kitchen table math, the sequel: The "traditional" math sequence

Monday, April 12, 2010

The "traditional" math sequence

Well first there is algebra in 9th grade. But then take the entire sequence geometry --> algebra I --> algebra II --> precalc --> calc I --> calc II etc. (Maybe if you are advanced enough you get to do Calc III or linear algebra in the last term of high school or maybe the opening semester of college.) Why necessarily in that order? Is it necessary to have fully mastered all the theorems of geometry before moving on to algebra I?

Furthermore, there is a perspective lost in the entire sequence, because generally, a lot of fields from number theory to graph theory are not even considered until college, except as shallow "enrichment" activities. Are kids capable of doing hardcore number theory in eighth grade? Well take the International Math Olympiad (for young students) or your average state math competition. You can get straight As on your calculus exams and totally get creamed at these competitions by a bunch of ninth graders from a school with a more dedicated math team.

Formal linear algebra is mind-expanding, but it occurs to me it doesn't require college-level understanding to master its basic concepts (like transformations). Arguably, you /really/ understand what differentiation and integration is all about when you learn to map from one polynomial space to another. The most important thing about solving math is abstraction, and a rather powerful abstraction is the ability to rotate entire shapes by a simple entity called a matrix; ever had that annoying geometry problem where you had to tediously map all the points of some crazy polygon after it was rotated an arbitrary angle? Well now, what a perfect time to bring up a concept that would systematically map every point to its new point after the rotation...

Here is the place to use a calculator. Take a polygon with 20 points (or some inconvenient number). Students would then have to design a matrix that carried out said transformation. Perhaps an arbitrary transformation whose formula wouldn't generally be memorised (like with standard rotations). The calculator is the "reward": simply input all the points, the appropriate transformation and then report the results of the magic. Here, the calculator ensures that students actually understand how to use the abstraction being learnt (rather than say manually applying an algorithm to each point). On the other hand, it is not being used as a crutch for bad understanding.


There has been much said about "holistic" educations. Many ostensibly "holistic" curricula are not holistic curricula at all. One of my disenchanted teachers put it -- you can't just toss a bunch of department heads together into biweekly meetings and call the result "holistic". (Now mind you, this teacher was from a top Singaporean school and he had been "released" from duty by being too vocal about certain teaching practices.) There is so much reteaching of math concepts from applied subject to applied subject. And oh, all the forgetting! Imagine what could be done if all the inefficiency was removed...

See, I don't actually disagree with the actual idea of integrated math. Integrated math should be a vehicle for exploring important life concepts (like, ooh, equilibrium! optimisation under constraints! recursion! etc.) taught tightly with useful, complex, mind-expanding applications. You could pack a lot of learning into a year. It shouldn't be about using matrices as some sort of organisational table for keeping scores of random soccer matches.


Allison said...

That's not the standard US fare.

No one does Geometry before Algebra 1 in any school I've seen, so no, they don't think anyone thinks it's necessary.

The standard seq here is Algebra 1--> Geometry-->Algebra 2-->Pre Calc-->Calc (with no 1,2,3 distinctions.)

My high school 20 years ago had an honors track that did algebra 1 and the first 5 chapters of the geometry book in one year (roughly a semester each, first five got you through triangle congruences) followed by the rest of the geometry book and algebra 2 in year two, a pre calc course in year three, and then AP calc in year 4.

There is a huge conceptual leap needed to get to linear algebra. I've posted about this somewhere else on this blog. It's not that you need college level understanding, it's that you need college level sophistication. You need to have seen something abstract before. you need to have seen some ideas about operators and functions, and calculus in the only course where high school students have any chance of having seen it. HS geometry is just not formal enough, away enough from the page to get there.

Understanding "span" was something I could not do in my first semester of linear algebra, even though I had received an A in the course. My second time through the same material (MIT didn't allow my UC San Diego course to count) I was no better, and my grade much much worse. But somewhere after several years of classical mechanics, quantum mechanics, and stat mech, linear algebra was a piece of cake. When did that gel in my head? I've no idea.

I do remember that the course that taught me how to understand operators and transformations was not a math or physics course. It was the first CS course taught at MIT and UC Berkeley, a course taught in a dialect of LISP, where the concept of state and operating on that state is so elegant and clear that I think it's the most important course those schools offer.

Anne Dwyer said...

I totally agree with you here. Why didn't I see modular arithmetic until graduate school? And number theory is totally accessible to high school students (or younger for some topics). I'm taking a special topics class in Number Theory now. We are learning all kinds of interesting (and fun!!) things that tie in directly to what you can do with a math degree in real life. There are lots of interesting topics that lend themselves to learning basic proofs.

Anne Dwyer

le radical galoisien said...

Hmm, but on the Math Olympiad (or random state competitions) abstract operators are seen. They're not trivial SAT-style problems either.

I kind of skipped geometry and algebra I in my American HS. I had the impression that geometry was taught before algebra I. (This is with me failing my sec 2 Singaporean math course with a grade of 47, before makeup). Most of that stuff is taught by the PSLE.

In fact abstract operators, etc. would be an interesting way to analyse PSLE-style problems, especially those problems where you kept shuffling beads from one box to another with some resultant change in ratio. Sometimes iteratively. Which is where an operator representing a "shuffle function" would be useful. Gosh you work hard for those 5 points.

Okay it wouldn't exactly be a PSLE problem, but it would be inspired by them. A little step away from Olympiad.

Barry Garelick said...

I totally agree with you here. Why didn't I see modular arithmetic until graduate school?

Yes, why until grad school? I saw it in undergrad

SteveH said...

Some schools swap Geometry and Algebra II. I think the reasoning is that the kids will be more advanced (?) so they can get more rigorous in proofs. I tried to find out what textbooks they (a couple of high schools in our area) use, but got nowhere. I think they use the same textbooks which are poor in the proof area, so they don't gain much by the swap.

My son has the basics of matrices and transformations in his geometry book. He also had some exposure to solving simple systems of equations. I don't think they ever mention the term linear algebra.

"I don't actually disagree with the actual idea of integrated math."

But I'll bet you disagree with what they are doing to the term. It's like balance. I could create a math sequence that was integrated and balanced, but it would be nothing like what you see in K-12. I would have to call it something else.

Then there is the issue of lost opportunity. It's easy enough to introduce the concepts of nonlinear optimization and constraints. I can come up with all sorts of college topics that high school kids can understand, and if I taught high school math, I would find ways to do that. But I wouldn't change the content or sequence of tratdional math courses in high school (much). It would still be a bottom-up approach to math, not a top-down or thematic approach. I would still emphasize mastery of skills rather than general concepts. I suppose I could come up with a more rigorous thematic approach, but that might only be for the most motivated students.

It might be great to motivate kids by talking about where the tangent goes to zero on a curve and what that means, but I wouldn't build a curriculum around that. It's easy enough to teach a binary search algorithm to find the roots of a function and talk about error bounds, but how much time should you spend on that?

The ed world uses words and terms that nobody can disagree with, but things get changed completely when you get to the details. I think it's a combination of ignorance and deceit.

I should have kept the link from years ago, but a well-known education professor made the comment once that (in effect) K-12 educators define K-12 math, not college math professors.

jtidwell said...

In the Louisiana public schools, where I grew up, the sequence was Algebra I -> Geometry -> Algebra II -> Trig -> Differential Calculus. When I got to the gifted magnet school (LSMSA), the possibilities for senior-year math opened up: linear algebra, integral calculus, multivariate calculus, etc. Students generally succeeded in these classes -- we were used to thinking in terms of abstractions at that point -- but obviously the student sample was highly biased. :-)

MIT no longer teaches the intro CS course in Lisp, sadly. I loved that class. It was a magnificent way to learn how to think productively about abstraction -- and it was hands-on! If you didn't get it, you couldn't write a working program. I wonder if it would be useful to have high school students learn a little Python or Ruby in the course of a math class, and then grapple with some of these abstraction concepts (functions, arguments, recursion, etc.) in the remarkably unforgiving context that programming offers.

Anne Dwyer said...


I didn't see modular arithmetic as an undergraduate because I didn't major in math....I majored in Chem Eng. The only math we took was the 4 semester calc sequence.

Anne Dwyer

owen thomas said...

nice post, rad.

GCD and LCM, being made explicit
and worked with in some reasonable
and consistent way, would all by itself
be a revolution in elementary math ed;
i expect the consequences could be
pretty noticable if one were allowed
actually to keep track honestly.

clear up a lot of difficulty with "fractions"
for one thing. just like that. print up
a book people can actually read,
distribute it widely; bingo.

and that's without modular arithmetic;
there's *plenty* of number theory that
can be done by elementary kids as i know
from my own life: i got modular math
in sixth grade and it did me a world of good.

i got linear algebra way late and with pain
mainly because of the doctrine espoused
here by allison. gotta disagree emphatically.
i've taught some very useful linear algebra
to beginners with much less "math maturity"
than it takes to learn anything useful
about, say, calculus.

there's a practice in the standard pedagogy
that focuses on abstract properties that
generalize nicely to "infinite-dimensional"
spaces (so-called; the very *object* of
"linear algebra" from this point of view
is "vector spaces"); i've been saying ever
since grad school to anybody that would
listen that we'd do our math majors
a great deal of good to require freshman
*algebra* (not calc) and deal with
sigmas (summation notation) and
*finite*-dimensional vector spaces
(strong emphasis on dimension 3;
go ahead and try to guess why);
equivalence relations and their
"factor spaces"; stuff like that.
focusing on "matrix manipulations"
for low dimensions removes
the conceptual barriers allison
cites and indeed clears the way
for their introduction far better
than "limits" and other weird
calculus concepts have have
or could.

the "discrete" courses the computer-heads get
seem typically to have plenty of good stuff
the majors never see (till grad school);
basic set-theory and the hugely important
*combinatoric theory* (counting for the
careful) are relegated to probability
courses for business majors (that
forget everything they've learned
within a few years [because math
is mostly pretty useless in most
of their future endeavors as a rule]).

sets and cross-products
and functions (actually)
considered as
"sets of ordered pairs"
typically come up, if at all,
in a "discrete" course like i've mentioned
or a "transition to advanced mathematics"
course like the one it taught at dominican
(with the smith/eggen/st.-andre text
of that title; a fine book and a blast to
teach), along with (the *hugely* important)
topic of *mathematical induction*.

symbolic logic, though.
wow. start here and get miracles.
dolciani does this so i'm hardly
spilling any secrets. still it
might as well be illegal talking
about this stuff seriously to
big-money players; they'll
never hear it in a million years.
it's been tried and too many
people learned too much
with not enough bureaucrats
getting paid.

there's no *reasonable* grounds for
any of it unless "turf wars" count
reasonable... but if they *do*, everything
begins to make a whole lot of sense.

Anonymous said...

"Are kids capable of doing hardcore number theory in eighth grade? Well take the International Math Olympiad (for young students) or your average state math competition."

Two things to keep in mind:

1) The kids competing in math olympiads are not "typical." I'd venture to guess that most schools do not have enough of them at each grade level to justify creating a class. And I don't know how to do a class with all the math-brain kids, but some of whom are 4 years ahead of the others.

2) It appears that the *VAST* majority of K-8 teachers in the US do not understand how fractions work. They can teach algorithms, but don't really understand the underlying concepts. Even if you find kids who can do number theory (and graph theory and ...), these teachers *CAN'T* teach these kids.

High school can/should be different, but (1) would still apply, and while I'd expect the teachers to be better, I'd still worry than many of them wouldn't do so well teaching that far out of their traditional curriculum (e.g. having taught algebra and geometry for 10 years, how ready are they to teach graph theory?).

-Mark Roulo

Allison said...


this Allison is Greifer-Allison, if you remember. Though maybe I'm better off if you don't. Anyway, nice to see you, so to speak.

I know MIT doesn't teach it anymore, but Cal stil does. And so do a few dozen other schools. Cal's best cs teacher, Brian Harvey, taught that class for decades. His TA Matt Wright and he wrote a book that would be great for high school students, called Simply Scheme. That book was used as a pre-cursor to their 001 course, for students that had no prior programming experience. No OOP but functional programming and recursion. The Little LISPer is the best inquiry driven constructivist book ever written. It would be good in high school too.

Allison said...


the shuffle operator is, yes, an operator. But that's a lot less abstract than transforming to find eigenbases, or going from SO3 to SU2.

The shuffle operator is a good example of how functional programming clarifies operator theory more than, say, differential equations do. So again, Simply Scheme, or a LISP/Ruby/Python/functional programming language course would be a great benefit before tackling eigenbases.

jtidwell said...

Allison, I can't honestly say I remember you, but we might have overlapped at MIT. Nice to hear someone else remember Scheme and 6.001 so fondly!

I think I have a copy of "Little LISPer" somewhere. Is that the one that's a series of Socratic dialogues between two characters? ...No, that one's about Java. Never mind.

My first introduction to linear algebra was when dealing with motion vectors in physics. It made a ton of sense in that context, and I found it a very elegant way of expressing a physical system. Idly, I wonder how poorly high school physics is being taught these days, if so many kids have failed to learn any higher math.

I hear from a chemistry-teacher friend that most kids come into high school chemistry with extraordinarily poor preparation. They have trouble reading and constructing really basic data graphs, for instance, and can't do dimensional analysis to save their lives. He has to teach them all that.

le radical galoisien said...

Umm, why the need to teach kids about SU3, etc. that early? I mean, it's not the kids need to learn the finer mathematical details of particle physics at that age?

On the other hand, there are a lot of natural processes that are both imaginable to HS students and good to describe by linear algebra.

What about whole concept of iteration. Iteration that is not blind "trial and error" but rather gets you closer to the answer with each step with each additional term in the sequence. You can even avoid the quadratic equation (especially when it pops up in physics or chemistry problems) by linear approximation and then iteratively "correcting".

(Now that I look it up ... this seems to be a special case of Newton's method. Hmmm.)

owen thomas said...

i had the amazing good fortune...
plus i "cheated" by overlooking
some so-called "prerequisites"...
to attend dan friedman's
introductory lectures on scheme
(CS 311 if i recall correctly)
when i was an undergrad
at indiana u.

the guy was *good*.
i got to watch a lot of teachers
over the years on the way
to my doctorate... but this
guy... that wrote _the_little_LISPer_,
by the way... there were some
people on this thread talking
about what a great tool
LISP is for clarifying one's
thinking about how code works... well,
he stands out even among the standouts.
[the TA... AI's we called 'em there...
was eric tannenbaum, another
talented guy and a pleasure
to work with (don't call me
"ET"). dropped out of grad school
i think. anyhow, next time i
saw him he was recruiting
for TI (texas instruments,
the dark overlords of the
mathed universe).]

anyhow. great book. great lecturer.
i love IU more than i'll ever be able
even to to begin to say. scheme rocks.

great thread here.

Allison said...

My point was not that you need to teach linear algebra to youngsters, but that you should do something else first to teach some operator theory. I guess I misread the original post; I thought you were suggesting we should be doing linear algebra (continuous transformation on vector spaces, etc.) before calc.

I strongly suggest that instead, we stick with discrete math and CS theory as a way into operator theory, and that that can be done with advanced students before/during their intro to calc. Some number theory, even as far as proving the chinese remainder theorem is taught to the students who go to math camps like Ross and PROMYS and Hampshire. Those kids are very bright, but many more are very bright and would respond well to seeing that they can *prove* something and know what that means.

These do make it much more manageable but I wouldn't call it linear algebra.

Allison said...

Jen Tidwell,

We travelled in the same circles, though you were a few years after me, I think. I lived in Sr Haus and was course 8 until I decided that 18 hurt less. I was close to the zbt crowd that later inhabited Curl, etc.

Your comments about high school science are tip of the iceberg. As Catherine says, it's worse than you think. Physics and chemistry in high school are disasters because the students come in so weak in simple algebra and geometry. They don't know enough baby trig to understand parabolic motion; they don't know enough algebra to recognize inverse relationships. And yes, Mark's point that we can talk about this nice advanced stuff, but where would we find the teachers to teach it is dead on. The problems start in elementary math ed, and compound with every year.

Allison said...

Dan Friedman's book is terrific.I can only imagine he's as good as Sussman and Abelson, and taking the class from him would be fantastic.

The Little LISPer is a dialogue; I guess you could call it socratic, though my impression of Socrates from Plato is that he never REALLY wanted to teach anyone as much as he wanted to AHA! CAUGHT YOUR ERROR! Youre WRONG and I'm RIGHT ! :)

The Little LISPer gives examples (something I don't recall Socrates doing) and encourages inductive reasoning on them, as well as deductive at times. It has very very few actual English sentences that are making arguments about what you see/get. It's almost entirely a set of questions about the right output for the given input.

It is the first book I've ever seen like it, and is just joyful.

I guess I'd say it's more like the game of Myst than any other book I've ever read.

le radical galoisien said...

Exposure to the (useful) use of matrices should start early. I think part of the underteaching of linear algebra concepts is how it developed late and so is represented late. But linear algebra should not be started in college, I think; its foundations should develop much much earlier. And it makes calculus make so much sense.

Take two-dimensional rotation operations. There's no reason why that shouldn't start in say, fourth or fifth grade. I mean, you use coordinate systems in fourth of fifth grade. "Baby trig" should also start a little earlier.

le radical galoisien said...

Part of the issue with teaching calc is that you rapidly fly through different functions and their derivatives -- and really, I feel like kids would get a whole lot more out of it if it was "stretched out", and began earlier. You don't even have to formally define a derivative until a little earlier, or what dx means, etc. You can introduce it in very broad, informal terms, and then make it more rigourous as they get more advanced.

Take the fact that d/dx sin x = cos x. I didn't REALLY appreciate this fact until I noticed one day a few years ago in a table in the back of my physics text book that near x=0, sin x increased rather linearly such that sin x = x. I thought, "that's rather odd".

It took looking at a clock (a handy unit circle already divided into units of 30 and 6 degrees) to realise that d/dx sin x near 0 is cos 0 aka 1. But it struck me that cos x was falling quite slowly, such that sin x = x works quite well until 15 degrees (x=0.262) and even at 25 degrees (x=0.436) the approximation's off by only 0.013.

This seems tiny but in a table, with values listed for every interval of 0.01, it suddenly seemed a whole lot more impressive.
So I had to look at the rate of the rate of which it was falling: -sin x. And of course, when x is near zero, cos x is falling at a rate of near-zero (explaining why the linearity is maintained for quite a while), but this rate of fall will increase in magnitude as x progresses.

But that's rather curious isn't it? Having your second derivative defined as the negative of yourself. You can go in a little loop, that goes on forever...

Oh hey y'' = -ky.
Oh hey periodicity.

And then it REALLY clicked. The trouble is, no teacher ever mentioned that to me. It was all mechanical, procedure-memorising stuff. You memorise how the derivatives of sin x and cos x go in a little loop, you prove that d/dx sin x = cos x as an exercise, and you have to graph stuff and its derivatives but you don't even go over the fundamental rate-increasing / rate-decreasing behaviour of such functions, or that any time you have a relation y such that y'' = -ky, you get a periodic relation and this is the basis for biological clocks.

Biological clocks are rather cool -- simply have a few genes in group A which activate transcription for genes in group B, whose protein products inhibit either the transcription of group A, or the protein product itself, depending on the type of periodicity "desired" (or selected for). Then maybe set up secondary loops in the same fashion (inhibiting or activating transcription as necessary) to fine-tune the period length to say .... 24 hours. Then set up a way for light input to train the clock and a way for the clock to interact with behavioural output. A simple case of periodicity emerging from y'' = -ky, arising through natural selection.

You get the interesting phenomenon where inhibiting a clock protein or its transcription (say, PER, or CLK) can slow down or speed up a clock, depending on what phase of the clock you're inhibiting it in. This is not because the protein has both inhibitory or stimulatory interactions...

le radical galoisien said...

For periodic genes (about 10% of the human genome exhibits regulation by the clock circuit), you can arguably say that rate of transcription (dependent on the state of a genetic locus), rate of translation (dependent on the amount and state of mRNA) and rate of protein activity (dependent on the amount and state of said protein) are rather analogous to acceleration, velocity and position.

Sadly, no teacher or professor in either my physics or chemistry classes have ever used that analogy. =(