Questions tagged [undergraduate-education]

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

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85
votes
16answers
15k 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, ...
70
votes
20answers
18k 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 ...
66
votes
21answers
17k 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 ...
52
votes
4answers
4k 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 ...
49
votes
13answers
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 ...
49
votes
14answers
11k 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 ...
46
votes
8answers
11k 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 ...
45
votes
12answers
30k 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) ...
42
votes
16answers
20k 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, ...
39
votes
15answers
2k 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 ...
36
votes
4answers
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 ...
34
votes
11answers
4k 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 ...
34
votes
4answers
986 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, ...
33
votes
7answers
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 ...
32
votes
10answers
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 ...
32
votes
11answers
1k 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 ...
32
votes
3answers
3k 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. ...
32
votes
4answers
1k 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 ...
31
votes
13answers
4k 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 ...
31
votes
1answer
1k 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 ...
31
votes
3answers
1k 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, ...
30
votes
13answers
2k 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 ...
30
votes
12answers
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 ...
30
votes
5answers
5k 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 ...
30
votes
9answers
6k 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 ...
30
votes
4answers
3k 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 or Hodge et al'...
29
votes
7answers
925 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 ...
29
votes
7answers
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 ...
29
votes
6answers
858 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 ...
28
votes
10answers
1k 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 ...
28
votes
8answers
2k 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 "...
28
votes
3answers
686 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 ...
27
votes
18answers
3k 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 ...
27
votes
6answers
2k 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 ...
27
votes
4answers
2k 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 ...
26
votes
11answers
2k views

Impressive examples where a “proof by picture” goes wrong

There are many proofs where the whole idea can be expressed by a picture and often naturally translated into a correct formal proof. Often one has to argue with students that a picture is not a proof ...
26
votes
10answers
8k 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(-...
26
votes
6answers
611 views

Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?

I read a paper in my "Research Issues in Mathematical Education" class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach. I take five minutes to explain the ...
26
votes
4answers
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 ...
26
votes
4answers
779 views

The Undergraduate Responsibility Gradient

We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they ...
26
votes
1answer
650 views

Do “gateway tests” work?

Here is an overview of the practice of "gateway testing", which explains it much better than I could: https://sites.lsa.umich.edu/michigan-math-in-action/2015/09/24/25-years-gateway-testing-at-...
25
votes
12answers
3k views

Should we teach functions as sets of ordered pairs?

The context of this question is an "introduction to proofs and mathematics" class for freshman/sophomore math majors. Most textbooks for such a class say something about functions between arbitrary ...
25
votes
9answers
3k views

How can I teach my students that other disciplines are important too?

I'm sorry if this is the wrong place to ask this but I don't know any other place to ask. I'm a research professor, but I enjoy teaching and put a lot of time into my classes. I mainly teach ...
25
votes
6answers
1k 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 "...
24
votes
5answers
1k views

What is the proper way to ask a “find the domain” question?

A function is not really a function unless it's defined everywhere on its domain. So consider these three questions: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the square root function $f(x) = \...
24
votes
9answers
2k views

Teaching students to find and correct their own errors

Many students have a fairly good grasp of the topics they are learning but fall down because they miss fatal errors in their work. Some don't check for errors at all, while many simply can't find them....
24
votes
4answers
1k views

How should LaTeX be taught to university students?

There are several groups of people that would benefit from learning LaTeX in college. Future teachers can use it to write exams, scientists and mathematicians can write papers, and everyone can write ...
23
votes
11answers
1k 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 ...
23
votes
4answers
2k views

How to Teach Adults Elementary Concepts

I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school ...
23
votes
4answers
456 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 ...