[Maths-Education] Something more on Manipulatives in Math

[Walter Whiteley]

I am late onto the list, but I wanted to share some reflections on  
manipulatives as they are used in higher mathematics - in teaching,   
in research and in communication of mathematics.  It is possible that  
the materials I am going to describe are more closely embedded as  
representations of the mathematics being learned, than some  
'manipulatives' for numbers mentioned in the previous thread.   Think  
perhaps of coin tossing as a representation / manipulative for  
'random events' and 'probability'.

I am a researcher in Discrete Applied Geometry (as well as in  
mathematics education).   For my work with structural engineers,  
mechanical engineers, biochemists, biophysicists, computational  
geometers,  ....  I regularly use physical (and virtual)  
manipulatives to help me understand problems, to imagine (and image)  
possible methods and reasoning, and to communicate the problems and  
conclusions.   I am almost notorious for using models in my talks,  
and discussions (a recent visitor commented she could not come to  
talk at our seminar without at least one model!)   So manipulatives  
are not just for elementary school, or for elementary mathematics.   
They can be part of the practice of mathematics at all levels, and  
like the use of technology, should be learned as part of the  
processes and methods of mathematics.

Of course, the reasoning associated with handling the manipulatives,  
and later with rehearsing the motions, feel, and senses of that (in  
our minds eye) evolves over time and experience.   There is an  
interaction with virtual manipulatives, but to date the virtual does  
not completely substitute for the physical.  My own experience /  
general proposal is that the ideal way to use the physical materials  
in the practice of (my types of) mathematics is to weave back and  
forth among physical, virtual, and algebraic and not discard any of  
these as a source of insight and understanding.   People who are  
going to use math later, need to learn how to use manipulatives well  
to aid their reasoning and problem solving.  The same applies to the  
use of virtual manipulatives of the higher sorts (have you ever  
played with visualization tools for biomolecules!)   GSP, Cabri and  
Cinderella which were developed for teaching geometry, have become  
key tools in the research practices of people working in geometry.    
Done well, the use of these representations is part of preparation  
for continuing exploration, research, and applications of the  
mathematics.

Manipulatives are important enough to my research group that we will  
pay substantial $ to have them.  (Some current uses of 3-D printers  
are an example, in interdisciplinary work.)  There are some nice  
stories in the recent biography of the geometer Donald Coxeter (the  
King of Infinite Space, by Siobhan Roberts) which describe his early  
travels with his custom made kaleidscopes, carefully packaged in  
pieces inside felt covers.  [We actually have these at my University,  
donated by Coxeter.]   Coxeter used these manipulatives for his work,  
and for his communication with people at all levels of math.   I am  
currently working to build some kaleidoscopes (Kaleido-spheres - the  
mirrors of platonic solids)  for the teaching / learning of  
reflections and isometries in space.   Having 'thought about them'  
and having used 'virtual software' for them, I have still found it  
important to experience them myself, as well, even at my age.   At a  
recent working group on Geometry, Space and Technology (see the wiki  
below), with mathematics educators, it was striking how important it  
was to have mirrors, hinged mirrors, and combinations to actually  
invite explorations.  My earlier efforts with teachers, using  
software and elastics to 'mark mirrors' you 'saw' as symmetries of,  
say, a cube, were simply not enough to full engage the way inserted  
mirrors actually do, for all of us.  Such options do not 'afford' the  
cognitive process which can become the powerful reasoning techniques  
when they are engaged.

I also use spheres when teaching spherical geometry (like we use the  
'manipulative' of paper and paper folding for plane geometry).   It  
is essential, even in university and in grad courses, to have  
physical spheres (and encourage learners to use spheres at home) in  
order to get some sense for what is happening, what will happen, and  
why.  (I also recommend Spherical Easel, a free java program which is  
like Cabri and GSP, but for the sphere.  Great for exploring, after  
the hands on, and for generating pictures for assignments.)    I  
cannot resist mentioning a story from a math educator studying a  
class learning spherical geometry a few years ago.  His observed that  
watching the gestures of the students indicated whether they had  
grasped the sense that a 'great circle' on a sphere is straight.   
When they ran their finger along the sphere, in a certain way, you  
knew that a key cognitive connection had been made!

That suggests a further use of manipulatives / physical  
representations in math.  Engaging / occasioning gestures and the  
kinds of kinesthetic reasoning which consistently appear in fMRI  
scans of people 'doing math'.  Kinesthetic reasoning is generally  
under reported and under valued as a basis for doing mathematics, as  
the gestural analysis (of analysis) of Rafael Nunez indicates.  Our  
capacities in 'kinesthetic reasoning' do change / grow over time -  
and can engage many levels of reasoning, if we pay appropriate  
attention.   As a very simple pointer, recent studies of 'mental  
rotation' (a well-studied part of spatial reasoning) indicates (a)  
the use of pre-motor motion planning even when nothing physically  
moves; (b) differences in this planning when the objects being moved  
are a pair of hands (both sides of the brain planning the motion) or  
tools (the dominant hand is actively planning) or abstract objects we  
do not usually 'move'.   There is a lot still to learn!

Walter Whiteley
Mathematics and Statistics,
York University, Toronto
Graduate Programs of Mathematics, Education, and Computer Science.
http://www.math.yorku.ca/~whiteley/
http://wiki.math.yorku.ca/index.php/CMESG