1, 2, 3... not the best place to start? (Part A - background and history)

NOTE - This post may include some mathematically technical material. I will attempt to give warning by bolding parts that need some extra background.

How did you learn math? Almost certainly, you would say it started with counting.
STEP 1: You learned a list of words - "one two three four five six seven eight nine ten" perhaps, or maybe "一　二　三　四　五　六　七　八　九　十" or something similar, and practice pointing at different things while saying these words.
STEP 2: Soon after, you begin to apply that list to actual things you see - you realize that the word "five" can be applied both to "five apples" and "five people." This is actually a pretty major step - there are many young children who can count blocks "one, two, three," but don't know how many blocks there are. This is where children learn one of the most important properties of math, abstraction - that a single concept (like "five") can apply to many completely different areas. From there, the natural direction to go is
STEP 3: Comparing numbers (6 is bigger than 4 because you have to count past 4 to get to 6)
STEP 4: Addition (you have 1 2 3 blocks here, and 1 2 3 4 5 blocks here - how many altogether?),
STEP 5: Subtraction (you have 1 2 3 4 5 6 blocks, and 1 2 3 4 are in a special group - how many are not in the special group? Or, if you've already learned addition - what number added to 4 makes 6?) and then
STEP 6: Multiplication is taught as doing addition of the same number several times.

But then along comes division. And EVERYONE struggles with division, at least the first time they see it. "It's just the opposite of multiplication," right? So if multiplication is repeated addition, does that mean division is repeated subtraction? (The answer is yes, kind of, but it's not very intuitive). Division is also the first time that you kind of have to leave the whole numbers - adding and multiplying whole numbers give whole numbers, and if you always make sure take the smaller from the larger (which is simple enough to check), same with subtracting. But you would have NO idea if dividing 46 by 4 would give you a whole number or not. So this is where decimals and fractions start becoming necessary. But fractions are confuuuuuusing... Why do we use two numbers to represent one number? Why can't you just add the numerator and denominator? Why do we use such big words? What happens when you divide by zero? Why do we need to learn this? Can we go to recess early?

A bit more background about myself - I love math, and I'm going to study pure mathematics at university... but I've been very interested in developmental math. How do people learn math, and what gets in their way? This interest first came up when I went to Mathcamp, a 5-week summer math program in the U.S., in the summers of 2009 and 2010. Some of the visiting speakers there talked about how people learn math (or just about the psychology of math in general) - talks by Dan Zaharopol, Josh Tenenbaum, and Catherine Havasi were particularly inspiring, and some of the ideas in this post can be attributed to them. And over my 12th grade year, I researched math anxiety and its causes, giving me an even bigger interest in the topic. Which leads me to this post - I'm not trying to push any particular viewpoint, I'm just speculating - what if a completely different approach to math, from a very early age, would address a lot of problems that currently face students learning math?

Myth: counting is the foundation of everything we learn about mathematics.

FACT: It's not.

It's true that a counting process can be used to form a rigorous basis for all mathematics - once the natural numbers have been constructed, it's simple enough to construct the integers, rationals, reals, and complexes (complices?) via equivalence classes of ordered pairs and Dedekind cuts or Cauchy sequences... but I'd like to propose that perhaps this isn't the (only) natural way to learn mathematics.

A bit of history... the Greeks were very good at math. Except for calculus and some ideas about trigonometry and limits, pretty much all elementary, middle, and high school math was known to the Greeks. But they had two completely separate sections of math - geometry and number. They did find ways to connect the two sometimes (such as through the Pythagorean theorem, connecting a right triangle to an equation about numbers), but they started with each separately, and kept the two separate. Why? Because the two are completely different. Their study of number was a discrete process - where discrete basically means "can be counted." Their study of geometry was a study of continuous things - things that "can be measured." And for most of history, the two were kept completely separate. Again, people found many ways to connect them, but you would always learn them separately, then show that they could be joined together.

It wasn't until about a century ago that this changed. Basically, people went crazy proving and discovering all sorts of wacky things, it was getting harder and harder to understand what was going on, and finally someone decided "this is too messy. We need to have a common ground to base all mathematics on, so we can all agree on what's actually happening." And so, almost arbitrarily, some mathematicians chose to make number the foundation for geometry, and proved that all basic facts about geometry could be explained just using facts about numbers.

Was this a wise decision? I don't know. But it certainly had major impacts. For example, one of the most famous mathematical works of all time, Euclid's Elements, which basically sets foundations for geometry, stopped being popular around that time. In the past, Elements was an important part of Western culture, both in mathematics and out; major non-mathematicians such as Thomas Hobbes, Bertrand Russel, and Abraham Lincoln all worked through and loved the Elements. According to John Stillwell, "Perhaps the low cultural status of mathematics today, not to mention the mathematical ignorance of politicians and philosophers, reflects the lack of an Elements suitable for the modern world." And it's quite likely that its drop in popularity came from people thinking "yeah, geometry's great, but let's start with learning about numbers first."

And from the little I know about developmental psychology, it seems like the choice to start with number impacted a lot more than just the popularity of a particular book.

(continued in Part 2A)