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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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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An Activity Index

Ofactoryccupy Math looks at many different situations where mathematics (or the lack of mathematics) are important to people. One of our many threads is activities and information for teachers and parents. This post provides an index to these activity posts and then an index to some of the informational posts that might give helpful background. One thing to keep in mind — if there is a type of activity you might like to see, let Occupy Math know (e-mail

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Figuring out when you can do a puzzle.

This week’s Occupy Math looks at a type of puzzle where you want to fill a rectangle with a shape. We will be using the L-shaped 3-square polyomino, used to fill a 5×9 rectangle below, as our example shape. The goal is to figure out every possible size of rectangle that can be filled with this shape. If you are constructing puzzles for other people — e.g., your students — knowing which problems can be solved gives you an edge. The post will not only solve the problem for our example shape, but also give you tools for doing this for other shapes. The answers, and the tools, are at the bottom if you don’t feel like working through the reasoning.


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Grid Games: Reinforcing Geometry with Puzzles

This post is the next in Occupy Math’s series of activities for teachers, students, and parents. There is a good deal of basic algebra and geometry that can be practiced with puzzles on grids. These puzzles are intended for grades 6-10, depending on how much the students know about polygons and finding areas. You can use standard graph paper to supply the grids for these puzzles. We start with the following puzzle. If we assume the grids have spacing one, draw a square whose corners are on grids and that has an area of exactly five.


A student will have a natural desire to draw a square with vertical and horizontal edges — but the squares that can be drawn that way on this grid have areas of 1, 4, 9, and 16 — that is all. How do we get five? Read on to find out.

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It all adds up! (A puzzle)


In this post we look at the problem of finding a sum of consecutive numbers that has a specified value. This is one of Occupy Math’s new activity posts. Let’s look at an example. “Find a sum of consecutive whole numbers that add to 100” could be solved by 18+19+20+21+22=100. This sort of problem is good for practicing arithmetic while also building the logic muscles. Best of all, if you are a parent or teacher, this post will show you how to find exactly which of these problems have answers, which don’t, and for the ones that do have answers, what all the possible answers are.

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The Riddle of the Sphinx

sphinxThis post is about an activity that helps students practice recognizing which numbers are factors of others. It is intended for grade five and above. The activity has several different forms and we will comment on which are harder as each variant is described. In order to run this activity, the parent or teacher will play the part of the Sphinx. If you have an Egyptian head dress or other prop, that helps set the mood.

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Coloring Book Math: An Activity


This post is about an activity — so it begins with the ages the activity is intended for.

  1. Ages 3-6: just color the pictures, have fun!
  2. Ages 7-9: color the pictures but also try to answer the first question. A parent or teacher should help, and maybe look at this article on rotational symmetry.
  3. Ages 10-12: color the pictures and use them to answer all the questions if you can. Students may need help from a parent or teacher.
  4. Ages 13+: just color the pictures. Answer the questions if you’re interested.

This week’s Occupy Math is the first post in a new category: materials and activities. In response to a reader comment, Occupy Math is working up math-related activities. This one is a coloring book intended to introduce symmetry. A colored version of one of the images from the book appears above. The first page of the book asks five questions; all the other pages are intended to be colored. This post discusses and answers the five questions for a teacher or parent that might be using the activity.

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Prime Numbers and Teaching Fractions


This week’s post is a follow-up to a post objecting to the way fractions are taught post from a while back. This is also Occupy Math’s third post about prime numbers. It should be much more down to earth than the first and, unlike the second, there are no insects or implications for ecology. This post explains why students who know about prime numbers will find it easier to do arithmetic with fractions. Being able to find the prime factors of numbers gives those numbers character: prime factors are like personality traits. Another way to say it is that numbers with a prime factor in common form a sort of a tribe with common characteristics. Also, there are some games you can play with prime numbers near the end of the post.

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Secrets of the Guild: How Occupy Math found one.

The phrase secrets of the guild was used by Bertie Wooster to describe the reason that his manservant, Jeeves, could not divulge the ingredients of his morning pick-me-up. This post is intended to give pieces that could be used for discovery learning about basket curves, a modified type of petal curve. Occupy Math has looked at petal curves before, to make flowers. This post highlights a “secret” in the form of a not-completely-obvious pattern that permits students to make complex and interesting pictures.

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