#### By. Jan Cohen, Founder, UrbanMathTrails

Math trails in nature are creative and authentic activities that stimulate student engagement and foster enthusiasm for math and the outdoors. They arouse interest by stimulating spontaneous observation among children of the connections between math and the elegant geometric shapes and patterns found in the physical world, including plants, animals, the landscape, and other features of the natural environment. Math trails encourage children to discriminate, reason, communicate, and solve problems while exploring, discovering, and celebrating the beauty of math and its presence all around us. And, since math trails focus on the process of formulating, analyzing, predicting, and explaining, children absorb and apply mathematical concepts naturally.

No tools are needed, other than perhaps a tape measure, so children are free to roam, observe, consider, reason, and internalize a new appreciation of the math embedded in their natural environment. Potential topics to explore are wide-ranging, including measurement, sorting and classifying, ratio and proportion, symmetry, geometry, probability, statistics, algebra, and many more, depending upon the age group, interest, season, or geography. Any park, garden, woodland, or other natural habitat is suitable.

UrbanMathTrails designs math trails for all sorts of institutions in a range of venues and conducts professional development workshops for educators and students who want to design their own. Any teacher, teaching team, or group of students can design a math trail. How? Find a location of natural interest and variety. Walk, observe, and as you consider the environment, think about the mathematical connections that can be made. Compare shapes, patterns, and sizes. Make estimates. Measure degrees, heights, areas, volumes, slopes, and distances using standard and non-standard measurements. Compare estimates to these measurements. Evaluate similarities and differences. Find combinations. Find ratios. Assess probabilities. Collect data and calculate mathematical averages and outliers. Devise methods or plans, make conjectures, test hypotheses.

Nature obeys rules, which students explore and express on a math trail, and brims with opportunities for fascinating mathematical investigations. The following are just a few examples to stimulate your thinking and, hopefully, inspire your planning of a math trail:

- A single frond of a fern closely resembles the whole fern, in miniature, wherein the same catalog of pattern repeats itself. This is called self-similarity. Look at some trees and bushes and explain their patterns of self-similarity.
- Most plants exhibit some type of symmetry. Examine leaves and flowers. What type of symmetry do they illustrate? Why do you think symmetry is so prominent in the natural world?
- Flowering plants also exhibit numerical patterns. Count the petals on several flowers. Does the number adhere to a number in the Fibonacci Sequence, 0, 1, 1, 2, 3, 5, 8? How many other plants can you find that conform to the Fibonacci Sequence?
- A treeâ€™s age can be estimated by measuring its trunk. Measure the circumference of a tree. Â Assuming the circumference of a tree grows at about 1/2 to 3/4 inches per year, how old is it?
- Tree rootsÂ normally grow just below ground; however, roots are frequently exposed through erosion. What volume of soil or mulch would be required to cover roots at a depth of two inches?
- One important way to identify trees and plants is understanding phyllotaxy, or the arrangement of leaves around the stem. There are three basic types of leaf arrangements: alternate, opposite, and whorled. Observe leaf growth arrangements on stems and stalks and keep a tally. What is the most common arrangement?
- Streams and ravines are part of the diversity and beauty of natural areas with foot bridges sometimes constructed to cross over water. Bridges have several geometric details that can be described, including parallel and perpendicular lines, angles, and measurement of their dimensions. In how many ways can you describe the footbridge mathematically?
- Find a deciduous tree. Notice that the tree will repeatedly branch into smaller and smaller branches from the ground upward. How many times does each stem split into a smaller one? (On a young tree, it may split four or five times. An older tree may split as many as 11 times.) Look at other trees in the area, evaluate their branches, and assess whether they are all of a similar age.
- See any spider webs? Spiders create near-perfect circular webs that have near-equal-distanced radial supports coming out of the middle. How many radial supports do you count in your spider web? Estimate the interior angle formed between the radials.
- Is it a sunny day? If so, you can estimate the height of a tree using its shadow. First, measure your shadow then measure your height. Measure the tree’s shadow. Set up a proportion: (your shadow length / your height) = (tree shadow length / x). What is the approximate height of the tree?
- See any dried mud patches? Describe the crack pattern. Do you see any polygons? Is the cracking pattern uniform? Do you see any right angles? Straight angles? Acute angles? Any concave shapes? What other geometric observations can you make?

In my experience, good math trails share four primary practices. First, the format varies, using written directions, tables, and/or pictures to stimulate different ways of thinking, appeal to different types of learners, and channel mathematical interest and understanding through the presentation. Second, the content varies, allowing students to interact with math in different ways, investigate various applications and possibilities (not just a single answer), provide the opportunity for different insights, and practice different skills. Third, math trails are tested in advance to ensure clarity and accuracy, identify needed prompts, and assess time and resource requirements. Finally, an answer sheet is developed for each trail so students can validate their thinking on-site and learn from their errors.

The following do’s and donâ€™ts may be secondary considerations, but are important in math trail design.

**Do’s**

- Organize teams strategically for collaborative problem solving.
- Ensure realistic walking time.
- Keep required tools to a minimum.
- Design problems not like those in class.
- Allow students to work at their own pace.
- Leave space in the trail guide for students to compute, keep tally, explain, and sketch.
- Consider taking a trail break for group discussion.
- Plan time afterward for sharing observations, strategies, and reflection.

**Don’ts**

- Avoid spending too much time in one spot.
- Avoid long distances between stops.
- Avoid crowded and unsafe locations.
- Avoid all single solution and closed-ended questions.
- Donâ€™t answer questions for or impose conclusions on students.

Math trails are a natural fit and essential element of outdoor education, providing opportunities for students to discover explanations of natureâ€™s patterns and underlying logic, shape, quantity, and arrangement. Regardless of age or skill level, math trails get children out of the classroom and into the great outdoors to help them develop mathematical and environmental literacy while having fun. Careful planning of a math trail enables children to engage with mathematical experiences in the real world and gain first-hand knowledge of how math can be used to interpret the world in which they live, while illuminating new mathematical concepts that complement and extend what they learn in textbooks and the classroom.

**Author Bio**

Jan Cohen is the founder of UrbanMathTrails, an education consulting firm that serves various institutions in the application of math in new contexts. Drawing upon an extensive background in math education, finance, and architecture, and a passion for the outdoors and the arts, Janâ€™s programs inspire children to discover math in the environment around them. Prior to establishing UrbanMathTrails, Jan was a middle school math teacher for 16 years, following senior positions on Wall Street for 20+ years.