15 votes
Accepted

How does a math Olympian fare in undergraduate maths courses?

Mathematics is many things. Solving tricky problems is part of it, but there is also learning theory and new concepts and ways of thinking, and even developing routine. I would expect a person with ...
Tommi's user avatar
  • 6,751
15 votes

Any examples of calculus sequence that deemphasizes calculation tricks?

As you undertake the journey, you should read about the Calculus Reform movement of the 1990s. A unifying idea of the efforts was to increase conceptual understanding by using symbolic computational ...
user52817's user avatar
  • 10.3k
10 votes
Accepted

Special topics for introductory probability

A classic application of Bayes' Theorem is in medical testing, and the difference/conversion between "what is the probability I test positive, given I have the condition" vs. "what is ...
Kevin P. Costello's user avatar
10 votes

How can we explain intuitively the convergence and divergence of these two series?

Look at a simpler example first: $(1.000000000001)^n$ compared to $0.9999999999^n$. Do they accept that the first sequence tends to $\infty$ and the second to $0$ even though it would take quite a ...
KCd's user avatar
  • 3,258
8 votes

Where can I find new types of problems regarding graduate level mathematics?

Nowadays, you will be able to find many research mathematicians' websites with lecture notes for advanced courses, many with exercises/examples, many with solutions/discussions. I myself certainly put ...
paul garrett's user avatar
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8 votes
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Where can I find new types of problems regarding graduate level mathematics?

As per my information , standard books from renowned authors and publications are the only sources of problems but still I want to know some sources where I can get difficult problems which can be ...
Justin Skycak's user avatar
8 votes

Loaning students calculators during exams

To summarize what I arrived at, based on the top answer on my prior parallel question on SE Academia, OP's #2 is the way to go -- keep a small supply of calculators on hand, but assess a usage penalty ...
Daniel R. Collins's user avatar
7 votes

Bridging the gap between students' intuitive problem-solving abilities and expressing ideas through formal writing

I've worked with some of these types of students in the past. One trend I noticed was that these students often have experience with coding, which they tend to enjoy and excel at (since the computer ...
Justin Skycak's user avatar
6 votes

Special topics for introductory probability

One example of elementary probability is the so-called Birthday problem which asks for the probability that in a room of $n$ people two will share the same birthday. Sometimes formulated as a paradox ...
mdewey's user avatar
  • 161
6 votes

How can we explain intuitively the convergence and divergence of these two series?

For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width). The $n$th prime is asymptotically $n \ln n$ (...
Justin Skycak's user avatar
6 votes

Special topics for introductory probability

You might already be aware of this one, given how famous it is, but the first thing that comes to my mind is the Monty Hall Problem. It doesn't require any fancy mathematical machinery, just a basic ...
Justin Skycak's user avatar
5 votes

Special topics for introductory probability

Bertrand's Paradox is an old saw. The point is that trying to randomize an experiment is tricky since there can be different points of view.
MaxW's user avatar
  • 151
4 votes

Beautiful planar geometry theorems not encountered in high school

Take any triangle, and draw any number of cevians from the top vertex to the base, with any spacing between the cevians. In each sub-triangle thus formed, inscribe a circle. Now rearrange the order ...
Dan's user avatar
  • 821
3 votes
Accepted

Strategies for Designing Challenging Yet Feasible Quiz Problems for Upper-Level Math Courses

I get a lot of clarity for designing assessments by writing out a list of learning objectives for the course. To get started, I write out three lists: Things I want my students to have seen (only ...
TomKern's user avatar
  • 3,982
3 votes

How can we explain intuitively the convergence and divergence of these two series?

Intuitively, to me, it means that if you take the positive number line, put a blue dot at every prime, and a red dot on all the the numbers of the form $n^{1.000000000001}$, then eventually, very far ...
Arthur's user avatar
  • 401
3 votes

Loaning students calculators during exams

Allow the students to use Desmos Test Mode. The app allows students to lock a phone or other android/iOS device in single-app mode and use the Desmos graphing calculator that we know and love in an ...
Chris Cunningham's user avatar
3 votes

Loaning students calculators during exams

My intuition is that further penalizing already disengaged students will likely lead them to become even less engaged. It also sounds like some of these students might not know how to use scientific ...
TomKern's user avatar
  • 3,982
2 votes

Any examples of calculus sequence that deemphasizes calculation tricks?

Take a look at this: https://calculus.ucmerced.edu/wwh-calculus-project/project-mission-and-goals You may find other examples that the California Learning Lab is funding at: calearninglab.org
Bamdad Samii's user avatar
2 votes

Loaning students calculators during exams

In my opinion, all of your decisions should be based on the following question: What is the pedagogical goal? In my classes, I generally try to focus on the mathematics and on basic (basic) "how ...
Xander Henderson's user avatar
  • 7,520
2 votes

Special topics for introductory probability

I would look ar some of the basic six sigma literature and at doe. It is connected to all kinds of factory snd other process improvement. Very clear business connection. I would eschew the Bayesian ...
Guest poster's user avatar
2 votes

Problem of the Week for College Students

I don't understand the aversion to contest problems. If you want contest-type questions, then why not just pull from well-known contests (AMC, IMO, Putnam, etc) where a variety of interesting, ...
Justin Skycak's user avatar
1 vote

Special topics for introductory probability

[Additional to previous answer--can't edit, sorry.] dt688: I would be very wary about being too difficult or particular, when teaching in a corporate environment. I.e. if GMers are your target ...
Guest poster on another device's user avatar
1 vote

How can we explain intuitively the convergence and divergence of these two series?

Consider the fact that $\sum_{n=1}^\infty n^x$ converges if $x<0$, diverges if $x>0$. Clearly the transition from just a little bit negative to just a little bit positive makes a big change to ...
Simon Crase's user avatar
1 vote

Is there any proof of the fundamental theorem of algebra that can be introduced to undergraduates who have just completed Calc III?

Just reposting: Proof (can be found in Lang’s analysis book and Vinberg’s algebra book) Let ${ f(z) }$ be a complex polynomial of positive degree. The approach is to show that: ${ \min _{z \in \...
Karthik's user avatar
  • 121

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