# Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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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 ...
267 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?
2k 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'...
354 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 ...
109 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 ...
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 ...
129 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'...
159 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 ...
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 ...
1 vote
214 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 $a/b > 0.$
384 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 ...
4k 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 ...
209 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 ...
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 ...
82 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?
736 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 ...
7k 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 ...
664 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 ...
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 ...
255 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 ...
474 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 ... 165 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 ...
278 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 ...
356 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
201 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 ...
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. ...
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 ...
971 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 &...
463 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, ...
850 views

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^...
369 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 ...
4k 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 &...
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 ... 85 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, ...
590 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 ...
790 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&...
1 vote
323 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(...
173 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 ...
208 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 ...
714 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 ...
240 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 ...
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). ...
755 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 ...
237 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 ...
283 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 ...
339 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 ...
384 views

### Transitioning proof based math courses online

I'd love to learn from anyone's recent experiences teaching online proof based math courses, especially those that have a large group of students who will be working asynchronously. My usual proof ...
327 views

### What is "mastery" in a mathematical topic?

This question was prompted by looking at Khan Academy's website to see how a comprehensive lecture series could be done and often I see the word, "mastery". To me, I'd think mastery is ... 