2x2, I presume? I'd be delighted to be wrong!

Rotation matrices?? In sixth grade???? Are you kidding me?????

I didn't know what a 'matrix' was until I took linear algebra, in second year of UNIVERSITY.

Learning stuff before you're ready for it is unfortunately a hallowed tradition in our schools. I studied set theory in grade one, and so did many of you.

Other 'hallowed' traditions include;
- learning stuff way after you are ready for it,
- going over the same course material year after year ("I don't care that you went over this last grade."),
- cliques, and
- zero-tolerance for topic du-jour for the students.

There was nothing about grade school set theory that I was not "ready" for, although it was never integrated into the rest of the curriculum. It was just this fun but mysterious ritual played out at the beginning of every year before being forced to do long, tedious sheets of long division in the months that followed.

I can't see what's wrong with rotation matrices. It'd certainly be useful to teach them *before* teaching complex numbers.

Maybe I'm bucking the conventional "back to basics" wisdom, but I tend to think that if kids didn't have to learn how to emulate a base-10 ALU with a pencil and paper--or anyway weren't drilled to do it fast--there would be a lot more opportunity for teaching interesting mathematics (including Conway's Life!) at an early age.

Is he seeing rotation matrices for angles other than 90, 180, and 270 degrees? If so I am deeply disturbed; if not, less so.

I didn't see matrices until I believe the 7th or 8th grade. But we did see 90 degree rotation matrices shortly thereafter.

But then again, if the point is teaching geometry,
why show rotation matrices at all? The axis-angle
representation makes a lot more physical sense.
Or it could be a great way to introduce the kids to quaternions, which are much simpler objects to
understand and handle than matrices.

On the other hand, if it is just a way to
show that matrix algebra is actually useful
for something....

Franco
(who was taught quaternions in 6th grade in Italy)

Cripes, he didn't even pick a good Clarke story -- and he wrote an awful lot of them in the 1950s.

Egad, I am envious.

Didn't seem to set you back.

I sort of think that the point of education is not exactly the things you learn you learn, but what the things you learn do for you. I am curious how the educators think teaching rotational matrices (whatever they are) will benefit 6th graders. Teaching advanced geometry to 6th graders definitely has no practical value in the vast majority of cases.But they may believe it improves thinking skills or understanding of the world.I can see that. But if improving thinking skills is really what they are after, why aren't concrete thinking skills explicitly taught? If we really want kids to analyze problems, develop hypothosis, test hypothosese, etc... then it shouldn't be to hard to come up with basic lessons, algorythms to follow, mental techniques to practice. I believe something like this would achieve more than teaching rotational matrices.

This is not an area that I know a lot about, but it's something that occured to me today after reading an essay on Dewey.

Sounds like a pretty enjoyable class you were in. I thought schools weren't hard these days? Oh wait, your child probably goes to private school...?

A confusing typo and some bad misspellings in my previous message. There should only be one "to learn" in the first sentance. This is how you spell hypothesis and hypotheses.

What school is this? Given how mediocre my school was, when I have kids, I'd move so that they can go wherever the 11-y-o goes...

Sixth Grade.

Hmmm... let me think about this.

There was the teacher, who beat my hands bloody after I did not spell her name properly on a test. (It was Sleyekhuis, or something like that. )

Then there was the science teacher who shocked me with the Variac(TM) when I would not stop insisting that there were sources of light that did not give off heat.

And then there was the shop teacher, who threw one of my friends into the trash barrel because he was acting gay.

In between dodging the random strikes of lightning I managed to transform a few overheard comments into my own idiosyncratic-and-thus-worthless form of the integral calculus. That one only got me a sweatsock stuffed in my mouth, and a sweaty jock draped over my head.

Damn! I wish my parents had been Berkeley professors.

Just to say what French students learn...

At 8 and 9 years old students learn the Cartesian coordinate system.

At age 12 they learn the rudiments of geometrical proofs (prove that this quadrilateral is a rectangle).

My oldest child is 12, so can't say much more than that. Apparently the 12-year old will begin to learn algebra (solving equations with variables) later this year, or at least his teacher has promised it.

But no matrices yet, rotation or otherwise.

Big deal. Now when the US starts learning from the Cameroi system of education, *then* I'll be impressed.

I'm totally opposed to teaching a rotation matrix to 11-year-olds.

Look, only the first Matrix was any good (and even that featured such lines as "I know kung-fu"). So what you are essentially doing in a Rotation Matrix is forcing 2 of every 3 kids each year to watch a really, really, really crappy movie.

That's just wrong.

glopk:

You were taught quaternions in 6th grade? I had a lecture on quaternions in a graduate robotics class and I *still* don't completely understand them! There's something about 4-dimensional complex numbers that's rather intimidating...

That said, being a 3D graphics programmer who regularly works with matrix math, I would have loved to learn rotation matrices in 6th grade...

But to understand rotation matrices in their full glory, you need a grounding in sines and cosines. Surely a sixth grader has not already covered all of trigonometry already?

I think Brad's point is that elementary school kids in good public schools are working harder on more advanced work than their parents did.

My kids go to an excellent public elementary school in Dekalb County GA, and I completely agree.

I think they're working the kids too damn hard, to be honest.

well, this explains a conundrum from the recent election, viz, why after decades of expanded educational opportunities Americans - per the PIPA studies - in astounding numbers don't know basic facts (eg, WMD were in fact NOT found in Iraq). it's because they've been too busy learning about rotation matrices and "quaternions" (???) to read newspapers.

Interesting point, CW. One thing we always hear about is how America is lagging other countries in science in math. Maybe we're trying too hard, too early, and without enough class heterogeneity.

I was part of a county-wide program in the Howard County (MD) public schools which took mathematically-adept students in middle school, and fast-tracked them. I was through to basic Differential Equations by the end of high school. Given that I started college as a math major (and ultimately became a computational linguist), I have a use for mad mathematical skillz. But the vast majority of my peers from back in grade-school probably don't. OTOH, they really should've gotten better classes in history and civics, and some training on how to interpret statistics and think about what "experts" are telling them. (A friend of mine who went to a public school in Montgomery County, which offered the International Baccalaureate program, said that all IB students had to take a class called "Theory of Knowledge", basically an intro to epistemology.)

(n.b.: I meant, above, "heterogeneity" among classes -- i.e. lots of different classes teaching to different levels of student, not having lots of different levels of student within the same class.)

I've got a masters in Policy Analysis, and I don't even remember what a rotational matrix is, or maybe I didn't learn it in the first place...

A true story: I too was taught about sets in first or second grade. One day when I came home from school, my mother asked me what I had worked on that day, and I said "sets." She instead heard me say "sex," and promptly called the principal to complain.

Since nobody picked up on it, I just wanted to repeat my suggestion of teaching rotation matrices before complex numbers (or, stated otherwise) not teaching complex numbers until you have taught rotation matrices. None of this is going to help if you haven't taught cartesian coordinates of course.

Generally, complex numbers are taught as some kind of mysterious unintuitive thing. But they really make intuitive sense once you've gotten the notion of how to represent rotations, and you understand that matrices can be multiplied, and therefore squared, and therefore have a square root.

Then the identity matrix is 1 and the 90 degree counterclockwise matrix is i (and its square, 180 degree CCW is negative 1). Write every other matrix you like using these two as a basis, and suddenly you are doing complex arithmetic whether you know it or not. (Likewise with quaternions, and their basis.)

In my opinion, kids who like 3D video games and have even the slightest spark of curiosity about how their toys work are already well motivated for this discussion of cartesian geometry at an early age.

The actual way complex numbers and matrices were taught to me was so backwards (anyway in terms of spatial intuition) that it took me till I was out of college to feel I had a solid understanding of what was going on, and not just a cookbook of disconnected rules.

My first exposure to matrices in the form of a seemingly ad hoc representation used in Gaussian elimination. My dislike of solving large systems of linear equations on paper made me allergic to matrices for years before discovering their beauty and simplicity.

In general, the problem with school math education is that it tends to be taught as a disconnected set of algorithms by teachers who (with some exceptions) are tone deaf to the beauty. It's as if you were drilled endlessly on how to solve Rubik's cube as fast as possible by an accomplished Rubik's cube coach who has difficulty offering a non-circular justification for why there is any point to solving Rubik's cube in the first place.

Finally, I think the real shame is that the concept of mathematical induction is left as an advanced topic or some hastily covered unit. It's an easy concept to apply, but one that meets with initial mental blocks. It's also one of those things that lead to "aha!" moments the way long division somehow fails to.

Dear Brad,
Hey! I had an "open classroom" in Sixth Grade too! At least for the first several months. We got to design our own lesson plans. But my teacher, Dr. Feinberg, got in trouble with the School Board over his hippi-esque style of instruction, and after a court battle, they fired him right in the middle of our year. A very tentative new teacher came in who was forced to teach in a political firestorm. Poor Mr. Moore! It was all so distressing. But anyway, I was curious: where was your sixth grade class?

I detested math in school and bitterly resented that I had to learn more than basic addition/subtraction/multiplication/division and how to do basic fractions. To paraphrase The Smiths: Hang the blessed DJ/Because the music that he constantly plays/Says nothing to me about my life. I wanted to learn about literature and art and music, not waste my time with algebra and so on; I had no intention of becoming an economist or an engineer or a scientist. And spare me the "But...but...it teaches you to think!". I could have learned that by analyzing Beethoven scores. Due to my not-even-disguised hostility towards the subject, the people who *did* want to learn were shortchanged; the same with with the subjects I was interested in and people that were hostile to *them*. I hated high school because to me, it was a rehash of everything I learned in junior high.

To address cw's point, I think one reason why Americans lag in mathematics is they are never taught how or why it applies to the world.

1/ My 11-year old is learning statistics - in his geography class. Learning the math while one is applying it, seems to me to make a lot of sense.

2/ In France word problems will (purposefully) sometimes contain extraneous information. "There were ten people on the bus. Five got on. Three got off. The bus driver is fifty-five years old. How many people are there on the bus?" I think this kind of problem is very important, because it forces the child to try to understand what is relevant or not to solving a problem, which is essential for all but the simplest problems (which are the kinds Americans get).

I don't think that's all of it by a long shot, but unless you're an ambitious self-starter, I don't think you will get as good a math education in an American school as you would in many if not most other industrial countries.

I was in the 6th grade in Concord, CA, in San Franciso's East Bay. This was in 1961 or 62. We were in one of the first classes to learn "new math". In fact, Dr. Beagle, who was one of new math's inventors, came to observe our class once. My dad, who was a chemist, was always concerned with our math and science education and he hated new math. It was just a waste of time to ask for help if the problem wanted me to "find the truth set." Before he'd help, I'd get a big, long talk about how crummy this math was. It didn't do much good to point out that it just meant "what's the answer."

Lucky for me, I was good in math, and learing via new math didn't seem to hurt me much. The math I was learning in the 11th and 12th grades was what my dad had done as a freshman in college, and it was still pre-calculus.

My wife teaches fourth grade (California). I've
looked through her teachers math manual. This is
a big grade level for testing. In math, the fourth grade kids are being tested in algebra. To be sure,
not high levels of algebra, but algebra nevertheless. The present California public education/testing system is designed to fail.

A rotating matrix? Yes, that happens to me sometimes when I'm drunk while watching the movie.

Brad, you do a disservice by not saying what the rest of the curriculum is. I think that makes a big difference. An 11-year-old with some talent can easily deal with mathematical constructs usually reserved for college.

In fact, about age 11 is a great time to be doing more advanced stuff. In my day (50 years ago, say), there was a vast mathematical wasteland between about third and eighth grade. Once we finished arithmetic, they laid us off until algebra. We spent our time learning how many rods were in a mile and doing story problems with such concepts. It was like watching paint dry. I'm sure this kind of thing still exists, and it probably has much to do with the low-level of math achievement in this country. Although I think the greater problem may be that when the kid says he's doing rotation matrices, 99.9% of parents in this country say, "Why are they teaching you that!"

My kid on the other hand went to a very expensive school that started teaching math as a separate subject in the second grade, tracked to the talents of the students, and moved the good ones along as fast as they could go. At about fifth grade he was taking a seventh-grade (honors-type) class taught by a high-school level teacher dealing with problems in solid geometry and other fascinating topics. Then on through algebra starting in about sixth grade, finished calculus as a high school sophomore, and so on. No reason good kids can't do this kind of thing -- maybe not quite this fast, but far better than what we are doing.

What it takes is this -- schools with teachers who can teach advanced topics, schools that can be flexible, and parents who know how to say, "That sounds really fascinating, tell me more!" We don't have nearly enough of any of these.

So if the matrices are part of a coherent curriculum matched to the abilities of the class, great.

And the preceding was in part by way of disagreeing with Dave Mathews that simple algebra is inappropriate for fourth graders. I say, a good time to start.

just noticed that commenter names are case sensitive and that whereas I am "CW", another commenter is "cw". which wouldn't matter except that my post, which was intended to be sarcastic (I'd be shocked to find that the overlap between the misinformed discovered by PIPA and those who have any idea what a matrix is, nevermind a rotation matrix, isn't negligible), seems to have been interpreted by some as serious, and hence, argueably anti-intellectual, and might thereby inappropriately reflect badly on "cw". (of course, it also may reflect badly on me and/or my sense of humor, but that's my problem.)

Dear quaternions,

The first time we met was when I started to toy with 3D graphics, OpenGL and stuff. It's strange, I thought you were very advanced, (like, from a university curriculum and stuff ) but now I learned you met with someone else on a regular basis in the 6th grade!

I'm devastated.

Your sluggo.