Beliefs about Learning Geometry
Some previous discussions generated the following basic beliefs, we feel many people hold:
1. Learning 2-D geometry and visualization is easier for children than learning 3-D geometry and visualization;
2. Number sense (and algebra) are the important math, and geometry / space is less important (optional). (The math is numeracy reduction.)
3. Geometry is Harder than other math - it is about proofs. (Related belief - only geometry is suitable for teaching proofs, therefore that is what time on geometry must focus on.)
3(a). Using technology makes teaching geometry harder / learning geometry less valuable for mathematical processes (learning proofs).
4. Geometry / Space (and related technology) is too hard for teachers.
5. Geometry is not important for further studies in math, or math based subjects, or careers. (Look at entrance requirements for programs, look at the first few years of university studies, look at requirements in faculties of education.)
6. The use of technology (e.g. maple, mathlab, numerical techniques, automated procedures) for applying math makes learning geometry less important / more important for people not majoring in math;
1. 3D is the simpler visual and kinesthetic world for young children. That is where we are born, and move, and this is he way our cognitive processes are wired.
We need to collect together the 'evidence' from the literature which confirms this.
2. Hands on materials are needed to (initially?) support virtual / physical explorations. We speculated that, in many circumstances, it is optimal to go back and forth from physical models to computer representations (and other representations) and back to the models in a repeated cycle.
We need to bring together the evidence from the literature for this, and perhaps evidence for when our own mental / virtual representations are adequate for exploring or making sense out of the situation.
This type of cycling is what I (Walter) often do in my own research. Unfortunately generating a good hands on model for some of the virtual and algebraic models is time consuming, even material and machine intensive. So this might be pedagogically and cognitively correct, but it is even more demanding than 'just' using computer-based virtual representations, and clearly more demanding that using neither - just pretending that a vague mental model is sufficient for the learner, or for ourselves.
3. Perhaps primary to both 1. and 2., we believe that kinesthetic reasoning is engaged in many circumstances where the current 'self reporting' and 'analysis' suggests we are using visual reasoning. Kinesthetic reasoning, in a brain wired for eye-hand coordination, is in 3-D (at least locally).
Here are a few paragraphs from a recent e-mail to a new grad student:
There is some interesting educational and cognitive science research which does explore the connections of mathematics with the kinesthetic (and other senses). Some of this comes up with a google search of 'embodied cognition mathematics' and Raphael Nunez appears. In particular, Nunez has some interesting research about the role of gesture, and the ways in which this displays key metaphors in mathematics which may now be lost in the formally written material (but are still clearly part of the thinking even of the experts). Issues like time and motion live on in the hidden worlds of informal mathematical practice and communication, but do not appear in the texts, or the verbal portions of the classroom experience. There is also a bit of research suggesting the the teacher's gestures which accompany a grade 2 arithmetic class can impact student learning!
Personally, I have been working on, and trying to develop, visual and spatial reasoning as key portions of how mathematics is done, even though we hide this from students and even many teachers. Over the last decade, I have realized that much of what is excavated as 'visual mathematics' actually contains, or even is a pale reflection of, kinesthetic reasoning and practice. Kinesthetic practices are seldom captured in the artifacts of mathematical practice - the texts, the lectures, even the doodles and rough notes. They are a layer deeper even than visual practices, in terms of verbal self-reporting or consciousness. Video reporting is starting to bring gestures out as evidence of this type of thinking. So too are fMRI brain scans which confirm the role of brain activity otherwise associated with movement, or at least with planning movements, in mathematical problem solving. The area associated with eye-hand coordination is key to problem solving (and also to the analog number line we seem to be born with). The studies of 'mental rotation' (a basic spatial reasoning skill) are now confirming the interaction with 'pre-motion planning' as we compare 'objects' with mental rotation, including small differences in how we plan the mental motion of comparison if we are comparing two pictures of hands (we imagine moving both of them) vs comparing tools (we imagine picking one up and moving it to the other one, with our dominant hand)! I have not doubt, now, that imagined motion, sometimes 'felt' motion (with physical models), plays a key role in my own continuing research. (Roughly, I study what shapes and patterns are rigid, and flexible, in space and the plane, with applications to drug design, structural and mechanical engineering, robotics, and even origami!)
I am convinced there is a rich vein to be explored here. I am interested in the ways in which kinesthetic processes connect to the continuing 'mental kinesthetic' processes as we try some of the problem solving in a virtual / computer aided form. (The recent work on 'mirror neurons' - neurons which fire both when we are moving, eg. to pick up food and eat, and when we want someone else pick up food and eat, is suggestive that kinesthetic processes are quite active as we dream, as we image moving, as we problem solve, as we think mathematically. The key may be to develop some awareness, and some ways of developing and refining these processes, in ways that connect and integrate.)
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