Category Archives: Education

Thoughts on Educational Priorities

It seems like our education system currently has the goal of trying to get the bulk of students to some level of knowledge we consider minimal.  It’s like we aim at the middle 60% or so of the curve, presenting material that they should be able to understand, at a pace that works for them. OK, some folks below that won’t get it, and some folks above that could learn much more and more quickly, but we’re doing roughly the right thing for that 60-80%.

And schools spend extra effort/money trying to educate that bottom 10 to 20%.  No child left behind! Get everyone to that minimum!

But then these people go out into the workforce and it’s largely that top 10-20% that creates the value in STEM fields.  And the vast majority of the rest, the ones that we organized our education system around, well, most of them can’t really make effective use of what we taught them.  Yes, they learned algebra well enough to pass that exam in high school, but they didn’t really GROK it.  Now they’ve forgotten most of it, and even the part they do sort of remember they do not grok well enough to use it to improve any process.

Of course one problem is that we are forcing everyone to learn at the same pace in the first place.  We need to find a way to let the top 10-20% go fast and the bottom 10-20% go slow. Honestly, this would help everyone — even bright students have bad weeks, and find some things easier to understand than others.

But it also feels like we are investing the least in the people who offer the most potential for return.  Why not try hard to maximize what the really bright kids get out of school?  Instead of taking the attitude “they’re fine — those others need help,” spend some money on a good experience for the bright kids, too.

Jonathan Wai, a psychologist studying the correlation between early cognitive ability and adult achievement, was quoted in this article as saying “The kids who test in the top 1% tend to become our eminent scientists and academics, our Fortune 500 CEOs and federal judges, senators and billionaires,”

And if some kids are having a really hard time with even basic algebra, why torture them?  Let them focus on something else, something that comes more naturally to them.  No matter how much we pressure them, there is very little chance that they are going to make use of the knowledge outside of the classroom, let alone be the ones who change the world through STEM advances.  If we do manage to get them through an algebra course, and the last they think about algebra is the immense relief that they never have to do that again, haven’t we wasted our time, wasted their time, and done major damage to their self-esteem for nothing?

Don’t get me wrong. I think that everyone should give math their best shot, and go as far as they can — with solid comprehension and high retention. Don’t give up on those who struggle! But don’t insist that they all acquire certain skills, when we all know darn well that a lot of them are not going to remember it after the test. Really, what is the point in that?

I would like to see more effort and attention going toward the Diracs and Feynmans of the next generation, not just the ones who are trying to meet a requirement by demonstrating a skill they will never use in the real world.

For example, perhaps you could arrange to get the top students of all high schools in the state get together online under the tutelage of the best math and science educators for their level?  Have them dive much deeper into the material, interacting with their equally bright peers in small groups.  Young Einsteins and Eulers become friends during class, even though they live in different cities (or even states or countries!), and talk about what they are learning over Skype. Awesome!

My claim is that the top students could learn much more, and more deeply, without being held back (and often socially regarded as freaks) by the other students.  I know this to be true in part because one grade school I know of tried an experiment: they gave all the kids in the upper grades a long math test, and then divided them up into about a dozen groups based on aptitude.  Then they taught each group at its own pace.  Unfortunately they spent no money at all on the top students — they simply put them in separate rooms, alone, with a box of lessons.  Take the next lesson from the box, read the material, and work the problems.  Nobody to talk to — at all.  It was not a good environment: it was socially isolating, and the learning materials were not designed for that purpose.  Yet the top student did 22 months worth of material in 9 months — in spite of feeling bored and alone, and having nobody to ask questions of.

A less visible benefit accrued to the other students: they were less afraid of looking stupid.  Average students are loathe to venture a guess when they know that the “class brain” over there knows the answer for sure and is just holding back to give others a chance.

Imagine if you spent more money on the top students, not less!  Imagine if instead of teaching the top kids the same way you teach the average kids, you really dug in to each topic and had the kids grovel around in the concepts, soaking it all up.  What is algebra, really?  There are other algebras — let’s explore some!  What is a number, really?  Prove some interesting theorems.  Use a language like Idris or Agda to do some math. Imagine the top students coming out of that education system!  Wouldn’t you expect that the downstream economic benefit from having taught those top students much more, and with deeper understanding, would more than cover the extra cost of teaching them at their own pace rather than just shunting them off to be taught with the middle of the bell curve?

I doubt that this would ever fly, because society places value on equality.  Teach them equally, please, even if that means extra effort spent on those with the least aptitude.  Even if most will never use it.  Even if those most likely to change the world with it could have learned far more, and been much better prepared to create the future with that knowledge.

Our definition of “equally” should allow students with greater aptitude to learn quickly and deeply — rather than chaining them to the pace and content of the majority of students.

A Thought on the College Admissions Scandal

The world of US university education experienced an earthquake recently when a federal indictment exposed wealthy parents paying to get their underperforming kids accepted into universities.  I’d like to share my personal take on this, channeling my inner Milton Friedman.

Instead of trying to ferret out and prosecute what we currently consider corruption, why not make it legal and visible?  This will surely sound weird at first, but hear me out.

Each university would produce a ranked list of applicants, each with an associated score derived from metrics indicating their likelihood of success — AND NOTHING ELSE — and then let parents sweeten the pot.  If you really want your mathematically mediocre kid in Caltech’s physics department, you can pay extra according to some published schedule to bump their score.

All above board.  By definition it’s not cheating.  The base score is STRICTLY about potential and is subject to review and challenge.  The base score is never bumped because your parents went to the school, or because of your race or religion or anything else.  The only way to bump the score is with money.

Wait — isn’t that bad?  Don’t we want to force colleges to accept the applicants we think are most deserving?

I’m strongly in favor of getting those students with “the right stuff” intellectually a good education so that they are more productive (and pay more taxes).  So given that there is not enough education to go around, how do we get more of it?

Money.  Note that by accepting that bump money from parents whose kids wouldn’t otherwise rank high enough, we are obviously getting that one kid into school who wouldn’t otherwise go, at least not there.  Sure, some of them will fail — at least one of the kids identified in the scandal didn’t even want to go to college — but the more likely the kid is to fail, the more money the parents have to give the school in order to let them try.  And you use that money to provide more/better education the next year.  This isn’t a zero-sum game; more money means that more kids can get educated.  (And even those rich kids who fail will probably learn *something* from the attempt.)

Want to give a break to a particular kid you believe deserves it?  Fine, there’s a way — the same way open to everyone else.  Want to give an entire class of people a break, like poor inner-city kids?  Fine, there’s a way.  It’s all right there in the open, and it works the same for everyone.  Get some money allocated to the project and you’re off to the races.  The costs of the program are obvious.  No crooks getting rich off of bribes.  No encouraging parents who want the best opportunities for their kids to cheat.  No more spending money to investigate that crap.  Who would bribe a crook to cook their kid’s SAT results, knowing that they could be convicted of a felony (and publicly humiliated) for doing so, when they could just pay the university directly to let their kid in, knowing that the money would be used to educate more kids?

OK, I’ll bet now you’re worried that the school will just keep all of the money instead of using it to educate more students.  But that would be tremendously stupid of the school, since they only get the money for the kids they admit and educate.  More kids admitted, more money.  More money, more slots for more kids.  Any school that decided to take only rich parents’ kids would simply be giving up the opportunity to serve the rest, along with the associated revenue.  Some other school will gladly step up and take the money.

And universities would be required to be transparent about the average score of the kids they admitted, the bump money that they accepted, etc.  If a university accepts so much bump money that it drags down the average scores, then maybe you don’t want to go there.  Every university would strike its own balance, not accepting so much bump money that it would significantly degrade the school’s reputation.  Students could see where various universities land on this spectrum and decide whether they want to pay more in order to get a more “elite” experience.

In short, let rich people pay extra to get their kids in — more kids get educated that way.

If right now you’re thinking that people shouldn’t be able to buy *preferential* access to a university education, you’re still thinking that it’s a zero-sum game, that for some reason having more money does not allow us to educate more students.

I actually think we need far more radical reforms to our education system than this, but I don’t have any problem at all with rich people buying their kids an education that they couldn’t get on intellectual promise alone.  And I would argue that you don’t mind either if you think that it’s OK that star athletes are given preference in admissions — you know that schools do that because people donate more to schools with competitive athletic programs, right?  Not to mention ticket sales.  It has nothing to do with the sport being important in any educational sense; in fact you could easily argue that reifying athletes is anti-educational.  Those athletic kids are accepted ahead of others who are intellectually more promising because of the money they bring with them.  The only difference from the current scandal is that the money is not coming from their parents.  If that difference somehow makes it OK in your mind, I can’t relate to your thought process.

So let’s turn this education scandal around, explicitly making a place for money to influence admissions rather than forcing it underground, where it would continue anyway but with the money going to unsavory folks.  Accept the money in broad daylight and use it to educate more kids.  Let’s be completely up front about it with students and parents — the more you look like a good bet academically, the less you pay, but anyone can come here to study.  Once in, you earn your grades on merit, and *that* is where we will be looking for corruption.

Attitude and Aptitude

My son Carlos just got his A.S. degree in Engineering. The road he took to get it is pretty unusual, and nobody would have guessed that he would accomplish what he has. I think there’s a good lesson in the story.

First I should explain that Carlos is not my biological son. He was born in Honduras, a poor country, to parents who were poor even by Honduran standards. We’re not talking about driving a used car instead of a new one — we’re talking about no car in the family at all, feeling excited if for Christmas you get a new shirt and pants because you won’t be getting them any other time of year, and skipping the occasional meal because there is no food. When I first saw the house his family lived in, I was a bit confused — I thought it was a storage shed.

When Carlos talks to friends from his childhood days and tells them that he is a student at an American college, they think he is pulling their legs. You see, as a kid Carlos was a terrible student, ultimately expelled from the eighth grade for causing trouble. He didn’t see the value in what was being said, and wasn’t picking up much of it. It seemed like a waste of time, so he didn’t invest himself in it — and failed miserably.

Since his family was so poor, he decided to help out by working in the banana plantations.  It was back-breaking work, and the others on his team — all much bigger than Carlos — were expecting him to fail. He surprised them all by coming back the next day, and the next — soon he knew not only that job, but other jobs on the plantation as well. The money was a huge help to his family. Carlos really wanted to help his younger brothers and sisters, especially Delma, whose life he saved when she was a toddler — they have a special bond.

But it was really hard work. One day he was taking a break, sitting on a tree stump, thinking about the men working with him. Some were in their 50s, and the work was clearly a challenge for them. “If I keep doing what I’m doing, I’ll be working on a plantation in my 50s just like them,” he thought to himself.  So when the opportunity came to join his mother in the US, he took it.

Coincidentally, within days of his arrival he went to his mother’s graduation ceremony — she had earned her GED (high school equivalency). She was given her diploma and led to a podium with a microphone to say a few words; tears streamed down her face as she thanked everyone.

Carlos had planned to work with his brother in construction, but I told him that if he wanted to get his GED too, he could, and that it would help him. “Here in the US,” I told him in Spanish, “a young woman’s parents may not want her dating a guy who didn’t finish high school. And there are lots of jobs that require a high school diploma.”

The next day he came to me and told me yes — he would like to get his GED. “Great!” I told him. “Let’s see how much you learned in Honduras. What’s 7 times 9?”

He thought for a few seconds before admitting that he had no idea. In math, at least, he remembered very little. Probably he had never learned very much in the first place; he had never had a textbook of any kind. My heart sank a little — it had been a long struggle tutoring his mother through her GED, and Carlos seemed no better prepared.

But he began taking classes, and I set up a whiteboard and helped as best I could. Pretty soon he understood fractions and basic geometry. He tells his friends now that I taught him Spanish, just to see the surprise on their faces; of course he already spoke Spanish fluently, but he needed to learn the basic concepts of Spanish grammar and improve his use of it in writing. It must have felt very strange to have a gringo explain to him how his own language works.

Algebra was a problem. I tried again and again to explain what was going on, taking different approaches, but he just didn’t get it. He couldn’t even tell me what part he didn’t understand — it just wasn’t working. I resigned myself to the idea that that would be where he stopped with math.

But Carlos surprised me. Instead of giving up, he spent day after day with the books (finally he had books!), and before I knew it he was doing basic algebra. He passed the math test easily.

He ended up with his GED in far less time than it had taken his mother. A GED is really not the equivalent of a high school education, but it was still a huge accomplishment given how little he knew when he started. And remember that this is the same kid who was ejected from school in Honduras. The neighbors, who are well-educated, cheered him on; in particular a pair of math teachers across the street were quite fond of Carlos and helped him celebrate his success.

Carlos had enjoyed his taste of algebra, once he got the hang of it, so when we ran across a thick elementary algebra textbook in a used bookstore I asked him whether he wanted it. He did. He disappeared into his room and started working through it on his own. He worked through far more problems than a teacher would have asked him to do, and he mastered the material.

Carlos had to study English for a couple of years before the community college would let him take regular college classes, but eventually he got there. He took College Algebra, then Trigonometry, then Pre-Calculus, and did well in all of them. He liked it. On he went to Calculus I, Calculus II, and Calculus III. Most people can’t handle math classes at that level — who would have guessed that this eighth-grade dropout would do well in them. Then on to Differential Equations, Statistics, and Discrete Math. Carlos has now taken all but a couple of the math classes offered by the community college (and is signed up for one of the remaining courses this summer), along with a lot of physics and chemistry classes and of course the usual requirements for English, Humanities, History, etc. At first it was a huge struggle for him to write even a small paragraph, but now he writes several papers in English every semester and gets good marks. He can now use Windows and Linux and a variety of tools that run on them — even program a little in Python. And his GPA is a hair over 3.5 — not bad for someone who spoke no English and had essentially no education eight years ago!

So what the heck happened here? How can this guy who is remembered in Honduras as the class clown and a dropout have accomplished so much here in the US, and in a different language? Clearly he was smart enough to have succeeded in Honduras; why didn’t he?

I think in large part it boils down to attitude. Oh sure, Carlos was much better supported here, and the quality of education is clearly better. But I’m pretty sure that fixing those things alone wouldn’t have done the trick. In Honduras Carlos couldn’t relate what the teacher was talking about to his life — it all seemed irrelevant. But he came to the US, watched his mother’s success in education, began interacting with educated people with good engineering and teaching jobs and an understanding of all sorts of things, and he saw the relevance. He had worked for years doing back-breaking labor that paid very little, and realized it was a dead end. When he got here he quickly realized that there was a lot he could learn, that he would understand the world around him far better. He understood that the more he learned, the more he could contribute, and the more he would be rewarded financially. He had always wanted to contribute; it just hadn’t seemed to him that school had any relevance to that. Now it did.

So when Carlos had trouble with a subject, he didn’t give up — he just kept working at it, surprising me more than once with his determination. Carlos feels tremendously fortunate to have gotten this education, and he isn’t going to give up easily when there are problems.

There was in fact one semester in which Carlos stumbled: he had discovered YouTube and Facebook and was spending too much time on them. I knew nothing of the problem until he came to me one day, handed me the laptop, and explained what had happened. He was honest enough with himself to realize that he had screwed up and to know what needed to be done — he cut himself off. It was entirely his idea; he had come too far to let an addiction to the Internet steal his future. He ultimately had to take the laptop back, of course, to do his schoolwork.  And the Internet is his connection to much of his family and to his old friends; he needs to spend some time with that. But to this day he comes to me and asks me to cut off his access to the Internet for a week (I use a MAC filter on the router) when he has a lot to do.

Carlos is smart, but he’s not a genius — the magnitude of his success owes more to attitude than aptitude. I find him an inspiration. And I look at Carlos and feel sorry for the many American youth who achieve far less in spite of having so much more support. He tells me about classmates trying to throw together a paper the night before it is due, seldom coming to class, doing the bare minimum it takes to get a passing grade. They have none of the handicaps Carlos had, but they lack his desire and determination to learn, to succeed academically, and to give back to society.

I can’t help but wonder whether they would be doing better now if, before starting high school, they had spent a year or two hauling bananas…

The Boredom Hymn of the Repugnant

I frequently review books, and often find that they studiously avoid the use of charts, tables, or formulas, even when such devices would make the text much clearer. I’ve seen paragraphs that were essentially the rows and columns of some unseen table, written out in English prose — horrible. No doubt the publisher advised against any hint of mathematics so as not to scare off potential buyers. I can’t help but wonder whether the alleged aversion is exaggerated in the minds of publishers, to the detriment of readers everywhere.

Anyway, my annoyance with books written for these hypothetical chartophobes inspired me to hijack the tune to The Battle Hymn of the Republic (“Mine Eyes Have Seen The Glory …”) in order to create a song for all those imaginary potential buyers the publishers don’t want to scare off. Sing it, please, to these fine instrumentals:

The Boredom Hymn of the Repugnant

Mine eyes have just glazed over
‘Cause this book contains a graph.
How dare they put this in here,
In high school I sucked at math.
I want constant entertainment,
Not some complicated task.
In truth I’m a moron.

Sorry, moron did we LOSE ya?
Oh, we didn’t mean to LOSE ya.
No, we can’t afford to LOSE ya,
So we’ll dumb the book down.

Oh, my parents blamed my teachers
For my lack of aptitude,
While my teachers cursed my parents
When I was bored, asleep, or rude.
Books just can’t hold my attention,
‘Less there’s some babe in there nude.
In truth I’m a moron.

Sorry, moron did we LOSE ya?
Oh, we didn’t mean to LOSE ya.
No, we can’t afford to LOSE ya,
So we’ll dumb the book down.

Just kidding. Mostly.

How much stronger is a 0.9mm pencil lead than a 0.5mm?

I don’t know why 0.5mm pencils are more popular than 0.7mm and 0.9mm. I had 0.5mm pencils around the house, but when I saw my son break the lead three times in 15 seconds I decided to try 0.7mm. Much less breakage. At the time I didn’t even know that you could get 0.9mm pencils, but when I found out, I got one. I was afraid that they would be too thick, but in fact the lines they produce doesn’t really seem much thicker — it’s just smoother to write them. That’s probably because the worn surface of the lead is curved, and for the same amount of pressure a 0.9mm lead will penetrate less deeply into the paper. That reduces both the spot size and the drag of the pencil.

The big win, though, is that 0.9mm leads are much less prone to breakage than 0.5mm. I’ll spare you the math, but at a first approximation the strength of the lead goes up with the cube of the diameter, so according to the simple model I used (I make no claims to either accuracy or precision) it takes something like 6 times as much force to break a 0.9mm lead as a 0.5mm. Also, because the 0.9mm lead is not making as sharp a trough in the paper, there is less perpendicular force on the lead. I’ve been using 0.9mm pencils for a year or so now, and I have yet to break a lead.  This diagram shows their relative sizes:

Other advantages to using thicker lead:

– You don’t have to advance the lead as often.
– You can make a bolder line, if you want to. I do a lot of underlining and quick diagrams, and bolder lines work well for both.
– The mechanism seems less finicky. I used to have to disassemble 0.5mm pencils from time to time, but I never do that anymore.
– Lines are more easily erasable because the lead hasn’t made so sharp a trough in the paper.

Now I can understand that for somebody doing drafting, being able to make a really fine line is important. Such folk may even need a 0.3mm pencil. But if you are just doing homework or taking notes, you might want to try a 0.9mm.

By the way, I think it would be a really cool high school science experiment to empirically determine the relative strengths of leads of different diameters.

And you might be curious why the strength goes up with the cube rather than the square of the diameter (or radius). Clearly the cross-section of the lead, which must be pulled apart, is a disk, the area of which increases with the square of the diameter. If you were just trying to pull the lead apart by pulling the two ends in opposite directions, the strength would indeed increase with the square of the diameter. But while writing, a lead breaks in response to a perpendicular force — effectively making a lever — and the cohesion of the lead has more leverage the farther away it is from the fulcrum. A larger-diameter lead adds surface area mostly on the side furthest from the fulcrum, where it does the most good.