Saturday, August 11, 2012

Teaching Math Correctly

There's nothing new in this piece, but it certainly gathers all the right ideas and puts them in one place:
In the never-ending dialogue about math education that has come to be known as the “math wars”,  proponents of reform-based math tend to characterize math as it was taught in the 60’s (and prior) as “skills-based”.   The term connotes a teaching of math that focused almost exclusively on procedures and facts in isolation to the conceptual underpinning that holds math together.  The “skills-based” appellation also suggests that those students who may have mastered their math courses in K-12 were missing the conceptual basis of mathematics and were taught the subject as a means to do computation, rather than explore the wonders of mathematics for its own sake.

Without delving too far into the math wars, I and others have written that while traditional math may sometimes have  been taught poorly, it also was taught properly.   In fact, a view of the textbooks in use at that time reveal that they provided both procedures and concept.  Missing perhaps were more challenging problems, but also missing from the reformers’ arguments is the fact that not only are procedures and concepts taught in tandem  but that computational fluency leads to conceptual understanding.  
I'm beginning to believe less and less in the "math for its own sake" mentality.  I don't know that that can be taught, and to be quite honest, I'm not sure there are enough people out there who are willing to put in enough effort to get to the point where they can see the beauty, wonder, and interconnectedness of the mathematical mosaic.  I now lean towards the "teach to fluency" mode,  which is "necessary but not sufficient" to get to the "beauty and wonder" mode.  Oh, I'll take my students to "beauty and wonder" if they want to go there, and plenty do, but that's not where I focus my classes.

Too much "beauty and wonder" and not enough "fluency" puts the cart before the horse.


jeffsters said...

Agreed - teaching students "fluency" in math is the most important part. I would rather a student know how to multiply than why.

In my advanced calculus (aka mathematical analysis) we studied the theoretical foundations of Fourier Transforms, but we never worked any applications. On our exam we were given an application problem, i.e. find the Fourier Transform of (blank). No one got it right. We all new the theorems behind it but had never done one.

The professor learned a good lesson from this as much as we did. Just because you know the theory doesn't mean you know how to use it.

Learn math fluency and if you want to get down into the theoretical details then do it, but that's only for a small fraction of students.

David Foster said...

jeffsters...I've seen similar comments relating to physics: students were were getting "A"s on all the tests, but when given practical problems came up with absolutely bizarre answers.

One of them involved a marble or similar object being propelled through a curved tube. The question was whether when it left the tube it would continue to follow a curved path or not...

Elaine C. said...

My problem with the way skills based is often taught is that students have no flexibility.

Change the problem slightly, and the don't know what to do. Add an extra layer or two, and you get the deer in headlights look. If it doesn't fit the pattern, they just give up.

I need them to have both the manipulation skills AND the thinking skills... and any program that teaches only one of those just really isn't going to cut it.

Darren said...

Elaine, I infer an implicit assumption that "skills-based" means just learning how to do a certain type of problem and then repeating it. That's bad education no matter what the subject is. Knowing and having math skills is a little more complex than just applying an algorithm or rule.

Knowing multiplication tables, for example--makes for a wide variety of questions with easy applicability, don't you think?

Barry Garelick said...

Elaine, like many others, assumes that traditional math taught poorly is the norm. I would argue that much of reform math is taught poorly because it is inherent in the concepts used in reform math: "just in time" learning, group work, teacher as facilitator rather than instructor, and a belief that explicit instruction is injurious to the process of learning.