proofs Class

Beginning next semester I am teaching a course in proofs and mathematical problem solving at my local university. For some background, the university is primarily a commuter university and the ...
Ryan's user avatar
  • 153
14 votes
4 answers
943 views

Proofs that make theorems less clear

Teaching Theory of Computation for the first time, I encountered a phenomenon which perhaps is familiar to others in different contexts. I realize most MESE participants are not conversant with Th....
Joseph O'Rourke's user avatar