Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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64
votes
21answers
16k views

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 ...
41
votes
21answers
16k 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. ...
37
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6answers
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....
34
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17answers
6k 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 ...
33
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7answers
4k 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 ...
28
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10answers
4k 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!" (...
27
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7answers
2k 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 ...
26
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8answers
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 ...
25
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11answers
2k 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 ...
24
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7answers
3k 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 ...
23
votes
11answers
1k 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 ...
22
votes
1answer
414 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 ...
22
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0answers
531 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} = \...
21
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9answers
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'...
20
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7answers
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)...
20
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4answers
656 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 ...
19
votes
6answers
6k 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 ...
19
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3answers
593 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 ...
18
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5answers
733 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 ...
18
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4answers
309 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 ...
18
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3answers
330 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 ...
18
votes
3answers
561 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 ...
18
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2answers
3k 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 ...
16
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5answers
865 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 ...
16
votes
5answers
1k 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 ...
16
votes
8answers
1k 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 ...
16
votes
3answers
986 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 ...
16
votes
2answers
252 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 ...
16
votes
1answer
290 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 ...
15
votes
7answers
392 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 ...
15
votes
4answers
503 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 ...
15
votes
3answers
205 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. ...
15
votes
2answers
313 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 ...
15
votes
3answers
660 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 ...
15
votes
1answer
242 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 ...
14
votes
9answers
5k views

Why do inequalities flip signs? [closed]

Is there a mathematical reason (like a proof) of why this happens? You can do it with examples and it is 'intuitive.' But the proof of why this happens is never shown in pedagogy, we just warn ...
14
votes
1answer
603 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 ...
14
votes
3answers
1k views

Why do we prove things we already know?

As math majors and math educators we take for granted the importance of proofs and being precise. However with I have found that non-math majors are content with anything that looks reasonably ...
14
votes
1answer
841 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 ...
14
votes
3answers
249 views

How to measure the understandability of a proof?

Is there a way to measure the understandability of a proof? From a search in the internet I have only found methods for measuring the understandability of software or tests for measuring the ...
13
votes
3answers
560 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 ...
13
votes
4answers
727 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....
13
votes
3answers
217 views

Resource request: incorrect “proofs” for undergrads to correct/critique

I am teaching an intro to proof course for undergraduate math majors at a medium-sized american research university. I would like to provide my students with some incorrect proofs for the purpose of ...
13
votes
5answers
281 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 ...
13
votes
0answers
129 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 ...
12
votes
5answers
684 views

Descriptive Thinking vs. Formal Writing

Sometimes I come across some exam answers which describe a proof sketch or a counterexample very well but are not written formally. Such proofs show that particular student understand the general ...
12
votes
7answers
355 views

What different ways do people use to show students that $\mathbb{R}$ is uncountable?

In particular, if you use Cantor's diagonalization argument, do you ignore the repeating decimal annoyance? Or prove that it's not a problem? Is there another clean way that gives students intuition ...
12
votes
3answers
6k 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 ...
12
votes
2answers
738 views

Definitions/proofs that allow “useless” cases?

I often see students confused/mystified by definitions (and proofs) that allow/consider "useless" cases. A case in point is the definition of a DFA (deterministic finite automaton), which allows ...
12
votes
2answers
254 views

Effectiveness of proofs in secondary education

I'll have a department meeting in about 10 days and I want to bring the subject of proofs up. While most teachers do proofs in the blackboard, I want to argue that we should put problems to prove in ...