In a (now famous) critique of math teaching called the
Mathematician's Lament, the educator Paul Lockheart wrote, "The main problem with school mathematics is that there are no
problems." What are there? Exercises, opportunities to mimic, to perform an established algorithm quickly. The stereotypical hallmarks of a challenging math classroom - pages of problems, an increasing number of variables and symbols - demand surprisingly little thought.
Math is actually full of puzzlement, wonder, improvisation, proof, argument, modeling, and artistry. Teachers in a progressive math classroom assiduously avoid making problems simpler. Instead, most of their time is spent posing questions that cast doubt, that complicate, that expand the scope of inquiry. They rarely, if ever, ask students to imitate their own math moves. That said, our students develop facility through practice just as musicians do. But it is in support of, and not in place of, solving deep, complex open-ended problems.
The list of topics our middle schoolers study is only a very small part of the story of our program. That list is unsurprisingly like the ones of peer schools. It includes proportional thinking, surface area and volume, and exponential functions, to name a few. We use a research-based curriculum developed by Michigan State University called the
Connected Mathematics Project (CMP). The curriculum is problem-centered and inquiry-based, and uses all the best practices we currently understand. The sequence is created for sixth through eighth grade; we teach the sixth grade units in fifth grade. A summary of those units is
here and a rationale for this type of program is
here.
Many of our eighth graders skip ninth grade 'Math 1' and go straight to 'Math 2'. But as those who teach math know, the hierarchy of math topics is a fiction. Algebraic thinking, for example, doesn't begin with the introduction of symbols in seventh grade, it begins in kindergarten with pattern and prediction.
There are better measures of what this kind of classroom produces. A professor (and former LREI parent) once said we are training the kinds of students math professors dream of. On any given day in a math classroom, you might see students grappling with the unknown, leveraging past knowledge, combining old ideas to test new ones, debating their solutions, engaging in proof and modeling. Rather than giving children exposure and practice at a specific set of algorithms, we're teaching them to be confident and nimble. We're teaching them how to grapple with the confusing, the unfamiliar, and the vague. In other words, the kinds of problems produced by life.
Here is a sampling of perplexing problems:
- If a cone, sphere and cylinder have the same height and diameter, how do their volumes relate?
- Draw 2/3 divided by 1/4 (and 2/3 divided by 4. What's the difference?)
- Create a data set in which all the measures of center (the mean, median, mode and midrange) are different.
- How many different open boxes with whole-number dimensions can you make out of the same area of cardboard?
- Use a system of equations to graph your initials.
- How many triangles do you see?
In case you can't tell, I love talking about math and am always happy to do it more. Please consider coming to the meeting on Thursday morning. Always feel encouraged to reach out with questions.
Warmest,
Ana