20150910 THE FIRST INVERSION:

Not a sign, but an operation to change it.

Mathematics is so notoriously difficult to learn because education is so conservative and selfish. They teach like many hundreds years progress never happened. They squirrel away every helpful invention and discovery and turn it into an obstacle to learning. Mathematics, which is tremendously useful and empowering, they present as a scare.

Without introducing anything new, let's just rewire the teaching. I will use a simple and exciting narrative: people were inventing new operations, they tried to reverse them, and every time this led to a discovery of a new kind of number.

The most basic operations are counting (increment) and counting back (decrement). Teaching them as such (like some preschool educators do) to me is simply impractical. Before you explain them to kids, they know numbers, so increment inevitably turns into +1. Still increment and decrement are very handy for teaching, and they allow us to introduce unary operations.

Incrementing many times at once is what we call addition. In positional system it's not just another name. We have a method, which makes addition much easier. Not even much easier, but exponentially easier. Demonstration of how positional addition is easier than counting forward is breathtaking (since I wrote this, I spent another month and produced STEREO LEARNING to illustrate some of such statements).

Repeated decrementing leads to subtraction, and we face the problem of numbers for the first time. We can always add two numbers. Incredibly, there alway is a natural number waiting to be discovered through addition. Yet we can not always perform subtraction. We need negative numbers to make it possible.

This kind of reasoning becomes available after we exposed our children to big numbers, which school does not do. Well, without big numbers you can only leave the learners to discover their knowledge and conceptual thinking. I keep picking on conceptual thinking because it's such a shameless lie. On par with calling a daycare an Academy.

As soon as we invented negative numbers, we understand that natural numbers - the only kind of numbers we knew before - are positive. Or we may think that every natural number can produce a pair of integer numbers if you add it to 0 and subtract from it.

Next comes another unary operation, which can be rightfully called inversion. I often call it negation, which is worse because of asymmetry. At school they do not teach such things. I wonder if Steven Leinwand knew that inversion is an operation.

Inversion turns negative numbers to positive and back. One way to do it is subtracting from 0. Another, and more interesting way, is multiplying by -1. In fact, -1 is the only negative number we need to build the rest of them.

The true meaning of the minus sign is the operation of inversion. Now what do you do to subtract a negative number? Yes. You invert and add.

Is it so easy? Of course it's not. Soon I will return to subtraction to discuss the problems with it. Elementary math is actually a toxic swamp. Allowing children to "discover" it without guidance is inhuman.