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6
votes
3answers
122 views

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 ...
4
votes
3answers
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 ...
5
votes
4answers
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 ...
11
votes
6answers
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 ...
-1
votes
0answers
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. ...
-2
votes
0answers
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 ...
-2
votes
2answers
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? ...
3
votes
4answers
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 ...
2
votes
0answers
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 ...
7
votes
1answer
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 ...
4
votes
2answers
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 ...
17
votes
7answers
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 ...
0
votes
1answer
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 ...
11
votes
3answers
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, ...
9
votes
1answer
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 ...
3
votes
0answers
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, ...
2
votes
0answers
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 ...
3
votes
2answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
27
votes
18answers
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 ...
4
votes
2answers
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 ...
6
votes
6answers
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 ...
6
votes
2answers
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. ...
21
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)...
3
votes
2answers
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 ...
0
votes
3answers
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 ...
15
votes
5answers
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 ...
1
vote
1answer
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 ...
2
votes
1answer
115 views

Why is my 8th grade Algebra 1 tutoring student learning mean absolute deviation and standard deviation?

I’m tutoring an 8th grade student in Algebra 1, and he showed me that their class learned how to find standard deviation and mean absolute deviation using the following formulas: $SD=\sqrt{\...
1
vote
1answer
124 views

Best books for mathematical statistics self-study?

I'm hoping to start a masters in mathematics in the fall, and am hoping to find a good book on mathematical statistics to study so that I'll be able to take graduate level mathematical statistics once ...
2
votes
4answers
202 views

How can 17 y.o. high school students intuit that P(n, r) stops at $n - (r - 1)$, not $n - r$?

Every year, some 17 y.o. student makes the mistake of stopping $P(n, r)$ at $\color{darkorange}{(n - r)}$, rather than $\color{forestgreen}{(n - (r - 1))}$. Because they are in their last year of high ...
9
votes
1answer
366 views

Advice on Proof-based Math Topics for High Schoolers

I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent ...
0
votes
0answers
144 views

What books teach the formula for the # of k-permutations of n objects, with x types, and $r_1,⋯,r_x$ = the number of each type of object?

Some of my 16 year old students hanker after the formula for the # of k-permutations of n objects, with x types, where $r_1, ⋯, r_x$ = the number of each type of object. This is more generalized than ...
6
votes
1answer
216 views

How to convince a high school student that the $=$ symbol denotes identity?

In French language, arithmetic statements are often read, at the elementary school level, as , say, " deux et deux font quatre" , i.e. something like " two and two make four". Out ...
27
votes
5answers
2k views

What are some good examples to motivate the implicit function theorem?

I always had problems teaching the implicit function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to provide easily accessible examples ...
12
votes
1answer
214 views

tutorial active learning

This is a question I asked on [Academia.se]. It did not get an answer, so I am re-posting it here. In the country where I live, university students studying mathematics usually attend lectures, ...
26
votes
12answers
2k 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 ...
14
votes
3answers
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 &...
3
votes
1answer
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 ...
5
votes
0answers
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 ...
11
votes
2answers
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?...
3
votes
0answers
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, ...
16
votes
6answers
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, ...
1
vote
0answers
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 ...
47
votes
24answers
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. ...
17
votes
1answer
1k views

Is metacognition ever bad?

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 ...
9
votes
3answers
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 ...
1
vote
1answer
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 ...
10
votes
6answers
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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