Sunday, December 12, 2010

Geometry and Optical Illusions

Optical illusions show the need for proofs in Geometry

Yesterday I had two Geometry tutoring sessions. The first one in the day was only my second session with that student, while the second one was my first session with the other student [ a curious coincidence: (1,2) and (2,1); that is a symmetric pair in that relationship, with x = ordinal number of the session in the day; and y = ordinal number of the session with that particular student but I am digressing ].

One of the students showed progress in the sense that she was approaching the problems with more confidence than one week ago. Her mom told me she had also improved her grade in the last Geometry quiz she took. I attribute her improved performance to her increased confidence, which in turn I see as the result of our first tutoring session, because the confidence students have in their own understanding, and capabilities, usually increases when they benefit from the immediate feedback provided by the tutor subtly guiding them through their practice problems.

In the other session, the student asked some very good questions about proofs.
One question was to the effect of: “How do I know what properties, or theorems to use in a proof? Because at some point there are just too many of them, how do I know which ones to use in a proof?”
This is a very good question. It points to the core problem of looking for a path that connects the given statements with the desired conclusion. Many geometry students are confused by the uncertainty of the guessing involved in the process of finding a proof.
Elementary Arithmetic, an Algebra, are different from Geometry, and from Calculus, in that (among other things) most of their material can be presented as a set of prescribed, step-by-step procedures that specify what rules to apply, and in what order. They lend themselves to an algorithmic presentation more easily than Geometry or Calculus do.

Another good question this student had was about the very need for proofs, especially when the problem in question references a figure, drawing, or diagram. He said: “Why do I have to prove that those triangles are congruent, when you can just look at the picture and see they are congruent?”
Many geometry students ask this question. My short answer for it is: “What if the triangles in the picture are not really congruent because the picture is a tiny little bit off but we cannot notice the difference without a magnifying glass?”
We also talked about optical illusions, and how our sense of sight is charged with a heavy propensity to be “fooled,” “deceived,” or “misguided” by a whole variety of optical illusions, and so we can never trust pictures one hundred percent, not because of the picture but because the way our brain processes visual information.

Later in the afternoon, in between two tutoring sessions, I dropped by the bookstore looking for a CSET prep book, and I was gladly surprised when, by chance, I found a bargain priced book, precisely on the subject of optical illusions. It is this book right here, by Inga Menkhoff. It has plenty of quality pictures, with the corresponding explanations. I have always liked optical illusions.

Thursday, December 02, 2010

Random Comments From Students

A math tutoring day with nine sessions

After nine math tutor sessions in the same day I barely remember where I was during the day yesterday but I remember some random comments my students made during the tutoring sessions. The show must go on, so I have no time to go into detail about any thoughts or opinions I may have about the following topics. I just present here some of the comments I remember, in case you find them interesting.
Some paragraphs below are single comments made by students. Two paragraphs are short pieces of dialogue, as indicated.

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“I already understand everything we have seen so far but my mom wanted me to come take this tutoring session anyway.”

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“This problem is actually pretty easy! That’s weird. I was expecting it to be this huge big thing but it’s just that simple!”

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“Volume is everything that goes inside, it is the whole space!”

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Student: “Well, they say these lines bisect these angles, then this half angle should be 33 as well.”
Me: “Exactly! Very good. Now, could you do the same with the two other half angles? Can we write x and y for those two other half angles, as well?”
Student: “No! You cannot do that because you don’t know how much x is.”

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Student: “I’m sure I’m never going to use this stuff in my life, ever! This is so boring!”
Me: “What would you rather be doing?”
Student: “Anything!”
Me: “For example?”
Student: “Writing a speech.”
Me: “About what?”
Student: “About how math is silly, and stupid, and totally useless. They just make us waste our time struggling with these things.”
Me: “Imagine a world where nobody had to take math classes, only people who really wanted it.”
Student: “That would be awesome! But then you would be out of a job, and a whole bunch of other people would be out of a job as well. They just want the system to keep going the way it is because they want to protect their jobs. I don’t want to disrupt anything. I wouldn’t take away people’s jobs.”

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“Oh, decimals! Decimals always confuse me. I always get confused when I see decimals.”

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“These are my notes, sorry. I don’t know what was going on over here. He was going so fast, oh my God! I was just writing down everything as fast as I could but I didn’t understand anything. Well, I understood a little bit but not really.”

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“Do we add these, or do we multiply them? This is getting so confusing! It is so much! And he said this is just skimming the surface of it, can you imagine?”

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“At the beginning I was actually enjoying this class, and I was interested but now I don’t care anymore, I am not interested. I just want this to be over, I just want to get a passing grade in this class. The final is coming up, and I’m getting so stressed out!”

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“Why did they invent all of this? What for? Is it used anywhere? Does it have a purpose?”

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“Oh! So, that’s what remainders are! That is very good to know because there are a whole lot of problems that use those things.”

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“That is so cool! I never knew that! Is this kind of math used in the real world?”

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“What did I do now?”

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“Oh, no! My bad.”

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“First we move this bad boy over here to the other side. Then x equals 6, then x equals 3, then x equals 9, and this is what we add. Now we square the equation but there is no square root of 7, so that’s it. Am I good so far?”

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