You may want to watch the video before reading my thoughts about it. This way, you can make your own meaning out of this metaphor.
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Thoughts:
- It is healthy for humans to focus attention on the features of the environment directly helping in their goals.
- It is probably beneficial for the group to have some “attention deficit” people on lookout for “hidden bears” to warn the group of slight dangers. Attention deficit may not be a disorder, but a group survival mechanism.
- If the bear were real, most people would notice it. They would also cease to count passes. It’s healthy, hehe.

There is a lively discussion about division models going on in the Living Math mail list. A question came up: “Isn’t there a better written way to express what the actual relationship of 1 3/4 divided by 1/2 = 3 1/2?” I shared the funky idea Kelly, my student, used in a similar situation. It is all about extending the metaphor “division is sharing” into fractions, and the funk comes from using fractional people! Let’s consider an example.

Kyoma and Naruto, pictures by Harumi Uchiha, are sharing six pizzas. How much food will each one get?

The sharing is easy to set up, and the division problem directly follows the sharing metaphor:
6:2=3

Now let’s extend the examples into fractions! Half a person (insert giggles here) has a pizza and three-quarters. How much food will that one whole person get?

It is logical that if half the person got some food, the whole person will get twice as much food. Here are the formulas and a picture:

1 3/4 : 1/2 = 1 3/4 * 2 = 2 6/4 = 3 2/4 = 3 1/2

The fraction operations here follow the pizza slices nicely; if you have trouble, make a model out of paper and you will see what’s going on.

While lol-worthy, the fractional person sharing situation is not very realistic, even in the magical fighting world of Naruto where people sometimes do get sliced. But there are everyday situations involving this model. Here is an example.

Lucy had about a can and three-quarters of blue paint and she thought it would be enough to paint a room, but it only covered a half of it. How much paint would she need for the whole room?

The division problem in symbols: 1 3/4 : 1/2 = 3 1/2

Let’s try a similar problem with juicier numbers. Halves are too simple to manage, and this simplicity makes understanding of operations more difficult!

Chris was seeding his lawn. He had six little seed bags, which covered three-quarters of his lawn. For the next season, he wants to figure out how many bags he’ll need for the whole lawn.

Chris reasoned: if six bags covered three-quarters of the lawn, it comes to two bags per each quarter. Since there are four quarters altogether, it then comes to eight (2*4) bags for the whole lawn. If, for some strange reason, Chris wanted to write it down very formally (this is not a realistic desire, by the way), he could do this:

6 : 3/4 = (6 : 3) *4 = 8

This model shows, well enough, the “flipping of the fraction when you divide” procedure some textbooks stress. I’d like to note that division by fraction is NOT how most humans think of their day-to-day situations. It is more of an extension to make mathematics involved prettier and more streamlined. Division by fractions is rarely, maybe never done to make our day-to-day lives easier. It’s not practical, it’s beautiful, but appreciation of that beauty is inside the mathematics involved. Well, it is practical in the sense that it makes math theories easier and it makes advanced math go smoother. However, the examples about pizzas and paint would not involve division by fractions if real life humans set them up in their daily routines, or if the goal was just to do these computations. I still use such examples, though, but with a caveat that they are a part of a larger mathematical picture I have not even started to discuss. The beauty of the picture is there, but this post only shows a tiny little corner of the picture - not nearly enough to appreciate the beauty.

General rules:

• Find an Object to define.
• Move 1: Маке a definition of the Object.
• Move 2: Call "Too Wide": something that is not your Object, but is defined by your definition (a counter-example). Call "Too Narrow": something that is your Object, but is excluded by your definition.
• Move 1': Give a new definition that fixes the problem found in Move 2.
• Repeat Moves 1 and 2 until nobody is able to make a Move 2, or players like their definition well enough, or players get too emo to continue

Here is an exampe piece of a game script, played cooperatively. Game moves are in bold, unlike the extra chatter.

It helps to write down the moves for future reference. Re-reading such a "history of war" after a game brings much laughter. If you play the game at the beginning of some learning journey, such as a mathematics or an education course, the notes will be quite useful later, when you grow in the mastery of the material, and the corresponding sophistication of definitions. If several people play, the note taker should jot down their names, their game moves, and the more amusing of their other comments. However, you can play without keeping records, say, on car trips.

This game, according to the Natural Math ideal of humans as creators, is a great opener to a new topic. Trying to construct your own definition, while somebody (possibly yourself) is poking holes in it at every step - challenging, baffling, scary, exhilarating. Some teaching folks or textbooks open a topic with their own definition and follow with "a series of simple, repeatable exercises" explicating the definition; but that's not much of an adventure, is it?

I can't predict how your definition wars will go, because it depends on what you choose to define, and who is playing. I can offer these observations of what may come of it, as reasons why I love the activity:

• Players will develop some general strategies for building definitions. These strategies are indispensable any time you are creating any scientific work - and also any time you find yourself in the middle of a political, religious, parenting, or any important debate. People who can form powerful and elegant definitions have such an advantage in any conversation, sometimes I think it's unfair. For some advanced lists of strategies and traditions on defining things, try searching for "definitions" in sources on both science and rhetoric; Wikipedia's article on definitions is a good start. The whole of Wikipedia is a recording of a giant, (mostly) cooperative game of Definition Wars.
• Even young children quickly realize that many things in life are hard to define, and attempting to define them will cause group tensions and infinite iterations of the game. You can expect a final, perfect, agreeable definition of something like "rhombus" or "quadratic equation" to emerge in just a few steps of the game. Try an everyday object, like "book" - and you are forever trapped debating whether a novel your auntie hand-wrote and only three people ever read, or a yearly almanac published by a journal, are "books" or "counter-examples." Most math terms are easy to define, and for many people, this is a major appeal of the subject. There is a certain comfort in defining "parabola", compared to defining "love" or "religion". After playing the game for a few rounds, people learn to anticipate how easy or difficult each definition will be. Still, the difficulty level of defining even something mundane, like "door" may surprise definition veterans. Do you expect "multiplication" to be easy or hard to define?
• Definition war is a team-building activity, because it promotes close intellectual and emotional encounters among players. It also allows to plainly see players' mind and character traits: a flair for the bizarre, competitiveness, stressing analysis or synthesis, skills at criticism, attentiveness to emotional states of others, and so on. And, of course, your team will need to develop great conflict resolution skills if they are to survive the definition wars.
• This game is a research and diagnostic tool, because it exploses knowledge and thought processes of participants. 

I am reading or re-reading a few books about creativity disciplines and techniques, and thinking of the internet applications. Let me start a list of examples. I am brainstorming on the topic of “user generated content.”

Search for your “problem” keywords in quotation sites. Great minds can get us going. My search for “content” yielded this quote:
An author is a fool who, not content with boring those he lives with, insists on boring future generations.
- Charles de Montesquieu
What a great explanation of why content freshly made by each day’s “generation” of visitors can be more engaging for them that something authored once and for all! Though it was an accident that this quote even got pulled out, because it uses “content” in its different sense. We always want accidents for brainstorming, though.

Search Flickr, Google image search and other collections of visuals. My search for “user generated content” brought up this picture, by Dirk van Eunen:

Paint-your-own shoes. And now I am thinking of “make your own” math objects, such as symmetry craft kits.

Type a few random letters into a web search and look at the first site that comes up. Try to connect it with your current brainstorming. Just now, I typed “adks” and found some roof heating electric cable that prevents ice built-up. It got me thinking about measures that can prevent trolling and other “dead weight” build-up in social sites.

I rarely get stuck without ideas, in part because brainstorming techniques are now very intuitive and automatic to me. However, I frequently help other people who do get stuck, and that’s why I search for easy and powerful beginner tools, like these.