I have to say, reading Dick Tahta’s Is there a
Geometric Imperative is not as straight-forward as the other articles I have
read. Tahta (2010) summarizes the ways of framing a “coherent geometry course”
(p.1) as imagining, construing, and figuring.
Imaging, as it is defined, happens when people try
to visualize things that are not presented in front of them, or “seeing what is
said” (Tahta, 2010, p.1). The key concept here is that no one can guarantee
that the describer and the audience have the same images in their mind----and
indeed, it is almost impossible that they visualize the same thing. But how can
we see the similar things when we hear the descriptive words? I got the idea of
vocabulary and the perspective, but that is not all what Tahta is trying to
say. I will explain this in the following article.
Construing, which Tahta (2010) interpreted as “seeing
what is drawn, and saying what is seen” (p. 1), contains two parts. Seeing what
is drawn is the input, while saying what is seen is the output. Both are built
based on an assumption that the listeners who are listening to the describer have
the same vocabulary as the describer. And because what is described is an
image, the perspective is with the same significant level as the words used to
describe the image. It is true that an image must be different when viewers see
it through different perspective.
And finally, figuring tells us to “[draw] what is
seen” (Tahta, 2010, p.1). What I understood is that geometry is a subject about
images. It is of great importance to build a bridge between image and
descriptions. We human spend hundreds of years trying to figure out how to
express what we see on some intermediates.
I really like the imaging, construing, and figuring
ideas. It is not only the way used in geometry; rather, it is the way we used
to describe the nature. We use abstract intermediates, such as words,
definitions, and images, to symbolize and explain our surroundings.
This seems like a very interesting article in the it discusses geometry in very naturalistic and abstract ways. I have never really broken down the idea of geometry into its subparts, but I like how it is broken down here into imaging, construing, and figuring. If I were to break it down into my own subparts, I would say it is imagining, constructing, and then interpreting. For example, I would imagine an idea or shape I want to express onto paper/computer software, then I would actually construct it. This maybe on paper or computer software, like I just mentioned, or even with something like clay or wood, just anything where what I imagined can be seen outside of my own mind in a more concrete way. Then I would use different mathematical skills to interpret the construction. So if I were given a Pythagorean Theorem word problem I would first read the problem and imagine what I need to construct in my head. I then would construct what I imagined by drawing a picture, most likely a right triangle, and labeling everything I know. Lastly, I would interpret/solve whatever it was that I wanted to find out. I guess my break down is similar to the break down in the article you read.
回复删除To me, geometry is much more than a way of mathematical communication and the input and output of images. Geometry is also more than an expression of one's perception or an interpretation of someone else's visualization. However, it is a visual way of explaining mathematical phenomena that other avenues cannot easily or clearly represent. For example, although much of geometric theorems can be proven via algebraic steps, for a secondary mathematics learner, the steps quickly become complex and difficult to understand. If the same steps are visually represented with the help of colour coding, the geometric theorem can be easily understood by the intended learner. I do agree that geometry requires a certain level of imagination especially in spatial reasoning. To me, geometry is also like putting together a puzzle. There is no one correct way of putting together the puzzle. Nevertheless, the player(s) can formulate a method to efficiently complete the puzzle either by grouping together similar shades or constructing the frame first. At times, there is the need for trial and error. At the end, the puzzle tells a story, the story of the image it displays. For me, geometry is largely composed of an aesthetic nature that possesses the kind of beauty in mathematics that other avenues of mathematics are incapable of.
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