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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 ...
Benjamin Dickman's user avatar
39 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 ...
TomKern's user avatar
  • 4,022
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 ...
Sue VanHattum's user avatar
  • 20.2k
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 $\...
Tanner Swett's user avatar
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: ...
Steven Gubkin's user avatar
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 ...
Steven Gubkin's user avatar
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 ...
Justin Skycak's user avatar
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 ...
Sumyrda - remember Monica's user avatar
22 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 ...
Chris Cunningham's user avatar
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 ...
Nate Bade's user avatar
  • 1,941
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 ...
MvG's user avatar
  • 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 ...
Henry Towsner's user avatar
20 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 ...
bta's user avatar
  • 331
19 votes

Why do students like proof by contradiction?

Suppose you're in an unfamiliar city without a map. You're trying to get to a particular address, which you know is within five blocks of you, but you have no idea how the streets are laid out, ...
Daniel McLaury's user avatar
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 ...
John Coleman's user avatar
  • 1,506
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 ...
Joseph O'Rourke's user avatar
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 ...
fectin's user avatar
  • 279
16 votes

Should my 8th graders see a proof of the Pythagorean Theorem?

If you are in the United States, at a public school, then you should explain a proof because this is one of the common core state standards: http://www.corestandards.org/Math/Content/8/G/B/6/ I ...
Steven Gubkin's user avatar
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 ...
Benoît Kloeckner's user avatar
16 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, ...
Aeryk's user avatar
  • 8,025
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 ...
fedja's user avatar
  • 3,831
15 votes
Accepted

Proofs that make theorems less clear

"A well-chosen example illustrates ... and is entirely convincing." For me, all of what is usually called "generic proof" satisfy your criterion. Consider the Euclidean algorithm for finding the ...
Amir Asghari's user avatar
  • 4,428
15 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 ...
Benoît Kloeckner's user avatar
14 votes

Good, simple examples of induction?

Here is another one: $\color{blue}{\text{Prove that the power of $13$ can be writen as a sum of two squares}}. $ I will give two proofs of it. First one is more involved and includes the following ...
nonuser's user avatar
  • 400
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 ...
mweiss's user avatar
  • 17.3k
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/...
JDH's user avatar
  • 4,106
14 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 ...
Pedro's user avatar
  • 1,132
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 ...
The Dark Canuck's user avatar
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 ...
Justin Meiners's user avatar
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 ...
Dirk's user avatar
  • 2,992

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