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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
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
  • 3,869
9 votes

Best practices for Proof Revision/ Proof Portfolio?

I taught an abstract algebra course this way. Each student made an overleaf account which I required to be something like "Firstname-Lastname-MTH358-Spring-2024". They shared this folder ...
Steven Gubkin's user avatar
9 votes
Accepted

How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?

I think you will want to start by convincing the audience that $p(aX+b = ax_i + b)$ is equal to $p(X=x_i)$, probably with examples. I am not an expert statistician so please let me know politely if I ...
Chris Cunningham's user avatar
8 votes

How to convince a student without calculus that great circles are geodesics in a sphere?

Take a physical sphere such as a beach ball, and a string. Pick two points. Hold the string down with one finger at one point then stretch it to the second point. Next, holding the string tight at ...
user52817's user avatar
  • 10.8k
8 votes

How to prove, without the LOTUS formula, to student that $V[aX+b]= a^2 V[X]$?

This is a consequence of the definition of the variance (1) the linearity of expectation (2) and an algebraic manipulation (3): $$V(aX+b)\stackrel{(1)}{=}\mathbb{E}(aX+b-\mathbb{E}(aX+b))^2\stackrel{(...
Kostya_I's user avatar
  • 1,381
7 votes

Unique steps leading to a non-unique answer

Giving examples is not the only thing I would do in this case. I would also introduce new vocabulary. Having precise words helps when talking about precise concepts. If each step of an algorithm can ...
Stef's user avatar
  • 556
7 votes

Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing

I've worked with some of these types of students in the past. One trend I noticed was that these students often have experience with coding, which they tend to enjoy and excel at (since the computer ...
Justin Skycak's user avatar
6 votes

Better proof for a proposition when a proof is already available

Let me disagree with this dichotomy: "Either we can find much generalised proof ... or we can suggest a much simpler proof." I think there are other options, proofs that yield different ...
Joseph O'Rourke's user avatar
6 votes

How to convince a student without calculus that great circles are geodesics in a sphere?

Answer inspired by Michał Miśkiewicz's comment on the OP. To a high school student: Put an ant on a basketball. Draw a tiny arrow representing the direction it should walk. The ant always just puts ...
Steven Gubkin's user avatar
6 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

A blast from the past comment, for the consolation of your students, of a mathematician being marked down by one of the most influential mathematicians of his day: John Wallis in his Arithmetica ...
user1815's user avatar
  • 5,545
6 votes

What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

It's quite likely to be a consequence of the belief that they have to answer the question. When they can't work it out, when they've gone around in circles and got lost, but still they have to give an ...
Nullius in Verba's user avatar
6 votes
Accepted

Best practices for Proof Revision/ Proof Portfolio?

Here's a thing I did stepping in that direction. In a discrete mathematics course, I made use of the discussion forum in the LMS in this way -- require each student weekly to pick a distinct homework ...
Daniel R. Collins's user avatar
5 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

I am (one of the) colleagues David refers to in his post. The reason I am doing this lies in some of the answers/comments posted here already. For example, Humberto sais: "While technically it ...
Ferenc Beleznay's user avatar
5 votes
Accepted

Better proof for a proposition when a proof is already available

As an example how would you compare if a mathematician could able to prove FLT in less number of pages when comparing with the proof given by Andrew Wiles since after giving the full credit for the ...
Justin Skycak's user avatar
5 votes

Demarcated "Proof Idea"

I'm not sure if this is what you're looking for, but "How to Read and Do Proofs: An Introduction to Mathematical Thought Processes" by Solow is an intro-to-proof-writing book that frequently ...
Aeryk's user avatar
  • 8,011
5 votes

Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

Take a class, and ask the professor. As the class starts, try to get an idea for what kinds of statements the professor is assuming to be true without justifying them, that you do not immediately see ...
Chris Cunningham's user avatar
5 votes
Accepted

How to convince a student without calculus that great circles are geodesics in a sphere?

Recall that a great circle is the intersection of the unit sphere and a plane passing through the origin. A key point is that for short arcs of great circles, the ratio of euclidean distance between ...
Mikhail Katz's user avatar
  • 2,242
4 votes

Better proof for a proposition when a proof is already available

For educational purposes, what you want is a proof that has different ideas in it. This is similar to teaching: you want pupils to present different ways to solve a given problem and then compare and ...
Tommi's user avatar
  • 7,154
4 votes

Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

I did the exact same thing you did, except in physics instead of math; I would in fact rederive the relevant equations from scratch every time I used them so as to really hammer into my head where ...
linkhyrule5's user avatar
4 votes

Should I really just "shut up and calculate"? On learning at a good pace without sacrificing rigour

Mathematics is supposed to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic ...
Cort Ammon's user avatar
  • 1,225
3 votes

How to convince a student without calculus that great circles are geodesics in a sphere?

Maybe this would work? The spherical law of cosines says that, on a sphere of radius $R$, if you have a sphercial triangle with edge lengths $a$, $b$, $c$ and angles $A$, $B$, $C$, then $$\cos \tfrac{...
David E Speyer's user avatar
3 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

Typically you want to name things. This makes them visible and something you can discuss. So, while teaching, you do want to say that this thing her is induction, so we have to remember to check the ...
Tommi's user avatar
  • 7,154
3 votes

Identifying Trigonometrical proofs

The proof: https://youtu.be/p6j2nZKwf20 For context, here's the main idea of the proof. Using the definition of sine, we have $c^2 = \dfrac{2ab}{\sin 2\alpha}.$ Our goal is to show that the RHS is ...
Justin Skycak's user avatar
3 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

Peano's axioms without the axiom of induction has some models that do not correspond to the natural numbers that we have in our minds. In order to prove that every natural number $n$ other than $0$ ...
user52817's user avatar
  • 10.8k
3 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

In a pedagogical context I can see four types of situation where it may reasonably be required for students to explicitly state their use of the principle of mathematical induction: If writing proofs ...
Yiab's user avatar
  • 31
3 votes

Teaching strong induction instead of induction

Note that the "strong induction" is not even the most general induction scheme you can design. A high school level example is the proof of the AM-GM inequality for $n$ positive numbers where ...
fedja's user avatar
  • 3,869
3 votes

How to convince a student without calculus that great circles are geodesics in a sphere?

Determine the shortest route from New York City USA to Rome Italy using a piece of string on a globe. Both cities are south of the 45th parallel and yet the the shortest route deviates considerably to ...
Dan Christensen's user avatar
3 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

Let's not overlook an obvious application of induction is to turn it around and go from general to specific. Let's say we have established $H(1)$ and that $H(k+1)$ follows from $H(k)$ and we are ...
user52817's user avatar
  • 10.8k
3 votes

Does a proof by induction have to explicitly refer to the principle of mathematical induction?

To be a proof, an argument needs to be explicit about the logical structure. An induction proof won't be any different. Take a very standard task like: such as showing that, for all $n\in \mathbb{N}$...
Louis's user avatar
  • 131

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