Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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5
votes
2answers
205 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 ...
15
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1answer
249 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 ...
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....
9
votes
1answer
249 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
votes
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 ...
19
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6answers
6k views

Examples and applications of the pigeonhole principle

The Pigeonhole Principle (or Dirichlet's box principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble ...
16
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3answers
1k views

“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 ...
14
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3answers
1k views

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 ...
8
votes
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). ...
9
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6answers
632 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 ...
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 ...
12
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3answers
259 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. ...
5
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1answer
99 views

How to teach application of pumping lemma (automata theory)?

The pumping lemmata (for regular languages and for context free languages) are used to prove languages non-regular/non-context free by contradiction. But such proofs are often horribly botched by ...
2
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2answers
552 views

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). ...
18
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2answers
3k views

Example “bad proofs”?

As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ...
15
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1answer
631 views

Is there any evidence about the effectiveness of “table proofs” in pre-college mathematics education?

I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol ...
4
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2answers
754 views

Teaching fractions: the generalization problem

I've been thinking about how you would go about teaching fractions, and there seems to be a problem in that every basic fact needs to be proven/explained twice, using two different layers of ...
12
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2answers
748 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 ...
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 ...
12
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5answers
691 views

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 particular student understand the general ...
12
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7answers
361 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 ...
27
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7answers
2k views

Is there a good age/level to start learning mathematical proofs?

I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math. What is a good point to start learning proofs? what are ...
4
votes
1answer
54 views

What are ideas and strategies on improving at discovering counterexamples? [closed]

What are ideas and strategies on improving at discovering counterexamples? I originally posed this as an Example Question.
22
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0answers
535 views

What are some good simple examples that getting the right result is not enough? [closed]

When I want to point out to my students that getting the right result is not enough, I like to show them the example: $$\frac{16}{64} = \frac{1\hskip-.1cm- \hskip-.4cm{6}}{-\hskip-.2cm{6}\,4} = \...