"In all things of nature there is something of the marvelous." –Aristotle
Nature certainly is marvelous and yet, in remarkably short supply in San Francisco. Take my school: 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 are drawn to a fascinating garden cultivated over the years by our school librarian and resident gardener. We call it the Adventure Playground. It is with a renewed sense of adventure that I have been using this garden to consider the nature of school curriculum in general and math instruction more specifically. Let me explain.
David Perkins, Co-Director of Harvard’s Project Zero, describes a continuum between tame and wild ideas in school curricula. Tame ideas tend to be closed learning experiences with defined input and predictable output. Tame ideas are bountiful in many school curricula: five paragraph essays, standard algorithms for long division and textbook science, for example. Wild ideas, on the other hand, are unpredictable, opened 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.
Gardening presents us with an apt metaphor to further understand the interface between wild and tame ideas. There is, of course, a huge diversity of garden aesthetics around the world and across time, but let’s consider present day California. A good example of an extremely tame garden is a classic fertilized and irrigated lawn in the middle of arid California: little diversity and great predictability as long as you put in the effort and time to maintain it. 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 essentially left to its own resources. This might not even be considered a garden by many people. It could produce some food, but this might be difficult to find and potentially present many dangers to the harvester.
A garden designed to produce food as well as pleasing aesthetics would ideally fall somewhere between these two extremes. It can be characterized as a “taming of the wild,” all in moderation and with an eye towards preserving as much of nature’s wildness as possible. This garden represents the sort of “sweet spot” that many educators aim towards when designing classroom activities. Yet, just as there is a tremendous diversity of garden designs and purposes, so, too, is there a wide range of opinions of what the ultimate outcome actually looks like.
Extending the gardening metaphor to school curricula, educators work hard to produce a rich bounty of ideas and concepts. There is often an interesting mix of tame and wild ideas governing their instructional choices. Each person brings their own individual sense of tame vs. wild to their classroom. Educators are charged with “taming the wild” so that their students can make sense of complex ideas appropriate to their age. The question that always needs to be asked is: at what point have these 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, its representation in schools has a long history of static tameness. Take, for example, a classic rhyme to remember how to divide fractions: “Yours is not to question why, simply invert and multiply.” Critical thinking and conceptual understanding have effectively been weeded to oblivion from many math classrooms in favor of “lawn” curriculum of textbooks with very specific procedures on how to long divide with remainders, add unlike fractions or find the slope between two points on a line. It feels neat, defined and controlled. For many of us, it feels comfortable, like the math we learned ourselves in school.
At the other extreme, though, would be an entirely wild math curriculum where students had to make their own sense of the “jungle” of problems facing them. There is a certain romantic notion about the real-world, perhaps practical, nature of this type of math instruction. Students construct their own understandings and methodologies in math. This works very well for some students and proves itself entirely uncomfortable for many others. So while the real world of math is, indeed, a wild, wild place, it is the role of the educator to tame it just enough to allow the most productive learning possible.
Given human diversity, there is no such thing as the perfect degree of tameness or wildness in any educator’s math instruction. However, pressure is placed on math teachers from students, parents, colleagues and administrators to tame the curriculum to the point of sterilization. It can be tempting to tame the math concepts students learn to the point that critical thinking and logical decision-making are effectively removed. This makes math feel safe and accessible in many people’s eyes. But that does not make it useful nor does it necessarily provide students with the logical thinking skills crucial to their success.
In my own practice over the years I have seen the effects of overly tame or unduly wild teaching. For example, I have carefully parsed out a series of equations for determining slope of a line with given information without, perhaps, letting my student muck around in the patterns that slopes of all straight lines possess. Later, I would observe students freeze when faced with similar problems out of the context of the perfectly controlled input I had taught them. In effect, they were starving on a flawless lawn.
At the other extreme, I used to ask my 4th and 5th graders to “invent” different methods for multiplying and dividing multi-digit numbers. I asked them to do this with minimal computational instruction from me so as to maintain purity of thought and not overly influence their methodologies. While some students did have inventive ways of doing this arithmetic work, their successes rested more on their home experiences with their families than on their inventive minds in class. In too many instances, students simply gave up when faced with the responsibility to create all their understanding without sufficient guidance by me or other adults in their lives. These students, from my observations, were starving in a dark and lonely jungle.
These types of experiences have taught me that I have to tread thoughtfully in a zone between the excessively tame and the dangerously wild ideas that govern my instructional choices. And as my years of professional experience add up, I have been considering another related question: Is it possible that the longer I teach, the tamer I manage to make the curriculum for my students? Have I dropped ideas that are too wild and cannot be effectively tamed? If any of this is true, can I identify ways to “wild the tame corner” for both me and my students?
One are in which I have felt success balancing the tame with the wild is using 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, worth its weight in gold, consists of 3000 bananas. The market place where the stash can be cashed in is 1000 miles away. However, Camila must walk to the market, and can only carry up to 1000 bananas at a time. Furthermore, 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?
I consider this problem to be sufficiently wild because it is not solvable by any simple algorithm, yet it is also sufficiently tame so that many people feel they have some entry point to start working on it. It is wild enough to allow for several different approaches towards ultimately solving it, but tame enough that any one of these methods is usually accessible to a wide range of math learners. So while the necessary math skills are not particularly complex their application often inspires creativity. It does have a best answer (highest number) but actually there are many good answers that are perfectly acceptable approximations. Success at finding a solution is much more dependent on flexible thought and the ability to discern a pattern in the data than it is formulas. It is a wonderfully wild problem that can be cultivated using a relatively straightforward math skills in new and inventive ways.
E.M. Forster once wrote: “Spoon feeding, in the long run, teaches us nothing but the shape of the spoon.” It seems to me that if we succumb to the temptation of merely breaking down math (or any subject) to its bare components, we are teaching how to take care of a lawn rather than how to promote diverse gardens. Let’s look at math from an organic gardening perspective. Let’s wild the tame corner!