# All Questions

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### Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'd like to ask my students whether some real function is differentiable at a certain $x_0$. I prefer not telling them that they have to use the definition of the derivative, but to instead present a ...
205 views

### How do people 'master' a subject? What is the difference with 'memorizing' (not purely memorizing, but rather having it drilled inside the brain) it?

I'm in my 3rd year of pure math bachelors, and a thorny issue keeps on reappearing for me. I understand the theory, for a day, maybe two. I've reached the point where our lecturers no longer detail ...
1k views

### 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 ...
1k views

### Reference requests: Is there a text that is even more advanced than books on "advanced engineering mathematics"

Advanced engineering mathematics is a subject of its own, building up from simple notions of functions, series, integration techniques and brief review of linear algebra which leads to transform ...
170 views

### What do students gain — if anything — from learning proofs of corollaries, even when the more general result is thornier to prove?

Assume that students are fully capable of proving the more general, but labyrinthine result — because it lies within their zone of proximal development. the Inventor's Paradox doesn't appertain. ...
50 views

### High school student Teacher assistant limit

As a student asking teacher educators Is there a limit to how many teachers assistants you can be for, for a high school student (United States New Jersey)? If there is a limit, is it better to TA for ...
175 views

### 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? ...
301 views

### Applications of abstract algebra outside of mathematics and suitable textbook

The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra. What are some applications of abstract ...
99 views

### Worldwide standard textbooks vs textbooks from one's home country vs lecture notes by various people - pros and cons

So far I've had three types of professors in my undergrad studies when it comes to choosing the main text for the course: Type A: these are the professors who pick some standard textbook(s) in English ...
638 views

### Does there exist an international certification of mathematical skill?

I think that an international certification of math level would be really important also for non-math educators. Here in Italy primary school teachers actually teach math without being proficient at ...
290 views

### Is Trigonometry done differently in the US?

I'm Italian and I've watched some videos from Americans and noticed a weird thing. Let's talk about a linear trigonometric equation like this: $$\sin x+\cos x+\sqrt3=0.$$ I've seen Americans solving ...
582 views

### Hands on activities for a college history of mathematics course

I will be teaching a course in history of mathematics to juniors/seniors who are math and math education majors, many future school teachers. It should include highlights from antiquity to early 19-th ...
98 views

### Abstract math, examples and understanding or visualising

After reading some papers about special kinds of algebras and rings like Gorenstein rings, Dickson algebras, Cayley-Dickson construction, i want to ask do examples of general abstract objects in math ...
2k views

### How do I sketch a good gaussian curve freehanded, or by using only common sketching tools?

I'm a lousy artist. If I want my Gaussian curves to be accurately drawn when I use a whiteboard, or work with pen & paper, what are my options? Is there a way to use a straight edge, or compass, ...
192 views

### Should I upload slides before or after a class

I used to post my slides before a class. But I noticed that many students simply read it while in class instead of listening . So I am thinking not doing it in the future. But they can still get it ...
160 views

### For 15 year olds, are there exercises — with full solutions — on the Fence Post or Off by One error?

Which books contain practice questions — preferably with full solutions — to assist 15 year olds with the Fence Post or Off by One error? Most students at my institution have not heard of this name, ...
79 views

### Measures to quantify complexity of algebra equation

Like the title says, I am looking for ways to measure the complexity of an algebra equation. For now, I am focused on linear equations, but I would think any metrics could be generalized for ...
170 views

### Is copying working and explanation plagiarism in this context?

In my institution I have a friend who is part of this Math ambassadors club where they write blog posts and share them online at medium.com. However, there is an issue with my friends post.After they ...
140 views

### Mental Health in Mathematics

I am not sure if my question is relative to this meta but I still want to put forth my thoughts and concerns and questions because I think its not just me but others too who have similar issues. My ...
68 views

### Problem solving approach to learning and psychology

I try to have a problem solving approach to learning math. What i mean by this is if someone sets some questions or problems regarding the material i am reading how should i answer or what questions ...
16k views

### Given a 3 4 5 triangle, how do you know that it is a right triangle?

Without knowing the Pythagorean theorem, and in presenting reasons why the theorem might be true (without giving a full proof), is there any way to give examples of triangles that are intuitively ...
212 views

### Why has the chapter on second-order differential equations been moved to the website instead of being put in the book in Stewart Calculus 9th edition?

From the book The chapter on Second-Order Differential Equations, as well as the associated appendix section on complex numbers, has been moved to the website. It doesn't mention a reason in the ...
471 views

### Aspiring HS Math Teacher: Textbooks for learning Algebra, Geometry, Trigonometry, and Calculus?

I plan to study for 6-12 months to take a high school mathematics teaching license exam. This one to be exact. Florida Teacher Certification Examinations Test Information Guide for Mathematics 6–12 ...
3k views

### 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. ...
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)...
169 views

### How do you study subjects you're not that interested in

I'm an undergraduate who doesn't find analysis particularly interesting, but I'm taking a calculus on manifolds course next semester, so I'm reviewing measure and integration theory since my grasp on ...
263 views

### How can I visualize differential equations and Integration in real life?

How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We ...
720 views

### What are some recent, interesting, accessible pieces of mathematics

Mathematics can come across as a sterile, dead subject - a catalogue of techniques long-ago decided, and forever relearned by each successive generation of students. This is approximately true for ...
150 views

### 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 ...
115 views

918 views

### 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 &...
92 views

### Questions to help better understand the textbook

I am teaching a linear algebra class for math majors and non-majors out of the first 4 chapters of Lay's book. My plan is to have the students read a section prior to each class, have them answer a ...
136 views

### What do you think, is teaching on an actual board more efficient than using an online board?

I am a sophomore math undergraduate and so far all of my university courses have been online due to the pandemic. I am really curious what you guys think about the efficiency of teaching mathematics ...
759 views

### Are there examples of countries where the use of CAS systems or graphing calculators was deemphasized or discontinued?

In the last 30 years more and more countries introduced graphing calculators and then CAS systems to their high-school students. But are there already any examples of a trend in the opposite direction?...
163 views

### 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, ...
8k views

### Explanation for cutting a Möbius strip at one-third its width

Can anyone offer a concise, convincing explanation for why cutting a Möbius strip along a line, not midway but rather one-third of the width of the strip, and eventually joining back to itself, ...
354 views

### Is it more efficacious, productive to jump to perusing full solutions — before and without attempting to solve problems?

Too many students lack the luxuries of time and effort to mull exercises and problems. They must juggle MULTIPLE jobs to pay exorbitant tuition fees. Single parents or adult learners must prioritize ...
18k 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. ...
1k views

Metacognition seems pretty universally positive. I'm wary of viewing it as such. Aside from the obvious criticisms like "you can't learn to ride a bicycle by thinking about and writing a 200 page ...
384 views

### Explaining the intuition for why finding roots of polynomials is hard

I'm currently teaching a mini-seminar to high school students, most of whom have at most a background in Algebra/Algebra II (in the US high school system) about finding roots of polynomials. In ...
172 views

### Introducing direct substitution in an intro calculus course

I'm revisiting the materials I've put together for students taking a non-proof-based intro to calculus, and my goal is for them to have a clear but rough sense of a limit as a bound (basically enough ...
1k views

### Question formats for online tests, to deter cheating

I'm teaching calculus 1 online this term and anticipate being plagued by the perennial problem of cheaters. I have seen suggestions for how to arrange the testing time to accommodate for traditional ...

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