Questions tagged [proofs]
For questions about mathematical proofs in an educational context.
36
questions
41
votes
25answers
10k views
Good, simple examples of induction?
Many examples of induction are silly, in that there are more natural methods available. Could you please post examples of induction, where it is required, and which are simple enough as examples in a ...
81
votes
21answers
22k views
Why are induction proofs so challenging for students?
This forum already has many
good, simple examples of induction proofs, a great resource.
As I am soon to teach induction for the $n^\textrm{th}$ time—this time to some perhaps under-prepared ...
20
votes
2answers
3k views
Example "bad proofs"?
As a sidetrack in this question it came up that it is important to have students read texts (in particular proofs) critically. As examples it is nice to have correct proofs at hand (presumably in the ...
9
votes
6answers
741 views
How to teach Proofs
I was taught in 9th grade the two column proof, and it wasn't until 11th (when I saw some number theory) that I realized what a poor method that is. However, it is certainly effective in getting ...
29
votes
7answers
2k views
Teaching logic with a proof assistant
I am thinking about teaching a university-level "introduction to proofs" class (mainly for math and CS majors) making use of a computer proof assistant like Coq. I feel like there is a lot of ...
28
votes
8answers
3k views
Is there a good age/level to start learning mathematical proofs?
I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math.
What is a good point to start learning proofs? what are ...
29
votes
10answers
6k views
Is Euclid dead? or Should Euclidean geometry be taught to high school students?
Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
21
votes
7answers
2k views
What are some good low-prerequisite examples for the heuristic advice "If you cannot prove it, prove something stronger."?
One useful trick in mathematics is to prove something stronger instead of the question asked.
This works well in induction proofs (because strengthening the claim also strengthens the induction basis)...
17
votes
5answers
9k views
Good examples of proof by contradiction?
In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter ...
16
votes
1answer
892 views
Is there any evidence about the effectiveness of "table proofs" in pre-college mathematics education?
I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol ...
16
votes
8answers
2k views
How to teach Mathematical Induction mathematically?
I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I ...
14
votes
11answers
5k views
When do college students learn rigorous proofs?
I teach in a regional university.
In my department, students take their "proof course" (a course that sole focus on writing proofs) in the third or even fourth year.
All the courses before ...
12
votes
2answers
471 views
Teaching logic through "high school algebra"?
I am going to be teaching a discrete math class in the fall. One of the major goals of the course is a solid understanding of the basics of logic: the precise meanings of "and", "or", "not", "implies"...
6
votes
3answers
311 views
Should one justify formulae in middle school?
Consider two possible lesson outlines:
Check homework.
Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279
Calculate the area of a circle as an example.
...
12
votes
2answers
833 views
Definitions/proofs that allow "useless" cases?
I often see students confused/mystified by definitions (and proofs) that allow/consider "useless" cases. A case in point is the definition of a DFA (deterministic finite automaton), which allows ...
8
votes
2answers
527 views
Should students be given partial scores when they gave an incomplete proof by contradiction?
In a quiz, there was a question asking students to show something doesn’t exist. A lot of them gave proofs by contradiction.
Initially, I designed the marking scheme so that an incomplete proof by ...
22
votes
9answers
3k views
"A computer program IS a proof": Introducing rigor via programming
This provocative essay
Igor Rivin.
"Some Thoughts on the Teaching
of Mathematics—Ten Years Later."
Notices of the AMS, Jun/Jul 2014.
(PDF download link).
suggests that a discussion of Igor'...
47
votes
24answers
18k views
How to explain Monty Hall problem when they just don't get it
Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job. ...
13
votes
5answers
806 views
Descriptive Thinking vs. Formal Writing
Sometimes I come across some exam answers which describe a proof sketch or a counterexample very well but are not written formally. Such proofs show that a particular student understands the general ...
25
votes
5answers
1k views
Inability to work with an arbitrary mathematical object
This question is motivated by student responses to homework and quiz problems I have recently posed in an undergraduate real analysis course. I will share some examples and observations first, to ...
16
votes
1answer
337 views
How to assign grades to proofs: what do(es) the literature/experts suggest?
I am teaching an introductory course on proofs in mathematics in a mid-size American public university, and trying to develop some kind of consistent grading meta-scheme for grading proofs. I am ...
13
votes
3answers
319 views
Any suggestions on how to approach recursion and induction?
Much mathematics is intimately tied to recursion, be it in definitions (like factorials and integer powers) and proofs by induction. This is also very relevant in computer science and programming. ...
24
votes
1answer
665 views
Is there a Piagetian age at which proofs can be comprehended?
I am wondering if there is literature on the developmental age
(pre-adolescent?, adolescent?) at which the notion of a "proof"
can be understood? I am less interested in mathematical proofs
and more ...
14
votes
2answers
563 views
The use of "$\therefore$" and "$\because$"
In schools, many students learn the usage of "$\therefore$" and "$\because$" in proofs. Such three-dot notation are popular in many high-school books and exams, but are almost ...
13
votes
6answers
481 views
Unique steps leading to a non-unique answer
When asked to show a math problem has a unique solution, students sometimes think that if an algorithm leading to a solution has unambiguous instructions at each step (no need to make choices at any ...
11
votes
3answers
457 views
Why don't textbooks explain proofs' discovery?
This question concerns only proven statements. I don't know if research papers do, but most math textbooks don't. Counterarguments:
Space?
1.1. The increased length from explaining the discovery is ...
9
votes
1answer
168 views
Motivation vs. Rigor
This is such a vague topic that I hesitate to post.
I constantly struggle between the time-tradeoff between
motivating a topic, and delving into the rigorous details necessary
to fully "grok" the ...
26
votes
12answers
2k views
Why do students like proof by contradiction?
Every-so-often I come across proofs of the form
Assume $X$ is false.
Prove $X$ is true (without using that it is false).
This contradicts that $X$ is false.
Hence $X$ is true.
I've seen students ...
11
votes
1answer
387 views
Effectiveness of students seeing proofs - reference request
If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated).
I am wondering if any research in ...
8
votes
4answers
790 views
May we permit identities to be established by equivalent equations?
A trigonometry text like Sullivan's Algebra & Trigonometry often has a prohibition like this (Sec. 7.3):
WARNING: Be careful not to handle identities to be established as if they were ...
7
votes
4answers
425 views
Writing up a proof that assumes what is to be proven?
I was working on this question on math, where (among other things), the OP was asked to prove that $$x \oplus y=\sqrt[3]{x^3+y^3}$$ is associative.
After some prompting, the offered proof was $$\...
20
votes
3answers
666 views
How can I discourage proof by patchwork?
I have a student who is working in their spare time on proving or disproving a conjecture of the form
$$\exists x.\forall y.\phi(x,y).$$
Right now their strategy is to construct an $x$ and then show ...
16
votes
5answers
944 views
Rigorous proofs vs. illustrative examples
No one would argue against the idea/ observation that proofs are very important in mathematics. Some people are trying to make their notations on a blackboard during a lecture as consistent as ...
6
votes
5answers
305 views
Different approaches to proofs that "are the same"?
This question (and answers) on MSE got me thinking on simple examples of different ways of proving the same (hopefully somewhat interesting) result, as examples to be discussed on difference in ...
3
votes
2answers
369 views
How is it correct for a lecturer to prove and "explain" a proof while explicitly knowing students are not familiar with logic itself?
I often see a situation when professors use words "logic", "mathematical proof" and even prove logically while actually knowing that students are not even familiar with logic itself, i.e. no formal ...
2
votes
6answers
602 views
Undergraduate Math Seminar topic
** Edit
Thanks everyone for some great suggestions. I should have been more clear though. I am actually looking for a college level proof that pertains to algebra or leads to algebra in some form. ...
