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.)