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Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

4
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2answers
456 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 ...
14
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1answer
601 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 ...
28
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10answers
4k views

Is Euclid dead? or Should Euclidean geometry be taught to high school students?

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
4
votes
2answers
695 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 ...
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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5answers
858 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 ...
12
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2answers
734 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 ...
11
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4answers
328 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 ...
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
680 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 ...
25
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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 ...
20
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4answers
653 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 ...
12
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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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7answers
355 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 ...
26
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8answers
1k 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 ...
31
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17answers
6k views

Good, simple examples of induction?

Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
4
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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.
12
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3answers
253 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. ...
18
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4answers
308 views

Multiple Solutions Methods vs. Encouraging a Particular Approach

It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic. One ...
22
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0answers
522 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} = \...
27
votes
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 ...