CTK Insights

07 Oct

Review of "Zombies and Calculus" by Colin Adams

Colin Adams, author of the unique book "Zombies and Calculus", opens the book with a warning that "if you are squeamish you should not read the book." I venture an additional warning: if you lack a sense of humor, you should not read it either. As an afterthought, the author considers that, given the title of the book, there's little likelihood a squeamish person wold pick it up, in the first place. In my view, such thinking is a mistake - the first of two mistakes the author has committed. Many a student felt like a zombie in a calculus class. These might have shown interest in the book in the hope of finding an explanation to their experience. To these students I say, no, the book is about hard-core zombies - stiff-legged cannibals, entirely devoid of high mental functionality. The one thing on their mind is feeding on anyone yet alive and thinking. But, unlike in the movies featuring Milla Jovovich, in the book zombies have been observed to be tearing clothes off zombies of the opposite sex - in public and with an obvious intention. Unfortunately, the narrator - a college math professor - being followed by his former students and colleagues who now saw him only as a potential meal had not the time to further dwell on his observations.

One really needs a well developed sense of humor to read about a talented student who, with a chunk of her neck missing and the head at a weird angle, tries to grab her professor of a few minutes ago, and so had to be knocked down. On the other hand, every one would laugh at a delinquent student, oblivious of the surrounding dangers, who hands to the narrator (concerned with escaping a crowd of zombies in hot pursuit around the corner) his late homework. One again needs to summon one's sense of humor to read about the sad fate that befell the unthoughtful student short time after the encounter.

At the outset, I have mentioned that the author committed two mistakes. Here's the second one. Early in the story, several survivors of the initial attack, found themselves locked up in an office, with zombies moaning and banging into(?) each other just outside the door. Naturally there arose a question of active protection, and a girl removed her stockings to be filled with heavy objects that, when swung, could bring a zombie down. The narrator then cuts the stockings with scissors into two legs. However, as every grownup knows, stockings are already a two-piece item. Pantyhose is the one dress item that needs separation.

Stockings Pantyhose

The episode with the stockings serves to develop a conversation (and related mathematical tools) about speed, force, and the strength of human (i.e., zombie's) skull. As the story evolves, other mathematics comes in handy: equations of pursuit, logistic equation, predator-prey model, stationary point of a system of ODEs, Newton's Law of Cooling. Mathematics in the book is impeccable. However, from the incomplete list of topics, one may start suspecting (and justly) that the book should be more properly titled "Zombies and ODE" or "Zombies and ODE Modeling". (The author admits as much in the introduction by pointing to those who already learned calculus as his primary audience.) I do not count this as a third mistake, for this is a common marketing ploy to rely on a sound bite title to attract broader readership. And the book well deserves attention from not squeamish math instructors and a wider audience of intelligent readers, curious of a new literary genre that mixes storytelling with gentle mathematical instruction.

14 Sep

Review of Coffee, Love and Matrix Algebra

I am admittedly a compulsive reader. I either stop reading a book if I do not like it, or I continue reading until finished - with only obligatory interruptions. Gary Davis' book brought me an entirely new and tantalizing experience. Gary used daily tweets on twitter.com to announce new installation at his popular blog Republic of Mathematics (http://www.blog.republicofmath.com/). So there was no other way but to do the reading a chapter a day. Had it been my choice - and I can candidly say that after the fact - I would have gobbled the book in one setting.

The story is about a year long episode in a life of a college math department. Any one, I believe, who ever held a position in an academic department would easily identify the traits of Gary's protagonists as shared by some of their colleagues. The characters were authentic, evolution of events realistic; it took me a while to realize that the book was entirely a work of fiction.

Naturally, while there are similarities, not all math departments are the same; Gary's no different in this respect. It is painted with its own problems and peculiarities. Although a mathematics professor, Gary navigates his story with the skill of a professional writer. He narrates his story that takes several imaginative turns with confidence of a participant and kind humor of life's keen observer. That's a great story, masterfully and engagingly told. Read and enjoy.

01 Aug

Distance to the Horizon on the Fourth of July

I had the luck to celebrate the past 4th of July with our friends in their newly acquired home just above the marina in Atlantic Highlands, NJ. The view from their backyard was absolutely breathtaking. The ambient light that appeared to blur the background made the view even more enchanting.


Here is a map that would help you identify parts of the panorama.

NY-NJ bays

In the middle above the center line, across the Sandy Hook and Low bays the part of the Interstate 278 is the famous Verrazano Bride, past which there are visible Manhattan tall buildings; on the right, that's Brooklyn whose buildings seem taller, but only because of relative proximity.

The sunset was spectacular.

Sunset begins

From the beginning (above) and to the end (below).

NY-NJ bays

With the dusk, came fireworks. We were able to simultaneously see the bursts of illumination over Manhattan (that appeared just above the Verrazano bridge) and Brooklyn. (You can see these too if you click on the photo below.)

NY-NJ bays

And next came the fireworks just above our heads shot from the marina below.

NY-NJ bays

Next day I was downloading my 4th of July photos to my computer from the camera. In the zoomed-in version of the Verrazano bridge I noticed a short upward stick under the bridge which - to my great surprise - happened to be the Stature of Liberty.

NY-NJ bays

To inquire the distance to the stature has now appeared quite natural. This can be estimated with the following map.

NY-NJ bays

It is also possible to use the well known formulas for such estimates, I refer to the wikipedia. The distance is about 22 miles, Verrazano bridge being at about 2/3 of the way. Using the reversed formula it is possible to estimate how high above the marina have I been celebrating the 4th of July this year.

01 May

Making Escher Proud - in Dance

24 Apr

Environmental impact of power lines

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:

strangely vertical growth pattern

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:

The magnetic field induced by power lines causes the trees in their vicinity to sprout strictly vertically.

More research is necessary to support that conjecture.

17 Feb

Beating a Dead Horse: Mathematics Education Reform - Again

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.

29 Jan

Wizards, Aliens, and Starships

Wizards, Aliens, and Starships by Charles Adler

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.

He explains

... 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.

18 Jan

Kordemsky's Palindrome Problem

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:

Find a 10-digit number, with all digits distinct, whose quotient of division by 9 is a palindrome, i.e., a number that is read the same from both ends.

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:

Assume 10-digit n, with all digits distinct, is such that m=n:9 is a palindrome. Assume also that the 4th and 6th digits of m are both zero, while the fifth one is not 1. Then swapping the 5th and 6th digits in n gives another solution.

For example, 4059721386:9=451080154 and 4059271386:9=451030154, and another pair, 1503276849:9=167030761 and 1503726849:9=167080761. But this one 3921457806:9=435717534 and 3921547806:9=435727534 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.

It so happened that I have recently purchased a Raspberry Pi computer that came loaded with two versions of the programming language Python and full-pledged version of Wolfram's Mathematica.

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:

5871269304 6071359284 1653087429 1574086239
1643087529 5693087124 2037641598 7594086132
4015823796 7041269583 7051269483 7861359402

One of the two eight element sets:

7803456912 7803546912 7804365912 7804635912
7805364912 7805634912 7806453912 7806543912

And here's the only one with seven elements:

2893546107 2894356107 2894635107 2895364107
2895634107 2896453107 2896543107

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.

25 Dec

Math Associations on a Trip to Longwood Gardens, PA



A projectile / an arrow

A projectile

Circle packing

Circle packing

Ball packing

Ball packing



Hairy Ball Theorem

Hairy Ball Theorem

Two parabolas

Two parabolas



Up and Down

Up and down


Just watching this sunset was worth the trip


20 Dec

Beautiful Geometry

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.

Triangles on the same base and with the same altitude have equal areas

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

Sunrise over Miletus

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.

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