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I would like to add my two cents to the ongoing discussion about necessary mathematics and methods in mathematics education. My graduate degree is in mathematics education and my B.S. is in mathematics. I agree that drill and practice are a necessary component of mathematics. There are skills which much be second nature. The simplest and strongest case for this involves algebra skills. The ideas of calculus are not difficult - and can be downright fascinating - if the individual is not constantly bogged down in difficulties with manipulations of equations. When a student tells me, "I do not understand what we are doing in class," I always ask if their confusion lies in the concepts or the machinations to get the answers to problems. This is the difference between 'why' and 'how' in mathematics. "Solve for dy/dx" is not the answer to the question "What information about f(x) does the derivative tell us?" but the distinction between the two statements and the relationship between the two ideas are difficult for some students to make. Far more often than not, students are not asking conceptual questions, they are asking concrete questions. They are not concerned about what the answer can tell them, they are concerned with how to get the answer. Herein lies the rub. There is a fine line between drill and practice and drill and kill. There must be practice and there must be a point to the practice. Why solve 25 problems for y' if all the information contained in y' is not discussed, examined, explored, questioned? On the other hand, why do only 2 problems in the same length of time, supposedly in depth, if the algebra done to get the answer seems to arise as if from the ashes because the algebra steps taken to get there appear magical and not reasonable? I believe more time needs to be spent in both areas. A 36 hour day would work wonderfully in this regard. Class time can be spent discussing the concepts, the whys, in groups with others and with the teacher. Students need to practice the concrete skills more on their own. This is the time that has been lost - the time spent outside of the classroom practicing. I am not saying take all practice time out of the classroom, but I am advocating moving away from nothing but plug and chug when there is so much more to it. I believe that the material has been slowly watered down as individual responsibility for the material has waned. Johnny should be responsible for doing a certain amount on his own. He should not be responsible for making einsteinian insights into material on his own but he should be able to follow a logical series of steps. If practicing this in class is not enough time, then practice outside of class. Make time to learn the material. Do not lower the level of the material to a level where everyone can learn it in x minutes - make the students rise to the level of the material. The main ideas of algebra and calculus have not changed in hundreds of years but the way they are being taught has. We need to maintain their depth and richness (the why) for the students and move away from only cranking out answers (the sometimes monotonous and boring how). Eager to keep this discussion going, Andrea Motyka Director of the Math Center Washington College 300 Washington Avenue Chestertown, MD 21620 410.778.7862