Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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12
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4answers
308 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 ...
12
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3answers
156 views

Tasks that encourage argumentation

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 ...
12
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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 ...
12
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2answers
362 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"...
12
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1answer
227 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-...
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 ($a \neq 0$). ...
11
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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
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3answers
6k views

Good examples of proof by contradiction?

In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
11
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3answers
272 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 ...
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 ...
11
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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 ...
11
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1answer
372 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, ...
10
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5answers
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 ...
10
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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 ...
10
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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) ...
9
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6answers
633 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 ...
9
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3answers
226 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 ...
9
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1answer
250 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 ...
9
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2answers
261 views

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 ...
9
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1answer
145 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 ...
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 ...
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 ...
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 ...
8
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2answers
368 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 ...
8
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4answers
664 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 ...
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 ...
8
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1answer
503 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)...
8
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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 ...
8
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4answers
282 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). ...
8
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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\} ...
8
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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 ...
7
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6answers
2k views

Ockham's Razor & Mathematical Proofs

Occam's Razor (also written as Ockham's razor from William of Ockham (c. 1287 – 1347), and in Latin lex parsimoniae) is a principle of parsimony, economy, or succinctness used in problem-solving. It ...
7
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6answers
924 views

is it appropriate or beneficial to mention weird results in math?

Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level? Examples of these results include the ...
7
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4answers
365 views

Writing up a proof that assumes what is to be proven?

I was working on this question on math, where (among other things), the OP was asked to prove that $$x \oplus y=\sqrt[3]{x^3+y^3}$$ is associative. After some prompting, the offered proof was $$\...
7
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4answers
192 views

Showing the Necessity of Proving the Impossibilities

"It's impossible because I tried but couldn't do it!" I need situations which shows that this kind of reasoning above is not working! I do have an example, but look for more: It's so hard to cover a $...
7
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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....
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 ...
7
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1answer
130 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 ...
7
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0answers
118 views

Is there any example of a “forwards/backwards” induction?

I like to make the "dominoes" analogy when I teach my students induction. I recently came across the following video: https://www.youtube.com/watch?v=-BTWiZ7CYoI In this video, a sequence of ...
6
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5answers
927 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, ...
6
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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. ...
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 ...
6
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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 ...
6
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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 ...
6
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2answers
258 views

A question from a young student to mathematicians

I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single ...
6
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5answers
646 views

Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula?

According to a recent experiment conducted by user Steven Gubkin, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is ...
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 ...
6
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1answer
269 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 ...
6
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0answers
96 views

Questions similar to Wason Selection Task

The Wason Selection Task (described by Pete Clark here) is a great problem for getting students to grapple with all of the intricacies of logical implication. I will be teaching a discrete ...
5
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3answers
880 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 ...