Monthly Archives: May 2016

Math 42 Spring 2016 Student Showcase

This spring, I taught Math 42: An Introduction to Elementary Number Theory at Brown University. An important aspect of the course was the final project. In these projects, students either followed up on topics that interested them from the semester, or chose and investigated topics related to number theory.  Projects could be done individual or in small groups.

I thought it would be nice to showcase some excellent student projects from my class. Most of the projects were quite good, and some showed extraordinary effort. Some students really dove in and used this as an opportunity to explore and digest a topic far more thoroughly than could possibly be expected from an introductory class such as this one. With the students’ permission, I’ve chosen five student projects (in no particular order) for a blog showcase (impressed by similar sorts  of showcases from Scott Aaronson).

  • Factorization Techniques, by Elvis Nunez and Chris Shaw. In this project, Elvis and Chris look at Fermat Factorization, which looks to factor $n$ by expressing $n = a^2 – b^2$. Further, they investigate improvements to Fermat’s Algorithm by Dixon and Kraitchik. Following this line of investigation leads to the development of the modern quadratic sieve and factor base methods of factorization.

  • Pseudoprimes and Carmichael Numbers, by Emily Riemer. Fermat’s Little Theorem is one of the first “big idea” theorems we encounter in the course, and we came back to it again and again throughout. Emily explored the Fermat’s Little Theorem as a primality test, leading to pseudoprimes, strong pseudoprimes, and Carmichael numbers. [As an aside, one of her references concerning Carmichael numbers were notes from an algebraic number theory class taught by Matt Baker, who first got me interested in number theory].

  • Continued Fractions and Pell’s Equation, by Max Lahn and Jonathan Spiegel. As it happened, I did not have time to teach continued fractions in the course.  So Max and Jonathan decided to look at them on their own. They explore some ideas related to the convergence of continued fractions and see how one uses continued fractions to solve Pell’s Equation.

  • Quantum Computing, by Edward Hu and Chris Long. Edward and Chris explore quantum computing with particular emphasis towards gaining some idea of how Shor’s factorization algorithm works. For some of the more complicated ideas, like the quantum Fourier transform, they make use of heuristic and analogy to purvey the main ideas.

  • Fermat’s Last Theorem, by Dylan Groos, Natalie Schudrowitz, and Kenneth Berglund. Dylan, Natalie, and Kenneth provide a historical look at attacks on Fermat’s Last Theorem. They examine proofs for $n=4$ and Sophie Germaine’s remarkable advances. They also touch on elliptic curves and modular forms, hinting at some of the deep ideas lying beneath the surface.

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Math 42: Concluding Remarks

As this semester draws to an end, it is time to reflect on what we’ve done. What worked well? What didn’t work well? What would I change if I taught this course again?

Origins of the course

This course was created by my advisor, Jeff Hoffstein, many years ago in order to offer a sort of bridge between high school math and “real math.” The problem is that in primary and secondary school, students are not exposed to the grand, modern ideas of mathematics. They are forced to drill exercises and repeat formulae. Often, the greatest and largest exposure to mathematical reasoning is hidden among statements of congruent triangles and Side-Angle-Side theorems. Most students arrive at university thinking that math is over and done with. What else could there possibly remain to do in math?

Math 42 was designed to attract nonscience majors, especially those not intending to pursue the standard calculus sequence, and to convince them to study some meaningful mathematics. Ideally, students begin to think mathematically and experience some of the thrill of independent intellectual discovery.

It is always a bit surprising to me that so many students find their way into this class each spring. This class does not have a natural lead-in, it satisfies no prerequisites, and it is not in the normal track for math concentrators. One cannot even pretend to make the argument that number theory is a useful day-to-day skill. Yet number theory has a certain appeal… there are so many immediate and natural questions. It is possible to get a hint that there is something deep going on within the first two classes.

Further, I think there is something special about the first homework assigned in a course. Homeworks send a really strong signal about the content of a course. I want this course to be more about the students exploring, asking questions, and experimenting than about repeating the same old examples and techniques from the class. So the first several questions on the first homework are dedicated to open-ended exploration.

There are side effects to this approach. Open ended exploration is uncertain, and therefore scary. I hope that it’s intriguing enough (and different enough) that students push through initial discomfort, but I’m acutely aware that this can be an intimidating time. Perhaps in another time, students were more comfortable with uncertainty — but that is a discussion for later. I’m pretty sure that many fears are assuaged after the first week, once the idea that it is okay to not know what you’re going to learn before you learn it. In fact, it’s more fun that way! (One also learns much more rapidly).

My approach to this course is strongly influenced by my experiences teaching number theory to high schoolers as part of the Summer@Brown program for the past several years. During my first summer teaching that course, I co-taught it with Jackie Anderson, who is an excellent and thoughtful instructor. I also strongly draw on the excellent textbook A Friendly Introduction to Number Theory by Joe Silverman, written specifically for this course about 20 years ago.

Trying final projects

I am still surprised each time I teach this course. I tried one major new thing in this course that I’ve thought about for a long time — students were required to do a final project on a topic of their choice. It turns out that this is a great idea, and I would absolutely do it again. The paths available to the students really opens up in the second half of the course. With a few basic tools (in particular once we’ve mastered modular arithmetic, linear congruences, and the Chinese Remainder Theorem) the number of deeply interesting and accessible topics is huge. There is some great truth here, about how a few basic structural tools allow one to explore all sorts of new playgrounds.

A final project allows students to realize this for themselves. It also fits in with the motif of experimentation and self-discovery that pervades the whole course. The structural understanding from the first half of the course is enough to pursue some really interesting, and occasionally even useful, topics. More importantly, they learn that they are capable of finding, learning, and understanding complex mathematical ideas on their own. And usually, they enjoy it, since it’s fun to learn cool things.

For various reasons, I had thought it would be a good idea to offer students an alternative to final projects, in the form of a somewhat challenging final exam. In hindsight, I now think this was not such a good idea. Students who do not perform final projects miss out on a sort of representative capstone experience in the course.

There are a few other things that I would have done differently, if given the chance. I would ask students to be on the lookout for topics and groups much earlier, perhaps about a third of the way into the semester (instead of about two thirds of the way into the course). My students did an extraordinary job at their projects this semester. But I think with some additional rumination time, more groups would pursue projects based more on their own particular interests. Or perhaps not — I’ll see next time.

Several of my students who really dove into their final projects have agreed to have their work showcased here (which is something that I’ll get back to in a later note). This means that students in later courses will have something to refer back to. [Whether this is a good thing remainds to be seen, but I suspect it’s for the better].

Interesting correlations

In these concluding notes, I often like to try to draw correlations between certain patterns of behaviour and success in the course. I’m very often interested in the question of how early on in a course one can accurately predict a final grade. In calculus courses, it seems one can very accurately predict a final grade using only the first midterm grade.

In this course, fewer such correlations are meaningful. Most notably, a large percentage of the class took the course as pass/fail. [I can’t blame them, as this course is supposed to draw students a bit out of their comfort zone into a topic they know little about]. This distorts the entire incentive structure of the course in relation to other demands of college life.

There is a very strong correlation between taking the class for a grade and receiving a high numeric grade at the end. I think this comes largely from two causes: students who are more confident with the subject material coming into the course decide to take it for a grade, and then perform in line with their expectations; and taking a class for pass/fail creates an incentive structure with high emphasis on learning enough material to pass, but not necessarily mastering all the material.

In sharp contrast to my experience in calculus courses, there is a pretty strong correlation between homework grade and with overall performance. While this may seem obvious, this has absolutely not been my experience in calculus courses. Generally poor homework grades correlate extremely strongly with poor final grades, but strong homework has had almost no correlation with strong performance.

I think that a reason why homework might be a better predicter in this course is that homework is harder. There are always open-ended problems, and every homework had at least one or two problems designed to take a lot of experimentation and thought. Students who did well on the homework put in that experimentation and thinking time, reflecting better study habits, higher commitment, and more grit (like in this TED talk).

Finally, there is an extremely high correlation with students attending office hours and strong performance in the course. It will always remain a mystery to me why more students don’t take advantage of office hours. [It might be that this is also a measure other characteristics, such as commitment, study habits, and grit].

Don’t forget the coffee

This was my favorite course I’ve taught at Brown. On the flipside, I think my students enjoyed this class more than any other class I’ve taught at Brown. This is one of those courses that rejuvenates the soul.

While returning home after one of the final classes, I flipped on NPR and listened to Innovation Hub. On the program was Steven Strogatz, a well-known mathematician and expositor, talking about his general dislike of the Calculus-Is-The-Pinnacle-Of-Mathematics style approach that is so common today in high schools and colleges. The program in particular can be found here.

He argues that standard math education is missing some important topics, especially related to financial numeracy. But he also argues that the current emphasis is not on the beauty or attraction of mathematics, but on a very particular set of applications [and in particular, towards creating rockets].

While this course isn’t perfect, I do think that it is the sort of course that Strogatz would approve of — somewhat like a Survey of Shakespeare course, but in mathematics.

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