Monday, September 6, 2010

Spirolaterals II

Here I have finally figured out how to post a Excel file on my blog. I know, I am not as quick with the technology as I thought I was. When you open it on sheet one, you can change the turn, numerator, and denominator to see different patterns. There is a large image of the graph on another tab of the workbook.



Sunday, September 5, 2010

Review of Chocolate Key Cryptography

So this is an idea that has been bopping about in my mind for a while, and while last year was a crazy hectic year for so many reasons, I have worked to make this year less crazy. I want to do some reviews of articles that I read in Mathematics Teacher from NCTM. I could write letters to the editor I suppose, but the feedback loop on those is too short for this web 2.0 iphone instant message world we live in.

So I give you my thoughts on Chocolate Key Crytography by Dale J. Bachman, Ezra A. Brown and Anderson H. Norton in the September 2010 issue Page 100.

The title is intriguing especially in Mathematics Teacher because it promises something that I know very little about called the Diffie-Hellman Key Exchange, so initially I am very interested. Of course it comes with the requisite Mathematics Teacher cute graphics, but at least none of the painful pictures of students with faux engagement painted on their faces.

The article is broken down into several sections. The first covers what cryptography is, including a helpful breaking down of the word into cryptos and graphos for those of you new to the English language. Followed by an explanation that the internet is important and that concepts about cryptography can be taught to anyone, even high school students.

They then go on to describe a fundamental problem of key creation in cryptography. Essentially, you and the person that you want to communicate with have to pick a number together with out meeting. How can you do that? The authors have created a metaphor for this problem that uses M&M's and at that point one of their colleagues probably told them about how math teachers and Mathematics Teacher love M&M's, hence the article.

The article goes on from there to discuss some of the more interesting mathematics of the problem that includes group theory. The multiplicative group of integers [; Z_{100}^{*} ;]. There is also a lot of discussion of how this is not hard and anyone can do it.

Why should we teach it? You may ask, and if you didn't you should ask yourself why you didn't. Well this shows that math has an application in the real world. Finally an answer to the internal question of when will ever use this. Of course, this doesn't answer when will ever use this. Because your students are not really "using" this. Someone else used to create the internet, and as cool as that is, it is something that is done, finished, kaput.

Overall I give it 3 [; \frac{dy}{dx} ;][; \frac{dy}{dx} ;][; \frac{dy}{dx} ;] out of 5. It is a nice application of math, but the authors don't give enough explanation or scaffolding to help a teacher present the ideas in the article to their classes. Their tell us that anyone can understand it, but their description is not detailed enough to help, and then they scare away a significant portion of the high school teachers of the world with the discussion of group theory. They also don't do a good job helping teachers to fit this into the curriculum. How could this connect with other topics and questions that high school students have to understand? Still I like cryptography, and I would like to see more computer science and discrete math covered in schools.