Wednesday, December 11, 2013

Math Should Be Taught Like Literature or Art

Well, that got your attention, didn't it?

(Where this is going: we need to teach more probability and statistics.)

Teaching math is based on the idea of progressing from one skill to another. What we are learning today is based on what you were supposed to be learning yesterday, and you will need to understand both if you have any hope of understanding what we are doing tomorrow.

Teaching literature and the arts - and to some extent history, the social sciences, classics - endeavors to train students to see the world a certain way: to look for connections, trends, things others might not notice. What we call the culture wars arise because conservatives believe that the current dominant model in the schools is an attempt to train children to see things in ways that ultimately favor liberal politics and that is screwing up the country: identity studies of many sorts, power imbalances, focus on narrative.  Liberals counter, with some justice, that we used to train children to see things in ways that favored conservative politics, and that's already screwed up the country.

I could make nice lists in each column for what viewpoints have been privileged, for how long, and who benefited, but I think that my readers know that I side about 70-30 with the conservatives on that and we'll say no more about it here.

What I am noting is that other than some information in each genre about poetic feet, or third-person omniscient, or archetypes, or notions of beauty, literature and the arts don't teach progress. You could teach freshman english to seniors, and senior english to freshman, with only some adjustment in presentation. Creating a habit of mind is what is sought, not only in reading but in seeing the world.

In school math, we draw an imaginary stalk leading from arithmetic through algebra and geometry on to calculus, at which point the field breaks open like a flower and there are a dozen directions to go. That is sorta true, but not really.  If we accept that "spreading out" metaphor, we see that it actually extends far back down the stalk. Even as far back as middle school, geometry, symbolic logic, and fractions don't really depend on each other that much.  Some, but not as heavily as advertised.  Because the same kids who are good at one are mostly the same kids who are good at the others, we can get away with the fiction that it still requires a progressive approach.  No. Each line may require some progressive approach, but the connections are already weakening.

Only 10% of students should be taking Algebra II and Calculus.  You can play with the percentages, but it's something like that.  And we know who that 10% is going to be by 8th grade.  There will be late bloomers and diamonds in the rough, or hard workers who were able to disguise themselves as capable of the abstract reasoning, and we can create systems to ease transitions from one group to another, but it's not many who are going there.

The math that the other 90% are going to need, day in and day out, to understand the lives they are going to be living, are probability and statistics.  A lot of this only takes a knowledge of arithmetic, plus some algebra and geometry. There is some progress of knowledge - and certainly both probability and statistics can get into some complicated math - but it isn't really progress that is needed.  It is habit of mind.  It is familiarity and comfort.  What does this graph say?  What does it hide?  Is a 50-50 chance good or bad in this instance? Where is the data from? What's a sample size? Is this trend dramatic or unimportant? Students need to live in this world for all of their highschool years, not so much to progress to some higher level of being a statistician, but to increase their comfort in living in this world.

Because right now, the people who know how to make graphs and screw with them are manipulating the people who don't, and it's one of the most powerful forces in our society.


lelia said...

You are so right.

james said...

I knew kids who were good in Algebra and stunk up the place in Geometry, and vice versa.

Geometry is a nice place to teach rigorous logic, because the pictures often help guide the mind and sometimes show whether you got the chain right or not. Whether you ever use geometry again in your life doesn't matter so much as the training in thinking precisely, so you can tell the difference between fuzz and proof. People will often prefer the fuzz, but at least they can have the possibility of knowing.

Some things (fractions/ratios) are critical building blocks for later work, but you are correct that you can modularize the studies at lower levels.

I've been pressing "How to Lie With Statistics" on acquaintances for years.

Dubbahdee said...

You have covered half of what I have been harping on for years.

The other half is business math - accounting, inventory, financial analysis, etc. Like Prob and Stats, some algebra is certainly required here - but even basic accounting will be exponentially more useful to most people than advanced algebra.

Texan99 said...

Having lived with net present values and the time value of money for my whole professional life, I always find it interesting to read the history of usury. What odd notions most cultures have had about money, and how poor they kept themselves because of those notions.

Even today, an amazing fraction of the population thinks there's something fishy about renting money, and can't quite grasp the difference between $10 today and $10 in ten years. On the other hand, they often know instinctively which end of that bargain to avoid when it's their own money at stake.

Christopher B said...

My only issue with your prescription is that a lot of what passes for teaching mathematics has migrated to attempting to teach a state of mind. The problem is it's being done before the proper preparation of the mind, almost as if we would attempt to teach literary concepts before teaching students to read.

Even more of the math people use every day is basic operations plus a heavy dose of fractions. I'm not sure that we are doing a good enough of job of getting students comfortable enough with those concepts that they would be able to utilize statistics and probability.

pat said...

I'm not quite certain what your point is.

I taught calculus and statistics for many years. I'm always surprised that feminists resist the notion that boys are better at math than girls. They need only ask their girl friends, mothers or daughters. Women just aren't as good at math. Period (as Obama would say).

Feminists never object to the similar finding that women are better at verbal skills than men. But they seem hyper sensitive about their math difficulties.

Many women seem to feel guilty about their math difficulties. They seem to think - because they've heard all the rhetoric - that women in general are just as good at math as males but that they are somehow an exception. This seems needlessly cruel. Let's tell them the truth.

Women are of course half of humanity. If they are not as good at math as males - that's just the way it is. As Doctor Pangloss would say 'whatever is, is right'.

Actually, human beings are all bad at math. I used to ask for volunteers in my classes. I asked for someone who thought they were good at math. Then I would ask my victim - "What is the seventeenth root of 9,324" - or some other crazy set of numbers. Then I would show that my cheap pocket calculator which had less brains than a cockroach could provide the answer instantly. In terms of mental ability, math was the first ability that we could automate. Math is very easy - unless you're a human.


Donna B. said...

I always thought I was bad at math, until I took chemistry in college. The math involved in that course, I had no problem with. That's when I discovered that math is a language itself and I hadn't learned much of it.

A language with multiple dialects... I understand statistics just a little bit and I am perpetually puzzled when it comes to most graphs.

I don't worry about it much now. I raised a mathematician and if I need some advanced math done, I just call her.

Texan99 -- "time value of money" is a phrase that irks the livin' daylights outta me. It has been used to legally skirt the word "interest" and to make buying structured settlement annuities something different than a loan. In fact, "time value of money" is synonymous with "JG Wentworth" to me.

Texan99 said...

And yet it's a helpful and descriptive phrase, as used by people whose professions require them to deal with daily. It's a shame if it's become associated with hucksters on TV. It's impossible to deal in investments or financial instruments or debt without addressing the fact that the value of money is completely indeterminate without specifying the time or times it will become available.

Donna B. said...

Those hucksters on TV used that phrase to scam people. And the legal community went along with it. It is a useful and descriptive phrase... until its use as a scam becomes legally enshrined.

A member of my family who suffered a traumatic brain injury and was awarded a structured settlement was scammed many years ago by JG Wentworth. Our family (including several members who are attorneys) have tried multiple times to get him out from under those contracts.

Perhaps you can explain to me how the "time value of money" differs from "interest".

Texan99 said...

Interest rates are are very particular manifestation of the time value of money, whose more general expression is discount rates, which often are implicit and which are subject to change at all time depending on new circumstances. That's why, for instance, the prices of treasury bills float even though their interest rates are fixed.

Take a structured settlement, for instance. They're usually set up like a fixed-term annuity (though they might also be an annuity for life). If someone starts with a lump sum and opts to receive payments over time the form of annuity, then it's possible to recast the stream of payments so that they include an implicit interest rate. But usually we refer to that implicit rate as the "discount rate" instead. For one thing, the discount rate that was used in setting up the stream of payments at inception is not necessarily the same discount rate that will be applied by someone who offers to buy the stream of payments for a new lump sum.

Let's say a tortfeasor is saddled with a $100 judgment and offers to pay it off as 20 annual payments of $10. The total payment stream is $200 in gross face amount, but it's worth about $100 in present value with an implied discount rate of about 7% (by the rule of thumb that a principal amount doubles over 10 years at 7% interest). A 7% rate may make complete sense at inception, but in later years the appropriate discount rate may decrease to 4% or increase to 11%, depending on all kinds of factors such as changing inflation rates. At that point, someone offering to buy the stream of payments may not offer $100; he may offer more or less, depending on what he thinks the appropriate discount rate is.

There are two big problems with paying lump sums for annuities or structured settlements. One is that the people who own them tend not to be financially sophisticated; after all, they usually acquired them by virtue of having been injured, rather than by buying them in an arms-length financial transaction. The other is that the settlements often were set up to be paid over time in recognition of the fact that they would be substituting for lost long-term income. If the owner receives a lump-sum, he will be put to the necessity of making other arrangements for his long-term security, and experience tells us that he is unlikely to do so, or he wouldn't have cashed in his structured settlement for a lump sum in the first place. But if the owner is financially sophisticated enough to address these problems, then the only remaining issue is whether the buyer paid the right price, which is a question of applying a "fair" discount rate. Unfortunately, unless the owner receives a range of competing offers, it's hard to judge what a fair price is. His only defense is to refuse to sell, and we're naturally queasy about the short-term financial pressures that might push him to do so.

Still, I don't think it's fundamentally different from offering to buy any large asset from a potentially disabled person in dire straits. Unless the owner is subject to a guardianship, I'm queasy about regulating what offer he's permitted to accept. We hope he won't take the first offer that comes along, but I'm not very strongly inclined to blame the offeror for making the lowest offer he thinks he can get someone to accept, any more than I blame any potential buyer of any asset for driving a hard bargain. It's ultimately up to the seller to say no if it's a bad deal.

Which brings me back to the importance of people understanding the time value of money. Otherwise it's very difficult properly to evaluate an offer to pay a lump sum for a stream of future payments. Most accountants can give useful guidance in calculations of this kind.

Texan99 said...

I messed up that present value, since it's over 20 years rather than ten. I should have said 10 annual payments of $20.

Donna B. said...

Texan99, I think you misunderstand my dissatisfaction with the JG Wentworth type of buyers of structured settlements. And... from your explanation, I think you may misunderstand some types of structured settlements, ie those that are guaranteed for the life of the recipient vs guaranteed payments and/or lump sums.

Do you represent JG Wentworth or any of the other companies that relentlessly pursue the recipients of structured settlements?

Texan99 said...

I've never represented a company like JG Wentworth. Much of my practice was taken up with pricing discounts to future income streams in the business context, but it's different there, because the playing field is pretty even. When you bring consumers into the mix, it doesn't always work the way a dispassionate schema suggests it ought to.

So you may be right that I didn't understand exactly why you objected to JG Wentworth. It sounds like you and your family had a terrible experience with them. I'm so sorry. What happened?

Donna B. said...

texan99 -- the experience was horrible. The pursuit of the annuity payments was relentless and threatening. The terms of the contract (and the conditions under which the contract were written) were/are unbelievable.

And by relentless, I mean we still get phone calls after 15 years.

The sticking point was the phrase "time value of money". The end result meant that the purchase of the payments could not ever be considered a loan and could not ever be paid off before term (even when the offer was for 100% of the amount of the payments purchased).

The family member who sold the payments was in a psychiatric facility when he was approached by JG Wentworth. It's a long and ugly story.

hugh lygon said...

there is no argument against turkheimer's results except that he faked them.

heritability = 0 + lower mean = POVERTY LOWERS THE MEAN.

the very idea that poverty doesn't affect test scores is absurd and can only be entertained by a moron.

i remember you bitching at steve for your results.

well there weren't any. JUST as i predicted.

hugh lygon said...

and considering i made the high score in the country on the first soa exam on the date i sat for it, you can trust my mathematical sophistication.

Texan99 said...

I had the idea that JG Wentworth deals were a simple lump sum up front in exchange for an irrevocable assignment of a stream of future payments, sometimes for a term, and sometimes for life. Do they sometimes structure them as loans, using the payment stream as collateral? I suppose there might be legal issues about the straightforward assignability of the income stream that prevent treating it as an outright sale. If it's structured as a loan, it may not be prepayable. We tend to assume that all loans are prepayable, but I know to my sorrow they are not. My parents got stuck in one like that once, and it was an education for me.

Doing business with anyone residing in a psychiatric facility sounds squarely within the concerns I addressed above about how some business decisions call out for the intervention of a guardian. There might even have been (way back when) issues of rescission on grounds of duress.

Are the calls 15 years later still an attempt to buy the annuity, or are they harassing the seller over some kind of enforcement of a deal with terms that are still being performed? It sounds perfectly awful. One thing I'm still not clear about is why they would be repeating "time value of money" as some kind of mantra. If they structured the acquisition as a loan, it does sound as though "the time value of money" were simply a euphemism for what we'd more normally call "interest." In cases like that, usury laws often permit the recasting of various discount mechanisms as interest for purposes of determining whether the implied rate is excessive.

Donna B. said...

The calls today are attempts to buy the remainder of the guaranteed for a certain time period payments. They are not interested in buy those guaranteed for life because they have no control over the life.

Previously, every time there was any contact with them (change of address, etc.) the exchange was threatening. "You better not be thinking about trying to stop these payments." "You better not be trying to sell your other payments to somebody else." "We'll come after you if you do anything that delays us getting the payments."

The situation is somewhat better now, since court approval is required before payments can be sold. What I haven't heard of is any court disapproving, though I'm sure that's happened.

David Foster said...

A liberal-arts approach to math should also include the concepts of mathematical modeling (of physical systems, business processes, etc)..I think it would be possible to do this in a meaningful way without getting into the analytical derivation of calculus and queueing theory, etc.

5705freedaysout said...

Throws so many questions up imo
If we forget about all we started out with, I would question what color red is to anyone that had never been taught to see it in a particular way which might infer they did not have a visual impairment, then move back to Maths and ask if we were taught 1plus1 equals 2, why 1 apple and 1 orange should equal fruit?
5 apples still equal fruit which is singular.

1 plus 1 equals 2 only because were are told it does in order to make things easy.
If we went on the basis that 1 plus 1 equals 1 we would be able to get our heads around that, but getting our heads around 1 plus 1 equals 3 may be harder.
When we get into negative numbers my eyes tend to glaze over.
One piece of good art and 1 piece of bad art should balance out as zero art, but it is a little biased toward the viewer and the interpretation of good art and bad art.
How many people visit an art gallery and phone before hand to ask if it is good art or bad art?