40
votes
Accepted
Students confusing "object types" in introductory proofs class
I personally use terms like "type disagreement" and "type error". This agrees with the notion of types within computer science (https://en.wikipedia.org/wiki/Type_system).
When I ...
- 4,825
39
votes
Accepted
Why do inequalities flip signs?
I'm slightly concerned that
Is there a mathematical reason (like a proof) of why this happens?
is a purely mathematical question, but since you write "we just warn students" I will assume that ...
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33
votes
How can I explain why we need proofs to someone who has no experience in mathematical thinking?
More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot ...
- 21.3k
32
votes
I want a "true" proof by contradiction of an implication $P \Rightarrow Q$
As you've noticed, there are (at least) three potential ways of proving an implication $p \Rightarrow q$:
Assume $p$, and conclude $q$.
Assume $\neg q$, and conclude $\neg p$.
Assume both $p$ and $\...
- 631
31
votes
Accepted
Math Proofs - why are they important and how are they useful?
Proofs are important because proofs are just understanding how we know that something is true. This is what mathematics is all about!
What if all you care about is using the results of mathematics: ...
- 25.8k
30
votes
Accepted
Explaining why volume of cone is a third of cylinder
This is an experiment which can lead you to guess that the volume of a cone is approximately $\frac{1}{3}$ the volume of a cylinder with the same base and height. It is not a proof in any sense of ...
- 25.8k
26
votes
Accepted
Does a proof by induction have to explicitly refer to the principle of mathematical induction?
The appropriate level of granularity for a proof depends on the audience.
If you're taking an "Intro to Proofs" class and your homework is to do some proofs by induction, then yeah, you ...
- 12.8k
25
votes
Accepted
Inability to work with an arbitrary mathematical object
I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?"
When you say "Suppose $A⊆R$ is nonempty ...
- 21.8k
23
votes
When do college students learn rigorous proofs?
From day one.
In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus ...
22
votes
Why do some linear algebra courses focus on matrices rather than linear maps?
Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of ...
- 1,951
22
votes
Math Proofs - why are they important and how are they useful?
I am an engineer. I have not done a mathematical proof since leaving school. Despite that, I believe that proofs are the second most important skill that any student will learn during their entire ...
- 531
20
votes
Why do we care about multiple proofs of the same theorem?
I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances ...
- 369
20
votes
Accepted
Should students be given partial scores when they gave an incomplete proof by contradiction?
There's no abstract reason that an imperfect proof by contradiction should categorically fail to get credit.
A proof should generally get partial credit based on how much knowledge of the relevant ...
- 11.6k
18
votes
Why don't textbooks explain proofs' discovery?
In many cases, standard theorems in a theory were first proved as special cases before the theory itself was even invented. For example, Lagrange's theorem in group theory predated the invention of ...
- 1,536
18
votes
Taxonomy of bad proofs
Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, ...
- 8,069
18
votes
Accepted
Proof by contradiction - more than one case
(1)
Here is a $3$-case proof from Larry Cusick's webpages:
Theorem. There are no rational number solutions to the equation $x^3 + x + 1 = 0$.
Proof. (Proof by Contradiction.) Assume to the ...
- 30.1k
18
votes
Math Proofs - why are they important and how are they useful?
I am an engineer.
Proofs are important to "get" engineering, but are not directly used. I see three aspects of learning proofs as important: Logic, Process, and Ontology.
Logic is the ...
- 278
16
votes
How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?
This is indeed tricky, and it seems to me the most effective way (in far more general, similar situations) is to show them the problem would be to have them apply their method to another, close ...
- 9,120
16
votes
Accepted
Why are proofs written in flowery language incomprehensible?
To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some ...
- 9,120
16
votes
Accepted
Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour
I looked at some of your posts on MSE before answering. Well, I wouldn't say that you are a "Jack of trades" yet, but you are certainly way above what one would expect from somebody 2 years ...
- 4,254
15
votes
The use of "$\therefore$" and "$\because$"
Is there any better alternative to the three-dot notation?
The usual general advice is to use words instead of symbols.
The best notation is no notation; whenever it is possible to avoid the use of a ...
- 1,890
14
votes
Good, simple examples of induction?
Here is another one:
$\color{blue}{\text{Prove that every power of $13$ can be written as a sum of two squares}}.
$
I will give two proofs of it. First one is more involved and includes the ...
- 420
14
votes
Should my 8th graders see a proof of the Pythagorean Theorem?
I think an important aspect of this question -- one that I don't think has been mentioned yet in the other answers -- is the verb "see", as in "Should my 8th grader see a proof". While "Explain a ...
- 17.4k
14
votes
Should students be given partial scores when they gave an incomplete proof by contradiction?
I agree completely with Henry, but let me try to mention some specific, practical advice that you or others may find helpful.
I strongly encourage you to adopt a more wholistic manner of marking/...
- 4,086
14
votes
How can I explain why we need proofs to someone who has no experience in mathematical thinking?
One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the ...
- 141
14
votes
How can I explain why we need proofs to someone who has no experience in mathematical thinking?
Most answers describe proof as an improved correctness checking tool.
But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and ...
- 240
13
votes
Accepted
is it appropriate or beneficial to mention weird results in math?
I would be careful with the type of result for which one needs a lot of new math to digest the explanation.
For example, I would avoid talking about $ 1 + 2+3+.. = -1/12$ because there is basically ...
- 3,001
13
votes
Accepted
When do college students learn rigorous proofs?
In my experience (U.S.), that's on the boundary between 2nd and 3rd year -- either the end of sophomore year or the start of junior year.
Two years ago I did a survey of Associate in Science (2-year) ...
- 26.6k
13
votes
How to teach the Pythagorean theorem in a satisfying way to high school students?
I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion.
I will try to come back and write a more elaborated ...
- 17.4k
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