Day 14: Old Books

On Saturday, we got our first real tourist exposure to Portland. We went to the arboretum to see some massive trees, to the rose garden because Portland is the Rose City, and to one of the highest points in the city to see this:

The mountain in the picture is Mount Saint Helens. That’s pretty cool.

In addition to being the Rose City, Portland is also known as beervana, so we went to a brewpub for lunch, after which was the real highlight of the day: Powell’s City of Books. Powell’s is a book store that takes up a full square block. The rooms are color-coded so you know where you are, and it takes a map to get around. The best part is that they carry both new and used books, which means you can find some great things if you have the time to look. I made a beeline for the math section, which was three full aisles of books. I had something in mind when I went there, and I actually got lucky and found it pretty quickly:


I don’t know how long the Schaum’s Outlines series has been around. This one was published in 1958. They still publish these books today, but the current editions pale in comparison to older editions. I have a Calculus edition that is 50 years old, and I’ve found some great, challenging problems that I’ve used over the years in my multivariable calculus class. I was hoping to find a similar resource that might prove useful in our upcoming work on integrated math, and this book looks like it will fit the bill. 

Consider, for example, this question, which is one of the last ones in the section on ellipses: “Prove: The locus of the midpoints of the chords of an ellipse drawn through one end of the major axis is an ellipse.” (pg. 351)  Compare that to this one, the last question on ellipses in the third edition of Schaum’s Outline of Precalculus, which was published in 2013 : “Use the definition of an ellipse PF1 + PF2 = 2a directly to find the equation of an ellipse with foci at (0,0) and (4,0) and major axis 2a = 6.” (pg. 1087).

The first question above is a challenging one, requiring an understanding of both the algebra and geoemtry of an ellipse as well as a more general familiarity with geometry. The second question is little more than a plug-and-chug question, one only a little more challenging to, “If AB + AC = D, and B=2, C=3, and D=6, find A.” It does require a geometric understanding of ellipses, but there is no request to prove any relationship, recall other material previously presented, or synthesize ideas anywhere. 

While I understand why textbooks have moved from questions like the former to ones like the latter over the last fifty years, I think we’re really lost something in the process. I was thrilled to find this book, along with a few others, and I can’t wait to start finding some integrated math inspiration in them.


Days 12 and 13: Making Thinking Visible

The twelfth day of our journey was pretty uneventful. We drove from Richland, WA to Portland, OR. Most of the drive was along the Columbia River, including the Columbia River Gorge, which led to some beautiful scenery but not a lot of activity. It felt great to get to Portland, though, and settle in with friends we hadn’t seen in over a year.

The next day, Friday, they had some obligations so we took a trip to the Oregon coast. The coast is amazing, with beautiful beaches surrounded by cliffs that afford amazing views: 

It was also pretty chilly, with temperatures in the mid-60s and a cool breeze blowing. Still, we toughed it out and set up a couple of chairs and some sand toys here:


For the first time since we started our trip I was able to sit down and open a book, so of course I chose something school-related. Our department is reading a couple of books this summer, including Making Thinking Visible by Ron Ritchhart, Mark Church, and Karin Morrison. While I don’t know that I’ll share my thoughts on every chapter of the book, the first one had some interesting points that I think are worth summarizing and synthesizing.

As one might expect, the first chapter of the book is dedicated to establising the basis for the arguments to follow. In this case, this means defining  what the authors mean by “thinking.” The authors quickly and convincingly establish that “understanding” is not a type of thinking, as Bloom and his students contended, but is in fact the goal of thinking. This allows them to establish, from their research, six kinds of thinking that lead to understanding:

  1. Observing closely and describing what’s there
  2. Building explanations and interpretations
  3. Reasoning with evidence
  4. Making connections
  5. Considering different viewpoints and perspectives
  6. Capturing the heart and forming conclusions (pg. 11)

While I’m not sold on the phrase “capturing the heart” I think this is a pretty reasonable list. The authors go on to add a couple of other kinds of thinking, and then a second list of six other types of thinking that are useful in problem solving and decision making. I won’t reproduce those lists here, but I will say that I did find them compelling as well.

The most interesting part of the chapter for me was in the discussion that followed the list presented above. A team of sixth grade teachers at the Interational School of Amsterdam was involved in the authors’ research. This team of teachers decided that if these were the thinking skills they wanted their students to develop, then they should make these skills explicit. These teachers had their students create porfolios that demonstrated the ways in which they had used each of these skills.

I love this idea, but I think I would take it even further. Why not post these skills on the wall of the classroom? As the students are working through a problem and question, I think that these skills should be visible as reminders of the things they might try. I can’t think of a better way to make these skills visible than to literally make them visible, to discuss them regularly, and to reinforce their importance when working with individuals or groups. Understanding is my goal for my students, and if this is what it takes to get them there then I think I should be as explicit about it as I can.

I’ll be working my through this book slowly over the next few weeks. I’m interested to see how my thinking on this evolves as I continue reading, and if anything really grabs me then I’ll share it here. 

Day 11: Children’s Toys Are Surprisingly Complicated

One of the things we’re trying to do in our travels is see science museums and children’s museums. On our last big road trip a few years ago, we stumbled upon a small math and science museum that’s in an elementary school in Grand Junction, CO. Despite the size and the fact that it was clearly run mostly by volunteers and on a shoestring budget, it was one of the best science museums I’ve ever seen. Ever since then, I’ve been interested in visiting science museums of all sizes. They all do a great job of presenting complex ideas in a simple, accessible way, which is appealing to me as a teacher, but I’ve also seen some unique ways to present the ideas, which helps expand my own thinking about the idea. (I think my kids don’t like going to these museums with me anymore because I often spend much more time at each exhibit than they do.)

On Wednesday morning, we had planned to visit the Mobius Science Center and Children’s Museum in Spokane, WA. The science center is quite small and we arrived right after a summer camp got there, so we decided to head to the children’s museum instead. It was small, and a little light on exhibits for the price, but I still managed to find a couple of interesting things. One of them was this: 

You can probably imagine what happens, but you can click here to see this in action. I think we could build a significant part of a physics class on this simple toy. Energy is the first thing that occurred to me here. There’s clearly potential and kinetic energy involved, and if we idealize the motion (no slipping) then we can talk about both rotational and translational kinetic energy. In addition, energy is not a vector quantity, which means that it has no direction, so the direction changes here would certainly challenge students to understand the difference between vectors and scalars. We could also talk about forces, inclined planes, and kinematics (position, velocity, and acceleration) in an interesting way. I love finding simple toys like this and discovering how they can engage 16-year-olds (and for that matter 41-year-olds) as well as they engage 6-year-olds.

Day 10: The Unverifiable Claim

We left Glacier National Park, headed for Spokane, WA. With just a few hours of driving and no specific plans for the day, we decided to stop in Coeur d’Alene, ID. There were two things that made this city sound interesting: a huge, child-designed playground in a park downtown, and this:

 To my wife, this sounded like a neat thing to see and do with the kids. To me, this sounded like math. 

How can they assert that this is the “world’s longest floating boardwalk?” Wouldn’t that require a comprehensive survey of cities, towns, and even villages around the globe? How can they possibly know that this is truly the longest one in the world? Don’t get me wrong, it was a neat boardwalk and the lake it’s on was beautiful. It’s the claim that bothers me. More importantly, it’s claims like this that I want to bother my students as well. 

This is related to the earlier claim that Montana is the fifth windiest state in the country. That claim is justifiable, but it’s possible that the math and science used to justify the claim are dubious. I’d like thm to be able to explore the validity of claims like that and decide for themselves whether they’re backed up by solid evidence. Here, the claim itself isn’t justifiable. That doesn’t mean it isn’t correct, it just means there’s no way to confirm with certainty it’s correct. I’d like my students to identify claims like this, too. In the past, I’ve always thought about claims like this as being found in various media sources, and particularly in opinion pieces. It was a good reminder to see this and know that these claims are easy to find in the world around us, not just in the news.

Day 9: A Detour, and a Meta-Detour

It seems like every article I read in the last six months related to our trip mentioned Going-to-the-Sun Road in Glacier National Park. If the article was discussing top attractions in national parks, the road was mentioned. If the article was about top scenic drives, the road was mentioned. If the article was about not-to-miss destinations, a review of a new RV, or even likely GOP presidential contenders, the road was mentioned. With this much pressure on us, we felt like we had to include Going-to-the-Sun Road on our trip. 

Going-to-the-Sun Road is a 50-mile stretch of two-lane road that cuts across Glacier National Park. It features incredible mountain views, waterfalls, and the chance to see multiple climate zones as you climb several thousand feet of elevation to Logan Pass and the Continental Divide before descending down the other side. There are plenty of hikes, visitors centers, boat and bus tours, and other things to do in the park, but Going-to-the-Sun Road is the quintessential experience, the thing to do if you only have time for one thing to do. So of course the road was closed when we got there.

The road is impassable through the winter and typically opens in early June. I knew we were cutting it a little close when we made our plans, but it looked like everything was going to be fine so I was pretty surprised to pull up to the park entrance in St. Mary and be told that the road was closed from there to Logan Pass. Weather wasn’t the culprit in this case (snowfall totals were below average this year), but road construction was running a little behind schedule and had delayed the opening of the road for about a week. Disappointed and a little thrown off, we went to the visitors center, where we talked with one of the rangers about alternatives.

Rather than heading west across the park on Going-to-the-Sun Road as we had planned, we headed south and went to different part of the park. In the Two Medicine area of the park, we took a short hike to a waterfall and spent some time looking at, and skipping stones on, a glacial lake. We then continued south, going around the bottom end of the park and seeing a couple of interesting spots along the way before ending up at the other side of the park at its west entrance. From there, we were able to drive Going-to-the-Sun Road up to Logan Pass, ultimately driving 30 of the road’s 50 miles before heading back down and out of the park. All in all, it ended up being a pretty good day.

A few days after our visit to Glacier National Park, it occurred to me that our detour was a good metaphor for thinking about what we teach and why we teach it. The metaphorical detour I’m thinking about isn’t the one that happens when a class doesn’t go as planned or current events take you off track for a day. I’m thinking about a much bigger detour and our response to it.

Suppose that you, the teacher, arrive at school for opening faculty meetings, and your department chair says to you, “I’m not sure if you knew this, but this year you won’t be able to teach _____.” The blank represents a specific set of content that you have taught for years, are comfortable with, and believe you know the value of for our class. For the English teacher, perhaps it’s The Great Gatsby. A history teacher might lose ancient Greece, a biology teacher might lose the nervous system, or a math teacher might lose exponential functions. In each case, the teacher is still expected to help students develop the knowledge, skills, and habits of mind necessary to move on the the next class, the next level, etc. 

If you were handed this unexpected detour, what would you do? Is there any content area so signigicant that it’s loss would mean the course should just be canceled? Assuming that’s not the case, what skills and habits of mind are being lost with this detour, and what might you do to overcome it? With this detour forced upon you, what might you encounter that you otherwise would have missed?

I’ve had some time to think about this, and I know how I would adapt if certain things were removed, but that’s not really the point. I’ll look to my colleagues this fall to challenge me by suggesting things I not be allowed to teach. Even if it’s only an academic exercise it will still be an enlightening one. With a little luck, I’ll discover something so good that I decide to go ahead an remove the topic from my class for the year. I’d also welcome suggestions in the comments, and I’d love to hear responses if anyone else tries this.

Day 8: Which State is the Windiest?

We were driving from Billings to St. Mary, MT when we passed a sign that said “For Wind Information Tune to 1610 AM.” I excitedly reached for the radio, sure I was going to be getting a report on wind conditions around the state, only to be disappointed when I discovered that it was actually just a promotional station for the local energy utility and wind energy. I only listened for a minute, but it was long enough to here them comment that Montana is the fifth windiest state in the country. My disappointment was wiped away by a new sense of excitement and curiosity – was this really the fifth windiest state, and how do we know?

My wife/research assistant was up to the challenge and quickly found an article that listed Montana as fifth windiest. We then detoured into related trivia (“Guess which state is windiest/least windy/biggest/etc.”), but this fifth-place ranking kept bothering me. How was this ranking determined? Was it based on an average daily or hourly measurement of windspeed or on some continuous measurement? Where was the wind that was used to determine the rankings measured? How was the data collected? How were anomomlies like tornadoes and hurricanes factored in? If sampling was used, what was the confidence level, and how do you ensure a fair comparison when sampling from states of different sizes? 

 At this point, two things should be clear. First, Montana is a really big state, and driving across it gives one a lot of time to think. (As an aside, we later heard people refer to Montana as the “Texas of the north” because of both the landscape and the size, and it’s a pretty accurate description). Second, there’s a rich mathematical topic here, one that can be mined in many ways. I’ve already alluded to the various statistical issues involved, and the discrete/continuous question is a great one for both precalculus and calculus students to ponder. Beyond that, the very idea of ranking something can challenge students to apply even pretty elementary math (means, for example) in some pretty complex ways. 

Suppose that I built part of a course around the question, “What is the best lunch the dining hall can serve to athletes on game days?” I’m pretty sure this meets Grant Wiggins’ definition of an Essential Question, and even if it falls short in some way it’s still a rich question that can be explored in multiple ways and is relevant to students’ lives. To answer the question, students would need to learn about and consider a variety of factors, including physiology, nutrition, sports psychology, and individual preferences. They would likely need to build some sort of metric for assessing and weighing the factors they identified, and they’d need to collect data and apply their metrics to it to come up with a ranking. 

As I said, the math itself wouldn’t be terribly complicated in this example, but it would be a great project that pushed the students to more deeply understand the question and how to tackle it. I’m intrigued by the possiblity of finding questions like this that necessitate the exploration of various mathematical topics at all levels of the curriculum. What question might we ask that would let us get at logarithmic functions in a more interesting and engaging way?

Day 7: Discovering Geometry at Devils Tower

This is Devil’s Tower National Monument in northeastern Wyoming: 

It is, well, a tower of rock that rises up from the land and looks nothing like the surrounding mountains. It has all the makings of a great interdisciplinary topic:

  1. It’s awe-inspiring. 
  2. Scientists don’t know why it exists but have three competing theories. 
  3. There are various Native American stories that explain its existence, and it still holds a sacred place in the local cultures today. 
  4. There is a voluntary rock climbing ban during the month of June because of its cultural significance, but we saw four separate climbers the day we visited. 
  5. The name itself is the result of a mistranslation made hundreds of years ago (which is also true of the Badlands). 

Also, there’s some good math to be found here. We encountered this sign right after we drove in: 

Why tetrahedrons? I don’t know, but it’s a question worth investigating. In addition, there was this in the visitor center:
Here’s a closer view of the tower so you can see the columns:


I know that the reason they form in these shapes has to do with chemical bonding angles as the rock cools, but I took Engineering Geology twenty years ago so I don’t remember all the details. So there we have it: two ways that math appeared in unexpected places on my trip today, plus the potential for an interdisciplinary topic that includes a non-trivial way to tie math to other fields (which is not always the easiest thing to do).