# Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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### 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 ...
3k views

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 ...
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 ...
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 ...
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 ...
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!" (...
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)...
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 ...
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 ...
291 views

### Teaching logic through “high school algebra”?

I am going to be teaching a discrete math class in the fall. One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of "and", "or", "not", "implies"...
262 views

### Should one justify formulae in middle school?

Consider two possible lesson outlines: Check homework. Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279 Calculate the area of a circle as an example. ...
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 ...
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'...
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. ...
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 ...
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 ...
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 ...
255 views

### Any suggestions on how to approach recursion and induction?

Much mathematics is intimately tied to recursion, be it in definitions (like factorials and integer powers) and proofs by induction. This is also very relevant in computer science and programming. ...
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 ...
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 ...
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 ...
320 views

### Effectiveness of students seeing proofs - reference request

If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated). I am wondering if any research in ...
121 views

### Motivation vs. Rigor

This is such a vague topic that I hesitate to post. I constantly struggle between the time-tradeoff between motivating a topic, and delving into the rigorous details necessary to fully "grok" the ...
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 ...
351 views

### Should students be given partial scores when they gave an incomplete proof by contradiction?

In a quiz, there was a question asking students to show something doesn’t exist. A lot of them gave proofs by contradiction. Initially, I designed the marking scheme so that an incomplete proof by ...
250 views

### Different approaches to proofs that “are the same”?

This question (and answers) on MSE got me thinking on simple examples of different ways of proving the same (hopefully somewhat interesting) result, as examples to be discussed on difference in ...
I was working on this question on math, where (among other things), the OP was asked to prove that $$x \oplus y=\sqrt[3]{x^3+y^3}$$ is associative. After some prompting, the offered proof was \...