Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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42 votes
25 answers
10k 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 ...
3 votes
2 answers
542 views

How to internalize solutions/proofs to theorems and exercises?

In particular, my question is about abstract mathematics such as group theory, analysis, topology, etc. where most textbooks are filled with exercises which require proof, and how to go about ...
20 votes
18 answers
5k 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 ...
5 votes
4 answers
392 views

Doctorate and examples of difficult solved problems

Okay. My questions are: How do some people do doctorates in mathematics and spend so much time like three to six years trying to answer one or two open problems? How do they have the patience, ...
4 votes
4 answers
611 views

Third isomorphism theorem: how important is it to state the relationship between subgroups?

In texts which present the third isomorphism theorem: $$(G/N)/(H/N) \cong G/H$$ the relationship between the entities is often seen presented in the form: Let $H$ and $N$ be normal subgroups of a ...
16 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 ...
30 votes
13 answers
6k 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!" (...
10 votes
4 answers
432 views

Should students in a first university linear algebra class be taught to write simple proofs?

I am teaching an introductory linear algebra courses for undergraduate students in math, computer science, or data science at a liberal art university. Most of the students have not decided their ...
17 votes
4 answers
5k 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 ...
3 votes
2 answers
210 views

How to understand the book and the material to the deepest possible level?

I'm a first year mathematics major and I have a problem with my learning process. In my university, I only have books and questions that the university published, so I have to learn the most of the ...
8 votes
1 answer
144 views

Proofs with Independent Parts

In my Intro to Proofs course I have an activity (more details below) where the goal is to write a collaborative proof of a statement. What makes it work as a whole class activity is that, to prove the ...
4 votes
2 answers
785 views

Questions about proofs

To prove, e.g., the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$, I remembered working, in high school, in the following way. Expanding the LHS gives \begin{equation} (a^2+b^2)(c^2+d^2)=a^2c^2+a^...
8 votes
2 answers
256 views

Seeking short algebraic proofs that an Algebra2 student can appreciate

There are many many elegant algebraic (more generally, non-geometric) proofs, but fewer of them are both accessible and interesting to a pre-calculus student. Three nice examples of what I'm looking ...
-2 votes
4 answers
343 views

How can students prognosticate to rewrite the same sum backwards, then add the same sum twice?

This comment doesn't fulfill my students or me, because it doesn't demystify this trick of writing $S_n$ forward, then backwards, then adding. What would spur students to action these unnatural steps? ...
5 votes
4 answers
1k views

How long would it take someone to master the topics in the book "Book of Proof" by Hammack and similar?

If someone never had any experience with mathematical proofs and had only classes like Calc I-III (which he passed, without paying any attention to the proofs present in the textbooks), how long would ...
5 votes
2 answers
3k views

Should proofs include a third “context” column?

Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. ...
21 votes
7 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)...
1 vote
1 answer
181 views

Math outside of undergraduate studies and proofs

I read sometimes mathematical works of others outside my undergraduate studies. I think i can not follow the understanding of the proofs of theorems sometimes. What should i do? Should i read other ...
9 votes
1 answer
387 views

Advice on Proof-based Math Topics for High Schoolers

I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent ...
26 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 ...
14 votes
3 answers
950 views

Is there a measurable learning goal related to understanding proofs of important theorems?

I believe that good math courses are structured around measurable learning goals. For example, "can correctly replace a line integral with an equal double integral using Green's Theorem" or &...
47 votes
24 answers
18k 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. ...
18 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 ...
17 votes
5 answers
10k 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 ...
2 votes
1 answer
359 views

Solving math problems and learning

Should i solve math problems by writing the answers to papers or notebooks with pencils or should i try solving them in my head at undergraduate studies at university? Also, sometimes after learning ...
14 votes
5 answers
3k views

I want a "true" proof by contradiction of an implication P => Q

When teaching proofs by contradiction of an implication P => Q, one starts by assuming both P and (not Q), and then reaches a contradiction. The problem is, most elementary proofs of this type are &...
6 votes
4 answers
3k views

How to keep students' attention while teaching a proof?

I taught the proof inclusion-exclusion principle to CS students yesterday. While the proof is not too long, it does involves quite a bit notations. I could feel that most students lost interests a few ...
-3 votes
1 answer
79 views

Abstract math and making proofs

What is abstract math about? I think we can not visualise probably what we read. Or can we? I am talking about the theorems, definitions and proofs in areas of math like Riemannian geometry, ...
2 votes
4 answers
581 views

Trials and instructions on proofs

So, in more advanced courses and fields in math like undergraduate and more advanced, how does one make a proof? He makes trials of everything he knows about the problem? What are the instructions to ...
7 votes
5 answers
692 views

Example of why proof by exhaustion is inelegant

There's a nice example of why people dislike proof by exhaustion on the Wikipedia page. The problem statement is "prove that all years in which the Modern Olympics are held are divisible by 4&...
32 votes
4 answers
1k 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 ...
1 vote
2 answers
245 views

Should I do all the proof practice problems in How to Prove It, an intro to proofs book?

Like the title says. I am self studying intro to proofs(How to prove it by velleman) so I can start an introduction to analysis. I am wondering if I should complete all the exercises in the textbook(...
18 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 ...
23 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). ...
2 votes
1 answer
156 views

Improving exposition of a proof about polynomials over infinite fields

This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero ...
24 votes
1 answer
685 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 ...
13 votes
3 answers
520 views

How to teach the Pythagorean theorem in a satisfying way to high school students?

I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons: The chosen proof feels like magic and I don't feel like I have a better understanding of ...
7 votes
2 answers
169 views

Is $\overline{AB} \cong \overline{BA}$ usually taught as an instance of the symmetric property of congruence?

I have been tutoring a wide range of math subjects for many years. Recently, I began tutoring a girl in high school geometry (in California, for context). This semester of the course is starting with ...
4 votes
3 answers
219 views

Differing Choices of $\delta$ in a Limit

In conceptually motivating the $\epsilon-\delta$ definition and proof of a limit, I realized a new way of choosing the $\delta$. For example, consider $\lim_{x\to 4}\sqrt{x}=2$. In the "standard ...
9 votes
2 answers
462 views

Can some lovers of math truly never create something previously unseen?

Can someone truly love math, and master and remember discovered calculations, counterexamples, proofs; but still fail to invent anything new (e.g. incapacity to prove anything unseen, calculate ...
14 votes
6 answers
521 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 ...
11 votes
4 answers
594 views

How do you teach students about the concept of a proof?

I get this question a lot from new students who are taking their first proof-based math class. They are struggling because they don't have that fluency with proofs, to begin with. They don't know ...
8 votes
2 answers
413 views

Is there any example of a "forwards/backwards" induction?

I like to make the "dominoes" analogy when I teach my students induction. I recently came across the following video: https://www.youtube.com/watch?v=-BTWiZ7CYoI In this video, a sequence of ...
5 votes
1 answer
265 views

How important is it to come up with or learn an elementary solution?

Note: by "elementary" I mean "without using more advanced theory and tools". Students are sometimes required or encouraged to solve very difficult problems using limited number of ...
7 votes
2 answers
568 views

Logic and proofs in secondary school

Inspired by the question When do college students learn rigorous proofs?, I became curious when pupils in secondary schools learn about proofs, what kinds of proofs they are, how rigorously they are ...
5 votes
0 answers
188 views

A video game for teaching the concept of a mathematical proof

Two students in my game development course would like, as a final project, to develop a video game for teaching mathematics. In contrast to the many other games for teaching maths, which focus mainly ...
-4 votes
2 answers
328 views

Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$ [closed]

Numerous online resources parrot the usual proof by contradiction of the irrationality of $\sqrt{2}$. These all rely upon the assumption that the rational form (say, $a/b$) is in its simplest ...
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 ...
4 votes
1 answer
78 views

What are ideas and strategies on improving at discovering counterexamples? [closed]

What are ideas and strategies on improving at discovering counterexamples? I originally posed this as an Example Question.
29 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 ...