Formalism and Intuition: more problems in teaching math

topOne of the big issues that interests Occupy Math is the teaching of mathematics. In earlier blogs we have looked at teachers being blamed for things that are not under their control, problems with teachers being chained to high stakes standardized tests, math teaching strategies that implement fads without understanding them, teaching topics in silos, and the problem of thinking of math as a form of ritual magic. Parts of this last topic are examined in finer detail in today’s post, where we look at the difference between formal math and understanding math.

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Image of the Week #113

Occupy Math’s image this week is one of the hybrid (2nd/3rd power) Mandelbrot sets.  This one looks like the coast of somewhere.


Math is not Science!


Occupy Math is a member of the College of Physical and Engineering Sciences at the University of Guelph. The fact that the Department of Mathematics and Statistics is in this college makes it seem as if mathematics is one of the sciences — but it is not. Math is often considered to be part of the natural sciences, and it is central to and remarkably useful to the natural sciences, but the techniques, methods, and philosophy of math are different from those of natural science. Technically, based on its techniques, mathematics is the most extreme of the humanities.

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Different flavors of Mathematics



Many people encounter arithmetic and algebra in school but don’t know that there are other types of math. People that know there are other types of math often don’t know that there are dozens of types, even more than the big categories shown in the picture above. If you go on to university, then you also get calculus and possibly differential equations, a tiny sample. This week, Occupy Math looks at the different kinds of math there are. It’s interesting to note that mathematics, as a discipline, is arguably larger than science in the number of different topics it covers. If you would like a post explaining why math is not itself a science, please let Occupy Math know. We have continuous math, like the number line or geometry, we have discrete math that deals with whole numbers or collections of individual objects, and almost all kinds of math have grown, creating more and more general forms of the math known in ancient times.

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Möbius strips, doughnuts, and Klein bottles.

mobiusantsLook carefully at the M. C. Escher picture to the left. If an ant keeps moving forward, he will end up on both sides of the strip he is walking on. This suggests that the strip has only one side (neglecting the edges of the strip) which is pretty cool. The funny walkway is a Möbius strip which does have only one side. Making one-sided objects is only a small part of what the math that makes Möbius strips can do. The same trick can also show you that 1980s video games were being played on the surface of a doughnut, give you a way to understand four-dimensional objects like Klein bottles, and even let us describe the shape of the observable universe.

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Why are there irresolvable arguments?


One of the nice things about the modern world is that arguing over matters of fact has become less of a thing. With the internet at your fingertips, it is possible to look up a lot of things in a flash. On the other hand, it is still possible to argue motive and interpretation and we are still having a lot of arguments. In this post Occupy Math looks into a type of stalemated argument that cannot be influenced by logic or fact. This type of argument is closely connected to a primary foundation of mathematics. One nice thing — once you notice the wrinkle that shows why some arguments cannot be resolved, you get a tool for understanding that the other person may not be an idiot. The problem is, rather, that their assumptions are different from yours.

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