Through daily problem solving sessions, students are tackling abstract algebra content comparable to that of an undergraduate course.
Ring Theory is a college-level course in abstract algebra, typically taken as part of an undergraduate mathematics major. When planning our algebra sequence in the 2015-16 school year, we decided to wait until this year to offer introductory abstract algebra so that we could offer a full college course that is comparable in level and content to the that offered at most undergraduate institutions. This year, after a year of math classes at Proof, we felt our most advanced algebra students were ready to undertake that challenge; they have been thriving in a college level ring theory course this block.
What is ring theory? Rings are algebraic structures used to model numbers or functions. For example, the integers, which consist of the whole numbers 0, 1, 2, 3, ... and their negatives -1, -2, -3, ... are one of the most basic examples of a ring. The two basic operations we can perform in a ring are addition and multiplication. Other than the integers, our second fundamental example is the polynomials with real coefficients. Students are learning that ring theory is not only a beautiful topic in itself; it has deep connections to number theory and geometry. We get a rich body of examples by looking at rings that originate from problems in algebraic number theory and algebraic geometry.
On a typical day in class, students are hard at work typing or writing up solutions to problem sets. Several times a week, I give mini-lectures on new material. In four weeks, we have covered rings, homomorphisms, quotient rings, the isomorphism theorems, and Euclidean Domains. We are about to start in on understanding how factorization works in general, and diving into principal ideal domains, unique factorization domains, and factorization of ideals.
If you walk into our classroom, more often than not you'll hear student voices talking to each other about the problem sets, reviewing concepts, discussing questions with me, and, quite frequently, making silly puns and jokes about ring theory terms. For example, one student asked,
Student 1: How do we play card games in this class?
Student 1: I deal.
Student 2: We play a variety of card games in this class.
NB: Ideal and variety are both ring theoretic terms.
In addition to a midterm and a final exam, every week the students have vocabulary quizzes to make sure they are keeping up on the week's vocabulary terms. On this week's quiz I asked the students what they wanted In the Classroom blog readers to know. In addition to what I have already included, the number one answer that I got was that students would like to tell you that they are having a lot of fun.
-- Sachi Hashimoto