Oftentimes, when I have to grade homeworks, quizzes, and exams in my math classes, I find that the quality of the assignments is rather low. Oftentimes, the mathematics is correct, but it will read like a wall of equations. This is okay for “drill” problems which just test whether you can do a computation (e.g. a plug-and-chug problem), but for many other problems, I would prefer that people give reasons for their work.
The directive “Show your work!” (or a variant thereof) has been a part of the instructions for many of my math assignments, ever since I was in middle school. During those years, it means that they wanted to see all the numbers and every step of the algebraic process. I suppose that at the elementary levels, that is important, because one way to test knowledge of pre-algebra and algebra is to show EVERY algebraic manipulation (in some sense, like what I did yesterday on the blog).
However, as you get to more advanced math courses, it sometimes becomes more about the concepts than the computations. For example, translating a problem into an equation is an important task in mathematics, but the process of doing that is sometimes overlooked or just given as a formula without sufficient justification (in my opinion).
This really became salient to me in my sophomore year of college, when I took Math 439 (Mathematical Methods in Biology). Our professor, after grading the first homework, sent an e-mail to the class, which said (paraphrased): “The mathematics on your assignments were mostly correct. But I was disappointed in the quality of the assignments. You need to use words to justify your reasoning in these assignments–this is an advanced level class.” (Thankfully, my assignment was not of the worst quality.)
It seems, from the assignments that I have graded, that oftentimes on non-plug-and-chug problems, students will just charge head-first into the problem without considering the tools that they might use. For example, I graded a problem on the exam which ended up being a NIGHTMARE to grade. The most elegant way of solving it was to apply Green’s theorem after directly computing one of the line integrals, but most students attempted to just compute all three line integrals directly. It was painful to sort through all of the algebra errors that they made.
Sometimes, a problem which requires some computations is actually more conceptual in nature. I feel that unless a problem says “explain your reasoning in words” or something of that ilk, the students will continue to show work only in the sense of a wall of equations.
Showing work only by means of equations, without any words in the explanation, smells like their major conceptual hold on math is how to plug and chug. It is much more elegant and beautiful than that, if they would try to learn the concepts and how to apply some of the theorems to make work easier…
Today is the one-hundred and forty-ninth day of M.M.X.I.V. That makes twenty-one weeks and two days.
היום ארבעה וארבעים יום, שהם שישה שבועות ושני ימים לעומר
Today is the twenty-ninth day of M.A.P.L.E. That makes four weeks and one day.