Tuesday, December 3, 2019

A Math Trail to Every Classroom By Jan Cohen, Founder, UrbanMathTrails

A math trail through nature is a simple concept: use math to enhance student exploration, communication, understanding and appreciation of the spatial forms, patterns and quantitative underpinnings of the natural world. When students discover the connections between math and nature through observational learning, discovery and application, they embrace concepts and principles and retain knowledge and to a degree not achievable through traditional textbook-classroom-centered education.

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 12 to 34/ 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
https://www.urbanmathtrails.com for additional ideas and information.

North Carolina NCCAT participants

North Carolina NCCAT participants
At the Wayah Bald Fire Tower

Mary Jane

Mary Jane
On top of Silers Bald