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Tools and Conceptualization

There is some evidence that our conceptualization of mathematical concepts, including conceptualizations in geometry, are influenced by the tools we use, including the interfaces in our software tools. Those who grew up with Compass and Straightedge will likely have different concepts than those who grow up with dynamic geometry programs, in transformational tools in particular.

Current work along this line has examined tools such as controllers for robotic manipulators.

The same possibilities arise with any tools, including physical manipulatives and virtual manipulatives - as well as the use of paper representations / projections etc.

Further Links on Manipulatives

Some initial reflections: Objects, Tools and techniques of mathematics

Gestures and Learning

The use of physical manipulatives (and some carefully designed virtual manipluatives) may connect to the literature on gesture / learning / memory in mathematics.

There are some interesting articles on how students use of gestures aid retention even in grade 12 arithmetic.

 Susan Wagner Cook, Zachary Mitchell, Susan Goldin-Meadow: Gesturing makes learning last

In this experiment, the authors taught three groups of students in Grade 2, an arithmetic process:

  • one group were taught using spoken words and asked to describe their processes using spoken words;
  • one group were taught using gestures and asked to describe their processes using only gestures;
  • one group were taught using gestures and spoken words and asked to describe their processes using gestures and spoken words.

They all learned the material in an immediate post-test. The critical finding was that in a test about a month later, in the classroom where the students did not know it was part of the experiment, there was a significant difference in the retention: both groups using gestures retained the material.

There is every reason to anticipate that this is also true for spatial reasoning. There certainly is research which notes students' (and teachers') use of gestures when recalling polyhedral features, or when solving problems involving spatial transformations.

With the findings that when carrying out mental rotation also involves neurons for pre-motor planning, it is possible that virtual manipulatives afford the use of pre-motor planning as well. It would be valuable to see if students use gestures when using virtual manipulatives, or when later carrying out problem solving.

Posting on Manipulatives

Here is a modified version of a posting I made to a Mathematics Education List:

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.


Geometry for Teaching

Back to Big Ideas in Geometry: procedures, concepts, patterns

Back to Spatial Reasoning Page

Link to CMESG Working Group on Geometry, Space and Technology pages