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Several years ago, I almost got into a fistfight with a friend while attending a speech by Malcolm Gladwell at the NCTM annual conference in Salt Lake City. What began this near melee was Gladwell’s assertion that Pablo Picasso’s innovations somehow “peaked” in his youth, while Cezanne was the example of someone who gradually refined his work until it achieved the excellence it was remembered for. I took exception to this assertion, mostly because the description of Picasso’s career is completely untrue: Picasso’s work was innovational through the many decades that he worked, and he continued to innovate up to the end of his life. The narrative Gladwell had created was based on the auction value of each artist’s works: apparently, early Picassos were more value than later ones, therefore his work early in his career was more innovative.

Unfortunately, this correlation that Gladwell created was a complete crock: the value of an artist’s work is based on a lot of complex variables, including the availability of the pieces from that period (I wouldn’t be surprised to find that PIcasso’s early work is scarcer than his later work), the buyers’ tastes (believe it or not, different styles of art go in and out of fashion like everything else, affecting their auction prices) and, last on the list, whether it was an example of the artist “at his best.” Clearly, Gladwell was out of his depth when it came to valuing artwork. If the facts don’t fit your story, don’t change the story; alter the facts.

Of course, this kind of irresponsibility is rampant among journalists of all stripes; our latest example is by Annie Murphy Paul, who has written yet another book on learning called “Brilliance: The Science of Smart.” I’m not going to criticize something I have not (and will not) read, but I did see Paul’s (or is it Murphy Paul?) latest column on the Time website to know that when something looks and smells like BS, it most likely is. Let’s read one of her choice quotes:

“Indeed, many experts who have observed the wide gap between the math scores of American and Chinese students on international tests attribute the Asian students’ advantage to their  school’s relentless focus on memorizing math facts. Failure to do so can effectively close off the higher realms of mathematics: A study published in the journal Math Cognition found that most errors made by students working on complex math problems were due to a lack of automaticity in basic math facts.”

Where to begin? How about with the relentless use of the word “many,” which is used “many” times. “Many experts” are cited in the opening sentence – well, how many is it? Who are these “experts?” Did these “experts” really lay all the blame on the differences between Chinese and American students’ achievement in mathematics on math facts? And why is there no link to the article in Math Cognition, a journal which even I, a class A nerd of all nerds, has never heard of? Perhaps because the study was from, wait for it, 1999? Maybe it was the fact that the students were not actually working on complex math problems at all, but multi-digit addition problems? Maybe because the study looked at 3rd through 6th grade, and made no mention of how their data compared to Chinese students? This is a clear cut case of cherry picking data to fit a narrative. How Time allowed Murphy Paul to publish this packet of prevaricated nonsense is a mystery to me.

I love visiting classrooms as part of my work as a teacher coach and curriculum designer, and one of the great perks is sitting down with students and asking questions while they are working on an assignment. Today was one such day: I was observing a 5th grade class where the teacher was concluding a unit on statistics, and the students were busy making corrections on a quiz they had taken the previous day.

I placed myself next to Julian, a young man whom I had watched for the previous 2 years while he was in the 3rd and 4th grade. Julian tended towards distraction, and on many occasion when I saw him drifting off into free flight, I would tap him on the shoulder as I strolled by, turn towards him and use the Robert DeNiro “I’ve got my eyes on you” signal from “Meet the Parents.” Julian smiled each time I used this little trope, and it became part of our bond.

I looked over Julian’s quiz, made some suggestions for corrections, and then asked him a few questions about the meaning of mean, median, range and mode. My questions moved into the hypothetical “meta” realm, where I asked things like “what if I only knew the median for this data; what would I know? What wouldn’t I know? We chatted for about 15 minutes, during which time I developed a new lesson idea: suppose you had the statistical information about 5 people, including the mean, mode, range and median. What kind of data could you assign to each person to get this set of statistics? Could someone create a completely different set of data which would yield the same set of statistics?

As I stumbled out of school at the end of the day, I was stopped by Julian’s mother. “I just wanted to thank you for the time you spent with Julian today,” she said, “he really appreciated the time you took to explain things to him.” I told her the truth, which was that I hadn’t “explained” much of anything, and that we were just having a little conversation. Nonetheless, she thanked me again.

All of which got me thinking: what makes “teaching” teaching? In between checking the homework, explaining the assignments, creating and grading the exams and all the other elements of teaching mathematics, what it all comes down to is those moments of intellectual intimacy, when two minds co-mingle and understanding emerges.

I don’t know if Singapore Math is any more “effective” than TERC or Chicago Math. I have my doubts that Khan Academy is going to have any staying power, and I am not yet convinced that technology will make kids any better at mathematics. What I do know is that the 5,000 years of history has shown that the most effective form of teaching takes place when someone taking an interest in a child, a conversation takes place, and eventually, understanding develops. It is the intellectual version of the “I and Thou” relationship about which Martin Buber has written. It is what we have all experienced at one time or another, and even though I am not a strong believer in the existence of a supreme being, when those moments take place, God is in the house.

As my loyal readers have probably caught on to by now, I am a die-hard skeptic of any curriculum that seeks to make math “fun and easy.” Actually, I have no problem with the fun part (although it is debatable what constitutes “fun” in mathematics, especially to a 13 year old), but anyone whose intention is to make math “easy” is beneath my contempt. By its very nature, mathematics is difficult and  frustrating, and anybody, ANYBODY who attempts to convince me otherwise is probably not actually doing or teaching mathematics.

Let’s look at just a few things that makes mathematics fundamentally annoying:

Ambiguity and Inconsistency: Those who are ignorant of mathematics claim that mathematics is inherently consistent and rule driven. These are the same people who claim to have an inability to fathom mathematics because they consider themselves “creative,” “people oriented” and “rule breakers.”  Mathematics is, however, by its very nature ambiguous, and it has a long history of creative minds that piled on idea after idea to construct this towering edifice of inconsistency and seemingly improbable ideas (really, there is a square root of -12? Are you kidding me?)  These ambiguities and inconsistencies surface over and over again in the field of mathematics, and anybody who isn’t willing to admit to and confront them is doomed to failure.

Lack of Precision: Yes, there is only a single correct answer to the problem 8 x 7 (according to a story by Robert Kaplan, it is not 53 or 57.) However, 8 x 7 is not mathematics, but its important  and superficial sibling known as “arithmetic.” A real mathematics problem asks us to consider whether there are more points between 0 and 1 or -? and +? on the number line (they’re actually the same, but that’s another matter.) Furthermore, much of the actual mathematics we use is probabilistic, whether it is involves predicting the value of my IRA when I retire to determining what the number of number of hours I’ll need to master the bassoon part on Ravel’s Bolero. Mathematics is not infuriating because it is so precise; rather, it offers up precision where none is available, and even when invoked, its precision can be startlingly imprecise.

Unclear and Inconsistent Methodology: Yes, there are standard algorithms for computing everything from the quotient of two multi-digit numbers (better known as “long division”) to calculating the sum of an infinite series. But this does not fall into the purview of actual mathematics; it is merely an advanced form of arithmetic. True mathematics involves interpreting or inventing procedures that apply not only to particular cases, but also varying those procedures so that they can be streamlined or made “elegant.” There is no “correct” way to perform “long division;” however, there are some ways that are more efficient, transparent and elegant.

Many years ago there was an outcry over a talking Barbie doll which, among her repertoire of mindless statements, rendered an opinion that “Algebra is hard!” Lost in the outcry over of a polystyrene doll uttering such a remark was the actual veracity of her statement. Algebra is hard, and so is much of real mathematics. Let’s just admit it and move on. Let’s remember, however, that just because something is hard, doesn’t mean it isn’t worth doing.

Why does it take a former political science professor to tell us what is patently obvious in the field of mathematics education? Andrew Hacker, a professor emeritus at Queens College and co-author of the “Higher Education? How Colleges Are Wasting Our Money and Failing Our Kids — and What We Can Do About It,” tells us in a front page article in the Sunday New York Times that we should jettison our current high school math sequence in its entirety. In an article entitled “Is Algebra Necessary?” Hacker argues that forcing all students to march lockstep through the algebra – geometry – trigonometry tunnel is a waste of time and results in many students leaving school dispirited and disinterested. To me this is not a terribly daring position, and anybody who would disagree with Hacker’s proposition probably belongs to the faction that believes that “tradition” consists of subjecting our children to the same mind-numbing educational system that we and our elders suffered through because “it was good enough for us, so it must be good enough for them.” Judging from the negative responses, you might have thought that Hacker was demanding the downfall of the Occident, for the consensus seemed to be that we should fix things by getting better teachers (or better students, as I often quip.) I myself have taught algebra for many years, and even though I attempted to make a go of it, I can only agree with his assessment.

To begin at the beginning, let’s answer Hacker’s question, “Is Algebra Necessary?” with another question: “Is Cubism necessary? Are Shakespearean sonnets necessary? Is avant garde jazz necessary?” No, Virginia, none of these are necessary, and when it comes right down to it, neither is algebra. But that doesn’t mean any of them are irrelevant or unimportant.

But before I begin arguing the point, I think it is necessary to distinguish between “algebra” the noun, and “algebra” the verb. Algebra the noun consists of the symbols written down in textbooks, is riddled with lots of Xs and Ys (I once considered applying for the right to copyright the use of X and Y in algebra texts, in the hope I would get rich by receiving 1/100^th cent for every time either one appeared), and involves lots of “rules” that came down from some central authority, the most egregious of which is “FOIL.” This is the “algebra” that Hacker describes and that the majority of us who have attempted to learn and teach now decry.

If we understand “algebra” as a verb, then it could be very useful to most people. Algebra the verb, which I’ll refer to heretofore as “algebraic thinking,” represents a non-inconsequential step in human understanding. For what is algebraic thinking but the transition away from “case based” arithmetic, and into a generalized form which can be applied to any instance? Algebraic reasoning allows us to do such things as identify patterns and relationships (which can be compressed using symbols, but I’ll leave that to the textbook writers) that can be applied to a variety of situations.

For example, arithmetic tells me if I purchase 5 pounds of bananas for 60¢ per pound, I know the total price will come out to be $3.00. Algebraic thinking, on the other hand, tells me that if I multiply the number of items by the price per item, I will always get the total price, regardless of whether the items are bananas, magic beans or credit default swaps. The power of algebraic thinking is that I don’t have to go through the trouble of developing a new method each time I want to solve a certain type of problem: much like the venerable Swiss Army Knife, one tool, better known as an algebraic equation, can handle it all.

Algebraic thinking also gives me access to tools that can help understand the past, present and/or future through the use of patterns that can be recorded and generalized. For example, if I found myself afloat in a leaky rowboat and noticed that the water was rising at the rate of 2” every hour, it is safe to assume that after 8 hours, my boat will have taken on 16” of water, which could be especially disturbing if the boat is, in fact, only 12” deep. Algebraic thinking gives me the power to work out a variety of scenarios: if my boat has already taken on 10” of water, I might want to consider bailing very quickly, while if it was only at the 3” level, I would have a bit of time to play a few hands of Solitaire before taking action.

Finally, having access to algebraic thinking puts me in better control of my life and can directly influence how I make daily decisions. As I write this, I am sitting on line with 300 other people to see “Shakespeare in the Park,” an annual ritual which involves queuing up at 8 am to wait until tickets are distributed some 5 hours later. By my last count, there are 300 people in line ahead of me, some, but not all of who will get 2 tickets apiece. The theatre holds about 1800 people and I just learned that 500 of those tickets are going to be distributed to residents of Brooklyn. Using algebraic thinking, I can put together a reasonable prediction that staying on line for the next 2 hours would be a reasonable course of action. (As it happened, I did end up getting a voucher which would entitle me to get two tickets shortly before the show begins; what time I should arrive to do this is yet another algebraic problem.)

So what are we to make of Hackman and his call to eliminate algebra the noun from the high school curriculum? Of course, some of Heckman’s assertions are debatable; when he cites that “most of the educators I’ve talked with cite algebra as the major academic reason” for students dropping out of high school, he is clearly confounding personal anecdote with actual data. On the other hand, as one of those teachers who has himself observed students suffer through factoring polynomials without an actual explanation of why this may be relevant to anything they might pursue later in life, I’ve come up with bupkis over and over again. To this I would wish a fond farewell to algebra the noun: nobody is likely to miss you.

A teacher asked me to bring pentominoes to her class, for they were reading a book that involved a character who walked around with pentominoes in his pocket. As one of the students was working on arranging the 12 pentominoes into the shape of a rectangle, he sighed under his breath and commented, “man, I really stink at this….”

Whenever students tell me they are not good at something, I immediately remind them that while a lot of mathematics is tricky and challenging, it can also be mastered through practice. This has led me to change my language: I never tell them they don’t understand something; it is always, “you don’t understand this yet.” I find the power of “yet” to be immediate and optimistic: it conveys the idea that they will eventually understand, and just because it is not immediate does not mean it will never arrive.

To this student, I replied, “You just started working on this like, what, 5 minutes ago? How can you tell whether you’re no good at this?”

The student shook his shoulders, and his frustration continued.

“I am really no good at this!” he complained.

“You can’t tell yet whether you’re not good at this. Give it a lot more time.”

“How long?”

“Well, to really know if you’re not good at something takes a long time. Usually 10 or 20 years…”

“What? How can that be?”

“Because, the real test of whether you’re no good at something is to work at it for years and years and to see no improvement. That’s the only way to know that you’re no good at something.”

“So…?”

“Basically, you’re too young to stink at something. You’ll need to spend a lot more time on this to really know if you’re no good at it.”