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I submitted the following article on orienteering for mathematics learning
to a Pennsylvania homeschool newsletter and the internet homeschool mail
list. It relates to Logo so I'm posting it here. In addition to the mathematics
mentioned in the article, there is a definite fractal dimension (ouch)
to orienteering. Forging a path through boulder fields and thickets is
anything but a linear activity. I haven't formalized it yet, but I have
found that I must calibrate my step counting to the terrain at the start of
a course in order to accurately account for the fractal nature of such paths.

Orienteering = Mathematics

By Dale E. Parson

I was glad to read Tansy Wagner's recent article and story on orienteering
in the Spring, 1992 issue of Pennsylvania Homeschoolers (Issue #39). Our
family uses orienteering--use of map and compass to navigate unfamiliar
rural and woodland terrain--both for recreation and as an introduction
to mathematics for our two kids, Sierra (6) and Jeremy (3). I am writing
to mention some of the mathematical aspects of the sport.

Having taught map and compass use in Boys Scouts twenty-some years ago,
my interest in orienteering for homeschooling was renewed after learning
about turtle geometry, a body-oriented form of geometry. Unlike Cartesian
geometry with its global, disembodied frame of reference, the `turtle' of
turtle geometry is an analog of a child's body in motion. In the Logo
programming language, a child (or adult) draws pictures by writing programs
that move a graphical turtle forward or backward some number of steps, or turn
it left or right some number of degrees. Children often start Logo programming
by walking out the effects of the moves and turns that they will program for
the turtle. Changes in location and direction are always in relation to the
child's or turtle's departing position and heading, not in relation to global
coordinates.

The problem with Logo for young children is that it is too textual. To write
Logo programs they must type special keywords into a keyboard, requiring
spelling, vocabulary and typing skills. While mastering these skills is
important at some age, young children have far less aptitude for them than for
some ways of interacting with a computer, such as drawing colored shapes
using a `mouse' or other graphical input device. Our kids have spent many hours
drawing computer pictures, becoming familiar with basic geometric shapes in the
process. But typing Logo commands is not an obvious way to make pictures
for a young kid.

Enter orienteering. As with turtle graphics, the basic actions in orienteering
are changing heading and going forward some number of steps. Heading actually
does relate to a global coordinate--magnetic north--but north can be found
easily in relation to any point, is similar to the top of a piece of paper
(a map) or a computer screen, and is simpler than Cartesian coordinates.
Orienteers do not merely practice by walking out their program: making the
moves and turns is the game itself. It is fundamentally mathematical, and not
simply an application of mathematics. In consulting a map, making a decision,
setting a heading and walking some distance, an orienteer is actually doing
turtle geometry. Families can take walks in the woods and make entry into
mathematical concepts without even trying. The formal introduction of ideas
can be done a little at a time, but the emphasis is on exploration, detective
map work, and fun.

What are some of the mathematical concepts? The most fundamental is that of
a mapping, a projection of entities and relations in one domain onto entities
and relations, that in some way are equivalent, in another domain. When a kid
recognizes from a path marked on a map that at the coming road junction,
seen with their eyes, they must take a left turn to stay with their course,
they have performed mathematical mapping. Orienteering actually requires
several varieties of mapping. Geometric mapping relates identical angles,
rotated directions, and scaled distances between the map and physical world.
Symbolic mapping relates graphical symbols such as the one for a "rootstock"
(the symbol is a little brown X) to physical objects (tree roots sticking
out of the ground). Colors, such as blue for water, help symbolic mapping.
Kids take to both readily.

Mapping and symbol manipulation are general mathematical activities related
to all branches of math. Orienteering includes a lot of more obvious
connections to geometry. Use of angles--angle between north and next direction,
angle between top of map and north to name two--occurs at most map
consultations. Use of distance occurs continuously during motion. Use of
scaling, which in turn uses addition, multiplication, and division, occurs
in relating map distances to physical distances. Step counting requires
calibrating your feet by counting your steps over some known distance, and
then using this calibration to determine when you have covered some desired
distance in unknown terrain. Step counting also uses scaling. And subtraction
becomes important when you want to start figuring how far you have to go.

Despite scientific training, I was never much of a fan of the metric system
for everyday activities until I started orienteering. It is SO easy to take
5 centimeters (or .05 meter) as measured on a map with a metric ruler,
multiply it using the 1:10000 scale printed on the map to get 500 meters
(by sliding the decimal point right 4 places for the four zeroes in 10000),
then converting 500 meters to .5 kilometer (since 1000 meters = a kilometer).
Once you have walked a measured kilometer a few times you can do map-to-world
scaling by sliding a decimal point around in your head and you will know what
to expect. Try doing that with the 1760 yards in a mile.

The competitive aspect of orienteering brings time into the picture, and with
it more physics than we want to talk about here. Velocity and acceleration
are two big ideas.

A reminder, though: we want to keep it fun. Manuals on teaching it to kids
recommend avoiding heavy compass use, with concentration on relating graphics
on the map to things and directions in the world. Most beginners' courses stay
on trails and do not require compass work. There are even short "string"
courses that allow a toddler to toddle around a set of checkpoints guided
by a string. We find that the kids have fun playing with the compass, but
we do no push it at these ages.

In addition to the Franks mentioned in Tansy Wagner's article, more information
can be obtained from:

U.S Orienteering Federation
P.O. Box 1444
Forest Park, GA 30051

There is an internet orienteering mail list at
	orienteering@graphics.cornell.edu

mail subscription requests to
	orienteering-request@graphics.cornell.edu


Dale Parson, Bell Labs, dale@mhcnet.att.com

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|    "These words are too solid, they don't move fast enough           |
|     to catch the blur in the brain that flies by, and is gone..."    |
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|     Suzanne Vega                                                     |
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