Why Johnny can’t X

David Brin, “Why Johnny Can’t Code,” Salon (2006)

Mike Davidow, Why Johnny Can’t Read and Ivan Can (1977)

Tommy Dreyfus, “Why Johnny Can’t Prove,” Educational Studies in Mathematics (1999)

Rudolf Flesch, Why Johnny Can’t Read (1955) and Why Johnny Still Can’t Read (1981)

Thomas Frank, “Dark Age: Why Johnny Can’t Dissent,” The Baffler (1995)

Konstantin Kakaes, “Why Johnny Can’t Add Without a Calculator,” Salon (2012)

Walter Karp, “Why Johnny Can’t Think,” Harper’s (1985)

William Kilpatrick, Why Johnny Can’t Tell Right from Wrong (1992)

Morris Kline, Why Johnny Can’t Add: The Failure of the New Math (1974)

Myra Linden and Arthur Whimbey, Why Johnny Can’t Write (2012)

Opal Moore, Why Johnny Can’t Learn (1975)

Douglas Rushkoff, “Why Johnny Can’t Program,” Huffington Post (2010)

Arthur Trace, What Ivan Knows that Johnny Doesn’t (1961)

Quite an everyday occurrence

from Huygens and Barrow, Newton and Hooke, Vladimir Arnold, translated by Eric J. F. Primrose (1989):

Hooke was a poor man and began work as an assistant to Boyle (who is now well known thanks to the Boyle-Mariotte law discovered by Hooke). Subsequently Hooke began working in the recently established Royal Society (that is, the English Academy of Sciences) as Curator. The duties of the Curator of the Royal Society were very onerous. According to his contract, at every session of the Society (and they occurred every week except for the summer vacation) he had to demonstrate three or four experiments proving the new laws of nature.

Hooke held the post of Curator for forty years, and all that time he carried out his duties thoroughly. Of course, there was no condition in the contract that all the laws to be demonstrated had to be devised by him. He was allowed to read books, correspond with other scientists, and to be interested in their discoveries. He was only required to verify whether their statements were true and to convince the Royal Society that some law was reliably established. For this it was necessary to prove this law experimentally and demonstrate the appropriate experiment. This was Hooke’s official activity.

[…]

At that time it was easy to carry out fundamental discoveries, and large numbers of them were carried out. Huygens, for example, improved the telescope, looked at Saturn and discovered its ring, and Hooke discovered the red spot on Jupiter. At that time discoveries were not unusual events, they were not registered, not patented, as they are now, they were quite an everyday occurrence. (This was the case not only in the natural sciences. Mathematical discoveries at that time also poured forth as if from a horn of plenty.)

But Hooke never had enough time to dwell on any of his discoveries and develop it in detail, since in the following week he needed to demonstrate new laws. So in the whole manifold of Hooke’s achievements his discoveries appeared somewhat incomplete, and sometimes when he was in a hurry he made assertions that he could not justify accurately and with mathematical rigour.

[…]

Holding the chair at Cambridge, Newton earned considerably more (200 pounds a year), and the farm that he had inherited, which he leased out and where the famous apple tree grew, gave him roughly the same income. Despite the fact that Newton was quite well off, he did not want to spend any money on the publication of the book, so he sent the Principia to the Royal Society, which decided to publish the book at its own expense. But the Society had no money, so the manuscript lay there until Halley (who was the son of a rich soap manufacturer) published it on his own account. Halley took on himself all the trouble of publishing the book, and even read the proofs himself. Newton, in correspondence at this time, called it “Your book”…

An engine of discovery

from the preface to Cell Biology by the Numbers, Ron Milo and Rob Phillips:

One of the great traditions in biology’s more quantitative partner sciences such as chemistry and physics is the value placed on centralized, curated quantitative data. Whether thinking about the astronomical data that describes the motions of planets or the thermal and electrical conductivities of materials, the numbers themselves are a central part of the factual and conceptual backdrop for these fields.  Indeed, often the act of trying to explain why numbers have the values they do ends up being an engine of discovery.

A requirement for genuine expertise

David Foster Wallace in conversation with Dave Eggers, The Believer, 2003:

We live today in a world where most of the really important developments in everything from math and physics and astronomy to public policy and psychology and classical music are so extremely abstract and technically complex and context-dependent that it’s next to impossible for the ordinary citizen to feel that they (the developments) have much relevance to her actual life. Where even people in two closely related sub-sub-specialties have a hard time communicating with each other because their respective s-s-s’s require so much special training and knowledge. And so on. Which is one reason why pop-technical writing might have value (beyond just a regular book-market $-value), as part of the larger frontier of clear, lucid, unpatronizing technical communication. It might be that one of the really significant problems of today’s culture involves finding ways for educated people to talk meaningfully with one another across the divides of radical specialization. That sounds a bit gooey, but I think there’s some truth to it. And it’s not just the polymer chemist talking to the semiotician, but people with special expertise acquiring the ability to talk meaningfully to us, meaning ordinary schmoes. Practical examples: Think of the thrill of finding a smart, competent IT technician who can also explain what she’s doing in such a way that you feel like you understand what went wrong with your computer and how you might even fix the problem yourself if it comes up again. Or an oncologist who can communicate clearly and humanly with you and your wife about what the available treatments for her stage-two neoplasm are, and about how the different treatments actually work, and exactly what the plusses and minuses of each one are. If you’re like me, you practically drop and hug the ankles of technical specialists like this, when you find them. As of now, of course, they’re rare. What they have is a particular kind of genius that’s not really part of their specific area of expertise as such areas are usually defined and taught. There’s not really even a good univocal word for this kind of genius—which might be significant. Maybe there should be a word; maybe being able to communicate with people outside one’s area of expertise should be taught, and talked about, and considered as a requirement for genuine expertise.… Anyway, that’s the sort of stuff I think your question is nibbling at the edges of, and it’s interesting as hell.

Energies and perseverances

Thomas Jefferson to Dr. John P. Emmet, May 2, 1826, discovered in Nathaniel Grossman, The Sheer Joy of Celestial Mechanics:

[…] consider that we do not expect our schools to turn out their alumni already on the pinnacles of their respective sciences; but only so far advanced in each as to be able to pursue them by themselves, and to become Newtons and Laplaces by energies and perseverances to be continued throughout life.

Weigh and consider

Francis Bacon, “Of Studies,” originally 1597, enlarged 1625:

Studies serve for delight, for ornament, and for ability. Their chief use for delight, is in privateness and retiring; for ornament, is in discourse; and for ability, is in the judgment, and disposition of business. For expert men can execute, and perhaps judge of particulars, one by one; but the general counsels, and the plots and marshalling of affairs, come best, from those that are learned. To spend too much time in studies is sloth; to use them too much for ornament, is affectation; to make judgment wholly by their rules, is the humor of a scholar. They perfect nature, and are perfected by experience: for natural abilities are like natural plants, that need proyning, by study; and studies themselves, do give forth directions too much at large, except they be bounded in by experience. Crafty men contemn studies, simple men admire them, and wise men use them; for they teach not their own use; but that is a wisdom without them, and above them, won by observation. Read not to contradict and confute; nor to believe and take for granted; nor to find talk and discourse; but to weigh and consider. Some books are to be tasted, others to be swallowed, and some few to be chewed and digested; that is, some books are to be read only in parts; others to be read, but not curiously; and some few to be read wholly, and with diligence and attention. Some books also may be read by deputy, and extracts made of them by others; but that would be only in the less important arguments, and the meaner sort of books, else distilled books are like common distilled waters, flashy things. Reading maketh a full man; conference a ready man; and writing an exact man. And therefore, if a man write little, he had need have a great memory; if he confer little, he had need have a present wit: and if he read little, he had need have much cunning, to seem to know, that he doth not. Histories make men wise; poets witty; the mathematics subtile; natural philosophy deep; moral grave; logic and rhetoric able to contend. Abeunt studia in mores. Nay, there is no stond or impediment in the wit, but may be wrought out by fit studies; like as diseases of the body, may have appropriate exercises. Bowling is good for the stone and reins; shooting for the lungs and breast; gentle walking for the stomach; riding for the head; and the like. So if a man’s wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again. If his wit be not apt to distinguish or find differences, let him study the Schoolmen; for they are cymini sectores. If he be not apt to beat over matters, and to call up one thing to prove and illustrate another, let him study the lawyers’ cases. So every defect of the mind, may have a special receipt.

The mind is assailed, as it were

Edward Thorndike, “Reading as Reasoning: A Study of Mistakes in Paragraph Reading,” The Journal of Educational Psychology, 1917:

Understanding a paragraph is like solving a problem in mathematics. It consists in selecting the right elements of the situation and putting them together in the right relations, and also with the right amount of weight or influence or force for each. The mind is assailed as it were by every word in the paragraph. It must select, repress, soften, emphasize, correlate and organize, all under the influence of the right mental set or purpose or demand.

[…]

It thus appears that reading an explanatory or argumentative paragraph in his text-books on geography or history or civics, and (though to a less degree) reading a narrative or description, involves the same sort of organization and analytic action of ideas as occur in thinking of supposedly higher sorts.

[…]

It appears likely, therefore, that many children fail in certain features of these subjects not because they have understood and remembered the facts and principles but have been unable to organize and use them; or because they have understood them but have been unable to remember them; but because they never understood them.

It appears likely also that a pupil may read fluently and feel that the series of words are arousing appropriate thoughts without really understanding the paragraph. Many of the children who made notable mistakes would probably have said that they understood the paragraph and, upon reading the questions on it, would have said that they understood them. In such cases the reader finds satisfying solutions of those problems which he does raise and so feels mentally adequate; but he raises only a few of the problems which should be raised and makes only a few of the judgments which he should make.

Where the appearance of disorder reigned

Poincaré, “The Future of Mathematics,” 1908, in Science and Method, translated by Francis Maitland:

The importance of a fact is measured by the return it gives—that is, by the amount of thought it enables us to economize.

In physics, the facts which give a large return are those which take their place in a very general law, because they enable us to foresee a very large number of others, and it is exactly the same in mathematics. Suppose I apply myself to a complicated calculation and with much difficulty arrive at a result, I shall have gained nothing by my trouble if it has not enabled me to foresee the results of other analogous calculations, and to direct them with certainty, avoiding the blind groping with which I had to be contented the first time. On the contrary, my time will not have been lost if this very groping has succeeded in revealing to me the profound analogy between the problem just dealt with and a much more extensive class of other problems; if it has shown me at once their resemblances and their differences; if, in a word, it has enabled me to perceive the possibility of a generalization. Then it will not be merely a new result that I have acquired, but a new force.

An algebraical formula which gives us the solution of a type of numerical problems, if we finally replace the letters by numbers, is the simple example which occurs to one’s mind at once. Thanks to the formula, a single algebraical calculation saves us the trouble of a constant repetition of numerical calculations. But this is only a rough example; every one feels that there are analogies which cannot be expressed by a formula, and that they are the most valuable.

If a new result is to have any value, it must unite elements long since known, but till then scattered and seemingly foreign to each other, and suddenly introduce order where the appearance of disorder reigned. Then it enables us to see at a glance each of these elements in the place it occupies in the whole. Not only is the new fact valuable on its own account, but it alone gives a value to the old facts it unites. Our mind is frail as our senses are; it would lose itself in the complexity of the world if that complexity were not harmonious; like the short-sighted, it would only see the details, and would be obliged to forget each of these details before examining the next, because it would be incapable of taking in the whole. The only facts worthy of our attention are those which introduce order into this complexity and so make it accessible to us.

Mathematicians attach a great importance to the elegance of their methods and of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or a demonstration? It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts. But that is also precisely what causes it to give a large return; and in fact the more we see this whole clearly and at a single glance, the better we shall perceive the analogies with other neighbouring objects, and consequently the better chance we shall have of guessing the possible generalizations. Elegance may result from the feeling of surprise caused by the un-looked-for occurrence together of objects not habitually associated. In this, again, it is fruitful, since it thus discloses relations till then unrecognized. It is also fruitful even when it only results from the contrast between the simplicity of the means and the complexity of the problem presented, for it then causes us to reflect on the reason for this contrast, and generally shows us that this reason is not chance, but is to be found in some unsuspected law. Briefly stated, the sentiment of mathematical elegance is nothing but the satisfaction due to some conformity between the solution we wish to discover and the necessities of our mind, and it is on account of this very conformity that the solution can be an instrument for us. This aesthetic satisfaction is consequently connected with the economy of thought. Again the comparison with the Erechtheum occurs to me, but I do not wish to serve it up too often.

It is for the same reason that, when a somewhat lengthy calculation has conducted us to some simple and striking result, we are not satisfied until we have shown that we might have foreseen, if not the whole result, at least its most characteristic features. Why is this? What is it that prevents our being contented with a calculation which has taught us apparently all that we wished to know? The reason is that, in analogous cases, the lengthy calculation might not be able to be used again, while this is not true of the reasoning, often semi-intuitive, which might have enabled us to foresee the result. This reasoning being short, we can see all the parts at a single glance, so that we perceive immediately what must be changed to adapt it to all the problems of a similar nature that may be presented. And since it enables us to foresee whether the solution of these problems will be simple, it shows us at least whether the calculation is worth undertaking.

What I have just said is sufficient to show how vain it would be to attempt to replace the mathematician’s free initiative by a mechanical process of any kind. In order to obtain a result having any real value, it is not enough to grind out calculations, or to have a machine for putting things in order: it is not order only, but unexpected order, that has a value. A machine can take hold of the bare fact, but the soul of the fact will always escape it.

Since the middle of last century, mathematicians have become more and more anxious to attain to absolute exactness. They are quite right, and this tendency will become more and more marked. In mathematics, exactness is not everything, but without it there is nothing: a demonstration which lacks exactness is nothing at all. This is a truth that I think no one will dispute, but if it is taken too literally it leads us to the conclusion that before 1820, for instance, there was no such thing as mathematics, and this is clearly an exaggeration. The geometricians of that day were willing to assume what we explain by prolix dissertations. This does not mean that they did not see it at all, but they passed it over too hastily, and, in order to see it clearly, they would have had to take the trouble to state it.

Only, is it always necessary to state it so many times? Those who were the first to pay special attention to exactness have given us reasonings that we may attempt to imitate; but if the demonstrations of the future are to be constructed on this model, mathematical works will become exceedingly long, and if I dread length, it is not only because I am afraid of the congestion of our libraries, but because I fear that as they grow in length our demonstrations will lose that appearance of harmony which plays such a useful part, as I have just explained.

It is economy of thought that we should aim at, and therefore it is not sufficient to give models to be copied. We must enable those that come after us to do without the models, and not to repeat a previous reasoning, but summarize it in a few lines. And this has already been done successfully in certain cases. For instance, there was a whole class of reasonings that resembled each other, and were found everywhere; they were perfectly exact, but they were long. One day some one thought of the term “uniformity of convergence,” and this term alone made them useless; it was no longer necessary to repeat them, since they could now be assumed. Thus the hair-splitters can render us a double service, first by teaching us to do as they do if necessary, but more especially by enabling us as often as possible not to do as they do, and yet make no sacrifice of exactness.

One example has just shown us the importance of terms in mathematics; but I could quote many others. It is hardly possible to believe what economy of thought, as Mach used to say, can be effected by a well-chosen term. I think I have already said somewhere that mathematics is the art of giving the same name to different things. It is enough that these things, though differing in matter, should be similar in form, to permit of their being, so to speak, run in the same mould. When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same.

A well-chosen term is very often sufficient to remove the exceptions permitted by the rules as stated in the old phraseology. This accounts for the invention of negative quantities, imaginary quantities, decimals to infinity, and I know not what else. And we must never forget that exceptions are pernicious, because they conceal laws.

This is one of the characteristics by which we recognize facts which give a great return: they are the facts which permit of these happy innovations of language. The bare fact, then, has sometimes no great interest: it may have been noted many times without rendering any great service to science; it only acquires a value when some more careful thinker perceives the connexion it brings out, and symbolizes it by a term.

Not results, but powers

George Eliot on Thomas Carlyle, 1855:

It has been well said that the highest aim in education is analogous to the highest aim in mathematics, namely, to obtain not results but powers, not particular solutions, but the means by which endless solutions may be wrought. He is the most effective educator who aims less at perfecting specific acquirements than at producing that mental condition which renders acquirements easy, and leads to their useful application; who does not seek to make his pupils moral by enjoining particular courses of action, but by bringing into activity the feelings and sympathies that must issue in noble action. On the same ground it may be said that the most effective writer is not he who announces a particular discovery, who convinced men of a particular conclusion, who demonstrates that this measure is right and that measure wrong; but he who rouses in others that activities that must issue in discovery, who awakes men from their indifference to right and wrong, who nerves their energies to seek for the truth and live up to it at whatever cost. The influence of such a writer is dynamic. He does not teach men how to use sword and musket, but he inspires their souls with courage and sends a strong will into their muscles. He does not, perhaps, enrich your stock of data, but he clears away the film from your eyes that you may search for data to some purpose. He does not, perhaps, convince you, but he strikes you, undeceives you, animates you. You are not directly fed by his books, but you are braced as by a walk up to an alpine summit, and yet subdued to calm and reverence as by the sublime things to be seen from that summit.