Comments

Wow.

Another tour de force.

So much to chew on here......

Also, we have GOT to have a 'Greatest Hits' spot SOON.

Because we're building a collection.

-- CatherineJohnson - 28 Sep 2005

For some people math problems are like puzzles (I think that Catherine is discovering this trait within herself these days)

I think that's true, and it's come as a huge surprise.

For quite awhile, I thought I was doing what I was doing because I was determined to see Christopher succeed. (And 'success,' at first, meant simply: stop failing. Which was hard enough.)

Mama Bear motivation is vast & as far as I can tell unlimited, so it took me quite awhile to realize another motivation had surfaced, which turns out to be the fact that, apparently, I love math.

I'm still getting used to this idea.

-- CatherineJohnson - 28 Sep 2005

I'm fascinated by the idea that everyone, eventually, hits the wall.

This is one of the things that keeps me coming back to the feeling that Math is Different.

I don't think that people in other disciplines would say they eventually hit the wall (I suspect physics may be one, but for the same reason math is one).

In neuroscience research, for example, there are hard, hard questions people are banging their heads against.

But it's not the same, IMO.

Neuroscientists are up against the limits of their empirical knowledge, and possibly against the limits of their paradigms.

However, they're not having a problem really understanding their existing empirical knowledge and paradigms.

They're trying to figure out where to go next; they're trying to SEE what their data is telling them, to steal the secrets.

Some--a few--may be thinking whether the paradigms themselves need changing. (I have no idea; I'm not suggesting that's what anyone should or should not be thinking about.)

It's different.

-- CatherineJohnson - 28 Sep 2005

Wonderful post, Carolyn.

I think a big reason why people think of math as being purely innate is that the mathematically talented have trouble explaining why they understand something. When it's broken down step-by-step, we can see the work that goes into it, and can repeat with effort what others do with mere talent. But when you know the answer is right just because it looks right... well, you may just as well have explained it with voodoo. The triangles are similar because the Magic 8-Ball says they are.

I think this goes back to the question of automaticity and overlearning, as well. Some ideas come naturally to some people, and they 'get it' instinctively; others will never quite reach that same level, but can approach it by overlearning. Obviously, the person with the talent is at an advantage, but not an insurmountable one. It just takes practice - probably lots of it.

-- IndependentGeorge - 28 Sep 2005

Just talked to Ed about this (and about KDeRosa's Terminator Math essay from yesterday.

He says there's no wall in history.

Historians have different levels of output & brilliance, and you can see a point at which a great historian has peaked, and his work isn't as good as it was.

But there's no wall.

-- CatherineJohnson - 28 Sep 2005

"Sooner or later, you're going to hit the wall if you keep up with it. I don't think having a "math brain" that makes math easy for you is what's necessary for success in math."

Especially if people claim that kids hit the wall in third grade. There is a really, really big difference between doing well on trivial NAEP-type tests versus doing well in engineering school. Very few students should have any kind of difficulty until they get to algebra. Everyone should get the same rigorous math curriculum until about 7th or 8th grade, where you can begin to separate those who can or will from those who can't or won't.

I would strongly say that this discussion has absolutely nothing to do with what we should expect from our kids in elementary school.

I think it's impossible to separate interest in a subject (or help at home) from any sort of innate ability in that subject at the lower levels of learning. And, lack of interest (or innate ability) in a subject is NOT grounds for expecting less from students in elementary school.

Now, I have noticed that there are some people who have a real (innate?) problem with details. They don't like details and they have a hard time with step-by-step processes. It could be that subjects which require these skills are just more difficult and require a lot more concentration. Learning is more difficult when there are objective right and wrong answers. Isn't that part of what elementary math is supposed to do? Teach kids about details and hard work. This is not creative journal writing.

A teacher relative of mine once told me about different learning or thinking styles: Concrete Sequential, Concrete Random, Abstract Random, and Abstract Sequential. I don't really believe in this sort of mumbo-jumbo simplification (it could be just an outward reflection of what they like to do), but it all depends on what you do with it. If you are using it to find different ways to get a child to learn, that is fine. If you are using it as an excuse for expecting less from kids and putting them into lower phases in third grade, that is horribly wrong.

I think it is easy to get distracted by subtle philosophical or physiological questions and lose track of the main issue: the horribly bad math curricula and low expectations in grades K-8, all in the name of some sort of "authentic" or better understanding of math. My contention is that if lower school math is fixed, then you would magically find many more kids who were "good in math".

I have noticed this approach by schools. All they want to talk about is discovery, conceptual understanding, developmentally appropriate, and learning styles, rather than their grade-by-grade lowering of expectations. Everyone then starts arguing about these fuzzy topics and ignores the fact that content and skill mastery are reduced.

I think you have to be very careful talking about a wall (an exponential function) on potential below the college level. I think it is a very dangerous concept to use in connection with education. One cannot easily tell the difference between a wall and poor preparation. The wall becomes an excuse.

-- SteveH - 28 Sep 2005

I'm going to, today, 'blooki' Wayne Wickelgren's passages on kids learning math. I'm also going to unveil my collection of Late Bloomers in Math anecdotes.

Wickelgren was the guy who galvanized me into action; he said exactly what Steve is saying: he said that no one knows which kids are good at math & which aren't.

He also said--and this blew me away--that kids don't particularly enjoy learning math period, no matter how innately good they are at it.

You don't get much more counterintuitive than that.

-- CatherineJohnson - 28 Sep 2005

Now, I have noticed that there are some people who have a real (innate?) problem with details. They don't like details and they have a hard time with step-by-step processes.

Interesting.

-- CatherineJohnson - 28 Sep 2005

I think you have to be very careful talking about a wall (an exponential function) on potential below the college level. I think it is a very dangerous concept to use in connection with education. One cannot easily tell the difference between a wall and poor preparation.

I am in absolute agreement with this.

In fact, this is pretty much my Core Principle.

-- CatherineJohnson - 28 Sep 2005

I had a very interesting conversation with Ed on all of this, which I'll report later on.

One thing I'm certain is happening to people like Ed & me is the following sequence:

• We were 'naturally' good at grade school, middle school, high school. 'Naturally good' meaning school was easy. In Carolyn's words, we 'could learn anything they threw at you.'

• We noticed that, at some point, math stopped being easy & natural. This happened long before any other, language-based subject stopped being easy & natural.

• We also noticed that, for a small, select group of kids, math seemed to be, yes, easy and natural. We weren't in that group.

• We concluded we were supposed to be in the other group, and we took ourselves out of the game.

++++

This sequence wasn't about laziness.

Ed and I were hard workers; Ed did four hours of homework every single night in high school; then did the same at Princeton. He works practically around the clock to this day. He's one of the hardest working people I know.

I can plow the fields 18 hours a day, too, and have done so.

Nor is it not about being too much of a sissy to stand up to an academic hazing. I am a person who read every last word of Umberto Eco's THEORY OF SEMIOTICS in grad school, and understood maybe 5% of it, so don't talk to me about whiners who can't hack the work load & the frustration.

What happened in our case, and what I'd bet the ranch is happening in many other bright kids' cases, is that we had no idea math was different.

We had no idea math had a wall.

-- CatherineJohnson - 28 Sep 2005

I think it is easy to get distracted by subtle philosophical or physiological questions and lose track of the main issue: the horribly bad math curricula and low expectations in grades K-8, all in the name of some sort of "authentic" or better understanding of math. My contention is that if lower school math is fixed, then you would magically find many more kids who were "good in math".

More great stuff! It just keeps coming!

When I talk about a 'wall', perhaps I'd better define my terms. By wall, I don't mean something you can't surmount; I mean something that, for whatever reason, you really have to struggle to learn.

That doesn't mean you always have to struggle ever afterward in everything you do mathematically from then on; that's definitely not the case. It's often an isolated this-or-that that gives you trouble.

That doesn't mean you don't eventually learn the subject well; Bernie and I, oddly, both hit mini-walls when learning 'related rates', a calculus 1 topic, and now we can do them all day long.

Steve, I fear the notion of ability being the key determinant of a kid's mathematical future is a genie that is already out of the bottle. It's such a common notion that people either do or don't have 'math brains', and that you can tell if you're an ed school ignoramus whether your 3rd graders have them or not. I wanted to float the notion that something completely different -- a quirk of personality -- could be a stronger factor than one's 'math brain'.

-- CarolynJohnston - 28 Sep 2005

What quirk of personality are you thinking about?

-- CatherineJohnson - 28 Sep 2005

Innate math talent is THE belief of U.S. citizens.

It's worth reading THE LEARNING GAP, just to get a sense of how critical this is.

That was my big moment, reading Wickelgren; my paradigm shift.

When he said no kids like learning math, I thought: hey! Christopher doesn't like learning math, either! He must be good at it!

-- CatherineJohnson - 28 Sep 2005

What quirk of personality are you thinking about?

Liking puzzles -- getting reinforced by doing puzzles and getting the right answer.

-- CarolynJohnston - 28 Sep 2005

Puzzles in the general sense?

Like logic puzzles?

I've never liked puzzles!

-- CatherineJohnson - 28 Sep 2005

SNORT

-- CarolynJohnston - 28 Sep 2005

That Russian math problem you posted the other day is a classic logic puzzle!

-- CarolynJohnston - 28 Sep 2005

I do, however, love paradoxes & contradictions.

-- CatherineJohnson - 28 Sep 2005

Oh, yeah. I did like that Russian Math problem.

I have a new one to post! It goes back to....the 1800s??

Something like that.

-- CatherineJohnson - 28 Sep 2005

very well said, professor johnston.

vlorbik can't read

yes, that's "johnston". no more from me
about "the johnsons". only took me about
a month ... what can i say. i'm a slow learner.
heck, it took at least a week before i knew there were
two hostesses at KTM ....

-- VlorbikDotCom - 28 Sep 2005

"But there's a wall for everybody if you push it far enough."

I've certainly hit that wall at times, both in math and physics (not that there's really much difference, at least at a university level).

I think, though, that there are walls in other subjects too. Specifically, I think there are walls in art and "language arts" that are fundamentally very similar to those in math. People talk about them the same way too:

"I've never been able to draw [paint, sculpt, ....]."

"I'm not an artist."

"How can you stand up in front of hundreds of people to speak? I'm sure I'd just freeze."

"I'm not a writer. Where do you get all those ideas from?"

In each of these, as is the case in math, there is a small group that seems to be naturally (and seemingly effortlessly) good. But each of these is something that nearly anyone can become pretty good at with practice and good instruction. Most people will never be among the best in the world at any of these, just as most will never be that good at math.

But the biggest walls come early enough that we don't remember how hard they were to overcome. I remember the day I really "got" reading. It was in first grade and I ran home thinking about how amazing it was to not have to struggle. (More exactly, I ran home thinking, and I suspect saying to myself, "I can read! I can read! I can read!") I remember being in a rush to tell my mother about this amazing discovery.

Now, my birthday is in November, and this was in first grade, so I would have been 7 or close to it. Clearly, reading isn't something that came especially early to me, but I don't think I was ever categorized as poor at words.

I suspect that learning to speak is a similar wall, and again it's something that every child has a different experience with. Some kids learn to speak very early and some much later.

The difference with reading and speaking, though, is that there's a huge support network of pediatricians and teachers telling parents that, "Every kid picks these up at a different time; don't worry about it."

I hope I'm not being too insensitive here; I understand that some kids need intervention. But an inability to get past one of those walls is seen as a problem that needs intervention; no one says, "Well, I guess he just doesn't have a 'speech brain'. I'm sure he'll be good at something else."

-- DougSundseth - 28 Sep 2005

_yes, that's "johnston". no more from me about "the johnsons"._

It's a tough thing to get clear on -- Carolyn Johnston and Catherine Johnson -- hard to believe we didn't plan it this way just to annoy everyone.

I'm pretty sure some people think I'm Catherine's more boring alternate personality, and sometimes even I wonder!

-- CarolynJohnston - 28 Sep 2005

Carolyn, you're just a bit more stealthlike with your wit. I still haven't gotten over the infected Latins, poor things.

-- SusanS - 28 Sep 2005

V

heck, it took at least a week before i knew there were two hostesses at KTM ....

I'm pretty sure a lot of people don't know this.....

It's a well kept secret.

Which reminds me, I've been meaning to get a family photo scanned in--

-- CatherineJohnson - 28 Sep 2005

Doug

But the biggest walls come early enough that we don't remember how hard they were to overcome. I remember the day I really "got" reading. It was in first grade and I ran home thinking about how amazing it was to not have to struggle.

Now that's an interesting idea.

There are walls, but they come earlier.

Still and all, I just don't think writing, or science (EXCEPT PHYSICS!) have walls the way math has walls.

I spent 7 years funding autism research & hanging out with neuroscientists, and no one every mentioned any wall.

-- CatherineJohnson - 28 Sep 2005

Ed thought the wall idea was fascinating, because he's just read Bernd Heinrich's Why We Run. Heinrich holds, or held, the world's record for longest distance run in 24 hours; he's a super-marathoner.

But even he has hit the runner's wall.

I think he ran something like 60 miles (??) in the super-marathon he ran, but one time he hit the wall at 34 walls, and collapsed.

-- CatherineJohnson - 28 Sep 2005

Here's something else.

When you're doing creative nonfiction work, which in my case means trying to piece together a bunch of conflicting and/or seemingly unrelated ideas (and is incredibly hard) you can hit the wall the way a runner hits the wall, only mental.

BUT you don't have the sense that there's a whole group of people out there who already understand what you're trying to understand.

That's how I think of 'the wall' in math.

You're losing your mind trying to understand something that a fairly large group of people including mathematicians, engineers, physicists, math teachers, and plain old math hobbyists already do understand perfectly well.

In other fields, the 'point' of extreme difficulty is past the point of _everyone_'s understanding.

In math, people experience extreme difficulty understanding concepts even at the most elementary levels, not just at the Reimann's hypothesis level.

-- CatherineJohnson - 28 Sep 2005

I'm pretty sure some people think I'm Catherine's more boring alternate personality, and sometimes even I wonder

umm.....I'm pretty sure they don't!

-- CatherineJohnson - 28 Sep 2005

"I spent 7 years funding autism research & hanging out with neuroscientists, and no one every mentioned any wall."

Fair enough, my experience and training are more limited.

Still, brain research I've seen indicates that children go through periods of skill plateauing before sudden bursts of advancement, and this is exactly my experience in teaching my son to read. He will make exactly the same mistakes and struggle with exactly the same things for weeks when I can see that he's working hard. Then within a matter of days he'll be at an entirely new level.

It may well be the case that these walls aren't as high relative to average ability as those in math*, but they sure feel similar to me. As I see it though (and I've not had the experience you have, so take it for what you will), the average kid is never told to expect to be unable to climb (break through?) this wall. Instead he's told to keep working and he'll get it eventually.

Also, I think that the walls in reading (speaking, whatever) come very early and the slopes are much smoother later on. By contrast, in math the conceptual challenges just keep on coming.

"EXCEPT PHYSICS"

When in a particularly snarky mood (I work in an engineering shop; engineers can drive nearly anyone to snark) I've been known to declaim, "Engineering is just Physics for people who can't understand the theory and Math is just Physics for people who have completely lost contact with the real world." While this is both hyperbole and subject to replies in kind**, there is at least a bit of truth. Math, Physics, and Engineering are very tightly intertwined. You really can't succeed at either Physics or Engineering without quite a strong grounding in Math.

* The very first wall in art may be (at least in practice) as high as most in math. The percentage of students who go through years of art classes in primary and secondary school without ever being able to draw is remarkably high. How many people have you heard make some variation of the claim that, "I could never do that" when looking at a piece of art. I know that I've said that, though I'm at least a bit better now. I'll be a complete Philistine, though, and say that I don't think art is as important as reading and math.

** I'll leave those replies as exercises for the mathematicians and engineers who might read this; there's no reason to aid the other sides. 8-)

-- DougSundseth - 29 Sep 2005

I'll try to get this out quickly, since I think we're posting past each other.

"You're losing your mind trying to understand something that a fairly large group of people including mathematicians, engineers, physicists, math teachers, and plain old math hobbyists already do understand perfectly well."

I think this is exactly the experience of a child learning to speak or read. Everyone around you can do this thing without even really thinking about it; why can't you?

-- DougSundseth - 29 Sep 2005

I agree that math has walls all the way down. The first one I remember clearly was in 4th grade. I was required to memorize the multiplication tables. I hate memorization. I don't think I even got the concept of memorization. In that particular class we had to take our seat based on the test score we received on the last test. The "A" students sat at the front. It was a linear ranking. I had to move from the front to the very back because I didn't know that answers to the multiplication questions. They threatened to kick me out of class; my parents got involved and forced me to memorize the multiplication tables using flash cards; I moved back to the front. First wall hit and overcome, first trauma endured.

I had many more. Maybe an advantage I had over Catherine is that I never supposed there wouldn't be walls.

I agree completely with Doug that there are walls in all fields. But I think math is different because for the most part it is nothing but walls.

It's a large mistake to believe that there's a group of people over there who get it all. There isn't. Math is really like a set of mountain peaks, and I think this applies all the way down to grade school. You climb one--it takes a lot of work, but you do it--but that doesn't do anything for all the other peaks out there. They all have to be climbed one by one. Some people have climbed several, a few people have climbed a hundred, but nobody could possibly climb them all. And not just because they're lacking time, but because they're lacking talent. There are all sorts of different kinds of mathematics which require different talents of various sorts. No one has all the talents.

The whole concept that there are "math people" who can get it on the one side, and then the rest of us on the other who can't, is incredibly debilitating. It lets kids off the hook for being lazy when they should have continued on and persevered. It's a horrible concept and completely wrong.

And it lets the more mathematically talented off the hook because they think that just because they have some mathematical talent they don't have to work anymore. I've seen a lot of those, and they were all lying by the wayside. Everybody who does math or physics or engineering seriously has to work just as hard as Ed. Grothendieck was probably the best mathematician of the latter Twentieth Century and he was famous for working all the time.

-- BernieJohnston - 29 Sep 2005

Doug

"I spent 7 years funding autism research & hanging out with neuroscientists, and no one every mentioned any wall."

No! No!

I didn't mean you were wrong about 'early walls'--not at all!

I think the idea of walls we run into so early we don't remember them is very interesting.

I was talking about neuroscientists doing neuroscience.

What I meant was that neuroscientists--or any of the scientists I've spent a lot of time talking to--didn't talk about walls they themselves have encountered in trying to learn their disciplines.

I don't know for sure, but I don't think a group of scientists would have this kind of conversation (the one we're having here).

-- CatherineJohnson - 29 Sep 2005

It wasn't hard for them to learn their fields. Hard work, but not hard-hard.


Here it is:

What is hard in other fields is discovering new knowledge, not acquiring old knowledge.

In math, both are hard.

-- CatherineJohnson - 29 Sep 2005