# Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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### 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 ...
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### 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 ...
737 views

### Proofs that make theorems less clear

Teaching Theory of Computation for the first time, I encountered a phenomenon which perhaps is familiar to others in different contexts. I realize most MESE participants are not conversant with Th....
215 views

### Educational styles for writing proofs

Can someone please point to research papers that analyze different ways of expressing informal proofs from an educational point of view? I am particularly interested in proofs by induction but I would ...
264 views

### Should one justify formulae in middle school?

Consider two possible lesson outlines: Check homework. Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279 Calculate the area of a circle as an example. ...
129 views

### Research on the use of outlined / structured proofs in instruction

Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs? For the ...
250 views

### Different approaches to proofs that “are the same”?

This question (and answers) on MSE got me thinking on simple examples of different ways of proving the same (hopefully somewhat interesting) result, as examples to be discussed on difference in ...
1k 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 ...
299 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 ...
237 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 ...
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### If you do not 'read A to Z', then how can you discover the idea? [closed]

The following is from an article in The New Yorker on Y. Zhang and his proof on gaps between primes: Rutgers University Professor [Henryk] Iwaniec and his friend, John Friedlander, a professor at ...
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### Teaching strong induction instead of induction

After teaching induction and then strong induction (i.e. the version where you assume $\forall k<n, P(k)$ and prove $P(n)$), one of my students asked why we ever use ordinary induction, since ...
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### What am I supposed to be learning with long proofs of the main theorems in class?

It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important....
324 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 ...
562 views

### How to use false theorems or proofs?

I would like students to be critical and not believe that every proof they see is correct. Lecturers make mistakes and students should not think: "That must be a valid argument/proof/syntax because it ...
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### Why are induction proofs so challenging for students?

This forum already has many good, simple examples of induction proofs, a great resource. As I am soon to teach induction for the $n^\textrm{th}$ time—this time to some perhaps under-prepared ...
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### 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 ...
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### Teaching the use of pictures in proofs

Today I realized that many of my students in an upper-division undergraduate (projective) geometry class don't really understand how to use pictures in proofs. Some draw no pictures at all, making ...
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### How to measure the understandability of a proof?

Is there a way to measure the understandability of a proof? From a search in the internet I have only found methods for measuring the understandability of software or tests for measuring the ...
236 views

### Are there automatic solution checkers?

Suppose I would like to make a maths test that one can do by computer and that can be automatically checked by computer. Question: Is there some automatic theorem prover that can check the proofs ...
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### Is it natural for self-learners to forget most proofs of the theorems they learn?

When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it? I ask this question as I'm a self learner ...
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### Critiquing Proof Style During Class

I would like to spend a day with my students analyzing mathematical writing. One way I might accomplish this is to offer multiple proofs (some good, some poor) of the same simple statement and ask ...
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### Analogies for mathematical induction

What are the most successful analogies that are used to teach the concept of mathematical induction? To clarify, I am not looking for a formal explanation of the principle of mathematical induction, ...
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### 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 ...
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### 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 ($a \neq 0$). ...
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### 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 ...
124 views

### Has anyone written anything notable on the relation between mathematical progress and the simplification of proofs overtime?

The author of an answer on math.se remarked that much of mathematical progress is a factor of the accrual of both, theorems, which other mathematicians can use in their proofs, and more efficient ...
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### Uninsulting way to say “this will eventually be easy”

When presenting a proof, there are usually a lot of parts which look like "obvious", "routine" manipulation to me, and between zero and two genuinely insightful steps. I want to point out the ...
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For example, consider trying to prove that: If $A$ is a set and $F \subset P(A)$, then the relation $R := \{(a, b) \in A \times A$ such that for every $X \subset A - \{a, b\}$, if $X \cup \{a\} ... 3answers 564 views ### How can I discourage proof by patchwork? I have a student who is working in their spare time on proving or disproving a conjecture of the form $$\exists x.\forall y.\phi(x,y).$$ Right now their strategy is to construct an$x$and then show ... 1answer 292 views ### How to assign grades to proofs: what do(es) the literature/experts suggest? I am teaching an introductory course on proofs in mathematics in a mid-size American public university, and trying to develop some kind of consistent grading meta-scheme for grading proofs. I am ... 2answers 255 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 ... 2answers 286 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) ... 1answer 242 views ### Order of Topics in Introductory Proofs Class Beginning next semester I am teaching a course in proofs and mathematical problem solving at my local university. For some background, the university is primarily a commuter university and the ... 6answers 243 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 ... 2answers 203 views ### On problems which can be proved easier if we use a different induction step Say we have a property$P$defined on the natural nubers. Usually students are taught that to pove$P(n)$is true for all$n\in\mathbb N$you have to do the following: make a basis and use either ... 2answers 449 views ### Examples where it easier to prove more than less [duplicate] Especially (but not only) in the case of induction proofs, it happens that a stronger claim$B$is easier to prove than the intended claim$A$since the induction hypothesis gives you more information.... 4answers 366 views ### Intuition behind$\zeta(2) = \frac{\pi^2}{6}\$

The result $$\zeta(2) = \frac{\pi^2}{6},$$ tends to amaze young students because of its beauty. However, although in literature there are many proofs of this result, I find that none is suitable for ...
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### 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 ...
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### Proving theorems on one's own: how long should one persist?

I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found ...
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### Grading Computations vs. Grading Proofs: Is there a difference?

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 ...
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### “Proof” meaning in maths and society

When we ask students to prove a particular result in a math class, students often reply with examples. For example, if I state: if a number is even its square will be even, and ask the students to ...
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### 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 ...
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### Why do we prove things we already know?

As math majors and math educators we take for granted the importance of proofs and being precise. However with I have found that non-math majors are content with anything that looks reasonably ...
620 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 ...
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### “A computer program IS a proof”: Introducing rigor via programming

This provocative essay Igor Rivin. "Some Thoughts on the Teaching of Mathematics—Ten Years Later." Notices of the AMS, Jun/Jul 2014. (PDF download link). suggests that a discussion of Igor'...
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### Nice examples of proof by contradiction? [duplicate]

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). ...
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### 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). ...