34 votes
Accepted

How should I grade true-or-false questions if the student's writing is unclear?

I read them as TTTFT. But the fourth is very hard to tell, the first somewhat hard to tell, and all show poor writing, perhaps indicating a lack of engagement. You could tell him that such terrible ...
  • 356
28 votes

Combative students in proofs classes

You don't say in the question what kind of school this is. It must be a four-year school rather than a community college, but there is no indication of what its admissions standards are like. If this ...
  • 297
28 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

I think a more correct view is that proof is the LAST of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case. Before ...
25 votes

Combative students in proofs classes

There's probably no silver bullet. But one tool I use is in these situations (e.g., I teach discrete mathematics etc. at a U.S. community college) is to very closely align with a good textbook. In ...
23 votes
Accepted

Should an undergraduate math program contain a course on Lebesgue integration?

I think the existing answers understate how much a standard American math major does not see the Lebesgue integral. I'm going to poke around at a variety of college websites to see how they cover this ...
21 votes
Accepted

Concrete vectors spaces without an obvious basis or many "obvious" bases?

Some physical examples from physics: Consider two spaceships that meet each other in deep space with arbitrary orientations (pitch, roll, and yaw). Even if they take the origin to be the midpoint ...
  • 328
17 votes

Combative students in proofs classes

EDIT: I would like to clarify that my response below is not intended to be definitive. This is an extremely difficult problem to have. It is perhaps the most difficult problem one can have as a ...
14 votes

Is there a College Algebra book that was written by a world-class mathematician?

This question is close to one you've already asked, a subset of it practically. Should I append a cute Venn diagram, showing the relation? Which textbooks on College Algebra, Trigonometry, Pre-...
  • 149
14 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? No (assuming that "have a course" means "require people to take such a course in order to get ...
  • 141
14 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

I have taught Discrete & Computational Geometry to US undergraduates project-based, as opposed to assignment- and test-based. Some of the projects do involve proofs, but others are more ...
14 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

With the technological advances of the past couple decades, computational mathematics is now somewhat accessible to undergraduates. The wikipedia entry for computational mathematics lists out the ...
  • 6,504
13 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

Two more examples: The set of infinite Fibonacci-type sequences (those of the form $a_n=a_{n-1} + a_{n-2}$) (with point-wise addition and scaling) forms a 2-dimensional (real) vector space. E.g., ...
  • 7,703
12 votes

Where can I find public repositories of past math exams?

Quite a few universities publicly post the math exams their faculty write: UC Berkeley hosts an archive of their past exams, sorted by course. University of Michigan hosts past exams for some ...
11 votes

Why the fear of polynomial long division?

I like to include synthetic division as a topic in a college algebra or precalculus course. It is an opportunity to take a 20 minute digression to talk about Horner's method, which is used in several ...
  • 8,033
11 votes

What are best practices for building a dedicated space for mathematics majors?

Chalkboards. Hagoromo chalk. Some small (individual) study desks and some round tables - with acceptably comfortable chairs. A couple of couches and living-room-type chairs. POSTERS OF DIVERSE ...
11 votes

What are best practices for building a dedicated space for mathematics majors?

Along with what others recommended, I would add some simple food prep options (microwave, fridge, sink, coffee-maker, cabinets). We are physical beings. It will encourage usage of the space.
11 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

In other words, "knowing" math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew. You appear to believe that somebody ...
  • 211
10 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

I think you're on the right track with the polynomials. They're not wrong that $(a,b,c)\mapsto (x\mapsto ax^2+bx+c)$ is an obvious linear isomorphism from $\mathbb R^3$ to what I will call $\text{...
  • 5,539
10 votes
Accepted

How can I help/tutor a friend who is taking the same course as me?

I had two goals the whole time: do the homework as quickly as I could do it well, and teach him the concepts. If you really want to help, the first goal should be scrapped entirely, IMHO. The best ...
  • 566
10 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

When I describe undergraduate research to students majoring in mathematics, I ask them to browse the abstracts of the most recent MAA undergraduate poster session. Here is a link: Abstracts for the ...
  • 8,033
9 votes

Why the fear of polynomial long division?

(This answer is me speaking as a student. I taught both polynomial long division and synthetic to my first calculus class, but realized by the end of the unit that it was really unneeded for AP ...
  • 5,539
9 votes

Combative students in proofs classes

Some helpful feelings of mine about teaching: You can't force someone to learn. As an undergraduate professor, I am responsible for being a resource to provide my students with the information they ...
  • 2,224
9 votes

Teaching Solving Linear Equations before teaching evaluating expressions

I hope this does not come across as overly harsh: I do not think that thinking of teaching as "covering material" in a particular order is a useful framework. If your students are solving ...
9 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

That is a linear algebra course? So presumably before you get to this point of abstract vector space, you already did solution of systems of linear equations? For example, solution of matrix ...
  • 6,418
8 votes

Concrete vectors spaces without an obvious basis or many "obvious" bases?

Here's another example, and a comment. First the comment: I think this is a great opportunity to talk to students about a trend in abstract mathematics where we progressively transition more and more ...
8 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? Yes, and I find it bizarre that a university would not have one. Lebesgue integration (or measure theory more ...
  • 189
8 votes
Accepted

Is the Wronskian still assumed for graduate education?

I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true. Certainly, anyone who has studied a semester of ...
8 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

Not quite an answer to your question, but if you have students who are interested in mathematics but not interested in (generating their own) proofs, encourage them to go into mathematics ...
  • 2,224
7 votes

What's the Deal with Inverse Cotangent?

For teaching, I think this situation gives a good excuse to talk about how in mathematics, sometimes we get to embrace ambiguity. One thing that differentiates the two definitions is that they satisfy ...
  • 8,033

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