This is going to seem like a silly, and potentially obvious
question, but I've done a lot of thinking over the last month, and I’m becoming
convinced that the way someone answers this question defines their entire
outlook regarding mathematics education, including what content should be
included, how it should be taught, and how it should be tested. The question, and I believe it to be the most
important question in mathematics education today, is this: what is mathematics?

If your answer to this question centers around memorizing
times tables, learning the basic mechanics of algebra, or memorizing geometry
formulas for area and volume, then in my experience you probably lean toward a
lecture-based method of delivery in the classroom and test students using
problems that strongly resemble those that were in the assigned homework. While not explicitly teaching to the test,
you do expect that the problems on standardized tests will resemble those found
in the textbook from which you are teaching, and that, if you so decided, you
could teach to the test and your students would succeed in passing these tests.

On the other hand, if your answer to this question centers
around discovering patterns, regardless of where they may be found, then in my
experience you probably lean toward a more constructivist approach in the
classroom and test students using exercises that require them to use the
material covered in class in ways that may or may not have been done in
class. Teaching to the test is an impossibility
for you, since the point of the test is to see how well the students can use
the material in new ways…in short, to see how well they can problem solve.
Standardized tests are just one more opportunity for your students to problem
solve, and as such the preparation for them is a regular, ongoing part of the
course, not a separate entity on which to focus.

The answer to this question also seems to relate to how you
feel about the Common Core standards and the tests that are due to accompany their
implementation. The “mechanics” people
tend to not agree with CC, while the “patterns” people tend to see it as not
quite so big a deal. (Note: I’m not talking about the part where the test
scores are part of the teacher evaluations…that’s an entirely different
discussion.)

Now, to be clear, I’m not saying that the “mechanics” people
don’t try to teach the kids to problem solve.
Nor am I saying that the “patterns” people don’t understand the value of
being fluid with the fundamentals. What
I am saying is that for every mathematics educator I’ve met, one seems to take
center stage, while the other is either a follow-up (as the problems solving
seems to be for the mechanics folks) or a part of the process included along
the way (as it seems to be for the patterns folks). But the conversation we are currently having
in math education seems to be missing a discussion about the fundamental
question that underlies all of our beliefs.

So, how would you answer the question?

(Admittedly, this is rhetorical, but I’d love to hear some
of the answers.)

Hello Johnothan,

ReplyDeleteJust discovered your blog and have been enjoying reading the posts. Did your school end up implementing Standards Based Grading? Common Core? I am thinking about ways of integrating Exeter problems into my courses (9th grade Alg I/Geometry, 10th grade Alg II). We are implementing SBG and Common Core standards this year. One idea I've come across is grouping the problems by topic and following the progression of problems, and while I know that's not ideal, it might help me deal better with the standards side of things. In any case, I am certainly looking forward to reading future posts.

All my best,

Steve