Skip to main content
18 votes

Calculus problems arising from real research problems

Modeling Basketball Free Throws by Joerg M. Gablonsky and Andrew S. I. D. Lang, SIAM Review vol. 47, no. 4, pp. 775-798, 2005, https://doi.org/10.1137/S0036144598339555 Abstract This paper presents a ...
JRN's user avatar
  • 10.9k
16 votes

What are some of the open problems that can be suitably introduced in a calculus course?

It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.
Alexander Woo's user avatar
13 votes

Calculus problems arising from real research problems

Consider Betz's law, which was worked out around $100$ years ago by three different scientists independently (in Germany, the UK, and Russia). It determines the maximum power that can be extracted ...
KCd's user avatar
  • 3,536
12 votes

What are some of the open problems that can be suitably introduced in a calculus course?

It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here ...
Joseph O'Rourke's user avatar
11 votes

What are some of the open problems that can be suitably introduced in a calculus course?

You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.
Gerald Edgar's user avatar
  • 7,607
10 votes

What are some of the open problems that can be suitably introduced in a calculus course?

This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the ...
Chris Cunningham's user avatar
9 votes

How to get better at proofs

A great start to doing proofs is working through Daniel Velleman's How to Prove it: A Structured Approach, 2nd Edition.. I've used it many times in teaching, usually as a supplementary text.
amWhy's user avatar
  • 2,095
8 votes

Doing research projects when one's knowledge is limited: is it preferable?

I think it is very possible to have real, meaningful research projects at all levels. As an example, I had 2 of my students in Calc 2 work on a project which started with the idea "Can we define a ...
Steven Gubkin's user avatar
8 votes

Calculus problems arising from real research problems

Consider trying to find a curve that will let you link up two straight tracks smoothly, such as two straight parts of a roller coaster track. By "smoothly" meeting, we want the derivatives ...
KCd's user avatar
  • 3,536
7 votes

Grad school after doing an online bachelor's degree without support for undergraduate research

It seems like some of the other answers are aiming at PhD programs. I would suggest (as your question on academia.sx suggests) that you may wish to look at a Master's program (at a non-PhD-granting ...
kcrisman's user avatar
  • 5,986
7 votes

Seeking references for why it is good that students understand why mathematical rules work

The key term you are interested in is "conceptual knowledge" (more specifically, "conceptual understanding"). According to this document from the National Council of Teachers of ...
JRN's user avatar
  • 10.9k
7 votes

Seeking references for why it is good that students understand why mathematical rules work

I am totally convinced that, in most cases, it is good for students to understand why mathematical rules work. I could write a very long text justifying this belief but, from an academic point of view,...
Steven Gubkin's user avatar
6 votes

Are there any undeveloped areas of mathematics accessible (for both understanding and research) to an undergraduate?

In line with your identification of graph theory, I suggest you might look into what is now known as "Discrete and Computational Geometry." Although there is much to learn (there is a 1937-...
Joseph O'Rourke's user avatar
6 votes

How to get better at proofs

Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is ...
Daniel R. Collins's user avatar
6 votes

How to get better at proofs

Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.
Gerald Edgar's user avatar
  • 7,607
6 votes
Accepted

Ball rolling down curve simulator

The Energy Skate Park by PhET seems to come quite close: In the "Playground" section you can draw arbitrary curves and add a stopwatch. The simulation has to be manually stopped at the end ...
Jasper's user avatar
  • 2,699
6 votes

Calculus problems arising from real research problems

Consider the reflective property of a parabola: rays coming into a parabola parallel to its axis of symmetric will all bounce off the parabola (angle of incidence equals angle of reflection) and meet ...
KCd's user avatar
  • 3,536
6 votes

Calculus problems arising from real research problems

There should be a lot of interesting examples in ecology. The first one that comes to mind for me is population dynamics: Wikipedia Link The simplest model is exponential growth: $$ dN/dt = rN $$ ...
anjama's user avatar
  • 221
5 votes

How to get better at proofs

As a general introduction, the book Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi can be useful. For example, how to prove that a sequence converges? A detailed explanation ...
Pedro's user avatar
  • 1,800
5 votes
Accepted

Can or should students do research in standard major math courses

I think the term research causes a lot of confusion here. When mathematicians think about research, we immediately think about proving new theorems. But that doesn't seem to be what's asked here. ...
Henry Towsner's user avatar
5 votes

How to explain to undergraduate students what research means?

Here's a useful article. Gareth E. Roberts. "Conducting Mathematical Research with Undergraduates." 2008. PDF download Abstract (extract). This article discusses some of the key issues ...
Joseph O'Rourke's user avatar
4 votes

Course-based undergraduate research experiences in math

Yes, however there aren't lots of great resources. For instance, I have a friend who taught a combinatorial game theory course that turned into research for many students, but I don't think he wrote ...
kcrisman's user avatar
  • 5,986
4 votes

Doing research projects when one's knowledge is limited: is it preferable?

Perhaps a research project could be replaced by an independent study project. The idea is to learn more math, in a more independent way. The student would still have to produce something to show what ...
Sue VanHattum's user avatar
  • 21k
3 votes

How to get better at proofs

I learned more from Lakatos than any other source about what constitutes a proof, and: the roller-coaster ride adjusting definitions to clarify the proof claim, perhaps realizing that a hidden lemma ...
Joseph O'Rourke's user avatar
3 votes

Subject advice in Number Theory

I think it would be a good Bachelor topic to explain, at a high-level, the MRDP theorem (Matiyasevich–Robinson–Davis–Putnam), which settled Hilbert's 10th problem: A set of integers is Diophantine ...
Joseph O'Rourke's user avatar
3 votes

Doing research projects when one's knowledge is limited: is it preferable?

Opinion: I think they get enough exposure to writing reports in other subjects. Think the time is better used for drill and exams, than for writing a report. The one benefit might be learning to ...
guest's user avatar
  • 325
3 votes
Accepted

How to improve mathematical skills(University level)?

I can only help with (3). A. This behavior is not unusual and not just with intuitionists. It's good to be able to do things in your head, but you need to "know your head" and when you will have ...
guest's user avatar
  • 42
3 votes

Can or should students do research in standard major math courses

Do some applied projects: statistics or linear programming optimization or diffy Qs. Not pure math research. Doesn't need to be publication type finding a new law either. Could be analyzing a ...
guest's user avatar
  • 1,828
3 votes
Accepted

Why do the stages of rigorousness have specific timestamps?

I don't see any specific timestamps in the piece quoted. There are some vague time frames given, all marked by phrases like "generally" and "usually", which indicates that they're far from universal. ...
Henry Towsner's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible