Via Swans On Tea, a ranty blog post titled Sucky Schools - How To Repair Our Education System, which takes its structure and much of its tone from Paul Lockhart's "Mathematician's Lament" (which, unfortunately, is a PDF file). I'm fond of ranty posts about education reform, but both of these kind of lose me. Lockhart, in particular, strikes me as being an excellent example of the dangers of being too attached to a subject.
He writes with great passion and at great length about the fun and creativity involved in math, which is all very nice. Unfortunately, it also leads to paragraphs like this:
The saddest part of all this "reform" are the attempts to "make math interesting" and
"relevant to kids' lives." You don't need to make math interesting-- it's already more
interesting than we can handle! And the glory of it is its complete irrelevance to our lives.
That's why it's so fun!
See, right there, we part company. I'm just not that fired up about pure math-- and I make my living working in a highly mathematical science. Math is a tool for me, that's it. I have enough appreciation of it to be faintly impressed by some of the cute tricks he describes, but they're interesting in the same way that odd bits of historical trivia are interesting.
I don't really object to his proposed reforms (which basically amount to shifting the emphasis in math classes from rigid and formal definitions to problem-solving skills)-- after all, they're roughly comparable to the reformist introductory physics curricula that I like. I think his argument isn't terribly convincing to anyone who isn't already a mathematician, though. Really, the main thing I like about it is that some of the examples he gives look faintly similar to physics problems, and I think it would be lovely if somebody else were teaching students that approach as well.
This is a problem that I find myself grappling with all the time, as a scientist, educator, and blogger. Obviously, I think physics is intrinsically fascinating, because it's what I choose to do for a living. I don't feel I should need to explain the relevance of quantum mechanics, because it's just so darn cool that everyone should want to know more about it.
That's not the way the world works, though. Not everybody feels the way I do about physics problems, or finds magic in the quantum statistical behavior of very cold collections of atoms. I think it would be a better world if more people did, but I don't get to dictate that, and just assuming that everybody else will be fascinated by the same things I am doesn't bring in the grant funding.
The fact is, lots of people only care about things that they find relevant. I have classmates and colleagues who only read non-fiction, because they don't see any point in novels. I send grant proposals to program officers who want to know how my work will advance technology. I teach large numbers of students who care about physics only as a hurdle to cross on their way to an engineering degree, or medical school.
One of the best pieces of advice I got as a new-ish faculty member was "Don't assume that your students are like you were." People who end up as college faculty are a tiny minority of students, and most of them have always looked at the world in a different way than their classmates who go on to other occupations. I'm probably somewhat closer to the mark than many of my colleagues, at least as regards socialization as a student, but I still fall into the trap of assuming that things I find incredibly fascinating will also captivate my classes, and I get snotty comments to that effect on the end-of-term evaluations.
Lockhart seems to me to be falling into this trap. He assumes that everybody will be enthralled by math in the same way that he was enthralled by math, and that's just not true. I'm totally with him on high-school geometry, which was excruciating, but a more free-form approach isn't going to make me care about the angles of a triangle inscribed within a circle. Unless it turns out to have some relevance to a problem in physics.
His proposed reforms sound ok to me, because, let's face it, it would be hard to come up with anything worse than what we've got now. But reveling in irrelevance doesn't strike me as a good general approach-- there are lots of people out there who really do want math to be relevant. I'm one of them.
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I think that basing things on relevance is very difficult to pull off, even if we agreed that it was desirable (which we don't). What of mathematics and science is relevant to the average person's life? Algebra isn't. Geometry isn't. Newtonian physics isn't. Calculus isn't. Maybe a little bit of addition, subtraction, multiplication and division. Maybe working with decimals and percentages. That's about it. Add to those skills the ability to read and write, and you've pretty much exhausted the skills that are directly relevant to most people's lives.
I think that the point of higher education (that is, beyond 6th grade, which is the point where people have learned everything that is relevant) is to teach general skills of reasoning, which are relevant to everyone. The problem is, there is no way to teach general reasoning skills except as demonstrated through specific examples in a lot of detail (more detail than is relevant).
The other point to be made is that there are many people who do need calculus, geometry, algebra and physics in their jobs. How would those people ever discover that they might be interested in those subjects if they are not exposed to them before they know that they are interested?
Kind of an off point, but "algebra isn't relevant to people lives"? How are you defining algebra? Solving equations with unknowns is part of algebra as far as I know, and if that was a little more in peoples lives we might have a few trillion dollars less of bad debt in the U.S. right now. How do you tell if buying something in bulk is a better deal than buying individual smaller packets. You have to factor in how much you use, calculate (or know to look for) the price per weight and essentially solve a little system of equations. People don't think of it that way I guess, but that is algebra.
I think the lack of algebra and use of it in peoples live might actually be an interesting way to tell if someone is destined to be poor or not! Certainly rich people themselves or through their hires definitely use algebra to heighten the value of their assets.
I'm agreeing with Chad, regardless of Lockhart's enthusiasm.
Let me oversimplify the human brain as a filter. Unlike antique theory (unfortunately embedded in the the foundations of Education), your brain does not remember everything that gets in through the senses. Your brain filters out almost everything.
Your brain, to put this metaphorically, asks itself these questions of everything coming in as sight, sound, smell, touch, taste:
(1) So what. Does it mean anything? If not, throw it away.
(2) What's it to me? If not relevant to survival, food, sex, power, throw it away.
Those things that get through both barriers are MUCH more likely to be retained as memories. A small fraction that gets through one filter and not the other get retained. Those things stopped by both filters are never in the brain long enough to be remembered at all.
Hence, as I've explained in hundreds of pages of essay in Charter College of Education, foundationally we must have a reformed Education that uses what we now know from fMRI and other scientific study of the brain.
Want to teach Physics or Math? Everything you teach, using the right mix of sensory modalities, must be MEANINGFUL when first explained. It must be shown to RELATE to actual life.
And this is hard because the students are heterogeneous. No two have the same learning style. Hence "meaningful" and "relevant" vary from student to student.
That's why a teacher's job is so very hard. The INSTRUCTION (which I've summarized), the ASSESSMENT (to find out what made it through the filters, was retained, and can be used), and the MANAGEMENT (of time, the classroom, the lesson plans, the paperwork).
Any thoughts on this?
Markk,
Perhaps the average person should use algebra in everyday decision-making, but the fact is that they don't.
In New York State, you can look at the per-unit price, which is found on all price tags in all reputable grocery stores.
Yes they SHOULD use algebra. So it IS relevant. That is the point.
Per unit price is only part of the story as I said. You have to take into account whether you will actually use the stuff before it goes bad. That was what I was trying to get at.
Mortgage comparisons, how to split your savings and budgets, Should you drive to that gas station a mile further away? Does a tax cut with deficit and inflation really help your finances? Many more things in daily life need algebra to give good or even just barely acceptable answers. That fact that people believe algebra is not relevant when it so obviously is tells something about the teaching of math I would guess. Tinfoil hat folks would think this is deliberate slant (frame?) done by the upper class to keep the tools of critical thinking out of the hands of those that "don't need it". But that would be way too thoughtful and organized I guess.
I occasionally find myself wondering how it is that mathematics escaped being part of the Philosophy department.
"Application" took the form of bizarrely contrived and improbable "word problems" for most of my mathematics education, with the exceptions of "Introduction to Statistics" and "Applied Calculus". It's probably no coincidence that the latter two mathematics classes are the only two I recall actually enjoying.
The experiment to watch: Governor Arnold Schwarzenegger has ordered that, within 3 years, ALL 8th graders in California public schools pass Algebra 1. He didn't add any money for this to the budget (California being de facto bankrupt and in a crisis that risks dragging the USA into recession) nor give a clue on how to teach the teachers of algebra.
For me, as an ex-Math Professor who's been grappling with teaching Math to those who've flunked it once, twice, even thrice before in middle school and high school, this is some sort of employment guarantee. Once, that is, I've earned my Single Subject Math Teaching Credential in Spring 2009, through a No Child Left Behind system that purports to provide qualified teachers but, de facto, scares away all the former aerospace engineers who lost their jobs but know how to use Math in actual engineering.
The theory seems to be that it is better to have someone with a B.A. in Education and a Certificate from a teacher's college who doesn't really know Math or Science, to teach Math or Science, than to put someone who's used Math and Science their whole professional life. Because, in the latter case, a parent might sue the school district for putting an uncertified teacher in the classroom. I'm not making that up.
They don't usual have blind teachers teach art, or deaf teachers teach music, but they routinely have math-challenged teachers teach Math, and Science-challenged teachers teach Science. And, speaking as a loyal Union member (who's been active in at least 3 unions, and an officer) the teachers unions seem more dedicated to keeping bad teachers from being fired than in providing resources to good teachers.
Anyway, watch California. See if the Terminator has pushed Algebra into a generation here on the Left Coast.
Compound interest is killing many college students and grads. They really have no idea at all why the Bank of America can afford to pay a nice young man or woman to sit at a table and give away credit cards to someone with no assets and little income.
Many college loans operate like "interest only" mortgages in a context where the property (the degree) is worth much less than the balance of the loan except that walking away (foreclosure or bankruptcy) is not an option. The discussion of an article on this topic at IHE went on forever with horror stories where interest had tripled the debt owed for a degree, making it likely that the college loan would out-live the graduate.
"a nice young man or woman to sit at a table and give away credit cards" -- I don't object to Capitalism in action. I do object to colleges and high schools selling their student enrollment lists to the credit card companies.
I do agree that the typical student in the USA does not understand compound interest. Thus, they do not understand the very idea of savings or of investment.
I ask students (over the past 2 years) in urban middle school and high school classes:
"If I've earned a million dollars, which I have; and I've spent a million dollars, which I have; what do I have left?"
Invariably the low- to middle-class students call out "Zero" or "nothing."
"If I were a fool, that would be true," I say. "But I'm not a fool. I live in a million dollar house."
Of course, in the past few months I've had to add: "At least it was a million dollar house until the real estate bubble burst. Now it's just an $800,000 to $900,000 house. But the point is, things go up and down in value. Buy low. Sell high. When you go shopping, you like to buy things on sale, right? You may have noticed that some things like movies, food, gasoline, cost more now than last year."
The middle-middle to upper-middle-class students are able to volunteer some basics about stocks and mutual funds. The impovershed stare at them as if this was said in ancient Greek.
Because they don't understand compund interest, they seem to think that when you spend money, you get a consumer product which you consume, leaving nothing. They don't grasp that they've exchanged currency for a good or service that has value, let alone value that changes over time. They do not know what "wealth" is. No wonder the median family in the USA has close to zero net value (financially speaking). No wonder the USA is roughly $6 x 10^12 in debt.
See those Olympics? That host country owns about $1 x 10^12 in various bank accounts and piles of U.S. Treasury bills, and stock in US companies, beyond those that they bought outright (i.e. Lenovo from IBM).
As goes the math-challenged compound interest-ignorant, so goes the credit-card indebted (roughly $1 x 10^4 per USA family) and mortgage defaulting family. So goes state and federal governments that have been spending money that they dopn't have. And now the chickens are coming home to roost.
You know, JvP, requiring people to learn basic algebra before high school seems pretty reasonable. Surely seven years is long enough to get the hang of arithmetic?
Anyone know when algebra is introduced in other curricula? I think I first saw it in grade 7 in Canada. My final year in the Finnish system was grade 4, and it had not yet shown up by that time.
Because they don't understand compund interest, they seem to think that when you spend money, you get a consumer product which you consume, leaving nothing.
I'm picking a nit here, but these are two separate issues. They may be correlated in the population you see in your classes, but one does not follow from the other.
Failure to understand compound interest is a reason why so many people, and the country as a whole, are in so much debt trouble. It only takes a little bit of basic algebra--heck, these days a few minutes of Excel work would do the trick--to show that making the minimum payment on your credit card will result in your purchase costing much more than if you had paid cash, or paid the full amount due when you got the first bill. The (il)logical extreme is the negatively amortizing mortgage: your payment doesn't cover the interest accrued, so the difference is added to your loan balance (this is the premise of the so-called Option ARMs which are blowing up the credit markets this year). I cannot see how anybody who understands compound interest (and who has a material interest in seeing that the mortgage is eventually paid off) would be a party on either side of such a contract, because it should be immediately obvious that absent a major positive change in their financial situation (e.g., selling the house at a substantial profit) the borrowers could not possibly pay off that mortgage.
The second part, assuming that money spent is spent on something consumed, may be in line with the experience of your less wealthy students--in many cases the entire family income is spent on food/rent/medical bills, which are all consumed. You can be in this situation while understanding how compound interest works. Conversely, there are many apparently well-off people in this country who get that money can be "spent" on assets that (at least potentially) increase in value but who are in serious trouble because they charged too much on the credit card, got themselves into a toxic mortgage, etc., and don't have money in the bank.
# 10 | Johan Larson
I agree that everybody who can should know algebra (excluding the genetically/neurologically sufferers from clinical dyscalculia, those too young -- whatever that is, and those whose other human rights are being violted and thuis can't access education). What's weird about the Gubernator making the executive order is that the budget is so tight that programs which he promoted (Physical Education) are being cut right now. And that he did not ask for input from the Department of Education of his own state. Again, I agree with the objective. I fear that it will be achieved by dumbing dopwn the definition of what it means to "pass Algebra 1."
# 11 | Eric Lund | August 20, 2008 2:55 PM
"'Because they don't understand compound interest, they seem to think that when you spend money, you get a consumer product which you consume, leaving nothing.'
I'm picking a nit here, but these are two separate issues."
I accept as a friendly amendment what you clarified.
The correlation is strong. That's why I've sweated and wept in teaching in urban schools. America, regardless of Brown v Board of Education, has re-segregated its public schools according to race and income. My friends in other nations (Singapore, India) have told me that this is because the USA does not know how to handle the collapse of its empire, and has chosen to abandon its inner cities. Be that as it may, I find the situation profoundly unamerican, and a tragedy. I cannot say that it is for somebody else to fix. I had to throw myself into the front lines, into the trenches, of the war between education and ignorance. If I can't do that, almost nobody can. I don't "scale up" to solve the problem nationally. But as I succeed, I can honestly give hope to other teachers.
Your insistence on "relevance" probably explains why you're an uncultured barbarian scientist instead of a refined, learned humanities dude.