[Maths-Education] Re: ICT in mathematics
Walter Whiteley
whiteley at mathstat.yorku.ca
Wed Mar 9 13:38:05 GMT 2011
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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