kitchen table math, the sequel: math facts & letter facts

Thursday, February 1, 2007

math facts & letter facts

Susan J left this comment:
Maybe we need another term besides "math facts." They are much more fundamental than contingent facts like, say, dates when certain historical events occurred.

Do kids still learn "letter facts" like the names of the letters and alphabetical order?

That's a terrific point. The term "math facts" does lend itself to the view that "math facts" are on a par with historical dates.

I'm going to start using the terms fundamental versus contingent to talk about the math facts.

(I guess we could call them math fundamentals...on occasion.)


here's Barry:
Call them something exotic so the reformers at large will think it's higher order thinking skills. Call them theorems, because that's what they are. 3 + 2 = 5 is a theorem which can be proven using Peano's axioms. And like all theorems, once proven, it can be used without having to re-prove it, thus alleviating our kids from having to draw clusters and go back to first principles each and every time they add, subtract, multiply or divide lest anyone think they were doing things by rote.


Steve H
I've never liked this either. I think it's a carfully-selected term used to degrade the importance of the knowledge. when I told my son's Kindergarten teacher (years ago) that I supported an educational approach that emphasized basic knowledge and skills, she said that I would like one of the second grade teachers who required each child to give her a "math fact" to enter the room.

They are just clueless.

From his first grade teacher I learned about "superficial knowledge". I wanted to tell her that it is fundamental knowledge.

Basic knowledge and skills are the foundation of education. You build from the bottom up rather than the top down. Somehow they think that education should be thematic or top down; that kids can learn basic knowledge and skills by osmosis.

Basic facts and skills require hard work.

Hard work is a filter.

The only point I would add here is that achieving expertise requires hard work and expert teaching.

I've been trying to make my way through the research on expertise, with limited success. It is vast.

However, I've managed to do some mighty skimming.

As far as I can tell, expertise is expertise; the mechanisms by which one acquires expertise are (largely) the same.

These days I think about athletics and athletes when I'm confused by an issue in K-12 education.

There's a reason you never, ever, see an Olympic athlete who is self-taught.

That reason is that expertise requires superb teaching.

I'm not sure whether the strong form of this statement is correct.* Certainly I've seen many people who are self-taught (I taught myself to write).

There must be a literature somewhere on self-teaching.

Nevertheless, the weak form of the statement is certainly true.

Our kids need expert teachers who themselves take the development of expertise seriously. Hard work isn't enough.

I would rewrite every school mission statement in this country to reflect this understanding.

I'd take out all the "lifelong learners" and "critical thinkers" and "all children can learn" foofaraw.

I'd insert the word expertise.

__________________

* One of the reasons why I'm not sure is that procedural learning and knowledge, which I believe is dominant in athletics, seems to follow a slightly different set of rules than declarative knowledge, which is dominant in academic disciplines.

Procedural knowledge seems to be much more vulnerable to error. Once you learn a procedure - a golf swing, say - the wrong way, you can't unlearn it.

As far as I can tell, declarative knowledge is less vulnerable. If you mislearn a fact, you can overwrite it in memory. I'm guessing it's possible to learn from error in declarative knowledge.

caveat: I don't "know" these things. This is what I surmise based in many years' surfing literature on memory.

22 comments:

Barry Garelick said...

Call them something exotic so the reformers at large will think it's higher order thinking skills. Call them theorems, because that's what they are. 3 + 2 = 5 is a theorem which can be proven using Peano's axioms. And like all theorems, once proven, it can be used without having to re-prove it, thus alleviating our kids from having to draw clusters and go back to first principles each and every time they add, subtract, multiply or divide lest anyone think they were doing things by rote.

Anonymous said...

"Johnnie, let's practice math fundamentals."

I do agree that "math facts" might not sound so good coming from and adult, but then neither does it sound mathematically mature for an adult to justify "invert and multiply" by appealing to the axiom of fraction overlays.

When speaking to my children I say, "You need to practice your two times table."

I suppose the opposite of a math fact is a math conjecture?

SteveH said...

"Maybe we need another term besides 'math facts.'"

I've never liked this either. I think it's a carfully-selected term used to degrade the importance of the knowledge. when I told my son's Kindergarten teacher (years ago) that I supported an educational approach that emphasized basic knowledge and skills, she said that I would like one of the second grade teachers who required each child to give her a "math fact" to enter the room.

They are just clueless.

From his first grade teacher I learned about "superficial knowledge". I wanted to tell her that it is fundamental knowledge.

Basic knowledge and skills are the foundation of education. You build from the bottom up rather than the top down. Somehow they think that education should be thematic or top down; that kids can learn basic knowledge and skills by osmosis.

Basic facts and skills require hard work.

Hard work is a filter.

They want full inclusion where mastery cannot be required, so it must not be important or it can be delayed. (It never gets done.)

Ergo, no linkage between understanding and mastery. They are saying that any knowledge and skills you need can be learned sufficiently in context. Mastery is not important, facts become "mere" and knowledge, "rote". Only the fuzzy context is important, not the facts and skills.

They can't defend their educational philosophy as anything more than opinion, so they resort to massive PR.

It doesn't matter.

There is no process.

There is no discussion.

SteveH said...

I was thinking about the differences between the engineering profession and the education profession. I distinctly remember the time in engineering school where after many long and detailed calculations, we proceeded to multiply the result by a "Safety Factor" of 2.5. I thought this was amazing. Why go through all of those calculations when you have to multiply by such a big fudge factor?

I soon learned the reason. The education profession hasn't figured this out. Perhaps it's because they cannot or will not even attempt to quantify what it is they are doing except at a very minimal level. If so, they will never make any progress at all.

Catherine Johnson said...

Perhaps it's because they cannot or will not even attempt to quantify what it is they are doing except at a very minimal level.

This is what leapt out at me from the passage on teacher training in Russia: they had quantified the amount of time and number of practice sessions & homework assignments it would take for a second grader to reach mastery of the multiplication theorems.

Catherine Johnson said...

I wonder what happened to the "Recent Comments" sidebar?

Catherine Johnson said...

hmm....

Recent Comments show up on Safari, not on Firefox.

I'll have to reboot.

Barry Garelick said...

Quit fooling around with Blogger and write that letter!

Ben Calvin said...

Once you learn a procedure - a golf swing, say - the wrong way, you can't unlearn it.

I slightly disagree with this. It has been widely noted that Tiger Woods has broken down and reconstructed his golf swing at least once (I think it's twice) during his pro career.

Michael Jordan, as documented in David Halberstam's Playing For Keeps used a coach to remake his body through weight training in the middle of his playing career and had to completely rebuild his shot in the process.

In both cases their performance went down as they unlearned their old style and developed the new.

So maybe the two types of learning aren't that different after all.

Catherine Johnson said...

I slightly disagree with this. It has been widely noted that Tiger Woods has broken down and reconstructed his golf swing at least once (I think it's twice) during his pro career.

I was told the opposite story - and I want to get this right, because it's something I've been worrying about.

I was told, a couple of weeks ago, that Woods at some point decided to change his golf swing and hired an excellent coach to help him do so.

It didn't work, according to the version I heard, and his regular swing suffered while he was trying to master a new swing.

Is that wrong?

Catherine Johnson said...

ok, I just read a bit more closely

Did Woods actually switch to the new swing?

I'm going to have to find my various sources.

I don't think I've got this wrong, but the reality could be more nuanced.

I do know for a fact that stroke victims do MUCH better using "errorless learning" - i.e. learning that doesn't allow them to make any mistakes at all.

There was a wonderful study on teaching people who seem to have been quite impaired after a stroke how to use PDAs. (I think perhaps the PDAs were going to function as assistive technology, but I could be making that up.)

One group was given errorless learning in which the O.T. only let them hit the right key.

The other group had standard teaching in which the O.T. corrected errors.

The results were poignant.

The errorless learning group all mastered the PDAs.

Not one of the standard-treatment group was able to do so.

I was obsessed with errorless learning for awhile because of Jimmy - and I think to this day it would have been best if he'd had errorless teaching.

I'd say that Saxon uses something very like errorless learning in his books.

Catherine Johnson said...

Temple has told me another poignant story.

Back when the meatpacking plants were reforming there were a number of workers who were never again able to slaughter the cattle.

They couldn't change from the way they'd been doing it (this was in kosher plants, I believe) to the new more humane way.

The managers would be furious with them, and would see them as resisting, slacking, etc.

Temple knew they couldn't do it, and she used to tell the managers, "They can't."

Catherine Johnson said...

So...I'm pretty sure procedural learning is best done through errorless learning.

What I'm not sure about is declarative memory.

Catherine Johnson said...

Barry!

It's done!

And I had synchronicity!

I'll fill folks in later.

Barry Garelick said...

I don't agree that you cannot unlearn things. Children have many misconceptions about how they think the world works only to have to modify their knowledge once they receive new information. I'm no fan of Piaget but he does talk about the accomodation/modification stage of learning. It might be more difficult with procedures learned to automaticity than say concepts or facts, but to say you can never unlearn it seems wrong.

True confession: When I was little, I though gravity was "caused" by the spin of the earth. (Actually, the spin counters the effect of gravity slightly). I thought this way until senior year in high school when I took physics and learned the formal definition of gravity. It was a revelation to me. I guess I had ignored mention of it in previous science classes because it was a topic that I thought I knew everything about and so would just tune out.

Instructivist said...

"When I was little, I though gravity was "caused" by the spin of the earth."

There are misconceptions that seem impossible to eradicate (unlearn), especially among journalists. One of them is that astronauts are in zero gravity, when they are actually in free-fall while orbiting. Granted, circular motion is a tough topic.

SteveH said...

"I don't agree that you cannot unlearn things."

I agree, kind of. My son's piano teacher always talks about learning the music right the first time. Of course, this is just sensible. However, many do warn that you cannot learn a piece correctly after you've learned it incorrectly. They are trying to say that there is more to it than changing your mind.

For the piano (and other skills that need to be automatic), the skill becomes ingrained more deeply. Once that is done, learning something different is more difficult. The problem I have is that there are those who say that something magical is going on that can't be undone. OK, I've overstated the problem.

My son holds his pencil incorrectly because his Kindergarten teacher didn't want to force kids into only one solution. I've tried to get him to change, but he won't. Notice that I don't say "can't". He could change it if he really wanted to.

He can change the fingering on a piece of music after he learned it the wrong way. It just would have been so much easier to learn it correctly the first time. For his piano teacher, he would rather start fresh and do a new piece correctly than try to fix up the old one. But you have to do something if the old piece needs to be prepared for a recital. It can be done.

LynnG said...

There must be some kind of time element involved, I think. How long have you ingrained the procedure you are trying to correct?

I toss out a simple example. I've ridden horses since I was a kid in the Midwestern farm country where a backyard fence and a nothing fancy horse were pretty common. I grew up riding Western, then switched to English when I moved East. I can easily switch back and forth between the two styles. Here's what I can't do, I can't stop dropping my left shoulder in the canter. If I focus really hard, I can stop myself, but then there's really no focus left over for other important things (like steering). This is a huge problem as it regularly causes me to lose my stirrups. I've compensated in all kids of ways that lead to other problems. It is immensely hard to change this ingrained bad habit that I've reinforced over many, many years. This is quite different from a "style" or a discipline, which I can switch between easily.

Maybe, if my shoulder slumping had been caught earlier, I could have corrected more easily than I can now.

The connection to education may not be obvious, but I think once a kid gets to high school, it's a lot more difficult to alter ingrained, incorrect procedures (though not impossible), than it would be in the elementary schools.

Catherine Johnson said...

Children have many misconceptions about how they think the world works only to have to modify their knowledge once they receive new information.

Right, but that's declarative knowledge.

As far as I can tell, declarative knowledge isn't (so) vulnerable to error.

Anonymous said...

"I don't agree that you cannot unlearn things."

Corollary: You can't change reality, but it sure can change you.

Anonymous said...

I teach college chemistry, so have definitely seen the impact of weak math skills. I've also seen the impact of educationese -- which my students are over familiar with (good teaching makes learning fun all the time; I understand the concepts but can't do these problems). One argument I've found helpful is that there are things they have to know (say the periodic table), because it is a waste of time for them to figure it out from first principles every single time. I'll admit I avoid talking about "facts" or "memorizing", because to them that means dumping it into the brain for the exam then forgetting it immediately. Saying that they just have to "know" it seems to work better.

Also, while many historical dates can get to be trivial, others really are critical to a good education (think 1776). Students can't apply higher order thinking skills if they don't have any facts to apply them to!

Catherine Johnson said...

Hi Wade!

I understand the concepts but can't do these problems

YES YES YES

We are having a HUGE problem with this in our middle school.

Don't know how long you've been reading, but we've been butting heads with the math department until blood has been drawn on both sides....awful!

Our accelerated math kids are a mess; they're being extensively tutored and retaught by parents.

The district's SOLE response is "extra help."

Extra help means the kid goes in before school and spends 10 minutes having the concept explained all over again.

Extra help does NOT mean ACTUALLY GIVING THE STUDENT PRACTICE PROBLEMS TO DO.

The math chair told us, "If students need distributed practice parents can find worksheets online."

Well guess what.

I have a house FILLED TO BRIMMING with worksheets I found online.

They aren't the worksheets I NEED at any given moment to give Christopher practice the EXACT problems he is currently expected to be able to do on a test.