Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

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9
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1answer
122 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 ...
6
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5answers
332 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 $$\...
8
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3answers
211 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
200 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
249 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
633 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
392 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
908 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 ...
7
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0answers
113 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
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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
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2answers
316 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
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6answers
288 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. ...
7
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2answers
351 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
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2answers
365 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 ...
8
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2answers
291 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"...
2
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2answers
126 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
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0answers
90 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
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1answer
184 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'...
-2
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1answer
176 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
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1answer
86 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 ...
18
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5answers
733 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
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1answer
215 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-...
5
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1answer
265 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
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3answers
205 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
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0answers
98 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
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2answers
184 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
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3answers
153 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
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1answer
138 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
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2answers
194 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 ...
10
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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
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5answers
281 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
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1answer
361 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, ...
11
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3answers
255 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 ...
8
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4answers
654 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 ...
6
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3answers
245 views

Is induction or recursion easier to understand?

This is not really a new question, more a revisiting of @vonbrand's "Any suggestions on how to approach recursion and induction?" In an introductory programming class this past year, I asked the ...
19
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3answers
593 views

Constructive refutation of student misconception

Although @Gareth Shepherd recently posted Addressing fundamental math errors close to the issue, I experienced my problem of misunderstanding in class, where two good K10 students were asked to ...
9
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0answers
100 views

Literature on student understanding of assumptions

In a discussion with a physics lecturer he mentioned that one major area where students fail is understanding assumptions - for example, if we are interested in two objects hitting each other and then ...
5
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3answers
865 views

Why is this type of reductio ad absurdum not taught more?

I recently read in a book about a proof that Archimedes did. I don't remember the exact details and I don't have the book on me right now, but it involved proving an equality. So let's say proving ...
7
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4answers
191 views

Showing the Necessity of Proving the Impossibilities

"It's impossible because I tried but couldn't do it!" I need situations which shows that this kind of reasoning above is not working! I do have an example, but look for more: It's so hard to cover a $...
24
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7answers
3k views

Why do we care about multiple proofs of the same theorem?

I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
15
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4answers
503 views

How to explain what is wrong in this “proof” that $\sqrt N$ must be irrational?

Here is the problem that I asked undergraduate students of an introductory number theory course to prove: Prove that if $N$ is a nonsquare natural number, then $\sqrt N $ is irrational. Many ...
2
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3answers
405 views

When two equivalent algebraic statements have two “different” meanings

Suppose I want to prove $\sqrt{7}$ is not a rational number. I suppose it is and it brings me to a contradiction. Here how it goes line be line: First line. $\sqrt{7}=\frac{m}{n}$ Second line (...
7
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2answers
222 views

Ethics of looking at other proofs before submitting work

I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick ...
8
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3answers
236 views

Is proof-based exercise-oriented math course without solution an effective way to teach pure math?

In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of: notes written during the lecture ...
9
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4answers
570 views

Why are proofs by contradiction counterintuitive?

And how to make them intuitive? We are tasked to prove $P \implies Q$. So we assume $P$ and are trying to prove $Q$. We assume not-$Q$ ($\neg Q$) and derive a contradiction, establishing $Q$. There ...
5
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0answers
160 views

Catalog of undergraduate's misconceptions / problems while proving

Selden & Selden (2011) listed 41 difficulties their students had in an experimental proving course into 9 categories. Unfortunately I haven't found similar work. Thus, my question is: Is there ...
1
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1answer
77 views

Logic and arguing [closed]

When I was in school I studied mathematical logic and proofs, thinking on how to prove stuff on my own as practice. This can be useful to be able to influence others visa logical, undeceitful thought....
4
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0answers
175 views

How to promote more elegant and beautiful proofs by students?

Following the premise that mathematics is an art as well as a science, I want to encourage students to produce not only correct proofs but also to try to find a particular beautiful/elegant proof. ...
13
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3answers
217 views

Resource request: incorrect “proofs” for undergrads to correct/critique

I am teaching an intro to proof course for undergraduate math majors at a medium-sized american research university. I would like to provide my students with some incorrect proofs for the purpose of ...
8
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1answer
249 views

How do you assign a grade to a proof?

This question is very similar to one I posed two years ago: How to assign grades to proofs: what do(es) theliterature/experts suggest? I would like to ask the more general question of: what do you ...