Getting at the Big Math Picture

While Curriculum Night is an opportunity to look at the big picture, it is also an opportunity to balance these ideas and LREI's approach to learning against your own educational experience. This can often be complicated work as the memories and habits connected with how you experienced school may have been very different. One area in which this tension often occurs is with our approach to mathematics and the development of mathematical thinkers. I include below some “big picture” thoughts on math instruction at LREI and its connection to each student’s development as a learner.

In listenting to the presentations of our math teachers, I always hear a key goal of our approach to teaching mathematics and of our math curriculum in general articulated repeatedly. As students acquire important math knowledge and practice the skills that allow them to put these concepts to use, it is equally crucial that they also discover their capacity for creative and critical thought and the rewards of curiosity. We play at math to discover important truths about the world we live in and these discoveries provide us with another language for making sense of our experience.

As with any endeavor, this requires hard work and practice. It is not always easy and for some the challenges can be frustrating. But when one steps back, there is a clear sense among our students that math has a relevance beyond just being useful for some future purpose. That students can derive intrinsic pleasure from their mathematical work speaks to the in-the-moment purposefulness of our curriculum.

When I poke around in our math classrooms, I am always struck by the depth of thinking and ability to think mathematically that our students have established as habits. In my mind, this is precisely the goal that a math program should have for its students. What this translates into is students who have a solid foundational knowledge AND who possess the ability to understand and use mathematics as a symbolic and relational language for looking at the world.

I had a fairly traditional math experience in middle and high school. While that program of studies led me to calculus in my senior year, every prior year was a seemingly endless series of memorizing algorithms and practicing them all the while saying to myself, “What is the point of this?” It was not until my senior year when I was taking calculus and physics that things clicked. That was hardly surprising given that the calculus was derived by Newton and Leibniz to resolve core questions in physics. Why did I have to wait until my senior year of high school for such a clear and powerful illustration of the beauty and utility of mathematics? The general orientation of the LREI math curriculum that we use seeks to address this question in the most obvious of ways: students must experience math as a means for making sense of the world if we want learning to occur on a deep level.

While it is true that there is a dimension of math competence that requires students to be able to work a variety of algorithms to get to a “precisely right answer,” I would not define that as the purpose of math nor actually as math. Following algorithms is an operational activity, which when not connected to a deeper understanding of the elegance and purpose of those algorithms becomes simply an exercise in following rules. So how do we help learners to begin to see the world from a mathematical perspective? And in so doing, how do we help students to acquire a set of discrete skills that not only helps to guide them to the answers to one set of problems, but also becomes the foundation for the exploration of new problems?

To answer this, I think that it is also important to note that while we use in all three divisions elements of formal math prorams, we are not driven by them to the exclusion of other ways of approaching math education. There are some camps that describe some progressive aligned math programs as “fuzzy” and there are other camps that describe the more traditional programs as “routinized.” For me, these distinctions (like those in the debate between phonics and whole language in the literacy realm) are ostensibly political. They are about different camps staking out positions on a field that has more to do with securing textbook contracts than with childrens’ learning. What good progressive schools (and I place us firmly in this camp) have always done is to take a balanced approach. It is not an either/or. In adopting any program (in full or in part), we recognize that it has certain limitations and that for some students its approach might be more challenging. That is why our math teachers have worked in deliberate and thoughtful ways to consider how to supplement any program to best meet the need of all students.

One question that seems to come up a lot is that the formal "text" materials that we use don’t teach math, they don’t feel like a “real” textbook. I’m not sure that I or any of the school's math teachers would say that the purpose of any textbook is to teach mathematics. The texts are indeed supporting materials. They support a rigorous approach to math that posits the math classroom as the crucial nexus point where students, the teacher, and mathematical ideas come together to make meaning. Math instruction at LREI is structured around a collective learning experience. It is not a course of independent study.

So any good math program will always be “in process.” Our goal is to not only find ways to be more balanced in general, but to always endeavor to work with your family to be more balanced relative to the needs of your child. So at the heart of all this process rests a partnership between home and school. As we work to define this partnership, we are often are asked about our methodology, which is sometimes misunderstood as “the teacher facilitates and the student discovers” without any “real teaching” on the part of the teacher. This description misses a crucial component that makes our approach a structurally sound one. This misunderstanding only posits two legs (teacher facilitates and students discover); the third leg is that teachers do “teach” after the discovery has been made. They do define a generalizable conclusion that leads to an algorithm that encapsulates the discovery. This third leg is the “traditional” math curriculum that many of us experienced when we were in school. Ultimately, it is what our teachers do with our program structure that creates powerful opportunities for learning. There is no holy grail of math programs here; there is, however, the committed work of dedicated teachers trying to get it “right.”

Questions of truth and beauty have always been mired in the conflict between the relative and the absolute. And in fact, one can look at the history of math and science as one long melody on the proven fallacy of what was believed in a given moment to be “precisely right.” While it is certainly the case that at an operational level math requires a necessary degree of precision, it is often the getting to the solution that is far more interesting than the solution itself. It is the debate over these varied paths that moves mathematicians and that makes of math a kind of poetry. I can think of no more meaningful road for our young mathematicians to travel along especially when they find themselves with an able guide and worthy traveling companions.

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