Rules
One of the things that mathematicians do is construct rules. Students learn the rules. They follow the rules. Why not let them experience what is like to be a mathematician and construct some rules?
There is a natural fit between this and the Insight method of teaching math.
The Mathematical Type of Rule
Mathematicians construct a particular type of rule. Consider the rule for turning a word into its plural. Mathematicians didn't make up the current rule. They don't make up a new rule. Well, mathematicians don't deal with plurals, we are now talking about what linguists do. But the style is the same. The mathematical style, and what linguists actually do, is construct rules desribe the current usage. So the rule is going to say when to add s and when to add es to follow what people already do.
Another example. There is a rule for determining the area of a rectangle. It is length times width. It isn't the case that the first mathematician could have invented a different rule, like adding the height and base. (And it is very important that students understand the the rules are not arbitrary.) There is an area of a rectangle, and the rule has to fit reality.
Most of what mathematicians do is expressed as a rule. There is a rule for determining the number of ways that 8 things can be combined into groups of five. There are rules for solving equations. There are rules for determining area, volume, etc. There are rules for converting from one measurement system to another. A computer program is nothing but rules for what the computer should do.
Teaching them to Build Rules
Letting students construct rules is natural for my problem-solving approach to math. They are given, for example, the problem to construct a rule for determining the area of a rectangle. In doing so, they will learn about area and rectangles. But, for that exercise to be much of a success, the students need to succeed and actually construct the rule.
In fact, constructing rules is difficult.
My idea is this. We teachers take it for granted that students know what the rules are, what they look like, and how they function. So the problem in constructing a rule for the area of a rectangle is one of content. Similarly, if you had the problem of constructing the rule for the volume of a cone, and assuming you did not remember the formula, you would have the problem of content -- how the heck do you figure out the volume of a cone.
But it seems likely to me, at this time, that young children do not have the skills of constructing rules, even when they perhaps know the underlying content. So the problems in this section focus on rule-construction skills, not content.
Beginning Exercises
This is a new idea for me, so I don't have a lot of suggestions. One good exercise for starting out, which I have used, is constructing the rule for making a word into a plural. Another good exercise, probably, is constructing the rules for adding in a Roman-type numeral system when the answer has to be in a standardized format.
In both of these exercises, the students know the underlying behavior they are trying to describe, though unconsciously and not in rule format. SO the issue is putting that knowledge into rules. That's difficult enough.
Checkers
I have also done the exercise of constructing rules to play checkers. The idea is for them to write down rules, then follow the rules as they play. The rules have to be straightforward and unambiguous. You can't have a rule "make the best move", because rules can't include the thinking of the person following the rule. You can have a rule be "move the most forward piece" because there might be two pieces equally far forward, and it might be ambiguous which way to move the piece.
At the start, the winner is the one whose rules go the longest. Someone loses when they don't have a rule for a situation, or their rule is ambiguous. My six-graders got good enough to be able to at least get to kings (when a whole new system of rules is needed), and some could play almost a whole game. (I am not sure we actually finished a game.) One student's rules, to my surprise, beat my rules, which I had thought about. And, I should add, that taught me something about the strategy of checkers.
As noted, part of the exercise is constructing rules, but most students do not have a good underlying understanding of checkers. So the students, quite appropriately, spent some time just playing checkers.
The rules might look something like this. Move the farther piece forward. Move it right if there is a choice. If there is more than one piece that is farthest forward, move the rightmost piece.
Transformers
One exercise that worked very well with a fifth-grade class was listing "transformers". I define a transformer as something that sits around, not doing anything. But when it is given something, it transforms that something into something else.
I gave the class my definition of transformers, then asked them to list transformers.
There are really a lot of different transformers. But people don't think that way, so it is difficult at the start to list transformers. The class will slowly develop skill.
Some things are not quite perfect transformers, but close enough that you want to categorize them as transformers. The soda machine doesn't really turn dollar bills into cans of soda. But from the outside it looks like it does. (And from the inside view, many transformers might not be perfectly direct in how they transform things.) So you don't want to criticize the soda machine if it is the first transformer someone suggests.
In our class, we got into the idea that the brain is a transformer. And of course, my exercise was devoted solely to transforming their brains.
I have to say that I cannot imagine seeing the world without the concept of a transformer. It is a fundamental concept. So I rank this exercise as very important.
At the end, I had time to just note that almost all math formulas are transformers. Those math formulas are also rules.
So, at least in theory, this exercise on transformers might help with the foundation of understanding how to build mathematical rules.
Computer Programming
As noted, computer programming is constructing rules. Perhaps the exercises in constructing computer programs would be good for learning to construct rules. But I suspect the learning points in the other direction -- if you can teach your students to construct rules, you might have an easier time with some of the hidden hurdles in teaching computer programming.
Foundations
If students have facility building rules, then you can give them problems where they have to construct rules and they don't have the underlying understanding of how to solve the problem without rules. In other words, you have opened up a powerful new method of teaching math.
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