# Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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### 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 a particular student understands the general ...
399 views

### What different ways do people use to show students that $\mathbb{R}$ is uncountable?

In particular, if you use Cantor's diagonalization argument, do you ignore the repeating decimal annoyance? Or prove that it's not a problem? Is there another clean way that gives students intuition ...
316 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. ...
229 views

### Resource request: incorrect "proofs" for undergrads to correct/critique

I am teaching an intro to proof course for undergraduate math majors at a medium-sized american research university. I would like to provide my students with some incorrect proofs for the purpose of ...
474 views

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 ...
1k views

### Algebraic Solving and Uniqueness Proofs

The following issue came up in my Intro to Proofs course and I wasn't sure how to explain my distaste of the student proof. Prove that the solution for $x$ in $ax+b=c$ is unique ($a \neq 0$). ...
830 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 ...
405 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 ...
771 views

### Proof by contradiction - more than one case

I am looking for some examples of when proof by contradiction is used in a problem with more than one case. In all the elementary examples, there are only two options (eg rational/irrational, ...
281 views

### Effectiveness of proofs in secondary education

I'll have a department meeting in about 10 days and I want to bring the subject of proofs up. While most teachers do proofs in the blackboard, I want to argue that we should put problems to prove in ...
368 views

### Collaboration on math homework assignments?

There is considerable evidence that pair programming, when executed properly, both increases the accuracy of the code produced and enhances the learning of both participants. I wonder if anyone has ...
169 views

I am looking for resources that have tasks such as the one below that encourage argumentation. I want tasks that 8th graders could do but would also be appropriate for high school students. I want to ...
204 views

### Words used in quantifier proofs

I'm creating a list of "gotcha words" that are often used in writing proofs (particularly quantifier proofs), but frequently in more than one possible way, and that beginners frequently misuse or ...
468 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"...
284 views

### How to write proofs on the board in the classroom

I'm teaching an introductory analysis course, and I am seeking some feedback on how proofs should be written on the board in class in order to maximize learning. I realize that there is an opinion-...
786 views

### How to arrive at infinitude of primes proof?

I know Euclid's proof of there being infinite number of primes. I want to let my brother (age 15) arrive at that proof by himself. He knows definition of a prime number (number divisible only by 1 and ...
1k views

### Book request: teaching proving and reasoning at an American university

I am a European postdoc who recently teaching at a large public university in the United States. I will have to teach a course for undergraduate students that introduces them to proving and reasoning ...
451 views

### Why don't textbooks explain proofs' discovery?

This question concerns only proven statements. I don't know if research papers do, but most math textbooks don't. Counterarguments: Space? 1.1. The increased length from explaining the discovery is ...
516 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 ...
339 views

### What topic can I use in an Introduction to Proofs course that would introduce students to a wide variety of proof methods?

What topics are appropriate for an Introduction to Proofs course which is: Aimed at Freshman who have taken integral calculus and nothing else Is designed to introduce them to formal reasoning and ...
382 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 ...
327 views

### Examples of proofs that use a cycle of implications to prove equivalence

I'm looking for some examples of proofs where it's easier to prove 'cyclical implications' $A\implies B\implies C\implies A$ than to prove $A\iff B$ directly. I can think of some (relatively) ...
401 views

### Using number theory instead geometry to introduce proof in Basic School?

It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, ...
259 views

### A basic game to make arguments about

I think a significant start to my development as a mathematician was playing card games (mostly Euchre) with my parents in my youth. After a particular round, my father would tell me, "Well, with your ...
363 views

For many years, I've been an instructor for lower level undergraduate math classes (precalculus through calculus III). During that time, I've noticed that the vast majority of problems I assigned ...
728 views

### How to teach Proofs

I was taught in 9th grade the two column proof, and it wasn't until 11th (when I saw some number theory) that I realized what a poor method that is. However, it is certainly effective in getting ...
289 views

### How to motivate students to do proofs?

I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this. How can ...
270 views

### Selling completeness, extreme value theorem, etc.?

There is a set of related topics in a freshman calc course that includes the completeness axiom for the reals, the intermediate value theorem, extreme value theorem, Rolle's theorem, and mean value ...
315 views

### How do you assign a grade to a proof?

This question is very similar to one I posed two years ago: How to assign grades to proofs: what do(es) theliterature/experts suggest? I would like to ask the more general question of: what do you ...
164 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 ...
752 views

### Why are proofs by contradiction counterintuitive?

And how to make them intuitive? We are tasked to prove $P \implies Q$. So we assume $P$ and are trying to prove $Q$. We assume not-$Q$ ($\neg Q$) and derive a contradiction, establishing $Q$. There ...
445 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 ...
315 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 ...
110 views

### Literature on student understanding of assumptions

In a discussion with a physics lecturer he mentioned that one major area where students fail is understanding assumptions - for example, if we are interested in two objects hitting each other and then ...
523 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 ...
778 views

### May we permit identities to be established by equivalent equations?

A trigonometry text like Sullivan's Algebra & Trigonometry often has a prohibition like this (Sec. 7.3): WARNING: Be careful not to handle identities to be established as if they were ...
299 views

### Is proof-based exercise-oriented math course without solution an effective way to teach pure math?

In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of: notes written during the lecture ...
724 views

### An alternative to "two column" geometry proofs

I'm a high school teacher in New York State (US), starting in on my first year of teaching Geometry. One of the things that really intrigues me is that the Regents exam (the state-mandated final exam)...
318 views

### Nice examples of proofs by cases?

The setting is undergraduate students in Computer Science, a course in Discrete Mathematics (first proof-oriented course they take, they had a mostly computation oriented first course in calculus). ...