Why is math so hard?

Why is math so hard?   What can we do about it?  Answer - give up!

More precisely,  when the approach we're using keeps on not working,  instead of doing more of it, maybe we should rethink the whole approach and give up the way that keeps not working.

I had four years of math beyond calculus and tutored many students, so I feel qualified to present an opinion on the subject.

I love math and science.  I'm very pro math.

But, I'm also pro reality.  Reality is good, even when it's inconvenient.

Most people don't like math. That's also reality.

Why is that?

I think the main reason math is so hard is that you can't get by being "mostly correct" - you have to be entirely correct.

An English paper with a grammatical error can still be excellent.  A math equation with an error is completely wrong.

Surprisingly,  if you are compulsively neat and organized,  math itself is easy, and, I love it because problems have correct answers that don't depend on what the teacher thinks.    On the other hand, I can't stand problems like "What did Hemmingway really mean by the image of the fish?" because the answer depends on the instructor.

The reason math is difficult is that most students today have no experience being the type of compulsively neat detail-obsessive workers that it demands in order to work at all.   Unless you get 100% of an solution's steps correct, the answer isn't "mostly right", it's totally wrong.

It is as if 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.

So, the implications of that strong assertion are enormous.  So everyone is playing "Let's pretend."

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" without it.  It cannot be made to work.

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 ever "do" math or science in any socially useful sense.

If my analysis is correct, then to get STEM education to work,  we need to have wide-spread specialized remedial courses in 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. 

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. "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.

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

OK.  Then let's be creative about this framing of the problem.   Let's stop pretending most students in K-12 today are ever going to be very good at STEM skills.    Period.  They are not.

Yes, in the long run, we should improve things, but in the short run,  don't bet on it working.

Working separately, as competing individuals,   we are very unlikely to win or even catch up and break even with the Chinese.

Isn't there ANY way to build a reliable system out of semi-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.   

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 treams of people working on problems together.

And, just like SEAL teams in the military,  maybe these teams should persist across years, learn to work well with each other, and then go apply for jobs as a team, not as an individual, and stay together on the job.

On the job front, one way to handle the logistics would be to incorporate the team as, say, an LLC and have the LLC take a job slot as a "consultant."  That can be done with today's technology.

The problem is the educational system, oriented almost entirely around work-units of size 1, that is, "individuals."

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.

Not only should that be "fair", it should be encouraged.

From the point of view of "business" or "commerce",    the only thing that is needed in a particular "slot" or "job" is some agent (person or company) 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:

• We could try to back-fill remedial high-quality self-discipline into our students and culture and also "learn math",  or
• 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 if 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 cancel out that sloppiness by working together.

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 should be allowed to 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.

Here's a crucial point:  if we demand students perform amazingly well as individuals and graduate high-school and college as "singlets" before considering them for inclusion in a work team, there are two guaranteed results:

• People who would make great team members and totally boost team energy and morale, say, but are mediocre working alone will never get a chance, as they'll drop out early as "failures". 
• The people who do succeed as singlets are then supposed to do a u-turn and work as team members, which they just spent 12 or 16 years learning to avoid.

After 16 years of socializing people as fiercely competitive individuals, expecting them to make good team members is, frankly,  unrealistic.   It's as unrealistic as expecting tenured faculty members, after 7 more years of "doing their own work" in the tenure process,  to end up being "collegial."


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.  

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?

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

A question that Science and Religion can come together around

As the war between sects of Islam heats up again, it becomes ever more important to find a common ground we can all move to while preserving our most crucial interests.

Sunni versus Shia,  Catholic versus Protestant,  Christian versus Muslim, Muslim versus Hindu,  it's pretty much the same old battle we've had for tens of centuries, but with frighteningly more and more advanced weaponry.

Then we have the battle heating up between all of the above and the institution of Science,  which may be losing ground far faster than it realizes.

This is not good, given that passions are high and willingness to destroy entire countries and cultures is already on the table.

So let me present an alternative.   A middle ground.   A place in which, as in the Harvard Negotiation Project's book  Getting to Yes, we can each protect our interests even if we have to relax our "positions" somewhat.

 If we take science's numbers and Drake's Law, to paraphrase Carl Sagan, there have been "billions and billions" of civilizations others than ours that not only made it to this point of technology, but did so before the Earth was even born.

Even if most of those self-destruct, at least ONE probably survived and has therefore been around for over 5 billion years. One is all it takes.

If we assume that (a) faster than light travel is possible and (b) they have the same tendency to put sensors everywhere as we do, then "they" have not only already been "here", but their sensors and probably their intervention agents are still here, busy at work around us.

For reasons of pure hubris, humans seem to want to avoid counting "1, 2, 3, ..." to get to infinity ("God"), but prefer to count "1, infinity" as if God is only one short step above mankind. That's unsupportable logic.

The far more scientific question, rather than investigating infinity with theologians, is to investigate the nature of "2", i.e., what's right here, all around us, that's higher than us but still way way lower than God?

You in the back row? No, "Congress" doesn't count as an answer.

The burden of proof, it seems to me, is on proving that we are NOT surrounded by a consciously managed framework, no more mysterious than our interventions to sustain the coral reefs.

In scientific terms, religion becomes mostly people sensing that framework and adding fanciful details.

A task on which both Science and Religion, as institutions, could and should agree on is figuring out what the shape and nature is of the real but non-mystical active and adaptive framework that surrounds us.

In fact, documenting such a framework might, in fact,  be a major step in defusing the perpetual and very destructive wars of different religions, or even different sects of different religions, over what are essentially minor cosmetic details compared to the massive framework we will find if we simply stop fighting,  develop and calibrate suitable tools for "looking" for such a framework, and take the time to look.

If nothing else,  good tools should prove their capacity by revealing a number of places where external, but very real, human agencies are messing with affairs we always suspected but couldn't prove.

* The image above is from Harvard's Kennedy School