I have been thinking about the divide between High School math and Elementary School math.  Teaching the upper years of Elementary school, I sit in the middle.  I see the kids leave my classroom after experiencing some very hands on math with inquiry projects, and then go onto math that is more abstract and distanced from from their "life".  I don't blame the teachers (honestly, I don't even know which way is better), it is more of a systemic problem than it is a local one.  Our educational institutions values pure mathematics over applied mathematics.

I have also been thinking about the things that we do in Elementary school that mess up thinking when they get to this level.  We give kids these metaphors (or visuals) to help them understand concepts and to master them at the level we teach them.  One such example is; multiplication is repeated addition.

We have all done it and said it.

4 x 3 is the same as 4 + 4 + 4 or even 3 + 3 + 3 + 3
4 x 3 is the same as 4 plus itself 3 times
4 x 3 is the same as 3 plus itself 4

Multiplication is easy, it is just repeated addition!  The just in that sentence, and I am guilty of it, implies that it is so simple and that is can be broken down into this way of viewing all the time.  The implication is this; if you follow this way of knowing multiplication, you will be fine and you will always have something to fall back on.  It is conceptual understanding.

However, the problem is that multiplication is not always repeated addition.  As students get higher up in math, the nature of it changes, and these metaphors expand.

π x 1/4 is the same as π plus itself 1/4 times...... what a second, does that make sense?

3 x 4 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3

So if we apply the same logic (and this is what kids do then they learn these rules, metaphors, axioms, images, etc.) we could say that:

π x 1/4 = (x)

which is the inverse of:

x ÷ π = 1/4
x ÷ 1/4 = π

I'm not an expert in math, but something about that does not make sense to me....

I have no idea how we can get around this as elementary educators.  I do know that this complexification of metaphors is a big problem.  I would guess (as this is not based on research) that this is one of the major reasons that kids drop out of math around grade 9.  The metaphors all get messed up, and the language they thought they knew, starts to change.

Perhaps, it is best to keep our definitions in Elementary school more open, to have kids understand that these ideas will change, and they will have to change with them.  Of course, that goes against the view of mathematics as a static and knowable language.

Math can't change, it is Math!  It is always true, and always right!

I question whether or not that is true.  I mean, Western mathematics used to reject the concept of zero, so obviously, it changes as our knowledge and ideas evolve.

Here is a question for you; does mathematics exist in the real world, and humans are just discovering it?  Or is math a human creation?  How you answer this question says a lot....

Just thinking... please correct any of my misconceptions....

Sorry for the rambling nature of this post, these ideas are still raw in my mind....