Teaching Trigonometry: concepts and historical
development.
There is no logical connection between the history of
mathematics and the way in which a curriculum might be organised to
support pupils' conceptual development today. History tells stories
about the past. Our classrooms are in the present with all the
different present contexts and contingencies, and our pupils are
children of our culture today. That being said, we can learn
certain things from history that may be used to provide ideas and
contexts that can be adapted for use in the modern classroom to
assist pupils' learning. But learning is not just about cognitive
activity, learning also involves the affective aspects.
Motivations and conventions
Pupils often ask questions like where did this come from? Or,
who invented algebra? Or, in the present example, why $360^\circ$
in a circle? These questions may require just a quick factual
answer, but they might lead to a longer explanation or even an
experiment or a demonstration. Very often, the origins of
mathematical ideas come from outside mathematics; they arise from
human problems that need solutions and thereby motivate the
creation of some mathematics that at the time is quite new. A great
deal of mathematics starts out as 'applied' mathematics, and the
'pure' mathematics of the classroom has too often been purged of
its humanity. The answers to pupils' questions can explain
conventions of naming or of notation, and offer the realisation
that mathematics evolved slowly over many hundreds, even thousands,
of years with contributions from a wide variety of people from many
different cultures.
Models, analogies and activities
The simple observational equipment that early astronomers,
star-gazers and shadow-measurers made and used are well within the
reach of any pupil who is interested in observing the skies or
checking the position of the Moon, stars or the shadow of the
Sun**. Living in an urban environment, it is not easy to see stars
near the horizon because of the glare from street lighting, and
with our weather not always easy to see the sky at night at all!
However, once we can recognise the Pole Star and realise that the
other stars revolve round the Pole, we share one of the fundamental
wonders of the universe with our ancestors. Exploring the language
we use to describe these events helps us to realise that even today
when we believe that Copernicus and Galileo showed us otherwise, we
still say 'the sun rises'.
The planets and the moon revolve around us, the stars revolve
around our solar system, the galaxies revolve in space. Rotation is
a deeply embedded and fundamental aspect of our existence, or so it
would seem.
Fundamental concepts
The summary below gives a brief list of the major events and
developments in the evolution of the ideas that led to trigonometry
as we find it in today's classroom. One thing we realise from even
a casual encounter with the history of mathematics is that it took
a long time for things to develop to the present, complex and
sophisticated state we have now. Do we make too many assumptions
about how quickly our pupils are expected to learn? The rotation of
the heavens and the recognition that the seasons corresponded to
special patterns of stars, led to the measurement of angle; the
shadow stick measured the passage of the day, and contributed to
the development of peg and cord geometry, the use of a right-angled
triangle and the comparison of measurement led to the concept of
ratio.
**[Make sure nobody looks directly at the sun without adequate
protection]
When in 1821 Charles Babbage invented the `Difference Engine' it
was intended to take over the work of making mathematical tables by
the techniques described in this article.