There seems to me to a number of problems involved in dealing with rapidly
developing technology and the impact on the curriculum.
The first lies in the need to be absolutely clear, when making changes to
the curriculum, that nothing of real importance is lost. Let me illustrate.
When the simple four function calculators first appeared and were accepted
in both primary and secondary schools, computation with fractions nearly
vanished. Too many pupils spent too much time vainly attempting to master
procedures which could be left to the calculator. Another casualty was "long
division" the procedure used to divide by, e.g., 3-digit numbers, and for
the same reasons. Few mourned their passing. However, in using these
techniques, pupils were required to do a fair amount of number juggling
mentally. The results of the failure to compensate for this loss are now
plain to see. Nobody wishes to see a return to the senseless repetitive
practice of these routines, but at last we have noticed the problem and are
attempting to deal with it - late in the day.
The second problem is associated with the need to provide a curriculum
consonant with the way young people's understandings of mathematics develop.
Again let me illustrate. When I started teaching in the early 60s, just
before the "modern mathematics" movement swept all before it, pupils had to
cope with a number of different images of, e.g., 'equation'. From their
early days, the symbol =, and the equation form, was simply a "do
something" signal, as in 2x3 = ? Indeed the syntax "computation = answer"
was difficult to eradicate. Next came the notion that solving an equation
was a process of working backwards, with little flags which flipped over to
show the inverse operation, or maybe function machines. In some places this
was referred to as the cover up method as in 3x+2=11, where you cover up the
term in x and ask yourself, "what added to 2 makes 11?". This creates a
problem with 3x+2=4x-1, since it is not possible to cover up both. A new
image, the equation as balance, was introduced. This image is useless for
dealing with inequations and quadratic equations and a new image was
needed - equation as propositional function. The modern maths course we used
took the view that all this was confusing for pupils and so we made straight
for "equation as propositional function" and talked about solution sets and
the like. It took a number of years for the profession to realise that this
was too sophisticated an idea for 11 year olds and many teachers reverted to
the older approach which required them to help pupils to realise that each
of the earlier images incorporated its predecessor.
The third issue concerns the image of mathematics which we create in the
minds of our pupils. Again let me illustrate. In Scotland, the curriculum
includes the standard method of expressing acosx + bsinx in the form
kcos(x-p) or ksin(x-p). Using a GC, one only needs to punch in acosx + bsinx
and then read off the amplitude and phase angle. However, there are issues
here about mathematics itself. The graphical approach (to getting an answer)
is powerful and pupils take to it very well. Yet how do we encourage them to
ask "does this always work?" So we have questions to ask, and to answer,
about the role of precision and of proof in mathematics. Should we project
mathematics as a collection of techniques to get answers (mathematics as a
big bag of wee tricks)? Should we allow the progressive deconstruction of
the subject into an increasing number of objectives, learning outcomes,
performance criteria and the like? I am not suggesting that these
necessarily flow from the use of technology, merely that we need to very
clear in our minds about what mathematics should mean to our pupils.
Finally, to the role of the GC itself. In conjunction with a view screen, it
is a very powerful teaching tool. There is also a role for the GC as an aid
to the exploration of mathematical ideas and to the formulation of
conjectures, to be proved, or otherwise. The discussion of these roles is
not usually kept distinct from considerations of the GC as a personal
problem solving tool.
As topic for possible discussion - should we continue to teach the
factorisation of polynomials?
David McLaren
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