In week 8 tutorial you will be allocated one of these Geometer’s sketchpad lessons to evaluate. You are to carry out the lesson and evaluate it (with a link to it included) on your blog. Ask yourself questions like: What part (if any) of the NSW syllabus would this lesson be appropriate for? What is the benefit of using the Geometer’s sketchpad software in this lesson? Could the lesson be taught any other way? If so, how? How could the lesson be improved? Are the instructions for the lesson clear?

Finding Relationships Between Area and Parallel Lines in Triangles

This lesson was alright. I couldn’t find the “Label Options” thing, so I don’t know if it was for a different Geometer’s Sketchpad than the one I was using. Apart from the slight differences, the instructions were very clear.

I guess the lesson is useful in cementing the idea of why area=1/2 the base times the perpendicular height. If students just remember the formula A=1/2bh they may not remember that the “h” stands for perpendicular height and may measure some obscure kind of height. This lesson proves that even if the triangle is incredibly stretched out, the height is still the same, and the formula tells us this because the base is staying the same, therefore if the height changed the area would change, but it doesn’t.

Trying to do this lesson without the help of Geometer’s Sketchpad would be very difficult. I think it is very handy in allowing students to see the ramifications of the formula for the area of a triangle in action. You really do need the software to move the point around and see that it doesn’t change. Drawing a billion different triangles within parallel lines with the same base would be tedious and not worth it.

OK, so I actually did the readings for last week (I read them on Friday actually but today I’m bored hanging around uni waiting for my carpooling buddies – perfect blogging opportunity). I actually found I do have some things to say.

Number1: Zero is not and never will be a counting number. According to me. If you have none of something, you don’t count it. If you’re doing that fence thing, you will have to realise that there is a post at the point 0 and add it on. It isn’t difficult. You definitely should not count the post as 0, because then you are saying that one post is zero posts!!! WHAT!!!!! No. I actually do say babies are 0 years old. Because I’m rounding down. To the nearest 2 years. Not because they are actually 0.

I don’t actually have the readings here with me so I might not remember everything I hated about the readings, which by the way were tedious and I did skim read them because they were dense and mainly the same as every other reading, except worse.

Number 2: North/South, up/down thing. I don’t remember what it actually said, mayhap I will edit this post later. But this ridiculous statement actually went on for like 2 pages, as I recall. It was too much.

This post is dedicated to Sarah P, Pru, Marjorie, Chris, Geoff, Dave, Nick and Ben. Love your work guys!

All of the articles in the course to date are starting to show similar ideas. Teachers need to discuss mathematical applications and use of mathematical language. Students need to see where problems can occur in the use and talk of mathematics and find ways to conquer these problems. Mathematics is a language in itself, and yet we try to explain it using another language. Problems arise when we use words that have relatively straightforward meanings in colloquial use, yet we use these same words to describe quite complex concepts. Conceptual complexity and apparent contradictions in mathematics language, and the artificial symbolic nature of mathematics, with its reliance on many non verbal ways of representing information, often acts to disturb students. Using background knowledge, using multiple teaching methods (for students from different learning styles) is helpful.

We need to be able to describe what we are doing in non-mathematical language as well as mathematical language. This use of students’ own understanding and explanations when dealing with new ideas is important as it connects the new knowledge with established structures that can later be refined to be more in line with conventional definitions. Though we should encourage students to talk in mathematical English, at times it is appropriate (and neccessary) to talk about examples in normal English to aid discussion and the deeper understanding that comes from such a free flowing discussion.

This week’s readings have a similar message to previous weeks; the language of maths conflicts with everyday use of the English language. They also offered similar suggestions for handling the difficulties that can arise for students. Mathematics “borrows words that already exist with everyday meaning” and in doing so changes the context of the word and therefore changes the meaning. cause a lot of difficultly and impatience for the untrained mathematical mind.

For example starting with a number and imagining that it is the area of a square, the square root gives the side length of that square, which is something that gives students a real example of how to view the square root. Teachers need to foster the use of correct mathematical language in the mathematics classroom at all times. Correct mathematical language should be prevelant in any mathmatical discussion. I also disagree that zero should be used as a number. For example, “zero years old” is not a useful description and would be taking things to extremes.

Students need to participate with the language, that is they need to be using it both orally and in its written form. It is usually agreed that mathematics cannot be learnt by watching, therefore it makes sense that using correct mathematical language cannot be learnt by simply reading and listening.

The Evolution of Math Teaching
• 1960s: A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
• 1970s: A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
• 1970s (new math): A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
• 1980s: A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word “potatoes” and discuss with your classmates.
• 1990s: A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.

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“The problems for the exam will be similar to the discussed in the class. Of course, the numbers will be different. But not all of them. Pi will still be 3.14159… ”

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Interesting Theorem:
All positive integers are interesting.
Proof:
Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, that’s pretty interesting! A contradiction.

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“The world is everywhere dense with idiots.”

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A circle is a round straight line with a hole in the middle.

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