The analysis
of construction problems in the dynamic geometry
Heechan Lew
hclew@knue.ac.kr
Mathematics Education
Korea National University of Education
South Korea
Abstract
In the 3rd century,
the great Greek mathematician Pappus systemized in his ?Collection§ the so
called ?analysis§ which Euclid also emphasized but, in his ?Elements§ not
appeared. The analysis, the oldest among mathematics heuristics in the history
of mathematics, assumes what is sought as if it were already done and inquire
what it is from which this results and again what is the antecedent cause of
the latter and so on, until by so retracing the steps coming up something already
known or belonging to the class of first principles. The synthesis as the reverse
of the analysis take as already done that which was last arrived at in the analysis
and arrives finally at the construction of what was sought by arranging in their
natural order as consequences what before were antecedents and successively
connecting them one with another. Greek thought the dialectic integration of
analysis and synthesis as a substance of mathematical thought. However, Euclid*s
Elements considered synthesis to reduce theorems from the foundation as a way
to guarantee the truth of mathematics. This lecture will show that the analysis
can provide the new direction for teaching Euclidean Geometry. Traditionally,
in teaching Euclid geometry the synthesis to arrange final results by mathematicians
is far more emphasized at the expense of the mathematical discovery process
by students. It makes normal students think mathematics a very difficult subject
because they cannot appreciate easily the deductive or axiomatic proof represented
by the synthesis. Furthermore, this lecture shows that it is because of the
lack of proper dynamic tools that the analysis known well by Greek mathematicians
like Plato and Euclid has not emphasized in schools since Greek era. Also this
lecture shows that dynamic geometry like GSP and Cabri developed since the late
of 1980s is an almost unique circumstance for the revival use of the analysis
which provides an alternative teaching method for normal students to develop
their proof abilities in Euclidean geometry.
