William Whewell
once contended that math education should build upon geometry, which focuses on logic and concrete objects, rather than upon analysis (algebra, calculus), which focuses on abstractions and symbol manipulation. Oddly enough, another blogger -- James Chastek at
Just Thomism -- seems to have encountered the same post today and had
this to say:
We’ve clearly fallen into exactly the fault that Whewell wanted to
avoid, and have dumped the geometrical approach almost entirely while
dedicating years to teaching analytic methods.
The fundamental pedagogical mistake is in this approach is that it
teaches abstractions before teaching what they are abstracted from.
Not content with this error, we went in the 1970s to teaching New Math; viz., abstract set theory
in grade school! A lot of geometry taught today is actually analytical geometry: The focus is on formulas for calculating lengths, areas, volumes of sundry geometric figures. Few indeed are those fortunate enough to prove geometrical properties with straightedge and compass and reasoning from the postulates, as TOF was, Lo!, these many years ago in geometry class from Sr. Amelia.
That was sophomore year. Freshman year was given over to Algebra. In my section -- Freshman 7 -- we covered both Algebra I and Algebra II, the latter being normally reserved to Junior year. If I read Whewell, Siris, and Chastek aright, we should do Geometry first and postpone Algebra until later.
C. S. Lewis was thankful to Robert Capron, the mad headmaster at ‘Belsen’, for geometry lessons and nothing else. Cf. Surprised by Joy:
ReplyDelete‘I can also say that though he taught geometry cruelly, he taught it well. He forced us to reason, and I have been the better for those geometry lessons all my life.’
The chiefest sin of modern education against the intellect, say I, is that it not only does not teach its patients how to reason, it positively frowns on the practice.
I honestly remember very little of my school math-- but almost everything my mom used, and the example of "how to calculate the height of a flag pole by its shadow," which somehow made me remember that A^2+B^2=C^2, and which I actually used last weekend. (building a bunk bed for the girls; room is exactly 8 ft tall, how tall can it be and be assembled on the floor then lifted up?)
ReplyDeleteI think shopping math is algebra, though-- price divided by amount, then multiplied by the amount of the container you're comparing price with, if the result is smaller than the price for the second container, the first one is a better deal.
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Teaching our eldest division. She got the idea that it's just adding the number over and over again, but is struggling with how exactly that works. Especially if a butterfly goes past. :)
Has anyone considered that New Math might've been a Soviet PSYOP? Or an elaborate practical joke perpetrated by the Russians on Western observers. "No, man, it's all about that modular arithmetic. Oh you don't teach modular arithmetic? *Tsk.* I'm so sorry."
ReplyDeleteRichard Skemp had doctorates in maths and psychology and used both in a book I recommend "The Psychology of Learning Mathematics." More recently a maths professor told me we should stop teaching maths the way we do. He said that below graduate level there is nothing that maths teaches that can't be taught better in a class on computer programming.
ReplyDeleteMaybe this explains why my experiences with "math" during my formative scholastic years were so universally bleak. The only break came when I took geometry during one year in high school. Yeah, I could do angles and sides and perpendicular lines and stuff! But then Algebra II came along...and the second half of that covered trigonometry.
ReplyDeleteSines and cosines and tangents? Say what? I asked my teacher why I had to learn this stuff and what it was good for--he just shrugged and said "you'll need it later on". Now, I have found that I can concentrate and learn hard stuff--if it makes freakin' sense, and I have some motivation. But if I try to concentrate on memorizing nonsense, my propensity to stare off into the distance and daydream usually wins. It turned out that the prospect of a bad grade was just not enough to get me to memorize this gibberish, and apply it in a rote fashion. So I failed Algebra II...and was on the point of failing it the second time through when I got the good news that my SAT scores qualified me for admission to the U of C, and THAT MY GRADES MADE NO DIFFERENCE. Oh the joy, the freedom of telling my math teacher he needn't ever see me again.
Because it's relevant, I'll mention blushingly that I went on to get a Ph.D. in Philosophy; this may (or may not) prove that I am not a complete dullard.
Fast forward a couple of years, when I have despaired of getting a tenure track teaching job, and tell a friend that I have realized one of the core truths of life: I have to earn a living, somehow. But it wasn't happening. He said, "Why don't you go into computers?" (This was about 1982.) I replied, "No way, that's math. I can't do math." He laughed and told me that computers have nothing to do with math, that they run on logic. "Oh yeah?" Said my young self. "I can do logic. I love that stuff."
So took some basic programming classes at the college where I was also a nugatory "Adjunct Instructor" in Philosophy. And found that I loved programming logic. You might have to solve a math problem--but you only had to do it once! After that, the program would do it over and over for you! And that's how I abandoned Philosophy and got a Real Job, just like Wittgenstein advised.
I still feel that my math blindness wasn't inevitable, that there might have been something for me to learn, if I had the right teachers.
Coincidentally, a few days ago, I got to wondering how I could calculate the height of a distant mountain without actually walking all the way to its base. (That would let me use good old Pythagoras, but the imaginary mountain was too far away. I really do think about random stuff like this. You might say it's an incurable propensity. I was probably trying to avoid work.) It seemed to me that there ought to be a way to do it, involving triangles and measuring angles, so I searched the web. Turns out there is a method, and it involves trigonometry. Say what?
What if my teacher had introduced me to trig by telling a story about an expedition that had to calculate the height of a distant mountain? I still wouldn't know where sines and cosines come from, but at least I'd have known they were useful!
The ancient Indian word: soh-cah-toa might help.
DeleteOn a right triangle, h is the hypotenuse, o is the side opposite either of the other angles, and a is the side adjacent to that same angle. So the sine is the ration of the opposite side over the hypotenuse, cosine=a/h, and tangent=o/a. That is, the trig functions are the ratios of the various sides of a right triangle to one another.
So if you sight to the base of the mountain and to the peak, you can get the angle subtended. Find the tangent of that angle. Since t=o/a, then at=o. So if you know how far away the mountain is (a), you can calculate the height of the mountain (o). Surveying used to be a major occupation, esp. in colonial times.
I agree that math is much easier if it is motivated. Supply the kiddos with problems that mean something, then work out how those problems can be solved. In the medieval curriculum, the trivium focused on logic (rhetoric).
Wow. It never occurred to me that "sine", "cosine", etc. meant anything. I thought they were just random numbers that appeared in a table, and that you were supposed to use to solve something. They're ratios of sides of a triangle? O.K. Maybe I just wasn't paying attention, but I don't remember that ever being pointed out to me. Now it makes sense. Hey, 60 years too late...but thank you.
ReplyDeleteI must seem exceptionally dumb to the math literate.
I don't remember it being mentioned, either.
ReplyDeleteOf course, our math class was taught by the same teacher who literally said it was not his job to teach, and copied the example directly from the book on to the board (complete with any mistakes) then sat down to play on the computer. He refused to help anyone unless they had asked the entire rest of the class, which meant I did most of the teaching and then he'd scream at me if I couldn't figure it out from the book.
(He was the football coach--but actually had a degree in math.)
Yeesh. How horrible. I HATED geometry. Algebra was fun.
ReplyDeleteDitto. There is a grain of truth in the "differentiated instruction" approach, even if a lot of people (ab)use it as an excuse for refusing to learn.
DeleteIt's pretty common for those who love algebra to hate geometry, and vice versa. The problem is that by ignoring the geometry-loving population until 7th or 8th grade (or even until 11th grade in my school), these folks become convinced that they are idiots who can't do any math, and often they don't have the prerequisites to take geometry!
ReplyDeleteMeanwhile, the algebra-loving contingent is just stuck doing busywork without much new content. After one masters the multiplication table and long division, there's just not much going on. You can veg your way through, as long as you do all the busywork homework problems.
And that's also why Trigonometry and Calculus should be introduced well before high school. (Beyond the obvious consideration -- that high school trig and calculus are gradekillers, for precisely the kids who need good grades to get into a good university. You almost never see a STEM kid as valedictorian, because the way you become valedictorian is by avoiding gradekilling classes of true difficulty and scholarship.)
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