Questions tagged [proofs]
For questions about mathematical proofs in an educational context.
6
votes
0answers
91 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 ...
11
votes
9answers
4k 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
306 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
201 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
votes
2answers
328 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
304 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 ...
6
votes
2answers
200 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
votes
2answers
119 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
81 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
140 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
142 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
83 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
642 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
200 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-...
6
votes
1answer
247 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
199 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
89 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
138 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
145 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
127 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
184 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
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
271 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 ...
7
votes
0answers
281 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
votes
3answers
231 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 ...
9
votes
4answers
636 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
votes
3answers
223 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 ...
18
votes
3answers
541 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 ...
10
votes
0answers
87 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
votes
3answers
857 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
votes
4answers
187 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
votes
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
votes
4answers
479 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
votes
3answers
399 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
votes
2answers
214 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
votes
3answers
221 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
votes
4answers
554 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
votes
0answers
156 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
vote
1answer
74 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
votes
0answers
171 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
votes
3answers
209 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
votes
1answer
244 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 ...
14
votes
5answers
862 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
...
13
votes
3answers
530 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....
8
votes
1answer
210 views
Educational styles for writing proofs
Can someone please point to research papers that analyze different ways of expressing informal proofs from an educational point of view? I am particularly interested in proofs by induction but I would ...
6
votes
3answers
247 views
Should one justify formulae in middle school?
Consider two possible lesson outlines:
Check homework.
Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279
Calculate the area of a circle as an example.
...
11
votes
0answers
112 views
Research on the use of outlined / structured proofs in instruction
Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs?
For the ...
6
votes
5answers
238 views
Different approaches to proofs that “are the same”?
This question (and answers) on MSE got me thinking on simple examples of different ways of proving the same (hopefully somewhat interesting) result, as examples to be discussed on difference in ...
24
votes
11answers
1k views
Why do students like proof by contradiction?
Every-so-often I come across proofs of the form
Assume $X$ is false.
Prove $X$ is true (without using that it is false).
This contradicts that $X$ is false.
Hence $X$ is true.
I've seen students ...
10
votes
3answers
240 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 ...
