# Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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### Proof that $e$ is finite

Define $e-1$ as the yearly compound interest obtained from a one dollar investment with 100% gross annual interest: $$e := \lim_{n\to\infty} (1+1/n)^n =: \lim_{n\to\infty} a_n$$ Nine year olds can ...
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### Teaching students how to check the validity of their proofs

The question Teaching students to find and correct their own errors and its answers address mainly calculation problems of the types typically found in secondary school and the lower levels of ...
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### How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?

The mainstream way to show $V[aX+b]= a^2 V[X]$ is by using LOTUS. However, LOTUS seems to me too powerful and out-of-reach for a last-year high-school student. Therefore I was wondering if we could ...
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### When teaching an upper-level proof based course, what criteria do you use to determine which and how many problems to assign?

When teaching an upper-level proof based course, what criteria do you use to determine which and how many problems to assign? And, as a corollary question: How do you determine the level of success ...
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### About a difficult exercise for 12 years pupils

You have to go from a point $A$ (start) to a point $B$ (arrival) by crossing a river $(d)$ and traveling as little distance as possible. Pupils first do a search by trying several paths $1,2,3,4$ and ...
579 views

### Best practices for Proof Revision/ Proof Portfolio?

I'm teaching a class small enough that I'm considering encouraging proof revisions (i.e. students taking a second try on proof based homework problems after getting feedback) for the first time. I'd ...
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### Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing

Seeking guidance on how to assist students who possess a solid grasp of problem-solving concepts, allowing them to intuitively arrive at solutions, yet encounter difficulties when it comes to ...
321 views

### What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

So in an advanced mathematics course for engineers, there are often problems of the type: Prove claim A. Given equation A, show that you can obtain equation Z. I am frequently faced with a problem ...
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### Does a proof by induction have to explicitly refer to the principle of mathematical induction?

I teach high school math. Some of my colleagues insist that a proof by induction should explicitly refer to the principle of mathematical induction, i.e. it must include the words "by the ...
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1 vote
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### Idea of using LLMs to help communicate ideas in math

What do you guys think about the ideas presented in this short text: https://github.com/yougetyourmanwww/AI-for-math/blob/main/AI.md The text is about how LLMs like chatGPT can be used for when doing ...
1 vote
214 views

### Identifying Trigonometrical proofs

How can we identify trigonometrical proofs from geometrical proofs, do we have purely trigonometrical proof of Pythagoras theorem as claimed by two high school students ? https://www....
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### Better proof for a proposition when a proof is already available [closed]

What is a much better proof in mathematics, is it need to be a much more advanced one compared with the proof already available or a much simpler one? I think you can challenge a proof in two ...
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### Utillizing Lakatos' "Proofs and Refutations" in Secondary Education

These days I am reading Imre Lakatos's Proofs and Refutations and I can't stop thinking how one could utilize it in the classroom (mostly high school). Some stray half-baked ideas I have had so far ...
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### How to convince a student without calculus that great circles are geodesics in a sphere?

how to convince or demonstrate to a high school student who does not know differential and integral calculus that the geodesics of a sphere are arcs of great circles?
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### Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

This question is about getting realistic expectations for how a university student actually does and should learn maths. I'm becoming increasingly suspicious that my approach is detrimental, but I don'...
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### Demarcated "Proof Idea"

Michael Sipser's textbook Introduction to the Theory of Computation (now 3rd ed.) includes for each major theorem, a demarcated Proof Idea of length a paragraph to more than a page, prior to ...
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### Infinite descent method

We have plenty of examples in mathematical induction for advanced level mathematics students. Can we introduce infinite descent method as extremely opposite approach to mathematical induction and is ...
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### 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 ...
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### Comparison of texbook for "how to write proofs"

I posted this question in the math stackexchange https://math.stackexchange.com/questions/4681694/comparison-of-textbooks-on-how-to-write-proofs and one person suggested that I cross-post it here. I'...
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### Student finding it difficult to recall theorem exactly

I've been trying to teach my sister school maths, and one difficulty I find is, she is unable to state precise formulation of theorems, and sometimes confuse the assumption and the implication. This ...
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### What implication arrows, if any, should I require in teaching?

Q: Solve $x+5=0$. A: $x+5=0\implies x=-5$. This answer would be given full marks. Isn’t it better to tell students to use $\equiv$ or $\iff$? Because that is what lets them say $-5$ is a solution to ...
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### A high school level proof that $a/b > 0$ [closed]

Is there a high school level proof of the following? If $a,b > 0$ then $a/b > 0.$
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### Proof by Contradiction vs. Proof of Negation

In constructive mathematics we make a distinction between "proof of negation" and "proof by contradiction". You can read a great account of the difference in this blog post of ...
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### 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 ...
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### How to explain the concept "Without loss of generality" (through examples)?

This is not a precise question. I am curious to know how do you present to your students the (imprecise) concept of "without loss of generality", and how to use it correctly/incorrectly. I ...
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### Does induction really avoid proving an infinite number of claims?

I am teaching calculus $1$ this semester, and I saw the following motivation for using induction by another teacher: Since we can't go over "manually proving" all claims $1,2,\ldots$ and ...
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### Fitch Style Deduction in Non-Logic Classes

Has anyone experimented with using Fitch-style proofs as a teaching aid in courses outside of logic specifically and if so, how was the technique received by students?
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### How can I 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
193 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 ...
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### 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 ...
372 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? ...
1 vote
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 &...
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### 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, ...
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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 (a^2+b^2)(c^2+d^2)=a^2c^2+a^...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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&...
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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 ...