Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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15
votes
2answers
321 views

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 ...
6
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3answers
259 views

Is induction or recursion easier to understand?

This is not really a new question, more a revisiting of @vonbrand's "Any suggestions on how to approach recursion and induction?" In an introductory programming class this past year, I asked the ...
26
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11answers
2k views

Impressive examples where a “proof by picture” goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...
26
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8answers
2k views

Teaching logic with a proof assistant

I am thinking about teaching a university-level "introduction to proofs" class (mainly for math and CS majors) making use of a computer proof assistant like Coq. I feel like there is a lot of ...
13
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3answers
218 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 ...
9
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0answers
103 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 ...
15
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4answers
514 views

How to explain what is wrong in this “proof” that $\sqrt N$ must be irrational?

Here is the problem that I asked undergraduate students of an introductory number theory course to prove: Prove that if $N$ is a nonsquare natural number, then $\sqrt N $ is irrational. Many ...
4
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2answers
215 views

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 ...
6
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2answers
238 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 ...
5
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3answers
878 views

Why is this type of reductio ad absurdum not taught more?

I recently read in a book about a proof that Archimedes did. I don't remember the exact details and I don't have the book on me right now, but it involved proving an equality. So let's say proving ...
24
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7answers
3k views

Why do we care about multiple proofs of the same theorem?

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
9
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4answers
588 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 ...
2
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3answers
407 views

When two equivalent algebraic statements have two “different” meanings

Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line: First line. $\sqrt{7}=\frac{m}{n}$ Second line (...
7
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2answers
222 views

Ethics of looking at other proofs before submitting work

I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick ...
8
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3answers
248 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 ...
4
votes
2answers
483 views

Good exercises in proof by induction, very early in freshman calculus?

At some point very early on in a freshman calc course, we use proof by induction to show that the derivative of $x^n$ is $nx^{n-1}$ if $n$ is a natural number. For the small minority of our students ...
5
votes
0answers
164 views

Catalog of undergraduate's misconceptions / problems while proving

Selden & Selden (2011) listed 41 difficulties their students had in an experimental proving course into 9 categories. Unfortunately I haven't found similar work. Thus, my question is: Is there ...
6
votes
5answers
253 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 ...
23
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11answers
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 ...
8
votes
1answer
217 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 ...
16
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8answers
1k views

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 ...
4
votes
0answers
175 views

How to promote more elegant and beautiful proofs by students?

Following the premise that mathematics is an art as well as a science, I want to encourage students to produce not only correct proofs but also to try to find a particular beautiful/elegant proof. ...
8
votes
1answer
254 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 ...
6
votes
3answers
267 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. ...
22
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1answer
450 views

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 ...
14
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0answers
138 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 ...
9
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1answer
242 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 ...
11
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4answers
330 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 ...
36
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6answers
4k views

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....
11
votes
4answers
762 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 ...
11
votes
1answer
326 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 ...
13
votes
3answers
575 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 ...
12
votes
2answers
168 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 ...
16
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2answers
257 views

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 ...
20
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4answers
669 views

What do mathematicians call a proof?

In mathematics education, "proof" is widely used for many kinds of argumentation. For instance, one example could be called a proof, if it is paradigmatic in the following sense: The argument can ...
6
votes
4answers
368 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 ...
14
votes
3answers
254 views

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 ...
18
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3answers
334 views

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 ...
6
votes
5answers
922 views

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, ...
11
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6answers
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 ($a \neq 0$). ...
16
votes
5answers
881 views

Rigorous proofs vs. illustrative examples

No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as ...
21
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9answers
3k views

“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'...
7
votes
1answer
129 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 ...
33
votes
7answers
4k views

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 ...
8
votes
3answers
193 views

How important is building up intuition for a theorem before trying to prove it?

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\} ...
18
votes
3answers
578 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 ...
16
votes
1answer
297 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 ...
10
votes
2answers
289 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) ...
12
votes
2answers
260 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 ...
10
votes
6answers
245 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 ...