A view in the quintic Mandelbrot set.
This post is inspired by an excellent blog post on why algebra is an awful experience for many students. This problem with dry, boring algebra has led to a call to stop teaching algebra to many students. This seems to Occupy Math to place the cart before the horse. If students are having trouble succeeding because of math, the math should be taught in a more effective fashion. Getting rid of it is like saying “horses are too much trouble — everybody walk and carry heavy loads on your back!” Beyond this, Occupy Math finds the idea that people should only study algebra if it enhances their economic well-being, and that for many students it will not, to be racist, classist, and contrary to having a democratic society. The students that are designated to “succeed” by not having to study math are disproportionately inner city, brown, and from the bottom end of the economy. The whole idea of making math the property of the privileged few is the opposite of what the coming complex, perilous world needs. This post discusses some ideas for making algebra less of a pain and more of a routine mental tool.
Leafspiral! A conjugated double cubic Julia set.
This post looks at how students just starting university can think math is hell while the professor looks at the papers he is grading in stunned disbelief because the students appear to have learned nothing in their previous twelve years in school. Occupy Math usually teaches a first year university class, either calculus or calculus mixed with physics. The material is not too hard in an absolute sense, but for many of the students it is quite hard. This post is about the reasons for the different perspective: the math professor thinks the material is pretty basic, but students — who finished a complete high school course of study in mathematics! — think it’s hard-to-awful. It turns out there are several reasons for this disjunction of experience. They include many factors including really bad design of the high school classes, active parental sabotage, administrative handicapping of teachers, and a failure to teach effective study techniques.
A conjugate Julia set at high iteration number — very spikey spirals.
One of the big problems with teaching arithmetic is that we carefully structure the learning process to be boring. Drill is effective, but only if the students do it, and some will not. In this post we present a challenging series of puzzles (and techniques for creating more puzzles) to wash some of the boredom out of learning arithmetic. The picture below shows the solution to a viral puzzle called four fours. The puzzle asks the student to make the numbers 0 to 20 out of four copies of the digit four. The term digit is used judiciously because it’s not quite the number four. Notice that “44” counts as two fours. The number of other symbols used is unlimited and sum, difference, product, quotient, power, square root, factorial, and parenthesis are all available. The solutions shown are far from unique. Consider 4+4+4+4=16 or 4×(4/4+4)=20, for example. This post extends the problem and provides keys for five fives and six sixes as well as proposing additional activities.
A very interesting spiral in the conjugated cubic Mandelbrot set.
There is a crisis in the price of textbooks. Occupy Math paid eighteen dollars for the calculus book he used for his first two courses in calculus. The calculus book used for the first-year for-majors calculus course at Occupy Math’s institution costs more than ten times that, though the price varies from year to year. Worse, after you buy many textbooks, you have to cough up a hundred dollars for access codes to let you get to the on-line material that comes with the book. The professor may well assign parts of the on-line material, from exercises to pre-generated homework problems, so access to the on-line material is not optional. With a book, roommates or study-buddies might buy one copy and share. The on-line materials track one person’s progress and grades, so sharing is impractical. This post is about the mechanisms behind insane pricing and some resistance strategies.
A conjugate Mandelbrot set, rich in spirals.
Occupy Math works with digital evolution on a number of projects including evolving parameters that generate interesting fractals. The advantage to doing this with a computer, assuming you can come up with an automatic function that at least sort of measures “this looks good”, is that you can sort through billions of fractals per hour. One of these is shown at the top of the page. The disadvantage is that people are much better than any of the automatic functions we have found so far at spotting cool fractals. If we use people, though, they burn out way before looking a even a paltry million fractals. This is the phenomenon of user fatigue. This post is about a way to let computers and people collaborate on a project, drawing on the strengths of each. Computers can evaluate huge numbers of fractals to find ones that might look good. Humans have a much better ability to judge which fractals are actually beautiful.