Tutoring for the CSET

Matrix Multiplication, Fields, and Other Abstract Concepts

Lately I have been tutoring a few CSET takers. The CSET tests for math knowledge equivalent to what is expected from a math major. Consequently, the CSET covers a lot of topics, and some of them are quite abstract. One of the common CSET preparation guides out there starts with one of its most abstract questions right at the very beginning. The question asks to identify, among five possible options, one valid argument showing that the set of all invertible 3-by-3 matrices is not a field. So, this particular question checks whether the student knows what a field is (as an algebraic structure), and also that matrix multiplication is not commutative.
Many CSET takers who are not math majors have the goal of teaching middle school math. They are usually surprised by the amount of math in the CSET. At least two of them have made this remark to me: “Wow! Maybe I do not want to teach math. This is quite a lot, and very complicated.”
Individual reactions to that first question (on the set of invertible square matrices not being a field) vary widely from student to student.
One of them told me: “Let’s just skip this one. I do not want to waste time on this. If I see a question like this one in the test, I am just going to take a guess and that is it.”
However, some other students have the sort of driving curiosity that do not allow them to let go so easily, because they want to know what the question talks about. So, another student kept asking me question after question, during a couple sessions, until she understood the algebraic concept of “field”. As we went through several examples and counter-examples of number types, sets, functions, operations, and properties, along with the abstract names and notation, she kept saying: “Wow! This is mind-blowing! I never thought they would expect me to know so much stuff.” But she kept asking questions all the way until she made sure she understood that first problem in her practice test.
I very much enjoy tutoring for the CSET, precisely because the wide variety of abstract topics it covers. Just like Linear Algebra, and Vector Calculus, the CSET reminds me of my college years.