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20

Let's start by saying that I strongly advice against such a dual-exam. Even if you and everyone involved in the planning think it is fair, students might think differently. In this way, you open up the floodgates for grade complaining. They might not succeed, but even the fact that some might try will cost you a lot of time and possibly reputation. Now, in ...

17

At my (community) college, I believe many teachers skip the (𝜖,𝛿)-definitions of limits. I give the main definition, and teach it with an example I call the cookie crispness index. But I don't assign problems in that section, and I don't test on it. I figure actually working with it is for an analysis course. Of the topics that I do test on, implicit ...

12

(My answer is just a guess and not based on any formal research.) I suspect the absolute value function may be difficult to understand because it involves "negative numbers that aren't negative." One way to define the absolute value of $x$ is: $$|x|=\left\{\begin{array}{rl}-x, & x<0\\x, & 0\le x\end{array}\right.$$ I think the $-x$ confuses ...

10

The apparent conflict between points of view expressed in the OP is illusory. There is no real conflict. The mathematics education researcher quoted in the OP is arguing that students find the definition difficult to appreciate and master. The mathematicians quoted in the OP are arguing that, once mastered, it provides clarity. I have questions about the ...

10

Mathematics constests are a kind of game or puzzle, like chess, poker, sudoku, etc. Not all mathematics adapts well to the context of a competition in a limited amount of time. While it's true that research mathematicians can be very competitive (in Yau's recent autobiography, he several times describes mathematics as a competitive activity - note though ...

8

I don’t think it is remotely reasonable to include the areas you mentioned, since (in the U.S., at least) all those that you mentioned are at least 3rd and 4th year undergraduate subjects. Indeed, functional analysis is usually a 2nd year graduate course (often not even required for a Ph.D. in math), and mathematical physics topics such as quantum mechanics ...

7

I think this is a bad idea, because it invites students to try to game the system: those who own a fancy calculator suddenly have to decide if they’re better off using it and taking an exam they think will be harder, and those who don’t have to decide if it would offer enough of an advantage to be worth getting one and taking the other version. Worse, you ...

7

Is 'at the same time' an option? I mean, by junior year, math majors should be taking at least two math classes per semester, right? When I was an undergraduate at Penn State, these two courses were the only 300 level math courses, both designed to be taken first semester junior year. The introduction to abstract algebra used "Numbers, Groups, and Codes", ...

6

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples: Lagrange's theorem in abstract algebra: The order of a subgroup $H$ of a finite group $G$ divides the order of $... 6 I'd say the whole$\epsilon/\delta$dance. Just because it involves inequalities, while students just have trained equalities all their life. To just work with (sometimes very rough) bounds, when you were drilled to get exact answers goes against the grain. 5 I am writing this based on pure observation (e.g., entering year four of teaching this topic to secondary school students, and having co-taught a minicourse for teachers on absolute value functions$^\star\$). There are a lot of definitions/interpretations of absolute values: the (abstract) axiomatic one; the piecewise or "case-based" one (provided by Joel ...

5

Related rates problems show the most issues on AP grading. (No reference, just my own observation) Topic involves word problems, geometry and some multistep problem solving. I don't think the concept of epsilon delta is so difficult as the tediousness of the algebra along with students openly or unconsciously questioning the value add for science and ...

5

I teach at community college. I often publish the homework problems at the beginning of the semester, listed by section. I have never had a student work ahead (that I know of). And I have had a few students who loved math, asked deep questions, and were interested in doing extra. I don't think that is likely to happen, except in very rare instances. One ...

4

You mention the desire for a book with content and exercises, but less fluff. But also, seem not to have done a lot of comparison research (other commenters recommend this to you, also). Two specific texts to look at would be: Schaum's Outline and Granville, et al. Both have a short, simple presentation. And there is definitely a student market for books ...

4

I would like to mention this recent opinion article in the AMS Notices (vol.66, no.7; PDF download) by Colin Adams (author of The Knot Book). His main point is that we should try "to impart a love of mathematics." He ends with this anecdote: I remember seeing a lecture by a professor who had won a variety of teaching awards. As I always am, I was ...

3

You may wish to view the MAA's report Transitions to Proof by Carol Schumacher, Susanna Epp, and Danny Solow: https://www.maa.org/sites/default/files/Transitions%20to%20Proof.pdf It has some suggestions for these kinds of courses, and gives references to some relevant books and articles. Ultimately, the course design is very open-ended, so it's hard to ...

3

I teach a lot of contest math and I have very mixed feelings. Contests reward repertoire, perspicacity and speed. These are all certainly useful at undergrad, but the emphasis given to them feels wrong. On the other hand, there is a decent overlap between students who can assail these problems and students who will develop the other qualities and attitudes ...

3

Although I admire the enthusiasm of the OP, I'd like to make a point similar to @guest's post: There are quite a few calculus textbooks out there. Each thumbnail below is the cover of a calc textbook. I would be wary of investing too much time in such a crowded environment without investigation of how it is already populated. Google images link Also, there ...

3

In terms of issues affecting most students I believe the concept of a variable and that of a function are still the most difficult concepts for calculus 1 students, even though the concepts are introduced in precalculus. Writing a full and correct mathematical sentence is a topic most students struggle with.

2

At my school (a community college in California) the curriculum is set up so that students first take an algebra course in which essentially every function is a linear function, and only later do they do anything at all with nonlinear functions. Nonlinear functions are just more complicated, so they require more thought about the logic. Thinking about logic ...

2

There are already answers suggesting you to make an exam that doesn't require a calculator at all, so I won't elaborate it here. But if you are concerned about students with different backgrounds and different expectations about what they know after it ends, I see no issue with splitting it along these lines. I have had a few classes in a similar setup in ...

2

Wanting your students to learn how to solve problems, instead of having them memorize some formulas and do basic arithmetic by for two hours, is a great goal imo. The first step towards that goal is making the formulas available to your students, just as they would be in the real world, but preventing them from accessing the actual solution of the problem (...

2

I figured that I can expand my comment into an answer, even though it is still more a comment than an answer. You say that math contest questions "involve a lot of small tricks that one hardly ever needs in math research", and because of this you find contest questions useless for future career as a mathematician. But I do not think that math contests are ...

1

As someone who probably took undergraduate courses more recently than most, I can attest to the fact that applications are necessary. If not given, students very often get lost - not understanding the big picture. They may be able to apply specific operations to specific problems, but it is the applications that allows them to see the forest instead of just ...

1

Be it undergraduate or professor, you need hands-on examples to "see" why some concept or technique is worthwhile. Sure, you can take it that more advanced people are better able to come up with their own examples and applications, or have a richer experience to which new material relates. So examples, applications, cross-connections are certainly needed for ...

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