This is to simply document my observation which I've been mulling over for a long time until very recently.
A couple of streets that I daily drive over are lined with trees whose branches seem to exhibit strange growth pattern. While their older branches point unremarkably each other way, the younger ones sprout pretty much vertically:
And here is another sample:
Now, at Spring time, the situation became even more salient
I've been preoccupied with this phenomenon until one day I realized that its weirdness was so eye catching that I stopped looking at the more normal trees:
And then all became crystal clear, if not scientifically proven. The vertical growth happened only in the proximity of power lines, all on one side of the street. Same trees on the other - wire-free - side of the street, grew in a natural, sort of a more random way.
So here's my discovery or rather conjecture:
More research is necessary to support that conjecture.
I find the opening paragraph in a recent article by Marcus du Sautoy (The Mathematical Gazette 97 November 2013 No. 540, 386-397) revealing:
During my year as President of The Mathematical Association the government began a review of the curriculum across all subjects taught in school. Given the constant tinkering with the education system by every government, this is probably a sentence that any MA president could write during their tenure.
The same of course holds for every president of the United States and all relevant organizations (NCTM, NEA, MAA, etc.) The fact is that the educational reform is an unceasing undertaking that in mathematical education started at the end of the nineteenth century and that is still going on strong. In the US, there were some discrete moments of new announcements - not to mention the New Math (1960s), there has been a series of standards from NCTM (Curriculum and Evaluation (1989), Professional (1991), Assessment (1995), Principles (2000), Common Core(2010)). It's a given that the effort will not stop any time soon. The CCSS have already gathered a plethora of critics.
There is one person whose name always comes up when I think of the stream of mathematical reforms that come one on heels of another - Diane Ravitch. I never met Diane and all I know about her is that at some point in time she changed her mind about reforms and switched the camps so that one camp praised her courage, the other lamented her betrayal. And when I think of changing one's mind, I am reminded of a curious episode involving a well known Russian physicist Yakov Fraenkel (1894-1952). According to legend, he was shown a slide with the graph depicting the outcome of an experiment. He brought it to light and immediately explained why the experiment went the way suggested by the graph. When one of his students pointed out that he was holding the slide upside down, Fraenkel turned it around and produced an explanation of why the graph had to be that way.
The relevance of my perception of the reform process to Diane Ravitch's story is that I feel that most math educators and after them politicians that make decisions come up with reform ideas viscerally or by the sixth sense. The arguments come later: articles written, committees are set, statistics is collected, theories come forth. All this seldom causes anybody to change one's mind. And I keep wondering why, when it comes to educational reforms, it takes making or changing one's mind, and not some kind of deliberate experimentation.
There were successful experiments. Two are well known and are to my liking: an unorthodox geometry course by Harold P. Fawcett at the Ohio State University starting in the 1930s, and somewhat earlier one by Louis P. Bénézet. The consensus I believe is that a teacher needs to be a Harold Fawcett to manage a course like Fawcett's or W. Eugene Smith who taught that course in the years 1945-1956. Here I believe lies the main reason for the continuous attempts to standardize mathematical education and keep it under control with standardized testing. Other, reasons are usually cited publicly, but I think this is the main one: teachers are not trusted to do the right thing by students on their own.
There is certainly a good reason for mistrust: as in every endeavor, there are excellent, mediocre, and outright bad teachers. For the latter it is easier to follow strict curriculum with testing than to adapt to their class and individual students' progress. Many may not be able to do that.
I strongly believe - what else can I say or how else can I argue - that the only way to find a solution to the real or perceived deficiencies in math education is to admit to that fact. In a country the size of the US, any attempt to teach uniformly, giving every one the same opportunity, so to speak, is simply impossible. Worse, such attempts will always shortchange the better students. As Steven Strogatz wrote in Notices of the AMS,
Though it’s taboo to admit it, I believe there are some kids who have a feel for math.
By extension there are students who do not have that feel, they may be even in majority.
I may be beating a dead horse, but here what Marcus du Sautoy wrote about in the article I mentioned at the beginning:
I often get the feeling that we are still stuck in Napoleonic France, just doing mathematics to serve the state. If only we could find a modem-day Humboldt to take the reins at the Department of Education ....
For the first time in Germany, the study of mathematics formed a major part of the curriculum in the new schools and universities. And mathematicians, freed from the need to model the physical world, began instead to explore mathematical ideas for their own sake. It gave rise to the creation of geometry that lives beyond our three-dimensional universe.
And before that
Just as in English a student isn't meant to grasp the full complexity of a Shakespeare play, we should be prepared to take the risky step of teaching big ideas that a student might not fully comprehend but rather they should be given a way to glimpse something of these great stories. Just as any course in English literature can give just a taste of the great works, a mathematical literacy course would not aim to be complete but to expose students to a sample of what is out there.
I only have one point of disagreement with du Sautoy. It seems to me (I apologize if I am wrong) that he talks of the middle and high school curriculum. I would start earlier by trying to engage little kids with big ideas, and stop ostracizing those who do not get them.
As a young boy, Charles Adler - nowadays a physics professor at St. Mary's College of Maryland - read a lot of science fiction. He often wondered which parts of the fiction were physically possible and which were unrealistic. His book is intended in part for children like his younger self who are curious to have answers to such questions. His curiosity served one of the motives for his decision to become a physicist.
... no science fiction writer can be really esteemed accomplished unless he or she has a thorough knowledge of basic physics, chemistry, biology, astrophysics, history (ancient and modern), sociology, and military tactics ...
But of course every writer takes a literary license to expand the bounds of plausible. The book is about the extent of violation of the scientific laws science fiction and fantasy authors commit in their books.
For example, the Great Hall at Hogwarts impressed Harry Potter as being lit by myriads of candles floating in mid air, but Adler suspects that during the shooting of the Harry Potter movies the hall has been lit by concealed electric lamps. Why? One obvious reason is that candles burn in an upward position and thus block much of the emitted light from going downward. But more importantly, only 0.8% of the light emitted by a candle falls into the visible spectrum. For a tungsten bulb this number - the luminous efficacy - is close to 13%.
Is teleportation possible? Well, if just a little bit, like 1%, goes wrong, there maybe release of energy equivalent to an explosion of an H-bomb. But also take into account that the "to" and "from" systems may be fast moving relative to each other, so that if moment is indeed preserved the impact may be disastrous even for small bodies. Do not forget the Heisenberg uncertainty principle: teleporters will have to control both the location and the momentum at the destination.
These are just two examples of the problems that a reader of science fiction and fantasy books may want to ponder about. My presentation was simplistically descriptive, but the book goes into every imaginable detail, with graphs, formulas, equations, and all kinds of calculations. In fact, there is so much (mostly elementary) physics in the book, that it could be easily and profitably used as a source of entertaining exercises in high school or introductory college courses.
Truth be told, at the outset, when I realized what the book was about, I was a little annoyed. Science is science and fantasy is fantasy, and one may not want to know that there might be something wrong with the concepts in the book one is enjoying. Should everything be laid bare? That's literature we are talking about, for crying out loud, not textbooks or manuals! But Adler's writing is lucid and engaging and it sucks you in. There are so many whys and whats that I eventually developed a feeling that reality may be by far more interesting then any kind of fiction. This is an unusual and worthy book.
B. A. Kordemsky (1907-1999) was a Russian doyen of popularizers of mathematics, compared in stature to the American Martin Gardner. He even defended (1957) a Ph.D. thesis "Cunning extra curricula problems as a form of development of mathematical initiative in adolescents and grown-ups." By that time he already published his now famous volume Mathematical Savvy (Matematicheskaya Smekalka) that underwent a dozen of printings and was translated in as many languages. I recently tweeted one problem from that book:
In his last book Mathematical Allurements (Matematicheskie Zavlekalki), published posthumously in 2000, he tells a story of a 7th grade girl who got tempted to solve that problem and found a solution, too. She informed Kordemsky that her solution was different from the one in the book. Kordemsky encouraged her to look further, for other solutions. Several of her classmates get involved in the search that eventually produced more than 120 solutions. I can imagine Kordemsky's delight in seeing his efforts at attracting young minds to mathematics being born fruit. The kids even came up with something unexpected: many of the numbers they came up lead to other solutions when some pairs of their digits get swapped. For example:
For example, and , and another pair, and . But this one and seems to fall under a different rule. Kordemsky points to more rules like that.
There is no telling how the kids found their solutions and theorems. On reading the story, as I already mentioned, I sent a tweet on twitter and a nice discussion ensued. I am grateful to Pat Ballew, Colin Beveridge, Dan Bach, Thony Christie, and Vincent Pantaloni.
My first ever Python program produced 626 solutions to Kordemsky's problem. An enhanced version combined those numbers into the sets with identical first and last three digits. It came up with 246 sets, of which only 12 were singletons, most came in pairs, but there were also triples, 4-, 5-, 6-, 7-, and 8-element sets. All pairs fell under the conditions found by the Russian children. Here are the twelve singletons:
One of the two eight element sets:
And here's the only one with seven elements:
With such a big number of solutions, the problem I believe should not be probably left as manual exercise. At this time and day, writing a short computer program should be a routine matter for many of the present generation of middle and high school students. All could get involved in finding and explaining the properties of solutions that allow grouping them into separate sets.
Personally, I draw a satisfaction from having written and debugged my first Python program, from having used a computer to suggest a meaningful exercise, and from figuring out - simple as it was - what made that rule found by the Russian children tick.
Eli Maor, the author of Beautiful Geometry together with Eugen Jost, wrote that their "book is meant to be enjoyed, pure and simple." Indeed so.
With 51 color plates - the art work by Eugen Jost - the book could be thought of as an art album with annotations by Eli Maor. Alternatively, it's a gentile, historical introduction by Eli Maor into disarmingly beautiful, elementary aspects of geometry and numbers, tastefully illustrated by Eugen Jost.
Here's, for example, an illustration of Proposition 38 of Book I of Euclid's Elements, Triangles which are on equal bases and in the same parallels are equal to one another.
Jost's illustration wisely underscores the fact that any side of a triangle may serve as a base lying on one of the parallel lines. In the accompanying text, Maor introduces the reader to the beginnings of geometry, Euclid's Elements, and gives Euclid's proof of I.38.
"Sunrise over Miletus" by Jost
is accompanied by a short biography of Thales of Miletus, description of several of his achievements, and the theorem that still bears his name: diameter in a circle subtends a right angle. Maor draws reader's attention to the fact that this is one of the first manifestations of invariance, an important mathematical and physical concept.
And this is how it goes: 51 chapters that combine pedagogically meaningful artwork together with informative, and often eye opening, text. The book ends with a short Appendix which lays foundations for several mathematical concepts mentioned in the text.
This is truly an enjoyable, simple book that meets if not exceeds the authors' expectations. It's a good seasonal present, too.
In a well known puzzle, a father willed to his three sons camels, with the proviso that of the inheritance should go to the oldest among them, with being due to the middle one and to the youngest. Shortly after the father's death, a wise man riding on his camel through the village noticed the three brothers in a quandary. He added his camel to the inherited thus getting a herd of animals. He gave of these (i. e. camels) to the oldest brother, to the middle one, and to the youngest. camel remained which he climbed up and rode away on. To the great satisfaction of all brothers, each of them received more than was willed by their father.
Among those who found interest in the puzzle, one may try solving it before reading the solution; another may read and enjoy the solution; somebody else may think up related problems. Paul Stockmeyer - a professor emeritus of computer science at the College of William and Mary - did just that (Math Horizons, Volume 21, Number 1, September 2013 , pp. 8-11), posing a generalized question and even listing several related research problems.
Are there situations that include two brothers? other problems with three? problems with four, etc. How many camels could be shared in such a manner? Below are several identities that answer some of the questions, if only partially. Rather obviously the problem involves the unit fractions:
Two sons, three camels:
Two sons, five camels:
Three sons, seven camels:
Three sons, eleven camels:
Three sons, eleven camels:
Three sons, nineteen camels:
Four sons, 1805 camels:
There are many more examples. But could an estate consist of 9 or 21 or of an even number of camels. Are there divisions where the oldest son is willed one third of the inheritance and not one half? What about one fourth?
This is amazing how many questions may be asked starting with such an elementary problem. Probably not all problem submit to such extensions, but many do. I've been recently referred to the site of an Online (more accurately, by correspondence) Russian Math Olympiad honoring Leonhard Euler. Here are some examples from this year's olympiad:
- In the expression replace each star with one of four arithmetic operations to make the result zero.
What comes to mind? First, an obvious generalization (that gives away a solution) is to end the product with Then it's natural to ask if there are other cases. I am not sure of the answer yet. What I am sure about is that a solution - if exists - is never unique. Prove that.
- In the island of knights (that always tell truth) and knaves (that always lie), fellows stand in a circle, all facing the center, and each declares that the ten to his left are all knaves. How many knaves are actually there?
Is the relation inherent for the solvability of the problem?
- Draw five staright lines in a plane that intersect in exactly seven points.
What number of intersections is possible for lines?
Martin Gardner refers to his latest, and - perhaps - last, book as a rambling (and also slovenly) autobiography, disheveled memoirs. It is anything but. It is the most sincere, unadulterated biography I ever read. He wrote it in 2009-2010 on an old typewriter while at an assisted living facility. "At ninety-five I still have enough wits to keep writing," he observed. At the time his library and files that once occupied the whole third story of a big house, have been reduced to a single file cabinet. So it must have been that he largely relied on his memory when bringing up names, episodes of people's lives, when describing events he witnessed or took a part in. The numbers of all those are huge. Martin Gardner lived an interesting life - he made it such.
For those who like me knew Martin Gardner from his books as a mathematics populariser, a pseudoscience debunker, a conjurer and a math magician, details of his life and personality exposed in the book help create a more complete picture of his fascinating person. It may surprise the reader that God and religion played an important role even in Gardner's adult life. So much so that he devotes the whole of a one page Preface to the Gods of the Old and New Testaments, mentions the question all throughout the book, explains his conversion in the "I lose my faith" chapter, and apportions the last chapter in its entirety to clarify his views. He ends his Preface with a paragraph that provides an inkling as to what they are:
The best-known remark of stand-up comedian Lenny Bruce was that people are leaving their churches and going back to God. What follows here is a rambling autobiography of one such person - me.
Gardner classified himself as a philosophical theist - a person who finds it comforting to believe in "a transcendent intelligence, impossible for us to understand ... somehow responsible for our universe and capable of providing, I do not know how, an afterlife."
Gardner wrote about H. G. Wells and G. K. Chesterton:
I have written elsewhere that if you can understand how I can admire both men, one a devout atheist, the other a devout Catholic, you can begin to understand my brand of theism.
But religiosity formed just one thread in Gardner's life and, consequently, the autobiography. In the book, Gardner emerges as an entertaining writer, well informed journalist, life's keen observer... very human, very open, very knowledgeable.
Martin Gardner had tremendous influence on several generations of young minds; his autobiography will help his fans appreciate how that came about. This is a book no one who ever heard his name would want to miss.
In the previous post I proved an identity in radicals:
which followed the method used in another post where another identity in radicals
has been derived. The latter post pointed to an earlier and perhaps a more exciting one where we established an unexpected identity
This was verified by simply taking the cube of the two sides of the equation. In a private correspondence related to the previous post Bruce Reznick suggested a different way of showing that
with rational and In Bruce Reznick's words:
So we look for that is, and Hard to solve directly, but we can multiply the first by and subtract times the second to get
which your friendly computer algebra system (otherwise useless) will tell you equals If then and so and and
Same with "". This method works quite generally.
And would not it? With the prior knowledge that the latter result seems not that surprising, one can think. Bruce continues to show with another example that expectations can be easily shuttered in mathematics.
Again, our previous experience may suggest the existence of rational and such that
If we proceed as before, but now squaring both sides of the equation, and equating rational and irrational components, we get and so
If then so This is not what we expected, but is at least true:
What about the other factor? If then so and This gives
Strangely, this reduces to exactly the result due to the first (and different) factor:
In terms of quadratic fields, , what we found may be expressed formally as but which, perhaps, helps enhance intuition of what goes on in Bruce's example but also adds to the mystery of or