Tall, Dark, and Mysterious

9/17/2005

What passes for success

File under: Sound And Fury, Those Who Can't, Queen of Sciences, Know Thyself. Posted by Moebius Stripper at 6:06 pm.

I realize that I don’t post many stories about my positive experiences with students. Part of it is because those experiences don’t require the sort of catharsis that the horror stories demand. Part of it is that my darkly comical style of writing doesn’t lend itself to expositions of, like, students actually learning stuff from me. And part of it is because sometimes, the success stories depress me even more than the failures.

I’ve been tutoring a high school kid for the past two months. The kid’s in grade 12; when I met him, he was doing math at a grade two or three level. This is not an exaggeration: he couldn’t add or multiply single-digit numbers without a calculator. And this wasn’t just rustiness, as this inability extended to not being able to compute things like 6+0, 5*1, or 3*0. In other words, he didn’t know what numbers were. Not surprisingly, he couldn’t solve linear equations, add fractions, or make heads or tails of the most simple word problem.

I met with him every other day, two hours at a time. And, to his credit, what he lacked in mathematical skill, he more than made up for in persistence. He worked diligently, if not terribly successfully, on his homework. We spent a lot of time on the basics - fractions, simple algebra, the meaning of equations. We also spent a lot of time - far, far more than I’d have liked - on how to use the fucking graphing calculator to perform tasks that every student should know how to do with a pencil and paper.

He’s two thirds of the way through the course now, and he’s pulling an A-minus.

And he’s learned a lot in this class, and I’m glad that his effort will almost certainly earn him the C he needs by the end of October. But little of what he knows falls into the category of “mathematics”; the bulk falls under the umbrella of “tricks that will get a student through a grade 12 math class in BC.” He’s vaguely familiar with basic algebra now, but he still falters when simplifying an expression (”Am I allowed to cancel out the x’s in (x+5)/x?”). He cannot immediately identify which methods to use in solving an equation - he’ll use the quadratic formula if someone tells him he’s dealing with a quadratic equation, but he needs that push. Meanwhile, he’s a whiz at using his graphing calculator: he’s mastered all of the fancy features, and can program all of the rules into it. (He’s managed to fit, in the calculator’s memory, examples of every type of question he might see, along with solutions.) After each test, he purges both his calculator’s memory and his own of everything we’d studied in the previous chapter.

Two weeks ago we wrapped up the trig unit, which vexed him even more than the previous four chapters had. Following some introspection, he was able to identify the source of his frustration: why, he wondered, did those bastard curriculum designers make him do this shit? Didn’t they know that the TI-83+ could neither do proofs, nor provide exact values of sin (pi/3) and the like? During one particularly trying session in which I was explaining, for the third time, how to find those values, he declared that he wasn’t going to learn “that triangle shit”, and I found myself, for the first time ever, raising my voice with a student. He relented. He got an A on the trig test. The next week, reminiscing, he asked, “trig - we did that already, right? Was that the stuff with the logs?”

Last week we started on the combinatorics chapter, and I braced myself for the damage. Combinatorics, unlike most of the rest of the course, requires some creativity: formulas are few, and variations on themes are many. Every question must be read carefully, and even the simplest ones require students to think about how to set them up. However, my student took to this section surprisingly well, and found it to be a lot less stressful than the others. He even asked me some questions that were related to the material, and not just to how to pass the next test.

The last section of the combinatorics unit covered the Binomial Theorem, which provides a shortcut for expanding expressions of the form (a+b)^n. I always thought that this material was the easiest part of the combinatorics chapter: it’s completely algorithmic, and demands no creativity.

It does, however, demand a knowledge of basic algebra.

“Here’s what we’re going to do in this section,” I explained. “We’re going to find a shortcut for expanding things like (a+b)^n - n is a whole number.”

“Which whole number?”

Sigh. We’ve been through this, many times. He always gets tripped up by questions that require him to generalize anything, even slightly. Has to do with skimming over the directions, and failing to understand that variables…can vary. Last month’s lesson didn’t sink in, apparently.

I explained it again. “N can be any whole number,” I said. “We’re going to introduce a formula that will let us expand (a+b)^1, (a+b)^2, (a+b)^3…and (a+b)^n for any whole number n.”

And away we went. We had worked out the first several lines of Pascal’s Triangle on the previous page, and I made sure that that was accessible as I had him expand the powers of binomials the long way.

He started having trouble with (a+b)^0. “Zero?” he asked. Even though he’d raised things to powers of zero in the chapter on geometric series. And the one on exponents. And a few others.

“What’s anything raised to the power of zero?”

He reached for his calculator and tried a few values. Frustrating, but at least he knew that he could figure out the answer by trying some values. He hadn’t known that in July. “One,” he declared.

The expansion of (a+b)^1 proceeded without incident. He was just as quick to give an answer for the next one: “(a+b)^2 = a^2+b^2,” he said.

We’d gone over this one a dozen times in the previous month. He’d made that mistake a dozen times. Each time we went over the basic rules of algebra, and tried plugging in numbers for a and b that showed that (a+b)^2 doesn’t always equal the same thing as a^2+b^2. But that was last month. He was supposed to know this still?

And it wasn’t over. You mean ab and ba are the same? So we can collect them? And we can’t collect ab and a^2? Why not?

Anyway, enough. My point is: in the BC high school math metric, this is A-minus work. A good graphing calculator and a mediocre short-term memory are sufficient to achieve excellence in high school math classes; most students will take the path of least resistence and develop little else. Grades of C or higher are required for the college program this kid’s trying to get into. Grades of D in this course are sufficient to gain entry into the precalculus class I taught at Island U last year. If a failure to grasp basic algebra doesn’t stand in the way of getting an A-minus, then what on earth does one have to know in order to get a C? Or a D? Why not just eliminate the middle man and admit anyone into a college math class, rather than wasting everyone’s time on this extended TI-83+ how-to seminar? This sort of thing is part of why I usually have little sympathy for students who claim that, no, really, they do know the material, it’s just that they do badly on tests. In my experience, it’s more frequently the opposite: they do far better on tests than their actual understanding of the material reflects. And then they sink like stones when they take a college class with me, in which I design the tests and don’t allow them to use their graphing calculators.

Well, not all of them sink. I had a student last year, who earned a C on the first test and then quickly adapted to my expectations. He had the aptitude to do so; it’s just that he’d never been required to use it to full capacity. He pulled an A-minus on the final exam, and wrote me a note at the end. I transcribed it, and I go back and read it every now and again when I’m particularly frustrated:

I hope I did as well on this exam as I think I did. I know it took me awhile to get going this term, but I worked really hard and I finally feel like I get this stuff. I always used to do well in math class in high school, but you actually made me think. Even though I got good marks in math class before, I feel like this is the first time I really learned math.

Thank you.

I guess that’s success. I expect a thank-you from my current tutee, who certainly would not have achieved the required C without me, and who will almost certainly get or surpass it next month. But I doubt I’ll feel all that good about it.

27 Comments

  1. Your story of the tutee is at once saddening and hopeful.

    He started down the path of thinking for himself.

    He is possibly at the beginning stages of grade 5 (or maybe 6?) math now. You have given him hope that he will not flunk every math class he takes in college–partly by aiding and abetting his dependance on the calculator, and on memorized answers.

    Could you have done better, given his desires, abilities, and time allowed?

    That note from the last student is quite heartening. From my own time teaching, I know that there aren’t enough students like that in the world.

    - karrde — 9/17/2005 @ 6:59 pm

  2. I don’t get the BC math curriculum. I am in the Engineering Science program in University of Toronto, and there are people from BC who successfully qualified for EngSci (not the easiest thing in the world to do, as far as Canadian high school students are concerned - you need 90+ average) without knowing what dot and cross products are. Perhaps expectedly, most of them dropped out of the program after a short while.

    (nota bene. EngSci is, purportedly, the hardest engineering program in Canada…)

    - Tom Yue — 9/17/2005 @ 8:37 pm

  3. i’d have to say that it’s nice if a high school student knows what dot and cross products are, but that’s a pretty darn high expectation. those are operations on vectors instead of numbers, and a real understanding of that requires a student to understand the abstraction called “operations”, and the fact that they don’t always apply to numbers. for a student below the age of 14 or 15, that’s quite a leap: the highest-order reasoning skills don’t develop in most people before that. that means they only had several years after reaching 15 to come to that level of abstraction, during which time they also had to learn trig, logs, conics, polynomials, induction, and possibly calculus. and even those topics don’t require abstraction of such basic ideas as “operation”.

    i think math-adept adults (including me) find it hard to understand how advanced abstraction is. we see a pattern, we look for abstraction; it’s natural. kids under 14 or so see a pattern, they think it’s a pattern, and they’re lucky if they can quantify it with algebra.

    i think you can expect high levels of symbolic manipulation skills from top high school students, and even strong abilities to graph and prove. but my intuition tells me that vector operations are a higher level of abstraction that very few of them will have.

    i’m sure plenty of them do have that ability to abstract, but that expectation feels a little high to me.

    - Polymath — 9/17/2005 @ 10:31 pm

  4. Polymath, you’re forgetting that what happens with a lot of students is that they just memorize an algorithim. And for a dot or cross product, that’s really quite simple and straightforward. There’s no concept of operation, it’s just memorize this formula and plug in numbers.

    - vito prosciutto — 9/17/2005 @ 11:24 pm

  5. I think Tom may be talking about now knowing what dot and cross products mean — when do you do them? What are they used for?

    I am still doing dot products all the time at work… because I’m doing weighted averages. The function for this in Excel is SUMPRODUCT. But Excel won’t tell me when I need to use this — I have to know when it’s appropriate.

    - meep — 9/18/2005 @ 3:08 am

  6. He reached for his calculator and tried a few values. Frustrating, but at least he knew that he could figure out the answer by trying some values. He hadn’t known that in July. “One,” he declared.

    I’d like you to know that I think you have done a seriously good job there, and I applaud you heartily for this.
    Giving a student the confidence to realise that s/he can work something out for themselves (rather than immediately becoming despondent) earns my utmost respect. Getting to that stage takes a lot of effort.

    And that skill alone will (imho) benefit that child more than a C grade in HS maths.
    So, yes, well done.

    - Ronald — 9/18/2005 @ 5:14 am

  7. Polymath says:

    those are operations on vectors instead of numbers, and a real understanding of that requires a student to understand the abstraction called “operations”, and the fact that they don’t always apply to numbers. for a student below the age of 14 or 15, that’s quite a leap: the highest-order reasoning skills don’t develop in most people before that.

    At the risk of sounding highly arrogant, I don’t buy such claims at all, in part because I know I understood such vector operations by age 12 and I’m pretty sure I had the capability to understand them earlier. What you say sounds very Piagetian, and as far as I am aware Piaget’s estimates of the ages at which different stages of cognitive development are reached were usually on the high side.

    I’m no expert in cognitive development, but it seems fairly clear to me that there are no major transitions in cognitive structure after age 4 or 5 or so, since we’re able to remember things after that age but generally not earlier. From that point on I tend to think that one develops incrementally, and there are no sudden “phase transitions” where one suddenly becomes capable of abstract reasoning. (Do you remember waking up one morning suddenly capable of understanding how to think logically?)

    Of course, some people never achieve the degree of abstract thought needed to understand much of mathematics. Other people achieve this quite early. I don’t really think we have any understanding of why this is so.

    - M.R. — 9/18/2005 @ 8:14 am

  8. The fact that a student doesn’t know vector algebra by the end of grade 12 math should be the worst of my grievances with the BC math curriculum. (Mind you, I do think that students in grade 12 physics should have at least a passing familiarity with them.) If I had to restrict myself to two issues, they’d be these: one, the overreliance on graphing calculators (duh); two, the fact that there are tons of topics in the course, and none is developed in any depth at all. What with all the speculation about the causes of teenagers’ short attention spans, I’m surprised at how seldom that “schoolwork that doesn’t require them to have longer ones” comes up.

    For example, here’s the chapter on sequences and series: first, a definition of a sequence, followed by some examples. Recursively- versus explicitly-defined sequences. Arithmetic sequences, geometric sequences, arithmetic series, geometric series. (Yes, we’re going to give you the formulas for all of those. Yes, we’re going to derive the formulas for evaluating the two types of series. No, we’re never going to ask you to prove either of those on a test, so go ahead and just program them into your calculator’s memory.) We see a little bit about infinite geometric series, so we introduce limits briefly. Then we get to a chapter devoted to a single topic - compound interest. And here’s where everything falls apart spectacularly. This chapter teaches students how to evaluate two types of questions:

    1. Bob has $1000. He invests it at 4%/a compounded monthly. How much money does he have after three years?

    2. Bob wants to have $300 by next year. How much does he have to invest at 6% compounded biweekly?

    That is all. God forbid should students have to solve for time (logarithms were last month), or interest rates (taking nth roots? Didn’t you say we were allowed to forget about that crap?). Also, there’s nothing about annuities, which are a natural application for series. Nor do we even see that the more often you compound interest at a given nominal rate, the more money you get. And we certainly don’t even consider the idea of continuous compound interest, because 1) then we’d really have to explain limits, and 2) the number e was a whole month ago, and damned if we’re going to explain why it has a special button on the calculator!

    Tom, the problem with the BC kids in your class wasn’t that they didn’t know vector algebra. That, they could have picked up quickly if they had a decent foundation for learning math. But they didn’t, and that was the problem.

    —————

    And I do recognize the progress that this kid has made with me, and I’m proud of both of us. I just wish that I could have spent this time actually teaching him math, rather than trying to optimize the amount of math-like stuff that could fit in his brain while he was writing the tests.

    - Moebius Stripper — 9/18/2005 @ 9:14 am

  9. Like M.R., I am not an expert on cognitive development, and I tend to buy into Piaget’s theories, again realizing that usually there aren’t sharp defining boundaries.

    I really do hope that computers will be the savior of the math curriculum. It all depends on how they are used. Computers certainly can become a crutch, but if people actually learn how to *program* them, and learn how make them do *exactly* what they want them to do, then they tend to be a revelation about how to do math instead.

    Instead of giving students a graphing calculator, give them a computer and get them to *create* a graphing calculator. Of course, languages and development technology needs to be carefully chosen so that students are not confronted with loads of superfluous issues.

    In order to program a computer, you have to learn what a variables, functions, and algorithms are. You have to get comfortable with manipulating values in systematic, algorithmic ways. But these concepts are not pre-requisites, rather, these concepts can be learned through computer programming. Computer programming is a great way to interact with abstract, esoteric concepts in a wonderfully concrete fashion.

    Not to mention that today’s $1000 desktops can do more raw computation in a few seconds than a human could possibly hope to accomplish in an entire lifetime. :-) You really have to be pretty mathematically sophisticated to make effective use of this incredible power.

    - Leon — 9/18/2005 @ 11:39 am

  10. i think we have to remember that this group of posters/readers is a self-selecting group. we might all have been able to do higher-order thinking at a younger age, but not everyone can. i’ve taught 14-year-old students who were working their butts off for months and getting C+’s until all of a sudden over the course of 3 weeks it all fell into place, and they got B+’s for the rest of the year. that felt like a developmental shift to me. i don’t think piaget had it totally right (from the little i’ve read about him), but i don’t think he was that far off base when he said that many tasks are cognitively too difficult before a certain age. in fact, one definition of very high intelligence might be that the brain can handle higher-order abstraction much earlier.

    mind you, i’m not saying that a young kid can’t, say, reason from the contrapositive correctly. but i’m saying that they won’t be able to understand the abstract notion of contrapositives, and why they have to have the same truth value as the conditional they came from.

    similarly, a 9-year-old might be able to do flawless long division using remainders, but not really understand why adding multiples of n to any number won’t change the remainder upon division by n.

    it’s abstraction that’s harder than we adults remember it.

    this is all, of course, based on my own admittedly non-brain-expert observation in classrooms. i would love to hear from a professional developmental psychologist about this (especially one who knows enough math to understand how abstraction like this works).

    - Polymath — 9/18/2005 @ 11:59 am

  11. i’ve taught 14-year-old students who were working their butts off for months and getting C+’s until all of a sudden over the course of 3 weeks it all fell into place, and they got B+’s for the rest of the year. that felt like a developmental shift to me. i don’t think piaget had it totally right (from the little i’ve read about him), but i don’t think he was that far off base when he said that many tasks are cognitively too difficult before a certain age.

    This is an interesting remark. Do you think the sudden improvement was because some developmental trigger turned on, and that this would have happened (mostly) independently of their “working their butts off” and your teaching? Your earlier remarks about how some tasks are difficult before a certain age made me think this is what you were suggesting, but now I’m not so sure. In reality I expect, as with pretty much everything, both biology and environment play some role. The sharper the transition the less plausible it seems to be that it can be brought about by diligent effort coupled with good teaching, and that troubles me.

    My worry is that if one assumes that development works this way — and again, I have little familiarity with the research in the field, especially more recent stuff that hasn’t worked its way into intro psych textbooks, so I don’t know how much is really known about this — then sub-par performance from students will become more acceptable. On the other hand, if such developmental stages can be “helped” by good teaching and hard work, then we can be more optimistic that improved teaching techniques at an earlier age can leave fewer students in the position of the guy Moebius Stripper is tutoring.

    - M.R. — 9/18/2005 @ 2:11 pm

  12. Well, I do not know Piaget, and I am not a psychologist, but I here is what I firmly believe. Almost all kids are born with the ability to do mathematics, to do it well and to understand it. This is a basic human skill. And this includes abstract reasoning and logic at an early age. However, most primary school teachers are mathematically illiterate and so is most of our society. So these kids never get trained to develop those skills because they rarely deal with somebody who has developped them himself! Most kids simply hate math because they had bad math teachers and because they were never taught math. If kids were taught math properly (and our society did not constantly send the message that hating math is the right thing to do), most kids would love math, and would be capable of abstraction.

    Sorry, I do not have facts to back this up. This is only my belief.

    - oxeador — 9/18/2005 @ 2:18 pm

  13. Young teenagers are perfectly capable of abstraction. They divide people up into abstract categories such as “jocks,” “losers,” “Goths,” etc. and observe that people in each of these categories follows a different set of rules. They manipulate social relationships that have many more than three dimensions, in their heads. Why couldn’t a young teenager make similar observations about and do operations on mathematical objects such as vectors, especially with some guidance?

    - Wacky Hermit — 9/18/2005 @ 3:04 pm

  14. For what it’s worth, when my wife was teaching elementary school math, she made sure that the kids (1) had a thorough grounding, (2) memorized what would be useful in later math classes, and (3) always always approached problems in several different ways, because she herself did NOT have good math teachers and was really put off math by a high school geometry teacher. But she knew the importance of math and worked really hard to make sure her dislike of math did not rub off on her students.

    Fortunately, she had me around to tell her what in her math books was garbage and what wasn’t. As far as I can tell, the “new math” was merely teaching kids the proper names for concepts, with little else really new at all — and teachers used the changes to form excuses as to why Johnny didn’t learn math! The so called new math never was implemented properly in the first place, so it’s hard to fathom why so many people said it failed. Of course, the same thing can be said for the whole language movement — it was never properly implemented, so it is totally false to say it failed. BTW, there is NOTHING in the whole language concept that says that phonics can’t be used, and in fact, good teachers of “whole language” use phonics where appropriate.

    - Rex — 9/18/2005 @ 4:24 pm

  15. As far as I know (this all based on PSYCH 101 a decade ago, bear in mind) the crux of Piaget’s theory wasn’t that one can’t master certain skills before a certain age, but rather that there are “prerequisites” to certain skills that have to be met before those skills can be attacked. Which dovetails rather nicely with the lion’s share of students’ mathematical weakness, at least as I’ve observed them: good luck teaching students who never learned basic algebra how to prove trig identities. Also, for those of you who didn’t see this: a few months ago, I solicited input from cognitive psychologists when I was particularly frustrated at my failure to get students to store their new math knowledge in a place less fickle than short-term memory. There are some great comments in that thread; worth checking out.

    Polymath: mind you, i’m not saying that a young kid can’t, say, reason from the contrapositive correctly. but i’m saying that they won’t be able to understand the abstract notion of contrapositives, and why they have to have the same truth value as the conditional they came from. I disagree, and I have the experience to back myself up. These sorts of syllogisms are really easy to make concrete. Consider the statement, “If it’s raining out, then I will bring my umbrella with me.” Say I don’t bring my umbrella with me. What can you conclude? And so on. Or even better: “if I had a million dollars, I wouldn’t be working at this job.” I’m at work today. Conclusions?

    But as I wrote earlier, students have trouble with simple if-then statements, never mind contrapositives. A large number of my probability students last year were unable to apply the concept of “if A and B are mutually exclusive, then P(A or B)=P(A)+P(B).” I could tell that they’d memorized that rule, though, because they’d make some statement about whether A and B had anything in common, and then they’d evaluate P(A)+P(B) no matter what they’d found.

    Wacky Hermit - oh, that’s excellent.

    Also, an aside about the kid I’m tutoring: he’s trying to get into a program that will require him to take a math class. One. And this math class makes the stuff I’m teaching now seem like rocket science: apparently anyone who can plug numbers into their calculators will get an A in it, and the textbooks helpfully tell students which numbers they need to plug in, and how to plug them in. So my student doesn’t see the point of having to learn grade 12 math. And you know, neither do I.

    - Moebius Stripper — 9/18/2005 @ 4:52 pm

  16. Going back to Wacky Hermit’s point — most people can do easily what is important to them in their lives. One of the most important things in life is relating to other people — (another one of these things is seeing things and recognizing them) — is it any wonder that most people can characterize others and say something about their personalities, and how well they get along with certain people?

    The number of layers one must go from there, the natural tasks of life, to proving the Pythagorean theorem is nontrivial (i.e. != 0).

    And yes, even smart people have difficulties with certain abstractions. Infinity is my favorite one - it’s definitely not a real-life experienced concept (unlike splitting people up into geeks and jocks). I’ve tried to teach some very smart middle-schoolers about the uncountability of the reals, and it just didn’t gel. A couple years older kids got it very well. And I’m talking the kids who are the best in math… and thinking back to age 12, it took me several years before I finally understood the implications of the Cantor diagonal argument. At which point, I was ready to tackle fractals.

    My measuring stick all these years has been =Godel, Escher, Bach= — the first read-through, all I got was propositional calculus and enjoying the dialogues and finding the word games. It wasn’t til later that I understood the self-referential abstraction of the Godel Incompleteness Theorem proof. There were other things in there that didn’t require reaching certain levels of abstraction in thought but simply the accumulation of more knowledge (like in grad school, wherein I learned some neurophysiology).

    What Leon said about making kids do numerical programming to learn math better reminds me of when I taught myself to program way back in the 80s. No web, no Windows, just DOS for me then (and then Macs, which put me in a whole different world). I remember, in particular, an assignment my Dad gave me: make a program to draw a circle of arbitrary size, with and without using the CIRCLE command (this was BASIC). And I did.

    Thing is, computers and calculators will not save math education. The teachers do need to know what they’re teaching about, and if they don’t even understand the math that underpins the computer exercises, they will be exercises in futility. My dad understood math, and so understood what would be good questions to ask and good principles to test. It would be best if the school systems threw away all the technology, handed the kids a bunch on paper and pencils, and make them practice the basic skills before allowing them to do ANY word problems.

    - meep — 9/19/2005 @ 1:51 am

  17. I grew up in a Piagetian household, so I’ll add two watered down factoids to this.

    1. The aquisition of “formal operational” thinking is not as if a switch has been turned on. However, there is an age (about 11) where one can start reasoning at a formal level in the logico-mathematical domain, but it takes several years (until the age of about 15) for someone to be consistently at this level. During this intermediate phase (termed “transitional operational” by researchers), the student may be good at X and bad at Y — even though to an adult X and Y might seem equally difficult.

    2. Unlike all the previous stages, the transition to formal operational thinking is not automatic. Students will not become formal operational thinkers unless they struggle at working at hard problems that require abstract reasoning.

    Mmmmm… jargon. (Do you want to guess what my dad’s research was in? Yep.)

    - Rudbeckia Hirta — 9/19/2005 @ 3:00 am

  18. one more bit (in response to Wacky Hermit and meep):

    The thing about people being able to do what’s important to them also helps in the transition to formal operational thinking. You know the low-level students that I have often found myself teaching? If you want to skew the test results so that the women score better than the men, write a bunch of tricky word problems about food, calories, serving sizes, exercise, and dieting. (”Activity X burns Y calories per hour. Use the Nutrition Facts label for food Z to determine how many cups of Z can be burned off by 75 minutes of X.) Students who care about this sort of thing will be able to solve it (and get the right answer!) using some sort of crazy method they invented themselves — even if they can’t do the basic proportion problems that you’ve been yapping about at the board for 2 weeks.

    - Rudbeckia Hirta — 9/19/2005 @ 3:09 am

  19. As for Moebius’s student, I think the Piaget stuff is irrelevant, isn’t it?

    The kid’s like 17 or 18, which should be plenty of time for abstraction to be in place, if it can exist at all.

    And it’s certainly not for lack of a) intelligence on the teacher’s part or b) lack of effort.

    I think the only conclusion here is that this kid’s mind isn’t wired for math. Hence, all the ‘workarounds’ like memorization, calculator skills, etc.

    It might be possible to teach him a kind of math that is more concrete (or taught in a concrete way, wherever possible), but to teach calculus to a kid with such poor abstraction skills (innate, in my opinion) is pointless.

    As a person with poor abstraction skills, I think it’s rather cruel to require these skills from a child where they cannot exist. ‘The system’ rewards workarounds that just waste time and don’t teach anything.

    Give the kid a good book or something. Or, if you’re a genius, developed concrete ways to teach abstract things.

    - John B — 9/19/2005 @ 12:17 pm

  20. RH, maybe you know this - is there an age past which people won’t be able to acquire formal operational thinking skills if they haven’t already? I remember reading that children who have not acquired a language by age eight (I think - is this the sort of thing a linguist might know?) then they’ll never be able to. Is there something similar with math?

    (My own family Piaget anecdote: my mom, whose degree is in psychology, is thirteen years older than her youngest sister. When Mom was taking her undergraduate psych courses, she used her sister as a subject, and was impressed to see that my aunt was going through all of Piaget’s stages in exactly the right order.)

    Also, regarding people being able to do what’s important to them - that’s why I like teaching financial math to low-level students, even though I find that stuff dreadfully boring. But people’s ears perk up whenever I start talking about how much money they could save by investing a few bucks a week in an RRSP.

    John B - if I drew the conclusion that you’re drawing with every student I had who struggled mightily with the subject, I might as well stop teaching. As karrde commented above, it seems like my student is currently at grade 5/6 level - which I’d say is about right. That’s three years of progress we’ve made in the past ten weeks, which I think indicates that my student actually very much possesses the ability to do math. That we haven’t progressed further suggests only that I am not a miracle worker, and I already knew that. Also, I’d like to repeat that this kid really did have some intuition for the combinatorics material - a unit that most people find very difficult.

    The reason for all these workarounds like calculator reliance and memorization have nothing to do with innate mathematical wiring. This kid isn’t going to win a Fields Medal, to be sure (heck, he’s never going to take a more difficult math class than the one he’s taking now), but he’s doing exactly what everyone else in the world does when faced with a task they find unpleasant: he’s taking the path of least resistance. It’s easier to memorize than to try to understand abstract material. And never before this summer has he ever had to learn abstract material - his calculator skills got him through grades three (ugh) through eleven. Why fix what you don’t realize is broken?

    - Moebius Stripper — 9/19/2005 @ 4:25 pm

  21. MS,

    i’m not sure we actually disagree. the point i was trying to make was exactly that “if i hear it’s going to rain, i’ll bring my umbrella, and i didn’t bring my umbrella” is a concrete example that even young children can understand. you can do example after example like this (i did several for my geometry class recently), and they totally follow the logic. and i said over and over that a conditional and its contrapostive have the same truth or falsity. but then my true/false question reads: “the contrapositive of a false statement may be true or it may be false” and several kids don’t get it. they say things like “the contrapositive of a statement is always true”.

    this is basically my point. success on concrete understanding does not necessarily predict success on abstract understanding (like your P(A or B) troubles). and it’s the abstractions that require a both a certain sophistication of thinking (much of which comes with age) and good teaching with hard work.

    and i think that readers of this blog may have an inflated notion of how easy it is to understand abstraction. try, say, teaching a 9-year-old how to add the numbers from 1-10 by showing the reverse-and-add-twice-then-halve method. then do it with 1-15 and even 1-20. i predict that the 9-year-old will be able to duplicate the procedure for 1-12 or 1-17, and might even be able to discover the basic rule, but only the brightest 5% (or something) will be able to follow the same proof using n for the largest number. i would guess that there is a high correlation between reading this blog and having been in that brightest 5%.

    - Polymath — 9/19/2005 @ 9:00 pm

  22. As my parents are visiting tomorrow, I shall ask my dad about the cut-off date question.

    (Why, yes, it would be TOTALLY FINE with me if my father with the doctorate in math education stayed at my house, nearly 2 miles away from my classroom while I teach on Thursday and Friday.)

    - Rudbeckia Hirta — 9/20/2005 @ 3:26 am

  23. To some extent, for me understanding seems to come with time, after I’ve done the memorisation and apply blindly step. I did reasonably well with maths at school (and in NZ at the time you couldn’t rely on calculators in exams) and then did engineering at university. But I still remember that day in my last year of high school, after 2 1/2 years of learning calculus, suddenly realising why differentiating a function and setting it to 0, then solving for x, produced the right result regardless of whether you wanted a minimum or a maximum. My teachers weren’t into constructionism, we were explicitly taught that differentiating told you the rate of change and that when the rate of change was 0 that was probably where the graph was changing direction. I memorised it, but I didn’t get it.

    And again at university - memorised drawing frequency graphs. Didn’t understand them. Two years later, teaching a friend who was a year behind me about them, I suddenly grasped how they worked.

    Now in my adult career, I expect that when I learn something new there will be quite a bit of time when I am just applying memorised rules. And something in the back of my brain will work away on it and in a few days or a few months or a few years send a note to the consciousness explaining how it actually works.

    So your student may suddenly understand things some time down the track.

    And congratulations on teaching him to test out rules on his calculators. I always felt proud when I had persuaded a student I was tutoring to do that without prompting from me.

    - Tracy W — 9/20/2005 @ 10:03 pm

  24. A disgruntled teacher drops a 100-pound anvil onto a calculator from a height of 12 inches…

    Via John, whose blog would be a daily read of mine if he posted that often but really no pressure buddy at all seriously none, here’s

    - The White Peril 白禍 — 9/25/2005 @ 1:38 am

  25. With modern research on learning, Piaget’s theories have been discounted in many areas. Mathematical achievement isn’t necessarily a given for most children, no matter the quality of the teaching.

    Formal operations: several years ago I filed away a quote from a book on giftedness. Apparently many studies have confirmed that some 50% of adults never reach the formal operational stage. My own observations in everyday discussions tends to confirm this.

    Last, from a decidedly non-mathematical person, is a question. Why do educational systems still insist on forcing higher mathematics on students who can barely handle the basics and will never have any use for higher math? We’re in the 21st century and education is still controlled by outmoded theories about learning.

    - Catana — 10/6/2005 @ 6:05 am

  26. Tracy W (are you still here? I should have responded to this comment awhile ago - it’s a good one) - I do hope that one day, things click for my student. I know that I’ve occasionally started understanding a difficult topic months or even years after I first saw it…but then again, I had to remain somewhat curious about the topic in order to even still be thinking about it. My student has made quite clear that he plans to never ever ever do any math, ever again…though maybe he’ll have to eat those words.

    Catana - do you have any data about mathematical achievement (at what level?) not necessarily being a given? Because I’ve had positive experiences with one self-directed system of learning math that has positive results more or less across the board. In school math classes in this province, though, I often feel that students who learn math, do so in spite of the classes they’re taking, not because of them.

    As for educational systems forcing higher math on students who can barely handle the basics - I think that the issue is that those systems have no real way of even measuring whether a student knows the basics. Nor do they know much about either pedagogy or mathematics. How else to explain the disjointed curricula? However, I maintain that every member of a democratic society needs, and is able to acquire, a basic understanding of statistics and quantitative analysis. Anyone committed to fostering media literacy should agree - understanding statistics is essential to being able to critically assess the quantitative aspects of the news.

    - Moebius Stripper — 10/6/2005 @ 10:21 pm

  27. Sorry, no data on math achievement, but it stands to reason that, given nature’s uneven distribution of talents, there are going to be some with no math ability at all. I admit I might have learned a little more with good teaching, but dealing with numbers in any way, shape or form is something that my brain doesn’t handle. I’m very analytical, but only on a verbal level.

    You’re quite right that failure to distinguish between the able and the not-able is just one of the features of the educational system. I’m fraid my question was purely rhetorical.

    - Catana — 10/7/2005 @ 12:14 pm

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