Questions tagged [undergraduate-education]

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

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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....
18
votes
3answers
956 views

Good problems that uncover difficult points in a theory

There is a great quote of Yitz Herstein: The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver." A number of such ...
19
votes
1answer
1k views

How much time to spend on a single question?

When I was self-studying as an undergraduate, I would spend up to two weeks working on a single problem or trying to understand a proof in Rudin's Principles of Mathematical Analysis. I realize now ...
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
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 ...
14
votes
4answers
1k views

When should I say “nothing is as it seems”?

"Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true ...
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 ...
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, ...
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 ...
17
votes
4answers
833 views

How can I choose a free calculus textbook?

As I have been recently informed, it is a good idea to consider free calculus textbooks for college and university courses. However, this feels risky to me, because: I don't know anyone who is using ...
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 ...
19
votes
6answers
2k views

Is there a good way to explain determinants in an elementary linear algebra class?

Many colleges offer an an elementary linear algebra class for sophomore math, science, and economics majors. Such a class typically covers a chapter on determinants, including the following aspects: ...
18
votes
7answers
944 views

How can I motivate the formal definition of continuity?

In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition $f$ is continuous when its graph can be drawn without lifting the pen. ...
16
votes
4answers
2k views

Requiring students to know all the proofs on an oral exam

I'm asking this question as a student, wondering what various pros/cons to the given formula for oral exams could be. Let me give some context first. I am a first year mathematics student at a ...
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 ...
15
votes
5answers
1k views

Is Peer Instruction suited to mathematics classroom?

Peer Instruction is a method developed by Eric Mazur in Harvard, designed with a student-centered approach in mind. In a nutshell, the core of the method is that when presented with a problem, ...
16
votes
6answers
10k views

Ideal Undergraduate Sequence

What is the perfectly (maybe unrealistically) ideal undergraduate sequence for a undergraduate majoring in pure mathematics who takes 2-3 mathematics courses per semester assuming a strong AP Calculus ...
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 ...
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 ...
13
votes
1answer
2k views

Is MacLane and Birkoff's “Algebra” suitable today as either an undergraduate or graduate text in abstract algebra?

I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's Algebra at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the ...
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'...
19
votes
8answers
870 views

What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty ...
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 ...
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 "...
16
votes
7answers
2k views

What is the ideal course sequence for an advanced student of mathematics?

Suppose that you meet a student who: has a firm grasp of algebra and trigonometry and is at least moderately intelligent has read a book such as Love and Math by Edward Frenkel so has some ...
21
votes
2answers
605 views

Exam philosophy

I'm curious if anyone knows of any books, studies, or other resources on the philosophy of creating and grading mathematics exams. After working as a graduate TA for 4 years and dealing with a wide ...
20
votes
9answers
2k views

The definition of natural log and e

I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my ...
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 ...
20
votes
2answers
671 views

Emphasizing the discrete in early undergraduate education?

From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's Fantasy Math curriculum (update: see also his material on discrete calculus) ...
19
votes
2answers
748 views

Should geometric algebra be presented early on in undergraduate education?

The Cambridge University GA Research Group’s website along with the “Geometric Calculus R & D Home Page” should serve as a good introductions to geometric algebra, along with the Wikipedia ...
13
votes
2answers
694 views

What are some activities/projects I can assign to calculus students from bio/chem/physics majors to specifically motivate their interest?

(This question was proposed during the area51 phase.) It's common for chemistry/biology/physics majors to be required to take certain calculus courses. At my school, chem/bio students must take up ...
13
votes
4answers
436 views

What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...
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 ...
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 ...
21
votes
8answers
665 views

Is it good to have solutions of homework published?

At a course at the university, the students have to do homeworks every week which will be graded and discussed in exercise groups. Is it a good idea to put "official" solutions of the homework on ...
20
votes
3answers
411 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 $\...
16
votes
1answer
845 views

Standards-based grading in calculus

A friend of mine recently tried a standards-based grading (SBG) approach for her Calculus II course. (You can read about Kate's experience on her blog.) I find this approach to evaluation very ...
10
votes
3answers
2k views

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

For example, I am confident that very few students majoring in pure mathematics can write a complete proof to the Abel–Ruffini theorem (there is no algebraic solution to general polynomial equations ...
18
votes
5answers
868 views

How to nurture a good student?

When you encounter a very bright student in a first-year (college/university) class (and who is therefore bored), what do you do? Leaving them to their own devices can be problematic. It can lead to ...
11
votes
6answers
977 views

How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
20
votes
10answers
837 views

What makes cosets hard to understand?

I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the ...
20
votes
8answers
3k views

What is a good reason to change calculus texts?

Our college is switching to an Early Transcendentals calculus text, and this seems like a good time to consider which text we are using in general. Larson, Stewart, Thomas, Briggs/Cochran, etc are all ...
16
votes
6answers
3k views

is the reciprocal function continuous

I'm curious the views of those who teach calculus. As you know the continuity of a function at a point is defined in terms of the limit in the typical course. I'd like to ask a pair of questions: ...
15
votes
0answers
822 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 ...
19
votes
9answers
2k views

Why do students have problems with showing that something is well-defined? How can this be improved?

I see a lot of students struggling when they have to show that something is well-defined. I have the feeling that this is often not understood. Two examples: When defining a sequence $x_n= g(x_{n-1}...
11
votes
3answers
530 views

Appropriate ways/sayings to discourage undergraduate students' overreliance on calculators

Main question: How do I, in a medium- to large-sized undergraduate class setting, appropriately and effectively discourage students from relying too heavily on calculators? There have been several ...
10
votes
2answers
705 views

Mathematical thinking skills for engineering students

A few months ago, I asked a question on teaching engineers mathematical thinking skills over at MSE. I also asked it a little later at The Mathematics Teaching Community, but traffic on that site is ...
9
votes
3answers
418 views

How many problems do we have to do as undergraduate mathematicians in order to learn a subject?

I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. We know that the professors often assign us a list of problems to solve as homework, ...
8
votes
5answers
502 views

How long would it take to teach proper limit calculations?

This question arose from discussion of this question. How long would it take you to teach typical undergratuate (calculus) students the difference between the following two calculations? $$\lim_{x\...
6
votes
4answers
188 views

Topics for Discovery-based Projects

Are there any existing sources for topics or projects which are: Suitable for first and second year undergraduates taking introductory math classes (possibly with some prompting or a couple pages of ...