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I have watched the discussion on the use of GCs with some interest, from =
Scotland, whose education system is quite different from England's and =
where the attitude to using GCs is much more favourable.
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 =3D, and the equation =
form, was simply a "do something" signal, as in 2x3 =3D ? Indeed the =
syntax "computation =3D 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=3D11, where you cover up the term in x and =
ask yourself, "what added to 2 makes 11?". This creates a problem with =
3x+2=3D4x-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
David McLaren,
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I have watched the discussion on the use of GCs with some interest, = from=20 Scotland, whose education system is quite different from England's and = where the=20 attitude to using GCs is much more favourable.
There seems to me to a number of problems involved in dealing with = rapidly=20 developing technology and the impact on the curriculum.
The first lies in the need to be absolutely clear, when making = changes to the=20 curriculum, that nothing of real importance is lost. Let me illustrate. = When the=20 simple four function calculators first appeared and were accepted in = both=20 primary and secondary schools, computation with fractions nearly = vanished. Too=20 many pupils spent too much time vainly attempting to master procedures = which=20 could be left to the calculator. Another casualty was "long division" = the=20 procedure used to divide by, e.g., 3-digit numbers, and for the same = reasons.=20 Few mourned their passing. However, in using these techniques, pupils = were=20 required to do a fair amount of number juggling mentally. The results of = the=20 failure to compensate for this loss are now plain to see. Nobody wishes = to see a=20 return to the senseless repetitive practice of these routines, but at = last we=20 have noticed the problem and are attempting to deal with it - late in = the=20 day.
The second problem is associated with the need to provide a = curriculum=20 consonant with the way young people's understandings of mathematics = develop.=20 Again let me illustrate. When I started teaching in the early 60s, just = before=20 the "modern mathematics" movement swept all before it, pupils had to = cope with a=20 number of different images of, e.g., 'equation'. From their early days, = the=20 symbol =3D, and the equation form, was simply a "do something" signal, = as in 2x3 =3D=20 ? Indeed the syntax "computation =3D answer" was difficult to eradicate. = Next came=20 the notion that solving an equation was a process of working backwards, = with=20 little flags which flipped over to show the inverse operation, or maybe = function=20 machines. In some places this was referred to as the cover up method as = in=20 3x+2=3D11, where you cover up the term in x and ask yourself, "what = added to 2=20 makes 11?". This creates a problem with 3x+2=3D4x-1, since it is not = possible to=20 cover up both. A new image, the equation as balance, was introduced. = This image=20 is useless for dealing with inequations and quadratic equations and a = new image=20 was needed - equation as propositional function. The modern maths course = we used=20 took the view that all this was confusing for pupils and so we made = straight for=20 "equation as propositional function" and talked about solution sets and = the=20 like. It took a number of years for the profession to realise that this = was too=20 sophisticated an idea for 11 year olds and many teachers reverted to the = older=20 approach which required them to help pupils to realise that each of the = earlier=20 images incorporated its predecessor.
The third issue concerns the image of mathematics which we create in = the=20 minds of our pupils. Again let me illustrate. In Scotland, the = curriculum=20 includes the standard method of expressing acosx + bsinx in the form = kcos(x-p)=20 or ksin(x-p). Using a GC, one only needs to punch in acosx + bsinx and = then read=20 off the amplitude and phase angle. However, there are issues here about=20 mathematics itself. The graphical approach (to getting an answer) is = powerful=20 and pupils take to it very well. Yet how do we encourage them to ask = "does this=20 always work?" So we have questions to ask, and to answer, about the role = of=20 precision and of proof in mathematics. Should we project mathematics as = a=20 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=20 increasing number of objectives, learning outcomes, performance criteria = and the=20 like? I am not suggesting that these necessarily flow from the use of=20 technology, merely that we need to very clear in our minds about what=20 mathematics should mean to our pupils.
Finally, to the role of the GC itself. In conjunction with a view = screen, it=20 is a very powerful teaching tool. There is also a role for the GC as an = aid to=20 the exploration of mathematical ideas and to the formulation of = conjectures, to=20 be proved, or otherwise. The discussion of these roles is not usually = kept=20 distinct from considerations of the GC as a personal problem solving = tool.
As topic for possible discussion - should we continue to teach the=20 factorisation of polynomials?
David McLaren