Blake Ross, one of the developers of Firefox, the new web browser taking the Internet by storm, is only 19! He started as a web designer at 10. Good on him. I use speedy Firefox to post on this blog, not to mention read lots of others (yes, and legacy media sites too)– and in the three months or so since downloading it, I haven’t had (touch plastic!) one virus.
Still, young genius has its downside. As Ross, now a sophomore at Stanford, tells the Seattle Post-Intelligencer:
“All my computer science professors are expecting straight A’s, even in classes that have nothing to do with the Internet.”
My advice to Blake: quit school. Hey, it worked for Bill Gates. He had a long run with Internet Explorer until Firefox came along.
I don’t come out about this very often, but I was a prodigy — first college at 9, programming professionally by 14 — and it’s a hard road sometimes.
Ditto Charlie: I was first published while still in high school, and had been programming for four years by the time I got to college. I don’t have any regrets, but I’d sure encourage prodigies to explore other aspects of life and now allow people to expect them to focus on their gift 24/7.
It’s also important to remember that a lot of “prodigies” are autistics of one level of function or another. Some of them don’t need our worship—they need our help.
I saw that there was a post in the thread and bet myself it was Charlie. Bingo. I would think that being a prodigy could be socially difficult, especially if you get into the professional world.
Then there was an ex-professor of mine who was asked to recite a poem on the first day of kindergarden. So he began:
A Gentle Knight was pricking on the plaine,
Y cladd in mightie armes and siluer shielde,
Wherein old dints of deepe wounds did remaine,
The cruell markes of many’ a bloudy fielde;
Yet armes till that time did he neuer wield:
His angry steede did chide his foming bitt,
As much disdayning to the curbe to yield:
Full iolly knight he seemd, and faire did sitt,
As one for knightly giusts and fierce encounters fitt.
What does a kid who can recite The Faerie Queen in kindergarden do for companionship?
Damn, and I thought I had it bad. At least I had hit puberty by the time I entered college. (Barely.) And while I could read in kindergarten, it was strictly New English, not Old(e)
I would like to know, though, what he means when he says, “All my college professors are expecting straight A’s.” What business is it of them what grades he gets in other courses? Where is it going to hurt him if he just gets the grades he’s satisfied with and be done with it?
I actually just finished up a Ph.D. at Stanford in EE but I confess to not knowing much about the undergraduate lifestyle here, having received my BS somewhere else (U.T. Austin, where a genuine/counterproductive social life wasn’t difficult to come by.)
I would not counsel him to leave school however. Just get the B.S. and be done with it. With his credentials he won’t have to worry about a job.
I also use Firefox. My favourite feature is the “Themes” option on the “Tools” menu, where you can download customized appearance packages for the browser.
And of course, I can now check the Drudge Report without getting popups and adware installed on my computer.
Firefox has been a blessing to me, too. Spyware was eating up my computer. Now I run Spybot right after I use Explorer for those few videos that require it, and that’s the only time it finds anything. No wonder Microsoft is giving up on it.
would like to know, though, what he means when he says, “All my college professors are expecting straight A’s.” What business is it of them what grades he gets in other courses? Where is it going to hurt him if he just gets the grades he’s satisfied with and be done with it?
There’s a peculiar thing about being a prodigy: after a while people get used to it, so that if you aren’t blazingly brilliant at something, you’re a disappointment.
Regular old hand arithmetic didn’t come easily to me, even though math concepts (or later, geometry, logic, all that theoretic stuff) did, but that wasn’t interpreted as a sign that there was something that didn’t come as easily as reading, science, and drawing had — it was interpreted as a sign that I Just Wasn’t Trying.
(Now that’s called ‘dyscalculia’, but to be fair this was about 1961 … the notion that there was such a thing as subtle differences in specific brain functions was not a common one.)
The point is, once you’re A Prodigy, you’re expected to be, well, a prodigy: extraordinary in all things.
Well, that’s true Charlie, if you’ve not been around lots of Really Smart People (TM). Those who have been/worked with/taught the very, very sharp know that everyone has their limitations. I have been a teacher at Mathcamp (http://www.mathcamp.org), and know whereof I speak…
But really, the smart kids nowadays do have so many more opportunities to connect to their intellectual peer group. When I was a kid, I satisfied myself with talking about stuff with my Dad, and hanging out in the library. When I got to middle school and high school, I found there were programs and schools run with our types specifically in mind (CTY at Johns Hopkins, TIP at Duke.. I went to the North Carolina School of Science and Math, a state-run boarding school for 11th and 12th graders.) And what’s really neat is that when you’re in an environment where everyone is expected to be smart, there’s no particular expectation of grades or achievement. I would’ve thought the profs at Stanford would be used to prodigies by now, but I guess not. I suppose the standards at Stanford have slipped.
You can’t beat Firefox for stability. Combine it with an IBook and Mac OS X and you’re rock solid.
BTW Ross is on this month’s cover of WIRED.
Ron
Duh, actually, I misread the quote I was referring to. I assumed that for some reason they were placing extra academic burdens on him. But I see your interpretation clearly once I read it in context. They just assume he’s going to be good at everything, whether or not it is directly related to the work he’s so good at.
I use firefox but I am not a prodigy. I am not sure what one has to do with the other, but I am just your average, ordinary underachiever who wasted her life.
sigh.
Being surrounded by all these smart people is intimidating.
I suppose the standards at Stanford have slipped.
At most elite colleges, gradeflation is so rampant as to render A’s meaningless. Everyone at Harvard graduates cum laude. The effect is to force the truly brilliant students– a minority at any school except maybe CalTech or MIT– to grade-grub, which is more often than not the opposite of investigating ideas and achieving unique insights.
Anyone with experience of summer math/science camps for smart kids? Are they worthwhile, fun for the kids? Any comparisons of individual programs, esp for elementary-school age kids, would be welcome – thx in advance, t
One more personal request: Charlie, or anyone other mathematically-gifted person, were you a late talker as a small child?
Tom Sowell did some research on this phenomenon after his child, a late talker gifted in math, was misdiagnosed by the speech therapists’ lobby.
Sowell found a meaningful correlation between late talkers and a family history heavily biased toward the heavily analytical and quantitative professions as well as musical talent. Describes our soon-to-be 3 year-old and his family bgrd to a T….
Thibaud — well, I’ll be damned. I was a very late talker (and to this day my mother can’t be prevented from telling people that my first complete sentence was “mother, what’s a deciduous tree?”) and just thinking about it, I’ve got:
father: professional trombonist, sat in with Glenn Miller in high school
younger brother: professional trombonist, CPA, now VP of Internal Audit at Aetna
younger sister: professional double reed player, now a geek doing training materials for Cisco
me: occasional professional singing gigs, musical comedies in regional theatre before I joined Actors Anonymous.
Everyone know the story about Einstein and the soup?
Meep — when were you in the S&M School?
I scored at the level of an average high school senior in math in sixth grade; I had taught myself long division in second grade but had so little occasion to use it that I had to relearn it all over again when it came up in school.
I only took a couple of courses ahead of grade (math & science), and definitely found that the harder part was dealing with older students and not the schoolwork.
At what age did you start talking, Charlie?
At what age did you start talking, Charlie?
Three-ish, I understand. Hard to be sure, as my mother is rarely a reliable source on anything.
Hi Charlie,
I never knew you were a prodigy; but it certainly makes sense. It all fits.
Thibaud,
I’m quite familiar with Sowell’s book on Late Talkers. Mental retardation was certainly a misdiagnosis for his son, but some variety of high-functioning autism would not have been, and I think Sowell has derailed a lot of people’s helping their autism-spectrum kids acquire speech, social skills, play skills, etc., with his well-intentioned book.
There’s no harm in getting a normal-but-delayed kid some help in kick-starting his speech (and you can do it without labeling him, btw), but there’s potentially a lot of harm in denying help to a young child with developmental problems.
Most of these kids’ first words aren’t requests for information about deciduous trees. Our Charlie is a very unique guy.
I see no reason to label this kid a prodigy as opposed to just a smart kid, unless the timing of his skills is closer to Charlie’s. As a professional in computing since the ’60s, I would argue that there are few hard problems in practical programming (as opposed to the theoretical side of computer programming) and the best ages for programming match are very young (younger than for mathematicians and physicists because less knowledge is required). I could teach any smart, motivated high school kid and most probably junior high kid to program pretty well.
Like many here, I was a very smart kid growing up without many other smart kids around. Until high school, which was in a university town where I was one of the faculty brats, and we had an honors program (not very good ones, by today’s standards, except in biology).
I learned boolean algebra before I learned regular algebra (which I learned in 31 hours on an experimental “teaching machine”), was doing number bases for fun in class (something that today serious computer geeks do all the time) in 7th grade, and had to walk to the university to take college course while in high school. I entered college (University of Kansas, it never occurred to me to go elsewhere) a junior, and I got straight A’s in humanities (other than German) and got an A+ in a University Honors English class on a pop quiz about a book I had not read. Oddly, at KU I didn’t get straight A’s in math, engineering or physics, which were my favorite topics, but a few years later did at UCLA.
KU had (and probably has) a program (Summerfield Scholarship) where they take the 40 top high school students in the state by standardized tests and grades, run them through days of testing on campus in Lawrence (including a bunch of psychological tests – MMPI, Thematic Aperception, Rorschacht, etc in addition to various achievement and IQ). They select 20 for the scholarship. I was one of the 20. The other 20 get a lesser scholarship. Within 2 years, at least half of that 20 had flunked or dropped out of school (I did the latter, and joined the Navy as an enlisted man). Go figure.
[An aside. They gave us the MMPI. I decided to see if I could beat it, and produced results that were essentially flat across the categories with a good score on the "he's not cheating" controls. Later I took it for real and the difference was dramatic. I conclude that the MMPI is not as cheat proof as they think. One of our folks intentinally made up a dragon story about every TAT image, and I think an incest story or about every Rorschacht, just for the heck of it. Smart people mess with the system.]
But I was not a prodigy.
Family wise for this non-prodigy: I was a natural musician (clarinet, flute), my mother won a college math award (and was also a flautist) and in WW-II was selected by the government as one of the top 100 women mathematicians in the country – to work as an engineer because the men were off fighting. My father is a winner of the Australia Science Prize (first non-Australian to win it, I think) and he and his brother are the only siblings in history to be in the National Academies.
My daughter went to the Center for Academic Precocity at ASU and taught herself calculus at around 12 years old while the grad students watched through a one way mirror, and is a neuroscientist graduate of Johns Hopkins. I don’t think she would be labeled a prodigy.
Ivy League grading is interesting. According to my daughter, they use a B+ average curve. This, to me, makes sense if the students are actually selected by capability (not legacy or AA), since it would be sort of silly to flunk out someone who is only in the top 5% of ability because many others were higher.
Johns Hopkins, on the other hand, uses a C-centered curve, and as a result, in the science areas, has an incredible attrition rate – over 75% in my daughter’s major. The result, especially for pre-Meds, is terrible, since most med schools don’t discount the different grading standards, and thus there is rampant cheating, including by ethnic group, lots of suicide, and now significant use of nootropic drugs (especially by neuroscience students, of course) such as ritaline, whereas in my day the nootropics were nicotine and caffeine
Hopkins has been asked many times to join the Ivies, and is now doing so (last I heard). They have to raise their curve center (fI guess to B+) as a REQUIREMENT of joining. Those who went through just before are supposed to get a special letter with their transcript explaining this.
Another comment… the level of errors in my posts is due to two factors:
(1) a feeling of time pressure causing me not to review when I should (my apologies)
(2) intermittent ADD (email via my blog email address if you are curious how this can exist).
Also, boredom in school (and later business meetings) produced this
All of this, of course, posted on Firefox.
Most of these kids’ first words aren’t requests for information about deciduous trees.
Uh, that was supposed to be my first complete sentence, not my first words. Also, as I say, it’s my mother’s story and must be taken with a grain of salt.
In this case, I swear I remember asking the question, and it was in our second house, which would put it when I was between 6 and 9 years old. God knows that 40+ years later I don’t trust my own memory particularly either, so I won’t say one way or the other. But I have been assured independently by many family and family friends that I was a later talker.
This assurance is regularly followed by the statement that, once started, I never stopped.
I see no reason to label this kid a prodigy as opposed to just a smart kid, unless the timing of his skills is closer to Charlie’s.
On the other hand, I never thought I was unusual — it was just that everyone else kept talking about it. I just wanted other people to either keep up or stay out of the way.
The point is that it doesn’t matter if Blake is in some sense a “real” prodigy; what matters is that he’s being treated as one.
John:
…regular algebra (which I learned in 31 hours on an experimental “teaching machine”)
Was this a version of the Skinnerian teaching machines? If so, did it have any of the failings that are usually ascribed to these devices (e.g. memorization without understanding).
Look, John and others, I want to make sure that I’m clear about something: I neither take any particular pride at this point in having been a “prodigy”, nor feel that the label is in any way useful. (What could be much sadder than being a 50 year old child prodigy?)
“Gifted” is perhaps a little better, but the real issue in my mind comes in two parts:
(1) our schools operate on a “factory” or “industrial engineering” model, in which students are the workpieces and “success” is modeled not by how effective the individual outcomes are, but rather by how effectively the factory produces a steady flow of barely acceptable work products. This means, with self-adapting “workpieces”, that some of us manage, in some of the areas of our interests, to “pull ourselves ahead” in the production line.
I think this model is massively flawed, and — now that I put it that way — I think needs massive revision.
(2) I think being labeled a “prodigy” has some good effects — I actually was permitted to attend college classes at 9 because of the label, and that the source of some of the happiest moments I had in childhood — but it also has some really deleterious effects that come from a combination of the “gifted child” expectations I talked about above, and flat-out resentment on the part of others.
Charlie – good point. I faced the high expectations too, and didn’t reach them.
It is my opinion that any high IQ kid (say Mensa IQ minimum) with the appropriate mindset (and, as I suspect we’ll find with FMRI in the future, specific but not uncommon brain wiring – the engineer “schematic”) can be taught to be a programmer at that level if he has the intense interest.
Charlie, since I don’t know much about prodigy-dom, what effects (other than emotional leftover from being treated as one) does being a prodigy have as an adult. Super-high intelligence? Just another smart guy? Do you have any asperger’s syndrom symptoms (not uncommon among good computer programmers, apparently)?
Is there a higher level of psychopathology (I believe there is among high IQ people in general)?
My advice to the kid. Do the right thing with the VC’s (remembering that VC’s are smart sharks that probably make Hollywood producers look like naive good guys), and get your money before programming turns into a low wage job (due to outsourcing). I speak as a person who has made most of his income from programming for 40 years and most of his retirement fund from as a result of activities related to the VC’s.
My son isn’t a prodigy, but he participated in the Duke TIP program. He did an architecture class for two weeks at Kansas University and really enjoyed it.
BTW, not everyone graduates cum laude from Harvard, certainly not in my son’s class. Harvard restructured its grading curve when Summers became president. You have to work hard to get an A. I think the previous post is correct that they standardize on a B+.
Also, last week I just read a letter to the editor from the Dean of Admissions at Harvard. Legacies and non-legacies have the same average test scores.
Firefox rocks!
Was this a version of the Skinnerian teaching machines? If so, did it have any of the failings that are usually ascribed to these devices (e.g. memorization without understanding).
Morgan, in the case of basic algebra (not to mention arithmetic and some other topics, eg, English spelling), I don’t think the distinction is all that useful. (In the case of basic algebra, I’m not even sure it exists! Knowing that (x+1)(x-1) = x^2-1 is essentially a matter of following an algorithm, like addition or long division; algebra as a math topic is just the observation that these algorithms have analogies with other parameters.)
Understanding in this context might lie in understanding the Peano axioms, or the properties of a ring and a field, but solving equations is largely a learned skill of manipulation.
Which mathematicians, increasingly, do with machine assistance anyway.
Wow, Charlie. But oh how I’m not surprised!
I certainly was no prodigy, just ahead of my peers by a few years in everything. It was decided not to skip me in grades because the thinking was that it was more important to be with kids my age than to challenge my mind. So I’d get straight A’s and an F in effort and get in trouble at home for the ‘F’.
And hang out with the older kids anyway.
I remember one time in gradeschool I had finished the assignment and started to color in my coloring book. The teacher made me stop coloring because, she said, it was unfair to the other students who were still working on their assignment. And this was back in the fifties!
How things have changed! Not.
As for Firefox, I’ve tried it and gone back to IE. It’s just a bit different and needs getting used to for me is all. It’s certainly a fine browser, no question about it.
But I do think the paranoia over IE is overblown. I’ve been using it for years and have not gotten any virii nor any adware and have had no stability issues. I must be doing something wrong.
WichitaGirl–
There’s no harm in getting a normal-but-delayed kid some help in kick-starting his speech (and you can do it without labeling him, btw)
How? Am wary of the speech pathologists, especially because it’s very likely that much of our child’s lateness in speech development has to do with speaking Russian at home with his parents while hearing English outside the home. We want him to be fluent in his mother(‘s) tongue. We have yet to meet a speech therapist or other professional who has any insight into or experience with bilingual households, least of all Russian-English ones.
I’m not worried about him but I sicken at the thought of all the pseudo-science and bad psychology that surrounds the schools these days. Have considered home schooling him but rejected this, as he’s on the shy side and needs playmates.
Charlie (CO):
I’m not sure the distinction exists, either – but educators sure do. They are constantly pushing “understanding” on kids, day after day, month after month, year after year. And at the end the kids still use their fingers to multiply.
That’s why I asked (looking for disconfirming evidence like a good scientist). The topic interests me largely because I think that education might be improved by focusing a good deal more on knowing and a good deal less on “understanding”.
I’d love for my kids to learn algebra, geometry, trigonometry, calculus, and statistics in a few weeks or months. Think of all the time they could spend doing other things (or more of those things, if that’s their muse). That’d be a hell of a competitive advantage.
Thibaud,
If you have means for private speech therapy, you needn’t do it through the schools. Take him to a speech therapist and have them give him some standard tests. Ask them to help you develop a program to help him catch up if he needs that.
Even if you take him to a school district speech therapist, all they will do is give him tests. If he tests below some level, he will be eligible for speech services. They won’t, they can’t, give him any diagnostic label other than scored-below-such-and-such.
The worst that can happen, by this scenario, is that he spends some time having a nice speech therapist give him some quality time. Whereas leaving a serious problem untreated during the preschool years is something to be regretted forever.
To my mind — a type 2 error is better than a type 1 error. And I speak as one who made a type 1 error.
Words of wisdom. Thanks, WG
Charlie(C):
Understanding in this context might lie in understanding the Peano axioms
I would think understanding would lie in recognizing that the variables could be treated as numbers and that solving an equation consisted of isolating the variables on one side of the equation. The rest is technical skill in manipulation. As to the Peano axioms, Edmund Landau wrote in the introduction to his book Foundations of Analysis, where he starts with the Peano axioms and carefully developes the various kinds of numbers — integers, rationals, reals, complex — that every mathematician should work through the material once, and then forget it. Which is how most mathematicians probably treat it, along with axiomatic set theory.
Speaking of set theory, wasn’t new math a disaster? Made the subject boring. The only way it could have been worse was if they had tried to foist axiomatic logic or the theory of categories and functors onto kids as the basis of mathematical understanding
Morgan asked: “Was this a version of the Skinnerian teaching machines?”
I don’t know because I don’t know the details of a Skinnerian system. The best way to look at this system would be as a programmed learning text (which is how I learned boolean algebra, by the way). At the time, my father was interested in unusual education methods and there was a company in Albuquerque called “Teaching Machines Inc” and it was their machine. I came away with an absolutely complete understanding of algebra, but I don’t know if that was my skill at deducing principles or if the system taught them. I think it was both.
I would also, by the way, argue that systems designed simply to instill facts are very useful, and in fact should be used in education – at least for the kids for which it works. My daughter learned all arithmetic while 4 or 5 from a really simple little computer program that ran on an Atari 800. It may have been Skinnerian – it slowly advanced the difficulty, and made it a game. When you hit a milestone, you were rewarded with this cute little fishing game (hmmm… maybe THAT’s why my daughter is such an avid fisherman – a side effect). We gave it to her and then didn’t really interfere at all. She just did it. It did not teach principles, but frankly, a lot of what is taught as principles to kids learning arithmetic should, in my opinion, be taught at a later age – they bounce right off of most of the students – they memorize but don’t internalize the principle. So I am in favor of rote learning of many things, and of using games and other devices to make it fun.
Agree with the above. Rote learning is not just a good thing for small kids; it’s the only way to get on your feet intellectually. Whether it’s math or poetry or history, memorization is critical to mastery. There simply is no shortcut; as the Russians say, povtereniye, mat’ ucheniye: repetition is the mother of learning.
I suspect that our primary and secondary students’ declining educational performance in the last few decades has much to do with the hostility to rote learning and memorization seen in the education theories promoted since the 1960s (or earlier).
John Moore:
Thanks. Google tells me that Teaching Machines, Inc. produced Skinner-inspired machines.
All the teaching machines dropped out of the schools in the mid to late ’60s. I’ll have to see if I can find out why. It coincides with the rise of the Cognitivist school of thought within academia, which seems to have been generally negatively disposed towards anything Behaviorist.
I wonder if they threw the baby out with the bathwater.
I also agree with everyone arguing that rote knowledge is an essential part of learning. Whatever the field, you need to pound in the pegs of knowledge before you can hang the web of understanding.
Ugh, that’s a terrible metaphor.
Charlie, I’m one course away from a Math/CS degree at UCLA and have never heard of that stuff. On the other hand, with the exception of analytical techniques needed for engineering, and for a very deep understanding of psychoacoustics (a friend and I conceived and designed the world’s first real time polyphonic digital synthesizer, and there were some artifacts such as aliasing that ultimately have deeper issues than appear on the surface).
I believe there are two concepts that are sufficient to understand ordinary algebra:
1) as mentioned earlier, the idea of variables
2) the idea that if two representations are separated by an equal sign, the same transformation on both sides doesn’t change the equality.
An addenda to #2 is then: solving an equation is a process of finding the right transformations, and you can’t go wrong by playing with them (other than dividing by zero in a hidden way, or something like that). Which means there is a third, less important function: you can use regular algebras on infinities.
Am I missing something else?
Calculus likewise requires very few principles – fundamentally, the concept of limits, and practically, how that gives you derivates and integrals (and the normal student doesn’t need to understand limits in order to “do” calculus).
Beyond that… it gets harder.
Comment on “proud.”
I am proud to be smart. And it’s absurd, because it was handed to me – I did nothing to achieve it. It’s like being proud to be tall, or handsome, or something. Silly.
My father considers it a responsibility – he told me once that I owed the world the “benefit of your intelligence” – something I have not achieved due to a somewhat balancing handicap. It is something he has done.
I think the value of rote learning depends on the subject and the student. Arithmetic : the student need to memorize the multiplication tables, methods, etc., there is no way around it. History : the student needs to memorize dates and names just to get a sense of place and time and to be able to talk about the subject. Algebra: no, I happened to find algebra easy because the principles were self evident. I have a fairly porous memory that hasn’t improved with age, so subjects that need rote memorization, like organic chemistry, biology, and arithmetic, I found relatively difficult. Subjects founded on principles from which the rest followed, were easier.
On the other hand, I think structure in teaching is important. Expecting the student to rediscover on his own things that took millenia of work by the best minds to discover — an odd approach that was in voque at some point — is expecting too much. The path needs to be smoothed, the student trained to the task, and hints given.
I meant to say that you cannot use regular algebras on infinities.
Regarding rote learning. I know that the educational establishment became very opposed to it not long after I escaped their clutches. And they were wrong.
Rote learning is useful for other things – so a skill in rote learning should be taught. For example, when I was in High School, I wanted an FCC license that would let me be a broadcast engineer, and also operate my dad’s research radar to look at weather. There was a huge book with all the possible questions and answers. Many of the questions did not involve concepts (what is the name of the oscillator shown in the schematic, what is the maximum power an AM station can run under X condition, and lots of other regulatory questions). So I made my own flash cards – one for each question with the answer on the back. I then sat in class with these and just tried each one, the wrong answers going into the stack to be done next time. Very powerful technique, very useful, instills zero understanding. I passed the test after 2 weeks of this.
I have a very low opinion of that establishment, even as I have a high opinion of some teachers. Mankind has been teaching for a long time, and the way educational theories have been derived in modern times is to use insufficiently powerful “scientific” techniques to prove various techniques, and then dropping them on the whole country in one big thud. The economics of textbook publishing (and the economics of being the person who writes them) gives an incentive for this process to continue.
Arizona also has a variant on the voucher system – you can contribute up to $600 to any school that meets the criteria, and is a full tax credit. This means the public schools are facing real competition from the private sector (I contribute and I have no school age children), and drives a little bit of results oriented thinking, although the folks doing the thinking have been mistrained in the first placce.. It also creates a diversity of pedagogical techniques. Qualifying schools, by the way, can be religious (my contribution goes to the Catholic school my daughter graduated from).
John Moore:
2) the idea that if two representations are separated by an equal sign, the same transformation on both sides doesn’t change the equality.
I would simplify this a bit further. If variables are in some sense numbers, then the equals sign is the assertion that both sides are the same number. Doing the same thing to both sides preserves this assertion.
Charlie, I’m one course away from a Math/CS degree at UCLA and have never heard of that stuff.
Then I didn’t do a very good job of saying what I was talking about — I don’t know of any analysis or linear algebra course that wouldn’t at least hit the defintion of a field. It’s just any system that has the same sort of algebraic properties as the real numbers — an addition operator, a multiplication operator, additive and multiplicative inverses, and closure (plus some pocket change, I’d have to look up the precise defintion. Dammit, man, I’m a logician, not an algebraist.)
But you’re making sort of my point. I’m for some reason doomed to be one of those people who really wants to know the foundations of things, which undoubtedly had something to do with why I did philosophy as an undergrad, too. But the point is exactly what you’re saying: if you know that you CAN do those things, and do them SAFELY, ie, everything you do results in a true sentence even if it’s not the sentence you wanted, then the rest is just manipulation practice.
Ditto calculus, except that the whole concept of a “limit” is a very subtle one that most calculus courses just sort of hand-wave over. Cauchy, Weierstrauss, Dedekind, Riemann, and God knows who else struggled for about 250 years to go from Leibnitz and Newton and infinitestimals to something that made sense without defining “infinitely small things” as a new kind of axiom. (Now, it turns out that in about 1960 Abe Robinson showed that you could introduce, as an axiom, those infinitely small things, pretty much just as Leibnitz had, and everything worked great. But that’s one of those “logician” topics.)
As far as “understanding” goes — it kind of depends on what you call understanding. Learning set theory is cool, and the “new math” was trying to get people to understand what Frege, Whitehead, Russell, Cantor, etc were doing around 1900 to put things on a solid foundation. The problem being that knowing that you can define “1″ as “the set of all sets of cardinality 1″ doesn’t help a damn bit learning to compute a tip. It takes Whitehead and Russell something like 237 extremely dense and ugly pages to finally be able to state and prove “1+1 = 2″.
And you don’t even want to see what it looked like once they did.
But then sometimes a little memorization is necessary to understanding. Our whole system of orthography is based, under it all, on using letters to represent sounds — and don’t drag in “ghoti” on me, that’s just coming from some historical accidents combining different orthographies and changes in language. Once upon a time, “cough”, “cuff” and “safe” didn’t end with the same sound.
I learned to read in about 1959 in the mountains in Colorado, from someone who had been born in about 1890 and who was enough of a dragon that the school board wouldn’t have dared mess with her. (I loved her, by the way: Mrs Heilmann.) She taught phonics by God, and after a few days, I could decode anything into the right sounds pretty reliably. And you only have to learn, memorize, sixty-odd orthemes to do it. Once you have done so, you understand reading. The memorization makes the understanding possible.
I was a prodigal at a very early age.
Okay, so I’m doing this in backwards order. Sue me.
Charlie, since I don’t know much about prodigy-dom, what effects (other than emotional leftover from being treated as one) does being a prodigy have as an adult. Super-high intelligence? Just another smart guy?
I’m not sure how to answer that. My IQ is up in the 3.5σ range, say 170 on one of the scales. (I don’t remember which one, just the number.) High enough that when I took the Mensa test to keep a friend company, I started getting phone calls saying “Are you sure you don’t want to come to a meeting?”
I’ve never had anyone ask me that at a job interview, but I’ve had plenty of people say that I was “intimidatingly smart”, or “overqualified”. Or say “Charlie, you’re really smart, but….”
And I sure don’t see myself as being in any obvious way “smarter” than, say, the people I talk to on here.
But my point about there being “nothing sadder than being a 50 year old child prodigy” isn’t that it’s inherently sad — I can see how I might have given that impression, but it’s not what I meant — just that its no more important as a 50 year old than, say, having been the high school football star. If that’s what defines you at 50, you’re Ed Bundy.
Do you have any asperger’s syndrom symptoms (not uncommon among good computer programmers, apparently)?
I score pretty high on the various Asperger’s screening tests I’ve seen, but right at the borderline for them to claim a “diagnosis”. I just thought I was a geek.
Is there a higher level of psychopathology (I believe there is among high IQ people in general)?
That’s an interesting question too. I believe that there is a higher incidence of depression in particular (Jamie, you reading this?), but from what I know about the pathogenesis of depression, that could be an effect of the childhood of a “gifted child” rather than intelligence per se.
I was a prodigal at a very early age.
See, that is something I was slow at.
Charlie…
All right, I’ve heard of fields. I enjoyed linear algebra and understand the idea of a vector space.
But it has been forever, and I don’t remember a lot of that course.
As someone who is more the engineer than the mathematician, the idea of the generalized Fourier transform (which appears when you think about vector spaces) was exciting to me and I have used it to understand some physical phenomenon that an engineer using the “sin/cosine” Fourier would be lost with.
But basically, my career and reading interests have been away from mathematics. It takes little math to do almost anything in the computer field. I have done supercomputer benchmarking, where analyzing the results takes a bit, but nothing serious. Only in electrical engineering have I used it – like that music synthesizer (the other guy, Rick Coupland did most of the work, but we discovered a number of the odder stuff in conversation – like why a binary representation based on the 12th root of 2 allowed us to multiply without using a multiplier or multiple shifts – somethihng I couldn’t even explain today).
My main intellectual interests are in the policy area – especially foreign policy, in ideology, and in warfighting techiques and technology. Activites are blogging and other informal debate, talking with people in different professions and life situations, and adventure (chasing tornados – used some math to read some meteorology texts, travel, search and rescue, disaster relief).
My math is long gone – only the multiplication tables, a few automatic memory techniques, and various handy deep concepts remain. My biggest problem in higher math was memory – remembering the gazillion various theorems so I knew when to use them. Concepts were usually easy, but that isn’t enough. In that sense, rote memory of theorems probably would have been a good thing.
PeterUK – we deduced that along time ago but everyone was polite enough to not say so
It might be hard being a 50 year old prodigy but it is even harder being 62 year old prodigal.
It might be hard being a 50 year old prodigy but it is even harder being 62 year old prodigal.
I would think practice would give something of an advantage.
But I do think the paranoia over IE is overblown. I’ve been using it for years and have not gotten any virii nor any adware and have had no stability issues. I must be doing something wrong.
Syl, one question: are you running a PC or a Mac?
Charlie(C)
showed that you could introduce, as an axiom, those infinitely small things
I believe the proper axiomatization, if there is one, is still an open question. Robinson showed how to construct/model systems that satisfied most people’s ideas of how infinitesimals should behave, but there are many such systems. There is no single definition of “infinitesimal” (the axiomatization problem). In contrast, the reals are a complete ordered field, and up to isomorphism there is only one such field.
I believe the proper axiomatization, if there is one, is still an open question. Robinson showed how to construct/model systems that satisfied most people’s ideas of how infinitesimals should behave, but there are many such systems.
You’re better up on it than I am, then. I read Robinson 20 years ago; what most impressed me, though, was the fact that Leibnitz’s approach was one such system, but that it took damn near 300 years to catch up with him.
Leibnitz was one smart dood.
I was really just pointing out, though, that if you look carefully into limits, the whole mechanism of delta-epsilon proofs and so forth then leads you into Dedekind’s cuts, and then you have to wonder why 0.99999… is exactly equal to 1.00000… . People trying to teach Calculus One skim past that whole issue. Rightly so, but an example where “understanding” may be less helpful.
There is no single definition of “infinitesimal” (the axiomatization problem). In contrast, the reals are a complete ordered field, and up to isomorphism there is only one such field.
Now you see the logician in me coming out. My reaction is, “uh, yeah? More than one consistent system? Cool!” To me they’re all just formal languages and models anyway, so why not?
…the idea of the generalized Fourier transform
You mean non-zero multiplicative linear functionals on Hausdorf locally compact Abelian groups? 0_-
You mean non-zero multiplicative linear functionals on Hausdorf locally compact Abelian groups?
Oh yeah! Well, well…
Oh, never mind.
Interesting http://www.stephanietolan.com/gifted_ex-child.htm
on the subject with a bibliography
Goodness. I am not sure how I can fit in such exalted company, though I hasten to add that in my time I was studying Fourier transforms myself, before I abandoned it all for biologyÖÖÖ
No prodigy though. Just hard work.
chuck, Charlie(CO), John Moore,
Regarding the Reals, there’s no doubt that although the “real numbers” represent one unique embodiment of the colossal insight of Newton and Leibnitz, the final word isn’t necessarily in (nor should it be) on what the proper axiomization of that insight should be. Is .999… really the same as 1? Not according to this:
http://www.math.fau.edu/Richman/html/999.htm
I think you’ll all enjoy that one.
thibaud,
I have a son who’s diagnosed with “Autism”. There’s no clear distinction betweeen “Autism” and “Asberger’s”. Having studied the field thoroughly over the last so many years I’ve concluded that a lot of what people consider “nerdy” is really just mild autism. Now that I understand it, I realize that I and many other smart folks I know are mildly autistic.
The problem is that the word “autism” carries such a heavy connotation that parents are reluctant to accept that diagnosis. I think that’s a huge mistake. I’m now convinced that Newton, Einstein, and Bill Gates were/are all mildly autistic, and probably U.S. Grant as well. Autism is not well-defined, but the symptoms can be ameliorated, maybe even eliminated through early treatment. The earlier the better!
My son has been enormously helped by his treatment and had we not begun the treatment he might never have spoken. Now people who meet him for the first time are skeptical that there’s even a problem. I fought the diagnosis for a long time and therefore fought applying the proper behavioral treatment. Thankfully I relented. Email me at yahoo and we’ll continue this discussion.
Syl,
I spent two entire days over the Christmas break removing what turned out to be over 6,000 viruses and malwares and the like on my (other) son’s Windows machine because he’d lent it to a friend who used IE. He had only had it six months. The mean time to infection on a Windows machine using IE has been clocked by a security firm: 10 minutes.
Trust me, there are some very good technical reasons not to use IE on Windows. Essentially IE gives the ability to reach deeply into Windows in a way that no third-party software like Firefox could ever do. Some details can be found here:
http://bookofhook.com/phpBB/viewtopic.php?t=387
snort!
We are seriously geeking out now. Roger, are you still paying attention?
chuck, no fair dragging the big character group duality guns into what was a reasonable discussion of Fourier theory. I’ll have to start talking about zeros of primary summand functions on compact solvmanifolds. And then you will be very sorry.
Enjoy yourselves now, for soon Catherine and I are going to take over this thread and convince you all that math education is in big trouble again.
http://www.mathematicallycorrect.com/
But first a word from WichitaBoy.
I’ll have to start talking about zeros of primary summand functions on compact solvmanifolds. And then you will be very sorry.
Heh. I always thought it was funny that I should have spent so much time learning to say things that hardly anyone could understand. Language development is usually supposed to go the other way.
Enjoy yourselves now, for soon Catherine and I are going to take over this thread and convince you all that math education is in big trouble again.
This could turn into the longest thread in history.
chuck, no fair dragging the big character group duality guns into what was a reasonable discussion of Fourier theory. I’ll have to start talking about zeros of primary summand functions on compact solvmanifolds. And then you will be very sorry.
Nor will Chuck be the only one.
Enjoy yourselves now, for soon Catherine and I are going to take over this thread and convince you all that math education is in big trouble again.
This could turn into the longest thread in history.
Simply enumerating everything wrong with math education could make it the longest thread in history.
All right, now you’ve done it.
Is .999… really the same as 1? Not according to this: (linked….)
First of all, 0.999… is clearly not the same as 1 because 0.999… is a double, while 1 is an int.
More seriously, though, this is where the logician in me comes in. Whether or not 0.999… == 1 depends on what rules we decide to play with when we start the game. Leibnitz and Newton came up with a cool set of rules that let them do cool things — like derive the basic laws of mechanics, and determine the mass of the planets.
Lesser men came along, trying to understand Newton and Leibnitz, and spent 300 years catching up. In the process, we found out there were some weird side-effects: the wheels were starting to come off with Hilbert’s program. The Russell destroyed Frege’s book, and then things got real weird with Cantor, and then Gödel sort of finished off the whole thing.
So what? Calculus still works, and all things considered, Abe Robinson’s nonstandard analysis can be a lot easier to explain. Or you can do Weierstrauss, Dedekind, Cauchy (oh my!), and look at something like the 10×0.999… proof as being an application of those rules. Is it convincing? Who cares? It is convincing that a knight can only attack by going up two and over one? Not very — unless you’re playing chess.
And that, by the way, means that this thread has gone from Firefox, to my childhood, to the Formalist position on the epistemic status of mathematics.
I think we can stake that out as the biggest thread drift in fewest steps in the history of Roger’s blog, and very possibly the Web itself.
0.999….
The question is what the … means. If it is the sequence of numbers .9, .99, .999 , etc. then it could be considered an extended real that differs from 1 by an infinitesimal. Normally though, 0.999… would be considered a notation for the real number, 1 in this case, that is the limit of the sequence.
This *is* getting seriously geeky, and I have the sneaking suspicion that WichitaGirl might end up putting us to shame.
Terrye
Don’t you ever feel intimidated, your plain spoken truths are much appreciated. You wanna talk wasted lives, you talk to me. I’m the one that is intimidated, though I did once score 99% on the Armed Forces Entrance Exam
I’ve been using firefox for 1 hour, like it so far. I run windows and ie on a pc. In the last eleven years I’ve had two virus’s, which were caught before doing damage. Spyware, yes that had been a problem of sorts until I purchased Lavasoft Ad-watch.
Roger has certainly gathered a group here, and no I am certainly not so presumptuous as to include myself. Thanks to all of you for making my life richer.
WichitaGirl:
Boy, there are a lot of links on that page. Can you narrow it down to a couple that give a good idea of what is going on? I saw references to “whole math”, “fuzzy math”, and “new-new math”, which already makes me want to nuke the schools. Gitmo for math educators? I’m having an episode. Who makes this stuff up?
Oh, math education must be in serious trouble if the college freshmen I see are any indication. I always calculate my course grades on percentages. So I tell my classes that if they are ever curious about their specific grade at any point in time they can just calculate the percentages. You wouldn’t believe how many questions I get! I don’t think they really understand except in the vaguest sense that 90% is good and 60% is not so good.
I flunked most math classes after trig, but read at a high level at a very early age. I’ve always been a contrarian! ;> And, like PeterUK, an early prodigal…
I’m guessing that “fuzzy math” here doesn’t have anything to do with Lotfi Zadeh and fuzzy logic or fuzzy systems, right?
mudmarine:
What’s great here is the *experience* of the commenters. I know some math, but lots of the folks here have tons more experience out in the world than I do. Terrye and yourself offer plenty; I think Terrye is among the most interesting commenters here, certainly above my level.
I’m guessing that “fuzzy math” here doesn’t have anything to do with Lotfi Zadeh and fuzzy logic or fuzzy systems, right?
I suspect fuzzy-wuzzy math is a more correct term. But it could be worse.
What happened to Jaime Escalante. Via WG’s page, and telling.
I am currently playing with just installed Firefox on my MacÖ. and unfortunately some sites do not come up as advertised. For example, Netflix comes up as a uniform, nice, grey page with ìWelcome, Katherineî on it and thatís that. Wellsfargo site does not seem to be wholly there, either. Looks like I will be doing the business bit using IE after all (I do have a firewall). On the bright side, I can sign up with the TypeKey using Firefox, something that IE on the Mac would not let me do.
I installed Mozilla on the recommendation of someone here,forgive me for forgetting,but until recently,for technical reasons, had a dial up connection and Mozilla was very jerky.Works very well with broadband,I’m glad Roger posted this.My only niggle is the favourites dropdown remain in random order,a reason I went back to IE.Anyone know how to cure it?
BTW Has anyone noticed on Mozilla Rogers picture seems to have an aura round it? Uncanny.
Number one
While I was not a child prodigy, as an adult I have a fair claim to being the queen of people who commit Type 1 Errors.
Number two
Thibaud: WichitaGirl is right. With Type 2 errors you and your child have had an interesting experience, you’ve learned new things, you’ve met new people, and you’ve laid out some money you could have used for other things.
With Type 1 errors the guilt is permanent, and killing.
Number three
Thibaud again: Don’t be wary!
I mean it!
I’m the least wary person on earth, and I’m here to tell you it works.
(I went to see a therapist a few years back because of a bad situation at one of my children’s schools. When she got to the questions about paranoia I said that not only was I not paranoid, I was not even wary, that I liked to talk to perfect strangers on the street or in subway cars or any place at all. Then I asked whether there was a disorder characterized by not-enough-paranoia and she said, Yes, bipolar. I’m not bipolar as a matter of fact, but I’m within hailing distance.)
Also, years ago I wrote a book on long, happy marriages, and I found that the one constant among 100 different couples was that all of them, every last one, believed that the world held answers for their problems. Some of these couples had faced horrific tragedies, including multiple losses of children, and they still believed the world could help them with their agony.
And the world did.
I think ‘wary’ is probably one of those you-put-it-out-there-you-get-it-back things.
When you send out trust, people help.
None of this is to say that there aren’t some toxic special ed helpers out there; there are.
But usually therapists & special ed teachers are the salt of the earth: these are people who volunteered to work with the smallest and most vulnerable among us. Some of them are miracle workers.
This is not to say that you & your son are in the ‘special ed’ realm. There are a zillion late-talking children who turn out to be completely normal, my brother being one. There are also quite a few seriously abnormal late-talking kids who also, for utterly unknown reasons, simply right themselves. It’s shocking how many of these kids I’ve come across.
Number four
As I mentioned, I was not a child prodigy, but I did teach myself to read at age 4, a fact I try to work into conversation if at all possible.
I have been inordinately proud of this one prodigy-like feat for my entire life, and will still be bringing it up when I’m 90, assuming I still remember that it occurred.
Number five
I spell great, too.
Number six
I found Thomas Sowell’s Late Talking Children so riveting that I stayed up most of one night reading it.
However, I agree with WichitaGirl that he probably didn’t do anyone any favors with the book. Most of the kids–including Sowell’s own son–sounded like high-functioning autistic people to me.
He has another book out I just discovered: The Einstein Syndrome: Bright Children Who Talk Late. I’m ordering it.
Number seven
It’s not rote learning!!!!!!
It’s procedural learning!!!!!
They’re not the same thing!!!!!
Number eight
Every single person on this thread needs to read Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning.) by Liping Ma before you decide once and for all that rote learning preceeds conceptual understanding.
OTOH, I met a woman from China today, whose husband has a Ph.D. in mathematics and who was also, like Liping Ma, sent to the country for re-education during the Cultural Revolution, and she said that as a child she did huge quantities of procedural (procedural!) learning and practicing before gaining conceptual understanding.
Number nine
She also said I am the first parent ever to mention any concern whatsoever about her child’s math curriculum.
Number ten
Now that we’ve done our Scots-Irish poll, we obviously need to do our autism poll, and our what-age-am-I? poll.
I’ve been intrigued for a long time by what seem to be the high numbers of families-with-autism in the comments section. I don’t know what to make of it. (I don’t think it’s a coincidence that we have a high number of families-with-math here as well.)
Apparently the rogerlsimon commenters are an alliance of:
Jewish neo-cons
Scots-Irish belligerents
Individuals with Autism and/or Asperger’s syndrome, AND Beleaguered Parents of Individuals with Autism and/or Asperger’s syndrome
Number eleven
I would have put PeterUK’s age at 40.
Number twelve
Are child prodigies autistic?????
Wow.
Somehow, it had never occurred to me to make this connection.
I see Einstein as having been autistic (there’s some interesting unpublished research on this); I see child math whizzes as autistic.
And yet I hadn’t generalized out to most or all prodigies.
Number thirteen
With all the autism going on around my house, I keep hoping for a prodigy to turn up, and frankly I feel I am owed. Why does Thomas Sowell get to have all the fun?
But no such luck.
Still, I have high hopes for Andrew, the younger of my two autistic sons. He’s 10, he’s never spoken a word in his life, and . . . now he’s talking.
He started talking a month ago. His first words have been things like ‘gewww’ for ‘juice.’ Nothing about deciduous trees as of yet.
The teachers can’t believe it; nobody can. This is not supposed to happen; the window is supposed to have closed.
Well, I don’t believe in windows; I never have. My proudest moment as a mother was the day I told the new Director of BOCES TEACCH, who had just told me he ‘had to be realistic,’ that “In our household we do not believe in realism.”
That turned his water off.
Terrye
OK, that is brilliant.
It’s time for someone to work out a Theory of the Droll.
Good place to start for fuzzy math
The proper term for fuzzy math is constructivist math.
I like E. D. Hirsch’s address to the California State Board of Ed as a place to start.
http://www.coreknowledge.org/CKproto2/about/articles/CAStBrd.htm
NYC Hold is also a terrific site:
http://www.nychold.com/
Just found another article by Hirsch on constructivist education that looks good:
http://www.taemag.com/issues/articleid.16209/article_detail.asp
chuck
“What’s great here is the *experience* of the commenters.”
Well, you’re right. It would be enlightening to know the bio’s and present circumstances of all the player’s here. I suspect I would be amazed. I do find it interesting that when ‘smart’s’ come up, math is the matrix, at which I am terrible by the way. And, no, thanks for the comment, but I do not comment very much, I am intimidated, truly and reasonably. I certainly agree with your take on Terrye.
PeterUK
Highlight a bookmark/favorite – right click – sort by name.
ìApparently the rogerlsimon commenters are an alliance of:
Jewish neo-cons
Scots-Irish belligerents
Individuals with Autism and/or Asperger’s syndrome, AND Beleaguered Parents of Individuals with Autism and/or Asperger’s syndromeî
Not being any of the above I think it is time for me to take my toys and go away.
Catherine,
Number eleven
I would have put PeterUK’s age at 40.
I will take that as a complement,I’ve always bee a slow developer.
Don’t know whether it was prodigal or prodigy,I might have misheard my parents,it’s certainly been hard work living up to it.
Terrye,
I don’t think anyone here would swap you for Barbara Boxer or Clare Short,or both.
Mudmarine,
Thanks.
Oh Catherine, thank God you’re here.
I had reviewed the links on the Mathematically Correct page, and decided that what probably best reveals what’s wrong with math ed in the states is the book by Liping Ma. More on that in a bit.
But, okay, the Math Wars (of which Catherine and I, with elementary/middle school kids, are on the front line) center around some newish math curricula that have been developed for elementary and middle school aged students. The idea is that traditional math curricula only involves rote learning, too much doing and not enough understanding, and that what kids need is more conceptual learning and repeated exposure to those concepts (called ‘spiraling’ by those In The Know, this actually means that the kids briefly visit a topic like, oh, long division, and leave it again without having mastered it, the idea being that they’ll ‘have it again next year, so it’s okay’).
My first exposure to this was when my son (the same high functioning autistic one) was in 4th grade. He had had 3 years of a truly excellent math program, Saxon math, about which we used to and still do rave, but in 4th grade they switched to a curriculum developed at the University of Chicago school of education (brrr) called “Everyday Math”. My son’s performance, which had been absolutely on the straight and narrow, became quite erratic quite quickly; I was told it was because verbal kids do best in Everyday math, and that my son is not a strong reader. It is true that verbal kids tend to do better in e math, and this by itself is not a bad thing; but kids doing everyday math are actually being harmed. They are not ever mastering things like long division, because the day when they will master it is being pushed infinitely far into the future; and they are not working on certain topics that they ought to be working hard on at this age, like manipulation (addition and subtraction of fractions).
Instead, right now my son is doing a section on statistics AGAIN. This is spiral number 4. Pie charts, stem and leaf plots, medians and modes (they don’t do means in everyday math; can you guess why?).
There is another similar offender in the middle schools now; Connected Math. A kid who goes through both these curricula will be, god knows, very conceptual, but he won’t actually be able to *do* math to save his life, and he will have real problems in high school, where one does algebra and not Connected Algebra or Everyday Algebra.
Liping Ma’s book was actually her thesis in, I think, math ed. She did a study in which she worked with a set of representative primary teachers from China and the US, and gives them each 4 problems. The ones I remember:
– create a word problem that illustrates the use of the computation: 1-3/4 divided by 1/2. (The Chinese teachers could do it; the US teachers not only couldn’t create a word problem, a few of them couldn’t do the computation!).
– a student comes to you and says he has discovered a mathematical rule: if you increase the perimeter of any polygon, then its area also increases. How do you respond? (The Chinese teachers worked some examples and discussed the principles, both, generally acquitting themselves well. The US teachers… I’m sorry to say that many of them assumed the principle was right, and tried earnestly to find a proof of the result in a reference).
Liping Ma’s conclusion: the culture of primary math teaching is different in China. Chinese primary math teachers are specialists, who’ve done specialist training, are respected for their work and given time each week to develop themselves and mentor each other.
In the US, primary school teachers don’t specialize. Secondary math teachers take courses in college math; this isn’t the same skills one needs to teach math. One (dare I say it!) needs pedagogy and a thorough grounding in the teaching of math; what Ma calls PUFM (Profound Understanding of Fundamental Math).
The role of self-respect, and the lack of respect given primary teachers in the US, is not mentioned by LipingMa, but I think it’s a silent problem in elementary ed in general, and in math ed in particular.
Terrye — Don’t be intimidated. Someone here has to have some goddamn sense.
Coming back at this:
Charlie – good point. I faced the high expectations too, and didn’t reach them.
John, one of the things about the “prodigy” or “gifted child” thing is that nobody reaches them, or at least feels like they reach them. A big effect of being known to be “gifted” is that anything you do that’s brilliant is mere what’s expected, but any slip from utter brilliance and adult behavior is — often — an immense disappointment.
Catherine,
As long as we’re doing polls, how about a poll of bipolars (oft tied to intelligence).
Charlie,
I suppose that’s true. I have also noticecd a lot of gifted folks (including my daughter) often pop out of 12th grad with the feeling that they can do anything because they are smart. Then they run into stuff where smart + memorization + practice are needed much more than before.
Re: the Chinese teachers.
I think we probably have some of the worst teachers in the world, for the subjects they teachh. I’d rather have a math grad student who isn’t completely devoid of sccial abilities teaching my kid than anyone with a degree in education. My mother, the mathematician, decided to become a school teacher after having been a housewife (and killer bridge player) for quite a while. She then did 20 years as an algebra teacher in the public schools. Her comments about teachers and administration (and she was a kind person) were something else. A friend who warked hard with the school boards and teachers of his kids’ schools had the same impression.
My best teacher ever was a biology teacher. He was a real biologist, and did field biology and wrote papers. But my high school’s advanced math program had us take the University’s standard calculus course, taught by a high school teacher but tested at the U, instead of the obviously better approach of having us take the *honors* college calculus. That did a lot of damage. It indicated some pretty screwed up thinking by whoever set it up.
America needs teachers who are subject matter trained a lot more than teachers who are pumped full of the latest nonsensical educational fad theories and techniques. The US is close to the bottom in K-12 education, and then at the top above that. Hmmm.
America needs teachers who are subject matter trained a lot more than teachers who are pumped full of the latest nonsensical educational fad theories and techniques.
In the upper grades, I think this is true. I suspect that dealing with young children requires different skills with less need for professional knowledge. My own most memorable teacher, like Charlie’s, was an older lady who taught 3′rd grade. Among other things, she made me learn those pesky multiplication tables, although for years I was under the mistaken impression that 7*8 = 57
Don’t know how that slipped through. Anyway, I don’t think she was educated at the higher levels, although she might have been, I just think she had high standards, knew what needed to be taught, and made sure that all the students came up to snuff.
I wanted to come back to this too; I think it applies to the math education thread, but I’m not clear how to say it. So I’ll make something up.
(1) our schools operate on a “factory” or “industrial engineering” model, in which students are the workpieces and “success” is modeled not by how effective the individual outcomes are, but rather by how effectively the factory produces a steady flow of barely acceptable work products. This means, with self-adapting “workpieces”, that some of us manage, in some of the areas of our interests, to “pull ourselves ahead” in the production line.
I think this model is massively flawed, and — now that I put it that way — I think needs massive revision.
Okay, so here’s what I was trying to say.
First, we don’t define “success” in the schools in a useful way. We start kids nominally on 1 September, in kindergarten; we expect they will go from workstation to workstation — that is, from grade to grade — on 1 September every year until they’ve been through 13 workstations and graduate. We haver an idea of what they ought to be getting from each year, but the definition of “success” isn’t so much that the kid is actually succeeding at that topic, as it is that the kid hasn’t “failed”.
In industry, in a “six sigma” or TQM environment, what that means is that we’re setting the tolerance to be very wide; we care more about how long the process takes than what the quality of the output is. In fact, we have a grading scheme that makes A,B,C,D all “passing” grades; we don’t ask for mastery of the topic before we kick kids on down the production line.
Second, the reward structure is all screwed up. (Yes, that’s the technical term.) The rewards to the workers (teachers) aren’t coupled to the quality of the outputs. In fact, if you want to get reqarded in the school systems, you need an M Ed or an MSW, and to become a councellor or a principal, or otherwise move up in administration.
Again, we’d expect from this to see all the best people moving away from teaching, and towards administration. The ones who remain as long-time teachers are either incompetent, or saints.
It’s frankly amazing how many saints there are, but there are a lot of the other ones, too.
Third — this is a bugaboo for WB, and rightly so — the people doing “education” as an academic topic get rewarded not for improvements in their outcomes, but for all the new papers and innovative ideas they can publish. To publish an “innovative” idea it has to be different, and you have to get it adopted.
Notice anything missing there?
Now, if I were the usual politician, I’d stop there. It’s a problem, something Ought To be done, that’s enough.
But I’m not, thank god.
Okay, what’s wrong and what can be done.
(1) For crying out loud, decide that the basics — like “write a literate sentence in which subject and verb agree in number”, “read a liteate blog”, long division — are, and start to measure the ability of people in school against those basics.
Don’t tell me you’re having a certain number of failures. That’s not acceptable. If it takes three years to teach first grade, it does.
(2) Teachers that do those things really well get paid more. Really *really* well, paid LOTS more.
I know I’ve ranted about the one-room school before, but it’s worth summarizing — I looked at the one-room school at the Adams County Museum. 25 seats, about 400 square feet, desks one each, a blackboard.
You ever look at what they were expected to learn? Not just reading, writing, and ciphering — no, they had to learn history, they memorized poetry, they could do things like work out an amortization schedule for a mortgage by hand.
(Why did it work better? My guess is that since the older kids were supposed to help the younger kids, and/or work on their own while the teacher worked with others, there was a lot of tutoring going on. Tutoring is a helluva way to learn a topic.)
If you paid $9000 per student per year — along the lines of what NYC pays — that’s $225K.
Pay $50 a square foot for the classroom rent per year (util. incl.), that’s $20K. Spend another $1K per student per year for materials, and that leaves $180K per year for the teacher.
Want to be that if we said “you can make $180K but your kids have to all graduate able to read, write, and do arithmetic” we might get a better quality of teacher?
I like Charlie’s analogy. I think one thing needs to be modified – the workpiece is (loosely) the mind of the student – not the student him or herself. I think the modification is necessary because there is a pernicious “teach me” (versus an “I will learn”) attitude among too many students. You can lead a horse to water, but you can’t make him drink – he drinks because the reward structure is set up to encourage it. I recommend reintroducing competition into the classroom – interschool competition might help. Ending social promotion should help here as well, as will attracting and retaining excellent teachers who know how to manage the student’s reward structure day-to-day.
The other thing I think we need to add is that there is a need for discipline in the classroom – not having a way to remove disruptive students degrades the process – in some cases catastrophically.
Finally, to extend the analogy, we ought to be looking for improved production techniques – ones that work faster, at a lower cost, and produce high-quality output with more consistency.
I agree wholeheartedly with developing definite standards for mastery of material and testing to ensure mastery. I also agree with the idea that social promotion should end. I don’t give a damn that some studies have shown that held-back students “have poorer outcomes” than socially promoted ones – you don’t fix the problem by ignoring it. Find a way to rework the materials – waste is (societally) expensive.
we have a grading scheme that makes A,B,C,D all “passing” grades;
Yeah, and what are these grades? When I went back to school for my PhD I was part of the team teaching introductory Calculus. Came the big final, and we all sat around the table grading papers, a perfect score was 200. How did we pick the cutoff for a D? Well, we couldn’t fail everyone, so the passing score was taken to be 40, just so enough students would pass. It had nothing to do with whether of not it was an adequate score (it wasn’t). I was shocked but everyone else was quite cynical about it.
It’s not that the students don’t appreciate being pushed. That summer I taught complex variables and was a complete nit-picking bastard grading the homework. In the second to last week of class one of the students finally got a perfect score. He sat there looking stunned while all of his friends leaned over to look at his paper and oohed and aahed. That got to have felt good.
Viz, one room school houses, my third grade year was spent in a two room school house; the town had run out of school rooms in the middle of the baby boom, so two classes were moved to a building out in the fields on the edge of town. There was a sense of intimacy and scholastic isolation that is hard to get in larger schools.
I agree wholeheartedly with developing definite standards for mastery of material and testing to ensure mastery. I also agree with the idea that social promotion should end. I don’t give a damn that some studies have shown that held-back students “have poorer outcomes” than socially promoted ones – you don’t fix the problem by ignoring it. Find a way to rework the materials – waste is (societally) expensive.
I’m not just saying that “social promotion: should end — I’ve got problems with the whole concept that you measure “progress” by moving from grade to grade on a fixed schedule.
Charlie:
I think I’m clearer on your point now – the current system is set up to be ideal if all students move at the same pace (across students) within each domain, and the learning rate is known for each domain. That’s the only way it could be “ideal” to promote everyone to the next level in every subject after the same amount of time. Mastery is not closely tied to grade, but curriculum is.
Is that closer to what you are thinking, or am I still missing something?
Morgan, that’s it exactly. Extending the industrial engineering metaphor — if not flat out stretching it — the underlying assumption is that children are identical, and that it takes me as long to learn piano as my sister.
Having set the world record for number of years in the John Brimhall First Year Piano book, I know that’s not true.
It’s interesting to think what the consequences of this would be, though. One that occurs to me right off is that you’d expect that to increase the variance: at the end of several years, the “fast” students would be much farther ahead of the “slow” students than they started out.
Does anyone know anything about the history of “classes” and “grades” like we do them here?
Hmmm. I appear to have promised to stretch the industrial metaphor and then never seem to have gotten there.
Clearly I need more coffee.
Charlie,
I think that the industrial metaphor also applies to the production of teachers. They are workers on the assembly line, and as such they don’t need training in the subject but only in screwing in some widget.
…screwing in some widget.
Please God, give me the strength not to use this as a straight line.
Please God, give me the strength not to use this as a straight line.
We’re waiting. The spirit is willing, but the flesh is weak.
… big damn widget.
This has taken a turn to which I’ll contribute something else from Peter F. Drucker’s Adventures of a Bystander.
[Monroe] Deutsch was then quite willing to talk about the university and particularly about the University of California, which was his whole life. But he never talked about himself. He had been born into one of San Francisco’s wealthy families and decided early to spend his life in public service. But being pathologically shy, he was unable to endure public exposure, let alone run for office. And so he had invented for himself the role of eminence grise of the California university system, or rather he decided it would become a “system.” He made himself “Provost” and went underground., literally as well as figuratively. Then he designed the multi-campus university which California became after World War I when UCLA was first started; and he designed California’s multi-tier system of university, state colleges, and junior colleges, which would maintain the scholastic excellence and exclusivity of the university and yet enable every high-school graduate in California to attend a tuition-free state institution of higher learning. Deutsch also largely designed the ingenious system that gave the university fiscal autonomy by guaranteeing it a fixed sum from the state for each student admitted. He pushed for state commitment to higher education as California’s first political priority, and commitment to excellence as the university’s first educational priority. “I didn’t know that there would be a Hitler and that we would suddenly be able to hire fifty or sixty first-rate scholars and teachers,” he said; “but I started fifteen years ago to make the state and the university ready for manna from heaven, should it even rain down.”
I find it ironic that when I voted in the affirmative on Proposition 13 I was living in Deutsch Hall.
I find it ironic that when I voted in the affirmative on Proposition 13 I was living in Deutsch Hall.
Uh, why? I don’t recall that Prop 13 said anything about education funding or the university system, just that revenue couldn’t be increased faster than valuations without a vote. I do remember the anti-13 politicians saying they would “have to” reduce education, police, fire, close the parks, and sell the redwood forests for lumber, but in fact total revenue grew faster than inflation, on average. (I’m sure that broke down during Jimmy Carter’s “Administration”, but on average.)
The real issue is that other things than the university system got higher priorities on that revenue.
That should be should ever rain down.
Sigh. …should it ever rain down.
Knowing that (x+1)(x-1) = x^2-1 is essentially a matter of following an algorithm, like addition or long division; algebra as a math topic is just the observation that these algorithms have analogies with other parameters.
My tenent’s daughter was an average student, but she bogged down completely when algebra was introduced. I simply told her “arithmetic is algebra, only done with numbers” and it fell into place for her and she got her first A in it.