Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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1answer
139 views

Math outside of undergraduate studies and proofs

I read sometimes mathematical works of others outside my undergraduate studies. I think i can not follow the understanding of the proofs of theorems sometimes. What should i do? Should i read other ...
6
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2answers
3k views

Should proofs include a third “context” column?

Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. ...
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4answers
974 views

How long would it take someone to master the topics in the book like Book of Proof by Hammack and similar?

If someone never had any experience with mathematical proofs and had only classes like Calc I-III (which he passed) without paying any attention to the proofs present in the textbooks, how long would ...
13
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3answers
904 views

Is there a measurable learning goal related to understanding proofs of important theorems?

I believe that good math courses are structured around measurable learning goals. For example, "can correctly replace a line integral with an equal double integral using Green's Theorem" or &...
3
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0answers
154 views

Doctorate and examples of difficult solved problems

Okay. My questions are: How do some people do doctorates in mathematics and spend so much time like three to six years trying to answer one or two open problems? How do they have the patience, ...
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2answers
591 views

Questions about proofs

To prove, e.g., the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$, I remembered working, in high school, in the following way. Expanding the LHS gives \begin{equation} (a^2+b^2)(c^2+d^2)=a^2c^2+a^...
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1answer
345 views

Solving math problems and learning

Should i solve math problems by writing the answers to papers or notebooks with pencils or should i try solving them in my head at undergraduate studies at university? Also, sometimes after learning ...
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5answers
3k views

I want a "true" proof by contradiction of an implication P => Q

When teaching proofs by contradiction of an implication P => Q, one starts by assuming both P and (not Q), and then reaches a contradiction. The problem is, most elementary proofs of this type are &...
6
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4answers
3k views

How to keep students' attention while teaching a proof?

I taught the proof inclusion-exclusion principle to CS students yesterday. While the proof is not too long, it does involves quite a bit notations. I could feel that most students lost interests a few ...
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1answer
68 views

Abstract math and making proofs

What is abstract math about? I think we can not visualise probably what we read. Or can we? I am talking about the theorems, definitions and proofs in areas of math like Riemannian geometry, ...
2
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4answers
568 views

Trials and instructions on proofs

So, in more advanced courses and fields in math like undergraduate and more advanced, how does one make a proof? He makes trials of everything he knows about the problem? What are the instructions to ...
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5answers
634 views

Example of why proof by exhaustion is inelegant

There's a nice example of why people dislike proof by exhaustion on the Wikipedia page. The problem statement is "prove that all years in which the Modern Olympics are held are divisible by 4&...
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2answers
217 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(...
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1answer
155 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 ...
7
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2answers
159 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 ...
12
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3answers
421 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
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3answers
213 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 ...
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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). ...
11
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4answers
531 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
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0answers
157 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
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1answer
260 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 ...
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2answers
318 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 ...
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3answers
358 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 ...
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2answers
266 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 ...
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2answers
542 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 ...
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6answers
2k 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 ...
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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 ...
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2answers
558 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 ...
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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
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1answer
775 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, ...
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2answers
206 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 ...
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2answers
203 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,...
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4answers
1k 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
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5answers
343 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
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2answers
474 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
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2answers
242 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 ...
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1answer
244 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
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1answer
744 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
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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 ...
7
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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 $$\...
9
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3answers
291 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
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2answers
234 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
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2answers
272 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
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5answers
674 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
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7answers
478 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
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6answers
973 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
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2answers
353 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 ...
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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
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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 ...
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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. ...