25
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 ...
- 6,111
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 ...
- 5,345
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 ...
- 1,228
5
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 ...
5
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 ...
- 6,111
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 ...
- 10.2k
3
votes
What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?
The answer to "what to do?" is to discourage it as much as you can, arguing from the position of authority, if normal explanations of why it doesn't work fail (but certainly you should try ...
- 3,599
3
votes
Does a proof by induction have to explicitly refer to the principle of mathematical induction?
From a traditional standpoint in teaching high school math, clarity and pedagogy are paramount. A proof by induction traditionally includes an explicit reference to the principle of mathematical ...
- 1,937
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 ...
- 6,730
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 $0$ ...
- 10.2k
2
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 ...
- 21
2
votes
What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?
I think the easiest way to deal with behavior like this is to tweak the grading method to disincentivize it (and of course have a conversation with the class where you explain the behavior, why it's ...
- 6,111
1
vote
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 $n\in \mathbb{N}$...
- 111
1
vote
Does a proof by induction have to explicitly refer to the principle of mathematical induction?
Does a proof by induction have to explicitly refer to the principle of mathematical induction?
There's no obligation of course, but it seems to be good pedagogy, especially at ealy(er) stages, to ask ...
- 59
1
vote
What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?
Consider two concentric coplanar unit squares (their sides do not have to be coincident).
Show that the area of the region inside both squares is greater than $\frac{3}{4}$.
It's easier if we show ...
- 771
Only top scored, non community-wiki answers of a minimum length are eligible