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┬áthe mountain.┬áThis 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?


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?


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.


Day 25: Crater Lake

We got up on Thursday morning, packed up, and headed south to Crater Lake for the day. It had been a week since we had been to a National Park, so I was definitely ready for some stunning beauty. As I’ve said before, my photography skills do not begin to do the scenery justice, but suffice it to say I was not disappointed.

 As we toured the area, I was struck by how little I remember about volcanos. Even though I studied them in high school and college, all I really remember is that errurptions can spew lava, ash, or both. I don’t remember how they’re formed, how they behave, or anything else about them. But, I knew these things at one point, and I know how to find out more about them quickly if I want or need to. It was a little strange at the time to realize how much I didn’t know, and I wish I had addressed this in advance instead of on the fly in the visitors center, but it was still an amazing experience.

The parallels here are obvious. We talk a lot at school about focusing on what students need to learn/understand for the long term and what they can look up as needed. I clearly don’t need to know all this stuff about volcanos for my day-to-day life, so on one hand I think this falls into the “look up” category. On the other hand, I needed to have learned enough about volcanos at some point to know what I did not know. The question is, where do we draw the line between what we want our students to understand long term and what we just want them to know enough about to look up later?

Except I think that’s the wrong question to be asking. I’m not interested in my students’ long term understanding, I’m interested in their long term mindset. I want them to look at something 5, 10, 25 years from now and think about it logically, asking questions, looking for gaps in the information, and identifying possible assumptions. So the question really is, what do I want my students to know enough about to look up later, and what content and pedagogy is best-suited to developing this mindset? To me, that has to be the driving question for our integrated mathematics currriculum development.

So, what if I never had learned about volcanos at all? Would my experience at Crater Lake have been any different? I don’t think it would have been. I’m a curious person, one who likes to understand how things work and how they relate to other things I know about. I would have read the information at the visitors center and would now know exactly as much as I did before I got there. If it’s true for me with volcanos, why couldn’t it be true for my students with, well, any number of things?

Day 24: What If My Approach Is Wrong?

When we were planning our trip I was very excited about going to Bend, OR. It has the reputation as being an outdoor mecca, the kind of place where you can climb a mountain in the morning and kayak a river in the afternoon, so I wanted to make sure we got to enjoy as much of it as we could. Unfortunately, a heat wave had settled in over the Pacific Northwest a few days before we got there. It was 100 degrees on Wednesday, which was far warmer than anything we were used to and not at all conducive to doing everything I had hoped. Still, we did get ouside for a while, giving the kids their first-ever opportunity to wade in a river, which in turn gave a me a little more time to sit and think (and to do some teaching).

My thoughts turned again to the “What if?” questions. As I said in my last post, I’d been thinking about how to pose the right kind of “What if?” questions, the kind that challenge students to deepen their understanding as they try to answer them, when another question occurred to me: what if I’ve been thinking about this the wrong way?

The way I think about teaching math is based on the idea that unlike the other disciplines, which have a manifestation in the physical world, mathematics can exist purely in the mind. One of the marvelous things about math is that we see it manifested in the world around us, or at times motivated by the world around us, but mathematics would exist even if we never observed it. Because of this, we can know things in mathematics with a certainty that we can never have in other disciplines. Scientists, for example, never prove something true; the best they can do is repeatedly fail to prove something not true. In mathematics, however, logic allows us to prove something is true. A proper liberal arts education, then, must include an understanding of pure mathematics simply because pure mathematics exists.
The implication of this argument is that while it’s fine to teach applications of math, we must teach pure math. The corollary to this is that we should focus on pure math, including applications only after we’ve taught the theory. This belief is evident in most major textbooks and most high school classrooms today. A second corollary is that applications should only be the focus of classes for students who aren’t “strong enough” to be able to do pure mathematics, which is another belief that is evident by looking at most high school curricula.  So that leads me back to my question, and to related questions:

  • What if we didn’t focus on pure mathematics first?
  • What if “strong” high school math students were those who could apply the math well instead of those who could prove things?
  • What if students used ideas first and proved them later?
  • What if our primary goal was to graduate students who are mathematically fluent?
  • What if we replaced one term of our existing curriculum with a course in critical reasoning or (non-symbolic) logic?
  • What if our goal was for students to take AP Statistics instead of, or in addition to, AP Calculus?

Let me be clear: I don’t have answers to any of these questions, they’re just the questions that occurred to me when I started considering the theory-application dichotomy. What I like about these questions is that they’re along the same lines as the question, “What if water didn’t evaporate?” that got me started on this train of thought. I still have a lot of thinking to do, and a lot more of The Falconer to read. The next few weeks should be very interesting.

    Day 23: Thinking about “What If?”

    We left Portland this morning and headed south. After a brief stop at Silver Falls State Park so we could hike behin a waterfall, we headed on to Bend for a couple nights. In case you’ve ever wandered what the back side of a waterfall looks like, here’s your answer:   

    This was our first significant driving opportunity in a few days and so my first time to ponder some things. One of the books my department is reading this summer is The Falconer by Grant Lichtman. I’d never read it but suggested it based on the recommendation of those I respect and on how I thought it might speak to some of the things we’ll be thinking about in our curriculum rewrite. I’m three or four chapters into the book and am really enjoying it. Something I read recently has been bothering me, and I want to try to play out an idea here.

    In the fifth chapter, “Step 1: The Art of Questioning,” Lichtman describes a fictional classroom in which a teacher challenges his students to write the questions for an upcoming test on evaporation. The students pose a number of questions, all of which have factual answers that can be found in a textbook. The teacher points this out and challenges the students to ask questions that might not be so easy to answer, suggesting they consider questions that start with, “What if…?” A student comes up with the question, “What if water didn’t evaporate?” which inspires other students to ask similarly-challenging questions. In the story, the teacher points out that these questions are great because (a) there’s more than one way to answer them, (b) answering them requires a deep understanding of the topic, and (c) they naturally lead to other questions and lines of inquiry.

    This idea of asking “What if?” isn’t new to me, but I like the idea of using it to frame assessments. In fact, I think that using “What if?” is a good way to develop essential questions to frame a topic or investigation. The problem I’m having is how to pose the right kind of “What if?” questions in a math class.

    I can pose a “What if?” question in physics and we can develop experiments to test our hypothesized answers to the question. While it’s not my field of expertise, I would imagine that a “What if?” question in a history class can be played with by looking at other things that were happening and how they might change. With math, though, this doesn’t seem quite as easy. 

    At one end of the spectrum of questions is something like “What if we make the coefficients of these equations negative?” which is really just a factual question disguides as a “What if?” question. At the other end of the spectrum is something like, “What if a plane isn’t flat but is the surface a sphere?” which leads us to spherical geometry, a fascinating subject that’s incredibly complex. What I think we need in our integrated math curriculum is “What if?” questions that sit in between these two ends of the spectrum so that they’re interesting and challenging but can still be explored deeply.

    What I’m wondering is, can we develop “What if?” questions that hit the right level but are theoretical? I can come up with good “What if?” questions that involve applications of math; in fact, I’ve come across a couple great examples recently. A couple months ago a video was posted on Numberphile th at asked the question, “What if baseball was played in a hyperbolic geometry?” Part of the answer is that you’d need about the same number of infielders but 10^94 outfielders (you can click here to see the full video). More recently, mathematician Alex Bellos asked the question, “What if pool was played on an elliptical table instad of a rectangular one?” You can see the answer to this question here.

    These are great questions that ask students to understand something deeply, but they’re also questions that challenge us to apply the math in a physical context. Are there similar questions that exist entirely in the theoretical realm? It turns out that this question is actually less interesting than others that it implies, which I think would make Lichtman happy, at least. I’ll try to address them in my next post.

    Day 22: Tour of a LEED Platinum Building

    Portland was one of the major destinations on our trip because we have a few friends there, including one who is in the Army and is currently posted there. Portland doesn’t have a military base, so he works out of the Federal building downtown. When he mentioned in passing that his office was in one of the greenest buildings in the country, he’d barely finished his sentence before I asked about the possiblity of a tour. It turns out that when you run a LEED Platinum building you get a lot of tour requests, so the building manager was more than happy to oblige.

    My school is in the planning phase of a new math and science center, and the current goal is to have it certified LEED Gold or better. Platinum would be ideal, but the architects mentioned at one point that reaching that level is hard unless your building is much larger than ours will be, and after the tour I understand why. I’ll share below some of what I learned on our tour, but the biggest take-away is this: I was blown away by the creativity of the architects and engineers who put this building together. While the expertise they showed in designing this building is something that can only be gained through education and experience, the solutions they developed require a perspective that we can begin developing in even our youngest students.

    As for the things I learned about this building:

    1. One of easiest ways to make your building greener is to take advantage of your environment. In the Pacific Northwest that means collecting rainwater for re-use. They have a 165,000-gallon cistern in the basement that stores collected rainwater. The collected water is then used to flush the toilets, water the plants, and replenish the mechanical cooling system. When the plants are watered, the runoff is funneled to the cistern as well, meaning the water gets re-used more than once. All told, they collect and use almost 600,000 gallons of rainwater each year, significantly reducing both costs and the need for water from the city’s water system.
    2. The roof is covered with solar panels, but this isn’t the only way they take advanatge of the sun to reduce power needs. Just outside the windows of the building are “lightshelves,” which reflect sunlight up to 16 feet into the building to increase the amount of natural light available. This is paired with smart light sensors that will automatically reduce the brightness of lights based on the amount of sunlight coming in, further reducing electricity consumption.As great as the sun can be for light, it can be a real problem for climate control in the building. To reduce the direct sunlight that can heat a building up, a system of “fins and reeds” was attached to the outside of the building. From the outside, it looks like this:

      From the inside, the combination of the lightshelves and the fins and reeds looks like this:
      The GSA gives a lot of tours to school groups, so they had posters made up that explain how this works:
    3. One of my kids’ favorite features was the elevator. Similar to the regenerative braking feature of some cars, the elevators in the building generate electricity as they descend. This was interesting, but the much more fascinating feature was how the elevators are dispatched. When you walk up to the elevator bank, rather than “Up” and “Down” buttons, there’s a number pad. You enter your destination and then the panel tells you which elevator to take (the elevators are numbered). By using this “destination dispatch” method the number of trips is reduced, thereby reducing overall usage and inefficiency.When the building was originally built it had eight elevators. In the renovation that brought it to Platinum status they were able to reduce this to six elevators because of the more efficient usage. In addition, only two of the elevators go to the upper floors for the same reason. This reduced construction costs and allowed them to regain space for other purposes. Here’s a picture of the panel that replaces the call buttons:

      And while the picture doesn’t truly convey just how disconcerting it is, here’s what the inside of an elevator with no floor-number buttons looks like:

    4. Finally, something in which I have significant personal interest: the heating and cooling system. I teach primarily in a building where my classroom temperature is an obstacle to learning about 75% of the time, so I was very interested to learn about climate control when energy efficiency and minimal environmental impact are primary goals. In this building, all spaces are heated and cooled through a radiant system that is installed in the ceiling. I’m familiar with radiant heat in the floor, but I’ve never seen it in the ceiling and I’ve never seen radiant cooling at all. The ceiling tiles are not the typical acoustical drop-ceiling tiles you’re accustomed to seeing, but instead look like this:They are definitely more visually appealing than standard acoustical tiles, but their primary purpose is to radiate heat in the winteer and absorb it in the summer using pipes that run through the ceiling. From another of their educational posters, here’s how the system works:

      In principal it sounds great, but my friend who works in the building says that there are drawbacks. For example, temperature for the entire building is set from a central office and cannot be changed. Despite the design features of the building some spaces get warmer in the afternoon than others, and there’s nothing the occupants can do about that.

    Overall, I was very impressed with the building, and I have a completely new understanding of what what it takes to design and build an environmentally-friendly building. As I said earlier, I also have an even greater respect for those who design these facilities. Most importantly, I feel affirmed in my belief that we should be teaching our students to ask questions, view problems from multiple perspectives, and to consider the impact of an action on others, for these are the attributes that are leading to impressive problem solving.

    Day 21-ish: Looking for Originality

    Somewhere in the last few weeks I lost the ability to count, so the numbering on these trip-related blog posts is off. This probably wouldn’t bother a normal person, but it’s driving me nuts so I’m going to go ahead and correct it and move on. This post is about the 20th, 21st, and part of the 22nd days of our trip, which were a Saturday,  Sunday, and Monday, but I’m just going to call it Day 22. 

    Saturday was a city day for us. We went into downtown Portland and did all usual downtown things: went to the open-air Saturday Market; had lunch from food trucks; window shopped; and looked for postcards and souvenirs. We also went back to Powell’s Books, but I was unsuccessful in my quest for additional references we could use in our curriculum redesign.

    On Sunday, we went to the Childrens Museum, and on Monday we went to the Oregon Museum of Science and Industry. (Before we started our trip we became members at our local science museum. Through their reciprocal membership program we’ve gained free admission to every museum we’ve visited so far. By my count, that’s a 200% return on investment so far and the trip’s only half over). I liked both museums. They were good and well thought out, plus they had some things I haven’t seen before. The Childrens Museum included an outdoor area where the kids could dig in the dirt, build a lean-to, and play in a fountain. The Museum of Science and Industry had a chemistry lab where the kids could conduct experiments, like seeing how acidic lake water becomes after acid rain. Despite this, I was left wishing for more.

    I’ve been seeing a lot of repetition in the physics areas of these museums. Demonstrations of physics principles have usually involved electricity, magnetism, or optics. When they do include principles of mechanics, they’re all pretty much the same: simple machines, leverage, or center of gravity. I’ve seen very little that’s new or innovative in a couple of weeks. Fortunately, I have a solution for this.

    The physics portion of our precalculus/physics course follows the AP Physics 1 curriculum. Our students take the AP exam in early April, which gives us 2-3 weeks between the exam and the end of the year. We could use this time for a project in which the students have to create an exhibit for a science museum that demonstrates something related to the principles they’ve studied during the year. We’d get student engagement and authentic assessment, and we could even enhance the real-world aspect of this by having some faculty children test the exhibits and incorporate their feedback into the grading. I’ve collected enough examples on this trip that I can show them what’s already been done so that they’re pushed for truly original solutions, too. We have to figure out how to ensure they didn’t just Google science museum exhibits, but if we can overcome that then I think we’ll have a great end-of-year project.

    Days 19 and 20: The Highlights Tour

    We packed up on Thursday morning and left Seattle, headed west to Olypmic National Park. We stayed that night in Pacific Beach, a small town located, well, on a beach on the Pacific. It was a quick stay; the next day we left the Olympic Peninsula and returned to Portland. At some point I’m going to write about Olympic National Park and about climtes and ecosystems, but that wasn’t really the focus of these two days.

    As we were packing up in Seattle on Thursday morning, my wife was lamenting our departure. She was really enjoying our trip, she said, but she was feeling like it was the highlights tour and she really wished we had more time to spend in some of these places. I have to admit that I’d been feeling the same way. I’m thrilled that we’ve seen and done as much as we have, but I also wish we’d had more time to spend in Seattle and in places like Rainier and Glacier. 

    My focus on these two days was on her comment, this feeling, and the fact that it applies to far more than just our trip. I’ve been struggling a lot with this same decision as a teacher. Too often I feel like my classes are just highlights tours of the math when I’d really rather slow down and spend a lot of time on one thing. I can’t help but think that the students don’t always love the highlights tour that class can be, and it certainly does seem to send the message that content is more important than understanding.

    Two years ago I did slow down for the first time, and it was a fabulous experience. The students were engaged, they were able to direct the learning, and in the process they had a much deeper understanding of the material. I did it again last year, and in the coming year I’m going to introduce some colleagues to this approach. Slowing down has its costs, of course. Spending a third day in Seattle would have required us to cut something out, and so it would be with my classes. I had to cut some topics two years ago and to this day I wonder if I did the right thing, even as I know my students are benefiting from the decision. 

    It should be clear at this point that my priorities are helping my students develop understanding and ask questions; connect the world around them to the theory; and take ownership of their learning. All of this seems to imply that we can’t take a highlights tour; there are some things you can understand quickly, but most ideas require dwelling on to fully understand them. Unfortunately, there are times when we have to do the highlights, whether, it’s because that’s all we have time for or because of some external reason like the SATs. Deciding when to dive deeply and when to do the highlights on our trip was easy because we had about six weeks. We have 2-4 years to do integrated math, so these decisions are going to be much harder to make, and the stakes seem much higher.