[Maths-Education] Re: ICT in mathematics

[Alexandre Borovik]

Dear John,

On 06/03/2011 03:41, John Mason wrote:

 > I can use interactive graphing to enable students to experience the dual
 > nature of a graph of a function: as an object and as the set of points
 > [x, f(x)] where x varies over the domain. Combining two functions using
 > 'cobweb' type diagrams, students can get a sense of what the composite
 > of two functions looks like. [....]

 > Surely this
 > applies to many uses of ICT for conceptual enrichment?

I wholeheartedly agree with you -- of course it does; I am not a 
Luddite; I started to teach entirely computer-based, computer-assessed 
courses at least 15 years ago; a detailed explanation of my views on ICT 
in (university level) mathematics teaching can be found  in 
http://www.maths.manchester.ac.uk/~avb/pdf/Mathematics_and_IT.pdf

But I would like to play the Devil's advocate role one step further and 
suggest that the use of ICT in school mathematics  may help students to 
pass school exams -- and damage their chances of understanding 
university level mathematics.

The "dual" nature of function mentioned by you is a wonderful example of 
a multi-layer structure of mathematical concepts. There is a more 
abstract layer of duality in the concept of function, the point-function 
duality, when a point x is seen as a functional evaluated at the 
function f. This duality is prominent in functional analysis or 
algebraic geometry, and, at the level of applications, in computer 
programming or, say, mathematical economics (in the latter it takes the 
form of the price/output duality of linear optimisation---among many 
other manifestations). Does experience of interactive graphing helps, at 
later stages of mathematical education, to understand this duality? Or 
hinders it, by tying the concept of function to concrete visual images?

I dare to conjecture that this is a general principle of pedagogy of 
mathematics: a next level of abstraction is better understood by 
students if the previous one has been learned by them by direct mental 
manipulations not mediated by use of technology.

For example, my favorite approach to graphs and composites of functions 
is to invite students to play with the absolute value of the absolute 
value function y = |x|, using just a pencil and squared paper. When you take

f(x) = ||x|-1|,

a graph of the composite f(f(x)) is easy to draw -- and, moreover, 
sketching of a graph of an iterated function like

f(f(f(...f(x)...)))

where the symbol f is repeated, say, 100 times, becomes an accessible 
exercise. The sketch of 100th iteration cannot be actually drawn -- only 
crudely sketched; but it is not difficult, however, to explain 
*verbally* how the graph looks like (and hand-waving is quite useful 
here, too). We have the all-important feeding of a visual image back 
into the realm of "verbal" thinking and locomotor intuition.

I had seen Mathematica programme (for Apple computers) first time in 
1990 or 1991, under curious circumstances: I taught a standard calculus 
course at University of California, and my teaching assistant, a very 
clever graduate student, was unable to change order of integration in a 
double integral without graphing first, with the help of Mathematica, 
the area of integration -- this was why she introduced me to Mathematica.

And this allows me to produce an example illustrating my conjecture:

If you wish to teach students to evaluate double integrals as an end in 
itself, then the use of interactive graphing is of course OK. (But why 
bother if computers calculate integrals better than humans?)

But if you wish to use change of order of integration as a propaedeutics 
for introduction of Lebesgue integral, then it is best done by hand.

Learning mathematics is about growing connections in the brain. And I 
love a simile from the boxing world: allowing a cut brow to slowly heal 
itself in a natural way is considered to be safer for future fights than 
application of stitches.

Best wishes -- Alexandre

-- 
Professor Alexandre Borovik * University of Manchester
Web:       http://www.maths.manchester.ac.uk/~avb/
Wordpress: http://micromath.wordpress.com/
Academia:  http://manchester.academia.edu/AlexandreBorovik