Brad DeLong's Semi-Daily Journal: A Weblog

I've just finished reading an excellent book on why numeracy (numerical literacy) is so important. The authors make a distinction between numeracy and skill at higher math (which most people can get away with), but provide several examples how a basic knowledge of dealing with numbers is vital.

The book is called "What the Numbers Say", and can be found on Amazon at http://www.amazon.com/exec/obidos/tg/detail/-/0767909984/103-5036252-5891040?v=glance

The people who make the most money in the U.S. are not the people with the best math skills. I would say that kids are just letting the market decide what their skills should be.

I teach MBA students that the reason they must understand the six functions of a dollar (rather than just calculate them on an HP or in Excel) is that they can intimidate the hell out of others when get a close approximation to a payment or a present value in their heads.

They seem to buy it.

To my horror, I have found an entire fairly-respectable natural science department using Excel to do their statistical calculations. Excel is known to have buggy stats algorithms. Stats programs generally have some, but Excel's bugs are longest-lived. The worse problem might be that Excel can only do some kinds of analysis, and choosing your analysis because there's a button for it is - so tempting, if you're bad at math.

Their main defense, when they can be convinced they need one, is that the corporate side of the discipline always uses Excel. "To chart their gov't subsidies?" I did not ask, because I find the subject important and want to seduce them slowly into the paths of greater reliability.... possibly by persuading actual statisticians to make R or S programs that can be run from a button in Excel.

I was charmed to find that NIST has reference sets of data and results for different stats procedures, even ranked from most to least wierd sets of data.

"The toughest math I tackle now is calculating a tip in a moving taxi."

Mr. DeLong I am totally disappointed in you on this one. I was expecting something related to physics, and calculating in a moving taxi as opposed to a stopped taxi, or how that compares with calculating in the standing position.

Seriously though, I don't think the majority of people use any more than basic math once they are through with school. When my kids asked me why they have to take math I tell them it is to prevent Alzheimer's and I am told that is actually true.

I arrived at the guesstimate of 15.40 USD, off by .32 cents.
Should I congratulate myself or should I take a refresher course?

A New York Times reporter who finds that the toughest math he tackles is tip-calculating is almost surely doing a very lousy job at his own retirement planning and general financial management.

Not to mention his day job.

Calculating the tip in a moving cab is more difficult than in a stationary cab, because in the moving cab the meter is still running.

This was foreseen by Isaac Asimov in his short story, Profession

On an off topic note, the type on this comment page is both too small and too gray. Feh. Please consider the eyeballs!

Here is the comment I posted on Slashdot. It's short.

I'm trying to figure out the downside here, and it's just not coming to me. So Americans are lousy at math. So what? Why should we care? It's not like we need them to be able to solve math problems.

The only reason we need Americans is that somebody has to be the first one through the door when the shooting starts. The last thing we need is for them to start learning how to do something useful and productive with their lives. Let other people do the math, and let the Americans be what they were born to be: door-openers.

Tim H. wrote:

"The people who make the most money in the U.S. are not the people with the best math skills. I would say that kids are just letting the market decide what their skills should be."

Well, I supose that's superficially true, since the people who make the most money in the US are a handful (say, 1000 out of a population of 300 million) of entrepreneurs, a handful (say 1000 out of a population of 300 million) of professional athletes and entertainers, and a handful (say, 1000 out of a population of 300 million) of exceptionally lucky heirs to family fortunes. For the rest of us schmoes who are unlikely to either found the next FedEx, be drafted by the Jets or suddenly discover that our fathers are the bastard sons of Sam Walton, we must ask whether or not math skills provide a leg up at getting a job in one of the professions that pays well but that does not have a vanishingly small number of members.

Off the top of my head, I can think of several such professions: Medical Doctor in a specialized field, Partner in a law firm, Partner in some other type of professional services firm (management consulting, accounting, etc.), Officer at a commercial or (preferably) investment bank, R&D Scientist/Engineer in a specialized field, Operating Executive at a large commercial corporation, and Top-tier B-2-B salesman.

Of these, only (some of) the sales jobs and (a tiny number of*) the law jobs don't require at some point mastery of some level of complicated, non-intuitive math. Good math skills, more than any other type of academic/intellectual skills, determine who scores big in the modern economy.

The tragedy of our educational system isn't that nobody learns good math skills. When the rubber meets the road aspiring surgeons and executives and corporate lawyers pick up the math they need. Rather, its that very few people acquire the kind of broad based understanding of math early on that allows them to make substantative new contributions to their fields later in life. I know I didn't. Its clearly because math isn't taught well in schools, though the reason why math education is so bad isn't so clear.

My personal theory is that math, more than any other traditional academic subject, is incredibly painful to learn when the pace of instruction isn't optimized to the student. If a math class moves too slowly, its mind-numbingly boring for students who pick up the subject quickly. If a math class moves too fast, its heart-crushingly frustrating for students who might well do fine at math, but need a little more time than average to grasp the material. Because just about any math class will have a mix of students of different abilities, its bound to dis-please most of them. When the pace of math instruction is just right for a student, the subject is tremendously enjoyable - engaging, playful, challenging, rewarding. But very few of us find ourselves in the sweet spot in math class.

* The Law is not a profession normally associated with excellent quantitative ability, and it may well be true that most licenced lawyers don;t have great math skills, but the population of lawyers who take home truly large incomes is dominated by corporate and securities specialists at large firms who do work that requires at least a working knowledge of modern finance, statistics, etc.

We won't import math whizzes.

We'll send the work to them overseas.

I wonder how many reporters the NYT and other media have covering Social Security, for example, whose math skill are as bad as McNeil's?

I wonder if the NYT editors care.

sd, I have an MS in chemical engineering, and I can tell you that engineers do not make big money. I also can't address lawyers' math skills, but I would doubt very much that doctors and most executives use any math beyond arithmetic. In my ex company, an average was the heaviest statistics the executives used, and ROI the most difficult formula.

"The typical lawyer or doctor [without math skills] makes 2 to 3 times what the typical
scientist does."

Source: "Don't Become a Scientist" by
Tenured Professor of Physics Jonathan Katz.

http://wuphys.wustl.edu/~katz/scientist.html

It doesn't help much to have you calling interest rate calculations "math". That is arithmetic. Retirement planning isn't even arithmetic.

American kids don't learn math for the same reason they don't learn a foreign language. They (correctly) figure that they will never get to use it. Rational choice theory in action.

It's not like math instruction is alone. Americans also don't read, write, or understand science well. We're just getting dumber.

I'm a carpenter and I make a very nice living (thanks to a union). During my apprenticeship we spent many evening hours in a classroom studying math--primarily geometry and trigonometry--but with a decent amount of calculus tossed in.

We use math everyday on the job and have to use it correctly. (The building falls down when you don't.)

In spite of that, a lot of my colleagues drive leased $40,000 pickup trucks. Sure, we're actually covered by a pension, but I plan to retire at 53 because I am putting the equivalent to their lease and insurance costs in the "bank" each month.

My point is that knowledge does not lead to the correct actions. (Of course in the case of the poltroons in the White House, ignorance almost certainly leads to incorrect actions.

Tim H. wrote:

"The typical lawyer or doctor [without math skills] makes 2 to 3 times what the typical
scientist does."

Source: "Don't Become a Scientist" by
Tenured Professor of Physics Jonathan Katz.

http://wuphys.wustl.edu/~katz/scientist.html

Holy shit that's bold. I followed the link and the quote that you provide here - in quotation marks no less! - does not appear anywhere in the article.

The closest thing I could find in the Katz article you referenced was this quote:

"Of course, you don't go into science to get rich. So you choose not to go to medical or law school, even though a doctor or lawyer typically earns two to three times as much as a scientist (one lucky enough to have a good senior-level job)."

Katz never says anything about the math skills needed in medicine or law. His article is an argument against pursuing a career in academic physics, a career which involves working in a not-for-profit organization and in a field with a glut of qualified people. He is saying that people with the intelligence and drive neccessary to succeed in academic science will have happier lives if they apply themselves to other fields.

Katz never says that you don't need math skills to become a doctor or lawyer because that's not a tenable position. A doctor must take two years of college-level chemistry, one year of college level physics and one year of college level calculus just to get into medical school. In medical school, he or she is constantly exposed to the scientific literature in the field, which itself is filled with moderately complicated statistics.

A lawyer can escape rigorous math if he or she chooses a branch of the law that doesn't touch on corporations, securities or public health and medicine. But very few lawyers make tons of money by doing so. Most well paid lawyers cut their teeth doing securities work, or M&A work, or medical malpractice work. The innumerate don't fair well in these specialties.

Nor is Katz's argument that you can't make a lot of money with a Ph.D. in say, pharmacology or molecular cell biology, because clearly people with that kind of training who leave academia to go to the private sector can make a mint. Ask senior research scientists for the big pharma companies if they feel they could have achieved their 7 figure net worths without good math skills. Ludicrous.

Science and math in the US are dominated nowadays by recent Indian and Chinese immigrants. Exactly these are the nations that are the challengers to US hegemony in the 2nd half of the 21st century. If only 1 out of 1000 of these scientists retains more loyalty to his/her old country than his/her new, the software that will run the US in 2050 will contain the kind of flaws/Trojan horses that are suspected to exist in todays voting computers (the math on that seemed sound enough on an economics forum) . Math certainly is not strategically important to the US. The fact that non-star scientist (the vast majority) are not well paid is not a reflection of its importance. BTW: the most mathematical applied science nowadays is economics, is this not supposed to be an economics forum ?

"It doesn't help much to have you calling interest rate calculations "math". That is arithmetic."

Do you understand the concept of subsets?

Actually, sd, that wasn't me that posted that. However, if there are large numbers of doctors, lawyers, or executives that use anything more advanced than occasional algebra in their work I managed to miss them for 53 years. 99% of their work would involve nothing more complicated than arithmetic, and they should be using computers for that.

Tim H.:

My sincerest apaologies. Still getting used to the new comment formatting. I'm truly sorry.

bhaim:

Holy shit that's bold, etc. etc.

Professor Katz says (paraphrasing - excuse me) if you have the intelligence to become a scientist instead become a lawyer (or doctor) and make more money.

The typical lawyer or doctor makes 2 or 3 times what the typical scientist makes.

The downside of Americans being bad at math is that lots of policy decisions are quantitative. If one politician makes proposals that rely on 2-1 adding up to 5, then reporters ought to notice and say that it doesn't work. Reporters who are proud of their own ignorance, who think numbers are boring, who are contemptuous of people who are comfortable with numbers, will be unable to do a competent reporting job on that story. The best they'll be able to do is to treat "2-1=5" and "2-1=1" as two different and equally valid opinions, and dutifully print quotes from people on each side of this abstruse mathematical dispute. Voters who can't tell whether 2-1 is 1 or 5 won't be able to tell which candidates are serious and which ones are trying to blow smoke in their eyes.

And yes, some of the political "disputes" that are going on right now really are as simple as the value of 2-1. The only reason there's the appearance of a dispute is that politicans have learned that using numbers is a good way of getting reporters' and voters' eyes to glaze over. If people were more comfortable with numbers, if they were used to mathematical reasoning being an ordinary part of life, then this sort of deception would be impossible.

Though the article was disappointingly poor, Donald McNeil is a sensational reporter. A reporter who has posted article on important article about health care problems in southern Africa and other regions. Health care is the strength of this reporter, math or no.

Katz is absolutely correct. And I say this as a tenured professor of physics.

BTW, people should read the essay before commenting on it. He specifically considers applied disciplines (like CS or engineering) separately from mainly academic disciplines (such as pure physics or biological specialties without biotech applications).

In applied areas the job and pay conditions are much better, even for profesors.

This doesn't mean math is not useful - I agree with Matt that the rigor of mathematical reasoning is useful in other areas, and often absent in political and policy discussions.

i give up.

I'm a holder of an undergraduate degree in mathematics. What I found best about my mathematical aptitude was it helped me get astronomical scores on the SATs and the GREs in the math area, which in turn helped me get into Princeton.

HOWEVER, excepting its aid in that regard, mathematics per se has been of little use to me (although I've yet to have clinical evidence of incipient Alzheimer's Disease!).

Taking one example of an area thought to be 'higher' mathematics, calculus (although many high schoolers are now taught it, and it's part of a freshman's curriculum if not previously taught it), my professors admitted that most integral equations cannot be solved by analytic methods, but rather 'brute force' numerical methods (i.e., by computer iterations).

The best thing going on in Math Education

http://www.artofproblemsolving.com/

The low cost online classes taught by people who know and love math.

"Leave graduate school to people from India and China, for whom the prospects at home are even worse. I have known more people whose lives have been ruined by getting a Ph.D. in physics than by drugs." -- Katz

Bold talk.

"A New York TImes reporter who finds that the toughest math he tackles is tip-calculating is almost surely doing a very lousy job at his own retirement planning and general financial management."

Many people do, after all. Isn't this, basically, why it is the Bushies get away with appalling fiscal policy?

"Leave graduate school to people from India and China, for whom the prospects at home are even worse. I have known more people whose lives have been ruined by getting a Ph.D. in physics than by drugs." -- Katz

Again, let me say that I agree with Katz.

Unless your abilities are clearly at the 1 in a million level you may very well be unhappy pursuing a career in theoretical physics. (This means 99% of high school valedictorians, accomplished econ profs, National Merit Scholars, software designers, etc. need not apply.)

I would not advise my son or daughter to go into theoretical physics unless they were that talented (and even then I would plead with them to look into other subjects like CS or biology).

We regularly see kids from China or India or the former Soviet states who were on their national math/physics Olympiad teams, or scored in the top dozen in the nation on the IIT or college entrance exams. Many of these kids still do not find a career in theoretical physics, or dislike very much being an "average" researcher in the field.

Take a look at Katz's CV - BS at 19, PhD (Cornell) at 22, tenure at 25. Yet he is not one of the leading intellects in the field.

Audiences and critics loved the line, presumably because they too rejoiced in knowing that they had never, ever used the quadratic formula again.

My father - who left school with one O-level, in art, at fifteen to become a welder's apprentic, and wound up with a PhD in engineering a few years ago, tells a story about that.

As I remember it - a couple of years down the line he was taking night-school classes, and trying to get a better job; one of them, doing the cursory interview, noted that he was taking some maths. "What's the quadratic formula?" "Uh, minus b plus or minus the root of b squared minus...", they figured he was at least intending to go somewhere rather than just marking time, and he got the job. Corny, I know, but that's what he told us...

Maybe the real reason Peggy Sue was unhappy is that her innumerate state kept her from moving out of a lower middle class grunt job.

I wonder what would happen if we asked the NYT to do an article on why the average person never has to do a book report once they get out of school.

The NYT does an article every few years along these lines of reporters braging about being innumerate -- Why???

Moreover, the article used two questions from a quiz that I assumed the reporter selected because he thought they were difficult, when in reality for anyone who is numerate they were elementary.

We ought to all send letters to the NY Times asking McNeil if he finds bragging about how dumb he is works well as a pick up line.

I work on video games. I use boatloads of math. And I'm not even a graphics programmer. I also read a lot of academic papers to keep up on new techniques in related fields and keep that fresh video game goodness coming. So, understanding math and the language of mathematics is important to my job. A solid understanding of Discrete and Linear math is pretty crucial to modern games. Also having a basic understanding of mechanics helps.

I think Katz is a little pessimistic about the prospects for advanced research. You have to be careful to pick a useful, well funded field (it looks like Katz's was High Energy Astrophysics, which doesn't count) but if you do, you have a good chance of getting an academic job. I joined a Math graduate program to study pure math, then switched to computational biology when I saw the writing on the wall, and now I'm doing a postdoc at a very well funded genetics research center, and I'm reasonably confident that I will get a good job afterwards. Of my former advisor's earlier students, few went straight into industry, but those who stayed in academia now have tenure-track jobs at Stanford, Berkeley, Princeton and MIT.

The typical lawyer or doctor makes 2 or 3 times what the typical scientist makes.

The typical lawyer or doctor also, to my understanding, works 2 or 3 times what the typical scientist works. At much higher-stress positions.

Linnen should either take a refresher course or rid himself of his slide rule.

His answer was actually off by 33 cents, not 0.32 cents as said in his post.

Alex,

Bioinformatics is in its growth phase. (This was the case for physical sciences just after Sputnik.) Eventually it will enter a zero or slow growth phase where the population of researchers is almost constant, so an opening appears only due to a retirement. In that eventuality (true of most mature disciplines), things will look very different. (Although, in disciplines with a lot of private sector applications, even the zero growth phase isn't so bad as industry can absorb a lot of the PhDs.) Imagine what your life would be like had you stayed in pure math and were fighting it out with other algebraic geometers right now.

Of course, there are always a few growth disciplines, and electing to pursue one of them is a good decision. But by definition, not everyone can be in the hot subfields, and some people may have a taste for working on the open, classical problems. (What if Andrew Wiles had decided that compiler design had better job prospects than number theory?)

Someone commented that doctors and lawyers work harder than scientists and suffer more stress. The latter may be true, but I doubt the former is.

Science is the most internationalized of all activities, and hence the one with the most globalized competition. A faculty job at a good research institution in the US is open to competition from smart people from all over the world. This is not yet the case in medicine or law, but I look forward to the day when I can hire the holographic avator of some smart professional in Chennai to represent me in court or look at my x-rays.

Greetings:

I use math every day. I am a PhD quantitative economist in private industry. I went through the public school system in the US of A. I graduated from second- and third-tier state universities; I am 50 years old and I make a reasonable living. I have developed algorithms and the program code that calculates 120-day hourly load forecasts for an electric utility, calculated on-the-fly in real-time, used by real-time traders and schedulers, in a 24x7 environment, who trade real money. The kids still cannot keep up. Sure, the days are long gone where I could visualize matrix algebra in many dimensions in my head (that may seem to be a foreign concept today, but it was very real at one time some years ago). You folks who say you cannot use math seem like you are trying to rationalize your own shortcomings.

Who are the best paid? That would be Bill Gates and a couple of his pals who started some sort of software company. Didn't they enter math competitions? And Buffett who probably does his calculations by hand on a piece of paper, but that would probably only involve arithmetic.

Some comments from a physicist / entrepreneur :

Just like a lawyer should never ask a question of a witness in court that he doesn't know the answer to, never do a calculation that you don't know the answer to. (If need be, break it down in steps, and back of the envelope is OK.) I have seen many people make huge math mistakes (many orders of magnitudes) using a computer or calculator and have no clue.

I also find I use basically all of of the math I know - calculus, quadratic equations, least squares (an application of matrix algebra), statistics, all of it. As the saying goes, to a carpenter with a hammer, everything looks like a nail...

This is actually a gem: "Mr. Snelgrove, I happen to know that in the future, I will never have the slightest use for algebra. And I speak from experience".

Number one, knowing the future from experience. Perhaps it has some connection with the paradox of free will.

Number two, a confusion between "not being able to use X in the slightest" and "X having no use".

A little example: it happens to assorted morons that they are leasing a car. There was a 20/20 segment about salesmen who pretend that they decrease the price while they are actually increasing it during the bargaining. The problem is that the contract does not state the price, but it gives the downpayment, monthly payments and the final value, and they only imply the actual price and interest rates.

The reporter that was responsible for the segment was asked how an average customer can figure that he/she is cheated. The answer was to contact an appropriate state office if you have some suspitions.

It is actually pretty easy to mentally calculate a decent estimate how the monthly payments should change if we simultaneusly change the value of trade-in and the downpayment, and it is simplicity itself to estimate that if the price is allegedly going down (or if the valuation of the trade-in is increased) then the monthly payment goes down.

One could also suggest to shell 20-30 bucks, buy a financial calculator and read the instructions.

If I may add something without the "how much dough do you make" argument hovering above, the things that have bugged me about math education in this country is how none of it is ever taught with any relationship to the real world.

A couple of comments have touched on that indirectly, but none has addressed it directly.

The first problem is that if you are good at math you can make more money in something other than teaching. The second is that is math is very similar to learning a language and everybody knows it ain't easy to learn a language at middle and high school age, much less later.

If I may be allowed to refer to my experience, I would not have been able to learn trig (applied, and only paritally at that--I can't plot a sailboat's course around world) if I was just being taught it. I had to be immersed in it by using and using reguarly in my apprenticeship and work.

Math needs to be taught as a language. A lot of pupils will say fuck it, but a fairly decent number will find it interesting and go further and be able to do things like figure out lease costs and a few will go on to do bigger things with it.

And perhaps a lot of people will not pull a Homer Simpson and think, as a former employer of mine (a CEO as he dubbed himself) that ten percent of a million bucks was $1,000.

I gather you Macro people have an answer to this, or at least a theory:

Some time ago, The (London) Economist opined that providing free higher education leads to socially inefficient overeducation. This makes sense in a short run perspective where job descriptions are stable. If you have x gas station attendant jobs to fill, you don't want x-k applicants without and k with college degree. On the other hand, optimal allocation of skills to jobs reduces market pressure to get higher education. Combine this with the recognition that education is costly even if it is provided as a public good (qua deferred returns and pains of feeling one's ignorance). So does the U.S., by optimally allocating education in the short run and by keeping structural unemployment low (through reduction of job security), undermine its educational base? Someone must have thought this through in some rigorous way.

steve: "I look forward to the day when I can hire the holographic avator of some smart professional in Chennai to represent me in court or look at my x-rays."

I think it was last summer I read in Business Week that Indian MDs are already examining X-rays outsourced to them from the US.

"my professors admitted that most integral equations cannot be solved by analytic methods, but rather 'brute force' numerical methods (i.e., by computer iterations)."

So studying the "higher" math may not be all that fruitful (I tend to doubt that, but I'm increasingly finding that people who can handle higher math well either can't or don't want to apply it well to where it might be actually useful) but numerical methods are still a major component of mathematics, critical for designing computer programs and various engineering applications, not to mention quite complex if you have to deal with interesting enough problems.......

Among the benefits of studying math is in learning to think abstractly and to construct a logical argument. Math is terrific preparation for the law (for one) for that reason. Every (legal) argument aspires to be a mathematical proof.

Matt Austern's comment is one of the best things I've read recently. I'll just add two of my pet peeves. One is when people equate math with calculation. The other is when (usually the same) people say they "don't think mathematically" but in some other way. That makes me want to kill. There is no thinking that is not mathematical; at best, such mental activity can be characterized as feeling and reflexes.

Thanks! And yes, equating math with calculation is wrong. Back when I was an undergrad at MIT, the folks in Course XVIII had a slogan: "I don't work with numbers; I'm a mathematician".

Are the opportunity costs of learning math worth it? I think the advantage to learning math is to get metaphors and ways of seeing the world that don't rely on words. That doesn't mean that it might not make sense to replace Geometry with statistics and/or economics as a subject matter in high schools.

Re: Doctors and Math
I am a practicing family physician and I use math all the time at work. A typical clinical challenge: Mrs. Tremble is to use no more than 6 Xanax per day (for her anxiety, irritable bowel syndrome, PTSD, OCD and general inability to handle the real world) and she gets one prescription per month. She last demanded a refill on Nov. 20, and today she is causing a ruckus in the waiting room, going into mild to moderate withdrawal. Questions: How many is she using per day? When is her next refill due? Extra credit: How do I get her out of my practice?