86
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Why are induction proofs so challenging for students?
The following list comes from a combination of reading various research articles and my own experience helping students in my Maths Learning Centre for the last seven years.
Some reasons why ...
- 9,053
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 ...
- 18.4k
39
votes
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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 ...
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38
votes
Why are induction proofs so challenging for students?
In my experience, the biggest issue is that students don't have a clear grasp of quantifiers, so they don't see the distinction between "for all n P(n)" and "consider an n such that P(n)". This leads ...
- 11.6k
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 ...
- 20.1k
32
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I want a "true" proof by contradiction of an implication P => 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 $\...
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31
votes
Why are induction proofs so challenging for students?
There is a fair amount of research on students' understanding of (and difficulties with) proof by induction. Some good places to start:
Palla, M., Potari, D., and Spyrou, Panagiotis. (2012) ...
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31
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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: ...
- 24.3k
30
votes
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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 ...
- 24.3k
28
votes
What am I supposed to be learning with long proofs of the main theorems in class?
Hmm apparently I will be the dissenter here. I think that long proofs taught in lectures are very much a good thing. This is particularly true for hard proofs. I will try and split the reasons why I ...
- 1,018
25
votes
Why are induction proofs so challenging for students?
For CS students specifically, there is another approach that would work better than the usual way induction is taught, namely by teaching structural induction, which goes like this:
If you want to ...
- 2,558
25
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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 ...
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23
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What am I supposed to be learning with long proofs of the main theorems in class?
I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes ...
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23
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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 are induction proofs so challenging for students?
Lots of good answers here (I've upvoted many). I'm won't try to add to the discussion about why induction is hard, but I can suggest some approaches that have helped some of my students.
Many have ...
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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 ...
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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 ...
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21
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Why are induction proofs so challenging for students?
I think the main problem students have with induction proofs is that the ordinary direct proof works by reducing a statement with unknown truth value to one that is known as true.
The bulk of an ...
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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 ...
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20
votes
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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
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 ...
- 301
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, ...
- 462
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 ...
- 1,506
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 ...
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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 ...
- 279
16
votes
Why are induction proofs so challenging for students?
As someone who took math courses but does not teach, I would claim that inductive techniques are taught with two rather separate approaches:
A step-by-step recipe to take a problem that tells you to ...
- 1,225
16
votes
How to use false theorems or proofs?
I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it.
For smaller mistakes that I ...
- 20.1k
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 ...
- 24.3k
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 ...
- 9,079
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, ...
- 7,500
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