Days 26 and 27: Seeing the Forest AND the Trees

We left Medford, OR on Friday and headed to the Redwoods National and State Parks in northern California. As a general rule, I’m not a fan of simplistic analogies, but this leg of the trip was evidence that sometimes even the most obvious analogies have value.

The Redwoods wasn’t part of our plans when we started thinking about the trip in March. Once we realized how close we were going to be to that part of California, it was a no-brainer to add it to the trip. Once you’re in that part of the state, you’re also pretty close to Mount Shasta, but that would have required a bit of a detour and we decided not to do that. What helped us make the decision to do the Redwoods but not Mt. Shasta was that we could experience the redwoods but only see the mountain. This ended up being my first teaching analogy. How many times has the material I’ve been teaching naturally pointed at some connnection? Sometimes I take us in that direction for half of a class, a couple days, or more, and sometimes I don’t. Thinking about this decision as a parallel to the trip has given me a slightly better framework for making the decision on when to take the tangent in class: are the students going to just watch and listen to me talk about something, or are they going to be able to experience it for themselves?

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By the time we got to the Redwoods, we had already seen some huge redwoods, sequoias, and cedars in Orgeon and Washington. I hadn’t expected to see these trees in other places, so once I had I thought that maybe we should change our plans and not head into California since we wouldn’t be seeing anything new. That was a silly thought, of course; it’s one thing to see a couple of these trees in a park in Portland and quite another to see huge groves of them along the Pacific coast. This brings me to my second analogy of the day. How often do my content decisions reflect the belief that as long as we see a concept a little bit, that’s enough? This is something I’ve been very aware of in the last couple years, of course, as I’ve tried to cut some topics so we can spend more time on others. Is it OK for me cut even more so that students can experience every topic in great depth? Are there times when just seeing one example of a topic is actually sufficient?

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Finally, there were the trees themselves. I keep thinking about the old saying about not seeing the forest for the trees. When we were in the redwooods, there were plenty of times when we couldn’t see the trees for the trees. That is, there were some trees that were so big and so close to us that they made it impossible to see anything else. In this analogy, the forest is a course and the trees are the topics that make up the course. More than once, I’ve looked at a textbook to start preparing to teach a new course and been struck by how incongruous some of the topics seem. They may be great topics, but they don’t seem to lend themselves to the “story” that the course is trying to tell about mathematics. This is what I mean by the forest getting obscured by the trees: at the end of the year the students may be able to tell you some of the things they learned but they don’t have a good sense of the big picture.

Having one tree be so big it obscures the other trees is a different kind of student experience. In the classes I teach, there are typically one or two topics that seem to challenge students more than any others. One good example of this is proving trigonometric identifies. A typical precalculus class might spend anywhere between two and five days on these “proofs” (which aren’t really proofs at all, but that’s another story). Then, the class will turn to solving trigonometric equations, which is the perfect time to apply these trigonometric identities, yet students are often unable to do so. What seems to happen is that the stress and challenge of the proofs have so obscured their vision that they can’t even see the same relationships in the equations. While I don’t want to extend the analogy so far as to say we need to cut down these trees that obscure students’ vision, I do think we need to be aware of them and look for ways around them. (But in the case of proving trigonometric identities, I’m just about ready to cut down the trees.)

This leg of the trip has been nice because it’s helped me step away from specific content and think about other aspects of my teaching. We spent a full day driving through the Redwoods, stoping for short hikes and incredible views. The next day was July 4th. We spent the morning at a town fair in Eureka, CA, which was a great small-town celebration. In the afternoon we drove to Redding, CA. The drive was beautiful, filled with mountain passes and winding drives through river valleys – a very different experience than where we were headed next.

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