H
ow do you simultaneously accomplish problem solving, making connections, communicating, and applying skills in a meaningful context? Try a math trail. Like the more familiar science trail, a math trail consists of a sequence of stops along a pre-planned route on which students examine mathematics in the environment (Cross 1997, p. 38). Math trails can be used by students of any age, preschool through college level. They offer concrete learn-ing experiences for any of the mathematics con-cepts taught in the school curriculum.Browsing the Web, you will find examples of math trails for the National Gallery of Canada in Ottawa (Peterson) and center city Philadelphia (Ledwith, Wexler, and Carver), and a virtual math trail for the San Francisco Museum of Modern Art (NCTM). A math trail can be much more than a mathematics field trip to an exotic locale, however. It can function as an ongoing tool for exploring dif-ferent topics in mathematics, how they are inter-connected, and how mathematics connects to other disciplines.
Bailey (1994) and Cross (1997) consider the many benefits of math trails. Because it takes place outside the classroom, a math trail creates an atmosphere of adventure and exploration. This
shared sense of anticipation and discovery natu-rally leads to communication of mathematical ideas that are the focus of the trail. Students observe, measure, collect, and record data in order to manipulate and interpret it back in the class-room. While completing activities on the trail, chil-dren use mathematics concepts they learned in the classroom and discover the varied uses of mathe-matics in everyday life. Best of all, you can create a math trail in your community, no matter how small, rural, or, on the surface, unlikely. Further-more, the use of these trails will not require field trip fees!
An individual teacher can design a trail. Cross (1997) describes a forty-five-minute trail to extend the concept of shape and space. Bailey (1994) describes three trails on school grounds: one each to explore color, number, and shape. A trail is more versatile, however, when designed by a team of teachers from different grade levels. Rosenthal and Ampadu (1999) describe a trail in Boston used by both seventh graders and twelfth graders enrolled in two different mathematics courses. The most flexible math trail is one for a whole school devel-oped by teachers at each grade level. At each site, teachers at each level identify activities to com-plete both at the site and back in the classroom and write directions for various levels, resulting in an adaptable resource that the whole school can use.
Designing a Trail
The examples used in this article are for an ele-mentary school curriculum, although trails can be designed for any grade level. (Ledwith, Wexler,
8 Teaching Children Mathematics /August 2004
Kim Richardson is a professor at Juniata College in Huntingdon, Pennsylvania. She is interested in imbedding the teaching of mathematics in other subjects in the elementary school curriculum.
By Kim Margaret Richardson
Trails
for the
and Carver describe a trail for a high school and Bailey [1994] describes one for preschool.) The first task is to select a locale. It can be anywhere in the community: a playground, neighborhood, gro-cery store, mall, nearby college campus, cemetery, church, or in and around the school. For this exam-ple, we will consider the environs of an elementary school and a team of teachers working to develop the trail.
Begin by walking around the building, inside and out, and selecting sites rich in mathematics for the trail. As you look at a potential site, think about all the topics you teach. Look for patterns, shapes, and things to measure, count, or graph; look for calendars, notices of events, license plates, clocks, or other things that use numbers. Pivot 90 degrees, then 180 degrees. What more do you see?
Take a photograph of each site and record pre-liminary ideas for how you might use the site. Be
Photograph by Alexander T
. McBride; all rights reserved
Photograph by Alexander T
Figure 2
Directions to site 7
Walk out the front door of the school and turn right when you reach the sidewalk. Begin counting the smaller cement sidewalk rectangles. When you reach the 13th rectangle, turn right and face the build-ing. (Alternatively, you could count the upper long rectangular windows until you reach the 8th one, or you could count the white columns and stop when you are between the 9th and 10th.)
sure to look at eye level, at ground level, and above the children’s heads. After identifying at least six to seven sites, load the photographs in a computer, create a map, and sequence and number the sites on the map (see fig. 1).
Begin to develop the sites more fully and write directions for getting to them (see fig. 2). Also think of “On Site” activities and “Classroom Follow-Through” activities at all grade levels (see fig. 3). You must link the questions and activities on-site to those in the classroom to enable students to use the data they collected or the observations they made. Finally, think of how you can connect the activities to other parts of the curriculum (see fig. 4). When you have completed one site, pass it on to col-leagues so they can critique it and add ideas.
Good questions cause students to notice care-fully the many structural and design elements that make up our environment. Such questions include “How many patterns are used on this brick building and how many colors of bricks are used?” and “Are these the only patterns/colors that were possible?” Good questions pique curiosity, in both the teacher and students, and lead to meaningful research in
the classroom. Examples include “What other pat-terns are possible using these colors of bricks?” and “Why might this pattern—and these colors— have been chosen over the other possibilities?”
Extending the Trail
After compiling the activities for each site, analyze and organize them to get an idea of when and how you might use the trail during the school year. Analysis may include the following:
• Keying activities to standards
• Writing directions appropriate for different grade levels
• Keying activities to the mathematics text that your school uses
• Designing activity sheets for students to use on the trail
A resource book for a school-based math trail is the result. A team of four or more teachers needs about a week to develop a full K–5 trail guide with activity sheets keyed to standards.
Photograph by Alexander T
Figure 3
Sample activities for site 7
On-Site Activities
• How many bushes are in front of the wall? Are they evenly spaced? If we wanted to add a bush, how far from the end bush would we plant it?
• List all the places you find rectangles. (For young children, draw a rectangle.)
• How many windows (or sidewalk blocks) are there? (Estimate first.) Are they all the same size?
• How many children fit around (or inside) the edges of one sidewalk square? (Estimate first.) What are the dimensions of the cement squares?
• How many yellow blocks and gray blocks are in the small wall?
• How many patterns can we find in the front of the building? Which pattern uses the most blocks? Which uses the least blocks?
• Measure the red and black bricks (standard or nonstandard measurement). Do they have the same dimensions? What about the white bricks?
• Estimate how many red and black bricks are on the front of the building. How could you get an exact count? Write out your plan, then execute your plan. (Students can do the same for the white brick columns.)
• List all the places on the wall where you find right angles. Can you find any angles that are not right angles?
• Sketch the front of the building. Be sure your sketch has the same number of windows as the real building.
Classroom Follow-Through Activities
• Color a template of a section of the brick wall with red and black bricks. Count by twos, fives, and tens.
• Label drawings of the sidewalk blocks, bricks, and yellow/gray wall blocks with the dimensions you collected on the trail (transfer data collected to drawings).
• What area is covered by the cement sidewalk squares, the red/black bricks, and the yellow/gray blocks?
• Using data collected on the trail, what is the ratio of yellow to gray blocks?
• What is the percentage of yellow blocks?
• If yellow blocks cost x and gray blocks cost y, how much did the blocks for this wall cost? (Students can do the same with the red, black, and white bricks.)
• What other patterns can you make with oblong blocks in two colors? Practice using blocks. Represent your pattern using letters, numbers, and sounds. How would the school look with a different brick pattern?
• Using the sketch you did of the building, think about the following questions: If you wanted to add another classroom to the building, how many windows would you need? Approximately how many bricks would you need? What color bricks would you need? Draw another classroom on the end of the building in your sketch.
Using the Trail
Using the trail can be as versatile as the trail itself. When you are teaching a particular topic, select one or more sites on the trail to support that topic, then have an adult (you, an aide, or a parent) take a group of children on the trail to do those specific activities. Each child has a clipboard, pencil, and prepared activity sheet with directions to the sites and questions to answer, data to collect, and draw-ings to make. The students work in pairs or teams, both to make the excursion manageable and because the trail is designed to stimulate commu-nication of mathematical ideas.
You might use only one site on a given day and explore it in detail, or you might select one or two activities from a couple of sites to complete. You would not approach the type of trail described in this article as a “unit” that you do in sequence in one day or over a period of days. First, you can maximize the values of the trail by repeated expe-riences. Second, if you consider this activity an “add on” to an already crammed-full curriculum, you will never find the extra time in which to do it. Rather, you should reuse the trail at different times of the year as you study different topics. The trail also should be used in the same school at different grade levels. The math trail is a resource to help children hear mathematics language in a meaning-ful context, use mathematics in problem solving, and connect mathematics to their world throughout the year.
Getting Started
In 1998, the “Math by the Month” department of Teaching Children Mathematics published a list of activities for exploring mathematics in the school environment (Lewis and Lewis 1998). The Web site for the National Math Trails Project (www.nationalmathtrail.org) also has a range of activities that teachers and students in K–12 have submitted. The submissions, which are indexed by grade level and topic, vary from themed pro-jects carried out in the classroom to complete multi-site outdoor trails. If you are not familiar with math trails, you can begin working by your-self or with your students, creating a one-stop trail to practice.
As mathematics instruction has moved from abstract, algorithm-focused, worksheet-based instruction to the meaningful problem-solving, hands-on teaching we see in classrooms today, math trails are a natural extension of the classroom
Figure 4
Examples of “making connections” for site 7
Language Arts
Write in your journal why you think only right angles are on this wall of the building. Write in your journal why you think the architect used white columns on the front of the building. Write in your journal what materials were used to make your house.
Bring to class the section of the newspaper that advertises the costs of building supplies. Apply those costs to the appropriate “Class-room Follow-Through” activities.
Read books about construction, bricks, and so on, such as Mud by Wendy C. Lewison (1990); Children Just Like Me: A Unique Cele-bration of Children around the World by Kindersley, Barnabas, and Anabel (1995); Sky-scraper Book by James Giblin (1981); Changes, Changes by Pat Hutchins (1971); and Round Buildings, Square Buildings, and Buildings That Wiggle Like a Fish by Philip Isaacson (1988).
Science
Learn about how bricks are made, and how to make and use mortar. Invite a mason to visit the class and give a demonstration. Learn how cement is made. How is it similar to or differ-ent from mortar? Learn about fire codes and regulations that may exist for materials used in the construction of schools.
Art
Design various two-color block designs; draw houses or commercial buildings using a variety of brick patterns and geometric shapes. Learn about architecture and building design.
Social Studies
Learn where in the United States bricks are made and why they are made there. What buildings are most often built of brick, what buildings might not be built of brick (for exam-ple, barns and skyscrapers), and why do some houses have a brick front but vinyl or wood sides and back? Learn about what building materials are used most often for houses and buildings in different regions of the country. Look at pictures of other schools. What do they have in common? How are they different? Learn when our school was built and what else happened in this community and in this coun-try that same year. Find out if this was the first school in this area. If not, what happened to earlier schools?
Cross, Rod. “Developing Maths Trails.” Mathematics Teaching158 (March 1997): 38–39.
Ledwith, Lisa, Marcia Wexler, and Ruth Carver. “Ger-mantown Academy’s Philadelphia Math Trail.” www. germantownacademy.org/academics/us/math/math trail/spring2001/index01.htm.
Lewis, Thomas R., and Catherine H. Lewis. “Take It Outside!” Teaching Children Mathematics4 (April 1998): 462–63.
National Council of Teachers of Mathematics (NCTM). “Illumination on Connections.” illuminations.nctm. org/imath/across/connections/vmt/vmtmake.html. National Math Trails Project (K–12). www.national
mathtrail.org.
Peterson, Ivars. “Math Trails in Ottawa.” www.maa.org/ mathland/mathtrek_5_29_00.html.
Rosenthal, Matthew M., and Clement K. Ampadu. “Making Mathematics Real: The Boston Math Trail.”
Mathematics Teaching in the Middle School 5 (November 1999): 140–47.▲
Figure 1
Fourth-grade NAEP item
Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all your work.
In what ways are the figures above alike? List as many ways as you can. In what ways are the figures above different? List as many ways as you can.
Correction
In the April 2004 issue of TCM, an incorrect graphic was printed in figure 1 on page 395 as part of the article “Lessons Learned from Students about Assessment and Instruction.” The corrected figure is shown below. We apologize for this error.
