Dick Tahta_Is there a Geometric Imperative

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.

Thinking mathematically by John Mason

John Mason in his book Thinking Mathematically demonstrated what we should do when encountering a conjecture (it is not necessarily a mathematical conjecture): (a) to convince ourselves; (b) to convince our friend; and (c) to convince our enemies. What he means by these is that we should describe the conjecture (as part (b) and explain it using formal language as proper as possible), and then see every statement as a conjecture and try to disprove it. These steps are significant in learning a mathematical concept.

But the below is what I feel valuable:

“Each version tends to become more abstract and formal, trying to be precise and to avoid hidden assumptions of informal language, but incidentally causing the reader to have to work harder at decoding the original insight and the sense of what is going on.” (p. 98)

This is why the math concepts are hard to read: the insights are covered under the intricate math signs. It might be helpful to demonstrate the true meanings of math concepts by showing students their origin. 

Thoughts of mathematical proofs

            The topics for this week’s reading are somehow out of my expectation, although they are interesting as well. I have thought some relevant topics about mathematical proof may be to explore why the reasoning process is a relatively boring subject for students, but Hanna and Barbeau’s article discusses the potential achievements students can make beyond the wordy mathematical proofs, while Zack showed his understanding and improved teaching skills through a series of classes dealing students’ counterarguments of a given plausible but wrong solution.

            Indeed, proof is a powerful and helpful tool for students to understand mathematical concepts. Zack’s improvement and understanding of children’s math thoughts strongly supports Hanna and Barbeau’s argument that teachers and students can improve their understandings through proofs. But proof itself may be scary for first-time learners if his/her instructor focuses more on the format of writing a mathematical proof. The situation may be more challenge if a student has a below average math skills because there is (almost) no formula for finding a proof. My own experience tells me that math mainly contains two parts: reasoning (proofs) and application. Things (do math problems) can be easy if my job is to simply use some rules; but it is always a challenge task to understand a proof because of its mathematical writing styles and the relationships between each step. Things can be out of control if I am asked to write a mathematical proof. That means I have to construct a persuasive reason and convey it into mathematical language.

            I believe this is the challenge part for some students, as conveying the reason into proper mathematical language always involve a certain level understanding of necessary mathematical knowledge. This requirement is hard to meet because (a) an average students may not master all the previous knowledge well, and (b) there are (is) limited (not as open as proofs should be in a class/test setting) ways for students to choose when conducting a mathematical proof because of their understandings and knowledge background. And thus a proof is somehow boring if students cannot figure out solutions by themselves.

            Anyway, what interest me in proof are the different-than-usual proofs of some familiar proofs. I am looking forward to explore more on this area.

Ps: I wonder where are the additional readings for this week as I am absent from the off-campus meeting.

The Marchall Island Mapping Models--------really drove me crazy....

Marcia Ascher showed me a series of elusive but beautiful findings of the Marchall Island maps. In the article Models and Maps from the Marshall Island: A Case in Ethnomathematics, the author demonstrated the mathematical thoughts in Marchall Island sailing tools. Local Marchall residents use mattings, rebbelith, and meddo to save secrets of their geological knowledge and to train their navigators.

Frankly, I found the matting part difficult to understand. The Marchall mattings are usually symmetric, and contain lots of triangles, sectors, angles, arcs, etc.. Local people use them to position in the ocean. However, even though I know that the arcs and swells are somehow similar to the shapes of waves in the Marchall sea area, I am still REALLY confused about how the tools work. The other two tools are somehow easy to understand: local Marchall people use intersects to identify islands so that whenever they see two islands, they can locate all the other islands.

I think this article is a good example of showing ethnomathematics, as it is difficult for me to understand the explanations in an off-background situation. I can understand every individual word; but when relating them together, I just lose my mind. I guess this is how Marchall people keep their secrets…….. 

Some thoughts of reading the article "problem posing in mathematics education" by Stephen and Marion

In Stephen and Marion’s article “Problem Posing in Mathematics Education”, five topics are mentioned: (a) some math questions are posed abstractly so that they do not make sense to students; (b) besides solving a question, there may be some other educational works that students might engage in, i.e., create questions using newly learnt math concepts; (c) a question with detailed descriptions may be reasoned similar to other questions, which means these questions reflect different sides of a coin. Students may benefit from exploring these questions; (d) based on part (c), students may learn more if they can construct new problems using the given conditions. This may broaden their understandings of the knowledge; (e) knowledge is constructed on social context, and so should be the math problems.

What we are doing as math teachers is to compress math thoughts of hundreds of years on students’ brains so that they can pass exams. The object here is so clear that solving the question is much more important than experience the process of learning. For those who can understand math concepts easily, they more or less follow this thinking process, and we believe that they can relate math concepts to their living experience. However, for those who cannot do math problems well, can teachers really “teach” the living experience they need? Even if we can teach the experience, are students willing to learn? Or they prefer to learn the “fast, accurate, and efficient” ways that can help them pass exams?

I spent hours writing the above paragraph, and I realize the situation here may be not because of the way we teach, but the object we teach and students learn. If the issue here is that math teaching and learning are exam-driven, students will no doubt focus more on methods of solving problems instead of methodologies used in mathematics (correct me if I use inappropriate words). The math education in Finland may be an example of showing how math can be taught in an exam-free educational system.