Friday, November 23, 2012

Agon -- the classic game you've probably never heard of

I don't know exactly what happened to Agon, but I'm pretty sure those damned orthogonalists had something to do with it. The game was all the rage in Victorian England where it was appreciated for its simple rules but surprisingly complex strategies. (the Victorians were also big fans of Lewis Carroll's Doublets, thus showing remarkably good taste in diversions.) The game was and is remarkably challenging and enjoyable but early in the Twentieth Century it faded away, perhaps due to the hegemony of orthogonal game boards.

You can find a complete set of rules on my game site. You can also buy boards there but obviously waiting for delivery would undercut the whole 'here's a game for this weekend' concept so I've included two JPEGs that you can print off if you can't find a suitable substitute (lots of games use a 6x6x6 hexboard so locating one shouldn't be that difficult).

The game is extraordinarily easy to learn. Each player starts out with six pawns and a queen spread out around the edge out the board.

A piece can either move around a concentric hexagon or go toward the center. The object is to get your pieces arranged like this (black wins):

'Capturing' is done in the style of many older games by placing two of your pieces on either side of the opponent's piece. I put the word in quotes because a captured piece is not removed from the board. Instead, it is moved back to the outer ring. Agon is therefore entirely a game of position. Novice chess players have a tendency to play for points and measure how well they're doing by how many of their opponent's pieces are lined up by the side of the board. Learning Agon can help break them of some bad habits.

I first came across Agon in David Parlett's Oxford History of Board Games -- an excellent resource if you're thinking about teaching a math class and not a bad read if you just enjoy games. Parlett is also a game designer of some note so he brings a lot of insight to the discussion.

Originally posted in West Coast Stat Views

Wednesday, November 21, 2012

Lesson Plan -- Homemade Pi

When I was teaching high school math, I would often take my classes to the computer lab so the students could use Excel (or an open source alternative) to apply some of the concepts they were covering. Here's an example that would be appropriate for 8th grade* and up. It employs the following skills and concepts:

Area of a circle


Area of a square 

Inscribed figures

The Pythagorean Theorem



Basic algebra

Basic computing

Monte Carlo techniques 

Start with the following 

The area of the circle is pi-r-squared.

The area of the square is 4pi-squared.

The proportion of the square that's shaded is (area of the circle)/(area of the square)

Do a little algebra and you get p = pi/4 or pi = 4p

If you picked points in the square at random, the number in the circle divided by total number would converge on p

Since the figure is symmetric along the vertical and horizontal axes, the shaded part of a quadrant should also be p.

Now pick a radius. I used   r = 3 here but make sure to mix it when assigning this project and use different radii (but not one -- you don't want a radius that equals its own square when presenting examples).

Have the students create x and y coordinates using 


Then use a conditional based on the Pythagorean Theorem that takes the value 1 if the point is in the circle.

Your estimate of pi is four times the average of that field.

As with all Monte Carlo based lessons, have the students start with a small sample and move up until they start getting reasonable answwers.

I realize this may seem like a bit much but remember:

1. These spreadsheet skills (functions, conditions, random number generators) should already be familiar to the students. 

2.  Kids have a way of surprising you (and sometimes in the good sense)

* Some people out there are probably saying this is too advanced for 8th graders. You know your kids best but I would encourage you to give it a try. They might surprise you.

Tuesday, November 20, 2012

Teaching yourself mathematics -- a footnote for future posts

I'm pretty sure I'm going to be making this claim repeatedly so I might as well take a few minutes to put it down in a linkable form for future use.

Of all the subjects a student is likely to encounter after elementary school, mathematics is by far the easiest to teach yourself. With the appropriate attitude and assumptions, adequate motivation and a simple and easily mastered set of skills the majority of students can take themselves from pre-algebra through calculus.

What is it that makes math teachers so expendable? Part of the answer lies in mathematics position on the fact/process spectrum. Viewed in sufficiently general terms, all subjects start with giving the student a set of facts and ideally end with the student performing some process using those facts. In subjects like history and to a slightly lesser extent, science, most of the early stages of mastering the subject center around absorbing the facts. On the other end of the spectrum, subjects like music, writing and mathematics involve a relatively small set of facts*. Students studying these subjects tend to focus primarily on process almost from the beginning.

Put another way, at some point all disciplines require the transition from passive to active and that transition can be challenging. In courses like high school history and science, the emphasis on passively acquiring knowledge (yes, I realize that students write essays in history classes and apply formulas in science classes but that represents a relatively small portion of their time and, more importantly, the work those students do is fundamentally different from the day-to-day work done by historians and scientists). By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians.

Unlike music and writing, however, mathematics starts out as a convergent process. It doesn't stay that way. By the time a student gets to upper level math courses like real analysis or applied subjects like statistics ** there are any number of valid proofs for theorems and approaches to problems, but for anything before, say, differential equations, most math problems have only one solution and students are able to quickly and accurately check their work. Compare this to writing. There is no quick or accurate way to gauge the quality of a piece of prose or, worse yet, verse. Writers spend years refining their editing skills and even then they still generally seek out other critics to help them assess their own work.

This unique position of mathematics allows for any number of easy and effective self-study techniques. One of the simplest is the approach that got me through a linear algebra section taught by the worst college level instructor I have ever encountered (and that, my covers some territory).

All you need is a textbook and a few sheets of scratch paper. You cover everything below the paragraph you're reading with the sheet of paper. When you get to an example, leave the solution covered and try the problem. After you've finished check your work. If you got it right you continue working your way through the section. If you got it wrong, you have a few choices. If you feel you basically understood the solution and see where you made your mistake, you might simply want to go on; if you're not quite sure about some of the steps in the solution, you should probably go back to the beginning of the section; if you're really lost, you should go back to the preceding section and/or the previous sections that introduced the concepts you're having trouble with.

Once you've worked through all the examples, start on the odd numbered problems and check your answers as you go. If you're feeling confident, you can skip to the difficult problems but if you make a mistake or get stuck you should probably go back to number 1.

Don't get me wrong. I'm not saying this is the only technique, let alone the best, for teaching yourself mathematics. Nor am I suggesting that we make a practice of dumping student in sink-or-swim situations. I think we should provide students with the best teachers and support system possible, but even under those conditions, the internal resources needed to teach yourself mathematics are tremendously valuable to all students and are absolutely essential to anyone who has to use sophisticated analytical reasoning.

Tragic postscript: In what I can only assume is an idiotic attempt raise standards, most books have stopped giving answers to odd-numbered problems. Under the old system you would assign odd problems when you wanted the students to be able to check their work and even problems when you wanted to make sure they weren't just looking answers up in the back of the book. It was a simple, flexible, and effective system that encouraged students to be independent and resourceful. No wonder it was such a prime target for reform.

* I heard a story (possibly apocryphal) of a professor who walked into an upper level math class, wrote properties of real numbers on the board, told the class that was all they needed to know to prove all the theorems in the book, then walked out.
** The relationship between mathematics and statistics is particularly complex, far too complex to discuss in a blog post.

Originally posted at West Coast Stat Views

Sunday, November 18, 2012

The Number-line Code

This one grew out of some time I spent recently helping a family first-grader with her subtraction homework. I made up some sheets with numbers and letters. My original thought was just to make the practice sheets more interesting by having the answer spell out a word, but I noticed my tutoree (who naturally has very good math genes) had, without prompting, started using the code key as a number line to figure out the answers.

The more I thought about it, the more I liked the idea of using this approach for a wide range of problems. For the very early grades, it teaches numbers and letters. It has great appeal for kids (particularly if presented with the right air of mystery -- I'd suggest a pirate motif). It reinforces the number line concept and the essential idea of looking things up in tables.

Here are some sample problems

For those learning to read, present it as a straightforward code:

PIRATES LOVE 7-15-12-4

For slightly more advanced students, replace the numbers with problems


10 - 3

10 + 5

6 + 6

6 - 2

For even more advanced students, make it a more explicit number line exercise






You can even play with functions by using the old SAT trick of using a circle to represent x+1 and a triangle to represent x-1 (or any other functions you can think of).

triangle 8
circle 14
circle 11
triangle 5

I'd suggest using this technique frequently enough to keep the familiarity high. It also offers extensive opportunities for teaching across the curriculum.

Anyone have any other ideas?

Thursday, November 15, 2012

Topologists at play -- the game of Sprouts

It's important to have students think deeply about math in both structured and unstructured ways (I have a guilty feeling that I ought to say more about this, but that will have to wait for a future post). It's the unstructured part that tends to cause problems. That's one of the reasons I liked to make games part of my lessons when I was a teacher.

Games (at least the kind I recommend) require a great deal of focus -- you have to think about what you're doing or you won't do well -- and they encourage exploration and a playful attitude to the material. All of these things help build mathematical intuition.

On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).

The rules are simple:

1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.
2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.
3. The lines can be straight or curved but they can’t cross themselves or any other lines.
4. Each dot can have at most three lines connecting it.
5. When you draw a line put a new dot in the middle.
6. The first player who can’t draw a line loses.

You can find a couple of sample games here.

Originally posted at Education and Statistics

Just what the world needs... another blog

Particularly from someone who already has a backlog of unfinished posts at another blog. There is a method to the madness here. I've been doing most of my blogging for a site called West Coast Stat Views (formerly Observational Epidemiology) which features analytic takes on various topics including education.

Back in my salad days, I taught high school math and English and college math and statistics and I accumulated a stockpile math games, puzzles, tips, exercises and the like. Some of these fit in at the stat blog but many don't. Besides I wanted to get this material to teachers, parents and math buffs, some of whom wouldn't have much interest in most of the topics covered on the other blog.

This site also gives me a chance to try a different approach to blogging. Since few of these posts will be topical, I'll be making aggressive use of the scheduling option to make sure that there's a steady stream of posts. Some of them will be recycled, others may be a bit on the short side, but every week you'll see at least one or two new posts.

Let me know what you think.