U.S. Math Education Needs a Page from Georg Cantor’s Book


Georg Cantor (1845 – 1918) was a German mathematician of international stature who, among other things, formalized set theory.  At first this theory was highly disputed and even Cantor’s mentor, Leopold Kronecker, became the main voice against him. Still, set theory prevailed and became foundational in the field of mathematics.

If you can remember back to the «New Math» introduced into elementary schools during the decades of the 60s and 70s, what made it «new» was largely Cantor’s set theory. It was new to the schools, the students, their parents and most of their teachers, but it had already been applied for approximately 100 years by mathematicians from around the world.

We are a strange lot, we Americans, and when something better comes along, if it is at all challenging or confusing, we reject it in favor of what we know. It happened with the metric system and it happened again with set theory.

After all, set theory was downright embarrassing. Most math teachers were not mathematicians and were woefully unskilled in the concepts of set theory. Some of them taught it as they interpreted it rather than as it really was. Parents were at a loss to help their young children with homework. Despite the emerging age of computer science, they helped to win the battle for regressing to traditional math like multiplication tables and long division by, say, a three-digit number, all things that could now be done more quickly and accurately by calculators. So within little more than a decade the new math had been supplanted by what looked and felt much more like traditional math, the «good old math».

Of course our children should still be taught to do the foregoing types of basic arithmetic problems but not to the extent that they become masters of computational techniques that they will never use again once they get their hands on a smart phone, calculator, tablet or computer. Sure, when I was in elementary school in the 1950s, if some real-life situation called for long division, it had to be done by hand. There was no other available choice. But we also had to write our communications out in long hand, put them in envelopes, and then stamp and mail them. Time changes all things. Today’s children need to understand math conceptually much more than computationally.

Back to set theory.

In his Book of Proof, R. Rhammack of Virginia Commonwealth University starts Chapter I with these words:

A ll of mathematics can be described with sets. [. . .] The theory of sets is a language that is perfectly suited to describing and explaining all types of mathematical structures.

In his post from 9/9/15 on phys.org, Opinion: «The Common Core is today’s New Math – which is actually a good thing», Kevin Knudson, a university physics professor said:

The New Math fell into disfavor mostly because of complaints from parents and teachers. Parents were unhappy because they couldn’t understand their children’s homework. Teachers objected because they were often unprepared to instruct their students in the new methods. In short, it was the implementation of these new concepts that led to the failure, more than the curriculum itself.

Read more at: http://phys.org/news/2015-09-opinion-common-core-today-math.html#jCp

Here are some basic symbols essential to set theory. If we don’t recognize or understand them, it’s okay. But our children need them and more.


The new math has been making a comeback in the Common Core, which itself is becoming more despised with each semester that passes. Some states have given up on it entirely.

Regardless of what we call the curriculum, we need to teach more conceptual math in elementary school. The brains of young children have been shown to be more receptive to new languages and concepts than are those of adolescents and adults. They are primed to handle math to the base 2 or 12, and to handle the relationships among sets which go to the heart of math theory. This kind of education while children are young will carry over into a better understanding of higher math when in due course they encounter it. If parents don’t understand it, well, that’s the way it goes; however, it is paramount that math teachers do.

Math and computer science are inextricably connected.If we continue to appear as country  number 35 on the world list of math achievement, we will have no choice but to keep granting work visas to foreign workers and teachers. Tech companies will continue to add operations in other countries where there are better equipped graduates to hire.

China has already hacked the Pentagon and the CIA and major US corporations, and yet the FBI didn’t have what it takes to hack an iPhone. Well, what can you really expect from No.35?



Commercialism on American Television is not just about Commercials

During the last half of 2000 the British produced a TV series in four episodes of just under one hour each called House of Cards. Browsing through Netflix, I came across it and decided to give it a shot. I loved it. Each episode was superbly acted by the lead, Ian Richardson; the story was tight, with no obvious flaws in logic or continuity throughout. Yes, the were some implausibilities, but what politically-based stories, non-fiction included, have ever been truly plausible. Just think Iran-Contra, Abu Ghraib, or the Clinton impeachment and acquittal.

The came the American version from the same book by Michael Hobbs, starring Kevin Spacey and the ageless Robin Wright. In the hand of producers and writers over here, what started out as a story somewhat similar to the British original soon began to cycle off on its own. In come the lobbyists, Chinese gamblers, side plots of alcoholism, Russian confrontations, murder (I believe four, but can’t swear by it), the President’s love for barbecue, ambassadorships for Ms. Wright, and many other distractions and glosses which seem mostly intended to extend the story.

This is where commercialism goes well beyond commercials. If a show is a hit, someone will undoubtedly pitch morphing a miniseries into a weekly soap opera that can go on for four or more seasons as House of Cards is doing. The alternative is to finish a good story where it should aptly end (good for the viewers) and then have to  schedule a new show in its slot that may flop before it catches on (bad for commercial TV).

Without trying too hard I believe we can all remember series that continued well past their welcome. I first felt this way about the series «Happy Days». It began as a true gem of original comedy but when it dragged on through its final three years it became a parody of itself. Henry Winkler’s Fonzie went for the charming protector and enforcer as necessary to acquiring unusual powers that could have been enough for him to apply for membership in the Justice League.

The thing is were so used to these practices that we come to accept them as they are instead of voting with a channel switch when enough is enough. If I have offended any American House of Cards fans, I’m sorry. I’m only expressing one guy’s opinion, which is what this medium was invented for.

If you are a big fan, maybe it would be a good time to follow the suggestion made by James Fallows in The Atlantic, issue of 02/12/2014:

Before You Watch the NewHouse of Cards, Do Yourself a Favor and See the Original. In which I (reluctantly) acknowledge the superiority of the British style of satire.

Let’s Make Math an Olympic Sport

What’s the most common American fan image after a victory by the home team? I’d say it’s this:


And very often, in sports, that’s what we are. Even in sports we don’t follow very much we want to see an American somewhere on the Olympic podium to receive a medal.

It feels good to win. It is good for our national and individual self-esteem and even for the bank accounts of professional players and gamblers who guess correctly. Losing isn’t good. We’re Americans so we win! But we lost?  . . .Our brains cant handle the dissonance. So we say, «It’s not a real sport (while still believing that golf is) or «It’s boring» or «The rules are stupid».  All of this thought and emotion being dedicated to activities that in the end don’t mean a thing.

Things that truly matter and affect our collective future don’t get nearly as much attention. Education is a perfect example. Yeah, domestically, schools are high on the list of essentials for parents with young and not so young children. Families move in to towns and neighborhoods only because «the schools are great.» University rankings trump cost and distance factors for families that have a little extra in the bank.

But what about international rankings? In which countries are students receiving the best education, especially in the areas of science, technology, engineering and mathematics (STEM)?

Looking at math for the answer, we find that we couldn’t show our ranking by using the fingers of both hands. So we take of our shoes and socks, adding toes to fingers. Nope, still no good. We must also add our four limbs, two ears and nose to get it right.

Twenty-six countries do better than we do. We’re number 27. ( Courtesy of PEW Research Center, Fact Tank, Feb, 2015).


This is downright scary! It shows why many good jobs are sent overseas and why so many foreign citizens get work visas to come here for jobs that our young grads just can’t do.

Jeez, who knows, this may even be more important than the Super Bowl, half-time show included.

In the next couple of posts I’ll try my best to see how we got here.












When Hope is Gone

This small volume of mostly poetry was published by the estate of Raymond Carver in 1989, one year after the author’s death. It was compiled by his wife, Tess Gallagher, also a poet and short story writer, who devoted herself to collaborating on what they both knew would be his final effort.


She also wrote the book’s introduction, and it was there, in a quote from Carver’s personal journal, that I found one of those magical lines of prose that make you stop reading to think long and hard about what you’ve just read.


When hope is gone, the ultimate sanity is to grasp at straws.

Sanity is an ephemeral state. It is circumstantial, cultural, temporal and not at all applicable in some universal way. Carver chose the sanity most comfortable for him and what he was facing then and there.

Some time afterward, he relaxed his stand and wrote the following well-known poem just before he died:

Late Fragment

And did you get what
you wanted from this life, even so?
I did.
And what did you want?
To call myself beloved, to feel myself
beloved on the earth.

-Raymond Carver, A New Path to the Waterfall

What at first appears to be an unimaginative and overly-simplistic list of what he wanted in his life, I could argue, after lots of thought, is really exhaustive.

The Serenity of Poverty


Colombian author Gabriel García Márquez, winner of the Nobel Prize for Literature in 1982, is most famous for his novel Cien años de soledad  (One Hundred Years of Solitude), perhaps the best single example of the magical realism that was at the heart of the Latin Boom in literature that took place during the last half of the Twentieth Century.

Five years before this novel was published in 1967, he published a book of eight shorts stories called Los funerales de la mamá grande  (Big Mama’s Funeral). The third story in this collection is La siesta del martes  (Tuesday Siesta), a seemingly simple story of a poor mother who takes her young daughter by train to a town where her son had already been killed and buried. She needed to find more meaning, more substance to make sense of her mourning.

García Marquéz spends little time before showing us her extreme poverty; it is at the heart of the story. They travel in a third class coach situated so the smoke from the engine will pass through its windows. He tells us of the mother:

«Viajaba con la columna vertebral firmemente apoyada contra el espaldar del asiento, sosteniendo en el regazo con ambas manos una cartera de charol desconchado. Tenía la serenidad escrupulosa de la gente acustombrada a la pobreza.»


«She was riding with her spinal column braced firmly against the back of the seat, and held a peeling patent-leather handbag in her lap with both hands. She bore the conscientious serenity of someone accustomed to poverty.»

I enjoy page-turners, but the main reason I read fiction is to find those rare authors who can make me stop abruptly because some sentence or phrase is just too good to pass by on the fly. In my opinion, García Márquez is one of the best at this. It’s a call to stop and think, forcing you to confront yourself and take sides sometimes.

«She bore the conscientious serenity of someone accustomed to poverty.»

I lived in Colombia, mostly Bogotá, for more than ten years. I met the poor and saw this serenity often while with them. Serenity without food but always with a cup of coffee to offer a visitor. Worn clothes but freshly creased by an iron (often borrowed). I marveled at it. Humor in conversations, hope for better days, and trust in God, always.

If you’re familiar with the Bible you may have come across the scene when Jesus, after his resurrection, enters the locked upper room and greets his frightened disciples. He gives them a gift, the gift of his peace. Why peace? I used to think. Why not strength, health, happiness, perseverance, prosperity or other attributes? We, as Americans, are guaranteed «life, liberty and the pursuit of happiness». What happened to peace?

This story helps me find the answer; no government can grant us peace, and yet peace is the only inner feeling that can get us through any adversity. García Márquez tells us the mother bore conscientious serenity, conscientious peace. She herself was aware of it, and she bore it proudly.

García Márquez’ novels and stories are filled with these lines that pack a punch. They always make you stop and think; sometimes leaving you angry, laughing, marveling, crying, or maybe even serene.

Short Stories That Pack a Novel’s Punch


Raymond Carver (1938 – 1988) died too young, as did Roberto Bolaño, whom I wrote about in Spanish a few days ago. Carver’s personal demon was alcohol and, though he tried, he just couldn’t lay off of it until he quit, cold turkey, in 1978. His death ten years later was attributed to lung cancer, so tobacco trumped alcohol in the end.

Besides short stories, he also wrote many poems, and I suspect he might have even preferred to be remembered for his poetry over and above his prose. But just as it is with nearly every life lost, those of us left behind are the ones to determine who and what our memories will carry forward.

Carver’s reputation and status among other writers of fiction is not something that other authors and critics all agree upon. He has been compared favorably to Hemingway, he has been judged unjustly because his original stories received  heavy editing to reduce word volume by Gordon Lish, his manager, and he he has been criticized for adopting the minimalist model only to use it to hide his laziness.

I have become enthralled by some of Carver’s short stories, and today, I would like to single out «So Much Water So Close to Home». If you become interested, be sure to look for this story in the recent edition of Carver’s works, Beginnings, as he wrote them, without any of Lish’s subsequent editing. Personally, I couldn’t expect more from any writer’s short story.

It starts out harmlessly enough when Stuart and three of his pokers and bowling buddies head off on one of their twice a year fishing trips to a distant lake. There they get caught up in a moral dilemma that is easy to argue from either side, but, in my mind, very difficult to judge impartially. It made me do more soul searching than many novels have done.

I won’t say more than this so as not to be a spoiler. As my final comment I will say that Carver’s genius really shows up when he decided to write the story in the first person, from his wife’s point of view. Give it a fair read and maybe you’ll agree.


Compulsivamente Rompiendo Moldes




El poeta y novelista chileno Roberto Bolaño Ávalos transformó la literatura escrita en español, moviéndola del realismo mágico del boom hispanoamericano al infrarrealismo , movimiento que él mismo denominó real vanguardismo

Denominada como se denomine, la escritura de Bolaño, la cual descubrí al azar, es magnífica.  Sí, puede ser pesada mientras el autor deambula por sus mundos etéreos sin advertirnos su próximo paso. Sin embargo, al lector captado por su prosa infundida ricamente de poesía, no le importan los sucesos inverosímiles o los tiempos que andan en forma de espiral. A mí me deja muchas veces con admiración, ojalá yo pudiera expresarme así. Desearía que pudiese haber dicho esto, por ejemplo:

Uno tiene que conocer gente de todas las clases, a veces es necesario empaparse de realidad.     -Nocturno de Chile, Roberto Bolaño

La escritura de Bolaño te secuestra alegremente en un ámbito fantasmagórico en que, por ejemplo,  un sacerdote del Opus Dei, también crítico de obras literarias, y después de una vuelta por Europa regresa a Chile para enseñalarle al mismo Pinochet los pormenores del marxismo (Nocturno de Chile). Tales cosas extrañas no desconciertan de ninguna manera. Una vez metido en el agumento de cualquiera de sus novelas, Bolaño me hace creyente, rendido al poder de su prosa e imaginación.

Maquetación 1

Bolaño tuvo una vida plagada de contratiempos y enfermedades. En 1968, junto con su familia, se trasladó a la Ciudad de México donde comenzó a escribir artículos para varios medios. Dos años más tarde decidió  regresar a Chile, días después del golpe de estado, y se incorporó a la resistencia. Lo arrestaron y pasó 8 días entre rejas. En cuanto pudo, volvió a México para dedicarse por completo a la escritura.  Se le atravesaron muchas cosas: no tenía documentos, y muchas veces le tocó trabajar en oficios como basurero, descargador de barcos, vigilante nocturno y lavaplatos, entre otros.

En 1993 fue diagnosticado con una condición hepática bastante grave. En lugar de tirar la toalla y prepararse para su eventual muerte, se dedicó aun más a la esctitura. Antes de este diagnóstico Bolaño había publicado dos novelas que no lograron mostrar los mejores de sus talentos. Pero a estas alturas se obsesionó con dejar un gran legado literario,sin dudar en ningún momento que fuera capaz de hacerlo. Consiguió un puesto en la lista de espera para recibir un nuevo hígado, pero no tuvo suerte y falleció el 14 de julio de 2003 antes de que llegara su turno.

Durante esa década desde el diagnóstico en 1993 y su muerte en 2003, escribía cada vez más compulsivamente para lograr su meta autoimpuesta. Gracias a su entrega total mientras se le acercaba la muerte, somos beneficiarios de un cuerpo de literatura invaluable. Gracias,  Roberto Bolaño Ávalos, y que en paz descanse.

If God is not in the Details, Where Next Should We Look

There is a famous saying, often attributed to Ludwig  Mies van der Rohe, the German-born architect who established his fame in Chicago, that «God is in the Details.» Others in the know claim it was Gustave Flaubert, renowned author of Madame Bovary and other novels, who said it. Personally, I love this uncertainty as to whom the quote should be attributed; it links together a literary giant with a architectural giant, and the latter’s work would have been impossible without the proper application of mathematics.  It is a joyful coincidence that literature and math are the two major themes of this blog.

I have a different slant on where God can be found, and I couldn’t be more serious when I say, for me, that God is in the vectors.

Let me explain.

My first exposure to linear algebra (LA) was in the early 1960s as a junior in high school. LA was an oddity back then, something that was useful if you needed a simultaneous solution to a given small number of equations with an equal number of unknowns. The trouble was, without the availability of computer power, once you got past systems of four or five such equations and unknowns, the machinations and iterations that were needed to be done manually became overwhelming. They also taught us about matrix determinants because if you calculate a determinant of a given matrix as zero it tells you immediately that there is no simultaneous solution for the unknowns of the given matrix.


Above is a typical example of a four-by-four matrix: four columns and four rows. Each column of numbers is called a column vector, and each row, a row vector. All of the sixteen numbers are the coefficients of the unknowns we hope to identify.  Although not shown in this example, the real (or complex) numbers used can also be negative and/or include zero.

One of the wonders of LA is that no matter how big the matrix is or how long the vectors are that make it up, there are only three possibilities for its solutions: one solution; no solution; or an infinite number of solutions. This has to sink in for a while; whether the matrix is three by three, like one of the nine little boxes in a Sudoku game, or 5 million by 5 million, the potential solutions remain the same: one; none; or an infinite number. Somehow, this leads me to contemplate alpha, omega, and everything-in-between.

Now, on to truly large matrices. Google has reportedly worked with matrices having upwards of one billion column vectors. Netflix is close to joining them in this mathematical exosphere.  I cannot picture one billion of anything in my minds eye, so let’s bring the number down to a matrix of 100 rows and columns for ease of illustration. Then, if we orthogonalize this matrix, we make the length of each vector = 1 and each of the one hundred vectors perpendicular (orthogonal) to the other 99. How can this be? How on God’s green earth can we end up with one hundred vectors with each one perpendicular to each of the others? We’re talking one hundred dimensions here, dimensions well beyond our ken, and yet these and much larger matrices are made to work every minute of every day to solve real word problems.

We can easily imagine 1, 2, and 3 dimensions: a point, a plane, a box or sphere, respectively. Einstein and others suggested that we could add a fourth, which would be time. When I plan to meet someone, it’s not enough to say, for example, «meet me on the 45th floor of the Hancock building.» GPS will get me to the correct latitude and longitude, and the elevators or stairway, to the height, but without the element of time we may never meet. Let’s choose to leave this last conjecture to the physicists and get on with physical dimensions.

There have been various attempts to drawn spaces of four dimensions and upward, but in my opinion they all fail. We just do not seem capable of visualizing physical spaces of more than three dimensions. An example of a three-dimensional space is easy to construct; three dowel rods, all at right angles to themselves, and we’re done. But now I give you a fourth rod, a fifth, a sixth. What can you show me now? The answer is nothing more that you showed me with the first three. Is it because we live in a three-dimensional world that our brains top out at three dimensions too? Or did our parents forget to select the nth-dimensional upgrade before putting their new-offspring order into the pipeline?

Whatever the reason, our brains seem to grasp three physical dimensions and no more. Yet Google does work in, give or take, one billion dimensions and the math works.  If this seems out of reach, we can take it down to matrices of ten dimensions. And, yes, the information we derive from them results in a valid solutions to the problems at hand. We get lost in the vectors, but since the math works despite our shortcomings, it must be God who reigns over them. God is in the vectors. 

These vectors humble us, just as they should. We are intelligent; we are the only being capable of altering the conditions of the world we live in and even of destroying it. Almost daily, technological discoveries, scientific breakthroughs, proofs of long-standing math conjectures or hypotheses and other achievements become responsible for our ever-swelling heads as a species.  And yet we are lost in other things with little hope of finding our way out. Can we ever understand «infinity» (the infinitely large and the infinitely small) or «eternity» (of course, first we need to agree that time really exists).

When all is said and done, though, I posit that we will never understand the secrets of large matrices and their vectors that arrange themselves in unfathomable arrays. I like to think of it as God’s Last Stand: the line in the sand separating humanity from divinity for ever. It is there to confirm the shout out to us from Psalm 46: «Be still and know that I am God.»

Postscript:  While linear algebra and some other math concepts are used here, this blog was never intended to be a math lesson. My goal was to give just enough information to make my theological point more easily understood by the reader. Those with a good understanding of math may say I left out certain operations and concepts. «Orthogonalize»  is perhaps the word most easily picked on. I chose not to explain it and instead decided to let it be taken for granted.The long explanation required would have detracted from my main theme. Did I succeed; did I fail. I’ll  leave that up to the reader. Finally, the classroom illustration used at the top of this blog shows Gilbert Strang, Professor of Mathematics at the Massachusetts Institute of Technology, whose books on linear algebra are generally considered the best.