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Doing without Understanding: "Why Do We Teach Math?"

I've had quite a couple of weeks of unexpected conversations. First, my wife and I took our son and a couple of his college classmates out to dinner. We've found that food provides a consistent vehicle to maintain some communication with our offspring. Otherwise, we might never see him. One of his friends, Nick, shared his experiences tutoring math in a South African township school over the summer. Nick's job was to tutor algebra to 11th graders, but he quickly found the needs ran much deeper. For instance, Nick was surprised to uncover that the students didn't understand that fractions described equal-sized pieces. The students lacked the most fundamental concept of what fractions mean, yet they were computing away and struggling their way through algebra. I, sadly, wasn't surprised. We see this lack of foundational understanding all the time right here in the United States.

However, I wasn't fully aware of how far students could carry doing without understanding until another discussion erupted in the software design class I teach at the Harvard Graduate School of Education. This year, as in many past years, I have a few students from the Teacher Education Program, folks who are transitioning from math-related careers into teaching. These students are working on a project to leverage technology to help students understand the base 10 system and number systems in general. In sharing the project's progress with the class, one of the students projected a chart showing different bases, from base 2 to base 10, with numbers displayed in the different bases. Some spaces in the chart were left blank, and he prompted the members of the design class to use the patterns to fill a few in.

About two-thirds of the students responded with curious stares. "What's a base 10 system?" one of
the students finally asked. I hadn't expected the question from these Harvard masters students, but the transitioning math teachers and I did our best to explain. I offered the analogy of a car's odometer. When one of the digits in the odometer passes 9 it rolls over to a 0, and the next place value increases by 1. An odometer displaying in a base 3 system would roll over after 2. There's a system, we explained, with 10 symbols in base 10 (0 through 9) for writing any number. In base 3 there are only 3 symbols (0 through 2) for writing any number. Our explanation elicited quiet murmurs of "wow" and "cool". Many of my students had never realized there was a system underlying the numbers they encountered and used everyday. That lack of understanding, however, hadn't prevented them from excelling in math and surviving quite well in the world. They could do, follow the directions and the procedures; they just didn't understand why it worked.

Another student inquiry pushed us to consider whether or not it mattered. "Why would anyone need to learn this?" Good question, one that prompted a more fundamental one: "Why do we teach math?" I just couldn't pass up the opportunity to engage this thoughtful, diverse group in muddling around at least a little bit in this topic. Ever the teacher, I answered the challenge with one of my own. "Have you ever asked this question about literature?" "No." "Great, then let me put it out there: Why do we teach literature?" A short period of quiet thought preceded some suggestions, one being: "We learn literature to understand different ways to communicate." Okay, we don't have students read and  analyze Tuck Everlasting, Hamlet, or other literary works because we may use them at a later date. The books are vehicles to learn something beyond the works themselves. We learn about how people think, feel, and express themselves. Cool.

So what about math? The answer wasn't so apparent. We haven't done a good job of identifying what mathematical thinking is and why it's valuable. Math has been something we just learn to do. Some in the class pointed to computer programming and the need to work in different number systems, like binary and hexadecimal. True enough that understanding base 10, base 2, and base 16 can have vocational value, but what if you don't want to be a computer programmer? How, I probed, would you think about setting the timing for traffic lights to make sure the cars don't get backed up? Would you turn to literature? History? Science? It's about understanding and articulating a system with consistent rules, like our number system. Over time we're going to forget a lot of the specific math content we learn in school just as we'll forget the details of Tuck Everlasting and Hamlet because we just don't use them. What are the more general learnings we want students to carry with them from digging into the math?

The answers, in part, have been articulated for quite some time in the NCTM process standards and the National Research Council's Adding It Up report. The Common Core State Standards document has re-articulated them quite well in its Standards for Mathematical Practice. We teach math as a vehicle for helping students to:

1. Make sense of problems and persevere in solving them;
2. Reason abstractly and quantitatively;
3. Construct viable arguments and critique the reasoning of others;
4. Model with mathematics;
5. Use appropriate tools strategically;
6. Attend to precision;
7. Look for and make use of structure;
8. Look for and express regularity in repeated reasoning.

These practices have broad applicability. Learning math allows us to practice and hone them. I'm glad to see them in the common standards adopted by so many states. Sadly, we know from experience that having the standards doesn't ensure they become embedded in instruction and, more importantly, assessment. We won't value these practices unless we have some ways of explicitly tracking their development.

We have much work to do. I'm hopeful though. Nick's students in South Africa and my students at Harvard were both pretty psyched to finally get under the hood of the math they were doing. It's a nice and powerful 'aha' when you realize, "Oh that's how it works. Cool!"

(For some accessible background on counting systems, check out chapter 2 of the wonderful book Here's Looking at Euclid by Alex Bellos. You can get a preview of the chapter on Google Books.)

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