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.
Enjoy.
Spirolateral.xls
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}^{*} ;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ubfMkmJACc26IZztGynEERJzpd91O1K4S-wN1kR59uzEw6aM4gYsBFegjwzlq_ZoL_Yuc214sm9tIO5ARvtdQ0fHDNdKQ8wrYFC9KeBklkYnFm894EjMCQsCR4=s0-d "Z_{100}^{*}")
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} ;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sfU6lInLFxMRF0BiMUt5tAH9QM6nkCpFFwO5ffi22kJqYFVnTtJjYJDu4URzSoKeLFFMWu1d53hoVRnPwg7tATfDzM-EYZqMwAaNvWGRUemiD_u8fgpy-ojlJ4bn0=s0-d "\frac{dy}{dx}")
Great writing inspires. I have been reading over the last several months a small book that I have found very inspiring. Everytime I pick it up I get new ideas, and I challenge my assumptions about teaching. The most surprising thing about it is the book is not about math, not about teaching, not about students. It is about drama. It is called The Three Uses of the Knife by David Mamet. It speaks to the nature and use of drama.
I know that there is a general sense among many teachers that class does not need to be a show everyday, and clearly that attitude taken to extremes creates teachers that are more like Michael Scott of "The Office" than Dan Meyer.
But Mamet forcefully explains that humans have a dramatic urge. People want to see themselves as part of a giant play with themselves as the hero. As teachers we have a responsibility to engage students, and to ignore this paradigm seems short-sighted at best.
Mamet is very insistent that there are good forms of drama and bad forms of drama. He uses school, politics, the evening news as examples of bad drama carried out every day.
In the next few posts I will reflect on certain sections on Mamet's work and describe what I think it means for teaching. We will start the section he calls The Perfect Game.
Imagine the perfect game. Is it your favorite team thumping their opponent? Of course not, it is your team starting out well, looking dominant and then all of a sudden things fall apart. They change many things and nothing seems to work. Then all of a sudden when things look darkest, a player redeems himself and for some previous screw-up in the game, and scores the go-ahead score, but then the referee calls it back, and again the players must find a way to take the lead, and they do, but the other responds to retake the lead, only to have a more miraculous play occur for your team, and so on and so on.
This is the prototypical three act structure. Humans look for this structure in their lives all the time, and if you watch sports, or follow politics you will see that these activities are recast by commentators in this framework all the time. Mamet would tell us that this recasting is for our pleasure.
Mamet describes this as "Yes, No, But Wait...", and it repeats again and again.
You can imagine this in a classroom, students come in an see a problem something familiar that they know and can deal with. They deal with it easily and look to you for their deserved praise. Then you give them another problem similar, but with some unexpected twist. They work forward trying many things. Many of those ideas fail, and the way seems lost. You don't help them (much, if any) then at some moment through an idea spread through the class based solely off of students previously failed idea (or through some cryptic magic incantation you said under your breath) and the students are off and they solve the problem, or they don't stymied by some other detail they have left out, or you have given them a new problem with a new twist. In this way, we model "Yes, No, But Wait...". And basically we should believe that this is what students want, crave, desire. They do not desire the answer, the algorithm, the process. They desire to be heroes.
Part of what Mamet talks about is our never ending need to dramatize our lives:
For we rationalize, objectify, and personalize the process of the game exactly as we do that of a play or drama. For, finally, it is a drama, with meaning for our lives. Why else would we watch it?That idea about rationalizing, objectifying, and personalizing is so important. Students need to do that to really connect with the content that we are looking for them to master. If they don't do it this more complex and deeper level, then they will never remember it.
Spirolaterals are an idea that I don't remember where I read about them, but the idea is relatively simple. You take the the decimal expansion of some fraction and using the digits create segments of those lengths then rotate some fixed number of degrees. I have an Excel spreadsheet that creates these pictures. If I figure out a way to post the file I will. Let me know if you know of a way to do that.
Here is 8/147 with a 90 degree turn at each step.
I have been reading a number of the posts on David Bresoud's blog at maa.org. They are slowly bringing me to the conclusion that my profession is in danger. More and more I see math education mirroring the american automotive industry. We are certain that there will always be a demand for what we provide. We are certain that people will desire a version of that product that is nearly identical to version we produced 20 years ago. This may not be true, and we have gone a long way to "reform", but there is still a long way to go. Bresoud articles seem to point out consistently our ridiculous adoration of calculus as the bridge between high school and college mathematics. He points out also how this leads to all kinds of systems and policies that lead people to take math they don't want and they don't need. The math they do want and do need is often under-supported, under-credited, and under-appreciated.
Those are my interpretations of his articles. You may have your own. One of his more recent posts, here, talks about the indicators of success in college. And of course points out what many know, the SAT and ACT are not particularly good predictors of college success. Which is what I understood their importance to be in the first place.
So what if SAT and ACT went away?
At first, I was excited by the prospect. So many pieces of ridiculous mathematics could be jettisoned from the curriculum. We would have freedom to create programs that make sense for today. It would be easier to rest control of the curriculum from calculus and refocus on statistics and discrete math. More and more students are going in biological sciences (need statistics) and fewer and fewer are becoming engineers (need calculus).
Then the dark side of it hit me. If math teachers could say you need this for SAT, then what is the likelihood we could keep our requirement status. Would you really need four years. How many students and parents would like their child to not HAVE to take math. Based on how many people regularly tell me that they hate math or were never any good at it, A LOT.
And what if happened quickly. Where would we be?
This is why I feel we have to teach math as if it were an elective. Every class. Every class needs to have a clear purpose. Students should not leave high school without knowing how to use excel, because if you are going to do real math in the real world you are going to use excel at some point. So screw the calculator and get the computer.
How much data is being produced today? Way more than can be analyzed currently, but they are looking for people to do it. Why has AP Statistics grown and grown. When will it plateau?
I guess I have a lot of questions, but I don't know how to rattle the colleagues I see around me to the coming danger.
In the video below I try to give some of the "patter" I use when I introduce the idea of "i" for the first time. I think that there is a great benefit in doing it this way because it helps strengthen their understanding of what real numbers do as well. It also emphasizes that numbers often get paired with operations. I actually could make that clearer. Anyway, feel free to steal this introduction to use in your own classes.
