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Chunking

This section is about a new verb: chunk. The action of chunking is a key to taking an image you can make in your mind's eye, and using it to solve the equation it represents. You probably have some idea of what it is from chapter , where it was introduced in the example. It is a tool we use to help rearrange the equation in ways to our liking. Before we can fully define chunking and how we use it, we must discuss a few concepts.

The first concept is the symbol. Math equations are composed of symbols. In the equation x + 2 = 3, one symbol is the variable, 'x'. Two of the symbols represent mathematical processes ('+' and '=' ), and the other two symbols are numbers. They are arranged in a particular order, giving the equation a particular meaning. If you change the order, it may change the meaning of the equation (x + 3 = 2) or it may not (2 + x = 3).

The next concept is an object. What exactly is an 'object'? How do you know something is an object, and how do you know that two objects are distinct? There are things in your environment right now that can be referred to as objects. You are wearing several^2.3, and if you are reading this on a computer screen, the display is an object. Anything that you can pick up and move around is an object.

There are also non-physical objects. These are not real in the same sense as tangible objects, but they can still be useful. In particular, and this concerns us, math symbols are objects. Let's look at this equation again:

$$x+2=3$$

Each of these five symbols can be considered an object. Whether a particular symbol is on a computer screen, paper, or in your mind's eye, you can consider them the same object.^2.4 In other words, the symbol 'x' on a computer screen is the same object as that 'x' on a printed page, and the same as an 'x' you imagine.

What about the equation itself? The equation as a whole is also an object. It is an object that is composed of other objects, the symbols. The same can be said for a jar of pickles, or a box holding kittens. The equation, jar, and box are objects that can hold other objects; when they are put in, the whole thing becomes a new, different, yet related object.

When you look at the equation above, you can focus on one symbol, an expression (group of symbols), or the equation as a whole. Your choice of focus will depend on what you need to do in that moment. When you choose to focus on an expression, or the entire equation, you are taking several objects and treating them as a single whole. This is fundamentally what chunking is. It is about working with an aggregate object that is composed of other objects. It implies a degree of flexibility in your thinking, since you are choosing to focus on an expression (such as x + 2) when it us useful to do so, on the equation as a whole (x + 2 = 3) when that is useful, or on one of the components individually (e.g., just x or 3) when THAT is useful.

Chunking is more about choosing where you put your attention than anything else.

The definition of chunk is ``the action of consciously taking a group of objects, and working with that group as if it is a single object.'' We would say something like ``Chunk the (some group of symbols)''. For example, in the equation

we can say ``Chunk the 4 + x'', which means we are going to work with the expression 4 + x as a group. In this book, chunks are pointed out by drawing a dotted box around them.

Footnotes

... several^2.3

Well, you're probably wearing several. I don't know your reading habits.

...^2.4

You can also consider them to be different objects. Neither view is more 'correct'. In this book, it is more useful to act as if a symbol like x is the same object regardless of where we see it.