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Questions tagged [undergraduate-education]

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

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100 votes
20 answers
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Unique candidate that fails

In the comments to David Speyer's answer here, he points out that "the distinction between 'if there is a formula, it is this one' and 'this formula works' is subtle." Does anyone have any simple, ...
Steven Gubkin's user avatar
87 votes
21 answers
26k views

Why are induction proofs so challenging for students?

This forum already has many good, simple examples of induction proofs, a great resource. As I am soon to teach induction for the $n^\textrm{th}$ time—this time to some perhaps under-prepared ...
Joseph O'Rourke's user avatar
76 votes
20 answers
19k views

Impressive common misleading interpretations in statistics to make students aware of

Statistics are used everywhere; politicians, companies, etc. argue with the help of statistics. Since calculations are needed for the interpretation of statistics, such things should be taught in ...
Markus Klein's user avatar
  • 9,438
74 votes
17 answers
10k views

How shall we teach math online?

Many universities, including mine, are now requiring we teach our courses online because corona. How shall we do this? Let’s brainstorm here. Some challenges: My school provides limited online ...
Stephen Herschkorn's user avatar
61 votes
4 answers
6k views

Is it worth grading calculus homework?

I am a young math educator. I've TAed four semesters of calculus for various instructors. Some instructors have required me to grade selected problems in homework sets. Another required me simply to ...
abnry's user avatar
  • 852
54 votes
13 answers
12k views

How do I motivate my students to go to office hours?

I'm currently TAing a Linear Algebra class where a significant portion of the class is struggling, oftentimes getting marked down on homeworks or tests because they misunderstand some concept (rather ...
user avatar
52 votes
15 answers
13k views

How can we help students learn how to read their textbook?

In most secondary and early undergraduate courses, students purchase expensive and carefully-written textbooks. These textbooks contain, roughly, three things: Exercises and Answers Reference ...
Chris Cunningham's user avatar
48 votes
8 answers
13k views

What do math majors (actually) do after graduation?

It's the time of year for prospective college freshman in the US to make campus visits, and I'm once again confronted with my lamentable ignorance when the students and their parents ask, "So what do ...
Mark Meckes's user avatar
  • 1,479
47 votes
12 answers
31k views

What should be included in a freshman 'Mathematics for computer programmers' course?

Many universities are changing up the way that they teach math service courses. 1-3 semesters of calculus and maybe a course in linear algebra are often included in majors (such as computer science) ...
Brian Rushton's user avatar
46 votes
16 answers
32k views

How is calculus helpful for biology majors?

It's common for majors in biology to take calculus courses, and many calculus textbooks (and calculus professors) try to cater to these students by including applications to biology. My question is, ...
Jim Belk's user avatar
  • 8,279
45 votes
18 answers
3k views

How to teach logical implication?

One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies ...
Benoît Kloeckner's user avatar
44 votes
4 answers
3k 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, ...
Brendan W. Sullivan's user avatar
43 votes
15 answers
21k views

Why do we teach complex numbers?

In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex ...
Simply Beautiful Art's user avatar
42 votes
5 answers
3k 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, ...
Torsten Schoeneberg's user avatar
41 votes
4 answers
5k views

Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra, Hodge et al's ...
J W's user avatar
  • 4,733
40 votes
13 answers
5k views

Beautiful planar geometry theorems not encountered in high school

I would like to impress college students (undergraduates in the U.S.) that there is more to planar geometry beyond what they learned in high school. I would like to show them beautiful theorems they ...
Joseph O'Rourke's user avatar
39 votes
16 answers
14k views

How to give exercises when students can use ChatGPT

I tried some math exercises we will give to students and ChatGPT does really well answering these. It excels at proofs and often gives details that were not our the example solution, and makes some ...
allo's user avatar
  • 496
39 votes
14 answers
7k views

How to make Calculus II seem motivated, interesting, and useful?

I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our ...
Frank Thorne's user avatar
  • 2,249
38 votes
6 answers
4k views

How can I give feedback that is not demotivating?

Background: To cope with online education, I taught linear algebra using a variant of the flipped classroom. I recorded videos and put them up on YouTube and students presented the content in these ...
Divakaran Divakaran's user avatar
37 votes
13 answers
3k views

Examples why university education is important for future high school teachers

At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically ...
Markus Klein's user avatar
  • 9,438
37 votes
10 answers
9k views

Combative students in proofs classes

When teaching my first discrete math class recently, I found a subset of about 5 out of 35 of my primarily computer science students who I struggled to reach. If these students simply struggled with ...
Opal E's user avatar
  • 4,043
37 votes
5 answers
13k views

Which cognitive psychology findings are solid, that I can use to help my students?

I read recently on this site that the growth mindset seems not to be real. I did not know that (I admit that I don't follow research into learning as closely as I would like). Can I turn that ...
Jim Hefferon's user avatar
37 votes
5 answers
2k views

Are there any benefits to having an entire course's homework problems available from day one?

I am designing a course for the upcoming fall semester, and I am tossing around an idea in my head. While planning which topics to cover each week and how to set the pacing of the course, I figured I ...
Brendan W. Sullivan's user avatar
37 votes
4 answers
3k views

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 ...
Patrick Lutz's user avatar
36 votes
9 answers
1k views

How can we help students learn to write about their mathematics?

As a guiding example, imagine an undergraduate Calculus II course where students have to complete a guided "research project" and write a "paper" about their work. This can be a shockingly new ...
Brendan W. Sullivan's user avatar
36 votes
4 answers
2k views

How to convey the meaning of "mathematical maturity"?

Some university-level courses have no specific prerequisites, yet are mathematically involved to the extent that someone with little to no experience in math will probably find themselves in over ...
Adam Bjorndahl's user avatar
35 votes
13 answers
5k views

Lecturers "(intentional) mistakes" as a teaching tool

I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind ...
Jyrki Lahtonen's user avatar
35 votes
11 answers
2k views

Epsilons and deltas in a first calculus course

In a freshman calculus course for non-majors; Is it to the benefit of the students to include discussion of epsilons and deltas? To what extent, if any, should they be used? For example, just to ...
Gamma Function's user avatar
35 votes
11 answers
5k views

Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III?

I am aware of three proofs of the fundamental theorem of algebra, using: Liouville's theorem The fundamental group of the punctured plane, or Multiplicativity of field extensions together with the ...
Brian Rushton's user avatar
35 votes
3 answers
4k views

What happened to the Moore method?

I always read about the Moore method with great enthusiasm. Somehow I always felt that it should be how we do it in an ideal world, but it is impossible to use because of time and other constrains. ...
András Bátkai's user avatar
35 votes
1 answer
2k views

Metonymy in mathematics

Metonymy is a figure of speech where a word or another expression is used to mean something other than its literal meaning. This phenomenon is not restricted to the "usual human languages" (such as ...
Joonas Ilmavirta's user avatar
34 votes
12 answers
7k 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 ...
Zuriel's user avatar
  • 4,275
34 votes
7 answers
6k views

What to do when you get "the empty stare"?

First, I am not a professor, but I was a teaching assistent for a couple of courses. One time I took over a few sections for a friend who was also a TA. The course was 'math for chemists' (I think it ...
Ruben's user avatar
  • 945
34 votes
5 answers
2k 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 ...
Brendan W. Sullivan's user avatar
33 votes
17 answers
7k views

Natural origins or learned habit: Why do students skip concepts before applications?

When teaching elementary mathematics, it takes a lot of time and effort to teach students that our goal is not to learn the examples, but to learn the concepts first, and then apply them to specific ...
Mahdi Majidi-Zolbanin's user avatar
33 votes
5 answers
6k views

What is a good method for drawing a Möbius band on the blackboard?

This week I'm going to give a talk on fiber bundles, and I found myself with an unexpected problem. Since I'm not using slides, I'll need to draw a Möbius band on the blackboard. Usually what I do is ...
Newman's user avatar
  • 441
33 votes
3 answers
1k views

What is the evidence about the effectiveness of remediation in math?

At many colleges in the United States, incoming students are required to take placement tests in basic skills such as math and reading. Those who score below a cut-off are required to take remedial ...
user avatar
33 votes
6 answers
3k views

What are the best practices for giving online tests?

Many of us our coming off our first semester of required-online classes; and at some of our institutions we are preparing for what is most likely a required-online semester in the fall. (That is: The ...
Daniel R. Collins's user avatar
32 votes
20 answers
6k views

How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?

Actually, there is no algebraic problem to show that $-(-x) = x$. This proof can be build upon the concept of the addition of the opposite like this: $- x + x = - x + [- ( - x) ]$, and thus by ...
Abdallah Abusharekh's user avatar
32 votes
5 answers
3k views

The best way to introduce trigonometric functions in a rigorous analysis course

This is something I have always had issues with. Generally, three approaches are used: The geometric path: this follows the standard way how you would introduce these functions in school. The problem ...
András Bátkai's user avatar
32 votes
6 answers
3k views

Allowing nonstandard mathematical language and/or notation

How important is enforcing standard mathematical language and/or notation? Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using ...
Nick C's user avatar
  • 9,639
31 votes
10 answers
12k views

Why do we teach even and odd functions?

I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-...
Nick Matteo's user avatar
31 votes
9 answers
7k views

What to do with students who think they "already know it," but actually don't?

Many students take calculus or algebra courses in high school, then later take college courses of the same name. There are various reasons for this, but in most cases the students in a college ...
Chris Cunningham's user avatar
31 votes
8 answers
3k views

How to react to students saying that they are allergic to applied mathematics?

I'm working in the field of applied mathematics (optimization and numerics) and I meet a lot of students saying that they are allergic to applied mathematics or that they hate it or some quotes like "...
Markus Klein's user avatar
  • 9,438
31 votes
7 answers
2k views

Mathematical education by country

Depending on the university, there are always slight differences in the syllabus and the structure of the standard material undergraduate students learn. But I also noticed that undergraduate ...
k.stm's user avatar
  • 419
31 votes
4 answers
7k views

When is it appropriate to warn about the difficulty of a subject?

I've been a TA across every class in the calculus sequence, under the assignment of professors with different teaching styles and curricula. It's often clear to me ahead of time when a certain subject ...
Feryll's user avatar
  • 419
31 votes
6 answers
917 views

Alternatives to University Lectures: Non-lecture Mathematics Classes

I am looking for resources for designing undergraduate mathematics classes that are not lecture-based. (Bonus points if the design is for an introduction to proof course). For example, Robert ...
David Steinberg's user avatar
31 votes
6 answers
3k views

What are non-math majors supposed to get out of an undergraduate calculus class?

When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 ...
user5249's user avatar
  • 311
30 votes
20 answers
8k views

‘Lies to children’ in mathematics and statistics education

In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as Lies to children: A lie-to-children is a statement that is false, but which nevertheless ...
Joel Ottar's user avatar
30 votes
10 answers
5k views

Getting students to actually read definitions

I'm teaching a second year "Introduction to Theoretical Computer Science" course, and one of the skills/habits I've tried to instill in the students is to actually read definitions, take ...
Arno's user avatar
  • 966

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