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For questions about teaching students at the undergraduate (university) level.

5
votes
1answer
91 views

Why emphasize moment generating function over characteristic function in a probability course?

I've noticed that some undergraduate introductory probability textbooks and courses emphasize or seem to prefer the moment generating function $m(t) = \mathbf E(e^{tX})$ of a random variable $X$ ...
3
votes
4answers
134 views

What are standard (or good) textbooks for undergraduate graph theory?

I'll be teaching graph theory this fall for the first time. The only undergraduate graph theory book I am familiar with is Doug West's book, which I like. But I'd like to consult some other ...
3
votes
2answers
252 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 ...
4
votes
1answer
106 views

Polar form before Cartesian form when introducing complex numbers

When I teach complex numbers to undergraduate engineering students, I invariably start, as appears to be customary, with $a + bi$ (or $a + bj$ for electrical engineers) and then follow up with the ...
4
votes
0answers
119 views

Learning math historically

What is meant by learning math historically (NOT learning math history only, but learning math with a historical development perspective)? I've seen some sources that to learn a math topic X, you need ...
5
votes
2answers
214 views

Trends in math education: Majors? Applied math? Statistics?

I would be quite interested in learning of trends (in the U.S. or internationally) of student interest at the undergraduate level in pure math vs. applied math, say, measured by majors at graduation. ...
10
votes
7answers
478 views

Why don’t all professors let students use notes, books, etc. on exams?

Last semester I had a teacher who let us use any type of information in the exam, for example the course notes, books, solved exercises, etc. The only thing he did not let us use was something ...
1
vote
2answers
86 views

Subject advice in Number Theory [closed]

At my University, we have the optional feature to write a project like a Bachelor Thesis. This semester have finished and I would like to work in the summer in project like this. So, I'm searching for ...
3
votes
2answers
92 views

Supplemental text for undergraduate real analysis

Context: I am an assistant professor at a small college in the US. Next semester I am teaching real analysis for the first time, and we are using Steven R. Lay's book. (It also happens to be the ...
9
votes
1answer
216 views

How to deal with poor students who don't take notes?

I want to set the context for me asking this question before stating it properly. I teach at college/university level. This question deals with first-year students, fresh from school. So think ...
4
votes
3answers
208 views

Teaching science and engineering students the field of inverse problems

There is a Mathematics Stack Exchange question on a good book on inverse problems for engineers. Here, I would like to ask for suggestions on how to approach teaching undergraduate upper-division ...
4
votes
3answers
193 views

Limit of questions that a student should ask in class without upsetting professor?

Days ago a professor told me to ask in class when I don't understand something and that he could keep explaining until it's clear for me. Being honest I'm a little slow to understand mathematics in ...
5
votes
3answers
292 views

Justifying the multi-variable chain rule to students

Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$. This is an exercise ...
19
votes
8answers
3k views

What is the point of teaching variance?

I am a teaching assistant for a sophomore engineering laboratory. We use standard deviation a lot during the semester. It is an incredibly useful concept that can be used in a lot of engineering ...
10
votes
3answers
214 views

Can or should students do research in standard major math courses

The following is an expectation for our "course-based research initiative". I'll include the complete wording so you can best understand my question. Designing a Research Proposal/Project ...
4
votes
0answers
136 views

Intuition: 5 regular polyhedra, 6 regular 4-polytopes, and then 3 regular d-polytopes

I have struggled to offer an intuitive explanation (to U.S. college students) why the number of regular polytopes in dimension $d$ is: $d=2$, number: $\infty$. $d=3$, number: $5$, the five Platonic ...
7
votes
3answers
215 views

How to resolve the new definition of subtraction and division seen in college algebra?

Here's the foundational thing that irritates me the most when teaching college algebra. Up through the secondary level, I think that instructors and students are trained to understand subtraction and ...
14
votes
1answer
186 views

How to teach ordinary differential equations to good students?

I am TA-ing a introductory course on ODEs and PDEs this year. At my university most introductory math courses can be taken at "basic" and "extended" levels. This one is the extended one. My students ...
4
votes
1answer
113 views

Reference request: undergraduate combinatorial topology

I teach at an American research 1 university. I am planning a course on combinatorial topology for undergraduates whose background is: multivariable calculus linear algebra at least one proof course ...
14
votes
3answers
326 views

What to do if all students lack prerequisites?

I am teaching a calculus class for business this summer (6 students) and all of them do not have the math background needed for the class. We are supposed to cover derivatives and integrals, but they ...
28
votes
4answers
518 views

How can I help a student who has a “wrong” kind of enthusiasm?

Alice (not real name) is a student in one of my Math 100 (calculus) classes. It's a course offered by my college as a dual credit course at a high school, so the whole class is about 17/18 years old, ...
6
votes
2answers
138 views

Effective Assessment that's Easy to Grade

A colleague of mine will be teaching 3 classes (pre-calculus and two sections of calculus, at the university level) next semester with an additional grader in only one of those classes (pre-calculus). ...
3
votes
1answer
182 views

Why do the stages of rigorousness have specific timestamps?

This is a reduced quote from There’s more to mathematics than rigour and proofs of Terrence Tao (emphasis mine): The “pre-rigorous” stage, in which mathematics is taught in an informal, ...
2
votes
1answer
126 views

Why aren't Bayesian Networks and Variable Elimination introduced earlier?

Throughout my undergrad, I dreaded probability. I hated it, I was horrible in it, I just never got it, and felt stupid when the professors used "summation/marginalization" equations out of the blue to ...
6
votes
2answers
253 views

Why most people think that :$(fg)'=f' \cdot g'$?

let $f$ and $g $ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?
3
votes
1answer
136 views

Effective computer lab layouts for a university math class

Many math classes benefit from occasionally being held in a computer lab. My question is about the pros and cons of different layouts and mechanics of a lab and "solutions" you have found to be ...
4
votes
2answers
87 views

Make a matrix algebra course (1st university year) more “project-based”

Among other courses, I'm teaching a (basic) matrix algebra course for 1st year university students (they are studying Economics, and the cursus leads them to management, finance, or econometrics in ...
8
votes
2answers
190 views

Computational Software for the whole curriculum and beyond

Our (United States, undergraduate) math program is considering the idea of putting more mathematical modeling and computation into all levels of our curriculum. One of the hang-ups is that we can't ...
2
votes
0answers
81 views

How to explain concepts of limit and continuity to non-mathematical students

How to explain fundamental concepts of limits and continuity to a non-mathematical background student? I am a PhD student in Mathematics working in Differential Geometry. As a part of my teaching ...
11
votes
1answer
140 views

Motivation for uniform continuity

What are some problems or theorems that motivate the distinction between continuity and uniform continuity? In particular, I would like: a) A useful, appealing theorem that applies to uniformly ...
11
votes
4answers
280 views

How would you introduce Frullani integral to students?

Some integration techniques are just "tricks", while some integrals are analytically significant in that they connect different fields of math or they embody higher level concepts. In the commonly ...
8
votes
3answers
158 views

Resources on solving systems of polynomial equations in multivariable calculus setting

Whenever I teach multivariable calculus I find students really struggle with both finding critical points and the method of Lagrange multipliers. I think that the reason is the same: solving systems ...
7
votes
2answers
193 views

How to teach Leibniz and Newton's notation

There has been many posts here and in MSE about different notations of differentiation. See for example this, this and this. However those questions only deal with the common misunderstanding about ...
25
votes
9answers
6k views

Should college mathematics always be taught in such a way that real world applications are always included?

I am teaching Linear Algebra this semester with the textbook Introduction to Linear Algebra by Serge Lang and most (perhaps all?) my students are not majoring in mathematics. As I was carefully ...
13
votes
4answers
250 views

Pedagogical advice/articles for graduate student teaching assistants

Are there any good pedagogical resources or articles that you would recommend to math graduate student teaching assistants (TAs)? Is there any sweeping advice that you would give a TA to improve their ...
2
votes
0answers
62 views

Is the annual system still in vogue anywhere in Europe, North America, Australia, or East Asia? And what about the 2-year B.A. / B.Sc. degree?

In Pakistan, we have until now had 2-year B.A./B.Sc. and 2-year M.A./M.Sc. university degrees after our 12th grade F.A./F.Sc. qualifications. Thus the former was a 14-year academic qualification, ...
18
votes
5answers
625 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 ...
19
votes
8answers
6k views

Why do no students know to change the limits of integration when doing substitutions?

I've TAed and tutored calculus for years and of the hundreds of students I've interacted with, it is always a shock when I tell them to change the limits of integration when they do substitutions. ...
17
votes
3answers
252 views

Polymorphic functions in vector calculus

While teaching multi-variable calculus for the first time in a while, I came across a tricky notational point in our textbook (Thomas' calculus - I'm not sure how widespread this notation is). When $\...
4
votes
0answers
102 views

Are there any high school level summer program that teaches Analysis?

All summer programs I know for highschool students focuses on number theory, combinatory, graph theory, logic, and all kinds of stuffs in discrete mathematics. (I am mainly interested in UK, US, ...
6
votes
2answers
230 views

How to teach real analysis?

I am recently going to make a series of videos about real analysis and measure theory. I wonder if anyone can give me some suggestions on how to arrange the material of the course. Should I introduce ...
18
votes
3answers
620 views

What are the minimum criteria when checking homework for completion only?

I know that some instructors collect homework and "grade that on the basis of completion" (e.g., item #2 on this answer). In fact, I tried this myself for several years, based on advice from my mentor ...
26
votes
3answers
661 views

Teaching undergraduates who expect a high-school-like learning environment

tl;dr: Some students expect to be told "what's on the test", to memorize and then move on. What can be done to change how they learn while teaching them what to learn? Context: Introductory, ...
6
votes
3answers
297 views

Online Accredited Mathematics Course

I am not sure this is the best or correct place to ask this question, but I thought I would give it a shot. I am currently deployed overseas (from the US) and prior to this I was attending a ...
12
votes
4answers
406 views

Real-world Markov chains

I will give a talk to undergrad students about Markov chains. I would like to present several concrete real-world examples. However, I am not good with coming up with them beyond drunk man taking ...
8
votes
2answers
142 views

Resource to supplement to Euclid's Elements

I am an instructor at a mid-sized American university, preparing to teach a two-quarter geometry course for junior and senior math majors. My plan is to use Hartshorne's "Geometry: Euclid and Beyond," ...
7
votes
2answers
151 views

Simple initial value problems - pros and cons of different methods

Consider the problem: Find $f(x)$ if $f’(x)=4x$ and $f(3)=12$ I have always done this, and taught it, as a two-step problem: First, find the general anti-derivative, $f(x)=2x^2+C$, and then plug ...
12
votes
3answers
142 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 ...
7
votes
2answers
118 views

“Personalized System of Instruction” (PSI) vs. “Individually Prescribed Instruction” (IPI)

This question may be a bit overly-broad for MESE, but I am hoping to find some responses that can help to fill in my understanding of two similar forms of instruction that had their heyday in the ...
9
votes
3answers
797 views

Why teach reference angles?

Reference angles are most useful for limited trigonometric tables with values from 0 to $\frac{\pi}{2}$. They can also help illustrate the periodicity of trigonometric functions, but I feel this is ...