Questions tagged [undergraduate-education]
For questions about teaching students at the undergraduate (university) level.
808
questions
93
votes
18
answers
18k
views
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, ...
- 23.6k
82
votes
21
answers
25k
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 ...
- 29k
74
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 ...
- 9,170
71
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 ...
60
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 ...
- 842
53
votes
13
answers
11k
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 ...
user37
51
votes
15
answers
12k
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 ...
- 20.8k
47
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 ...
- 1,469
45
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, ...
- 8,119
45
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) ...
- 11.3k
44
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 ...
- 8,989
43
votes
4
answers
2k
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, ...
- 10.6k
39
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 ...
- 4,596
38
votes
12
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 ...
- 29k
37
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 ...
36
votes
14
answers
13k
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 ...
- 466
36
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 ...
- 9,170
36
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 ...
- 3,947
36
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 ...
- 10.6k
36
votes
4
answers
2k
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 ...
- 511
36
votes
4
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, ...
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 ...
- 2,051
35
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 ...
- 10.6k
34
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 ...
- 2,136
34
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 ...
- 1,396
34
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.
...
- 4,159
34
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 ...
- 3,670
33
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 ...
- 935
32
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 ...
- 4,119
32
votes
10
answers
4k
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 ...
- 11.3k
32
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 ...
- 431
32
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 ...
- 23.6k
32
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 ...
- 10.6k
31
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 ...
- 419
31
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 ...
- 1,127
31
votes
5
answers
2k
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 ...
- 4,159
31
votes
6
answers
884
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 ...
- 3,750
31
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 ...
- 8,856
31
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 ...
user507
30
votes
10
answers
11k
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(-...
- 489
30
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 ...
- 20.8k
30
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 "...
- 9,170
30
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 ...
- 409
29
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 ...
- 732
29
votes
7
answers
2k
views
Good definition for introducing real numbers?
In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are.
What are the advantages or disadvantages in the different "...
- 9,170
29
votes
7
answers
1k
views
When $-x$ is positive
This recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative ...
- 11.5k
29
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 ...
- 291
28
votes
10
answers
2k
views
What are argument one can give to students on the definition $0^0$?
From high school to introduction courses in university, the expression $0^0$ is some (psychological) problems. High school students just apply it to their calculator and either the result is $1$ or ...
- 9,170
28
votes
5
answers
2k
views
How should normal subgroups be introduced?
One standard definition of a normal subgroup is
A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide.
There's a class of similar definitions (...
user37
28
votes
4
answers
3k
views
Students use WolframAlpha. Can we change calculus instruction to exploit it while discouraging 'cheating'?
(This question developed from a comment in the thread "Revisiting the chain rule".)
Students know that WolframAlpha and other software/computational resources exist and will make use of them as they ...
- 10.6k