I have sworn upon the altar of God eternal hostility against every form of tyranny over the mind of man. -- Thomas Jefferson

Sunday, October 11, 2009

Math Interlude:
Homeschool Math by Rotating Wheat Thins Boxes

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.

So what?

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.


  1. Your post reminded me of some of the things I was exposed to back in elementary school. I did not grow up in the US. In my native country, I attended a french school. Half a day was spent studying pretty much all the subjects in french, and the other half was spent studying, again pretty much all the subjects in my native language.

    Anyway, our math in french was called "modern math" even though the content was really pretty old. We did a lot of set theory and that's where I first saw the concept of "bijective sets" and "function" and how a certain operation with a set may or may not produce an abelian group. We saw lots of examples where the group was not commutitative.
    Now I wonder if this stuff is still taught somewhere in the world to elementary school students as part of a standard curriculum.

  2. Hi Dave,

    This is Allison from KTM. I can't find an email address for you on this blog or on your profile; drop me a line and I'll tell you more about your questions about MIT.

    I like your non commuting example for rotations. I too have started teaching my kids about that. We are just talking about rotations, but before that, I found an easier generalization of commutativity: socks and shoes. You cannot put socks on and then shoes and end up with the same state as if you put on shoes then socks.

    I think the general notion you have to not be "developmentally appropriate" is correct, since the people who claim they have any idea what's appropriate have been disproven over and over again. What's important is to try and teach something that stays true when you simplify it down to the child's age. If you can't simplify it for that age without saying something untrue or misleading, then don't teach it yet; think harder yourself on how to get there.

    My greatest difficulty personally is making the curricula coherent when I homeschool and teach these tidbits along the way. I wish I had time as a parent to think through the way I'd like to teach most of the subjects! I've thought a great deal about some, but not others. Do you find that your explanations are significantly better for your younger children, because you've improved?

  3. Hi, Allison!

    You wrote:
    >My greatest difficulty personally is making the curricula coherent when I homeschool and teach these tidbits along the way. I wish I had time as a parent to think through the way I'd like to teach most of the subjects! I've thought a great deal about some, but not others.

    Well... my wife and I have several acquaintances who are physicians. The way we tell the mediocre ones from the really top-rate ones is that the mediocre ones are the ones who are always absolutely certain about their diagnoses and feel no need to worry about the possibility of being wrong.

    The top-rate ones are the ones who are constantly concerned that they might be wrong and so turn over every stone to make sure they have not missed some little detail that might end up being vital to the patient.

    If I am ever deathly ill, I want one of those doctors who really worries about not doing the best possible job!

    It seems to me that the same thing applies to us homeschoolers. The ones who have a canned curriculum and just sit back and relax are probably not serving their kids as well as they could. The ones who are constantly thinking "But could I explain this better?" are likely to do better in fact. (Incidentally, almost all homeschoolers I know are in the latter group, not the former.)

    So, yeah, I most certainly know the feeling you are describing!

    But, that feeling, that we are not making the curricula as coherent as we could, or not figuring out how to include all the “tidbits” we would like, is probably a good thing – keeps us on our toes.

    And, even when we try to explain something to the kids, and manage to botch the explanation completely, that may not be all bad. The kids see that mom or dad is trying hard to explain something but that we did not quite manage to do it. And, we get another shot a few weeks or months or years later, anyway.

    You also wrote:
    >Do you find that your explanations are significantly better for your younger children, because you've improved?

    That relates to my kids’ age differences, etc., and I try to avoid going into detail here about the kids’ ages, genders, etc., so I am going to drop you a line via your e-mail as shown on ktm and address that in the e-mail (that will also give you my e-mail address). I guess I need to figure out a contact e-mail for this blog that I can somehow control spam on.

    I’m of course not worried about legitimate readers of this blog knowing details about my kids; however, since anyone in the world can read this, and there are a few crazy people in the world, I don’t want to post all of those details here.

    All the best,


  4. Hi there Dave! I stumbled on your blog today while googling for homeschooling inspiration and must say I enjoy your posts very much. I have a young science and math lover and hope it's okay to link to your blog so that we can visit it more often.

  5. OK! Clearly you know MUCH more about math than I managed to absorb in school... and I recently discovered that a good deal of what I absorbed in school has gradually leaked out of my "sponge" a couple of days ago when I tried to help my sister with her geometry homework. Yikes. But I did follow the last 6 paragraphs of what you've got here, and I absolutely agree with that! I read all kinds of things to my 3 year old, and while I know that he doesn't get it all, I also know that he gets some, and that in having some, it will make it easier for him to grasp more the next time around. I'll have to show your post to my husband and see if he can explain a bit more of it to me. I used to love geometry, but tonight I'm lost in all the x and z and so forth.

    I also thought you might like to know that the 2nd edition of the Classical HS Carnival is up.