Tag Archives: education

An approach to teaching statistics to 8th graders

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There are two main obstacles students have to overcome to learn statistics: one mathematical one philosophical. The math is somewhat difficult, and will be new to a high schooler. What’s more, philosophically, it is rarely obvious what it means to discover a true pattern, or underlying cause. Nor is it obvious how to separate the general pattern from the random accident, the pattern from the variation. This philosophical confusion (cause and effect, essence and accident) is exists in the back of even in the greatest minds. Accepting and dealing with it is at the heart of the best research: seeing what is and is not captured in the formulas of the day. But it is a lot to ask of the young (or the old) who are trying to understand the statistical technique while at the same time trying to understand the subject of the statistical analysis, For young students, especially the good ones, the issue of general and specific will compound the difficulty of the experiment and of the math. Thus, I’ll try to teach statistics with a problem or two where the distinction between essential cause and random variation is uncommonly clear.

A good case to get around the philosophical issue is gambling with crooked dice. I show the class a pair of normal-looking dice and a caliper and demonstrate that the dice are not square; virtually every store-bought die is not square, so finding an uneven pair is easy. After checking my caliper, students will readily accept that these dice are crooked, and so someone who knows how it is crooked will have an unfair advantage. After enough throws, someone who knows the degree of crookedness will win more often than those who do not. Students will also accept that there is a degree of randomness in the throw, so that any pair of dice will look pretty fair if you don’t gable with them too long. I can then use statistics to see which faces show up most, and justify the whole study of statistics to deal with a world where the dice are loaded by God, and you don’t have a caliper, or any more-direct way of checking them. The underlying uneven-ness of the dice is the underlying pattern, the random part in this case is in the throw, and you want to use statistics to grasp them both.

Two important numbers to understand when trying to use statistics are the average and the standard deviation. For an honest pair of dice, you’d expect an average of 1/6 = 0.1667 for every number on the face. But throw a die a thousand times and you’ll find that hardly any of the faces show up at the average rate of 1/6. The average of all the averages will still be 1/6. We will call that grand average, 1/6 = x°-bar, and we will call the specific face average of the face Xi-bar. where i is one, two three, four, five, or six.

There is also a standard deviation — SD. This relates to how often do you expect one fact to turn up more than the next. SD = √SD2, and SD2 is defined by the following formula

SD2 = 1/n ∑(xi – x°-bar)2

Let’s pick some face of the dice, 3 say. I’ll give a value of 1 if we throw that number and 0 if we do not. For an honest pair of dice, x°-bar = 1/6, that is to say, 1 out of 6 throws will be land on the number 3, going us a value of 1, and the others won’t. In this situation, SD2 = 1/n ∑(xi – x°-bar)2 will equal 1/6 ( (1/6)2 + 5 (5/6)2 )= 1/6 (126/36) = 3.5/6 = .58333. Taking the square root, SD = 0.734. We now calculate the standard error. For honest dice, you expect that for every face, on average

SE = Xi-bar minus x°-bar = ± SD √(1/n).

By the time you’ve thrown 10,000 throws, √(1/n) = 1/100 and you expect an error on the order of 0.0073. This is to say that you expect to see each face show up between about 0.1740 and 0.1594. In point of fact, you will likely find that at least one face of your dice shows up a lot more often than this, or a lot less often. To the extent you see that, this is the extent that your dice is crooked. If you throw someone’s dice enough, you can find out how crooked they are, and you can then use this information to beat the house. That, more or less is the purpose of science, by the way: you want to beat the house — you want to live a life where you do better than you would by random chance.

As a less-mathematical way to look at the same thing — understanding statistics — I suggest we consider a crooked coin throw with only two outcomes, heads and tails. Not that I have a crooked coin, but your job as before is to figure out if the coin is crooked, and if so how crooked. This problem also appears in political polling before a major election: how do you figure out who will win between Mr Head and Ms Tail from a sampling of only a few voters. For an honest coin or an even election, on each throw, there is a 50-50 chance of head, or of Mr Head. If you do it twice, there is a 25% chance of two heads, a 25% chance of throwing two tails and a 50% chance of one of each. That’s because there are four possibilities and two ways of getting a Head and a Tail.

pascal's triangle

Pascal’s triangle

You can systematize this with a Pascal’s triangle, shown at left. Pascal’s triangle shows the various outcomes for a coin toss, and shows the ways they can be arrived at. Thus, for example, we see that, by the time you’ve thrown the coin 6 times, or polled 6 people, you’ve introduced 26 = 64 distinct outcomes, of which 20 (about 1/3) are the expected, even result: 3 heads and 3 tails. There is only 1 way to get all heads and one way to get all tails. While an honest coin is unlikely to come up all heads or tails after six throws, more often than not an honest coin will not come up with half heads. In the case above, 44 out of 64 possible outcomes describe situations with more heads than tales, or more tales than heads — with an honest coin.

Similarly, in a poll of an even election, the result will not likely come up even. This is something that confuses many political savants. The lack of an even result after relatively few throws (or phone calls) should not be used to convince us that the die is crooked, or the election has a clear winner. On the other hand there is only a 1/32 chance of getting all heads or all tails (2/64). If you call 6 people, and all claim to be for Mr Head, it is likely that Mr Head is the true favorite to a confidence of 3% = 1/32. In sports, it’s not uncommon for one side to win 6 out of 6 times. If that happens, it is a good possibility that there is a real underlying cause, e.g. that one team is really better than the other.

And now we get to how significant is significant. If you threw 4 heads and 2 tails out of 6 throws we can accept that this is not significant because there are 15 ways to get this outcome (or 30 if you also include 2 heads and 4 tail) and only 20 to get the even outcome of 3-3. But what about if you threw 5 heads and one tail? In that case the ratio is 6/20 and the odds of this being significant is better, similarly, if you called potential voters and found 5 Head supporters and 1 for Tail. What do you do? I would like to suggest you take the ratio as 12/20 — the ratio of both ways to get to this outcome to that of the greatest probability. Since 12/20 = 60%, you could say there is a 60% chance that this result is random, and a 40% chance of significance. What statisticians call this is “suggestive” at slightly over 1 standard deviation. A standard deviation, also known as σ (sigma) is a minimal standard of significance, it’s if the one tailed value is 1/2 of the most likely value. In this case, where 6 tosses come in as 5 and 1, we find the ratio to be 6/20. Since 6/20 is less than 1/2, we meet this, very minimal standard for “suggestive.” A more normative standard is when the value is 5%. Clearly 6/20 does not meet that standard, but 1/20 does; for you to conclude that the dice is likely fixed after only 6 throws, all 6 have to come up heads or tails.

From skdz. It's typical in science to say that <5% chances, p <.050 are significant. If things don't quite come out that way, you redo.

From xkcd. It’s typical in science to say that <5% chances, p< .05. If things don’t quite come out that way, you redo.

If you graph the possibilities from a large Poisson Triangle they will resemble a bell curve; in many real cases (not all) your experiential data variation will also resemble this bell curve. From a larger Poisson’s triange, or a large bell curve, you  will find that the 5% value occurs at about σ =2, that is at about twice the distance from the average as to where σ  = 1. Generally speaking, the number of observations you need is proportional to the square of the difference you are looking for. Thus, if you think there is a one-headed coin in use, it will only take 6 or seven observations; if you think the die is loaded by 10% it will take some 600 throws of that side to show it.

In many (most) experiments, you can not easily use the poisson triangle to get sigma, σ. Thus, for example, if you want to see if 8th graders are taller than 7th graders, you might measure the height of people in both classes and take an average of all the heights  but you might wonder what sigma is so you can tell if the difference is significant, or just random variation. The classic mathematical approach is to calculate sigma as the square root of the average of the square of the difference of the data from the average. Thus if the average is <h> = ∑h/N where h is the height of a student and N is the number of students, we can say that σ = √ (∑ (<h> – h)2/N). This formula is found in most books. Significance is either specified as 2 sigma, or some close variation. As convenient as this is, my preference is for this graphical version. It also show if the data is normal — an important consideration.

If you find the data is not normal, you may decide to break the data into sub-groups. E.g. if you look at heights of 7th and 8th graders and you find a lack of normal distribution, you may find you’re better off looking at the heights of the girls and boys separately. You can then compare those two subgroups to see if, perhaps, only the boys are still growing, or only the girls. One should not pick a hypothesis and then test it but collect the data first and let the data determine the analysis. This was the method of Sherlock Homes — a very worthwhile read.

Another good trick for statistics is to use a linear regression, If you are trying to show that music helps to improve concentration, try to see if more music improves it more, You want to find a linear relationship, or at lest a plausible curve relationship. Generally there is a relationship if (y – <y>)/(x-<x>) is 0.9 or so. A discredited study where the author did not use regressions, but should have, and did not report sub-groups, but should have, involved cancer and genetically modified foods. The author found cancer increased with one sub-group, and publicized that finding, but didn’t mention that cancer didn’t increase in nearby sub-groups of different doses, and decreased in a nearby sub-group. By not including the subgroups, and not doing a regression, the author mislead people for 2 years– perhaps out of a misguided attempt to help. Don’t do that.

Dr. Robert E. Buxbaum, June 5-7, 2015. Lack of trust in statistics, or of understanding of statistical formulas should not be taken as a sign of stupidity, or a symptom of ADHD. A fine book on the misuse of statistics and its pitfalls is called “How to Lie with Statistics.” Most of the examples come from advertising.

From Princeton: dare to be dumb.

Let’s say you have a good education and a good idea you want to present to equally educated colleagues. You might think to use your finest language skills: your big words, your long sentences, and your dialectically organized, long paragraphs. A recent, Princeton University study suggests this is a route to disaster with the educated, and even more so with the un-educated. In both groups, big words don’t convince, and don’t even impress, like small words do.

Most people won't care what you know unless they know that you care.

Like this fellow, most folks aren’t impressed by fancy speeches. (cartoon by Gahan Wilson)

http://web.princeton.edu/…/Opp%20Consequences%20of%20Erudit…

People, even educated ones, want ideas presented in simple words and simple sentences. They trust such statements, and respect those who speak this way more than those who shoot high, and sometimes over their heads. Even educated people find long words and sentences confusing, and off-putting. To them, as to the less-educated, it sounds like you’re using your fancy english as a cover for lies and ignorance, while trying to claim superiority. Who knew that George W. was so smart (Al Gore?). Here’s George W. at the SMU graduation yesterday (May 18). He does well, I’d say, with mostly one-syllable words.

This is the sort of advertising that people notice -- and trust.

Lower yourself to be one of the crowd, but don’t go so far that you’re the butt of jokes.

Reading this study, I’ve come to ask why fancy language skills is so important for getting into  college, and why it adds points when writing a college paper. Asked another way, why are professors pleased by something that’s off-putting to everyone else. One thought: this is a club initiation — a jargon to show you belong to the club, or want to. Alternately, perhaps professors have gotten so used to this that it’s become their natural language. Whatever the reason, when outside of university, keep it simple (and) stupid.

Some specifics: at job interviews, claim you want to work at their company doing a job in your field. Only when dealing with professors can you claim your goal is capitalizing on your intellectual synergies, and phrase that means the same thing. Don’t say, you’ll do anything, and remember it’s OK to ask for training; poor education doesn’t hold-back American productivity.

Dr. Robert E. Buxbaum, May 19, 2015. Here are some further thoughts on education, and some pictures of my dorm and the grad college at Princeton back in the day.

Is college worth no cost?

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While a college degree gives most graduates a salary benefit over high school graduates, a study by the Bureau of Labor statistics indicates that the benefits disappear if you graduate in the bottom 25% of your class. Worse yet, if you don’t graduate at all you can end up losing salary money, especially if you go into low-paying fields like child development or physical sciences.

Salary benefits of a college degree are largely absent if you graduate in the bottom 25% of your class.

The average college graduate earns significantly more than a high school grad, but not if you attend a pricy school, or graduate in the bottom 1/4 of your class, or have the wrong major.

Most people realize there is a great earnings difference depending on your field of study with graduates in engineering and medicine doing fairly well financially and even top graduates in child development or athletic sciences barely able to justify the college and opportunity costs (worse if they go to an expensive college), but what isn’t always realized is that not all those who enter these fields graduate. For them there is a steep loss when the four (or more) years of lost income are considered.

risk premium in wages

If you don’t graduate or get only an AA or 2 year degree the increase in wages is minimal, and you lose time working and whatever your costs of education. The loss is particularly high if you study social science fields at an expensive college, and don’t graduate, or if you graduate in the bottom of your class.

A report from the New York Federal Reserve finds that the highest pay major is petroleum engineering, mid-career salary $176,300/yr, and the bottom is child development, mid-career salary $36,400/yr (click to check on your major). I’m not sure most students or advisors are aware of the steep salary difference, or that college can have a salary down-side if one picks the wrong major, or does not complete the degree. In terms of earnings, you might be better off avoiding even a free college degree in these areas unless you’re fairly sure you’ll complete the degree, or you really want to work in these fields.

Top earning majors Fed Reserve and Majors that pay you back.

Top earning majors: Majors that pay.

Of course college can provide more than money: knowledge, for instance, and learning: the ability to reason better. But these benefits are likely lost if you don’t work at it, or don’t go in a field you love. They can also come to those who study hard in self-taught reading. In either case, it is the work habits that will make you grow as a person, and leave you more employable. Tough colleges add a lot by exposure to new people and new ways of thinking about great books, and by forced experience in writing essays — but these benefits too are work-dependent and college dependent. If you work hard understanding a great book it will show. If you didn’t work at it, or only exposed yourself to easier fare, that too will show.

As students don’t like criticism, and as good criticism is hard to give — and harder to give well, many less-demanding colleges ,give little or no critical feedback, especially for disadvantaged students. This disadvantages them even more as criticism is an important part of learning. If all you get is a positive experience, a nice campus, and a dramatic graduation, this is not learning. Nor is it necessarily worth 4-5 years of your life.

As a comic take on the high time-cost of a liberal arts education, “Father” Guido Sarduchi, of Saturday Night LIve, describes his “5 minute college experience.” To a surprising extent, it provides everything you’ll remember of 4 year college experience in 5 minutes, including math, history, political science, and language (Spanish).For those who are not sure they will complete a liberal arts education, Father Sarduchi’s 5 minutes may be a better investment than a free 4 years in community college.

Robert. E. Buxbaum. January 21-22, 2015. My sense is that the better part of education is what you get when you don’t get what you want.

What is learning?

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It is common to spend the most of one’s youth in school — presumably learning something. The financial cost for primary education is a few hundred thousand dollars, borne by the state, plus 13 years or so of the student’s life. College learning costs another $50,000 to $200,000, borne by the student, plus another 4-6 years of life. The indication that you’ve learned something appears, in many majors by the ability to get a job that pays more than the school financial cost. But there is also a sense that you’ve learned something, and this is perhaps the only reward for students of film, religion, or archeology. My question is based mostly on this part: what is this learning. Is it the same as knowledge, a set of facts, or satisfaction — perhaps you could be as satisfied by ignorance or drugs. How do you evaluate the spiritual payback from 4-6 years of college? I don’t have all the answers, but ask to exercise my ignorance.

It would seem to me that an important standard of learning is that it should develop the mind and not corrupt it. But how do you recognize the difference? it seems to me one should leave with a set of mental skills should be new to you, recognizable to a normal outsider, and somewhat useful, as in the poem “Botany” even if you don’t use it. I’m not sure if the skills have to be true, by the way, or how useful they have to be. Perhaps developing a new confusion is better than having false notions — knowing that you doubt something.

sometimes education is the removal of false notions.

Sometimes learning can be the development of doubt.

If you’ve been educated in music, it seems to me you should be able to make sounds that appear pleasant to a normal listener; if you’ve been educated in mechanics, you should be able to make machines that work, and if you’ve been educated to think… perhaps then you should be able to walk into a discussion about something you once thought was true, and show that it is really false to an extent that others would accept it (and act upon it?). That is, my suspicion is that learning should involve an identifiable change –not only internal satisfaction, and I also suspect learning the new must involve unlearning the old.

Liberal Education may not be useful, or elevating

Education that isn’t useful isn’t particularly elevating

And that leads us to facts and methods: knowledge. Facts are good, they are the fuel and  substance of learning. Without facts there is nothing for the learning to attach to. But facts are often wrong — the ignorance of others, and even when right, they can be  deceptive. If you’ve learned the moon is made of rock, or out of green cheese, it’s pretty much the same unless there is a reason to think the fact you’ve learned is true, and unless you’ve a good understanding of what the fact ‘means.’ I can imagine a rock that is organic (a gall stone) and less solid than some (old) green cheese. The word rock or cheese must mean something to you to be a fact. Similarly in all subjects; if you learn that Shakespeare is a better writer than Poe, you should have a reason to believe it, and a clear understanding of the word ‘better’ in this context.

Turning to the knowledge of methods. It seems to me that learning a new method of thought, action, or argument is a necessary component of learning– one might even call it virtue, but this too seems to have limitations if it is not directed to use. A person is half-educated if he leaves school knowing how to do geometric proofs, but never doing any, or knowing how to run a great business, but never running one. A science graduate should at least be able to use the techniques learned to demonstrate that the world is made of atoms, and that the sun does not circle the earth and perhaps more. An argument can be made for traditional education areas of logic, rhetoric, mathematics, and dialectic. But these seem useless unless they are applied to a worthy end. One should do more with the new methods than to win drawing-room arguments.

There should be some satisfaction to accomplishments, but I'm not sure how it's learned.

Learning should provide satisfaction –in particular religious learning — but it’s nicer if it goes with doing good for someone (not only the poor) and the ability to earn an honest income. 

There should be a moral component of learning too, but here I feel less certain in describing it, or describing how it should be taught. Theodore Roosevelt said that “An uneducated man can steal from a rail car, “but an educated one can steal the whole railroad.” but perhaps stealing the railroad isn’t such a bad thing if it’s done legally. And as I don’t quite know when the honest stock deal is moral, I’m even more in the dark as to how to teach one to recognize the moral from the immoral in these situations. Two thoughts here: a student deserves some satisfaction from his or her learning and (from Aristotle’s Nicomachean Ethics) to be moral, the student has to earn an honest income. One who can not earn a living is bound to steal from someone sooner or later.

A final sign of learning, and perhaps it’s crown, is creativity, the ability to come to new understandings and develop new things. To do this productively requires some knowledge of the past plus an indescribable view of the future. A spark? A divine madness? Schools do not seem to be able to teach that, but it can help or hinder by either encouraging it, or beating it down. If you did not possess this ability when you entered school, you are unlikely to leave with it, even if you just did drugs, but school can teach one to direct the spark productively.

I’ve noticed that our high schools focus little on the above areas, perhaps because they are hard to test. Rather classes aim to the exams, and the exams test (as best I can tell), memorization, aptitude, and exposure. A surprisingly large fraction of our students leave diagnosed as ADHD. Still, strangely, our graduates do better than the Europeans.

Dr. Robert E. Buxbaum, December 29, 2014 (I taught in college). Here’s some advice I wrote for my 16 year old daughter in high school.

Einstein failed high-school math –not.

I don’t know quite why people persist in claiming that Einstein failed high school math. Perhaps it’s to put down teachers –who clearly can’t teach or recognize genius — or perhaps to stake a claim to a higher understanding that’s masked by ADHD — a disease Einstein is supposed to have had. But, sorry to say, it ain’t true. Here’s Einstein’s diploma, 1896. His math and physics scores are perfect. Only his English seems to have been lacking. He would have been 17 at the time.

Einstein's high school diploma

Albert Einstein’s high school diploma, 1896.

Robert Buxbaum, December 16, 2014. Here’s Einstein relaxing in Princeton. Here’s something on black holes, and on High School calculus for non-continuous functions.

My solution to the world’s problems: better people

Most of the problems of the world are caused by people. Look at war, it’s caused by people; look at pollution, people; look at overeating, or welfare, or gun violence. You name it, the problem is people. My simple solution, then: better people. Immigration is a simple solution for a county that can do it selectively (take in the best, leave the rest); it’s worked for the US and it doesn’t have to beggar the third world. Education is another way to help, but we’re not quite sure what sort of education makes people better. “An uneducated man may steel from a boxcar, an educated one may steal the whole railroad.” Theodore Roosevelt is supposed to have quipped.

Those who claim they are uncommonly moral and good at teaching it have barely any proof that they are. American schools produce financially successful people, but not particularly moral ones; Europe’s approach is different, but there’s no indication they’ve done better at moral education. We look to the 18th century, or the Greeks, but they were no more moral than us, as best I can tell. The Taliban, the communists, or similar fundamentalists claim moral superiority over the west, but from my perspective, they look even worse. 

I notice that people learn morality from one another — that is each person acts like his neighbor. I also note that people tend to act better when they are involved, and feel part of whatever country, city or group they are in. Targeted immigration might bring in better people–honest, hard working, non-violent — and these people might help improve and motivate the locals. And even if we don’t improve by interaction, perhaps lazy Americans will ride on the backs of the hard-working immigrants. But it strikes me that the disconnect between world problems of high unemployment, world hunger, and lots of open, US jobs is a moral problem that could be solved by targeted love. Allowing some increased mobility from country to country and job to job (plus better preaching?). If you can move you are more-likely to find a job or place where you feel fulfilled, and you are likely to do better and more there. Even the countries and jobs that are left might benefit by being rid of their malcontents. And we don’t have to take everyone.

From "Hispanic-hope."

From “Hispanic-hope.”, an interesting combination of Bible-study and immigration morality.

Living in America is desirable for most people from most countries. Far more people want to live here than we can accept. As a result, we are in a position to target the bright, honest, hard-working Peoples from virtually any country. These folks are helpful to industry and to the US tax base as these immigrants tend to work out — or get deported. In the short-term they might displace Americans or depress salaries, but even that is not certain. There is no fixed slate of US jobs nor a fixed amount of work need. Yesterday’s job taker is tomorrow’s job creator. Our country is built on immigrants, and has not suffered from it. We should not take those who hate the US, or those who hate freedom, or have no skills, criminals and the sick. Nor should we give citizenship immediately. But that still leaves plenty who we’d want, and who want to be here. Th. Roosevelt said, “you can not take in too many of the right people, and even one of the wrong type is too many.” I suspect this is true.

I suspect we’d have 90+% odds picking good people from a crowd. The Immigration system does a good job now, and the great colleges have done better for years. The past is usually a great indicator. If someone is well, and has worked for years, or has been here in school; if they’ve managed to stay productive and out of trouble, he/she is a good candidate. A first step would be a work permit, and in a few years they can apply for permanent residence or citizenship. Many of the most successful people in America are either immigrants or descendents of immigrants. The founders of Google and Facebook; the builders and the shakers. These people have the ‘get-up and go.’ You can tell because they’ve gotten up and gone.

Dr. Robert Buxbaum, June 16, 2014. I’m a child of an immigrant, went to public school, got a PhD at Princeton, have built my own company, and have (so far) avoided arrest, imprisonment or serious scandal. With the help of my Canadian-immigrant wife, I’ve produced three Buxbaum clones, my biggest contribution to improving the US and the world.

American education how do we succeed?

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As the product of a top American college, Princeton University, I see that my education lacks in languages and history compared to Europeans. I can claim to know a little Latin and a little Greek, like they do, but I’m referring to Manuel Ramos and Stanos Platsis, two short people, one of Spanish descent, the other of Greek.

Americans hate math.

Americans hate math.

It was recently reported that one fourth of college-educated Americans did not know that the earth spun on an axis, a degree of science ignorance that would be inconceivable in any other country. Strange to say, despite these lacks, the US does quite well commercially, militarily, and scientifically. US productivity is the world’s highest. Our GNP and GNP per capita too is higher than virtually any other country (we got the grossest national product). How do we do it with so little education?

One part of US success is clearly imported talent, Immigration. We import Nobel chemists, Russian dancers, and German rocket scientists but we don’t import that many. They help our per-capita GNP, but the majority of our immigrants are more in the wretched refuse category. Even these appear to do better here than the colleagues they left behind. Otto von Bismark once joked that, “God protects children, drunks, and the United States of America.” But I’d like to suggest that our success is based on advantages our outlook our education provides for our more creative citizens.

Most of our successful businesses are not started by the A students, but by the C student who is able to use the little he (or she) knows. Consider the simple question of whether the earth goes round the sun. It’s an important fact, but only relevant if you can use it, as Sherlock Holmes points out. I suspect that few Europeans could use the knowledge that the earth spins (try to think of some applications; at the end of this essay I’ll provide some).

Benjamin Jowett. His students included the heads of 6 colleges and the head of Eaton

Benjamin Jowett, Master of Balliol College, Oxford.

A classic poem about European education describes Benjamin Jowett, shown at right. It goes: “The first come I, my name is Jowett. There is no knowledge, but that I know it. I am master of this college. What I don’t know isn’t knowledge.” Benjamin Jowett was Master of Balliol College, Oxford. By the time he died in 1893, his ex-student pallbearers included the heads of 6 colleges, and the head of Eaton. Most English heads of state and industry were his students directly or second-hand. All learned a passing knowledge of Greek, Latin, Plato, law, science, theology, classics, math, rhetoric, logic, and grammar. Only people so educated were deemed suited to run banks or manage backward nations like India or Rhodesia. It worked for a while but showed its limitations, e.g. in the Boer Wars.

In France and continental Europe the education system is similar to England’s under Jowett. There is a fixed set of knowledge and a fixed rate to learn it. Government and industry jobs go largely to those who’ve demonstrated their ability to give the fixed, correct answers to tests on this knowledge. In schools across France, the same page is turned virtually simultaneously in the every school– no student is left behind, but none jump ahead either. As new knowledge is integrated, the approved text books are updated and the correct answers are adjusted. Until then, the answers in the book are God’s truth, and those who master it can comfort themselves to have mastered the truth. The only people hurt are the very few dummies who see a new truth a year before the test acknowledges it. “College is a place where pebbles are polished but diamonds are dimmed.” The European system appears to benefit the many, providing useful skills (and useless tidbits) but it is oppressive to many others with forward-thinking, imaginative minds. The system appears to work best in areas that barely change year-to-year like French grammar, geometry, law, and the map of Europe. It does not work so well in music, computers, or the art of war. For these students, schooling is “another brick in the wall. For these students, the schools should teach more of how to get along without a teacher.

The American approach to education leans towards independence of thought, for good or bad. American graduates can live without the teacher, but leave school knowing no language but English, hardly and maths or science, hardly any grammar, and we can hardly find another country on a map. Teachers will take incorrect answers as correct as a way to build self-esteem, so students leave with the view that there is no such thing as truth. This model works well in music, engineering, and science where change is fast, creativity is king, and nature itself is a teacher. American graduate-schools are preeminent in these areas. In reading, history and math our graduates might well be described as galumphing ignorants.

Every now and again the US tries to correct this, by the way, and join the rest of the world. The “no child left behind” movement was a Republican-led effort to teach reading and math on the French model. It never caught on. Drugs are another approach to making American students less obstreperous, but they too work only temporarily. Despite these best efforts, American graduates leave school ignorant, but not stupid; respectful of those who can do things, and suspicious of those with lengthy degrees. We survive as managers of the most complex operations with our bumptious optimism and distain for hierarchy. As viewed from abroad, our method is to greet colleagues in a loud, cheerful voice, appoint a subordinate to “get things done,” and then get in the way until lunchtime.

In any moment of decision, the best thing you can do is the right thing, the next bet thing is the wrong thing, and the worst thing you can do is nothing. An American attitude that sometimes blows up, but works surprisingly well at times.

Often the inability to act is worse than acting wrong.

The American-educated boss will do some damage by his ignorance but it is no more than  comes from group-think: non-truths passed as truths. America stopped burning witches far sooner than Europe, and never burned Jews. America dropped nobles quicker, and transitioned to electric lights and motor cars quicker, perhaps because we put less weight on what nobles and universities did.

European scholars accepted that nobility gave one a better handle on leadership, and this held them back. Since religion was part of education, they accepted that state should have an established religion: Anglican, in England, Catholicism in France; scientific atheism now. They learned and accepted that divorce was unnecessary and that homosexuality should be punished by prison or worse. As late as the early 60s, Turing, the brilliant mathematician and computer scientist, was chemically castrated as a way to cure his homosexuality. In America our “Yankee ingenuity,” as we call it, had a tendency to blow up, too (prohibition, McCarthyism, and disco), but the problems resolved relatively soon. “Ready, fire, aim” is a European description of the American method. It’s not great, but works after a fashion.

The best option, I think, is to work together with those from “across the pond.” It worked well for us in WWI, WWII, and the American Revolution, where we benefitted from the training of Baron Von Steuben, for example. Heading into the world cup of football (fifa soccer) this week, we’re expected to lose badly due to our lack of stars, and general inability to pass, dribble, or strategize. Still, we’ve got enthusiasm, and we’ve got a German coach. The world’s bookies give us 0.05% odds, but our chances are 10 times that, I’d say: 5%. God protects our galumphing side of corn-fed ignorants when, as in the Revolution, it’s attached to German coaching.

Some practical aspects of the earth spinning: geosynchronous satellites (they only work because the earth spins), weather prediction (the spin of hurricanes is because the earth spins), cyclone lifting. It amazes me that people ever thought everything went around the earth, by the way; Mercury and Venus never appear overhead. If authorities could have been so wrong about this for so long, what might they be wrong about today?

Dr. Robert Buxbaum, June 10, 2014 I’ve also written about ADHD on Lincoln’s Gettysburg Address, on Theodore Roosevelt, and how he survived a gun shot.

Is ADHD a real disorder

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When I was in school, ADHD hadn’t been invented. There were kids who didn’t pay attention for a good part of the day, or who couldn’t sit in their seats, but the first activity was called day-dreaming and the second “shpilkas” or “ants in your pants.” These problems were recognized but were considered “normal.” Though we were sometimes disorderly, the cause wasn’t labeled a disorder. It’s now an epidemic.

There were always plenty of kids, me included, who were day-dreamers. Mostly these were boys who would get bored after a while and would start to look around the room, or doodle, or gaze into space thinking of this or that. Perhaps I’d do some writing or math in the margin of a notebook while listening with one ear; perhaps I’d work on my handwriting, or I’d read something in another textbook. This was not called a disorder or even an attention deficit (AD), but rather day-dreaming, wool-gathering, napping, or just not paying attention. Sometimes teachers got annoyed, other times not. They went on teaching, but sometimes tossed chalk or erasers at us to get us to wake up. Kids like me took enough notes to do OK on tests and homework, though I was never at the top of the class in elementary or middle school. The report cards tended to say things like “he could do better if he really concentrated.”  It’s something that could apply to everyone.

Then there were the boys who would now be labeled HD, or “hyperactive disordered.” These were always boys: those who didn’t sit well in their chairs, or fidgeted, or were motor mouths and got up and walked about, or got into fights, or went to the bathroom; these were the class clowns, and the trouble makers — not me except for the fidgeting. Girls would fidget or talk too, and they’d pass notes to each other, but they didn’t get into fights, and they weren’t as disruptive. They tended to have great handwriting, and took lots of notes in class: every single word from the board, plus quite a bit more.

There are different measures of education, if you measure a fish's intelligence by the ability to climb a tree it will spend its life thinking it's stupid.

There are different measures of education, if you measure a fish’s skill level by the ability to climb a tree you’ll conclude the fish is ADD or worse.

Elementary and middle schools had activities to work out the excess energy that caused hyper-activity. We had dancing, shop, fire drills, art, some music, and sports. None of these helped all that much, but they did some good. I think the fire drills helped the most because we all went outside even in the winter, and eventually we calmed down without drugs. Sometimes a kid didn’t calm down, got worse, and did real damage; these kids were not called hyperactive disordered, but “bad kids” or “juvenile delinquents.” Nowadays, schools have far less art and music, and no shop or dancing. There are a lot more hyperactive kids, and the claim nowadays is that these hyperactive kids, violent or not, are disordered, ADHD, and should be given drugs. With drugs, the daydreamers take better notes, the nappers wake up, and the hyperactive kids calm down. Today about 30% of high-school seniors are given either a version of amphetamine, e.g. Adderall, or of Methylphenidate (Ritalin, etc.) The violent ones, the juvenile delinquents, are given stronger versions of the same drugs, e.g. methamphetamine, the drug at the heart of “breaking bad.”

Giving drugs to the kids seems to help the teacher a lot more than it helps the kids. According to a famous joke, giving the Ritalin to the teacher would be the best solution. When the kids are given drugs the disorderly boys (it’s usually given to boys) begin to act more like “goodie goodies”. They sit better and pay attention more; they take better notes and don’t interrupt, but I’m not sure they are learning more, or that the class is, or that they are socializing any better than before. The “goodie-goodies” in elementary school (mostly girls) did great in the early grades, but their good habits seemed to hold them back later. They worked too hard to please and tended to not notice, or pretended to not notice, when the teacher said nonsense. When it came time for independent or creative endeavors, their diligent acceptance of authority stood in the way of excellence.Venn diagram of ADHD

The hyperactive and daydreamers were more used to thinking for themselves, a prerequisite of leadership. The AD ones had gotten used to half-ignoring the teacher, and the HD ones were more openly opinionated and oppositional: obstreperous, in a word. Those bright enough to get by got more out of their education, perhaps because it was more theirs. To the extent that education was supposed to make you a leader and a thinker, the goodie-goodie behavior was a distraction and a disorder. This might be expected if education is supposed to be the lighting of a fire, not the filling of a pit. If everyone thinks the same, it’s a sign that few are thinking.

Map of ADHD variation with location for US kids ages 6-18, Scrips Research.

Map of ADHD variation with location for US kids ages 6-18, Scrips Research. Boys are 2-3 times more often diagnosed as ADHD; diagnosis and medication increase with grade, peaking currently in early college.

This is not to say that there is no such disorder as ADHD, or no benefit from the drugs. My sense, though, is that the label is given too widely, and that the drugs are given too freely. Today drugs are pushed on virtually any kid who’s distracted, napping or hyperactive — to all the members of the big circles in the Venn diagram above, plus to athletes and others who feign ADD to get these, otherwise illegal, performance enhancing drugs. Currently, about 10% of US kids between 6 and 18 are diagnosed ADHD and given drugs, see figure. The numbers higher for boys than girls, higher in the US than abroad, and higher as the kids progress through school. It’s estimated that about 25% of US, 12th grade boys are given amphetamine or Ritalin and its homologs. My sense is that only a small fraction of these deserve drugs, only those with severe social problems, the violent or narcoleptic: those in the smaller circles of the Venn diagram. The test should not be that the kid’s behavior improves on them. Everyone’s attention improves when taking speed. ADHD appears more as an epidemic of overworked, undertrained, underfunded teachers, and a lack of outlets, not of disordered kids, or of real learning, and real learning is never pretty or easy (on all involved).

Robert Buxbaum, April 18, 2014. In general, I think people would be happier if they’d do more art, music, dance and shop, and if they’d embrace their inner weirdo. It would also help if doctors and teachers would use words rather than initials to describe people. It’s far better to be told you’re hyperactive, or that you’re not paying attention, then to be called ADD, HD, or ADHD. There’s far more room for gradation and improvement. I’m not an expert, just an observant observer.

Toxic chemistry you can do at home

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I got my start on science working with a 7 chemical, chemistry set that my sister got me when I was 7 years old (thanks Beverly). The chemicals would never be sold by a US company today — too much liability. What if your child poisons himself/herself or someone else, or is allergic, or someone chokes on the caps (anything the size of a nut has to be labeled as a hazard). Many of the experiments were called magic, and they were, in the sense that, if you did them 200 years earlier, you’d be burnt as a witch. There were dramatic color changes (phenolphthalein plus base, Prussian Blue) a time-delay experiment involving cobalt, and even an experiment that (as I recall) burst into fire on its own (glycerine plus granulated potassium permanganate).

Better evil through science. If you get good at this, the military may have use of your services.

“Better the evil you know.” If you get good at this, the military may have use of your services. Yes, the American military does science.

Science kits nowadays don’t do anything magically cool like that, and they don’t really teach chemistry, either, I think. Doing magical things requires chemicals that are reasonably reactive, and that means corrosive and/or toxic. Current kits use only food products like corn-starch or baking soda, and the best you can do with these is to make goo and/ or bubbles. No one would be burnt at the stake for this, even 300 years ago. I suppose one could design a program that used these materials to teach something about flow, or nucleation, but that would require math, and the kit producers fear that any math will turn off kids and stop their parents from spending money. There is also the issue of motivation. Much of historical chemistry was driven by greed and war; these are issues that still motivate kids, but that modern set-makers would like to ignore. Instead, current kits are supposed to be exciting in a cooperative way (whatever that means), because the kit-maker says so. They are not. I went through every experiment in my first kit in the first day, and got things right within the first week — showing off to whoever would watch. Modern kits don’t motivate this sort of use; I doubt most get half-used in a lifetime.

There are some foreign-made chemistry sets still that are pretty good. Here is a link to a decent mid-range one from England. But it’s sort of pricy, and already somewhat dumbed down. Instead, here are some cheaper, more dangerous, American options: 5 experiments you can do (kids and parents together, please) using toxic household chemicals found in our US hardware stores. These are NOT the safest experiments, just cheap ones that are interesting. I’ll also try to give some math and explanations — so you’ll understand what’s happening on a deeper level — and I’ll give some financial motivation — some commercial value.

1) Crystal Drano + aluminum. Crystal Drano is available in the hardware store. It’s mostly lye, sodium hydroxide, one of the strongest bases known to man. It’s a toxic (highly poisonous) chemical used to dissolve hair and fat in a drain. It will also dissolve some metals and it will dissolve you if you get it on yourself (if you do get it on yourself, wash it off fast with lots of water). Drano also contains ammonium nitrate (an explosive) and bits of aluminum. For the most part, the aluminum is there so that the Drano will get hot in the clogged drain (heat helps it dissolve the clog faster). I’ll explain the ammonium nitrate later. For this experiment, you’re going to want to work outside, on a dinner plate on the street. You’ll use additional aluminum (aluminum foil), and you’ll get more heat and fun gases. Fold up a 1 foot square of aluminum foil to 6″ x 4″ say, and put it on the plate (outside). Put an indent in the middle of the foil making a sort of small cup — one that can stand. Into this indent, put a tablespoon or two of water plus a teaspoon of Drano. Wait about 5 minutes, and you will see that the Drano starts smoking and the aluminum foils starts to dissolve. The plate will start to get hot and you will begin to notice a bad smell (ammonia). The aluminum foil will turn black and will continue to dissolve till there is a hole in the middle of the indent. Draino

The main reaction is 2 Al + 3 H2O –> Al2O3 + H2; that is, aluminum plus water gives you aluminum oxide (alumina), and hydrogen. The sodium hydroxide (lye) in the Drano is a catalyst in this reaction, something that is not consumed in this reaction but makes it happen faster than otherwise. The hydrogen you produce here is explosive and valuable (I explain below). But there is another reaction going on too, the one that makes the bad smell. When ammonium nitrate is heated in the presence of sodium hydroxide, it reacts to make ammonia and sodium nitrate. The reaction formula is: NH4-NO3 + NaOH –> NH3 + NaNO3 + H2O. The ammonia produced gives off a smell, something that is important for safety — the smell is a warning — and (I think) helps keep the aluminum gunk from clogging the drain by reacting with the aluminum oxide to form aluminum amine hydroxide Al2O3(NH3)2. It’s a fun experiment to watch, but you can do more if you like. The hydrogen and ammonia are flammable and is useful for other experiments (below). If you collect these gases, you can can make explosions or fill a balloon that will float. Currently the US military, and several manufacturers in Asia are considering using the hydrogen created this way to power motorcycles by way of a fuel cell. There is also the Hindenburg, a zeppelin that went around the world in the 1930s. It was kept aloft by hydrogen. The ammonia you make has value too, though toxic; if bubbled into water, it makes ammonium hydroxide NH3 + H2O –> NH4OH. This is a common cleaning liquid. Just to remind you: you’re supposed to do these experiments outside to dissipate the toxic gases and to avoid an explosion in your house. A parent will come in handy if you get this stuff on your hand or in your eye.

Next experiment: check that iron does not dissolve in Drano, but it does in acid (that’s experiment 5; done with Muriatic acid from the hardware store). Try also copper, and solder (mostly tin, these days). Metals that dissolve well in Drano are near the right of the periodic table, like aluminum. Aluminum is nearly a non-metal, and thus can be expected to have an oxide that reacts with hydroxide. Iron and steel have oxides that are bases themselves, and thus don’t react with lye. This is important as otherwise Drano would destroy your iron drain, not only the hair in it. It’s somewhat hard on copper though, so beware if you’ve a copper drain.

Thought problem: based on the formulas above figure out the right mix of aluminum, NaOH, water and Ammonium nitrate. Answer: note that, for every two atoms of aluminum you dissolve, you’ll need three molecules of water (for the three O atoms), plus at least two molecules of ammonium nitrate (to provide the two NH2 (amine) groups above. You’ll also want at least 2 molecules of NaOH to have enough Na to react with the nitrate groups of the ammonium nitrate. As a first guess, assume that all atoms are the same size. A better way to do this involves molecular weights (formula weights), read a chemistry book, or look on the internet.

Four more experiments can be seen here. This post was getting to be over-long.As with this experiment, wear gloves and eye protection; don’t drink the chemicals, and if you get any chemicals on you, wash them off quick.

Here are a few more experiments in electrochemistry and biology, perhaps I’ll add more. In the meantime, if you or your child are interested in science, I’d suggest you read science books by Mr Wizard, or Isaac Asimov, and that you learn math. Another thought, take out a high school chemistry text-book at the library — preferably an old one with experiments..

Robert Buxbaum, December 29, 2013. If you are interested in weather flow, by the way, here is a bit on why tornadoes and hurricanes lift stuff up, and on how/ why they form. 

Calculus is taught wrong, and is often wrong

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The high point of most people’s college math is The Calculus. Typically this is a weeder course that separates the science-minded students from the rest. It determines which students are admitted to medical and engineering courses, and which will be directed to english or communications — majors from which they can hope to become lawyers, bankers, politicians, and spokespeople (the generally distrusted). While calculus is very useful to know, my sense is that it is taught poorly: it is built up on a year of unnecessary pre-calculus and several shady assumptions that were not necessary for the development, and that are not generally true in the physical world. The material is presented in a way that confuses and turns off many of the top students — often the ones most attached to the reality of life.

The most untenable assumption in calculus teaching, in my opinion, are that the world involves continuous functions. That is, for example, that at every instant in time an object has one position only, and that its motion from point to point is continuous, defining a slow-changing quantity called velocity. That is, every x value defines one and only one y value, and there is never more than a small change in y at the limit of a small change in X. Does the world work this way? Some parts do, others do not. Commodity prices are not really defined except at the moment of sale, and can jump significantly between two sales a micro-second apart. Objects do not really have one position, the quantum sense, at any time, but spread out, sometimes occupying several positions, and sometimes jumping between positions without ever occupying the space in-between.

These are annoying facts, but calculus works just fine in a discontinuous world — and I believe that a discontinuous calculus is easier to teach and understand too. Consider the fundamental law of calculus. This states that, for a continuous function, the integral of the derivative of changes equals the function itself (nearly incomprehensible, no?) Now consider the same law taught for a discontinuous group of changes: the sum of the changes that take place over a period equals the total change. This statement is more general, since it applies to discrete and continuous functions, and it’s easier to teach. Any idiot can see that this is true. By contrast, it takes weeks of hard thinking to see that the integral of all the derivatives equals the function — and then it takes more years to be exposed to delta functions and realize that the statement is still true for discrete change. Why don’t we teach so that people will understand? Teach discrete first and then smooth as a special case where the discrete changes happen at a slow rate. Is calculus taught this way to make us look smart, or because we want this to be a weeder course?

Because most students are not introduced to discrete change, they are in a very poor position  to understand, or model, activities that are discreet, like climate change or heart rate. Climate only makes sense year to year, as day-to-day behavior is mostly affected by seasons, weather, and day vs night. We really want to model the big picture and leave out the noise by considering each day or year as a whole, keeping track of the average temperature for noon on September 21, for example. Similarly with heart rate, the rate has no meaning if measured every microsecond; it’s only meaning is as a measure of the time between beats. If we taught calculus in terms of discrete functions, our students would be in a better place to deal with these things, and in a better place to deal with total discontinuous behaviors, like chaos and fractals, an important phenomena when dealing with economics, for example.

A fundamental truth of quantum mechanics is that there is no defined speed and position of an object at any given time. Students accept this, but (because they are used to continuous change) they come to wonder how it is that over time energy is conserved. It’s simple, quantum motion involves a gross discrete changes in position that leaves energy conserved by the end, but where an item goes from here to there without ever having to be in the middle. This helps explain the old joke about Heisenberg and his car.

Calculus-based physics is taught in terms of limits and the mean value theorem: that if x is the position of a thing at any time, t then the derivative of these positions, the velocity, will approach ∆x/∆t more and more as ∆x and ∆t become more tightly defined. When this is found to be untrue in a quantum sense, the remnant of the belief in it hinders them when they try to solve real world problems. Normal physics is the limit of quantum physics because velocity is really a macroscopic ratio of difference in position divided by macroscopic difference in time. Because of this, it is obvious that the sum of these differences is the total distance traveled even when summed over many simultaneous paths. A feature of electromagnetism, Green’s theorem becomes similarly obvious: the sum effect of a field of changes is the total change. It’s only confusing if you try to take the limits to find the exact values of these change rates at some infinitesimal space.

This idea is also helpful in finance, likely a chaotic and fractal system. Finance is not continuous: just because a stock price moved from $1 to $2 per share in one day does not mean that the price was ever $1.50 per share. While there is probably no small change in sales rate caused by a 1¢ change in sales price at any given time, this does not mean you won’t find it useful to consider the relation between the sales of a product. Though the details may be untrue, the price demand curve is still very useful (but unjustified) abstraction.

This is not to say that there are not some real-world things that are functions and continuous, but believing that they are, just because the calculus is useful in describing them can blind you to some important insights, e.g. of phenomena where the butterfly effect predominates. That is where an insignificant change in one place (a butterfly wing in China) seems to result in a major change elsewhere (e.g. a hurricane in New York). Recognizing that some conclusions follow from non-continuous math may help students recognize places where some parts of basic calculus allies, while others do not.

Dr. Robert Buxbaum (my thanks to Dr. John Klein for showing me discrete calculus).