No tools are needed, other than perhaps a tape measure, so students
may roam freely, observe, consider, reason, and internalize a new appreciation
of the math embedded in their natural environment. A myriad of topics may be
explored: measurement, sorting and classifying, scale, symmetry, geometry, data
and statistics, probability, algebra, etc., depending upon age, interest,
season or geography. Any natural habitat
is suitable.

Find a location of natural interest and variety. Walk, observe and consider the environment:

·
compare shapes, patterns and sizes;

·
make estimates and measure degrees, heights, circumferences,
perimeters, areas, volumes, slopes and distances using standard and
non-standard measurements; compare estimates to actual measurements;

·
evaluate ratios and proportion;

·
collect data, analyze statistically and assess
probabilities; and,

·
devise scientific methods, make conjectures, test hypotheses.

Nature
obeys rules, which students explore and express on a math trail, and nature 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 along the AT.

One important step when learning how 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. As you walk along the trail, notice the leaf arrangements on plants and keep a tally. What is the most common arrangement?

Streams
and ravines are part of the diversity and beauty of nature. This rustic foot
bridge includes many geometric details. Find several sets of parallel lines.
Estimate the angles of the posts. Estimate the length, width and height of the
bridge. How many seasons do you think were observed to determine the height and
span requirements of this stream crossing?

This tree fell down
after a recent storm. Did you know you can estimate a tree’s age by measuring
its trunk? Find a felled tree and measure its circumference. Given the circumference of trees grow at
about

^{1}⁄_{2}to^{3}⁄_{4}/ inch per year, how old was this tree when it fell?
Foresters use
diameter at breast height, or DBH, as the standard for measuring trees. Measure
the circumference of a tree at breast height, 4.5 feet above the ground. Find
the diameter (divide circumference by pi) and multiply it by the tree’s growth
factor to determine its approximate age.
Growth factor varies by species from about 2 to 7, but we can use 4 as
an average of many species. How old is
the tree? Keep a tally today.

A single frond of a
fern resembles the whole fern, in miniature. This is called self-similarity.
How many different types of ferns can you find? What other examples of
self-similar structures in nature do you see along the trail?

A
fractal is a term to describe self-similarity. It is a geometric pattern that repeats
itself at different scales in a specimen. What other examples of fractal geometry do see
along the trail? Look at some trees and bushes and explain their fractal patterns.

Patterns prevail
throughout the plant kingdom. Flowering plants exhibit numerical patterns. Count
the petals on several flowers. Do the numbers adhere to the Fibonacci Sequence,
0, 1, 1, 2, 3, 5, 8…? What are the next two numbers in this sequence? As you
walk along the trail, how many flowers can you find that conform to this
sequence? Look for pinecones. They are a striking example of the Fibonacci
sequence. Their scales are arranged in two intertwined spirals. Count the
numbers in each direction. Does the pattern conform to the sequence?

Most
plants exhibit some type of symmetry. Examine the symmetrical patterns of plant
leaves along the trail. What type of symmetry do you observe in each leaf? Why
do you think symmetry is so prominent in the natural world? What other examples
of symmetry can you identify?

Daisies have
rotational symmetry. As you rotate them in a circle, they always look the same.
Since the typical number of petals on a daisy is 42, it has what’s called
42-fold rotational symmetry. Find daisies, dahlias or sunflowers and count the
petals. This will tell you the number of distinct orientations in which they look
the same. What other flowers do you see that have rotational symmetry?

Orchids
are glorious examples of flowers that nearly all have bilateral symmetry. What
is bilateral symmetry? Why is it also called mirror symmetry? What other
examples of bilateral symmetry can you find along the trail?

See
any spider webs? They 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 the 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:

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?

Math trails are a natural fit and essential element of
outdoor education, providing opportunities to students to discover explanations
of nature’s patterns and underlying logic, shape, quantity and arrangement. Regardless
of age or skill level, math trails get students out of the classroom and into
the great outdoors to help them develop mathematical and environmental literacy
while having fun. They enable children to engage with mathematical experiences in
the real world, gaining first-hand knowledge of how math can be used to interpret
the world in which we live.

*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, UrbanMathTrail programs inspire children to discover math in the environment around them, exciting their mathematical imagination and electrify their mathematical senses. See*