Questions tagged [undergraduate-education]

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

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59
votes
4answers
5k 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 ...
26
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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
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3answers
1k 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 ...
18
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1answer
2k 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 ...
50
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14answers
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 ...
14
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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 ...
80
votes
21answers
22k 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 ...
70
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17answers
9k 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 ...
35
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4answers
4k 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'...
44
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17answers
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 ...
43
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16answers
29k 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, ...
31
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20answers
4k 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 ...
19
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4answers
1k 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 ...
52
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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 ...
19
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8answers
1k 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 ...
20
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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: ...
20
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2answers
789 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) ...
18
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7answers
1k 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. ...
28
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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 ...
21
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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 ...
15
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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, ...
9
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3answers
495 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, ...
3
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3answers
239 views

Applications of abstract algebra outside of mathematics and suitable textbook

The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra. What are some applications of abstract ...
72
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20answers
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 ...
88
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18answers
17k 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, ...
15
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6answers
13k 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 ...
16
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1answer
4k 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 ...
29
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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 ...
28
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7answers
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 "...
36
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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 ...
15
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7answers
3k views

What are the differences between popular undergraduate abstract algebra books?

I will be teaching a year-long undergraduate introduction to abstract algebra in the fall, and I am quite looking forward to it! I need to choose a textbook, and I don't have personal experience with ...
32
votes
6answers
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 ...
30
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8answers
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 "...
19
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 ...
21
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2answers
671 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 ...
21
votes
2answers
976 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 ...
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
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 ...
16
votes
1answer
1k 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 ...
16
votes
4answers
605 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 ...
14
votes
2answers
958 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 ...
27
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4answers
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 ...
25
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4answers
3k 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 ...
22
votes
8answers
824 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 ...
21
votes
3answers
527 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 $\...
17
votes
0answers
1k 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 ...
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 ...
10
votes
3answers
3k 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 ...
32
votes
6answers
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 ...
18
votes
5answers
940 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 ...