> Thanks for your insights about the weaknesses of such questions. > I would also wonder how to grade such questions, and students > might be puzzled as well how to answer such questions. Does > this alternative sound better? I would be sure that students > know what I mean by "algebraic methods" of solving quadratic > equations. > > Discuss each of the methods of solving quadratic equations > that we had mentioned by explaining some of the advantages > and disadvantages of each of these methods. Give an example > of solving an equation by each of these methods. Can you see > any advantages of the algebraic methods over the other methods? > > I think this version of the question still asks students to > think but requires the student to give meaningful answers > and requires students to know the mathematics to answer > the question.
It's certainly better, but it also runs the risk (mine as well) of being too lengthy to adequately answer. One of the things I learned in trying things like this (having students explain stuff as part of their assignment) is that often the best math students are the least interested in doing it, seeing it as an unnecessary obstacle to simply learning methods to solve problems. This especially applies to those talented enough and interested enough to pursue math competitions such as the AHSME and AIME (the math olympiad qualifying tests). Although a very small minority, this is not a minority I wished to alienate. It's not really a problem if done rarely (a fairly short essay answer for every assignment, or a more lengthy one like the above every few weeks), but if students have to write explanations for just about every problem they have to solve (like I've seen in some school texts), it becomes excessively tedious for them and the quality of work you'll get from them on these kind of assignments goes way down.
A possibly better approach might be a hybrid version that I've sometimes tried, where I outlined how to solve something (usually something they haven't seen before or something a bit harder than I'd expect them to be able to come up with on their own) and ask them to "fill in the blanks" in some way. It occurs to me that something I once tried as an assignment for a high school "math topics" class (that I once taught to two very strong students, where "very strong" means roughly U.S. math olympiad math qualifier level) might work well. See the handout titled "obtusity.pdf" that I posted at the following math-teach post:
Of course, this assignment is much more advanced than asking about quadratic equations, but the same principle could be used. Namely, the teacher (or textbook author) writes up a short and incomplete essay about two or three ways to solve quadratic equations and leave "extended blanks" in places where students are to come up with examples to illustrate a point, or to further explain a point that is made but not made very well.
Incidentally, writing an explanation is just one way of having a student "work the material" in non-traditional ways. In Fall 1996 I began teaching at a state supported math/science boarding high school and was teaching an honors algebra 2 class that was mainly filled with incoming Juniors gifted in other areas than math/science, such as art, drama, humanities, etc. During the first week I was covering polynomial multiplication and gave two explanations, one that I called the "left brain explanation" and another that I called the "right brain explanation". The left brain explanation was the use of the distributive law twice, as in (a + b)(x + y + z) is (a + b)x + (a + b)y + (a + b)z = ax + bx + ay + ... The right brain explanation was by using a rectangle diagram, where one side was divided into a, b lengths and the adjacent side was divided into x, y, z lengths. I then mentioned how something like (a + b)^3 could be exhibited geometrically by dividing the three dimensions of a 3-dim cube into a, b lengths and adding up the volumes of the 8 sub-rectangular solids these generate, and drew a rough picture of it on the board. This seemed to generate some interest with a few of the "artsy students" so, on the spur of the moment, I assigned as extra credit (up to 5 points extra on the first major test, I think) the following: Draw a neat and carefully labeled diagram of what I outlined, making use of different colors and/or shading to better illustrate the various parts, maybe drawing different views of the 3-dim cube (side view, back view, etc.). I said that only very well done drawings would get full extra credit points. And holy smokes, I sure scored a bull's-eye with that assignment! I got some absolutely excellent drawings from several of the "artsy students", showing they understood perfectly how (a + b)^3 decomposes the way it does rather than as a^3 + b^3. Several took other math classes from me, and one was one of my January "Special Projects" students (a week of no classes where faculty run "special projects" for students to sign up for) when my topic was math in science fiction, and he wound up making a poster of a hypercube showing the 16 parts that arise in the geometric explanation of the expansion of (a + b)^4 (which was motivated by one of the stories we read, Robert A. Heinlein's "And He Built a Crooked House"). Unfortunately, I don't have any tips on replicating things like this, even for myself. Sometimes things just work out well, but usually (for me, at least) trying to do things that you hope will generate high interest in certain students is like trying too hard to get someone you like to be interested in you (to continue the analogy I began in my previous post today) -- it almost never works, and in fact it usually backfires, resulting in the situation being worse than if you didn't even try.