I have a theory that if you cannot explain an idea in some form to a bright, attentive six-year-old, then that may be a sign that you do not really get the idea yourself.
So, here is a very simple math exploration that can be done even with a homeschooled six-year-old and that in fact connects to some quite advanced mathematics.
3-D Rotations Need Not Commute:
Get two identical boxes – we used a couple of Wheat Thins boxes.
Put both boxes on the table or floor in front of you facing towards you.
Now, the idea is to perform two rotations on the boxes, but in different orders.
First, take the box on the left and rotate it a quarter turn counter-clockwise towards yourself (i.e., 90 degrees counter-clockwise around the vertical axis): call this rotation Z.
Now, take the box to your right and rotate it a quarter turn so that the front face ends up face down on the floor (i.e., 90 degrees around an axis going from left to right): call this rotation X.
Now, let’s perform rotation X on the left box: i.e., rotate it so that the face which is now vertical and facing towards you is rotated forward and down onto the floor.
Finally perform rotation Z on the right box: i.e., rotate it counter-clockwise around the vertical axis a quarter turn.
In math we usually write transformations like this in reverse order: i.e., the first one performed in time ends up on being written on the right.
So, the left box ends up as X * Z * Box.
The right box ends up as Z * X * Box.
(By the way, the reason for the reverse order is that it seems natural, at least to mathematicians, to put the operation that operates first on the box to the immediate left of the word “Box.” Why does the word “Box” have to go on the right? It doesn’t, of course, but it is usually done that way.)
You’ll see that the boxes end up in very different positions.
In short, X * Z * Box is not equal to Z * X * Box.
Well… first, this is pretty weird. I would have thought they would end up the same! That such a simple geometry experiment gives unexpected results is rather a surprise.
Second, this invites various other experiments. What if we rotate by half-turns instead of quarter turns? What if we let X be a quarter turn and Z a half-turn. (By the way, I chose “X” and “Z” because the axes we are rotating around are what are usually called the “x-axis” and the “z-axis,” but I did not need to use those particular letters.)
Third, kids nowadays are expected to learn the “commutative laws” of addition and multiplication in early grade school. It tends to be hard for kids to see why these are really a big deal: how could things not commute!
Well, rotating Wheat Thins boxes by quarter turns is something even young children can do, and yet these operations do not commute. Commutativity can fail in fairly simple ways.
Finally, this ultimately connects with some quite advanced math, that is of interest both in pure mathematics and in applied fields ranging from computer graphics and robotics to elementary-particle physics.
Rotations are normally represented by matrices, but they can also be represented by “quaternions,” invented by the nineteenth-century mathematician William Rowan Hamilton: the fact that rotations can fail to commute is therefore a sign that matrices and quaternions will also have to exhibit this kind of non-commutativity.
Hamilton’s invention of quaternions (and their generalization to “octonions”) is an interesting story all by itself, and it connects to another simple math demonstration: the fact that you can rotate a teacup (with tea in it) by two full turns, holding it rigidly in your hand, without spilling a drop and without dislocating your shoulder (this is known variously as the “Philippine Wine Glass trick,” the “plate trick,” etc., but it is not magic, but a simple fact of mathematics).
More broadly, the group of rotations in three-dimensional space is what is knows as a “Lie group” (after the nineteenth-century mathematician Sophus Lie), and most Lie groups have this same property, i.e., that most members of the group fail to commute.
In physics, this failure to commute is one of the most important differences between the strong nuclear force and the electromagnetic force: the electromagnetic force is due to a commutative Lie group, the strong nuclear force to a non-commutative Lie group.
In short, there is a whole lot of math and science hidden behind a couple of Wheat Thins boxes!
So, what does all this have to do with homeschooling?
Well, this is about as simple a homeschool project as you can get in terms of necessary equipment and preparation time.
But, more than that, it illustrates a central point I am trying to make in this blog: ideas that are usually considered very advanced and complex in math, science, etc. can actually be introduced at a very early age.
Young kids cannot of course understand everything (indeed, neither can adults), but they can understand at least a bit about most things.
More than that, nobody can grasp complex ideas in one huge gulp: the idea in American schools – whether public schools or universities – that you can grasp algebra or calculus (or Lie groups) in just one nine-month period is a horrible mistake.
(In fact, I myself recently learned something about Lie groups – a simple proof of a theorem called the Baker-Campbell-Hausdorff theorem, which shows how the violations of the commutative law are almost the only thing that really makes Lie groups complicated. If not for the violation of the commutative laws, Lie groups would turn out to be rather like the surface of doughnuts – hyper-tori, as mathematicians say.)
This belief in teaching subjects in one huge gulp is connected to the “developmentalist” fallacy: i.e., the belief that kids are not ready to learn anything about many subjects until they reach a certain “developmental” level, and then, all of a sudden, the whole huge subject can be shoved down their throats.
Human beings do not learn that way.
One of the greatest advantages of homeschooling is that we can dump this dogma of “developmental appropriateness.”
We can talk to our kids about black holes, or have them see that rotations do not commute, in first grade. They can read about knights and castles, pharaohs and mummies, fossils and plate tectonics, early in grade school.
They will not grasp everything, but they will grasp much more than the dogmatic disciples of “developmental correctness” claim they can grasp.
So, get a couple of Wheat Thins boxes (or Cheerios boxes, or whatever you have in the pantry) and show your kids how simply rotating simple objects is much stranger than it looks.
And, tell them that understanding this strangeness is not only useful in robotics and computer graphics but that it also helps explain what holds protons and neutrons together inside the nuclei of atoms.