A mathematics trail is a walk to discover mathematics. A math trail can be almost anywhere - a neighborhood, a business district or shopping mall, a park, a zoo, a library, even a government building. The math trail map or guide points to places where walkers formulate, discuss, and solve interesting mathematical problems. Anyone can walk a math trail alone, with the family, or with another group. Walkers cooperate along the trail as they talk about the problems. There's no competition or grading. At the end of the math trail they have the pleasure of having walked the trail and of having done some interesting mathematics.

Math trails fit very nicely into the ideas of popularization of mathematics and of informal mathematics education that have been increasingly recognized as valuable adjuncts to improving mathematics education in the schools. The NCTM Principles and Standards for School Mathematics (2000) and Curriculum Standards (1989) call for recognizing broad characteristics of mathematics as Communication, Connections, Reasoning, and Problem Solving. Math trails are a medium to experience mathematics in all of these dimensions. Math trails anticipated the NCTM Standards and Principles. They exemplify a worldwide collection of projects that aim to popularize mathematics through out-of-school activities. By providing opportunities for doing mathematics out of school, these projects extend time spent thinking about math and math problems. They also tend to connect back into school. Many trailblazers are school people, and teachers often take advantage of the existence of trails by including them in their instructional programs. All of this makes for a stronger mathematics education program in general (Blane, 1989).

The earliest math trails appeared in England and in Australia. In 1985, Dudley Blane and his colleagues blazed a trail (Blane and Clarke, 1985; Blane and Jaworski, 1989, 114–116) around the center of Melbourne as a holiday-week activity for families. The trail’s mathematical ideas included investigating a circular pattern of bricks in the pavement (to discover the invariance of pi), studying the timetables in a train station, looking at the reflection of a cathedral in a pond (to estimate its height), trying to estimate the speed of water rushing down a spillway, counting the number of windows in a wall of a skyscraper, and looking for patterns in the numbers of post office boxes.

Australian mathematics educators constructed many more trails based on a variety of themes and venues, including preparing for prospecting in a gold rush town, acting as an apprentice keeper in a local zoo, and working on the ship works and sailing boats in a historical nautical village. Each of the Australian trails had a brochure that contained thought provoking, mathematically oriented questions. In many cases, the questions had no single correct answers as such. The tens of thousands of Australians who walked these trails attested to their popularity. Many walkers returned for a second round accompanied by their families. Because of the strong demand for Blane’s Melbourne trail, the organizers maintained it for several months longer than the planned one week.

Like any good idea, the idea of a math trail has spread and people have adapted it. Carole Greenes of Boston University (Massachusetts) created a historical mathematics trail in Boston centered on the Common and the Public Garden. Unlike Blane’s Melbourne trail, walkers on Greenes’ trail followed a human guide who knew the historical and mathematical aspects of the trail and who could give hints and suggestions to walkers who got stuck on a task or idea. Kay Toliver, an award-winning New York City schoolteacher, leads her students on walks while guiding them to discover mathematics in their school neighborhood. Student walkers do not write their ideas and solutions on paper, but informally discuss their discoveries on the spot and then take the discussion back to the classroom. Florence Fasanelli, Fred Rickey, and Richard Torrington developed an elaborate math trail that takes advantage of The Mall in Washington. It provides an opportunity for the thousands of people who visit The Mall every year to include a mathematical dimension to their sightseeing. These successful math trails show that the idea is robust and malleable enough to meet the needs and imagination of trailblazers in many different situations.

Excerpted from the Consortium for Mathematics and its Applications (COMAP) publication Math Trails by Mary Margaret Shoaf, Henry Pollak and Joel Schneider (2004).