## Friday, January 20, 2012

### The Concept of a Mathematical Function

The concept of a mathematical function should not be first introduced as a formula, but as an arbitrary (ordered) pairs of numbers. Pupils are conditioned to think of a function as a continuous line or a firmula. Later there are issues with statistics when function is shown to be function a set of distinct dots, representing ordered pairs of numbers, i.e. there is no formula at all. Moreover, in probability and statistics numbers appear to be showing at random!

The rule how you pair one number with another can be a formula, but also can be a completely random event. Math function is, first and foremost about pairing two numbers (or more in multivariable functions). Students should be aware that they can pair random chosen numbers, they do not need to calculate second number from first. The rule can be linked input or output, but that restricts the function in the way that you have to know input to get the other paired number, the output. Because, function can have a pairing rule "pick first number, then, ask another person to pick another number without looking at the first number, then pair two numbers". Rule is one thing. Paired numbers are another. I want to emphasize that function need not to be defined in a restrictive way by using words "inputs" and “outputs", which is more related to computer science. You do not have to know input to get output, in a function. Both elements of the ordered pair of an function can be completely random and independent from each other. Function is first and foremost a pair of ordered numbers. My examples show why the \"input\" \"output\" definition is restrictive and possibly misleading. In my view, the word pair, or more precisely definition "ordered pair" best describes the function. Then we can use word map, association of two numbers etc. Input and output really leads someone to think that there need to be formula or some dependency between output and input. But, it is not so. It can be, but that's too restrictive for function definition. As in my example, a function can be "pick an output that in no way depends on input". Or, pick one number, then cover it (hide it) then ask another person to pick another number. Pair these two numbers. Here, output in no way depends on input, yet this is a function.