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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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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Weighted Averages – a Useful Tool

We all know what the average of a bunch of numbers is: you add up the numbers and divide the total by the number of numbers you added up and you get something in the middle. This sort of average assumes that all the numbers you are adding up are equally important. There are many situations where the numbers are not equally important. This is why we have weighted averages — a weight is another number that says how important a number is in the group of values you are averaging.

The simplest example is when the weights are positive and add to one. This is something that often happens with grades. The teacher might say that “The final grade is 50% of your homework grade plus 30% of your quiz grade plus 20% of your final exam grade”. In this case the weights are 0.5, 0.3, and 0.2. If a student had 86% on homework, 72% on quizzes, and a 91% on their final then their grade would be 0.5×86%+0.3&times 72%+0.2&times 91%=82.8% for their final grade.

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Mathematical Thinking and the Law

TWTPhis Occupy Math is partly based on a More Perfect podcast entitled One Nation, Under Money. The More Perfect series tries to understand and explain US Constitutional law and this podcast is a doozy that looks at a very mathematical approach to the commerce clause:

The United States Congress shall have power “To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes.”

– Article 1, Section 8, Clause 3, United States Constitution

The application of mathematical thinking to this clause increased the power of the clause far beyond what someone reading it for the first time might think it had. Consider this while you’re reading the rest of the post: is the interpretation of the clause that arises anything like what the people that wrote the clause thought of while they were writing it?

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This week we delve again into the evolution of virtual robots. Last time this topic came up, the robots were symbots with continuous moves. This week we have the amoeba-like morphbots, like the one shown above. These robots are controlled by evolved computer code, very different from the simple neural nets that control symbots. We are going to look at how to correctly set up digital evolution and also at how the skill of the robots progresses the longer they evolve.

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Voronoi Tilings: Mathematics Serving Art

Ttile01his week’s post looks at a mathematical structure, the Voronoi diagram, that lets you generate some pretty cool images. An example appears at the beginning of the post. Occupy Math will also show you how to modify Voronoi diagrams to get even neater pictures. The idea is simple: you start by picking special points inside a square (or any shape) and then color the square based on which special point the point you are coloring is closest to. Regions that are the same color are all the points that are closest to the same one of the special points.

In the picture above Occupy Math is assuming the square wraps around top-to-bottom and left-to-right when computing “closest”. This means that if you use the image as a background it will tile correctly. Notice that a lavender tile of the diagram appears at all four corners of the square.

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Another Collection of Math Jokes

Since Occupy Math’s readers liked the last humor post, we will try again.

☆ The problem with math puns is that calculus jokes are all derivative, trigonometry jokes are too graphic, algebra jokes are usually formulaic, and arithmetic jokes are pretty basic. But I guess the occasional statistics joke is an outlier. (with thanks to Elizabeth K.)

☆ I see you have graph paper. You must be plotting something.

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AI Apocalypse worried? Collaborate with you Computer Instead!

In a number of Occupy Posts, we’ve looked at fractals. A long time ago, in the Goldilocks information post, we looked at the problem of having too much or too little information. Today’s post reveals one of Occupy Math’s secrets: how to let the computer look for interesting fractals on its own. The word “interesting” is chosen carefully because the fractals located this way are not beautiful or elegant (yet). They are just interesting in a very specific way. The use of this is to turn a berjillion fractals, most of which are not that good, into a short list that a human can select from or even brush up a bit. This is an example of a type of computer code where the computer is a computational collaborator.

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