At the beginning of every year I am reminded that there are little quirks I have about how I would like to see answers presented and I have this silly assumption that my brand new students should just know what I want. I wish that I remembered this in August and prepped myself and my students with some conversations. I am writing this brief post this morning while one of my classes is taking a test. I would love some feedback here in the comments or over on twitter where I am @mrdardy
I just graded my first problem set from my AP Calculus BC wizards. One of the questions asked for the point on the curve y = x^(1/2) that was closest to the point (4,0). Everyone correctly identified x = 3.5 as the critical x value and about half of them identified (3.5)^(1/2) or sqrt(3.5) [my LaTex skills are weak and I am in a bit of a hurry…] as their y-coordinate. However, about half of them gave me a three decimal approximation. For reasons I cannot completely justify it makes me nutty to see a less exact answer written as an extra step in their work. I know that their science teacher is not interested in radicals in their answers. I know that carpenters do not have radicals on their measuring tapes. I also know that all of my Calc BC kiddos had the exact value for y written at one point in their solutions and many chose to do a little extra writing to make their answer less exact. Am I being a crank if I make this a point of conversation? Along the same lines, I urge them not to rationalize but many cannot help themselves. I urge them to leave a line equation in point-slope form when that is one of the steps in their problem-solving process. I ask them to write x < 3 rather that 3 > x, especially in a piecewise function where I want to read through their domain in order.
These are tiny, tiny problems to focus on. I recognize this. But I also know that I will spend many months with this group of students and I want them to understand my thoughts since I am asking them to routinely explain theirs on paper to me. I would love to hear points of view about how important any of these habits are for kids of this caliber. I am completely open to recognizing that these are just weird little quirks of mine, but I also hope that there is some mathematical logic underlying all of this. Please drop a line to let me know what you think and how you have responded to students in your efforts to help them develop a clear and consistent strategy for communicating their mathematical ideas.