I've always been a big fan of games and puzzles as teaching tools. They develop logical thinking and problem solving, they remind us that learning is supposed to be an enjoyable activity, and more often than not, they have a dirty, little secret.

Doublets are a great example. Developed by Charles Dodgson (writing as Lewis Carroll), they were, for a while, the rage of the Victorian party scene. The rules were elegantly simple: take two words with the same number of letters; change the the first word to the second one letter at a time with the condition that each transition is also a word (think Scrabble rules -- no slang, no proper names). Though not required, the two words would usually have some logical connection.

Scores are determined by how many steps it takes. As in golf, low score wins.

Here's how a doublet player might go from FOOT to BALL:

FOOT

FOOL

TOOL

TOLL

TALL

BALL

Of course, FOOT BALL is an easy one. the doublets Carroll created tended to be far more challenging. Here are some examples from Carroll's Doublets: a word puzzle (available in cut-and-paste friendly plain text here)

Change OAT to RYE.

Get WOOD from TREE.

Prove GRASS to be GREEN.

Change CAIN into ABEL.

Make FLOUR into BREAD.

Evolve MAN from APE.

Now we get to the secret of doublets.

After you've used them as time fillers at the end of class and handed out the puzzle sheets and maybe even given some bonus points to the first student to solve a particularly challenging example, only then do you reveal the dirty little secret:

It's mathematics.

Specifically, it's graph theory.

That's right, you've tricked all of those poor, innocent kids into doing math and, worse yet, thinking they enjoyed it. You've introduced a sophisticated mathematical concept, reinforced it with a memorable example and laid the groundwork for future lessons.

Naturally, the patron saint of math teachers, Martin Gardner, was here first.

Originally posted in Education and Statistics

A blog of tips and recommendations for anyone interested in learning or teaching mathematics.

## Sunday, January 27, 2013

## Sunday, January 20, 2013

### Alexandria Word Searches

This grew out of a game I was marketing a few years called Alexandria (which I'll probably get around to discussing one of these days). The game involves forming words that snake around a hexagonally tiled board and I decided that the basic concept could put a new spin on traditional word search puzzles.

You form the words by moving from hexagon to adjacent hexagon. You can use the same hexagon in different words but not twice in the same word (an n-lettered word has to use n hexagons). This is easier to show than to tell.

In this example Washington and Lincoln overlap in numerous letters.

To hint or not to hint

There's a bigger discussion here about the amount of helpful information should be presented with a puzzle. My general feeling is less-is-more, but puzzles are supposed to be fun. That means making the difficulty high enough to be challenging but low enough not to frustrate.

For word searches, hexagonal or rectangular, there seem to be three schools of thought.

I. List the words in the puzzle. Maybe it's the pedagogical puritan in me, but this is my least favorite choice. It greatly reduces both the challenge and (in a teaching context) educational value.

II. Give a category. This might be my preferred technique when you're having a group do a puzzle. For example, you might break up a class up into small groups, give them something like the above puzzle, and have each group see how many words in the category they can find.

II. Give a category and clues for each word. Probably my preferred approach. These can be crossword style clues or if you're in a hurry, you can give the first letter of each word followed by blanks for the remaining letters

DIY

If you want to try making your own rectangular word searches, I'd recommend using a spreadsheet program. For hexagonal puzzles, try saving the blank grid as a separate file then fill in the letters and save each puzzle as an additional file.

Here are a few more examples

Serious Shakespeare

Not-so-serious Shakespeare

Metals

Nobel Writers

You form the words by moving from hexagon to adjacent hexagon. You can use the same hexagon in different words but not twice in the same word (an n-lettered word has to use n hexagons). This is easier to show than to tell.

In this example Washington and Lincoln overlap in numerous letters.

To hint or not to hint

There's a bigger discussion here about the amount of helpful information should be presented with a puzzle. My general feeling is less-is-more, but puzzles are supposed to be fun. That means making the difficulty high enough to be challenging but low enough not to frustrate.

For word searches, hexagonal or rectangular, there seem to be three schools of thought.

I. List the words in the puzzle. Maybe it's the pedagogical puritan in me, but this is my least favorite choice. It greatly reduces both the challenge and (in a teaching context) educational value.

II. Give a category. This might be my preferred technique when you're having a group do a puzzle. For example, you might break up a class up into small groups, give them something like the above puzzle, and have each group see how many words in the category they can find.

II. Give a category and clues for each word. Probably my preferred approach. These can be crossword style clues or if you're in a hurry, you can give the first letter of each word followed by blanks for the remaining letters

DIY

If you want to try making your own rectangular word searches, I'd recommend using a spreadsheet program. For hexagonal puzzles, try saving the blank grid as a separate file then fill in the letters and save each puzzle as an additional file.

Here are a few more examples

Serious Shakespeare

Not-so-serious Shakespeare

Metals

Nobel Writers

## Sunday, January 6, 2013

### Reasons to teach what we teach

[note: this is a math-centric post but most of the concepts can, on some level, be generalized to other subjects]

There's a curiously inverted quality to the education debate. We spend a great deal of time discussing revolutionary changes to the educational system and almost no time talking about what we should be teaching, as if the proper combination of reforms and incentives can somehow overcome the rule of garbage in, garbage out.

I spent a lot of my time as a teacher thinking about which parts of the mathematics curriculum were good and which parts were garbage and I came up with a list of reasons why a topic might be worth the student's time. The list isn't in order (I'm not sure it's even orderable) but it is meant to be comprehensive -- everything that belongs in the curriculum should qualify under one or (generally) more of these criteria.

1. Students are likely to need frequent and immediate access to this for jobs and daily life.

and

2. Students are likely to need to know how to find this (Samuel Johnson level knowledge).

(These are the only two mutually exclusive reasons on the list.)

3. This illustrates an important mathematical concept

4. This helps develop transferable skills in reasoning, pattern-recognition and problem solving skills

5. Students need to know this in order to understand an upcoming lesson

6. A culturally literate person needs to know this

Most topics can be justified under multiple reasons. Some, like the Pythagorean Theorem can be justified under any of the six (though not, of course, under one and two simultaneously).

Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.

If, however, a problem falls primarily under four, this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)

The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).

What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.

Long division and logic puzzles are, of course, extreme cases, but the same issues show up across the curriculum. Take factoring trinomials. A friend and former boss of mine wrote a successful college algebra text book that omitted the topic entirely. I had mixed feelings about the decision but I understood his reasoning: this is one of those things you will almost certainly never have to do outside of a math class (what fraction of trinomials are even factorable?).

You can justify teaching the factoring of trinomials because it illustrates important mathematical concepts and because it gives students practice manipulating algebraic expressions, but the way you teach this concept has got to reflect the reasons for teaching it. Having students memorize a step-by-step algorithm would be the easiest way to teach the students to answer these questions (and improve their standardized test scores) but it completely miss the point of the lesson.

The point about standardized test scores is significant and needs to be revisited a post of its own. By evaluating teachers and schools on standardized test scores, we put pressure on teachers to treat all subjects as if they fell solely under reason one. This is not a good outcome.

Even more important than how we should teach something is the question of what we should be teaching. Current curricula tend to be broad and shallow with a tragic evenhandedness that often grants the same amount of time to trivial techniques as it does to fundamental concepts. This is bad enough when a class on grade level and everything is going well but it's disastrous when a large part of the class is struggling. There is tremendous pressure under those circumstances to leave the stragglers behind (a pressure that actually increases under many proposed reforms).

In addition to being overstuffed, the current curriculum omits subjects that are arguably more important than most of what we cover. The obvious example here is statistics, a topic that everyone actually does need on a daily basis (as informed citizens and consumers if nothing else). Perhaps even more relevant is what we might call spreadsheet math (customized worksheets, recursive functions, graphs, macro programming). You could also make a case for discrete mathematics, particularly graph theory (I might even put this one up there with statistics and spreadsheets but that's a subject for another post).

Originally posted in Education and Statistics

There's a curiously inverted quality to the education debate. We spend a great deal of time discussing revolutionary changes to the educational system and almost no time talking about what we should be teaching, as if the proper combination of reforms and incentives can somehow overcome the rule of garbage in, garbage out.

I spent a lot of my time as a teacher thinking about which parts of the mathematics curriculum were good and which parts were garbage and I came up with a list of reasons why a topic might be worth the student's time. The list isn't in order (I'm not sure it's even orderable) but it is meant to be comprehensive -- everything that belongs in the curriculum should qualify under one or (generally) more of these criteria.

1. Students are likely to need frequent and immediate access to this for jobs and daily life.

and

2. Students are likely to need to know how to find this (Samuel Johnson level knowledge).

(These are the only two mutually exclusive reasons on the list.)

3. This illustrates an important mathematical concept

4. This helps develop transferable skills in reasoning, pattern-recognition and problem solving skills

5. Students need to know this in order to understand an upcoming lesson

6. A culturally literate person needs to know this

Most topics can be justified under multiple reasons. Some, like the Pythagorean Theorem can be justified under any of the six (though not, of course, under one and two simultaneously).

Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.

If, however, a problem falls primarily under four, this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)

The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).

What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.

Long division and logic puzzles are, of course, extreme cases, but the same issues show up across the curriculum. Take factoring trinomials. A friend and former boss of mine wrote a successful college algebra text book that omitted the topic entirely. I had mixed feelings about the decision but I understood his reasoning: this is one of those things you will almost certainly never have to do outside of a math class (what fraction of trinomials are even factorable?).

You can justify teaching the factoring of trinomials because it illustrates important mathematical concepts and because it gives students practice manipulating algebraic expressions, but the way you teach this concept has got to reflect the reasons for teaching it. Having students memorize a step-by-step algorithm would be the easiest way to teach the students to answer these questions (and improve their standardized test scores) but it completely miss the point of the lesson.

The point about standardized test scores is significant and needs to be revisited a post of its own. By evaluating teachers and schools on standardized test scores, we put pressure on teachers to treat all subjects as if they fell solely under reason one. This is not a good outcome.

Even more important than how we should teach something is the question of what we should be teaching. Current curricula tend to be broad and shallow with a tragic evenhandedness that often grants the same amount of time to trivial techniques as it does to fundamental concepts. This is bad enough when a class on grade level and everything is going well but it's disastrous when a large part of the class is struggling. There is tremendous pressure under those circumstances to leave the stragglers behind (a pressure that actually increases under many proposed reforms).

In addition to being overstuffed, the current curriculum omits subjects that are arguably more important than most of what we cover. The obvious example here is statistics, a topic that everyone actually does need on a daily basis (as informed citizens and consumers if nothing else). Perhaps even more relevant is what we might call spreadsheet math (customized worksheets, recursive functions, graphs, macro programming). You could also make a case for discrete mathematics, particularly graph theory (I might even put this one up there with statistics and spreadsheets but that's a subject for another post).

Originally posted in Education and Statistics

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