Did You Know Plants Make Their Own Food?

The other day I was out for a bicycle ride with my daughter when, out of the blue, she asked me if I knew that plants made their own food. I told her I did know that, and I asked her if she knew how they did it. She told me that they take air in through their leaves and use it to make “a kind of sugar stuff.”

My first response was to be blown away, because the daughter in question is four years old (she’d want me to say four-and-a-half, but who’s counting?). My second response was to ask her where she learned this. She told me, “It wasn’t from Magic Schoolbus,” which of course means she saw it on an episode of Magic Schoolbus. My third response was to fall into thought as we finished our ride. What I was thinking about what was this: the only difference between her explanation and one I might have given is that I would have used the word “photosynthesis,” and I would have mentioned that the plants take in carbon dioxide and give out oxygen.

I like to think of myself as a pretty well-educated and well-rounded person, yet when it comes to photosynthesis, that biochemical process so important that it is taught in-depth in both high school and college biology courses, I know as much about it as my four-year-old. Granted, the last time I took a biology class was more than 20 years ago, but since she hasn’t taken one at all I think she has me beat.

The obvious question is this: at this point in my life, how much should I know about photosynthesis? Put in a more subjective way, why is it that I don’t remember very much about this thing that was so important in class twenty years ago, yet I’m still able to be a productive member of society? Could we have spent less time on photosynthesis in high school biology to achieve the same outcome, and if so, what might we have done with that extra time?

Of course, I recognize that perhaps my teacher’s goal was not to for me to have life-long mastery of photosynthesis, but rather to develop certain life-long skills or habits of mind through my learning about photosynthesis. I think that there are topics we teach in every discipline that are used less for the content than they are as a tool for teaching a skill or mindset. But what I don’t think we do is make explicit for our students when the goal is the content and when the goal is a skill or mindset.

I’m going to try a couple of things this year to try to make this distinction more clear for my students. First, I’m planning to put some posters up on the walls that list the critical skills and habits of mind that students should be developing or employing, including some of the thinking moves described in Making Thinking Visible. Second, I’m planning to be up-front with my students about which concepts they need to remember long-term and which concepts they can forget about once we’re through using them, and I’ll explain why in both cases. My goal isn’t to have them remember what the graph of the sine function looks like twenty years from now, but if my approach helps them focus in on the things they need to remember for the next few years then it will have been a successful effort.


Days 34-38 – A Vaction from Our Vacation

We arrived in Steamboat Springs on Friday afternoon, happy to see some family and thrilled to have a place to settle for a few nights. This was the most vacation-like part of our entire trip, with lots of swimming, hiking, swimming some more, and sitting on the deck reading. For all the amazing things we’d seen and done over the last month, it still felt great to take a break from our travels. I had no plans to do anything math or science oriented during our stay, which was foolish. I should have known that I’d see something at some point that sparked my curiosity, and on the fourth day of our stay that’s exactly what happened.

By chance, we were given the opportunity to take a lesson on stand-up paddle boarding, something none of us had ever done but had seen plenty of when we went to the beach. I’m not a big fan of water-based activities, so I volunteered to stay on shore with our youngest child while my wife and the other two kids took the lesson. They had a great time, and our son loved it so much that he wanted to go out again the next day. Watching their lesson was intriguing enough, and it looked pretty straight forward, so I agreed to take him. 

He took to paddle boarding very quickly, and made it look easy:

I was able to do it, but like I said, I’m not as comfortable so I preferred to take a different approach:  

As I was paddling around the lake and just kind of playing around, it occurred to me that I was putting Newton’s Third Law to work. Newton’s Third Law states that for every action (force) exerted by one object on another, the second object exerts a force that is equal in size but opposite in direction. Thus, when I dipped my paddle in the water at the front of the board and rowed towards the back, the water responded by pushing my paddle forward, and since the paddle, me, and the board were all connected, the effect was to propel the board forward through the water.

We don’t usually think about these kinds of actions in these terms, and certainly not to this level of detail, when we’re doing them. I happened to find it quite comforting to think that way, and it gave me a framework for experimenting: what I rowed sideways? what if I didn’t paralllel to the board? etc.  It also gave me an idea. 

The description I gave above about how the forces are interacting is the kind of thing we might ask our physics students to write when solving a problem or doing a lab. The problem with this is that the situations are usually a little contrived in order to highlight the desired relationship. Students learn how to analyze these situations in the lab, and they learn how to analyze similar theoretical situations in problems, but they aren’t often able to appply the same analysis principles to something outside of the lab, like being on a paddle board. What if we put students in a situation, asked them to do something, and then asked them to analyze what they did? Rather than describing a person kicking a soccer ball, let the the students go kick a soccer ball. Rather than calculating the work required to push a box, let them push the box. By putting the students in the situation, they should have a better context for analyzing it, and ultimately for developing long-term understanding of the concepts. That way, if they’re ever stuck on a paddle board on a lake, they’ll be able to figure out what to do.

Day 33 – So This Is What It’s Like to be a Student

We woke up this morning in Vernal, UT, a cute little town at the eastern edge of the state that serves, among other things, as the gateway to the Utah side of Dinosaur National Monument. Dinosaur NM was our morning destination before heading on to Steamboat Springs, where we were going to visit family for a few days. 

I’ve seen plenty of dinosaur fossils in natural history museums over the years, but this was an entirely different experience. At Dinosaur National Monument, they have a large exposed fossil bed around which a building was built so that you can see what these fossils look like before they’re excavated. Here’s one example: 

As with most of my other photos, this doesn’t really do it justice, but hopefully gives at least some sense of what it’s like to see hundreds of fossils embedded in a relatively short expanse of rock. In addition to this fossil bed, there are also some individual fossils on display and a number of casts of fossils. One of the neatest things about this was that you couldn’t touch th casts because they were delicate, but you could walk right up to the rock and touch all the fossils you wanted. This reversal from what you might expect in a museum really seemed to make everyone there appreciate the experience even more.

The visitor’s center isn’t the only place where you could see fossils; we took a short hike and, with a little effort, found this: 

The trail leads right past this but there was no sign, so you had to be looking for fossils to find it, and we missed it at first. This was a good experience for us, too, because we realized that the way to find a fossil was to take our time and look for something out of the ordinary. The parallels to how we might develop a course are obvious, but it was nice to see yet another reminder that these habits of mind really do have long-term value.

We kept our visit to Dinosaur NM relatively short and were headed toward Steamboat well before noon. There was plenty to see that we didn’t bother with, including the entire Colorado side of the national monument, which doesn’t have any fossils but does have amazing canyons and spectacular hiking. Any other time, we would have headed there to continue exploring. But today, the anticipation of getting to see grandpop and nana, sleep in a house instead of a hotel, and feel settled for a few days dominated our thoughts.

It occurs to me this must be what my students feel like at the end of the year. It doesn’t matter how much they’re enjoying the class, there’s something more appealing to them just around the corner so they have trouble getting as engaged as they would be at other points in the year. I’ve always been aware of this intellectually, but I don’t think I’ve ever connected with this emotionally, certainly not as strongly as I did today.

We already do some things to accommodate the ebb and flow of student interest, of course. We might make a big, engaging project due right before a break, or save a lighter topic for the near the end of the year, for example. Having experienced this drop in engagement myself for the first time in many, many years, I’m left to wonder: how might we design courses to maximize student learning even when their engagement level is low?

Day 32 – The Leonardo

We had some more time to explore in Salt Lake City before continuing our drive east so we decided to hit a museum for the first time in a week. Salt Lake City has several good ones to choose from, but for me the choice was easy: The Leonardo.

The name alone captures the imagination of anyone who’s familiar with da Vinci’s work, with the fact that the name is not “The Leonardo Museum” but just “The Leonardo” adding to the intrigue. The museum’s form and function were inspired by da Vinci, but it’s not necessarily a museum about da Vinci. Areas in the museum included a hands-on building area; an exhibit on clean water and global water issues; an interactive area on mathematics; various puzzles and games; and a moving installation about poverty and homelessness among children. In other words:

When you first enter the museum, there’s a display that explains how to experience it (apologies for the bad angles on some of the photos):

The thing I was most struck by is what a phenomenal model this would be for a course, a curriculum, or even a school. What if we approached everything from the interdisciplinary, perhaps even transdisciplinary, approach that da Vinci took to his work, following our interests and ideas wherever they lead us? Just within the mathematics curriculum, the idea that we could use The Leonardo’s approach to building engaging, multifaceted courses that push our students (and ourselves) in new direction was enough to send me to Amazon looking for books about the man and his work. I think that some of these ideas blend nicely with other things I’ve seen and thought about over the last month, but they pull it together in a more inspiring and elegant way than I’ve seen before and so have given me something new to consider – which, come to think of it, is the reason The Leonardo was created to begin with.

Days 30 and 31: Thinking About Symmetries

We left Reno on Tuesday morning and drove…and drove…and drove some more. Seven hours later, we were in Salt Lake City. We drove through the Bonneville Salt Flats in western Utah, which was fascinating because it is the most bleak, desolate place I’ve ever driven through. People complain about driving through the desert of northeastern Arizona, or the endless, flat landscape of west Texas, but going through the Salt Flats gives a whole new perspective on the view out the car window. I wouldn’t say the drive was boring only because it was so completely different than anything I’d seen before.

It looks like this, of course, because virtually nothing can grow in the salty soil, so for 30 miles all you see is this barren white landscape with the hint of greenery in the distance. I feel like we could build an entire course, probably an interdisciplinary one, around this picture and the one I showed earlier from Lassen Volcanic National Park where there was a single, small tree growing above a sulfur vent, but I’m going to have to put that idea on the back burner for another time.

We spent Wednesday in Salt Lake City. In the morning we went to Temple Square, where the Mormon Temple and headquarters is located. While visitors are not allowed in the temple, there are a number of other buildings you can visit, including the Tabernacle. Every day, there is a 30-minute rehearsal/concert performed on the organ in the Tabernacle that is free and open to the public. The chance to hear an 11,623-pipe organ perform in one of the most acoustically sound buildings in the world. (They say that when the building is quiet, a pin dropped at the pulpit can be heard at the back of the room, 170 feet away.) 

Unfortunately for us, they were not performing in the Tablernacle when we were there because they were preparing for a funeral to be held later in the week. Instead, they held the concert in Assembly Hall on their still-impressive organ composed of 3,489 pipes. Both the space and the performance were beautiful.

As I was listening to the performance, I was struck by the symmetries visible in the organ’s construction. As I looked up, I realized it was more than just the organ that was symmetric:

By itself, this organ and the space that houses it are amazing to contemplate from a mathematical perspective, but as has been my habit over the last month, I can’t help but wonder not just what symmetries are present but why they’re present. I sense, if not a full interdisciplinary course, then at least a theme for part of the integrated math courses in the concept of symmetry. 

One of the things I’ve been thinking about this summer is how we can pull topics from our curriculum and shift our focus while still ensuring that our students are prepared for the SAT and ACT. It occurs to me now that symmetry may be a good topic because we can work to algebraic and geometric concepts through it while also extending our work in ways a more traditional curriculum wouldn’t allow for. It’s certainly something to explore….

Day 29: Insiders and Outsiders

We spent Monday in Reno, NV, where we did some hiking, went to the planetarium, and explored a little. Everything we did was on the recommendation of a friend of ours who had grown up in Reno, so as she said, we were effectively reliving her childhood. This was a very different experience than the ones we’d had in Portland and Seattle. A lot of our activities there were also based on recommendations of friends, but in that case it was friends who had recently moved to the area and had only recently discovered these things themselves. The differences in the recommendations between Portland and Reno were subtle, and we had great experiences in both cities, but if you looked closely there clearly were differences. 

It occurred to me later that this very much mirrors my experiences as a teacher. I teach math, but my degree is in engineering. In my experience, this makes me an “outsider.” I don’t have the same training, mindset, or background as many of my colleagues who have undergraduate, and in many cases graduate, degrees in mathematics, which makes them “insiders.” Put another way, insiders are people who grew up in the field while outsiders are people who moved into it. Because I’m an outsider, I’ve had to work a lot harder to master the subtleties of concepts, see my way through proofs, and communicate what it is to look at things like a mathematician.

As teachers, we make the same insider/outsider distinctions of our students. Within any given class, we have students for whom the ideas come easily and we have students who have to work hard to master the concepts.    

Day 28: Who Needs Carbon Dating?

On the morning of July 5th, we woke up in Redding, California. This is not by itself particulary interesting, but it makes me happy because (a) Megan Rapinoe, midfielder for the US Women’s National Team, is from Redding, and (b) later that day, Rapinoe and her teammates won the World Cup. I like coincidences like that.

We left Redding pretty quickly and drove an hour west to Lassen Volcanic National Park. I had never heard of Lassen before we started planning our trip, so I wasn’t surprised to learn that it’s on lists of least-busy national parks, easiest national parks to camp in, national parks with no lines, etc. If it’s not on your list of places to visit, I strongly recommend you add it. It was everything that all those lists named and more. 

Lassen is, as the name suggests, an area of past volcanic activity. While it has this in common with Crater Lake, it also has a significant difference: you can see a lot of active geothermal features like you’d see in Yellowstone, but you can do it without the long lines and large crowds that Yellowstone has. For example, there was Cold Boiling Lake:


At around 110 degrees the water isn’t exactly cold, but it’s also not boiling. The bubbles are from gases escaping from beneath the lake, a subtle reminder that this is still an active volcanic area. There was also this: 

The yellow in this picture is sulfur stains from active vents and hot springs. As you can see, the sulfur has made it difficult for plants to grow in the area, so I find the single, small tree growing in the middle of the hill fascinating.  Here’s an active sulfur spring:   

It’s difficult to tell from the picture, but the gray, dirty-looking area at the bottom of the picture is a boiling sulfur spring (I suspect it’s not as cold as the lake). While it’s facsinating to see features like this up close, it’s also a little disturbing to see steam rising farther in the distance, a somewhat less subtle reminder of just how recently this area had a volcanic erruption.

The Lassen volcano last errupted just over 100 years ago, in May, 1915. Because of this there is a lot of photographic evidence of how the area was changed immediately after the eruption and how it has recovered over the past century. Given the relatively recent eruption, the park was able to include this display along a nature trail:

 Notice the three small rocks, one directly in front of the display and the similarly-sized ones on either side of it. These rocks are each 100 years old, having formed in the last eruption. This stands in stark contrast to two other rocks that are just out of the frame but are shown in the display in the picture; each of those rocks is approximately 17,000 years old.
I find this absolutely fascinating, but I am no more able to explain why than I am able to explain why waking up in Megan Rapinoe’s hometown on the day she won the Wold Cup makes me happy. Beyond my inexplicable interest in this, though, a question occurred to me: how often do we create misconceptions in our students by the examples we choose in class? When we teach exponential growth and decay, we talk about carbon dating to establish the dates of fossils. In doing so, do we leave our students with the false impression that all geological things are really, really old? 

What other misconceptions might we help form by what we choose to focus on? I am not claiming that we should stop teaching carbon dating, or be sure to talk about young rocks like these (although I probably will this year since it’s on my mind). I am just wondering what more substantial misconceptions we might unintentionally foster. I’ll add this to the long list of things to keep in mind as I’m presenting material, asking questions, and helping to develop our integrated math curriculum this year.