Fifty is a big, round number and so Occupy Math decided to do something a little special — this is one of the images we’ve seen before, a cubic Mandelbrot set with quadrant convergence, but this time, instead of showing it a 56 iterations, Occupy Math has made an animation. This shows how the fractal develops, fits and starts and a final leafing out. Enjoy!
This week, Occupy Math looks at math tests — and some other tests — from the perspective of fairness. It turns out that questions that test the same skills can have extremely adjustable difficulty levels. There is also the issue of tests designed for failure. For that, we will look at some examples of cosmically unfair questions. On the issue of math tests, this post discusses the differences between easy and hard questions for the same topic. Occupy Math can probably dial the average grade on a test across a range of 20% by playing with the way questions are phrased. All this will give you some perspective on how to survive a test (it helps to be able to spot structurally hard questions) — but mostly the message is this.
Fairness is largely an illusion and enforcing it is close to impossible. Hope for competent teaching and mercy instead.
This is a hybrid Newton-Julia fractal using quadrant convergence. There are the “feathery worm” fractals.
In this week’s Occupy Math we look at a math problem that can be explained in less than ten minutes — that still stumps every mathematician who has ever looked at it. This problem is also a marvelous place for students to explore and look for patterns (some problems for students are near the end of the post). The problem is based on this rule: “If a number is odd, triple it and add one, but if it is even, divide it in half”. Not a hard rule, but it is the basis of an unsolved problem: “if you start with any positive whole number, do you eventually get to the number 1?” This question has several names. One of the most used is the Collatz conjecture (follow the link for many other names including the very common “3n+1 problem”). A number of prominent mathematicians, including the incredible Paul Erdős, have expressed the opinion that this problem is too hard for us in our current mathematical infancy. The theory also floated briefly, after multiple math departments were consumed by this problem, that it was created by the KGB (the Soviet Union’s spy agency) to stymie mathematics research in the west.
How can such a simple problem be so incredibly hard?
This is a triple-cubic Julia fractal with pure imaginary constants. These things have trilateral symmetry and this one also has a lot of busy detail. Enjoy!
Teaching is not an easy job to do well. Math teaching is somewhat more difficult than teaching many other subjects because its the bogey-man of school topics. This suggests that we should work as hard as we can on making teachers jobs less difficult and, maybe, we should spend a little more effort on cutting teachers who have math in their portfolio a break. Logical and obvious, right? Instead what happens is that we assume problems with teaching are the fault of the teachers, punish them, and try and create rules that force them to do a good job. Occupy Math is a professor of mathematics with a research specialty in the creation of sets of rules for games and controlling systems — he is sure this cannot work. A set of rules that forces good behavior does not exist in a situation more complex than a game of Candy Land.
Evidence suggests that simply treating teachers as professionals and letting them teach will yield better results.
This is another evolved Julia set using quadrant convergence. It has an interesting sort of supported spiral and fairly complex structure. Enjoy!
This post was inspired by a wonderful video about the effects on Yellowstone National Park of returning wolves to the ecosystem after an absence of seventy years. Wolves are keystone predators and kill and eat other animals. The effect of introducing them was to … sharply increase both the diversity and number of animals in the park and the number of green plants in the park; the wolves even moved where the rivers flowed.
As you will find if you watch the video, this is a wonderful example of a trophic cascade. The wolves showed up and the ecosystem’s health increased a whole bunch. This is also an example of a mathematical phenomenon called a nonlinear system. Systems we understand easily tend to be linear — the response of the system is proportional to the input. If you jump on the bed harder, you bounce higher, unless the ceiling introduces a painful non-linearity by impacting your head. Non-linear systems are far more common than linear ones, but harder to fathom.
The mathematics of ecology in highly non-linear. This is why ecology is more like whack-a-mole than science, sometimes.
This is another evolved, generalized Julia set — a paltry two complex parameters — but much to my surprise this fractal literally goes to infinity and beyond. For symbolic values of infinity.
You’ve probably been lied to. It might be a direct and intentional lie, as with high school students Occupy Math met at a girls math conference (not Occupy Math’s name for it). Teachers told these women that they couldn’t do math — often before grading any of their work. The lie might be implied, as when the person teaching you math in fourth grade was clearly scared of it herself. We also have a distressing cultural context which paints mathematical ability — or even just being smart — as being anti-feminine. Men are not immune to this sort of idiocy either. Occupy Math has encountered far too many students who, when faced with a difficult concept in math said “I can’t do math, I’m not smart!” With many of these students it turns out they could do the math and they were pretty smart. For some reason, however, running away was a much more comfortable proposition than, oh, saying “can you explain that again a different way?”
Occupy Math finds the term “math person” pretty frustrating. If you think you’re not a math person then you’re taking a whole hamper full of issues and putting a two-word sticky note on it. Do you mean you will never win a prize for mathematical research or do you mean you can never learn to add? “Not a math person” lumps together a huge spectrum of ability levels and training in one small place. Here’s the scoop: unless you meant the “not winning a prize for original research in math” extreme, you’re probably wrong about not being a math person. I don’t mean you’re good at math right now or that you’re not afraid of it. I mean that it is incredibly unlikely that you cannot do math at some basic (but useful) level and probably you can do more. In this week’s post we will look at both fear and the case for trying to overcome that fear.
At this point some of Occupy Math’s readers are thinking “But I don’t want to be a math person!”