Questions tagged [proofs]
For questions about mathematical proofs in an educational context.
139
questions
2
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1answer
133 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
253 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 ...
7
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3answers
271 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
206 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 ...
7
votes
2answers
423 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 ...
6
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
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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
317 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 ...
17
votes
7answers
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
719 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
189 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
186 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,...
20
votes
1answer
339 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
316 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
424 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
218 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
165 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
602 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
156 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
382 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
259 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
222 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
268 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
653 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
424 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
943 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
1answer
190 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
5k 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
340 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
451 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
407 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
417 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
411 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
146 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
107 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
357 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
242 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
95 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 ...
23
votes
5answers
968 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
255 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-...
7
votes
1answer
279 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
235 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.
...
2
votes
0answers
109 views
Strategies for learning proofs
What are the best methods for learning proofs? I'm tasked with learning two dozen proofs about the properties of continuous functions and real numbers in a week well enough to be able to present them. ...
3
votes
2answers
248 views
Is the Nomenclature of Triangle Congruency Proofs Consistent?
My Geometry class is doing triangle congruency proofs these days. In general, we find three pairs of congruent parts (sides or angles) in two triangles; we show that these congruencies reveal that the ...
12
votes
3answers
161 views
Tasks that encourage argumentation
I am looking for resources that have tasks such as the one below that encourage argumentation. I want tasks that 8th graders could do but would also be appropriate for high school students. I want to ...
5
votes
1answer
172 views
Tutoring Discrete Mathematics
A few weeks ago, I started tutoring a student in Discrete Mathematics (a subject I took a year ago).
I have previously tutored both pre-calculus and calculus, but never a proof based class. I have ...
0
votes
2answers
287 views
Is it possible to have taken intro to proofs, calculus 3 and differential equations and still lack the ability to do proofs?
Ideal Undergraduate Sequence
Main question:
I looked above and what I'm interpreting out of it is that one should be able to do proofs after studying some intro to proofs class, calculus ...
11
votes
5answers
1k views
Book request: teaching proving and reasoning at an American university
I am a European postdoc who recently teaching at a large public university in the United States. I will have to teach a course for undergraduate students that introduces them to proving and reasoning ...
13
votes
5answers
325 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
1answer
384 views
Using number theory instead geometry to introduce proof in Basic School?
It seems there is an overall agreement that Geometry is the right place to introduce proof in Basic School. However, number theory (arithmetic) looks like to be a more simple environment (consider, ...
