Questions tagged [proofs]
For questions about mathematical proofs in an educational context.
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Explaining why volume of cone is a third of cylinder
I came across this video explaining to kids why the volume of a cone is a third of the cylinder of same cross-sectional radius and height. Essentially the explainer presents pre-created cylindrical ...
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How to understand the book and the material to the deepest possible level?
I'm a first year mathematics major and I have a problem with my learning process. In my university, I only have books and questions that the university published, so I have to learn the most of the ...
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Should students in a first university linear algebra class be taught to write simple proofs?
I am teaching an introductory linear algebra courses for undergraduate students in math, computer science, or data science at a liberal art university. Most of the students have not decided their ...
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Proofs with Independent Parts
In my Intro to Proofs course I have an activity (more details below) where the goal is to write a collaborative proof of a statement. What makes it work as a whole class activity is that, to prove the ...
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Seeking short algebraic proofs that an Algebra2 student can appreciate
There are many many elegant algebraic (more generally, non-geometric) proofs, but fewer of them are both accessible and interesting to a pre-calculus student. Three nice examples of what I'm looking ...
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How can students prognosticate to rewrite the same sum backwards, then add the same sum twice?
This comment doesn't fulfill my students or me, because it doesn't demystify this trick of writing $S_n$ forward, then backwards, then adding. What would spur students to action these unnatural steps? ...
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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 ...
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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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How long would it take someone to master the topics in the book "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 ...
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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 &...
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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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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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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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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 &...
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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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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, ...
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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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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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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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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 ...
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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 ...
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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 ...
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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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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). ...
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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 ...
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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 ...
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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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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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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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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 ...
user13544
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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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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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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 ...
user13395
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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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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 ...
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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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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 ...
user13544
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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)...
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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 ...
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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 $$\...
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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 ...
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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, ...
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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 ...
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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 ...
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