Mathematics in Decision Making Learning Objectives for Large Sets

Originally Posted 12-22-2011

The first stage of planning for a course unit is to have a clear set of objectives. Let’s begin with the first section of Mathematics in Decision Making–the unit on comparing large sets.

It seems important to keep the list short, and to only highlight the truly major ideas. That way, this can be used as a guide in making classroom decisions. I don’t want this list to become a cumbersome detail by detail account of what each problem will be about.

What should the Students Learn?

I think the main objectives are these.

• To “count” a (finite) set really means making a bijection with a set of the form $\{ 1, 2, 3, \ldots, n\}$.

• It is possible to compare the sizes of two sets without actually counting either one.

• The difference between a finite set and an infinite set.

• What it means for a set to be “countable.”

• The decimal model for the system of real numbers.

• There are too many real numbers to count.

I suppose that somehow the third item isn’t really necessary to get to the fifth one (my ultimate goal). But I want there to be lots of places where we learn to sharpen our use of language, so I include it.

Of these, I think that item five is really the hardest. The others take time to absorb and are cognitively difficult, but understanding the basic model of the real numbers and its important properties (like $0.\bar{9} = 1$) is the part that has real technical teeth.

I think I’ll try to write a problem sequence that addresses the first two points in week one, the next pair in week two, leaves a whole week for just the fifth one, and then whatever partial week four I have to get the punch line.

I’ll spend some time in the next two days adjusting my colleagues problems and then share my draft.