[Maths-Education] Re: ICT in mathematics

[Walter Whiteley]

	I have been following the thread as it evolves.  I hope the following  
is still relevant.
My point of view starts as a practicing mathematician (applied  
geometer) but includes some research and writing on visual and  
kinesthetic reasoning in mathematics and mathematics education,  
including early childhood.

	 ICT is an integral part of the modern practice of mathematics, both  
by current mathematicians and by those who use mathematics in other  
disciplines.   I use some versions of ICT very regularly in my  
mathematical practices.  In particular, it is central to my  
collaborations with people outside mathematics (structural engineers,  
mechanical engineers, biochemists, biophysicists, material scientists,  
computational geometers, ... )  To not use the powerful visualization,  
data processing, and other affordances is to leave students unprepared  
for future use of mathematics, or even to engage with current uses of  
mathematics.   Let me give a couple of examples:

1. As I also think of statistics as one of the mathematical sciences,  
I do not know anyone who argues statistics should be learned, and  
applied, without major support of technology.  (These days that is R  
at the higher levels, often Excel at the lower levels.)  Major  
advances in data visualization are also relevant, and students need to  
learn to extract information from, modify and critique, and sometimes  
create appropriate visualizations of statistical situations to reason  
and communicate.   I don't know anyone who regrets the loss of  
statistical tables!  I don't know who would work with these  
visualizations without ICT.

2.  In geometry (and some other areas) it is possible, even desirable  
that key communication and reasoning be done visually (either  
externally supported or internally).   Key reasoning is embedded in a  
sequence of diagrams (with transformations).   In line with the  
theories of Sfaard - thinking is an internalization of the visual  
communication among peers and with experts.   Now - dynamic software  
such as GSP (what is licensed for all students here) permits some  
valuable changes in the external visual forms, and in the resulting  
reasoning.  It also supports development of constrained mental  
visualization, including transformations.  Initially, I found the  
software made visible to my students what I was already 'seeing' in my  
mind.  Now what is in my mind has also been modified.
       I regularly encounter students who are handicapped in their  
mathematical practice by their inability to extract information from  
and to communicate with themselves and others via visual forms  
(sometimes 3-D).   For example, I have encountered a number of  
students/teachers who 'hit a wall' in their first course of multi- 
variable calculus and their inability to 'see' the shapes being  
described.  This grew from a number of prior years (senior secondary  
school) where learning to visualize and reason spatially was  
considered unimportant.
	ICT can play a valuable role in supporting spatial reasoning, and  
words (and gestures) often refer back to prior visual abilities and  
experiences.   Of course, physical manipulatives can also play an  
important role in spatial communication, and in reasoning. (I have a  
cupboard of them for my undergraduate and in-service geometry classes.  
Students use them, and GSP, for in class work, for assignments, for  
presentations.)   These are the ways that young children have learned  
spatial reasoning (starting in their 3-D world).  That pencil and  
paper has reduced us to 2-D for too much of the elementary curriculum  
is an artifact of old technology we can overcome.  ICT is part of that.

I am generally concerned about references to a 'hierarchy' of tools  
and of communication.  I worry about descriptions which place visual  
reasoning and communication 'down' the hierarchy (as in van Hiele  
hierarchy).   There is now a recognition that these visual forms live  
(and thrive) at all levels of the reasoning hierarchy, and need the  
same kinds of teaching / learning support as verbal reasoning, and  
symbolic reasoning (which has its own key visual components - think  
visual patterns in matrices).

My claims and my concerns are informed by observing that my work (and  
my teaching) continues to become more centered on visual reasoning,  
including ICT supports, as my work evolves.  This is true for  
collaborating with other researchers, for supporting research  
students, for working with in-service teachers,  for teaching math  
undergrads, and for preparing pre-service teachers.

Walter Whiteley
Mathematics and Statistics, York University
Graduate Programs in Mathematics, in Education, in Computer Science,  
and in Interdisciplinary Studies.
http://www.math.yorku.ca/~whiteley/

On 9-Mar-11, at 5:00 AM, Alexandre Borovik wrote:

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> On 09/03/2011 09:07, Candia Morgan wrote:
> > Or if someone needs pencil and paper to solve a problem
> > they have learnt using pencil and paper not mathematics?
> > Different media allow different forms of mathematical
> > activity and thinking. Conventions and power
> > (including assessments) determine which of the
> > possible forms are valued.
>
> There is a natural hierarchy of human modes of communication. In  
> mathematics, the most important is the most ancient, voice.  Chalk  
> on a blackboard, penicil on paper are just assistive tools for human  
> speech.
>
> It is worth noting that, despite all the technological progress,  
> teachers are still using speech in teaching, but (at least  
> university teachers in this country) do not teach their students to  
> talk about mathematics. It is perfectly possible to get a good  
> university degree without ever opening mouth.  This is one of the  
> main flaws obstructing the cycle of reproduction of mathematics in  
> this country. (The situation is different on teh continent, where  
> many countries still stick to the tradition of public oral  
> examinations).
>
> 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
>
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