Ron Lancaster

Ron Lancaster

The math trail activity takes students from the classroom out into their environments, to discover the math that surrounds them. This Canadian teacher’s uncommon skill as a trail guide has enlightened students from San Francisco to Singapore.


What is your background?

I began teaching high school in Oakville, a town near Toronto, and eventually ended up teaching middle school and high school for seven years. It was just thrilling. I would spend the whole morning with grades 7-8 students and then in the afternoon I’d change gears and work with high school students. It was a wonderful mix; I think it would be fantastic for any high school teacher to make that kind of change.

What are you doing now?

Lately I’ve done a good deal of work in Singapore. I have designed math trails for students there in a large shopping mall called Suntec City. I have also created math camps, where I have gone into a school for a day and worked with around 120 kids to design a camp where they work at all kinds of puzzles and activities and smaller projects.

I have also worked with teachers in international schools to help them look at different ways of teaching mathematics through projects and activities, and using technology like graphics calculators and that sort of thing.

Tell me about your involvement with math trails.

I was doing some TV shows on mathematics for TV Ontario, which is similar to PBS in the US. Many were shot in the streets of Toronto. I was going to all these different sites and doing these math programs, and I kept looking into the camera and thinking, “This is really dumb — my students should be here with me, doing these things.” That got me thinking about the idea of actually taking students out of the classroom into an urban area, like downtown Toronto.

I have attended a math and technology conference for a number of years at Phillips Exeter Academy in New Hampshire. Vince Delisi, a mathematics teacher from Toronto, and I created some “math trails” for teachers at the conference. It was a natural name to use; it was a trail and there was some mathematics in it. We started taking teachers on these walks and I realized that I could do this with my own students.

In the mid 1990’s I started to take my students into Toronto for math walks. It was a wonderful experience to see them outside of the classroom doing mathematics in a different setting.

I actually began in the parking lot of the school.

What kinds of math problems can you find in a parking lot?

There are a number of math questions: finding the surface area of various cars and comparing them. Looking at tires, it turns out there’s a thing called an aspect ratio. If you were to look at a tire that says “D70-R14”, the 70 is the aspect ratio, and it is actually a measure of the amount of rubber on the side of the tire compared to the width of the tire. It is a wonderful activity to discuss this concept, have kids make measurements, and then look at the number of the tire and decode that number.

We also do some experiments where we put temperature probes into cars and looked at how the temperature changed throughout the day. Or students sit at the curb and watch how the tire valve moves as a car goes by, then trace the path that it took. The parking lot is a good place to start doing math trails.

What are some examples of different types of math that you might find in the community?

Picture a couple of escalators side by side. One of them is going down and the other one is going up. A standard trail question that I have used for a number of years is where one student gets on the down escalator and another gets on the up escalator. How does the distance between those two students change as they travel? If you picture that for a moment you can see that initially they’re getting closer and closer. At the halfway point they are the closest to each other, and then they get further apart.

The question is: what does that graph look like? Generally, my 7th or 8th grade students will give me a v-shaped graph, and that’s a good representation of it. But at high school level, it turns out that it’s not really an absolute function. It’s actually curved at the bottom. I have all sorts of other questions that I ask, like how would that graph change if the escalators were further apart? What would happen if the escalators were going at twice the speed?

What are you trying to communicate to students on a math trail?

A variety of things. One of them is an ability to look at the world mathematically, to be able to see mathematics everywhere, to appreciate mathematics and to do something about it. It’s not necessarily to show that mathematics is practical, but to show that you can look at the world through a mathematical perspective. Then you start to ask questions, and then answer the questions. I’d argue that this makes people better thinkers.

One of the other things I’m trying to do is to give kids a higher level of observation of their surroundings. I have parents come along with me on these trails — we’ll be walking by some skyscraper that has been in downtown Toronto for the past 20 years and they’ll say, “When was this building built?”

What effect do the trails have on your students?

One of the nicest things is that they start to look at the world mathematically. Over the years I’ve had them brings things in from a wide variety of sources; it might be a newspaper article or it might be a game they were playing or something that they heard on the radio.

How do you come up with your math trails?

I began by taking some rather lonely walks, where I tried to plan them out. Then I got other teachers involved, which I enjoyed much more. Sometimes I have students come out and plan the questions as well.

I have brought students to our math conference in Ontario and presented sessions on math trails with them. One year, my whole seventh grade class did a presentation with me and then the students took the teachers out for a math trail walk around Toronto. It was just amazing!

Can you adapt a math trail to any type of math that you are teaching?

I have quite a large bank of examples, questions and material that I’ve used in the past. There are some areas of math that are a little bit trickier to relate to trail questions, but for the most part I have been able to incorporate them into different grade levels.

Is it helpful to use technology with the math trail?

There are tremendous opportunities to use graphics calculators and machines like that to answer some of the math trail questions. If you use a graphics calculator, you can approach a problem from a variety of perspectives — maybe using a spreadsheet, maybe using a program, a graph, something like that.

I’ll give you just a couple quick examples. At the NCTM conference in Chicago this year, I co-presented a minicourse on Math Trails with two teachers, Carly Ziniuk from Toronto and Larry Ottman, from Haddon Heights, New Jersey. We designed a trail for teachers on the streets of Chicago. One of the really neat places on the trail was a structure for an entrance to the subway. The teachers went up an escalator inside this structure holding a motion probe pointed against the roof of the structure. As they went up, they collected data about the shape of the roof. I was fascinated watching this teachers grapple with this, because the data went all over the place. It wasn’t as if we were moving horizontally, at the same altitude. We were changing, and the roof was changing. Trying to sort all that out was really interesting. It wasn’t a simple thing to look at the data and figure out what was going on.

In San Francisco I had students in front of the Yerba Buena Center for the Arts, where there’s a great big sign shaped like a long wave. They were collecting data about the sign, looking at the shape of it. It turned out that the shape of the sign was a sine curve — a sine sign.

Are teachers using the Math Trail concept more and more?

Yes, and I guess the bottom line is that even if they never go do an actual math trail, even if they never end up developing one of these things, at least they’ve had a chance to think a little about taking mathematics out of the textbook and bringing in some other experiences.

If you looked at a lot of my trail questions, you’d find a lot of it is about making connections and noticing things. I think that is a very important skill in mathematics. You end up solving problems much quicker if you happen to notice something that every one else is missing.

Thank you!

You’re welcome.



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