Tag Archives: algebra

February 27, 2012, 8:00 am

Encountering abstraction with clickers

I had the great pleasure this weekend of leading a session at Math In Action, which is Grand Valley’s annual K-12 educators’ conference. My session was called “Classroom Response Systems in Mathematics: Learning math better through voting” and was all about the kinds of learning that can take place in a class where active student choice is central and clickers are mediating the voting. (Here are the slides.)

It always seems like a bait-and-switch when I do a “clicker” workshop, because although people come to learn about clickers, I don’t really have much to say about the technology itself. As devices go, clickers are about as complex as a garage door opener, and in fact they work on the same principle. There’s not a lot to discuss. So instead, we spend our time focusing on the kinds of pedagogy that clickers enable — which tends to excite teachers more than technology does.

The a…

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December 12, 2011, 7:45 am

Columnar transpositions: Looking at the initial cycle

Welcome to Math Monday! Each Monday here at Casting Out Nines, we feature a mathematics-themed article. Today’s is a new installment in an ongoing virtual seminar on columnar transposition ciphers.

Let’s return to our ongoing look at the columnar transposition cipher. In the last article, we introduced the notion of cycles. A cycle can be thought of as a cluster of points which are moved around in a circular nature by a permutation. All permutations — including the permutation implemented by a columnar transposition cipher — break down into a product of disjoint cycles, and we can determine the order of a permutation (the smallest nonnegative power of the permutation that returns it to the identity) by finding the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.

Since one of the main questions we are asking about CTC’s is about their order,…

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November 28, 2011, 7:45 am

Cycles, and the cycle decomposition of a permutation

Last week’s installment on columnar transposition ciphers described a formula for the underlying permutation for a CTC. If we assume that the number of columns being used divides the length of the message, we get a nice, self-contained way of determining where the characters in the message go when enciphered. Now that we have the permutation fully specified, we’ll use it to learn a little about how the CTC permutation works — in particular, we’re going to learn about cycles in permutations and try to understand the cycle structure of a CTC.

First, what’s a cycle? Let’s go back to a simpler permutation to get the basic concept. Consider the bijective function \(p\) that maps the set \(\{0,1,2,3,4, 5\} \) onto itself by the rule
$$p(0) = 4 \quad p(1) = 5 \quad p(2) = 0 \quad p(3) = 3 \quad p(4) = 2 \quad p(5) = 1$$
If you look carefully at the numbers here, you’ll see that some of…

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November 21, 2011, 7:45 am

A formula under the hood of a columnar transposition cipher

It’s been a couple of Math Mondays since we last looked at columnar transposition ciphers, so let’s jump back in. In the last post, we learned that CTC’s are really just permutations on the set of character positions in a message. That is, a CTC is a bijective function \( \{0, 1, 2, \dots, L-1\} \rightarrow \{0, 1, 2, \dots, L-1\}\) where \(L\) is the length of the message. One of the big questions we left hanging was whether there was a systematic way of specifying that function — for example, with a formula. The answer is YES, and in this post we’re going to develop that formula.

Before we start, let me just mention again that all of the following ideas are from my paper “The cycle structure and order of the rail fence cipher”, which was published in the journal Cryptologia. However, the formula you’re about to see here is a newer (and I think improved) version of the one in the…

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November 7, 2011, 7:45 am

Math Monday: Columnar transposition ciphers and permutations, oh my

I hope you enjoyed Ed’s guest posts on NP-complete problems on TV the last couple of Mondays. It’s always great to hear from others on math that they are thinking about. This week it’s me again, and we’re going to get back to the notion of columnar transposition ciphers. In the first post about CTCs, we discussed what they are and in particular the rail fence cipher which is a CTC with two columns. This post is going to get into the math behind CTCs, and in doing so we’ll be able to work with CTCs on several different levels.

A CTC is just one of many transposition ciphers, which is one of the basic cryptographic primitives. Transposition ciphers work by shuffling the characters in the message according to some predefined rule. The way these ciphers work is easy to understand if we put a little structure on the situation.

First, label all the positions in the message from \(0\) to …

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September 19, 2011, 8:00 am

Math Monday: What is casting out nines?

http://www.flickr.com/photos/artnoose/

Last week in this post, I asked for requests for math topics you’d like to read about. One person wrote in and asked:

Why don’t you enlighten us about the name “Casting Out Nines?” I learned a system in grade school with the same name –it was a way of checking multiplication and long division answers. Long before calculators.

A review please?

OK then. Casting out nines is an old-fashioned method of checking for errors in basic arithmetic problems (addition and subtraction too, not just multiplication and division). Here’s how it works, using addition as an example.

Let’s suppose I’m trying to add 32189 to 87011. I get a sum of 119200. But did I make a mistake? Do the following to check:

  1. Take the first number, 32189, and remove — “cast out” — any 9′s…

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October 25, 2010, 8:52 pm

Questions about an enVisionMATH worksheet (part 1)

The 6-year old had Fall Break last week, so no homework and no enVisionMATH-blogging for me. Tonight, however, she brought home a new worksheet for her weekly homework, and a couple of things caught my eye. I thought I’d throw those out there to you all, along with a question or two, as a two-part blog post.

For the first post, take a look at this (click to enlarge):


Questions:

  • In your own words, preferably those that a smart 6-year old could understand, what is the basic principle that this page is trying to get across?
  • What technique does this worksheet want kids to use when doing the Algebra problems?
  • What’s your opinion about the principle/technique you think the worksheet is trying to communciate? Reasonable? Natural? Likely to be useful, or used frequently later on?

 

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August 31, 2010, 9:33 pm

In the trenches with enVisionMATH

It’s been back-to-school time for everybody in our household (hence an excuse for the light posting). We started classes at the college today, and last week the 4.5-year old went back to preschool full-time and the 6.5-year old started first grade. (The 1.5-year old is rocking the local daycare.) One of the biggest changes for the kids is for our first-grader, Lucy, since she has real homework for the first time. It’s not much; the expectation is about 20 minutes a night, Monday through Thursday. Some of that homework is math, which I was very excited about — but then that excitement turned to alert caution when I learned my daughter’s class was using enVisionMATH.

I wrote this post on enVisionMATH almost three years ago, basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with…

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May 12, 2009, 1:02 pm

Four things I used to think about calculus, and what I've replaced them with

Show convergence of Riemann sum for all sample...
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I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and  as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.

When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:

  • I thought that competency in calculus consisted in…

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March 1, 2009, 4:40 pm

Keeping things in context

Part of Article 131 in the first edition (1801...
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I’ve started reading through Stewart and Tall’s book on algebraic number theory, partly to give myself some fodder for learning Sage and partly because it’s an area of math I’d like to explore. I’m discovering a lot about algebra in the process that I should have known already. For example, I didn’t know until reading this book that the Gaussian integers were invented to study quadratic reciprocity. For me, the Gaussian integers were always just this abstract construction that  Gauss invented evidently for his own amusement (which maybe isn’t too far off from the truth) and which exists primarily so that I would have something to do in abstract algebra class. Here are the Gaussian integers! Now, go and find which ones are units, whether this is a principal ideal domain, and so on. Isn’t this fun?

Well, yes, actually it is fun for me, but that’s because I like a…

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