Human Knots

I read a post from The Math Less Traveled entitled, The Mathematics of Human Knots. In this post, the author discusses the game “human knot”, in which every player stands in a circle and grabs the hands of someone else in the circle so that everyone is seemingly tangled up. The goal of the game is then to untangle the knot without any pair of hands letting go of each other. Many people believe that untangling is always possible, however after Dr. Heath’s lecture on knots (and reading this article), I know that this is “knot” true (I crack myself up).

The reason that accomplishing the goal of “human knot” is not always possible is because certain knots exist which cannot be untangled. The blog post provides the example of the figure eight knot:

Just from looking at this knot one can see that the knot couldn’t be untangled without going through itself.

This spring Professor Heath is teaching a course on knot theory. I look forward to taking it.

Math Games

Many of us have fallen victim to the distraction of online games. Whether playing solitaire or harvesting crops on FarmVille (which I can proudly say I have never played), there is probably something else that you should be doing. Well, tonight while I was playing online math games, I decided to turn procrastination into productivity by blogging about it.

The web site that stood out to me was The Truth Tree’s Math and Logic Board. Story problems are the main focus of this particular website (if you had told me I’d be doing story problems for fun 13 years ago I would have said you were crazy). Unlike many “math game” web sites, this one is fairly advanced, with problems ranging from logical riddles, volumes of spheres, counting, topology and more. The only downside is that the answers are not provided. There is, however, an index of similar sites on the page.

Women in Math

There are more men than women with mathematically intensive careers as a result of the choices made by women. This is according to a recent article posted on Math in the News titled Most Women Say No to Working in STEM. The article states that rather than discrimination keeping women from having Science/Technology/Engineering/Mathematics (STEM) jobs, many women simply opt-out of working in a mathematically intensive field. According to a study, while about half of the math undergraduates are women, only about 27% of those with PhD’s in math are female, and fewer still are professors. The article gave a possible reason for this.

Aside from simply choosing to study something outside of the STEM fields, many women still choose to stay home and raise a family rather than pursue their career. As a result, young ﻿female math professors are less likely to work towards tenure than their male counterparts. Furthermore, women are more likely to relocate to accommodate their partner’s work. The article concludes that perhaps more should be done by universities to support women (and men) playing a care-taking role.

A New Way to Teach Math

With the arrival of more advanced calculators (such as Wolfram Alpha), awarding points for correctness is becoming less feasible. What good does awarding points for a correct answer do if one could simply input the problem into a calculator and have the correct answer given to them? This issue is addressed in Teaching College Math’s blog post Shifting Assessment in a World with WolframAlpha.

In this blog post, a new form of assessment is suggested. Instead of asking students to arrive at an answer, simply give them the answer (along with the original question), and have them show how to (correctly) arrive at that answer. This takes the guess-work out of grading a problem with a correct answer and no work. This would also make problems more proof based and less computational, thus introducing proof concepts and integrating them into the work earlier in math curriculum.

I generally like the ideas presented in this blog. I think that having had the right answer before even starting the problem would have been helpful in seeing whether or not I was getting the right answer through my work. It would also force many students to take their learning more seriously, so they actually understand what is going on in their work, instead of simply knowing the shortcuts (in calculus, for instance). Finally, having been slowly introduced to proof techniques early on and building up to a class where proofs are the primary focus would have helped a lot (MATH 317 was not easy, for me at least).

Math and War

In the TED Talk Sean Gourley on the Mathematics of War, using mathematics to track and interpret war is discussed.

Sean Gourley, a physicist from New Zealand, began his project by assembling a team of scientists, economists, and mathematicians. They then used various media sources to obtain information on the war in Iraq, and then used a computer to filter all of it and pull out the bits in which they were interested. Using this data, the distribution of attack sizes in Iraq was produced and graphed. The vertical axis was frequency of attacks, and the horizontal axis was number of deaths. For instance, the ordered pair (47,1) would mean there were 47 attacks with 1 casualty.

They then did the same technique for other wars, and surprisingly, the same distribution emerged. Expanding their study further and further, every war produced a similar distribution. Furthermore, each war had a slope that was within .75 of the mean (which was -2.5).

Using this data, the team produced the equation $P(x)=Cx^{-\alpha}$, where $P$ is the probability, $x$ is the number killed, $C$ is a constant, and $\alpha$ is the slope of the line. The group theorized that this is a result of necessity when a group is fighting against a much stronger force. In order for their resistance to exist, it has to follow the discovered pattern.

Gourley concludes that we may be able to use this model to interpret the progress of a war, and in theory try to push it in the right direction, whatever that may be.