Questions tagged [proofs]
For questions about mathematical proofs in an educational context.
147
questions
1
vote
2answers
172 views
Should I do all the proof practice problems in How to Prove It, an intro to proofs book?
Like the title says. I am self studying intro to proofs(How to prove it by velleman) so I can start an introduction to analysis. I am wondering if I should complete all the exercises in the textbook(...
2
votes
1answer
146 views
Improving exposition of a proof about polynomials over infinite fields
This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero ...
6
votes
2answers
143 views
Is $\overline{AB} \cong \overline{BA}$ usually taught as an instance of the symmetric property of congruence?
I have been tutoring a wide range of math subjects for many years. Recently, I began tutoring a girl in high school geometry (in California, for context). This semester of the course is starting with ...
11
votes
3answers
299 views
How to teach the Pythagorean theorem in a satisfying way to high school students?
I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons:
The chosen proof feels like magic and I don't feel like I have a better understanding of ...
4
votes
3answers
198 views
Differing Choices of $\delta$ in a Limit
In conceptually motivating the $\epsilon-\delta$ definition and proof of a limit, I realized a new way of choosing the $\delta$.
For example, consider $\lim_{x\to 4}\sqrt{x}=2$. In the "standard ...
23
votes
23answers
5k views
How can I explain why we need proofs to someone who has no experience in mathematical thinking?
I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). ...
10
votes
4answers
438 views
How do you teach students about the concept of a proof?
I get this question a lot from new students who are taking their first proof-based math class. They are struggling because they don't have that fluency with proofs, to begin with. They don't know ...
5
votes
0answers
128 views
A video game for teaching the concept of a mathematical proof
Two students in my game development course would like, as a final project, to develop a video game for teaching mathematics. In contrast to the many other games for teaching maths, which focus mainly ...
5
votes
1answer
245 views
How important is it to come up with or learn an elementary solution?
Note: by "elementary" I mean "without using more advanced theory and tools".
Students are sometimes required or encouraged to solve very difficult problems using limited number of ...
-3
votes
2answers
280 views
Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$ [closed]
Numerous online resources parrot the usual proof by contradiction of the irrationality of $\sqrt{2}$. These all rely upon the assumption that the rational form (say, $a/b$) is in its simplest ...
8
votes
3answers
318 views
Transitioning proof based math courses online
I'd love to learn from anyone's recent experiences teaching online proof based math courses, especially those that have a large group of students who will be working asynchronously. My usual proof ...
5
votes
2answers
221 views
What is “mastery” in a mathematical topic?
This question was prompted by looking at Khan Academy's website to see how a comprehensive lecture series could be done and often I see the word, "mastery". To me, I'd think mastery is ...
8
votes
2answers
496 views
Logic and proofs in secondary school
Inspired by the question When do college students learn rigorous proofs?, I became curious when pupils in secondary schools learn about proofs, what kinds of proofs they are, how rigorously they are ...
7
votes
6answers
1k views
How to get better at proofs
As an undergrad student of applied mathematics, I have something to say that make's me ashamed of myself. I suck at proving things in mathematics and i know that if I don't get better in doing this ...
13
votes
11answers
4k 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 ...
14
votes
2answers
378 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 ...
18
votes
8answers
3k views
Why do some linear algebra courses focus on matrices rather than linear maps?
I hope the essence of the question is clear from the title. There are obvious advantages to making the linear map the central notion of a linear algebra course:
the notion can be illustrated with ...
12
votes
1answer
736 views
Proof by contradiction - more than one case
I am looking for some examples of when proof by contradiction is used in a problem with more than one case.
In all the elementary examples, there are only two options (eg rational/irrational, ...
7
votes
2answers
198 views
Do you teach different proofs or calculations of same question?
Recently I asked a question on math.stackechange about the most ways to differentiate the same function and it didn't seem to generate any interest - rather, the reason why I'd ask such a question was ...
4
votes
2answers
193 views
What would you recommend for the math thinking course for school?
We're going to make a new math course for kids as intermediary between middle and high school with math profile (for preparation to entrance exams to high school), and before the main part (arithmetic,...
23
votes
1answer
467 views
Taxonomy of bad proofs
I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ...
4
votes
5answers
329 views
Patterns that unexpectedly fall apart at large $n$
I am constructing a learning sequence for middle grade students designed to convince them that empirical arguments (arguments by example) are not sufficient in mathematics. To motivate this, I am ...
4
votes
2answers
457 views
Euclid Book 1 Proposition 4 [closed]
In Euclid's The Elements, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see ...
5
votes
2answers
225 views
Are there mathematical proof info-graphics?
I am teaching mathematical proof to kids (10th grade) and am of the opinion that proofs of theorems are a good place to start, where almost all of mathematics' important players come together.
On ...
1
vote
1answer
185 views
How to improve mathematical skills(University level)?
I am doing Ph.D in Mathematics, I feel I lack few of the skills, if I can improve those skills I think I can do better as a Math scholar. I need some suggestion on these following(below I am talking ...
8
votes
1answer
685 views
An alternative to “two column” geometry proofs
I'm a high school teacher in New York State (US), starting in on my first year of teaching Geometry. One of the things that really intrigues me is that the Regents exam (the state-mandated final exam)...
9
votes
1answer
160 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 ...
7
votes
4answers
397 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 $$\...
9
votes
3answers
280 views
How to motivate students to do proofs?
I am finding it difficult to motivate students on why they should how to prove mathematical results. They learn them just to pass examinations but show no real interest or enthusiasm for this.
How can ...
2
votes
2answers
225 views
Learning proofs in introductory analysis courses
I have browsed the website a lot and I encountered many similar questions but not a question that asks the same question as I intend to.
In introductory undergraduate classes in Analysis, usually, ...
6
votes
2answers
269 views
A question from a young student to mathematicians
I'm a young math student. And I live with the effort of always wanting to understand everything I study, in mathematics. This means that for every thing I face I must always understand every single ...
6
votes
5answers
664 views
Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula?
According to a recent experiment conducted by user Steven Gubkin, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is ...
15
votes
7answers
447 views
Should theorems be proved to students who are not majoring in mathematics?
My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not ...
7
votes
6answers
959 views
is it appropriate or beneficial to mention weird results in math?
Is it appropriate to mention weird/exciting results in math (or use as cautionary tales why one cannot apply mathematics naively) in say high school level?
Examples of these results include the ...
8
votes
2answers
261 views
Is there any example of a “forwards/backwards” induction?
I like to make the "dominoes" analogy when I teach my students induction.
I recently came across the following video:
https://www.youtube.com/watch?v=-BTWiZ7CYoI
In this video, a sequence of ...
14
votes
9answers
6k views
Why do inequalities flip signs? [closed]
Is there a mathematical reason (like a proof) of why this happens? You can do it with examples and it is 'intuitive.' But the proof of why this happens is never shown in pedagogy, we just warn ...
3
votes
2answers
347 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
546 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. ...
8
votes
2answers
462 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 ...
3
votes
2answers
451 views
Why are proofs written in flowery language incomprehensible?
Let's take an example in Wu-Ki Tung, Group theory in physics:
Theorem 3.4: Irreducible representations of any abelian group must be of dimension one.
Proof: Let $U(G)$ be an irreducible ...
12
votes
2answers
431 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"...
3
votes
2answers
154 views
Ideas for high-school proof class?
I have a math degree and have been hired to teach a proof class at a summer program. Our goal is to help the students learn the material they need for school (they take an algebra class separately) ...
6
votes
0answers
117 views
Questions similar to Wason Selection Task
The Wason Selection Task (described by Pete Clark here) is a great problem for getting students to grapple with all of the intricacies of logical implication.
I will be teaching a discrete ...
2
votes
1answer
493 views
What is the correct symbol to use for ending a counterexample?
I am familiar with the tombstone symbol, "$\blacksquare$", that is used to signify the end of a proof. However, it is my understanding that an example isn't technically a proof. For instance, one can'...
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votes
1answer
308 views
Proving basic Theorems and properties in high school [closed]
Why high school teachers do not emphasize knowing the proofs of properties and theorems in math. In my 40 years of teaching prospective high school teachers, I rarely found students who can derive ...
1
vote
1answer
99 views
Does studying elementary number theory improve one's proof skills and ability to understand algebra and analysis? [closed]
I'm taking a number theory course and don't know whether it's worth it. I currently can't understand algebra and real analysis and decided to take # theory to see whether this would help me prove and ...
24
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 ...
12
votes
1answer
274 views
How to write proofs on the board in the classroom
I'm teaching an introductory analysis course, and I am seeking some feedback on how proofs should be written on the board in class in order to maximize learning. I realize that there is an opinion-...
9
votes
2answers
440 views
Can some lovers of math truly never create something previously unseen?
Can someone truly love math, and master and remember discovered calculations, counterexamples, proofs; but still fail to invent anything new (e.g. incapacity to prove anything unseen, calculate ...
15
votes
3answers
250 views
How to teach students the value of concrete counterexamples?
I teach exercise sessions for a Linear Algebra course for 1st semester students in Europe. Students have to prepare some exercises at home. In class, I call on students to present their solutions.
...
