Tag Archives: mathematics

January 14, 2012, 2:22 pm

Programming for all?

Audrey Watters writes in Hack [Higher] Education that maybe it’s time for programming to join critical thinking and effective writing as part of the body of required knowledge for all university students:

But I will posit that all students should learn programming, whether they plan to become programmers or not. Many universities already require students take composition in order to graduate. Perhaps it’s time for programming — “the new literacy” — to become a requirement too?

I don’t mean that every student needs to learn C++ or Python or Perl or Java or Ruby. But I do think everyone needs to know how the Web works — how search engines operate, for example, and what’s “server side” and what’s “client side” and why the difference matters. Everyone needs to know some HTML (a mark-up, not a programming, language I realize). And with the move towards the fifth revision of the HTML…

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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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October 24, 2011, 7:30 am

Math Monday: TV Lawyers Solve NP-Complete Problem (part 2)

This is the second installment of a two-part article from guest blogger Ed Aboufadel. Thanks again, Ed, for contributing.

In Part I, we learned of an instance of the NP-complete problem subset-sum [1] that was solved by three lawyers on an episode of the USA Network show Suits [2]. The problem was to go through a set of deposits made to five banks in Liechtenstein and find a subset of deposits, where the total of the deposits was $152,375,242.18. Described as “simple mathematics” by one of the lawyers, the team solved the problem in a relatively short length of time. They couldn’t use a quick approximation algorithm for subset-sum, since they needed the sum to be exactly equal to their target amount. So, were they just lucky, smarter than the rest of us, or did they do something practically impossible?

Consider the following “back of the envelope” calculations. First,…

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October 17, 2011, 7:30 am

Math Monday: TV Lawyers Solve NP-Complete Problem (part 1)

For the next couple of weeks, Math Monday here at the blog will feature a guest blogger. Ed Aboufadel is Professor of Mathematics and chair of the Mathematics Department at Grand Valley State University, where I work. He’ll be writing a two-part series on a neat appearance of an NP-complete problem on network TV, adding yet another data point that mathematics is indeed everywhere. Thanks in advance, Ed!

On the new USA-network TV series Suits [1], Harvey Specter is a senior partner at the law firm of Pearson Hardman, and Mike Ross is his new associate. Mike never went to law school, but he combines a photographic, elephantine memory with near-genius intelligence to fake it well. Harvey is in on the deception, but none of the other partners know. During the eighth episode of the first season of Suits (broadcast August 11, 2011), Harvey and Mike, working with Louis Litt, a…

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October 11, 2011, 7:30 am

Is dependence on technology the real threat?

I came across this Seymour Papert quote over the weekend, the best part of which is below. In context, Papert is speaking about effecting real change in the content of school mathematics, and he focuses particularly on the teaching of fractions:

One theory [among educators about why we should teach fractions in school] was that manipulating fractions was actually closer to what people needed back before there were calculators. So a lot of school math was useful once upon a time, but we now have calculators and so we don’t need it. But people say that surely we don’t want to be dependent on the calculator. To which I say, Look at this thing, these eyeglasses, that make a dramatic difference to my life and the life of everybody who reads or looks at any tiny detail. Once upon a time we would have been crippled, severely handicapped. Now we’ve got these and we don’t need to go …

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September 28, 2011, 4:00 am

Midweek recap, 09.28.2011

Good stuff from the internet this past week:

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September 21, 2011, 9:00 am

Midweek recap, 9.21.2011

Interesting stuff from elsewhere on the web this week:

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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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September 15, 2011, 8:52 pm

Using clickers for peer review of proofs

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

Right now I’m teaching a course called Communicating in Mathematics, which serves two purposes. First, it’s a transitional course for students heading from the freshman calculus sequence into more theoretical upper-level math courses. We learn about logic, how to formulate and test mathematical conjectures, and we spend a lot of time learning how to write correct mathematical proofs. And therein is the second purpose: The course is also labelled as a “Supplemental Writing Skills” course at Grand Valley, which means that a large portion of the class, and of the course grade, is based on writing. (Here are the specifics.) It’s a sort of second-semester, discipline-specific composition class. (Students at GVSU have to have two of these SWS courses, each in different…

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September 7, 2011, 7:30 am

Midweek recap, 9.7.2011

From around the interwebs this past week:

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