Rails across a swamp - why math education is a waste of time in America

We've had a fascinating discussion of what constitutes core competencies and concepts in math, and now physics education.   I myself had benefit of one of the USA's top-10 high-schools,   and was an undergraduate Physics major, and just love math and physics.   I've also struggled to teach this skill and love to people I've tutored or taught.

On reflecting on the struggle,  and how to overcome it,  I think it might be worth pondering together whether we can separate out practices from concepts, and address them separately.

My own sense is that the concepts in physics and math are not that impenetrable or alien to children.  As college seniors we were allowed to bring a 3x5 notecard to an open-book exam once,  and it was not that difficult to put ALL college physics onto that card.  There's not that many equations or concepts -- almost everything is implications and derivations from them.  It's not nearly as full of separate islands of disconnected concepts and "facts" as history or biology or chemistry.  And, what concepts there are can be demonstrated in videos and experienced in person or on-line in a simulated experimental world.

To me, it's way way easier than "What did Hemmingway really mean when he used the image of the fisherman?"  or "Please disentangle the things Islam and Christianity agree on and those they disagree on."

So, why is it so "hard" then?

I submit that the problem is actually subtle, but also staring us in the face, so that, once seen, it becomes easy to continue seeing.

First, let me slightly oversimplify physics and math by modeling the activity as follows:    We know some things about the world, and there is something else we are now curious about.   It's hard, expensive, and possibly dangerous to simply go look or try stuff, so we want something easier, cheaper and safer.

So we circumscribe and transform what we're sure of in the real-world into a simplified abstract model, consisting of "facts" and "symbols" and "equations" and "rules".    Then, we can use some process to do two things, in an order that depends on our sophistication -- one is to "fit" the model's parameters to the real world, and the other is to "extend" the model, or play it out, or transform it so that it reaches to the part of the world we are curious about.    Either way, once the abstract model now is extended to that part of space,  we then can un-abstract it back into real stuff and see how well we did,  or build the thing, or look for the effect, or whatever our goal was.

For the part of life we have, for sociological and psychological reasons called "the hard sciences",  this process is relatively easy.   We CAN disentangle a part of the world from context,   build a model for it,  transform the model, fit one end to reality and read out the other end as our new insight about what "should be true" if our model is valid and no new physical laws have swung into play.  We conclude that "If we build the wing shaped like this, or the bridge shaped like that, it should work" and try it and it does and life is good.

The hard part of this has to do with the nature of the representation of facts and relationships as strings of symbols.  Right there we should stop and consider what we're doing, and historically we have not done that.

Let me oversimplify a huge controversy in the artificial intelligence community by saying that there are two ways to represent stuff:   as symbols and as images.    Each representation has some processes and these days,  hardware which can do the heavy lifting for you.

Images... I love images.   You can take a picture of George Washington on a dollar bill, say,  and randomly change half the pixels to white, or black, and you can still recognize Georgie boy.   We can say images are both packed with information ( a "thousand words") and robust against some types of "noise".   Images are, in that sense, "forgiving".    We can, in many ways, be sloppy about our collection process and our transformation process and still come out with the correct answer.

Symbols and symbol strings,such as equations.  This is a different story.  This type of representation and processing is exquisitely sensitive to noise or what one might call "error".     A single point-wise error in even one symbol can, and almost always will, make the remaining 5 pages of work totally useless.

I would submit for consideration that this is the crux of the "problem", the shoals upon which our students are smashing their conceptual boats and sinking.

It has nothing to do with the concepts of physics.  It has, in the end, very little to do even with the concepts of mathematics.  It has, however, a very great deal to do with the error-sensitivity of modeling and symbol processing.

The reason this is a problem is that our students today have no basis in the rest of their lives for being the type of compulsively neat quadruple-checking each and every step detail-obsessive workers that symbol-processing demands in order to work at all.

We are trying to lay precise rail-road tracks across a thinly-encrusted swamp of sloppy behavior, and, frankly,  that simply does not work, and cannot be made to work.   Unlike image processing, symbol processing does not allow a "close enough" attack to "mostly work".

It is a whole other discussion of what psychosocial context and cultural factors,  habits, training, experience, group memberships, expectations, value systems, sports experience,  and other things come into play in making it possible, or likely, or almost certain that a person can impose on an activity a high degree of neatness and structure.

What is clear, I submit, is that such capacity is an absolute, non-negotiable requirement for "doing math and science", at least mediated through symbol processing.

So, the policy and practice implications of that strong assertion are enormous. Either we bite the bullet and figure out what it takes to accomplish "self-discipline", or we should abandon all pretense that we can "teach math and science".

At this point,  self-discipline can probably be taught only by a continuous series of successful experiences using it to solve problems, which means we need to go way back before calculus, before stats, before algebra and get things mastery-learned to a higher degree of certainty starting with addition.

Put it a different way -- if the closest tolerance your machine-shop can give you is plus-or-minus ten percent on the dimensions you specify for parts,  you should abandon the idea that you are going to build a jet engine.

This is a problem that is completely distinct from questions of what shape the parts should be,  or engine design,  or propulsion, etc.

Either we should figure out processes that will work despite sloppy parts, (maybe a variant of image processing?) -- a sort of Loc-blocks of the mind that makes straight what we sort of get roughly straight,  or we should stop wasting all this time and energy on a pointless task.

There is no point trying to teach concepts to students who don't have sufficient self-control to keep a column of numbers straight on a page.   It may in fact be possible to teach the concepts, but it won't by itself result in them being able to "do" math or science.

If my analysis is correct, then we need to have wide-spread specialized remedial courses in whatever works to teach habits of structured work and self-discipline.   It's absurd to expect our math and science teachers to have to do that on top of teaching math and science. 

Shanghai

So, in other related news, yesterday the results of the world-wide OECD PISA test of 15 year olds were release, showing the USA near the very bottom of the industrialized countries, and Shanghai, the city of 20 million,   scoring at the very top.

See:

Shanghai students ranked best in the world at maths and science

Schoolchildren in Shanghai have been ranked the best in the world at mathematics, science and reading by the leading global study of secondary school performance. 

The counter-narrative told in the USA is that China scores so well because they are basically robots who are into mindless drill and obedience,  at odds with the innovative and creative spirit that has made this country great -- or some such thing.  Every Chinese version of a new car, plane, train, computer, etc is met with the comment that it is a "copy" or "clone" of creative work done elsewhere, most likely here.

Curiously,  they must have one heckuva copy machine, because their "clones" often work 20% faster, more reliably,  less expensively,  or otherwise seem to actually be an "improvement" over the prior version.

There is a mistake being made that confuses "rigor" with "rigidity" and freedom of action with anarchy.  By the "freedom" argument,   anyone trying to run the marathon, or play great basketball or football is hampered by the fact that their bones are "rigid" and they would be much better if only their bones were creatively flexible and not so darned rigid.

The reality is that what they would be if their bones were flexible is ... jellyfish.

For fluidity of graceful and powerful motion, some things should be rigid. 

To not damage Johnny's weak self-esteem,  we keep telling him in school that 85% is "just fine" until he manages to get out in the job market, which is now international in scope,  and discovers that "85% good" doesn't even make it into the "C" pile of candidates, let alone the "A" pile, let alone land a job, let alone allow him to do the job.

In mathematics,  the passing score for any concept should be 100% - -the only exceptions being questions that were poorly designed or ambiguously phrased. "Sort of knowing" something will not cut it.  Getting "most of the equation right" except for that one term there will not get most of the answer right.

This seems to come as a surprise to people, students and teachers alike.   You cannot build a high quality car or plane or computer-program out of parts that are 85% correct, or 95% correct, or 99% correct -- although 99.995% correct might work for machine parts for some things.

However, for mathematical equations and symbol processing systems,  even that is not sufficient. Each symbol, each step, each transformation, each equation must be 100% correct,  or the answer at the bottom of the page is pure rubbish.   Your satellite to Mars will attempt to land 200 feet below the surface, instead of on the surface, which is "pretty close", given the distance to Mars .... but not "close enough."

Either you go with "mastery learning" or you are wasting your time poking at mathematics skills that will never be useful to anyone.   Assuming we want to compete with China,   we need to go for "mastery learning."

And, to put it mildly, we are coming from behind on this issue these days.   Our students are behind. Their teachers were not taught well and are behind, overall, with some clear exceptions.   Given the rate of progress we've seen recently, and the total inability of those in Congress to reason together about, well ... frankly, anything,  it is arguing uphill to think we can get that turned around in less than a generation -- time we simply do not have.

But most of all our concepts of discipline,  structure,  order,  routine, rigorousness are the weak spot with the approach we've been using of "national collective power through individual genius capacity."

OK.  Then let's be creative about this level of the problem.

Working separately, as competing individuals,   we are very unlikely to win or even catch up and break even with the Chinese.  The arguments are above. They look solid to me.

So ... maybe we should put the "image processing" analogy on the table and look long and hard about it. 

Why is it that "image processing" is so strong and immune to damage from "noise" or point-wise error,  unlike the mathematical tools of "symbol processing" ? 

Isn't there ANY way to build a reliable system out of kind-of-reliable parts?

Yes, is the answer.   Creative redundancy.   There is a whole engineering discipline of making reliable "systems" out of unreliable and flaky components.

If we cannot make our INDIVIDUALS reliable,   that doesn't mean we are unable to make COMBINATIONS OF INDIVIDUALS reliable.     Individuals are like symbols;   teams are like images.

If we're going  to go that direction, then the dollars, priorities, and emphasis in education needs to change from attempting to maximize INDIVIDUAL skills and reliability to maximizing COMBINED skills and reliability of small TEAMS of people working on problems together.

I'll argue that it is obvious ("without proof" ) that two people, working together,  and cross-checking each other's work,  should be able to produce a math homework paper that has fewer errors on it than one person working alone.

From the point of view of "business" or "commerce",    the only thing that is needed in a particular "slot" or "job" is some person,   OR TEAM OF PEOPLE WORKING AS ONE,  who can take a problem and solve it in the time available. 

Already we see this in the concept "pair programming", where two people sit side by side at one computer, and together attempt to solve programming problems.    It turns out, if done correctly, this is something like 5 to 10 times more effective at generating workable programs than "dividing up the work" and having each person work in isolation on "their own piece of it."

So, here's the trade off.  To catch up to the Chinese in productivity in problem solving in math or science in the REAL WORLD,  which has, in fact, no constraint of "do your own work separately",     we have two possible approaches: 

(1) We could try to back-fill remedial high-quality self-discipline into our students and culture PLUS "learn math". ,  or

(2)  We could try to remove the "do your own work separately" constraint and start tackling problems as pairs of somewhat-sloppy but cooperating individuals. 

Neither of these is trivial or a cake-walk, but, of the two, the second seems more likely to succeed than the first.  At a mimimum, since we're that kind of place, we should explore some of each, have some schools try to go for structure, and others go for true-pairwork.

Both of these require a cultural shift to support them.

At the current time discipline is not popular.   On the other hand "groupwork" is a dreaded four-letter work in academia as well, as in "Oh God, ... I just found out this course requires group-work. I wonder if it's too late to drop it!"

My point is, if we want to be sloppy about our personal work habits, and we appear to take that as a cultural norm,  and we HAVE to be concerned about product reliability, which is demanded by the mission or a competitive marketplace,  then we have no choice that I have seen so far besides figuring out how to be less sloppy (in terms of errors in the work) when working together than when working separately.

And,  we need to start trying to figure out how to treat a work-dyad as an acceptable filler for a "job" that currently is intended for a work-singlet. (a.k.a. employee.)

How do you pay a dyad? What about health care? Do both people always have to show up for work or can only one show up on a given day? Who cares?   If they both work "from home" does anyone even need to know it's a dyad not a singlet? If we gave the dyad a "name" and a "social security number" would that help?

That's where the problems rotate into with the dyad approach.  

Again, I didn't say it was easy -- I suggested it was EASIER THAN THE ALTERNATIVE, given where we're starting. 

The dyad needs a name, and a resume, just like a singlet-employee.    Presumably, the dyad needs a single paycheck.   Desk space is a problem unless the dyad works "at home."

While we're at it, let's say the dyad IS allowed to "cheat" and have permanent full-time access to the internet during any portion of training, education, examination, or activity during the actual job. That's realistic these days.

So NOW the question is,  can  dyads of Americans,  with access to the world wide web, working just with each other and learning over time how to operate as a team,  trained as a team,  operating as a team,  outperform Chinese singlets working with what they learned and stuffed in their heads, without access to web?

My thought is,  yes.  

So,  we should redefine the OECD PISA test to reflect, not what we have assumed all along is the style of education and learning, but what is the 21st century style of operating.

We should allow small-units,  teams,   squads of  2 or more people, working together in the web-context, connected to each other and their friends on the web,   to compete with Chinese individuals working alone, and see who comes out better on THAT test.

That would more likely work if the teams, like SEAL teams in the US Navy,  or cockpit crews in commercial aviation,  trained AS TEAMS.

This is a whole new ballpark of opportunities and pitfalls, but the only ballpark so far that I've seen that has any chance of catching up with the Chinese -- already in motion and already beyond us.

Of course, it's sort of ironic that the Chinese, with a strongly stated national culture of collective action,  turns out to be succeeding best in the area of competitive individual action,   while the USA, by my analysis,  with a strongly stated national culture of competitive individual action,  may turn out to succeed best in the area of collaborative team performance.

I guess, when you're coming from behind, "whatever works" is a good philosophy.  We have a lot of sports where doubling-up on your opponent is a winning strategy, don't we? If they're allowed to use our tactic of "competition" we should be allowed to use their tactic of ganging up on someone, shouldn't we?

Especially if it would even the odds here and give us a "level playing field."

Besides .. it sure beats having to learn algebra for real.

Anyway, it IS "American as apple pie and baseball."   Baseball is a game where players learn how to collaborate with each other so that their TEAM can "cover the bases" and "compete".

Yeah -- Bill Gates, are you listening?  We need an international competition on CONNECTED-TEAM-ENTRIES into things like the OECD Exam.

You pick your team size,  you are allowed to be connected to the web, it's open book, open notes, open cell-phone,  ok to consult with each other.   Then we can see who can field a better "team".

And you know why that would be better? Because that's EXACTLY what international business competition is about.   You get to pick your size.  You get to pick your structure.  You get to pick your hardware.  You get to network.  NOW, let's see who can solve a real world business problem and take advantage of a real world opportunity faster and more effectively.

So, isn't that what our "schooling" should be preparing us for by running us through it over and over?