Math for awesome people

Cheating on the SAT

2011-12-02T10:56:00.001-08:00

The thing I can't get out of my head when I read this NYT article on kids paying other kids to take the SAT for them is simply this:

All they charged was $3600? Cripes, that's a bargain at ten times the price.

Don't get me wrong: you shouldn't cheat, both because it's wrong (which should be enough reason) and tactically too risky (in case the first argument wasn't enough).

But I mean, come on, people, do the math: a one-percent increase in salary over your life is easily a five-digit number even if you're kind of a slacker. Two significantly different SAT scores mean admission to schools of two significantly different calibers. And I doubt the salary increase we're talking about here is just 1%.

For those of you who are fans of the Drake equation, which uses best-guesses to try to figure out whether there's intelligent life out there, I challenge you to apply this reasoning to SAT prep.

In fact, you might even try to create an analogous equation governing this stuff, like I just did. (I hope you have more luck than I did; if so, please let me know.) But, equations aside, it's not really that hard to think about.

To figure out what a higher SAT score is worth, just do the following steps:

First, get a lifetime earnings calculator. (Google it; there are many.)

Then, use it to estimate the student's lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the initial SAT scores.

Then, take the average (expected) gain in SAT scores given a particular preparation method.

Then, use the calculator to estimate the student's lifetime earnings, given that he or she attends the best school to which he or she can gain admission given the final (expected) SAT scores.

The difference between the two lifetime earnings is the value of the higher SAT score.

And now that I've said all that out loud, I'm starting to realize that four-digit prices for SAT prep only make sense for providers who can offer only single-percentage-point gains with a high variance, as delineated in this article in the Wall Street Journal regarding the average benefit of SAT prep.

And you know what that means. Yep: I'm doing it wrong.

Ah, well, yet another learning and growth opportunity. Never a bad thing.

TED meets math

2010-12-15T21:27:00.001-08:00

...need I say more?

Enjoy: 20 Math TED talks!

What's wrong with math education

2010-11-15T16:14:00.000-08:00

If I could say it better than Tom Henderson, I would. But I can't. So here he is:

In a nutshell, what is the problem with math education in the US?

I have no idea. Let me instead describe the attitude that students have that is problematic, and you can reconstruct what must be wrong with it from that angle.

“Show me the steps.”

Many students want teachers to “show me the steps.”

They want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Seeing Grade Point Average, is largely determined by being able to Do The Steps.

But “The Steps” are cargo cult mathematics.

The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”

Now, the way that I explain mathematics, it sort of looks like I’m torturing the poor bastards. I handwave. I refer to certain groupings of symbols as “Alphabet soup” and write it down as a wild scribble with one or two symbols around it.

Because I’m trying to avoid showing The Steps and instead show them enough of The Idea that they can reconstruct what the steps MUST be.

Many students want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off. Why do they want to know that?

Probably because, for their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.

What’s wrong with math education in the US? What’s wrong is, Whatever it is that makes my students uninterested in learning any more math than is required to minimize feeling stupid.

So that we’re clear, lots of my students are totally awakened to the interesting weirdnesses of mathematics. But, it takes some doing, and I can’t do it by myself. Hence the podcasts and the lunatic twitter stream and the plans for TV shows and online games and godknowswhat else.

I’m trying to get across that if you are highly motivating, if you have a high degree of fire and “**** yeah!” and “What, that’s impossible, but true!”, you can get students to express interest in theorems named after dead Hungarians.

Professional diversity

2010-08-11T15:45:00.000-07:00

Today I received this email. It's posted here in its entirety, exactly as received, except that I've removed the email address.

I wish there were two different words for the services we each provide. Unfortunately, we're both "tutors."

With no further comment:

Hello Sir ,
We are from India and we provide Math Online Tutoring to US,UK and Canada. we are tutors to help students for math + we deal in SES tutoring and for after school education .We provide one-on-one Live Online Math tutoring,Math assignment help and homework help for grade/keystage/year K-12 as well as Math preparation for SAT & ACT test Guided by the tutor and backed by state aligned curriculum, assessments and individualized tutoring plan, it has the flexibility to offer 24X7 services, round the year. For our service we just tell you that thousands of students are improving their grades in schools and high stake exams. Tutors are patient, kind and professional. They are subject experts and care about the unique needs of each child.we are looking to work with you .

Thanks
Kevin : Email id : (redacted)@yahoo.com

Are you part of the solution?

2010-08-08T10:05:00.000-07:00

When it comes to education, I find that the world view of today's grandparents is strongly influenced by the idea of the US as a world leader in intellect.

Today's parents recognize that that is no longer quite the case.

Today's students don't even know what it feels like to grow up in a society where acquiring an excellent education is considered by all to be a child's primary job.

You can't reverse this trend single-handedly, but you can make a difference.

You don't have to have a child, and you don't have to be an educator. But you can do your part, simply by remembering that after food, shelter, and community, education is the highest-leverage activity in which any person can participate.

You have an impact not only every time you talk with a child, and every time you vote, but also every time you do a Friday NYTimes crossword puzzle in public, and every time you use correct grammar, and every time you take 30 seconds to look up a word in a dictionary, and every time you take a class -- on anything! -- just because you want to.

MIT offers best ROI

2010-07-28T13:46:00.000-07:00

Apparently a recent study claims that an MIT offers better ROI than that of any other undergrad institution.

As an alum, I suppose that's good news, though one wonders whether I should be happy about the value of my degree or disappointed about my not having exploited it more fully...

Then again, if one is doing good in the world, having fun, and paying one's bills, then that can hardly be bad. And perhaps it takes a huge ROI to portion it out in that way and still to have enough!

Regardless: "Go Tech!"

The cultured mathematician?

2010-07-09T12:14:00.000-07:00

David Brooks makes a powerful point in today's NYT: when it comes to getting smarter, books are better than the internet. (Well, I oversimplify, but go with me a moment here.)

He references Joseph Epstein's "Narcissus Leaves the Pool," which I duly referenced and read the relevant excerpt of. (Do. It's worth seeking out, and it'll take you only ten minutes.) The nut of it: a cultured person -- an ideal to which we should all aspire -- recognizes that the brain is very much finite, and so he or she focuses not on knowing as much as possible, but rather on knowing what to know, and consequently -- I love this next part -- what to forget. Finally! A cure for information overload, all without sacrificing one's love of knowledge.

Easier said than done, yes, but hey, at least the goal has a name now.

But I couldn't help thinking that that is in a real sense what I already do, at least when it comes to passing mathematical thinking to a few members of the next generation. Sure, as I've mentioned here before, I love pop culture math books. The best give a layman (including me; I'm a professional teacher but an amateur mathematician) a solid understanding of, and even intuition for, some area of math that anyone can find fascinating. But if I were to have to name a bible, a single book that informs my approach more than any other, I'd have to name Polya's "How To Solve It," because the core message and method of that book is not to know more stuff, but rather to use what you have as efficiently as possible.

And indeed it has the ring of truth, doesn't it? To lead a better life, be a better thinker. And to be a better thinker, don't seek to know more. Seek instead to make the best use of what you have.

Summer is here.

2010-06-08T20:00:00.000-07:00

The 2009-2010 school year is at its end.

To all of my clients (students and parents alike): let me extend both my congratulations and gratitude. We have not only survived, but also excelled through another year.

That's easy to say, but I don't mean it glibly.

Student CM is off to Deerfield, to join former student SD, who began there last year;
student CM (no, the other one) has demonstrated extraordinary progress in executive function, culminating in a self-driven charge towards a successful round of final exams;
new student TH is off to the races, tackling problems three to four years ahead of grade level;
long-time student JD continues to impress as not only an excellent mathematical thinker, but also as a deep thinker and good communicator overall;
student JL continues to ignore the downward pressure of expectations set by a learning difference, and instead employs dedication and hard work to run circles around those who would otherwise be peers, even in the face of taking two years of math concurrently.

Special mentions go to:

student JB, who pulled out of a precarious nosedive to end pre-calculus with a well-deserved B through a ton of hard, smart work;
and to new GRE student AW who fights test anxiety on a daily basis... and wins;
and to student VP, who continues to challenge me with incredibly insightful questions;
and finally to new student BL who is beginning to learn what it is to self-advocate as a student.

Also, let me congratulate those BodSAT students who turned in great performances despite being at the tops of their respective games when we began.

And finally, let me congratulate the Urban School Math Competition Team for another successful year, and specifically departing captain Amrit K, who served the team tirelessly and successfully for two years. Best of luck at UCSD, Amrit, and may the UMCT continue its proud tradition next year with Simon S at the helm!

As for me: I'm simultaneously humbled by my students' many accomplishments... and exhausted and ready for summer. :-)

Looking forward to seeing everyone again in the fall.

But first, some time off!

Excerpt on IQ testing

2010-05-07T00:38:00.000-07:00

I found this quite interesting:

"Children develop only as the environment demands development. In 1981, New Zealand-based psychologist James Flynn discovered just how profoundly true that statement is. Comparing raw IQ scores over nearly a century, Flynn saw that they kept going up: every few years, the new batch of IQ test takers seemed to be smarter than the old batch. Twelve-year-olds in the 1980s performed better than twelve-year-olds in the 1970s, who performed better than twelve-year-olds in the 1960s, and so on. This trend wasn't limited to a certain region or culture, and the differences were not trivial. On average, IQ test takers improved over their predecessors by three points every ten years - a staggering difference of eighteen points over two generations.

"The differences were so extreme, they were hard to wrap one's head around. Using a late-twentieth-century average score of 100, the comparative score for the year 1900 was calculated to be about 60 - leading to the truly absurd conclusion, acknowledged Flynn, 'that a majority of our ancestors were mentally retarded.' The so-called Flynn effect raised eyebrows throughout the world of cognitive research. Obviously, the human race had not evolved into a markedly smarter species in less than one hundred years. Something else was going on.

"For Flynn, the pivotal clue came in his discovery that the increases were not uniform across all areas but were concentrated in certain subtests. Contemporary kids did not do any better than their ancestors when it came to general knowledge or mathematics. But in the area of abstract reasoning, reported Flynn, there were 'huge and embarrassing' improvements. The further back in time he looked, the less test takers seemed comfortable with hypotheticals and intuitive problem solving. Why? Because a century ago, in a less complicated world, there was very little familiarity with what we now consider basic abstract concepts. '[The intelligence of] our ancestors in 1900 was anchored in everyday reality,' explains Flynn. 'We differ from them in that we can use abstractions and logic and the hypothetical ... Since 1950, we have become more ingenious in going beyond previously learned rules to solve problems on the spot.'

"Examples of abstract notions that simply didn't exist in the minds of our nineteenth-century ancestors include the theory of natural selection (formulated in 1864), and the concepts of control group (1875) and random sample (1877). A century ago, the scientific method itself was foreign to most Americans. The general public had simply not yet been conditioned to think
abstractly.

"The catalyst for the dramatic IQ improvements, in other words, was not some mysterious genetic mutation or magical nutritional supplement but what Flynn described as 'the [cultural] transition from pre-scientific to post-scientific operational thinking.' Over the course of the twentieth century, basic principles of science slowly filtered into public consciousness, transforming the world we live in. That transition, says Flynn, 'represents nothing less than a liberation of the human mind.'

"The scientific world-view, with its vocabulary, taxonomies, and detachment of logic and the hypothetical from concrete referents, has begun to permeate the minds of post-industrial people. This has paved the way for mass education on the university level and the emergence of an intellectual cadre without whom our present civilization would be inconceivable.

"Perhaps the most striking of Flynn's observations is this: 98 percent of IQ test takers today score better than the average test taker in 1900. The implications of this realization are extraordinary. It means that in just one century, improvements in our social discourse and our schools have dramatically raised the measurable intelligence of almost everyone.

"So much for the idea of fixed intelligence."

Author: David Shenk
Title: The Genius in All of Us
Publisher: Doubleday
Date: Copyright 2010 by David Shenk
Pages: 35-37

What's wrong with standardized tests for teachers?

2010-02-15T17:06:00.001-08:00

Well, I really mean to answer "what's wrong with using the results of students' standardized tests to evaluate teacher performance?"

On its surface, nothing. Indeed, student performance should be the primary metric of teacher quality. I personally am gauged by that metric in my role as SAT coach: what matters is the point gain.

The main problem, though, is that the standardized tests don't test everything that we want our students to know. In the same way that the standardized test at the DMV doesn't tell you whether you're a good driver (though it does catch some really bad drivers), standardized testing for schools doesn't tell you whether you're on track to be a productive member of society (though it does catch a number of people who really really aren't).

Also, if a standardized test samples say a random 10% of what one should know, then how well one does on such a test is, statistically, a good indicator of how much of the target material one knows. But if it's a non-random 10%, then you can count on prepared students to know that 10% very well. And possibly nothing else.

What makes a great teacher?

2010-02-13T18:48:00.000-08:00

This article from the Atlantic, drawing on decades of data from Teach For America, shows why it matters, what Arne Duncan is doing about it, and why you should care.

The cultural importance of education

2010-02-13T12:50:00.001-08:00

A friend of mine just pointed out this very interesting article called Understanding Asian Parents, in which the case is made that if you come from a place where education is directly and dramatically, obviously important to your health and well-being, you'll take it so seriously that you'll confuse and even alienate your American-born children, who don't see the importance of education in such stark terms.

I think it's a great read (albeit with some strong language) not only for people who find themselves trying to bridge a cultural gap with their parents (or with their teenaged children), but also for anyone interested in learning a little more about how culture informs education, and therefore why education can be so difficult to wrap our heads around in America, where a large part of the culture is by its nature a grand patchwork of backgrounds and perspectives.

Children construct knowledge; they don't swallow it.

2010-02-02T09:38:00.001-08:00

Well spoken, Ms. Engle. Your New York Times op-ed piece hits the nail on the head.

Misuse of statistics

2010-02-02T09:31:00.000-08:00

From a recent sales email from yelp.com:

"...on average these customers looked at 3 business pages for every spending decision they made. Said differently, for every 3 of these people who viewed your business page, on average 1 of them would turn into a paying customer."

False; this is a statistical fallacy. Namely: sampling bias. It discards all the people who looked at at least one page, and then didn't buy anything.

The three magic bullets

2009-11-23T11:22:00.001-08:00

Everyone wants an educational magic bullet. What worked for a public school in San Jose, and was picked up by the New York Times, turns out to be a pretty good summary of the formula that works like a charm in my experience:

1) High expectations for everyone,
2) Constant assessment, and
3) Family involvement.

Kids have enormous potential, but they need direction, challenge, support, and course-correction.

Bespoke education

2009-09-14T17:53:00.000-07:00

The current model of education was pioneered during the Industrial Revolution, when the idea of cranking out many carbon copies of some idealized "good student" was considered a good idea, because it was better than any available alternative.

However, "bespoke" education -- one-on-one individualized work with a tutor, a word that once meant someone who was a master not only of a particular subject, but also of teaching it in a one-on-one format -- is simply better.

This article explains how it's trying to make a comeback online. But this page on my website for students explains why one-on-one in-person work is, and always will be, the gold standard.

Summer homework: is it good or bad?

2009-08-31T16:45:00.000-07:00

The debate rages on at the New York Times site.

For me, it's pretty straightforward: two months off in the summer is responsible for the loss of one third of the school year (you miss two months, and you backslide two months' worth in addition), so I think that some academic work over the summer is essential.

However, I am always a stalwart enemy of busywork, and here is no exception. If your child's summer homework is mindless, object immediately. That serves nothing and no one.

Acing the SAT by managing your emotions

2009-08-26T12:20:00.000-07:00

Dan Pink's TED talk on motivation explains one of the reasons why SAT takers do poorly: when results really matter, linear thinking gets faster, but creative thinking becomes much harder.

It turns out creative thinking is a huge boon on the SAT, so managing this particular type of performance anxiety is incredibly helpful.

The elite-college admissions pressure keeps rising.

2009-08-25T11:44:00.000-07:00

There's a lot of lackluster test prep out there, so on one hand it bums me out that the industry has such a bad rap. On the other hand, our stats and feedback indicate that we are doing something very, very right. Sooner or later we'll eat the competition's lunch. In the meantime, we're not only sending a large handful of kids to great colleges... but we're teaching them to think more effectively, as well. Go us.

Stanley Kaplan, RIP

2009-08-25T11:40:00.000-07:00

I have plenty of criticisms of what's taught under the Kaplan name these days, but there's no denying that, love it or hate it, the test prep industry started with this dynamic teacher and entrepreneur (and, later, philanthropist). Rest in peace, Stanley Kaplan.

Math Doesn't Suck doesn't suck ...and yet...

2009-08-10T17:47:00.000-07:00

Danica McKellar's book Math Doesn't Suck (purchase here at amazon.com) did a number of things that are difficult and laudable:

* Target middle-school girls' math fundamentals
* Present those fundamentals in a logically consistent and compelling way
* Achieve top-20 status on Amazon -- a math book, for Pete's sake!

...so my hat's off to Ms. McKellar.

The only thing that drives me a little crazy about this book -- and I don't know whether to blame the author, her editor, or the culture at large -- is that so many of the examples play to the stereotype of middle-school American girl as possibly-capable but, well, vapid, at least in practice. With chapter subtitles like "Do You Still Have a Crush On Him?" and "When To Seriously Stop Raiding The Refrigerator" and "Choosing The Perfect Necklace" and "What Every Savvy Shopper Should Know," I just can't help thinking: this doesn't make math interesting; rather, it lures the reader into a discussion of math without ever leaving a tabloid tone.

On one hand, maybe this is exactly what the doctor ordered. If middle-school girls respond to the tone and therefore get more comfortable with math fundamentals (as well as with their own math ability), then I'm all for it.

But one of my takeaways is that while others might achieve success by sinking their teeth into a subject and mastering it, the best girls can hope for in math is to achieve modest success by leveraging their "inherent" love of shopping and boys.

I'm confident that that wasn't the intention, and it may be that no member of the target audience would come to that conclusion. But I did, and I found it offensive.

Let the reader beware.

I have an idea about a pyramid inside a cube...

2009-07-11T06:15:00.000-07:00

"I want them to understand that there is a playground in their minds and that that is where mathematics happens. So far I have met with tremendous enthusiasm among the parents and kids, less so among the mid-level administrators."

So says Paul Lockhart, the author of A Mathematician's Lament, who posits that math education is too frequently like a page out of Orwell's 1984.

Thankfully, Lockhart goes a step farther: "...and here's how to fix it."

I'm delighted to find that he views it (and does it) just about the same way I do: math as art, one student at a time, one teacher at a time.

Get in touch, or leave a comment, if you'd like to discuss.

SAT-prep charlatanism

2009-05-20T15:17:00.001-07:00

Let's see some regulation in this area, can't we, please?

For my SAT-prep company BodSAT.com, I'm running statistics comparing actual SAT scores before the seminar and actual SAT scores afterwards. The average gain is 180; the min and max: -30 to 280. Not as impressive as most marketing claims, except insofar that it's honest.

Math gives rise to art via engineering

2009-05-19T10:39:00.000-07:00

You simply cant tell me that this MIT solar car isn't a thing of beauty. So very lovely!

It's almost eough to make me wish I were a zillionaire living in the desert. Instead, perhaps I'll just change the background image on my desktop...

No, I haven't gone away.

2009-03-10T02:01:00.000-07:00

...I've just been swamped with a few projects. I'll be back soon! But first, a short vacation. Thanks for stopping by. Perhaps you'd like to tour the archives?

Team Codex goes down

2009-01-26T23:55:00.000-08:00

...as my team, Beginner's Luck, wins the 2009 MIT Mystery Hunt in 63 hours, 2 minutes. Check out some snarky and fun coverage of the event by G4TV's Attack of the Show.

Some call it the Brain Olympics

2009-01-15T10:53:00.000-08:00

...but I call it the MIT Mystery Hunt. (I also call it my January vacation.)

Wish me luck. If you see that "Beginner's Luck" won in 2009, then congratulate me, 'cause that's my team. If not, congratulate me anyhow, because I will have had a great time nonetheless.

You can find my writeup of my first Hunt (last year) here.

Enjoy!

Using game theory to beat the SAT

2008-12-18T18:59:00.000-08:00

Sure enough, game theory is a useful tool to have in one's toolbox when setting out to ace the SAT. This short article does a good job of showing how it works.

But don't be fooled. If you want to beat the SAT, your top two priorities are no secret: know your stuff and get enough sleep. That's where the real leverage is.

I don't care whether you're smart. I only care how hard you try.

2008-11-28T15:10:00.000-08:00

Carol Dweck hits the nail on the head. You might think that "you're so smart!" is a good thing to say to someone. You'd be wrong, though; that turns out to disincent people to work hard.

If you want to praise a student, a better choice is "you must have worked hard for that."

I've been busy!

2008-11-13T23:02:00.000-08:00

Gentle readers: It's been a busy autumn.

* The math team is doing quite well; we have 44 points out of 60 on the CAML moving into the third round (yes, that's very good, and yet we're only just getting going)

* SAT prep is going well as always, but the new development is that our administrative staff is kicking in the afterburners, so it's getting to be a smoother process for students and parents alike -- thank goodness for administrative staff

* My private students are really flourishing this year. Highlights from this week alone: an eighth-grader working on really grokking the idea of proof by contradiction with incredible success, and a sixth-grader using modulus and congruence to show that there are no perfect squares in the infinite sequences 5, 8, 11, 14, 17, ...

* Did I mention that the math team scored their first perfect score this week? I'm so proud of them.

Meanwhile, I'm having some trouble with some side-projects, not least of which trying to figure out how to decorate my apartment. Who knew that it could be so hard? I'd rather prove that the square root of seven is irrational, thanks. Ah, well.

***

Sometime soon I'll have time to comment on such things as the recent backlash against SAT prep in general. If only they knew that the proper objection is to the charlatans. (Sadly, that's most of the market. Sigh.)

Welcome to the team, Coach

2008-09-14T08:34:00.000-07:00

Just a quick announcement that Urban High School of San Francisco has started a math competition team this year, with yours truly as the coach.

It's clear from the first meeting that this is going to be fun.

(Other Bay Area schools, you're on notice: we're comin' for ya.)

Problem Of The "Every So Often"

2008-09-14T08:15:00.000-07:00

An interesting problem came up in conversation with a College Preparatory School freshman the other day, so I thought I'd share it with you:

100! (pronounced "one hundred, factorial") is the result of taking "100 times 99 times 98 times..." and so on, all the way down to "...times three times two times one." The question is this: when you write it all out, how many zeroes does that number end with?

I like this problem because it's clear that writing it out won't work; it would be boring, take too long, and you'd invariably make at least one mistake. And the number's way too big for a calculator to deal with. So... what next?

(By the way, January's Problem Of The Every So Often can be found, with solution, here, and I've posted the problems from a bi-annual puzzle event I host here.)

Math books I recommend you read

2008-07-06T14:54:00.000-07:00

Hello from vacation! New York City, as it happens. I'm spending the summer emphatically not working, but that doesn't mean I can't enjoy a little math every now and again.

Perhaps that's true for you, too. So, here are the titles of a few books I read last year, and which I recommend:

The Art And Craft Of Problem Solving is a great book by Paul Zeitz, intended to teach the reader how to think creatively in approaching difficult math problems. Lots of actual math to be done in this one. More a textbook than anything else, but a very good one. Also requires nothing but high school math... and a good helping of enthusiasm, which the book will help to increase. One can earn the (considerable) purchase price in value many times over without finishing even half of it.

The Equation That Couldn't Be Solved, by Mario Livio, is a great read: it's the story of the lives and work of Norwegian Niels Henrik Abel and Frenchman Evariste Galois, who created one of the major branches of modern math (group theory) while trying to solve a different problem altogether. The book begins by building up the problems Abel and Galois were trying to solve, and winds up giving the reader a powerful sense of what mathematics really is: the attempt by creative minds to find and explore patterns and symmetry. But it's the story of the two minds themselves that makes this story so good.

Prime Obsession, by John Derbyshire, is a fascinating account of the history (and the nature) of the so-called Riemann Hypothesis, one of the greatest unsolved problems in mathematics (and for which another (flawed) solution was recently published). It requires a willingness to consider some challenging puzzles. It's not a quick read; some pages took me ten minutes or more to really think through. But, it doesn't require anything other than high school math. It reads a bit like a mystery novel, but finishing it makes you feel like you've become a mathematician. It was so good (and truth to tell so rich with math) that I read it a second time a few months later.

The Numbers Behind NUMB3RS, by Keith Devlin and Gary Lorden, explains the math in the TV show "NUMB3RS" episode by episode, and is a surprisingly good intro into many branches of modern math for the interested but casual reader. Good starting book, particularly if you know and like the show. For me, it worked the other way around: I like the show considerably better now that I understand how the math they are showing is real.

Enjoy the rest of your summer!

Put a little math in your life!

2008-06-02T10:08:00.001-07:00

Brian Greene makes the point about science, and it holds just as true for math: "We rob science education of life when we focus solely on results and seek to train students to solve problems and recite facts without a commensurate emphasis on transporting them out beyond the stars."

Interested in communicating with a faraway friend without allowing anyone to eavesdrop on you? Of course you are; this problem affects a middle-schooler's daily life, and yet it is also the basis for modern commerce: communicating your credit card number over the Net without allowing a thief to eavesdrop is a non-negotiable requirement for our economy. How is this problem solved? First, you try for a while. Then let's talk about prime numbers and see what they can do for us.

Interested in taking a rocket to the moon? Well, if you want to drive a spaceship, you probably ought to understand how gravity works differently from your intuition when you're far from Earth. Let's talk about ellipses, and while we're at it, let's predict the next approach of Halley's comet. (And if you're a high-school sophomore or so, we can talk about inverse-square laws while we're at it.)

What if we picked up a signal for an alien civilization on our radio telescopes tomorrow? Do you think they'd speak English? Not a chance. Let's think about how we would communicate without a shared language, a shared culture, even a shared planet? Well, there's one thing we have in common with any race advanced enough to send us a signal like that: they know math. Let's think about how that might work...

One of the biggest rarely-asked questions in math is simply "why should I care?"

I love answering that one.

Abstract vs. concrete in math education

2008-04-25T23:06:00.000-07:00

As I see it, the issue that's missing here is this: a good teacher will use examples to illustrate a point, whereas a great teacher will convey the underlying point in a way that will prompt the student to generate the examples him- or herself.

Yes, the SAT is important...

2008-04-24T23:04:00.000-07:00

...but it isn't that important.

(But, just to put the plug in there, BodSAT.com is a good place to go if you're a superstar student and you want to hedge your bets.)

America, please catch up

2008-04-23T22:51:00.000-07:00

“In math and science, for example, our fourth graders are among the top students globally. By roughly eighth grade, they’re in the middle of the pack. And by the 12th grade, U.S. students are scoring generally near the bottom of all industrialized countries.”

This is not encouraging.

It reminds me of the study that forms the basis of The Teaching Gap: American math classrooms teach rote learning, whereas other countries (such as Germany and Japan) teach general problem-solving skills. As a result, their kids develop math skills faster, and reach higher levels.

If you don't think that this is a serious problem for the future of the U.S., then I'm sorry to say it, but you're confused.

College gets expensive in new ways

2008-04-12T09:55:00.000-07:00

This is the part of the current overarching financial picture that worries me most: parents simply unable to send their kids to college.

The Best Game Ever

2008-04-09T16:40:00.000-07:00

I've thought for a long time that it would be great if math got the same treatment as sports. I've also thought that that is possible. (More on this thought in future posts, as I work towards making it happen!)

But for now, a quick look at what happens when the little league gets treated like the big league. (I think that this is GREAT!)

The pros and cons of discipline

2008-04-09T11:50:00.000-07:00

Yes, one can always take anything too far. But if I had to guess, I'd say that the current average levels of discipline, structure, and expectation are below-optimal. (I base this on the relative maturity and accomplishment of those students of mine for whose life-structure I have some data.)

Here's the article that got me thinking about this in the first place.

"What good is math in the real world, anyhow?"

2008-04-03T20:03:00.000-07:00

How about optimizing the deployment of resources during a natural disaster? That sounds pretty useful to me.

"Record Low" Admissions?

2008-04-03T12:15:00.000-07:00

Interesting that the NY Times used the words "Record Lows" in the title of this article on the the continuing increase in admissions applications at the nation's top schools.

Remember, please, everyone, that scarcity lends itself to a higher valuation of the underlying commodity. In this case, I think I would use the word "overvaluation."

Not that the top schools aren't worth the tuition, mind you, but rather that the difference between what you get there and what you get at other schools might not be worth the -- I don't care if it's overused, I love this turn of phrase -- "irrational exuberance."

Let me underscore this point by repeating the article's quote of William M. Shain, dean of admissions and financial aid at Bowdoin: "Where we went to college does not set us up for success or keep us away from it."

DIY SAT prep

2008-03-31T20:03:00.001-07:00

This is not the way I'd recommend you prepare if you have the resources to hire expert coaches. But it is effective (as well as damn impressive) to study for the SAT with no one but committed peer students.

SAT 2370

2008-03-21T11:23:00.000-07:00

Not to brag, but...

...okay, to brag. One of Bodhisattva SAT Prep's first students just got her scores back from the College Board and "refreshed the page about 50 times to be sure."

Congratulations on your 2370, Laura!

...Readers who are not familiar with the March 2005 changes in the SAT will be surprised to learn that a perfect SAT score these days is a 2400. Therefore, Laura's score is equivalent to a 1580 on the old scale.

University endowments under scrutiny from Congress

2008-03-18T05:45:00.000-07:00

...and on the other end of the scale from underfunded public schools, we have the Ivies.

I'm not weighing in one way or the other on this one, but I thought it was worth noting.

Zeke and TEP hits it on the head

2008-03-17T17:55:00.000-07:00

Of course we've all thought it at one time or another. I've been thinking it aggressively for five years now. It's why I've set up my tutoring business the way I have. But Zeke Vanderhoek is actually getting it done, and my hat's off to him.

I quote: “The money, as funny as this may sound, is not about the money. The money is a signifier. Because money, in our culture, is a signifier of how jobs are valued, and right now schools are telling teachers that they are not valued. The great and talented people who go into teaching are incentive-ized in every possible way to leave the classroom for jobs in administration or jobs outside of schools altogether. What we are trying to do is reverse those incentives. We want the best teachers to keep on teaching, to be challenged and valued.”

I just wish I'd had what it took to do it myself.

Varsity Academics?

2008-03-10T02:18:00.000-07:00

Apparently I am making up for a lax February in posting so much recently.

But Will Fitzhugh seems to be onto something. Why isn't there a varsity math team, well, anywhere, so far as I know?

And for that matter, why do sports matter in schools in a way that academics seem not to?

ED in '08

2008-03-10T01:38:00.000-07:00

These folks are not messing around when it comes to raising awareness for education in 2008. Videos worth watching, I think.

Thank goodness.

2008-03-09T23:05:00.001-07:00

Glad to hear that the college admissions frenzy should be dying down in a few years. It's getting out of hand.

High schoolers of America, stop embarrasing us.

2008-03-03T19:39:00.001-08:00

Pretty bold words, I know.

Please, someone tell me that this article in Slate on "what high schoolers don't know" is inaccurate.

Otherwise, I may have to go scream from a rooftop.

Sleep more, dangit!

2008-03-03T14:37:00.000-08:00

Wired Magazine points out what we already know, even if we don't always act accordingly: we don't sleep enough. And it isn't just students, though as I've mentioned, they get hit especially hard. Here's the article.

This made my month.

2008-01-23T10:05:00.000-08:00

"An integer is called formidable if it can be written as a sum of distinct powers of 4, and successful if it can be written as a sum of distinct powers of 6. Can 2005 be written as a sum of a formidable number and a successful number? Prove your answer."

This problem, from the 2005 Bay Area Mathematical Olympiad, is intended primarily for bright and math-focused high schoolers. Imagine my delight to find four of my students -- three in middle school, no less -- not only fearlessly diving in, but actually solving it.

Congratulations to Cameron R., Sloan D., Seika N., and Jack D.!

2008 is going to be a very good year.

First class!

2008-01-22T23:23:00.000-08:00

The title can be read two ways, and that's no coincidence. The first BodSAT.com SAT class happened tonight, and the students seem top-flight. I'm looking forward to seeing what we can accomplish together.

SAT Class starts tomorrow!

2008-01-21T23:22:00.000-08:00

I'm excited to report that the SAT class I've started with my tutoring colleague Justin Sigars starts tomorrow.

Some months ago, Justin and I were complaining to each other about how so many companies have sprung up to make a buck promising SAT score bumps, without knowing half of what we know about teaching high schools students to think (in general) and to think about the SAT (in specific).

So we decided to put our time and money where our mouths were, and Bodhisattva SAT prep was born.

I can hardly wait!

(Oh, and yes, the MIT Mystery Hunt went well this weekend, but I'll save the story for another time.)

Weekend update

2008-01-20T02:30:00.000-08:00

Just a quick note to say that the Mystery Hunt is going well. I've certainly learned a lot already, and we're not finished yet.

Sometimes you have to reach beyond your grasp. Wish us (continued) luck.

Off to the races!

2008-01-17T09:25:00.000-08:00

Studying math correctly improves one's problem-solving skills. Many of us believe that the reverse is also true: strengthening one's problem-solving skills makes one better at math.

Puzzle-solving is therefore very important, in my view.

So, I'm heading out this morning to Cambridge, MA, where I will be competing in the 2008 MIT IAP Mystery Hunt, arguably the hardest weekend-long puzzle competition on the planet.

Wish me luck ...and get in touch if you'd like to be one of my team's remote solvers.

It's all in the basics.

2008-01-16T12:16:00.000-08:00

A good friend recently directed me to a computer guy's blog entry on the topic of interviewing, because it seemed to relate to learning math. How right he was! You have to, have to, have to deeply understand the basics before the advanced stuff can make any sense. Getting an A isn't enough. You have to understand the material as though it were the language of the country to which you wish to move. (Hey, that's not a bad analogy, either.)

(Read the relevant excerpt from the computer guy's blog here in PDF or text formats.)

Tutoring: in-person vs. offshore

2008-01-15T10:59:00.000-08:00

As the offshore tutoring industry continues to gear up and get huge, I've been coming around to the idea that different people have different needs, and it's good for everyone when many solutions are available for a problem. It's just like having McDonald's around the corner: it's no good when you want a gourmet meal, but if you want a Big Mac, then that's the very best place to go.

And then I read this article in SFGate, in which the following quote seems to have slipped past the editor:

Kevin said he feels more comfortable talking to his tutor -- sight unseen -- than asking questions in class: "No one is paying attention to me that much."

Is it just me, or is this a problem with this "service"?

Do you want fifty extra points on the SAT?

2008-01-14T12:43:00.000-08:00

That's how I get students' attention. How about a half-letter grade on your next exam? All you have to do... is get a good night's sleep.

Here's the link to the NY Times article on the importance of sleep.

Excuse me, how do you get to Carnegie Hall?

2008-01-13T19:41:00.000-08:00

Show me an exceptional math student, and I'll show you someone who has practiced correctly and practiced enough. I call foul on anyone who says it's just inborn talent.

Here's the link to the article on expertise in Scientific American. Warning: it's long. (Good, but long.)