In a recent guest column [update: error fixed] in the New York Times, a professor asks "do we need algebra?" The short answer is "hell, yes." A slightly longer answer is "yes, ..." followed with a long stream of Anglo Saxon vulgarities about the professor and the New York Times. The long answer is this diary.
Laymen's terms thus far:
1. A primer on the smart grid for laymen.
2. In laymen's terms: the 100 year flood.
3. A look at Flood Abatement.
Some months ago I started writing the Laymen's Terms series to try to make various matters accessible to non-technical people, matters relating to the environment, the nationâ's infrastructure, and our predicament in regards to energy. I have several more half written pieces awaiting completion. I was motivated by the growing gap between public understanding of these issues and what is readily apparent to engineers, and wanted to try bridging it.
The name "laymen's terms" for the series because the idioms "for dummies" and "complete idiot's guide" make me ill. (The use of plural for laymen is wrong, but makes it easier to track the articles on Google.) The older term, layman, simply means someone who has not taken the time to study a field or subject, and that is perfectly respectable. There is a good reason we pay people to take on work that requires in depth study of a subject. And it's for good reason that we pay other people, journalists mostly, to report on technical subjects and bring them into layman's terms. But, we have two different publishing outfits that present all sorts of topics with these epithets, implying that their audience is completely incapable of ever learning about things in depth. (Then again, books like "The Complete Idiot's Guide to Indigo Children", they have got to be all too accurately titled.) Or they imply that their audience has no obligation to learn beyond a superficial level.
The Dummy/Idiot books are enormously stripped down of complex subject matter and vocabulary. Other writers are also under strong pressure to do so, and I also have to guess how basic I have to make things in order to make this series of essays an effective one. And that is because the base of knowledge that defines layman's terms here in the United States is also enormously stripped down. It's unfortunate, but a fact we all have to deal with, and also struggle against, when there are forces at work trying to set the bar even lower.
This week Professor Andrew Hacker, of Queens College CUNY, decided to enlighten us about why he feels that it is time for schools to drop the emphasis on algebra in the high school years. I've never seen someone provoke so much fury among my circle of friends, not so much for the article but for the New York Times' decision to publish it. Professor Hacker's first argument against algebra is the large rate of students failing it at every level. In his words, "Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white." True enough. It is. For many reasons, some of which I'll mention at the end. He goes on:
"It's true that students in Finland, South Korea and Canada score better on mathematics tests. But it's their perseverance, not their classroom algebra, that fits them for demanding jobs." He of course gives no evidence for that assertion, because, to explain in layman's terms, it's a crock. Classroom algebra is what allows someone to learn difficult subjects in the sciences, in math, in engineering, in computing, without the need for perseverance. Other countries have more robust and predominant high-skill economies precisely because they do better at hammering algebra into their kids.
And then Professor Hacker ventures into fighting words: "Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job." I would not allow this kind of nonsense to be said in my presence. Anyone who said that in earshot would be treated to a chewing out he would not forget, because the algebra he is arguing against is IDENTICAL to quantitative reasoning. And literally so.
The word Algebra comes from the treatise Al-KitÄb al-mukhtaá¹£ar fÄ« hÄ«sÄb al-Äabrw waâl-muqÄbala (notice the al-gabrwâ?), meaning The Compendious Book on Calculation by Completion and Balancing, written by Muhammad al-Khwarizmi, a Persian mathematician living in Baghdad in the 9th Century. The book is the first detailed treatise about how to reason about the relationships between quantities while leaving them unspecified. That is, for example, how to take the relationship between [My Salary], [My Tax Rate], and [My Tax Bill], and then apply that same relationship to [Your Salary], [Your Tax Rate], and [Your Tax Bill], without specifying what they are.
Al Khwarizmi expands in his book in order to manipulate these relationships in order to get a definition of one quantity in relation to others, and how to reverse that same manipulation in order to get the definition of a different quantity in relation to the others (in other words, how to solve equations.) But Al Khwarizmi proves himself to be a genius, a giant worthy of being remembered and praised 1200 years later, because he does this entirely in the form of prose. His treatise on algebra doesn't contain more knowledge than my 7th grade algebra book, but the word count is a lot higher, and unlike my algebra textbook, I have yet to read it and don't plan to. That is because al-Khwarizmi had to work without the benefit of mathematical notation. Most of it had yet to be developed, and in fact al-Khwarizmi is responsible for the arrival of decimal number notation in the Muslim world and its propagation to Europe.
One of the earliest instances in the development of mathematical notations came with Robert Recorde, 600 years later, when he wrote the following in his book The Whetstone of Witte:
...to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle.
...to avoid the tedious repetition of these words: "is equal to", I will set (as I do often in work use) a pair of parallels (or Gemowe lines) of one length (thus =), because no two things can be more equal.
(Notice also this paragraph predates our conventions on quotation marks.) With such modest beginnings in 1558, mathematicians could take paragraphs like this, from al-Khwarizmi:
"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts."[15]
And turn them into this:
[images]
In modern notation this process, with 'x' the "thing" (shay') or "root", is given by the steps,
Let the roots of the equation be 'p' and 'q'. Then , and
So a root is given by
This is the algebra you learn in school. You learn it this way because centuries of experience show that this kind of notation is more readable and usable than what al-Khwarizmi had to use. And you learn it in a mechanistic fashion, that is, how to move variables from one side of an equation to another in order to solve for one variable. Then you're assigned to do it repetitively, dozens of times, with mostly the same letters assigned to the variables, and with no context provided. That is the whole point. Once you've developed the ability to perform algebraic operations regardless of the context, without fully engaging your mind, you''re ready to employ quantitative reasoning to new contexts, and engage your mind into the context, and not the algebra beneath.
Those contexts spread far and wide. Naturally, physics requires it. And there's a reason why so many computer languages were contrived to resemble algebraic notation, starting with Fortran ("Formula Translator") and continuing with most of the introductory computer languages (e.g. Pascal). The reason was that doing so would make computers instantly accessible to anyone with a grasp of algebra. This stuff opens doors. And we are churning out high school graduates and high school dropouts who cannot do this. I'll enlist one of my friends to describe the consequences, from her vantage point:
I really hope algebra continues to get taught in schools. My job would be a lot harder without it. For the folks who aren't [mr and mrs ocschwar] : I'm a college financial aid administrator. Much of what I do involves counseling students about how much money they owe the school and how to pay it. "Right now, you owe [your costs] minus [your financial aid]. If you want to add a gym membership, you'll owe [your costs plus your gym membership] minus [your financial aid]." And so on. This level of abstract thinking is a piece of cake for kids who've done even the most basic of algebra; they've gotten their brain around the idea that it's possible to do math using variables rather than actual numbers. I can always spot the kids who didn't take algebra, or failed it, because they need me to re-do all the math for them every time their costs or their financial aid changes--it's like it's an entirely new math problem for them.
So, on with the rationalizations for leaving students hamstrung like this:
"Toyota, for example, recently chose to locate a plant in a remote Mississippi county, even though its schools are far from stellar. It works with a nearby community college, which has tailored classes in âmachine tool mathematics."
"That sort of collaboration has long undergirded German apprenticeship programs."
So a remedial class that is tailored to do no more than teach what some factory workers need for a particular factory is the answer to failing algebra. Instead of high schools that teach algebra well enough to enable their graduates to apply their knowledge to the shop floor in the Toyota plant, we'll just train them in detail for the Toyota context. And if Toyota retools, they will get fired. How lovely. And of course Professor Hacker is wrong about Germany. No worker comes near the lathe or the milling machines at the BMW plant if he can't do algebra. German apprenticeship programs cover many trades, from truck drivers, cooks, chimney sweeps onwards. But the German high tech economy does not hire the non-algebraic.
"What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? Itâs true that mathematics requires mental exertion. But thereâs no evidence that being able to prove (x² + y²)² = (x² - y²)² + (2xy)² leads to more credible political opinions or social analysis."
Oh, actually, it does. Having taken entirely too long a gaze into the abyss that is Holocaust denial and 9/11 kookery over the years I can find specific cases of people falling for arguments that fall apart completely with only the most basic algebra. Lack of skill in algebra makes people fall for snake oil.
And then he drops this whopper:
"Many of those who struggled through a traditional math regimen feel that doing so annealed their character. This may or may not speak to the fact that institutions and occupations often install prerequisites just to look rigorous -- hardly a rational justification for maintaining so many mathematics mandates. Certification programs for veterinary technicians require algebra, although none of the graduates I've met have ever used it in diagnosing or treating their patients. Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn't figure in the clinical curriculum, let alone in subsequent practice. Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a professionâs status."
And I just have to laugh. Radiologists cannot do their job without knowing the mathematics behind the CAT scanner and MRI machine. Ophthalmologists have to know optics, a field of physics that employs particularly hard amounts of calculus. And all doctors have to calculate drug dosages. And there is another reason why medical schools want their incoming students to have learned calculus, as an old joke goes:
It's a lazy afternoon in a college classroom. The professor is at the blackboard, droning on about the proper method for finding a particularly difficult integral, and midway through the next equation, a premed student loses his temper, rises off his seat and asks "Professor, I am going to medical school. Why on earth am I here? Why do I have to study this stuff?" The professor replies "young man, calculus is very important in medicine. It saves lives every day." The student asks "how does it possibly do that?" And the professor answers: "by keeping certain people out of medical school."
Which is to say, some professions come with profound responsibilities. And a person, if he lacks the wherewithal to study a difficult subject and not ask "why do I have to?", he should aspire to a different line of work. Grinding through difficult subject matter, applying a mental method repetitively to a piece of paper, these are forms of scut-work no different from practicing a musical instrument, practicing an athletic skill, or a martial arts skill, or conjugating the same French verb 222 damned times. Our trouble with algebra comes in large part because in this one case, that of math, we lack the courage to tell young people "yes, this is boring, repetitive, and devoid of context, and yes, you have to do it."
You have to do it because there is no meaningful distinction between the "quantitative reasoning" that Hacker wants us all to learn, and algebra. Algebra IS quantitative reasoning, and quantitative reasoning IS algebra. It is taught in a particular form because that form makes quantitative reasoning feasible for minds that are not as brilliant as al-Khwarizmi's. It is taught in a particular form because that form shows up in so many other contexts, far more than just a single Toyota shop floor in Mississippi.
(Many thanks to the Wikipedia writers who rounded up the copy-pasta I placed above from the articles on al-Khwarizmi.)