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

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