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Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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11 votes
4 answers
2k views

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 ...
J W's user avatar
  • 5,033
2 votes
3 answers
271 views

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 ...
niobium's user avatar
  • 225
4 votes
1 answer
137 views

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 ...
SRobertJames's user avatar
2 votes
1 answer
227 views

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 ...
Stéphane Jaouen's user avatar
9 votes
3 answers
574 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 ...
Mathprof's user avatar
  • 1,215
7 votes
1 answer
690 views

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 ...
user1258481's user avatar
6 votes
3 answers
315 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 ...
Fraïssé's user avatar
  • 749
12 votes
11 answers
4k views

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 ...
Dan's user avatar
  • 1,245
1 vote
1 answer
138 views

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 ...
user23248's user avatar
1 vote
1 answer
212 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....
Janaka Rodrigo's user avatar
2 votes
3 answers
957 views

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 ...
Janaka Rodrigo's user avatar
3 votes
2 answers
107 views

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 ...
Vassilis Markos's user avatar
4 votes
7 answers
342 views

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?
Humberto José Bortolossi's user avatar
8 votes
5 answers
3k views

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'...
FShrike's user avatar
  • 478
6 votes
2 answers
369 views

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 ...
Joseph O'Rourke's user avatar
0 votes
0 answers
118 views

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 ...
Janaka Rodrigo's user avatar
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
5 votes
0 answers
177 views

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'...
underflow's user avatar
  • 151
0 votes
1 answer
179 views

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 ...
Cathartic Encephalopathy's user avatar
7 votes
4 answers
4k views

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 ...
Vivaan Daga's user avatar
1 vote
2 answers
224 views

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 $a/b > 0.$
Simd's user avatar
  • 311
4 votes
2 answers
561 views

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 ...
Steven Gubkin's user avatar
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
10 votes
1 answer
337 views

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 ...
Asaf Shachar's user avatar
12 votes
7 answers
3k views

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 ...
Asaf Shachar's user avatar
3 votes
1 answer
88 views

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?
Steve's user avatar
  • 1,594
7 votes
3 answers
771 views

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 ...
pyridoxal_trigeminus's user avatar
22 votes
19 answers
9k 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
4 votes
4 answers
705 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 ...
Prime Mover's user avatar
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
  • 303
3 votes
2 answers
303 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 ...
newmathstudent123123's user avatar
10 votes
4 answers
527 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 ...
user avatar
7 votes
1 answer
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 ...
Aeryk's user avatar
  • 8,039
8 votes
2 answers
312 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 ...
jackisquizzical's user avatar
-2 votes
4 answers
369 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? ...
user avatar
1 vote
1 answer
206 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 ...
plants's user avatar
  • 159
6 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. ...
jackisquizzical's user avatar
5 votes
4 answers
2k 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 ...
a a's user avatar
  • 153
14 votes
3 answers
1k 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 &...
Duncan's user avatar
  • 349
5 votes
4 answers
535 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, ...
plants's user avatar
  • 159
4 votes
2 answers
866 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^...
Dimitris's user avatar
  • 165
2 votes
1 answer
371 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 ...
plants's user avatar
  • 159
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
7 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 ...
user avatar
-3 votes
1 answer
91 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, ...
plants's user avatar
  • 159
2 votes
4 answers
602 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 ...
plants's user avatar
  • 159
7 votes
5 answers
869 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&...
Allure's user avatar
  • 170
1 vote
2 answers
458 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(...
vmusegameon's user avatar
2 votes
1 answer
202 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 ...
J W's user avatar
  • 5,033
7 votes
2 answers
293 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 ...
Geoffrey's user avatar
  • 898