Comments

Catherine

Go back to Joanne's blog and read the comments on the "engineering washout's" story. Many who commented label him a whiner and so do I. I had plenty of misgivings about some of the teachers I had at U of Michigan as an undergrad math major including a professor in advanced calc from Poland who couldn't speak English. It sucks when it happens, and I was plenty pissed, but somehow I stuck with it. And by the way, through the magic of internet, I looked up the professor and have a correspondence with him.

I'm also in contact with many mathematicians, quite gifted far beyond what I could ever hope to achieve, who said they had to work hard in school. Maybe someone like John Nash could get it without trying, though he paid his dues in other ways. But I agree with Bernie, and I believe others do too. It takes hard work.

As far as learning math, take a course and see what happens. My bet is you will learn something. (There are plenty of people around who will help you by the way!)

BG

-- BarryGarelick - 26 Sep 2005

I want to respond to this some more tonight.

Please take heart from my early history as a devoted underachiever. Talk about 'Not a 4' (and by the way, what the heck do they mean labeling Christopher in that way? That is complete crap. Please pardon my strong verbiage. It makes me angry).

Catherine, I am telling you: Christopher is a "4" if you and he want him to be, and you too are a born math learner. Not only that; you will learn, as you go, more about how to learn math, and the process of learning math will get easier. Your devoted cohort here will do all we can to help you learn math, and to help you learn to learn math (plus, we can tell you what the 'good books' are).

More later.

-- CarolynJohnston - 26 Sep 2005

I HAVE to 'blooki' Wayne Wickelgren SOON--remember, all those stories about people who didn't discover their math ability until late.

-- CatherineJohnson - 26 Sep 2005

Hi Barry--

He seemed like a whiner to me, at first, but I think he has a point.

Ed had the same experience at Princeton. He had been one of the best math students in his high school, which had a rigorous program (including SMSG geometry) and the Princeton advisor put him in the advanced freshman engineering class.

He managed to pass, but he was at sea every second of the course. He went to the professor's office constantly, and the professor tried to help him. But the guy had no way to explain things at Ed's level.

I've begun to think there's a specific challenge in teaching math (this may be wrong of course), which is that math is almost freakishly removed from language.

I realize this is going to put me way left of field of the math-is-a-language folks.....but math is very, very, VERY far from verbal language.

The other subjects I've self-taught could be pretty fully expressed, described, and taught in words. By the time I got to the point where I needed to learn more....'domain specific' content, including math (usually statistics) I had a lot to go on.

Also, I certainly shouldn't say that natural born math whizzes don't have to work; they do.

That's not remotely what I believe; geniuses work harder than anyone else. That's what makes them geniuses, ultimatly, though they have to be born with the right stuff to begin with (IMO).

-- CatherineJohnson - 26 Sep 2005

Carolyn, THANK YOU!

I have to say, contemplating college math courses has got me shaking in my boots.

I've thought about going to THE easiest college I can possibly find, and trying to hang in there.....BUT I think it's a distinct possibility that the 'easiest' college will have the hardest courses, if the teachers don't understand the content themselves.

The How to Ace Calculus book (which lots of people seem to love) says to do heavy-duty research on the professors, so that's what I'll do. I'll contact you NYU friend (and Ed will ask people) and find out who has a reputation for being able to teach math best.

-- CatherineJohnson - 26 Sep 2005

The author's HUGE mistake was taking the genius course.

That's exactly what happened to Ed, though he only took the course because he was told to.

But it wasn't the spot for him, and it probably did derail him from any further mathematical work, in any subject. He doesn't say that, but I suspect it's true.

The funny (not funny ha-ha) thing with Ed is that math was his strong suit. His math scores were always higher than verbal, which wasn't the case for me.

-- CatherineJohnson - 27 Sep 2005

hmmm

I just read this comment, and it's not reassuring:

What a whiner! I struggled through Engineering School with an undergrad 2.7 after 4 years. I also worked hard enough that the faculty bent the rules and allowed me to go for a PhD^?. This twit found out it was too difficult and actually quit before two semesters? Give me a break!

This sound like the classic case of someone who was never told that studying was hard. Working for grades is hard. Too much positive feedback and not enough reality.

Sorry, but Engineering is supposed to be tough. When mistakes you may make can potentially cost people their lives, then you want to make sure you do not let any of the riff-raff in.

As one of my professors was fond of syaing: "You don't get partial credit if the bridge falls down. You will get all of the blams."

First of all, I think this guy probably ought to be able to use the English language better than this, though maybe I'm wrong.

I don't think this guy could teach well with language skills this poor. Though, again, I could be wrong!

But what concerns me is what the Tech Central guy is getting at: does this guy really know engineering?

The fact that he struggled through with a 2.7 and rules were bent to give him a Ph.D.......what has happened here?

CAN this fellow build a bridge that doesn't fall down?

My sister-in-law, whose husband is an architectural engineer, says that new hires coming in with Master's degrees in engineering know nothing.

I'm sure all of these people worked their fannies off, but that's not what counts.

What counts is: do they know what they're doing?

My brother-in-law tells stories that would curl your hair. And these people have Master's degrees, perhaps with the same 2.7 averages.

I think the tone of the Tech Central piece obscures the argument, which is that engineering school as hazing doesn't serve anyone's purpose.

As a person who drives over bridges, I want students in engineering to be mastering the material, not bogus on-the-curve pseudo-grades that hinge upon the amount of partial credit that bored T.A.s choose to dole out.

What Kern is saying, I believe, is that, yes, people are working incredibly hard in engineering school.

But many of them may not be mastering the material. He was passing, and he wasn't mastering it:

I nearly fainted when I learned that I received a 43% on the Physics final. I nearly fainted again when I learned that the class average was 38%. A sub-50% grade on a science test is a curious creature, as much the product of grader whim as academic achievement. "Hmmm…looks like he understood a tiny bit of this question. I'll give three points out of ten. Or should I give four? Whoops…tummy rumbling…better make it three." Having allegedly mastered 43% of the course material, I was now deemed fit to take even harder Physics classes. I wondered: at the highest levels of physics, could you get a passing grade with a 5% score on a test? A 3% score? A zero?

-- CatherineJohnson - 27 Sep 2005

OK, now I'm seriously off the boat:

Seriously, it's a person's responsiblity to educate him/herself.

If it's a person's responsibility to educate him/herself, parents don't need to be spending $40,000 a year for the university not to do it.

I'm serious.

-- CatherineJohnson - 27 Sep 2005

Professors have jobs; they have a responsibility to teach, and to teach well.

I did when I was a professor, and Ed does now.

-- CatherineJohnson - 27 Sep 2005

However, the complaints about classes being left to TAs that are'nt up to the Teaching part of that job title may be valid. It depends on which school you choose. From what I've heard, there are prestigious universities where you'll get a similar situation in nearly every major, but there are also universities where the professors are expected to teach, and mostly do it well. So maybe I was lucky that circumstances precluded prestigious schools and I wound up at Oklahoma State U.

true

-- CatherineJohnson - 27 Sep 2005

Gone for Good: Tales of University Life After the Golden Age

One of Joanne's commenters recommends this book--sounds good.

-- CatherineJohnson - 27 Sep 2005

As I have said in other posts, when I was doing my grad work at a research university, there were 2 US students out of a total of 30 (Master's and Ph.D's). Pretty sad.

This is sure what I hear. I had an email the other day from an engineer saying that she/he'd seen job openings with 100 candidates applying, not one of them from America.

-- CatherineJohnson - 27 Sep 2005

OK, I have officially taken sides in the joanne jacobs thread; I'm with SuperSub, ragnarok, & Jeff.

I'm not with Mad Scientist.

Here's SuperSub:

zWell Mad Scientist, I just guess we have a significant difference in opinion on the function of universities. I believe that universities should educate students to better prepare them for whatever careers they choose, while you seem to think they are research centers where students learn through diffusion.

You want engineers that have just read some books and did some homework problems? I'd rather have engineers that have been actually taught by experienced engineers who know what it takes to make the world work.

Oh wow!

Tracy's good, too:

Also, despite what Mad Scientist says, if universities teach well, then their students can master more in the same space of time. That's really valuable when lives are on the line.

This is a far more optimal solution than Mad Scientist's. Students get all the benefits from working hard, and they actually spend their time doing something useful. It's not like there's so little to learn about engineering that professors can afford to waste time.

This is my theme: how about some efficiency?????

How about NOT having thousand-page textbooks that teach every concept under the sun?

The idea that engineering school should be 'Hard Work' and if you've got Hard Work you've got a good course means they're not doing their job.

Hard work is just the beginning.

If teaching a hard course were all there was to teaching engineering, I could teach engineering now.

I sure as heck wouldn't be explaining anything to anyone.

-- CatherineJohnson - 27 Sep 2005

Another great comment!

(I've had a hard day.....)

This one's from My Pal SuperSub:

I can tell you that pre-med programs actually do a heck of a lot better job teaching students than you're representing. My professors were always very connected with the students, willing to discuss material to clarify it for them, and had capable and English-fluent TA's. Seems they wanted to make sure students really knew the material, because in their cases people's lives really are on the line. They didn't want a self-taught genius working with some incorrectly formed assumptions.

This is the crux of it for me. Do graduates from engineering schools who have never laid eyes on an actual professor know what they're doing?

Why are people being passed through with grades of 43%?

-- CatherineJohnson - 27 Sep 2005

And one more thing!

Mad Scientist the guy with a 2.7 average for whom rules had to be 'bent' so he could get a Ph.D. is the same guy who keeps bringing up the image of bridges falling down!

-- CatherineJohnson - 27 Sep 2005

yikes!

-- CatherineJohnson - 27 Sep 2005

"Engineering professors are perfectly happy weeding out undesirables with absurd boot-camp courses that conceal the inability of said professors to communicate with words."

This commentary has so much baggage that it is impossible to separate the valuable feedback from the whining. The weeding out could just be because the material is tough. It could be because the TAs/profs are really bad. To flunk out, however, takes more than one or two bad teachers. It sounds like he is extrapolating far beyond his own data just to make himself feel better.

"But I'm not sure there are teachers out there who can teach me."

... or books. If you take enough math, engineering, or science courses, you develop a feel for things - like the gut feeling you get that it has to be easier than this (teacher or book). I remember one book I was using to learn some mathematical theory about surfaces. Impenetrable. Either that, or I wasn't ready for the material yet. Perhaps I wasn't ready for that book yet. (or ever) I have some books on my shelf, however, that are dearly loved and filled with notes, pieces of paper and scribbles. These were my learning books. So clear. So easy. Finding the right book can make all of the difference. I will say, however, that for some of these books, I had to finally decide to sit down and go through them slowly, page by page, equation by equation.

"I've never met anyone, apart from Bernie, who sees math as first and foremost a matter of hard work."

Can you see my hand raised? For anything in life that is difficult, you have to be willing to put in the effort. There are those who seem to have some natural innate ability (math, writing, dance, etc.), but they still have to work at it. It's kind of self-fullfilling. If you like something or get off to a good start in something, you will work harder. If you work harder, then you will like it more. Kids will then decide that they are good in math. If kids get a bad education in math, they will decide that they are just not good in math.

"I've never thought of Christopher as a 4," she said.

Can you hear me sputtering here? This is just so AWFUL! In fourth grade? A math-ignorant lower school teacher deciding on the future of your son? To be a 4; to get the (supposedly) good curriculum, you have to have some innate ability that is recognized by this ed school graduate? And, this innate ability shows itself at that early age? Can this teacher tell the difference between innate ability and those kids that got an early head start at home? Perhaps it is because of BAD CURRICULA and BAD TEACHING that kids look like 3's rather than 4's. How many of the 4's got help at home? Incredible!

"I guess what I'm saying is: Confessions of an Engineering Washout tells me that we have a math teaching problem at the professional level as well as the elementary, middle, & high school level."

His commentary tells me nothing, but the problems are in grades K-8. If those grades were improved, you would be amazed to see many more kids in high school college prep math who felt they were "good in math". We are not talking orbital mechanics (rocket science) here.

-- SteveH - 27 Sep 2005

I believe math is hard work, but does it feel like hard work if you're enjoying it most of the time?

The problem wasn't that this guy was working... I think it's that he was suffering. He doesn't sound as though he was enjoying anything at all about engineering school; he had just decided that that's what he was going to do. It wasn't fun for him.

-- CarolynJohnston - 27 Sep 2005

Steve, excellent comments. (The whole phase 4 thing makes me spitting mad too).

-- CarolynJohnston - 27 Sep 2005

This discussion has triggered so many reactions in me that I doubt I’ll be able to coherently relate any of them. But that’s never stopped me before…

The idea thread about math being easy for some people and hard for others is interesting. Throughout school, there are always kids who are “good at math.” I have no idea why. Perhaps their brains accommodate better “spatial reasoning” or something. Or, perhaps they just pay better attention, or they just enjoy solving problems (as distinct from telling stories or discovering foreign cultures or whatever). I think, though, that at some point, math becomes hard work (or confusing) for everybody. If, for you, that point is about the time you’re learning algebra, then I guess you’ve got a couple of choices: you can become a mathphobe or you start working hard at math. For all I know, you may become a mathematician through all that hard work. Others can get well into college before math seems confusing or hard. Again, that doesn’t mean you stop your math education at that point; it means that your continued math education requires hard work. You may choose to avoid that hard work by choosing a major that does not require math beyond your comfort zone (e.g. Finance). If you really want a career involving difficult math, it may actually be to your advantage to get into the math-as-hard-work mode earlier rather than later. I don’t know. I did well at math into my undergraduate engineering work. By my sophomore year, math was getting pretty hard for me. I did not respond by working harder at math. I did not go to my instructor’s office hours. I just tried to do well on my tests in sophomore math. Then, by junior year, I was done with math department classes. From that point, I was in engineering classes. There’s a lot of math in Theoretical and Applied Mechanics and Electrical Engineering classes, but it’s in the context of computing stress and strain or semiconductor doping or control systems. To me, that’s a different context in which to understand the math, and you learn what techniques to apply to a particular type of problem. That feels very different from learning the math in the abstract. Of course, even in engineering, there are areas where the math can overwhelm one. That happened to me in grad school. I was lost in my Communication Theory class: lots of determining theoretical limits and differentiating inside the integrals and stochastic processes (WAH!). I also made the mistake of taking a real math class again in grad school: Probability and Statistics. I was okay pulling red and black marbles out of a bag, but t-tests and beyond just seemed too hard. Again, I didn’t choose to work hard; I decided that I had reached my math limit. In my career, I steered clear of such hard math.

The concern about students coming out of college being unable to design a safe bridge is overdrawn. In the real world, people collaborate. College is great background, but you will learn the specifics of your job from lead engineers and other colleagues. Companies have software tools and design methodologies that new hires learn on the job. At worst you will be unproductive, but you won’t generally get the chance to design a faulty bridge on your own.

I agree that bad TAs are out there. I recall two classes where I had the highest scores in my section, but still got a B in the class. I always felt that showed what a lousy TA I had, and I should have been given the A for doing as well as I did with such a stooge.

The stuff about passing with 43% is impossible to gauge. You can make a test hard by including problems far different from what has been seen in homework, or you could include too much to work through in the allotted time. I remember getting take-home tests where the instructions were to pick five of the seven problems to do, and don’t spend more than 8 actual working hours on this test. I have successfully blocked from my mind the specifics of any of those problems, but I know they were very hard, and partial credit was my only hope to score any points.

Still, I don’t buy that engineering school is so hard that students who are great prospective engineers are being crushed and driven from the field. A springtime visit to West Lafayette, Indiana or Urbana, Illinois will reveal that thousands have survived to graduation.

-- DanK - 27 Sep 2005

Throughout school, there are always kids who are “good at math.” I have no idea why. Perhaps their brains accommodate better “spatial reasoning” or something.

It's fascinating.

It seems like this group is maybe.....5% of all kids?

Maybe even smaller.

Christopher's 4th grade teacher told us that in any given class--and I think our class is 150 kids--there will be 3 kids who are 'naturals.' (I wrote this down when she told me; wish to heck I knew where the notes were.)

When I ran this figure by my neighbor, the statistician, she said her dad would have put the figure in the same range. He was a high school principal for many years.

That reminds me: I need to get to my post on 'Our Best Students.'

I think what I've found is that about 5% of our high school seniors are competitive with the world's best students.

That would make sense, wouldn't it?

If we're doing a poor job of teaching math to the 'non-naturals,' the 3% to 5% who can learn math 'by smell' would be the students who can compete with the top students in Singapore, Japan, & China.

-- CatherineJohnson - 27 Sep 2005

To flunk out, however, takes more than one or two bad teachers. It sounds like he is extrapolating far beyond his own data just to make himself feel better.

That's the important thing to remember here: he didn't flunk out.

His grades were slightly above average, and remained so until he quit.

He's saying that engineering programs are washing out people who could actually be good engineers.

-- CatherineJohnson - 27 Sep 2005

_Can you see my hand raised? For anything in life that is difficult, you have to be willing to put in the effort. There are those who seem to have some natural innate ability (math, writing, dance, etc.), but they still have to work at it. _

I should put that more clearly.

'First and foremost' is the key phrase here--although it's entirely possible that actual mathematicians & engineers don't feel this way.

But everyone in my own walk of life thinks of math as something you're born with, or not born with.

Have you come across people commenting that no one would ever say, 'I was never good at reading,' but people have no qualms about saying, 'I was never good at math.'

Drawing is the exact same thing; people think of the ability to draw as almost magical.

They say, 'I can't draw.'

When I finally took a drawing course, a Betty Edwards course, I was shocked.

Not only is drawing easy (simple realistic drawing, that is), it's math.

That's all it is.

Math.

-- CatherineJohnson - 27 Sep 2005

That was funny, because we were taking a 5-day drawing boot-camp course, and in 20 minutes' time the instructor taught us how to do proportional drawings......and believe you me, there were maybe 2 people there who could do it, one of them being my neighbor, THE STATISTICIAN.

There were two retired ladies who got so frustrated they quit.

They just didn't have a CLUE how to hold up their pencil to whatever they were drawing, use it as a sight line, and THEN figure out the appropriate proportions in their heads AND put it down on paper.

I couldn't really do it, either, though I understood the principal. But I needed a LOT more practice, and on much, much simpler forms. (We were drawing the corner of a room, along with everything around it, which included an elevator door, a stair well door, etc.--way, way too complex to start with, though I brute-forced it through.)

btw, now that I think of it, that was an example of 'more' not being 'more.'

That exercise was too complex, and too hard.

I didn't actually learn anything about proportional drawing; I just brute-forced my way through.

If I'd had a series of proportional drawing exercises building from simple to complex, I would have acquired knowledge.

As it was, I made zillions of mistakes on each line I drew, figured each one out in an isolated & fragmented way, then retained nothing from the experience.

When you have to make many, many mistakes to draw one line correctly, it's extremely hard to remember what it was you did correctly. There's too much chaff.

-- CatherineJohnson - 27 Sep 2005

Can you see my hand raised? For anything in life that is difficult, you have to be willing to put in the effort. There are those who seem to have some natural innate ability (math, writing, dance, etc.), but they still have to work at it. It's kind of self-fullfilling. If you like something or get off to a good start in something, you will work harder. If you work harder, then you will like it more. Kids will then decide that they are good in math. If kids get a bad education in math, they will decide that they are just not good in math.

Yes, definitely. The Johns Hopkins folks did all kinds of research on the gifted & talented, and invariably what you see is that children with innate gifts will then work harder at those innate gifts, and become better.

They are also, invariably, supported by parents who recognize their gifts early, and put a lot of time and effort into providing lessons & opportunities for their child or children to develop their gifts.

Steve's right, I think, that when it comes to math we're losing kids .... I suspect we might even be losing kids who are gifted.

I know a little boy who loves math, but is also (just barely) classified as special needs. His mom is a mathphobe, and can't quite bring herself to think her son ought to be in the advanced class.

Now, it's true that, if he were in the advanced class, he'd need extra support. Language is hard for him.

But so what?

My feeling is: get him that extra support, and make it possible for him to push forward.

Nevertheless, I understand completely why she's fearful of pushing for this. She has less-than-zero confidence in her own ability to provide support (and that includes being able to evaluate any extra support she provides. There are plenty of useless tutors out there!)

-- CatherineJohnson - 27 Sep 2005

The stuff about passing with 43% is impossible to gauge. You can make a test hard by including problems far different from what has been seen in homework, or you could include too much to work through in the allotted time. I remember getting take-home tests where the instructions were to pick five of the seven problems to do, and don’t spend more than 8 actual working hours on this test. I have successfully blocked from my mind the specifics of any of those problems, but I know they were very hard, and partial credit was my only hope to score any points.

That's reassuring.

I'm used to more rational tests, tests that you REALLY do want to make sure a person can PASS!

-- CatherineJohnson - 27 Sep 2005

My freshman-level general physics course (for majors) had its curve announced at the beginning. (The class was graded entirely on in-class tests.) That curve was 80/60/40/20.

On the first test in my junior-level Electricity and Magnetism class I got (IIRC) a 26%, which was the second-highest grade in the class*. That year I shared an award for best physics junior in the school.

In the latter, I didn't learn much E&M, but I got a B in the class. Rationality can be concept foreign to grading curves.

-- DougSundseth - 27 Sep 2005

Doug--

Here's my question.

What did a grade of 26% mean, if anything?

Had you mastered the material you could have & should have?

btw, I should probably explain that I'm coming from a background in behaviorism & ABA training for autistic kids.

In behaviorism, you're constantly looking at mastery, defined fairly simply I'd say.....i.e. you don't get into a lot of questions of novel problem-solving.

In a behavioral training or teaching program, you want to know whether the student has learned the content 'to mastery.'

That is typically defined as a correct answer rate (and in behaviorism there are beaucoup correct answers; that's one of the main fields constructivism is reacting agains) of 80%.

I've felt that our schools here ought to set a goal of mastery, and I've had teachers tell me that what they want to know is this:

'Am I teaching for coverage, or teaching for mastery?'

(That's an anti-constructivist teacher stance, I'd say.)

With curves & grades of 26%, etc., where does mastery come in?

And are there alternative paths to ensuring that a student has mastery? (I'm thinking of the joannejacobs comment from a physician who wrote about how med school teachers made sure their students had mastered the techniques they were being taught. Med school doesn't seem to have a grade-on-the-curve concept. Med school seems to have a mastery concept.)

-- CatherineJohnson - 27 Sep 2005

"What did a grade of 26% mean, if anything?"

One or more of the following (I suspect all): the test was inappropriate for the class, the teacher was a fool, and that I hadn't learned the material.

"Had you mastered the material you could have & should have?"

I'll give that an unqualified "no". That same "no" applies to the material from the class as a whole. (And probably for most of the other members of the class as well.)

That professor was the worst teacher I can remember, and like most I have several anecdotes. (I may have more than most, since I was in nine schools K-8.) He taught directly from the book, with no evidence of preparation at all. When asked a question in class, his response was to read the appropriate section from the book. This was a class for Junior-level physics majors; we read the book before class. If we asked question, it was because the book failed for us. A reasonable person might suspect a different approach, or at least a few more examples, could have been useful.

Now E&M is hard (at least everyone I've talked to seems to think so), but Mechanics, which I was taking at the same time from a different professor, has the same reputation. I felt that I actually learned something in Mechanics. E&M was almost purely an exercise in partial credit.

As to mastery, I think that's a harder question, and I think there are several things that need to be addressed separately:

1) How much of the material forms a foundation for future pursuit of the subject? In the case of math, engineering, the hard sciences, etc., this may well be most of the material. If you have a systematic deficit (10% of the core material wrong 95% of the time, for instance), you've a serious problem.

2) Will the same subject be addressed in future classes? If this is a first exposure, intended mostly to provide context, the required level of mastery may be much lower.

3) How fault-tolerant is the subject? If a political scientist falsely remembers the capital of Nebraska as "Omaha", the consequence is likely to be minimal. It doesn't come up often, and if you make the mistake, it's not likely to cause more than embarrassment. If an engineer can't remember how to reduce algebraic expressions 20% of the time, he's not going to be employed very long (assuming that he doesn't go insane from sleep deprivation in college).

In short, I don't think a single number captures the issue very well.

80% may be adequate for general knowledge subjects. If you forget the 20% of the principal exports of Brazil, it's only a problem if you are importing from or exporting to Brazil. If you can regurgitate (if you will) the correct answer 80% of the time, you probably have a strong understanding of geography.

For the "deep core" issues, though, like basic math facts, I'd say the number for mastery is closer to 95%. All errors should be caused by mistakes rather than inability to answer the question if asked again. Any systematic error pattern should be remediated as soon as practical.

-- DougSundseth - 27 Sep 2005

Interesting.

I'm trying to think where I got the 80% figure.....and I hate to tell you this, but I'm pretty sure I misrememberd.

I think the 'real' figure (hah) is 90%.

-- CatherineJohnson - 27 Sep 2005

Yup, that's what my neighbor told me (and it's possible the figure used in my kids' autism school was lower, now that I think of it).

I'm remembering this now, because last year she and I were both obsessing over having our kids get 90% or higher on math exams, on grounds that 90% is the behavioral definition of mastery.

-- CatherineJohnson - 27 Sep 2005

I have to say, in practice--practice dealing with elementary school math--90% actually seems like a good figure.

So far, I haven't seen 90% grades trip Christopher up in moving on to the next material.

-- CatherineJohnson - 27 Sep 2005

"So far, I haven't seen 90% grades trip Christopher up in moving on to the next material."

To some extent, I think this might reflect the way math is taught. Since you need most of the earlier stuff to do the later stuff, you are implicitly being tested on the earlier material in most subsequent tests.

In order to stay at 90% on the newest stuff, I think you have to have a much better understanding on the material on which the new stuff is based. If you make a mistake on 10% of single-digit addition problems, you'll not come close to 90% success on basic algebra, since that requires correct answers to multiple addition problems for each algebra problem, in addition to testing new skills that can be sources for errors.

OTOH, if you can't remember state capitals (to use an earlier example) at better than 80%, that skill will not be implicitly tested on other problems in the future. You won't get error source adding upon error source in the same way as you would in math.

-- DougSundseth - 27 Sep 2005

To some extent, I think this might reflect the way math is taught. Since you need most of the earlier stuff to do the later stuff, you are implicitly being tested on the earlier material in most subsequent tests.

I was about to say that, and then I got lazy. (I swear.) Yes, the next material, one hopes, should rehearse the previous material (though you can't count on that in U.S. textbooks).

I'll have to go back through Russian Math to see whether every single chapter builds on the one before. Offhand, I have the sense that they do.

But the book also has tons of review if earlier material in every chapter & chapter section.

One very cool thing: they use the material on circles, circumference, etc. to lead up to creating quite complex circle graphs, which of course involve using fractions & percents & converting those to sector angles. Not to mention whopping big division problems. (One problem was to make a circle graph repesenting the oceans of the globe, whose areas were given in the problem.

So they've got data & representation & statistics & all that, but nary a bar graph in site.

Wonderful.

-- CatherineJohnson - 28 Sep 2005

oh wait

boy, i'm getting tired

you're saying that 90% will do for subjects less hierarchical than math

right?

-- CatherineJohnson - 28 Sep 2005

This is making me think that procedural fluency & drill are even more important than I've been assuming, because you simply can't get deep conceptual understanding quickly. At least, I can't, and I'm pretty sure most kids can't.

What you can get, quickly, is procedural fluency, with beaucoup practice & drill.

So.....when you're trying to get kids through an accelerated math program (which is a normal track for kids around the world) I'm thinking you're going to have to lean very heavily on procedural fluency, just to be able to keep moving.

Then you have to.....keep checking to see whether the conceptual understanding is 'coming in.'

I've got to take more notes on my own re-learning process.

I've noticed there can be quite a significant lag time between procedural fluency & conceptual understanding.

I've also noticed--and it's been strange--that ideas I was beginning to despair of ever understanding will 'come into focus,' so to speak, without my noticing the process.

What's funny about this is that it's a different kind of understanding.

I've mentioned before that my standard for whether I understand something or not is whether it seems like magic.

If it seems like magic, I don't understand.

I've also noticed that I seem to 'habituate' to some of the magic.

After I've used a procedure for awhile, it stops seeming like magic, and starts seeming normal and natural.

I get used to it.

I don't know whether that means I understand it.

But it feels as if I understand it.

+++++

hmm

That could just be Mondo Procedural Fluency......

sigh

-- CatherineJohnson - 28 Sep 2005

Catherine,

It’s probably different for different people, but I think a big reason for gaining understanding some lag time after attaining what you call procedural fluency is that during that lag time you move on to problems that build on the procedure in question. Seeing how the procedure is applied, you gain insight.

For me, a good example is using math in physics. Early in calculus you learn about how the derivative is the rate of change of a function. You look at a function like y = x^2, and you kind of get how the rate of change is increasing. You’ll take the teacher’s word for it that that rate of change turns out to be linear. Then, you move on to talking about second derivatives and inflection points. Okay, were those points in the original function or in the derivative? It’s the rate of change of the rate of change? Okay. I can follow the example. I can pass this week’s quiz. But I don’t really get it.

Then you start to do physics problems where the function describes the position of an object. That function’s derivative is the object’s velocity. Okay, that’s how fast its position is changing. I have an intuition of what velocity is, and it matches that stuff about the derivative being the rate of change. Then, the second derivative is acceleration. Of course. That’s how fast you’re changing the velocity.

So, you not only learn the physics, but you start to develop a conceptual understanding of that sterile calculus that you were struggling to memorize or extrapolate from the example problems. I think that very often, you don’t understand topic A until you’ve moved on to topic B or C, which depend upon topic A. It’s not just that you have more practice doing problems that include topic A. It’s also because you now see why people bother with topic A in the first place: to do topic B and topic C.

I’m groping for a good example that’s encountered in earlier. Perhaps means and medians is a good one. You learn how to find the mean and median of a given data set, but they seem interchangeable. They both give you the “average” or the “middle.” It isn’t until you’ve considered a lot of different data sets that you come to understand when one might be a more useful measure than the other or how a few new data points can change them. All right, I’ll admit that was a pretty lame example. Okay; I’ve got it: you don’t see why factoring is important until you start reducing fractions. No. I think kids understand factoring pretty well even before applying it to fractions. Bah! My whole thesis is about to collapse because I can’t think of a supporting example.

Sometimes it’s best to quit while you’re behind.

-- DanK - 28 Sep 2005

"I've noticed there can be quite a significant lag time between procedural fluency & conceptual understanding.

I've also noticed--and it's been strange--that ideas I was beginning to despair of ever understanding will 'come into focus,' so to speak, without my noticing the process."

It can take years. In the case of learning differential geometry (which I discussed at length in this post), some concepts have snapped into focus more or less unbidden -- that is, I wasn't learning things that built on it. It was as though some subterranean process had suddenly coughed up an answer (sorry, I know that's mixing metaphors).

-- CarolynJohnston - 28 Sep 2005