Author Archives: Mr. Chase

Google Ngram

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This is super fun. Google has just released this tool for playing with word frequency data from a huge amount of scanned literature (5 million books dating as far back as 500 years). You can read more about it here, including some nice research that’s already being done with the full data set that’s also been released. (also here)

For example, here’s a graph of the appearance of the word “homeschool” in the collective Google corpus.

You can also compare the appearance of words. For example, here’s informal evidence that we care less about ancient Greek mathematicians (BC) and more about European mathematicians (17th and 18th century) than we did 100 years ago.

Not very rigorous, I’ll admit. But it’s an example of what kind of interesting trends can be instantly teased out. As this article quotes Erez Lieberman-Aiden of Harvard University, “It’s not just an answer machine. It’s a question machine.” I think that’s a nice way to put it.

 

The Lecture

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The Lecture. Is it so bad? This recent post by Dr. John Fea at his history blog addresses this question (be sure to check out the links at the bottom too). And I’d like to weigh in on the issue too.

The trendy education gurus would tell tell you ‘lecturing is bad.’ As my students know, I use a mixture of lecture, guided practice, group work, games, etc. Even though I’d love to say I do mostly creative out-of-the-box activities, the truth is that I mostly lecture. I’m not sure that’s entirely bad, though.

When I was in college, I really enjoyed a good lecture–key word good. Dr. Fea was one of my history professors and he was a fantastic lecturer. He would get into it, he moved around, he was well-spoken, and he knew his stuff. I enjoyed many of my other professors for the same reasons (one of my philosophy professors, in particular). And I still enjoy listening to good lectures when I get the chance. A good lecture captures the audience like any other good performance would. Why do you think the lectures on TED.com are so popular?

I’m going to keep lecturing. In fact, I have a great lecture planned for my Precalculus class tomorrow and I’m excited to give it. I’ll be presenting a beautiful proof, and I’ve got a well-planned powerpoint to go with it. In fact, for this particular lecture, I tell the students they don’t even have to take notes…just soak it in!

Three cheers for the lecture! 🙂

Cryptography

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Sorry for not posting much recently. The reason is simply life’s busyness, partially due to work I’ve been doing for my cryptography class (I’m getting my masters). We just took the final exam last night, so perhaps I’ll be blogging more in the coming month. That being said, I enjoyed the class and learned lots about cryptography.

Much of my time recently was spent working on our final paper. I, with a colleague of mine (Mr. Davis!), chose to investigate secure variants on the Hill Cipher, which is a matrix multiplication encryption scheme first purposed in 1929 by Lester Hill. We implemented the modern versions of the cipher with computer programs and learned quite a bit in the process. Here’s a link to our paper and our presentation.

Also related to cryptography, I’ve been following a bit of the news on Kryptos, a sculpture at the CIA created by Jim Sanborn. On it there are four ciphers, three of which have been cracked. As of the writing of this post, the fourth has yet to be cracked. But just a few weeks ago (Nov 20), “Berlin” was revealed by Sanborn as a clue to the last unsolved section of the cipher. This information was released by the New York Times, but here’s the wired.com article where I first read about it. The clue decrypts 6 ciphertext letters of the last section. This is what cryptanalysts call a ‘crib.’ And most classic ciphers like substitution, shift, and the hill cipher easily succumb to a known-plaintext attack: If you give me some plaintext and the corresponding ciphertext, I can recover the key. It doesn’t sound like this last section is quite as easy as the classic ciphers, though.

One of the other project presentations in our cryptography class was on Kryptos. I also found a presentation on the recently cracked chaocipher very engaging.

 

Answer to Wild About Math Giveaway Problem

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A few weeks ago, I blogged about a calculator giveaway at Wild About Math. Since then, Sol has posted a submitted solution here (and here’s the direct link to the pdf solution by Nate Burchell).

Here’s the problem for those who didn’t see it:

One can create a triangle of consecutive positive integers as follows:

 1
 2  3
 4  5  6
 7  8  9 10
11 12 13 14 15
16 17 18 19 20 21
   . . .

Each row, R, has R numbers. Each column, C, has infinitely many numbers. Rows and columns begin at 1. We define a function F(R,C) for row R and column C such that F(R,C) gives us a value in the triangle. Thus, F(1,1) = 1, F(2,1) = 2, and F(2,2) = 3. Note that F(R,C) is only defined when 1 < = C <= R.

Part 1: Come up with a formula that computes F(R,C) in terms of R and C for any positive values of R and C when 1 < = C <= R. Show your work.

Part 2: Come up with a formula or algorithm that, given a positive integer n, determines R and C.

I also solved the problem and submitted a solution but I didn’t win. Here’s my own solution.