Questions tagged [proofs]

For questions about mathematical proofs in an educational context.

Filter by
Sorted by
Tagged with
13
votes
4answers
729 views

Proofs that make theorems less clear

Teaching Theory of Computation for the first time, I encountered a phenomenon which perhaps is familiar to others in different contexts. I realize most MESE participants are not conversant with Th....
12
votes
4answers
298 views

Collaboration on math homework assignments?

There is considerable evidence that pair programming, when executed properly, both increases the accuracy of the code produced and enhances the learning of both participants. I wonder if anyone has ...
64
votes
21answers
16k 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 ...
9
votes
1answer
123 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 ...
34
votes
17answers
6k 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 ...
15
votes
3answers
660 views

Is it natural for self-learners to forget most proofs of the theorems they learn?

When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it? I ask this question as I'm a self learner ...
6
votes
5answers
333 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 $$\...
16
votes
5answers
1k views

Should my 8th graders see a proof of the Pythagorean Theorem?

I've been teaching the Pythagorean Theorem in my 8th grade class, and I noticed something odd. In the book I'm using, the sequence goes something like this: Motivate the idea of distances on a grid ...
11
votes
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, ...
14
votes
1answer
843 views

Proving theorems on one's own: how long should one persist?

I've recently started learning linear algebra on my own. I always try to prove the theorems I encounter by myself, without looking at the book (only to check if my proof is correct), because I found ...
8
votes
3answers
212 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
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
votes
2answers
250 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
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
votes
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 ...
41
votes
21answers
16k 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. ...
7
votes
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 ...
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 ...
7
votes
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 ...
7
votes
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 ...
13
votes
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 ...
3
votes
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
votes
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. ...
3
votes
2answers
366 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
votes
2answers
293 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"...
19
votes
3answers
594 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 ...
2
votes
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) ...
20
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)...
6
votes
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
votes
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
votes
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 ...
7
votes
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 $...
1
vote
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
votes
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 ...
10
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 ...
12
votes
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
votes
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
votes
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. ...
18
votes
4answers
309 views

Multiple Solutions Methods vs. Encouraging a Particular Approach

It happens frequently in math that problems have multiple possible solutions. This might become troublesome, e.g. when students use some other approach, hence, not learning the current topic. One ...
2
votes
0answers
99 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
185 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
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 ...
28
votes
10answers
4k 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!" (...
5
votes
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
votes
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 ...
8
votes
4answers
655 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 ...
11
votes
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 ...
15
votes
2answers
313 views

Teaching strong induction instead of induction

After teaching induction and then strong induction (i.e. the version where you assume $\forall k<n, P(k)$ and prove $P(n)$), one of my students asked why we ever use ordinary induction, since ...
6
votes
3answers
246 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 ...
25
votes
11answers
2k views

Impressive examples where a “proof by picture” goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...