Nature is marvelous and in short supply in San Francisco. My school is bordered by a freeway to the east and many blocks of treeless streets in the other directions. However, as you pass through the front gate, you find a fascinating garden we call the Adventure Playground. I use this garden to consider the nature of school curriculum in general and math instruction more specifically.
David Perkins describes a continuum between tame and wild ideas in school curricula. Tame ideas are closed learning experiences (e.g. five paragraph essays, long division and textbook science) Wild ideas are unpredictable experiences. These can be harder to identify in most classrooms, though examples such as writers’ workshop, project based math and inquiry science certainly are alive and well in many schools.
There is a huge diversity of garden aesthetics around the world, but consider present day California. In irrigated lawn is a tame garden in arid California: little diversity and great predictability. It produces minimal benefit other than the satisfaction of having sculptured a landscape. At the other extreme, the very “wildest” of gardens would be nature left to its own resources. It produces some food which might be difficult to harvest and potentially dangerous to the harvester.
A garden designed to produce food as well as pleasing aesthetics would fall somewhere between these two extremes. This garden represents the sort of sweet spot between wild and tame that many educators aim for. Yet there a wide range of opinions of what appropriate curricula looks like.
Educators work to produce a rich bounty of ideas. They are charged with “taming the wild” so that students can make sense of things. The question that always needs to be asked is: have some ideas been tamed too much? Has the productive garden turned into the irrigated lawn?
Few subjects in schools inspire more “taming of the wild” than mathematics. While the field of mathematics is a vibrantly wild one, it has a long history of tameness in school. Take the classic rhyme to remember how to divide fractions: “Yours is not to question why, simply invert and multiply.” Critical thinking is often weeded from math, It feels neat, defined, and controlled. For many, it feels like the math we learned in school.
But there is a romantic notion about “real world” math that feels entirely wild. Students construct their own understandings and methodologies in math. Some are successful, while many struggle with this approach.
I have seen the effects of overly tame or unduly wild teaching. I have carefully parsed out equations for determining slope of a line without letting them muck around in the patterns. Later, I would observe students freeze when faced with similar problems out of context. They were starving on a flawless lawn.
At the other extreme, I would ask my 4th and 5th graders to “invent” different methods for multiplying multi-digit numbers. Some students did have inventive ways of doing this arithmetic work, but their successes rested more on their home experiences than on their inventive minds in class. These students were starving in a dark and lonely jungle.
We must tread thoughtfully in a zone between the excessively tame and the dangerously wild ideas. I have found a comfortable balance with Problems of the Week (POW’s). These problems are complex, messy, somewhat obscure but not impossible to solve if one persists. One of my favorite POW problems involves a camel crossing a desert:
Camila Camel's harvest consists of 3000 bananas. The market place is 1000 miles away. Camila must walk to the market and can only carry up to 1000 bananas at a time. Being a camel, Camila eats one banana during each and every mile she walks (so Camila can never walk anywhere without bananas).
How many bananas can Camila get to the market?
This problem is wild because it is not solvable by simple algorithms, yet it sufficiently tame so that many people have some entry point to start it. While the necessary math skills are not complex, their application often inspires creativity. It has a best answer but actually there are many good answers that are acceptable approximations.
If we succumb to breaking down math to its bare components, we teach how to take care of a lawn rather than promote diverse gardens. Let’s look at math from an organic gardening perspective. Let’s wild the tame corner!