In a recent Math Trail problem, Ledwith asks her students to think about the change in distance over time between two people as they travel in opposite directions on two parallel escalators. The expansive Galleria Mall in downtown Philadelphia provides the perfect location with escalators traveling side by side in opposite directions. Here the students can ride the escalators and put the problem into action.

When they arrive at the mall, students first discuss the problem among themselves to come up with a problem-solving plan. "It's a linear relationship!" one student claimed. "No. The people start far apart, get closer, then move apart so it can't be linear," observed another. Comments came thick and fast, "Quadratic?” "But the speed of the escalator remains constant so there will be a linear feel to part of the graph." “We can easily figure out the maximum distance the people are apart but what is the minimum distance between them?”

Once students agree how to approach the solution, possibly by using the Pythagorean Theorem, they go on to the next step, collecting data. "Have you measured the length yet?" “Are we sure the speeds of the escalators remain constant?” “How long did it take you to travel the length of the escalator?” One problem students don’t have is riding the escalators enough times to be sure they have all the data they need! Armed with adequate information, students then begin to formulate their solution. Besides making the necessary calculations, the students describe their process and findings in paragraph form. Because Math Trail activities can take more time than anticipated, it’s not always possible to completely finish discussing the solution at the location. However, if time permits, it’s optimal to solve the problem on site.

Ledwith thinks this is crucial. "Students must take the time to think about and discuss various approaches to the problem while in the environment.” This allows them to generate questions, gather and re-check data, and come up with their own methods for solving the problem right on the spot. Ledwith says, "It’s very exciting for kids. They develop an understanding of the value of thinking through the problem, rather than simply looking for a ‘right’ answer."

Then, the students take this problem-solving discussion back to the classroom, working for a couple of days to solve the problems. After this, Ledwith grades their solutions, and assigns each group one of the problems for which they post their solution on the school’s Math Trail project web site. Ledwith created the site because, "Our school wanted students to integrate technology into their work, so students write their own web pages for their solutions and link them to the main Math Trail web site. Posting on the web pushes kids to raise the quality of work they do. They see that their work is now part of a much bigger environment than just the classroom."

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