Learning Math “Matrix” Style

This past week I participated in the Department’s placement and advising of incoming LSA international transfer students.  Due to visa issues, these students are among the last to participate in summer orientation.  This year there were about 160 international transfer students, most of them from China.  I very much enjoy working with these enthusiastic, bright, jet-lagged students.

Usually, the process of evaluating where to place a transfer student begins with a review of the student’s paperwork – course descriptions, books, syllabi, old exams – and is followed by some pretty straightforward questions about the material.  Oftentimes students will present paperwork that makes it completely clear what first or second year math course they should take next.  About one in twenty students arrive having taken enormous amounts of what I consider to be graduate level math (e.g., measure theory or functional analysis), and their placement is more challenging (and fun), but still clear.

However, it is also very common for a student to arrive with a clutch of course descriptions/syllabi that are either  (1) so vague as to be useless (e.g. “This course enables students to understand calculus deeply by studying differentiation and integration.”)  or (2) look as if someone had designed the course by first cutting out the chapter titles from books about linear algebra, differential equations, calculus, and multivariable calculus and then randomly rearranging them into a list – such  a course description might start: implicit differentiation, Hilbert Spaces, eigenvalues, mean value theorem, radius of convergence, …

For these students the placement process is more difficult.  I usually begin by asking computational questions:  What is  the derivative of  \arctan(x)? What is indefinite integral of \sin(t)?  What is the radius of convergence of \sum_{n=1}^\infty  (\dfrac{x-4}{5})^n ?  What are the limits of integration when integrating over the bounded region that has boundaries x = 0, y =0, x+y+1 = 0?  What is the curl of F(x,y,z) = \langle xy, x \sin(yz), \exp(z^2xy) \rangle? A longish math discussion based on the student’s responses to these questions follows, and we eventually hone in on a course for the fall that will be appropriate for the student.

This quizzing on computational matters always bothers me – I don’t believe the value of taking a calculus class lies in learning that the derivative of  \arctan(x) is \frac{1}{1+x^2}, that the indefinite integral of \sin(t) is -\cos(t) + C,  or  that the power series \sum_{n=1}^\infty  (\frac{x-4}{5})^n has radius of convergence 5.   Yes, you should learn how to compute all of these things (and much more) in a year of calculus.  However, since WolframAlpha can also do these computations for us already, hopefully the students are learning something more than how to compute.

This year I experimented a bit by expanding my bank of quiz problems to include non-computational problems.  Since calculus at Michigan focuses on having students understand the underlying concepts, I decided to use old Michigan Calculus I exams.  The results were mixed, and I think I won’t do this again.  However, there was one obvious conclusion:  students who took math courses for which the course description was a laundry list of disconnected mathematical topics had basically no conceptual understanding of the mathematics they had studied.

This got me to thinking about the mathematical methods courses I was required to take as an undergraduate physics major.    Since I had already seen a coherent presentation of most of their content, I found these topic-a-day math methods courses to be shallow, unsatisfying, and easy (the latter was a good thing since I was trying very hard at the time to woo my eventual spouse).   However, I think my non-math major classmates probably struggled mightily, learned as much as the tests required, and long ago blocked the courses from their memory.  I understand the urge to design a course that is encyclopedic and covers much of the math a scientist working in the field right now might need, but I’m not so sure about the effectiveness of learning mathematics “Matrix” style .   I think yesterday’s Dilbert cartoon pretty well sums up my thoughts on this point.

So, what should the aim of an undergraduate math class be? Here is a partial answer.  For sure, a good math course should teach students how to do things that WoframAlpha can also do, but it must do more.  A good undergraduate course should also engage the students with the material, develop their problem solving skills, and have them grapple with the concepts that underpin the material.  And all math classes should help students develop their abilities to (1) think logically and abstractly and (2) express themselves rigorously and concisely.

Maybe, if we do it right, such students will even be equipped to learn the mathematics that explains the science not only of today, but the science they will encounter decades hence.  Maybe.


1 thought on “Learning Math “Matrix” Style”

  1. I think that’s an excellent statement of the goals of a mathematics class. I would add that if we really do it right, the students will find some joy in doing mathematics, and ownership of the subject and of their work. We should communicate that math is not just a sequence of procedures, nor a maze to be walked, but a country to be explored.

    BTW a good way to find old Calc 1 and 2 problems is to use the exam shops, which organize the last 14 years of exams by topic.

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