This weeks reading, "Is There a Geometric Imperative?" by Dick Tahta I found fascinating and challenging. It stretched my thinking past what I have experienced and thought of as Geometry and for that I am grateful. Tahta begins by talking about "time and space as basic areas of mathematical experience," and that "perceptions of the external world are internalised as mental images and these are the 'stuff' of geometry." I had never thought of mental images being considered as geometry, which lead me to believe there was a lot I needed to learn.
The main points made throughout the article are that geometry contains three powers: imagining, construing and figuring. I will endeavour to give you a glimpse into what each is.
- Imagining - The power of taking something that is described by someone and seeing it in your mind. Each individual is unique and therefore will have a unique form of imagining.
- Construing - The power of seeing something that is drawn and then describing what you saw. "All seeing involves a 'seeing', an interpreting of what our eyes are conveying to the brain." Having the context of a image/scene allows for a more accurate construing of it. Tahta acknowledges that it is a skill to take a two-dimensional picture and see it with three-dimensional depth, is something that should be practiced to experience improvement. Once again, individuals are unique and therefore construe images in different ways, or through a different "point of view."
- Figuring - The power of "drawing what is seen." This is obviously something that takes practice and should not be underestimated as a difficult skill. As with the other two powers, figuring can be viewed differently by different people given where they are when observing the object or situation.
I think that breaking apart geometry into three powers is extremely powerful. It is important to know that there is not just one main umbrella that fits all of geometry but there are distinctly different parts. Having said that, Tahta acknowledges that all three of these powers contain an element of personal difference and that the person engaging in the geometric activity will bring with them their own 'point of view,' and concept of 'space.' I have never thought of this when I have taught geometry. Tahta mentions the use of our semi-circular canals in our ears as an example of symmetry, which I find fascinating. This leads me directly into relating to the question asked this week about if I know enough about geometry to bring it into what we teach when it is relevant; I would have initially said yes, but I guess I would have to say no. I feel like I know enough but I do not think I look for it enough in the other subject areas I teach. I teach a unit on sound and I love the idea of bringing symmetry and the ear together! In teaching grade 4 we do a lot of work about seeing two-dimensional and three-dimensional objects in our environment which I hope is helping the students practice the power of construing. Relating geometry to the students' relatable 'space' is going to be a goal for me moving forward, as I feel geometry is extremely important and allowing the students' to see the applicability of it to their world at any given time I feel is imperative.
Phillipa
ReplyDeleteI enjoyed reading your reflections. One almost university experience of teachers in the early grades is that students surprise with what they can do, and what they try when afforded opportunities for spatial reasoning. A group of us have a book to appear in April for Spatial Reasoning in the Early Years (ages 0-8).
An interesting play on words is that 'figuring out' actually originally meant drawing a figure - and has now become a synonym for doing calculations. We need more of the original meaning of 'figuring things out'!
Imagining, construing, and figuring... I really like these three concepts. Breaking down a topic into smaller concepts like this can be very beneficial to both educator and student as they can focus their learning and better understand the individual skills required to tackle a problem. I was working with a grade 9 student recently who had such trouble construing a geometric shape: she couldn't quite describe it, or understand what it looked like from different angles. To help solve this, I loaded up the videogame Minecraft and created 3d models of the geometric shapes we were working with. She controlled her avatar in the game to be able to walk around the object and see them from all three sides. We used this to practice what its like to 'walk around' a geometric shape that you're imagining. I would love to revisit working with this student with the imagining, construing, and figuring framework in my mind.
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