Comments

My opinion is that it's not that difficult.

Regardless of whether kids are fast or slow learners, there should be certain knowledge to know and skills to master before a student can go on to the next grade. A grade of C- might allow a student to go on to the next grade without full knowledge or mastery, but this is a judgment call and teachers used to do it all of the time. If the level or cut-off is high enough, then most kids will be served (without separation by ability) through 5th or 6th grade. Then, you really have to offer separate paths as a lead-in to high school.

Whether the curriculum and teaching methods are good or bad, or whether the kids are separated by ability properly, are other issues. Ability tracking in high school happens. No ability separation, as in our public schools (K-8), is wrong. Separation by ability in the early grades should not be necessary, but if schools, like ours, are still trying to get kids to know their adds and subtracts to 20 in third grade, then something has to be done.

"Differentiated instruction is not the answer."

Differentiated instruction is only a way to get full-inclusion and no hard grade-level expectations to work. It's no guarantee of any sort of mastery. It's no substitute for content and skills.

-- SteveH - 27 Oct 2006

My opinion is that it's not that difficult.

You know - I think it is!

I think teaching ought to be brain surgery!

Or, at least, curriculum design ought to be brain surgery.

Engelmann is a superb creator of curriculum who has grasped aspects of learning & of the research on learning that have escaped the rest of us - including Gentile & Lalley.

If we all agreed that early tracking was fine, we might not have a problem.

But given the fact that Americans on the whole are egalitarian, early tracking simply doesn't "work" for us socially & politically.

I think Engelmann has probably figured out how to resolve this conflict through superb curriculum design.

-- CatherineJohnson - 27 Oct 2006

We could also adopt what I believe is the "true" Asian model.

Asian countries don't track kids in K-5, and everyone appears to learn to mastery.

However, I'm almost positive that the reason everyone learns to mastery in non-grouped classes is that parents take the slower learners to KUMON & similar supplementary classes.

(I'll post that article at some point....)

That could work for advantaged parents here; I assume it wouldn't work (as well) in disadvantaged populations, if only because of the logistical problems besetting single-parent families.

-- CatherineJohnson - 27 Oct 2006

btw, I don't know that the non-tracked Asian model depends upon parents purchasing supplementary education.

But I'd bet a not-insubstantial sum of money that it does.

-- CatherineJohnson - 27 Oct 2006

"We're not comfortable with the notion of hiving kids off into top-thirds, middle-thirds, and bottom-thirds, and then keeping them there for the 13 years of K-12. At least, I'm not. "

the parents of the top-thirds kids are. I am.

You are always posting about learning to the best of each child's capability--everyone does--but this isn't going to happen for the smartest kids absent tracking.

I'm not sure what I think about this post. In my experience learning and tutoring, the gap between slow and fast learners doesn't slow down. If anything, it INCREASES as the material gets more difficult. Certainly this tends to be true at higher levels like high school and college.

I don't mind the reality that school demands sacrifices. I merely find the "everyone benefits from diversity!" dreck to be more than a little bit tiring. Why not be honest? You want to benefit your child, so you're going to demand procedures that benefit him. Self interest is OK.

The debate is simply between two general theories:

Do you divide teacher time equally? (e.g. each group gets a hypothetical hour of instruction--i call this the "capitalist" approach)

Or do you divide student gains equally? (e.g. each group gets the %age of instruction needed to advance them to the same level as the other groups--I call this the "socialist" approach)

The capitalist approach benefits the top kids at the expense of the bottom; the socialist approach benefits the bottom kids at the expense of the top.

I find it frustrating when people claim the socialist approach has no negative side. It's just nonsensical. The negatives may be WORTH IT in the larger picture, but they're still there.

-- ErikHammarlund - 27 Oct 2006

Let me cover a few points here:

"What mastery to a high standard can do, in summary, is virtually bypass the effects of IQ for specified educational objectives. What is recalled about educational lessons is more dependent on how well the material is mastered than on such traits as rate of learning or general intellectual abilities." (Gentile & Lalley)

I think this is an overstatement. The high IQ kids who initially learn at a 3:1 rate and then relearn at a 2.1:1.5 faster rate will be far ahead of the lower IQ kids in short order. This hardly seems to bypass the IQ effect. No matter how you look at it, the smart kids will be learning more faster even under mastery learning.

Or, at least, curriculum design ought to be brain surgery.

We probably have more brain surgeons and rocket scientists than we have decent curriculum designers.

In my experience learning and tutoring, the gap between slow and fast learners doesn't slow down. If anything, it INCREASES as the material gets more difficult. Certainly this tends to be true at higher levels like high school and college.

This is a byproduct of not teaching to mastery.

-- KDeRosa - 27 Oct 2006

I think not teaching to mastery is silly. But I think the increasing gap is a function of the reality that higher level math is, well, harder. And higher level.

IMO, once you start getting into theoretical precalc (and even before that, in some advanced algebra) you enter the realm where some students just plain can't understand it without a learning speed so slow as to be a waste of their time.

The higher level which is attained, the larger proportion of students who will fall in that category. There is no easy way to "teach to mastery" when you are trying to grasp concepts of theory. You can master DOING something. But you cannot necessarily master UNDERSTANDING something, no matter how long you try.

So most students can probably learn to solve lim(2/x) as x approaches infinity. But some students--and we've all met them--simply can't get the concept of infinitely large (or small) numbers at ALL. So though they might be able to solve the lesson by "mastering" the rule that dividing by infinity results in zero, as soon as they need to think in a way which requires understanding and not calculating, they're hosed.

-- ErikHammarlund - 27 Oct 2006

You know, I've argued in this forum that old-fashioned tracking is bad. But all along, even I have agreed that kids ought to be separated by ability in math, even in the early years. I don't see anyway around it. I think it's because of the nature of math. It's layered and cumulative, and you need a strong foundation in place. As others have pointed out, kids master material at different rates.

But as far as reading goes, kids already are grouped by ability within the classroom. In our school, they have reading groups, and read material appropriate to their level. The strong readers move ahead rapidly to increasingly difficult books. It seems to me this works, even though you have children of a wide range of ability in the same classroom. I don't have a problem with this type of small group instruction. Even in the bad old days of tracking of my youth, kids were grouped within classes by reading ability.

I think with writing also, at least within the early elementary grades, kids can work up to their own level. If your kid is smarter than the other kids, then let him produce longer, more elaborate pieces of writing.

The emphasis in early elementary school is reading, writing and math. In science and social studies in the early grades, at least, mixed-ability grouping can work. These subjects are not emphasized as much and seem to depend more on studying and memorization. They are not as cumulative as math, at least not in the early grades.

Yes, maybe the smart kid would learn more social studies and science content if he were with kids of his own ability. But there are competing arguments against tracking that ought not to be dismissed as mere "dreck." Many people, including people who have children in the top third, value and understand the socializing function of schools.

So I propose grouping by ability in math starting in, say, first or second grade. Then I would gradually start grouping by ability in the intermediate and middle school years as the non-math subjects become more challenging and cumulative.

-- RobynW - 27 Oct 2006

the parents of the top-thirds kids are. I am.

You are always posting about learning to the best of each child's capability--everyone does--but this isn't going to happen for the smartest kids absent tracking.

Hi Erik!

Should I amend my post?

I probably should.

I don't want to put words in people's mouths or imply that they should feel a certain way...

-- CatherineJohnson - 27 Oct 2006

but this isn't going to happen for the smartest kids absent tracking

I suspect that's not true in the years ....... K-5?

Haven't read the rest of your comments, so you may have light to shed on when people really have to separate out into different abilities.

-- CatherineJohnson - 27 Oct 2006

The capitalist approach benefits the top kids at the expense of the bottom; the socialist approach benefits the bottom kids at the expense of the top.

That, I think, is wrong.

The premise is that fast/medium/slow are given when I think it's clear that they're given within some realms of cognition, but not others.

Engelmann has moved his grouping criteria to the realm in which fast/medium/slow is a much smaller difference.

-- CatherineJohnson - 27 Oct 2006

I think this is an overstatement. The high IQ kids who initially learn at a 3:1 rate and then relearn at a 2.1:1.5 faster rate will be far ahead of the lower IQ kids in short order. This hardly seems to bypass the IQ effect. No matter how you look at it, the smart kids will be learning more faster even under mastery learning.

I don't think it overstates Engelmann's position - or does it?

-- CatherineJohnson - 27 Oct 2006

In my experience learning and tutoring, the gap between slow and fast learners doesn't slow down. If anything, it INCREASES as the material gets more difficult. Certainly this tends to be true at higher levels like high school and college.

This is a byproduct of not teaching to mastery.

Right.

That's what I was about to say.

The Saxon books are an eye-opener.

I've been studying material no one ever taught me for a little while now, and when it's broken down into small increments, and when you have complete mastery of earlier material, it is bizarrely easy to learn.

-- CatherineJohnson - 27 Oct 2006

Everyone needs to read Hirsch in short order.

Hirsch & then re-read Willingham on flexible & inflexible knowledge.

Ed told me, today, about an ongoing dispute they have at NYU.

There's constant argument over whether it is or is not OK to put the Masters candidates in with the Ph.D. candidates.

Ed always says it's OK.

But in fact the Masters candidates don't do nearly the same level work as the Ph.D. candidates.

He's been confused by this, because in some cases they'll have a Masters candidate who is obviously extremely intelligent (can't say more than that)....and perhaps a Ph.D. candidate who doesn't necessarily strike everyone as innately more brilliant than the Masters candidate.

But the Ph.D. candidate does far better work. More sophisticated, more intelligent, etc.

For Hirsch/Engelmann/Willingham this is not a paradox.

The Ph.D. candidate has far more subject matter knowledge, and thus does "smarter" work.

-- CatherineJohnson - 27 Oct 2006

He's also seen this from another vantage point.

He directs the Institute of French Studies, and he's a historian.

The institute draws students from the history department and the French department.

Well, when you put a star French department Ph.D. candidate in a graduate course in French history, his or her work just isn't that good; it doesn't seem all that "smart."

I hope to heck no one from either department reads this, so in case they do I'm going to STRESS that this isn't a matter of innate intelligence & nor does Ed (or has he in the past) seen it as a matter of innate intelligence.

But people think of intelligence as a biological given and, to a large extent, a fixed-point given, not a range.

We also think of intelligence and "critical thinking" and "problem solving skills" as abstract skills that can be picked up and transferred from one subject matter to another.

Neither of those ideas is correct (in the view of geneticists & cognitive scientists).

-- CatherineJohnson - 27 Oct 2006

My favorite example of the non-transferrability of "critical thinking skills" was a column one of Ed's former colleagues wrote in the run-up to the election.

This guy is a very well-known historian who had started writing op-eds on contemporary politics, which he's not trained to do, and which he hasn't spent years of his life developing expertise doing.

So in the run-up to the election he said that George Bush woulld not be re-elected because he was incompetent and did not deserve to be re-elected and the American people never re-elect people who don't deserve to be re-elected, so therefore George Bush would not be re-elected.

This is a major, famous historian.

His "critical thinking skills" didn't transfer from American history (he's an American historian) to political punditry.

That's what cognitive science has been finding for years.

You can be a genius in one realm; that doesn't make you a genius in another related realm.

It's analogous to Michael Jordan not being able to switch to baseball.

-- CatherineJohnson - 27 Oct 2006

Engelmann does have ability groups, btw.

-- CatherineJohnson - 27 Oct 2006

Your comment about single-parent families sent me scurrying off to find statistics on how many families-with-children are single-parent vs. two-parent. I was expecting a lot more single-parent households than what I found. We always hear statistics to the effect that X% of children are growing up in single-parent households.

Turns out that:

• 35% of U.S. households have one or more children under 18
• 1% of the childed households are "non-family", i.e. children growing up in households without their parents or grandparents
• 67% of the childed family households are married-couple families with children; the others are single male or female with children

Source: U.S. Census Bureau 2005 American Community Survey table of HOUSEHOLDS BY PRESENCE OF PEOPLE UNDER 18 YEARS BY HOUSEHOLD TYPE

And:

• 99.2% of children live in family households
• 69% of children live in married-couple family households
• 7% of children live in male single-parent households
• 24% of children live in female single-parent households

Source: U.S. Census Bureau 2005 American Community Survey table of HOUSEHOLD TYPE AND RELATIONSHIP TO HOUSEHOLDER FOR CHILDREN UNDER 18 YEARS IN HOUSEHOLDS

-- GoogleMaster - 27 Oct 2006

IMO, once you start getting into theoretical precalc (and even before that, in some advanced algebra) you enter the realm where some students just plain can't understand it without a learning speed so slow as to be a waste of their time.

The higher level which is attained, the larger proportion of students who will fall in that category. There is no easy way to "teach to mastery" when you are trying to grasp concepts of theory. You can master DOING something. But you cannot necessarily master UNDERSTANDING something, no matter how long you try.

I'm looking forward to getting to these levels....

I have ZERO idea what's what at that point.

ALTHOUGH Saxon & Wilfred Schmid (and some folks around here?) say that you come to understand an advanced mathematical concept through the continual doing of math, if I'm putting that right.

As I understand this, and to the degree I've experienced it, I think of this as a kind of "naturalizing" of math concepts.....after you've done lots of math...

-- CatherineJohnson - 27 Oct 2006

But as far as reading goes, kids already are grouped by ability within the classroom. In our school, they have reading groups, and read material appropriate to their level. The strong readers move ahead rapidly to increasingly difficult books. It seems to me this works, even though you have children of a wide range of ability in the same classroom. I don't have a problem with this type of small group instruction. Even in the bad old days of tracking of my youth, kids were grouped within classes by reading ability.

Now that I've read Hirsch, who makes a VERY convincing case that reading is about background knowledge, I would put a little one in a Core Knowledge school in a heartbeat.

Content, content, content.

My inclination would be to ability group by placement-in-the-curriculum exactly the way Engelmann does.

-- CatherineJohnson - 27 Oct 2006

Hi Google Master!

I was talking specifically about disadvantaged black kids who have an extremely high rate of single parent families.

In other words, where an Asian country can rely on parents to give the slower kids enough outside work to keep up with the faster kids, and my own community could probably do the same, you couldn't set things up that way in urban schools.

-- CatherineJohnson - 27 Oct 2006

maybe the smart kid would learn more social studies and science content if he were with kids of his own ability

Again, I would offer maximum parent choice, so my ideal school, assuming it weren't a charter school "doing it's own thing" (which it would be)...my ideal public school would have a no-tracking option and an Engelmann Core Knowledge option where kids would move through subject matter content as quickly as possible, always learning to mastery.

Hard to believe I can have a name like Catherine Johnson and be a complete and total outlier on this stuff.

-- CatherineJohnson - 27 Oct 2006

Having been a gifted kid, with an equally gifted sister, I'll throw out some opinions based on our experience with "breathing in content".

IME it comes down to four things:

1)Being a voracious reader. I've been reading since before I can remember, and pretty much inhale books of all kinds. I was the kid in school who never paid attention in class because my nose was stuck in a book. This gave me a huge amount of background knowledge on many topics, which I would agree is essential for rapid learning.

2)Having an excellent memory. Once I've seen a concept twice (in some form or fashion) I own it. With all the reading I did, I usually had come across most topics before they were taught at some point.

3)The ability to naturally make connections across disciplines, as opposed to compartmentalizing knowledge (e.g. studying A Tale of Two Cities in English and never connecting it to the knowledge of the French Revolution gained in history). I was told that non-gifted students have trouble making these connections without help (we had a grand interdisciplinary setup in 7th grade to help promote this "horizontal learning", vs. "vertical learning" as they called it). I don't know if that is true, only that this ability gives you far more "hooks" to hang new knowledge on, and makes it rather effortless.

4)The sheer desire to learn, just because it is fun and exciting in itself, as that drives the voracious reading. This really should be number one, but I didn't want to renumber my list =).

Being able to use previous knowledge to deduce new info is useful as well (enabled me to pass several classes despite the fact I never attended and barely skimmed the text - just learned the info as I took the tests), but again, I don't know if that is gifted-specific.

As for re-learning, I'm still not sure I understand what is meant by that. Obviously I have forgotten things that I haven't used in a while (polar coordinates, for one) - giftedness, at least for us, doesn't negate the idea of "use it or lose it" - but how that correlates with the idea of re-learning I'm not sure.

-forty-two

-- KtmGuest - 29 Oct 2006

Guest describes my experiences with my kids. They were in a school that looked the other way if they were reading a book under their desk and the teacher was fairly certain they'd got the material. That is, they'd have to be attentive the first time through, but after that, most teachers were fairly good about letting them NOT pay attention to something they already knew.

This didn't happen for me growing up, we were expected to maintain that bright, attentive look even though we'd learned this same information days before and this was, in fact, the 30th repetition. I hated that. The boredom was excruciating. In HS it was somewhat better, if only because the teacher didn't have the time to stop you if you were writing long letters in your notebook when bored.

Children who can read well early on (first son learned to read in K, pretty standard here, then went on to read The Lion, The Witch and the Wardrobe over the summer, by himself, with full comprehension). So no, I don't buy the argument that elementary school is different, or less different. In a first grade classroom with non-readers through readers at 4th and 5th grade level readers, there are few "reading" activities that work for all -- except for reading aloud.

-- JenL - 01 Nov 2006

I certainly concur with the premise I think you are presenting: there is LESS distinction between students than is commonly thought... if you use your metric, that is, which seems to be the "what can they ultimately learn if they try hard enough and receive proper instruction?" metric.

But of course that's an unrealistic metric, as it seems to assume a resource-unlimited economy. You're ignoring opportunity cost.

So when I'm presented with three hypothetical math students, and one hour to teach them, what's my goal?

Here's an example which puts real numbers into play: Student A is brilliant at math. Student a "gets" everything, the proverbial 'mind like a steel trap.' If I were to teach the class as fast as Student A could learn it, it would be an enormous benefit to her.

Student B learns at an average speed in math. Student B can 'get' pretty much everything, but not nearly as fast as A. B is 5% slower than A.

Student C is slower than both A and B--10% slower than A, and 5% slower than b. He can still get things. But he won't make cognitive jumps on his own. He can learn everything... slowly. He spent a bit of the summer to be sure he was at the same level as A and B, and he is--for now.

What does your theory propose here? In this hypothetical, all the students are EQUAL in terms of mastery. For now. And there is "only" a 10% worst-case difference in learning speed. By this, I mean, that is A is capable of mastering 100 problems (and their attendant theories), then C is only capable of mastering 90.

After 10 weeks of instruction at C's speed, A has "lost" a week that A would have learned if the class were at A's speed. After a 180-day school year, A has "lost" 20 days of instruction.

Seeing as you detest losing even a single day, I think you will acknowledge there IS a cost to A: the cost of what A would have learned.

Now, normally I'd say "well, no problem, such is life." After all, school's for everyone (not just geniuses) and this is going to be prevalent everywhere.

But of course I'm talking about a TEN PERCENT learning difference. Now, i remember my own classes. Ten percent? Pshaw! I'd bet my mule there was a spread twice that.

30%? 50%? 100%? Sure! I mean, in second grade there were kids who could read, write, and understand very complex things, and there were kids who were still struggling to read 'cat in the hat'. by seventh grade, there were kids learning algebra at a rapid rate, and other kids still struggling with multiplication.

This learning difference did not magically appear at puberty.

And if you admit the difference might be in the order of, say, 30%... then by the end of a Student-C-speed 6th grade school, student A could have learned 8 grades worth of material. And I personally think 30% is way too small a number as it is.

As I said, I think this is OK. School is a public facility. i just don't see how folks can claim it's all "working out" for everyone, when we know that's not possibly true.

-- ErikHammarlund - 02 Nov 2006

ktm guest

I MUST get to work, but you describe my experience exactly, especially your first two points.

Voracious reading is pretty much the key to the kingdom (don't know if you've been around for the couple of posts I did about Fischgrund's (sp?) book on kids who scored perfect 800s on the SATs.

They were ALL, every last one of them, voracious readers.

Michelle Hernandez says the same thing about the stellar applicants they got at Dartmouth.

relearning

I can't say that I know precisely what is meant by this term - and it's possible a gifted person doesn't need to relearn, or perhaps relearns so quickly we don't perceive relearning taking place.

From a common sense POV, however, I can describe exactly what relearning is.

Instead of boring everyone with my tennis example again, I'll give my Manhattan example.

We moved here from L.A. in 1999.

8 years now.

I go into the city reasonably often, by my own lights....probably at least once a month.

At the same time, a few months can go by when I don't go in; then I go in a couple of days in a row....

If I had been commuting into the city for work, I would now "know it like the back of my hand."

Instead, I'm still a novice.

When I go into the city I "learn" - i.e. find out and remember - what subway line to take, which direction the traffic on Lexington travels, etc.

Then, when I come back home, I forget what I've learned.

If I went back to the city the next day I would relearn this information much, much faster, because "relearning" - i.e. hearing it again, finding my way again - would jog my memory.

Since I normally wait a month or two before returning to the city and "relearning" the routes, I end up starting from scratch.

I'm a very good learner, but after 8 years I'm practically a tourist in Manhattan still. (Not quite, but you get the drift.)

I was stunned to discover, in this passage, the fact that "slow learners" are slow for brand new material in brand new fields.

This idea had simply never occurred to me; I thought a fast learner was a fast learner & a slow learner was a slow learner.

In fact, probably none of us acts as if we believe this.

When I'm dealing with any expert in any field, including fields or jobs that probably don't require high IQs or speedy learning, I assume he or she will learn new material within the field very quickly.

I think I've mentioned our farmhand (memory gone - )

He may have had a borderline IQ, but that didn't make him a slow learner for farm work which he's been doing for years.

The implications of this material are profound.

-- CatherineJohnson - 02 Nov 2006

I'm not sure whether relearning is the same thing as relearning something you haven't used in a very long time (polar coordinates).

I suspect it's not, quite.

I suspect the concept of "relearning" is used to describe the process of completely mastering new material (whether in a novel field or not).

Not sure, though.

-- CatherineJohnson - 02 Nov 2006

The mixed-ability grouping issue intrigues me.

Off the top of my head, I don't see how it can work without the speedy thirds sacrificing learning.

On the other hand, you could also give the non-speedy kids outside "help," which might conceivably keep everyone moving forward at the same pace.

My sister's school did this for reading.

The slower readers actually came to school an hour early each morning.

She said it worked (from what she could see); when the school day started they were on the same track as the faster kids.

-- CatherineJohnson - 02 Nov 2006

I think this issue becomes moot down the line, once kids are old enough to do a lot of independent learning.

With the Keller method (I would KILL for Keller method courses here in Irvington) each student moves literally at his own pace.

When you're finished with a unit you take the test; if you pass the test at mastery level you move on to the next unit.

The professor gives lectures and does independent tutoring/meetings with students.

-- CatherineJohnson - 02 Nov 2006

I think you're still misreading the effect of the Englemann writings.

There really is no "magic pill". Slow and fast learners/rememberers will always be distinct.

What we CAN do is to minimize the inherent differences through appropriate education. In other words:

teaching to mastery,

which means little or no "relearning" on the spiral,

which removes an alternate avenue for the faster kids to "get ahead" of the slower kids.

What Englemann is saying is this:

It takes A 100 minutes to master a task. It takes B only 80 minutes to master the same task.

If A only spends 80 minutes, A will not master the task. A will have to relearn part of the task before reaching mastery--so if we assume A "wastes" 20 minutes doing so, A's total time to learn the task will be 120 minutes.

WITH mastery, the "difference" between A and B is only 20 minutes. WITHOUT mastery, it's 40 minutes.

WITH mastery, when A learns the next section of the course, A will have enough background information to process and learn it. WITHOUT mastery, A will have to spend extra time relearning... so if A only has 100 minutes on part 2, A won't master it. And the cycle starts again.

That is Englemann's point.

Note, however, that the inherent difference bewteen A and B is still there. It's just not exacerbated by bad teaching.

-- ErikHammarlund - 03 Nov 2006

The problem is that few in the education world feel that ANY mastery is very important. Take this comment from "Algorithms in Everyday Math":

"If taught properly, with understanding but without demands for 'mastery' by all students by some fixed time, paper-and-pencil algorithms can reinforce students’ understanding of our number system and of the operations themselves. Exploring algorithms can also build estimation and mental arithmetic skills and help students see mathematics as a meaningful and creative subject."

No required mastery by some fixed time and "exploring" algorithms guarantees that the reange between student 'A' and student 'C' will increase. If you don't master basic math algorithms, then there is little left to master.

" ... see math as a meaningful and creative subject."

But, apparently, not as a subject to master. Math appreciation.

The problems have nothing to do with grand concepts of learning. They have to do with basic assumptions, expectations, and competence.

-- SteveH - 03 Nov 2006

The problem is that few in the education world feel that ANY mastery is very important.

I would go so far as to say that ed schools actively oppose mastery learning; that's why an entire book bemoaning the absence of mastery learning has to be written.

At the simplest level, mastery has to mean "memory," and ed schools oppose memorization.

At a somewhat more philosophical level, ed schools oppose the teaching of content, and favor the teaching of abstract "skills."

Mastery can imply mastery of a skill, of course, but when it comes to content "mastery" has to mean learning - i.e. committing to memory - content.

-- CatherineJohnson - 04 Nov 2006

I glanced at our state math standards the other day, and I don't think I saw the word "learn" anywhere....it was all "understand."

Students are to "understand" concepts.

-- CatherineJohnson - 04 Nov 2006

I think you're still misreading the effect of the Englemann writings.

There really is no "magic pill". Slow and fast learners/rememberers will always be distinct.

I'm losing the thread - are you talking to me or to another person in the thread?

I don't think I've said slow and fast learners/ rememberers become one and the same....at least I hope not

-- CatherineJohnson - 04 Nov 2006

If A only spends 80 minutes, A will not master the task. A will have to relearn part of the task before reaching mastery--so if we assume A "wastes" 20 minutes doing so, A's total time to learn the task will be 120 minutes.

WITH mastery, the "difference" between A and B is only 20 minutes. WITHOUT mastery, it's 40 minutes.

WITH mastery, when A learns the next section of the course, A will have enough background information to process and learn it. WITHOUT mastery, A will have to spend extra time relearning... so if A only has 100 minutes on part 2, A won't master it. And the cycle starts again.

That is Englemann's point.

Note, however, that the inherent difference bewteen A and B is still there. It's just not exacerbated by bad teaching.

right

perfect!

-- CatherineJohnson - 04 Nov 2006

does anyone know how many groups Engelmann uses in real life?

does he end up with 3 in all?

-- CatherineJohnson - 04 Nov 2006

Three groups: high, middle, and low:

As a general rule for beginning levels of the programs, the lowest group should be relatively small compared to the middle and high groups. Students in the low group require more individual work and attention than students in the other two groups. Students should be divided into groups according to the number of personnel available, but guidelines for numbers of students in groups must still be followed. Low groups working in early program levels should have no more than 4 to 6 students in them. Other groups should have 10 to 12 students. These groups should be homogeneous; in other words, students in each group are at similar levels and learn at similar rates.

Students should not be locked into instructional groups for the rest of the school year. An individual student might start out with skills appropriate for a given group but learn faster or slower than the others. That student should have the opportunity to move to a group that is more appropriate for the student’s skill level and rate of progress. Moving a student to a new group should be communicated in writing so all personnel know what groups all children are in.

From Coaching Level A Manual

-- KDeRosa - 04 Nov 2006

Ken - thanks so much

that is a great help

Since I don't have time to look at the manual just now - do you know whether he "ends up" with three groups?

in other words, obviously he has these three groups to start; does he have these three groups throughout the sequence - i.e., after kids have started to acquire some proficiency in the subject?

-- CatherineJohnson - 06 Nov 2006

Catherine, I'm with you on teaching to mastery.

However, I'm also one of those who was bored silly in public school as were my two children. (And BTW, I may be your oldest poster, at age 66).

I think it is very hard on us smarter ones to not achieve mastery as fast as we can. At least two reasons:

First, if everything is always easy for us to understand, we don't learn when we are young how to achieve understanding. I'm talking here about vertical understanding, not horizontal "enrichment" which I'm very much against.

Second, remember the old rule that if something compounds at 5% a year, it will have doubled in about 14 years. That means that a student who is capable of learning only 5% faster than an average student could learn almost twice as much in 12 years of schooling as the average student. [For extra credit, figure out the factor for the student who learns 10% faster.]

And hi to Erik. I completely agree with you about inherent limitations to "getting" higher-level math. I'm great at computational math but there's lots of stuff in Courant and Hilbert I'll probably never grok.

-- SusanJ - 06 Nov 2006

Here's something from the world of computer programming that supports my point of view and my understand of what Erik was saying. Joel, the author, is the head of a company that develops software.

http://www.joelonsoftware.com/articles/GuerrillaInterviewing3.html

Start of quote: 15 years of experience interviewing programmers has convinced me that the best programmers all have an easy aptitude for dealing with multiple levels of abstraction simultaneously. In programming, that means specifically that they have no problem with recursion (which involves holding in your head multiple levels of the call stack at the same time) or complex pointer-based algorithms (where the address of an object is sort of like an abstract representation of the object itself).

I’ve come to realize that understanding pointers in C is not a skill, it’s an aptitude. In first year computer science classes, there are always about 200 kids at the beginning of the semester, all of whom wrote complex adventure games in BASIC for their PCs when they were 4 years old. They are having a good ol’ time learning C or Pascal in college, until one day they professor introduces pointers, and suddenly, they don’t get it. They just don’t understand anything any more. 90% of the class goes off and becomes Political Science majors, then they tell their friends that there weren’t enough good looking members of the appropriate sex in their CompSci^? classes, that’s why they switched. For some reason most people seem to be born without the part of the brain that understands pointers. Pointers require a complex form of doubly-indirected thinking that some people just can’t do, and it’s pretty crucial to good programming. A lot of the “script jocks” who started programming by copying JavaScript^? snippets into their web pages and went on to learn Perl never learned about pointers, and they can never quite produce code of the quality you need.

-- SusanJ - 06 Nov 2006

Interesting site SusanJ^?. As a programmer for about 35 years, and having taught college math and computer science a few (many) years back, I know exactly what he is saying. I'm just surprised that a student can get a degree in computer science nowadays without understanding pointers. Usually it's the Data Structures course that requires this - and weeds them out. But, it's not just the pointers. I had students who just couldn't handle details. Programming (and math) are all about basic skills and the ability to deal with exact details. For some people, their heads explode.

I thought this quote of his was most appropriate for KTM.

"Serge Lang, a math professor at Yale, used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t. The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff."

Just like with programming, the basics have to be really solid. I remember that it took me until the end of my Trig course in high school (Junior year) before I really understood and was good at algebra. You have to know it backwards and forwards.

-- SteveH - 07 Nov 2006

Hi Steve,

I'm glad you enjoyed the article.

I thought the quote you pulled was a good one about the importance of mastery to automaticity.

But I also think Joel is right about there being aptitudes as well as skills.

I've been a programmer for 36 years! I agree that programming requires attention to detail but also the ability to sense how the different parts of a logical structure interact.

-- SusanJ - 07 Nov 2006

I agree that there is an aptitude issue. Plus, you have to really like beating your head against a wall, especially for those bugs that seem to come and go depending on the contents of a stack.

Pointers and recursion are big things, but I usually saw issues in students before then. I taught CS back when Turbo Pascal was big. That made programming much faster, but it didn't help change their aptitude. Many just can't handle all of the details, so they write the program any which way and then try to debug it correct. Change something - see if it worked. Change something else - see if it worked.

Actually, Turbo Pascal made this worse. In the old card deck days, you really couldn't do this. It would take too long to repunch the card, submit the deck to the operations window, sit and watch your (low priority) status on the monitor, and wait for your printout. You had to be very methodical and analyze each line of code in your head. Turbo Pascal was so fast that students wanted to avoid the thinking part. They would fix the problems the program generated, but they wouldn't see the errors still hidden in the program. I used to take their programs (they thought were correct) and go through them line by line and show them all of the errors they missed. Turbo Pascal was a big help for compile-time errors - much faster to learn things. It just couldn't help them with the logic errors.

I usually could tell early on whether a student was a "programmer". Did they get the program done and did it work? Programming courses are killer courses and usually the first one weeds out about 70 percent. It could be the hard work or it could be the incredible amount of details, or it could be that they just didn't like it.

When you write a paper for another course, it's done when you decide it's done. The more you work on the paper, the better your grade. For computer programming, this isn't the case. You have to get the program done on time and it has to work (at least most of it). There is not a lot of partial credit and there is no direct relationship between the amount of work you do and success. People who are good in programming are the ones who are able and willing to do whatever it takes. Rule number one for computer science students: Get all of the programming assignments done (yourself) and make sure they are correct - even if it means staying up all night(s). There is no other course (that I know of) where homework is as important.

Desire is also an important factor. I didn't care at all about computers when I was growing up (before calculators). Then my girlfriend (now my wife) went to school for computer operations (she is now a Unix AdMin^?) and I decided (!) that I liked computers. I still remember my first programming workbook in FORTRAN. That was in 1972. I remember arguing about Nixon with other students while waiting for my printout.

-- SteveH - 07 Nov 2006

I think it is very hard on us smarter ones to not achieve mastery as fast as we can. At least two reasons:

First, if everything is always easy for us to understand, we don't learn when we are young how to achieve understanding. I'm talking here about vertical understanding, not horizontal "enrichment" which I'm very much against.

I agree.

I am VIOLENTLY opposed to deliberately holding back gifted and/or talented kids, which our district does as a matter of formal policy.

For some reason, I was never bored in school.

We didn't have much tracking, and I was always way out in front of the rest of the class, but I never felt bored....don't know why.

On the other hand, I DID spend quite a lot of time "talking to my neighbor."

-- CatherineJohnson - 07 Nov 2006

Second, remember the old rule that if something compounds at 5% a year, it will have doubled in about 14 years. That means that a student who is capable of learning only 5% faster than an average student could learn almost twice as much in 12 years of schooling as the average student. [For extra credit, figure out the factor for the student who learns 10% faster.]

I agree.

It bothers me no end that my district's administration refuses even to think about this.

They have a "philosophy" of "enrichment" not acceleration, and that's that.

There's no discussion.

(btw, I think Rudbeckia has an at least somewhat favorable view of enrichment, and I would have to defer on this to mathematicians, math brains, etc.)

However, when enrichment means, as it does in our district, having a fourth grader do a long division problem by means of "strategies" and skip-counting, I oppose it.

-- CatherineJohnson - 07 Nov 2006

On the other hand, I DID spend quite a lot of time "talking to my neighbor."

I was the one reading under the desk CONSTANTLY.

-- KathyIggy - 07 Nov 2006

I was the one reading under the desk CONSTANTLY.

I did a huge amount of reading a book inside a book.

I remember reading GWTW in geometry class.

The teacher made stop.

-- CatherineJohnson - 07 Nov 2006

Since I don't have time to look at the manual just now - do you know whether he "ends up" with three groups?

I believe that depends on the schools and the variation in ability levels within the school. In the upper levels, the variation between student learning rate reduces.

Still the teacher manuals recommend splitting up eaach class into low, middle and high. I suspect this holds even if the class is comprised of all gifted students. The teacher is supposed to gauge the pacing and mastery level of the class by focusing on the performance of the averqage group-- when they get it, it is time to move on. if teh lower preformers aren't getting it, more time can be spent with them alone. If the higher performers aren't getting it, it's time to reteach.

-- KDeRosa - 07 Nov 2006

Still the teacher manuals recommend splitting up eaach class into low, middle and high. I suspect this holds even if the class is comprised of all gifted students. The teacher is supposed to gauge the pacing and mastery level of the class by focusing on the performance of the averqage group-- when they get it, it is time to move on. if teh lower preformers aren't getting it, more time can be spent with them alone. If the higher performers aren't getting it, it's time to reteach.

ok, I'm going to have to read this

how does this differ from differentiated instruction?

does it differ primarily in terms of teaching to mastery and ongoing formative assessment?

also - how are the 3 groups functioning - are they doing lots of "group work" / "seat work" - i.e., what are the other two groups doing while one group is being taught by the teacher?

(I realize you may not want to be our permanent source on all this!)

-- CatherineJohnson - 08 Nov 2006

I'm positive he's right about the 3-groups business; Christopher's "Phase 3" teacher, Mrs. Panitz, told us that all classes divide up into 3rds.

The Phase 3 class had 3 thirds; the Phase 4 class also had 3 thirds.

Talking about that fact was what made her realize it made sense to move Christopher. He was at the top of the top third in her class; when she thought about the fact that the Phase 4 class also had thirds she suddenly realized that there was no reason to assume that the top kid in the top third of her class would be below the bottom third of the Phase 4.

Overlapping curves.

I wish I'd asked which third he was in in the Phase 4 class.

He might have been in the top one.

He definitely wasn't in the bottom third.

-- CatherineJohnson - 08 Nov 2006

The instruction is identical. The only differentiation comes in how fast each group learns.

In a typical homogeneous classroom, you still have the three groups, but the ability spread is close.

If the class is full of high perfromers, the class will move at a brisker pace than if the class were full of lower performers.

Within each class, however, the pace will likely be such that the pace is too brisk occasionally for the lower part of the class. It may also be that soem lower students do not pass the mastery tests given wevery ten lessons or so.

In both cases a remedy must be provided. That remedy is to reteach what was not mastered and retest. Ideally, this is done outside of the nrmal class time so as not to slow down the rest of the class.

If certain students require too much reteaching or fail to reach mastery too often, they likely need to move to a class where the pace is slower. Similarly, if a student is acing the mastery tests, he should be moved into a faster paced class.

This is why bigger schools work out better for mastery learning, because there will be a higher chance that a suitable class can be found for students that need regrouping that is close to an appropriate lesson for that student.

-- KDeRosa - 08 Nov 2006

That remedy is to reteach what was not mastered and retest. Ideally, this is done outside of the nrmal class time so as not to slow down the rest of the class.

oh, I see

have you looked at Princeton Charter School's description of its day?

-- CatherineJohnson - 08 Nov 2006

I’ve come to realize that understanding pointers in C is not a skill, it’s an aptitude. In first year computer science classes, there are always about 200 kids at the beginning of the semester, all of whom wrote complex adventure games in BASIC for their PCs when they were 4 years old. They are having a good ol’ time learning C or Pascal in college, until one day they professor introduces pointers, and suddenly, they don’t get it.

How odd. I remember encountering pointers, and suddenly, just not getting it.

However by that time I had encountered drawing frequency-graphs, and just not gotten it, and then next year wound up tutoring them to a friend. So I kept on using pointers and eventually got it. Or at least got it enough to get an A- on the course.

(This wasn't helped by a compiler that for some reason crashed whenever I tried to make a doubly-linked list using pointers, driving me and my tutors insane as we tried to figure out the bug in the code. Eventually my boyfriend of the time suggested running it on another compiler and it worked fine.)

-- TracyW - 13 Nov 2006

Serge Lang, a math professor at Yale, used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t. The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.

You see, if you can’t whiz through the easy stuff at 100 m.p.h., you’re never gonna get the advanced stuff.

wow

I love it!

-- CatherineJohnson - 21 Nov 2006

That is the entire point of Saxon.

The Saxon books teach you algebra (and arithmetic) inside & out.

They've been constructed on core principles of cog sci, and they work.

IMO.

-- CatherineJohnson - 21 Nov 2006

At this point I don't even care whether the math instruction at the high school is good.

Christopher will be doing Saxon regardless.

Certainly Algebra 1, but I may insist he do the "trilogy."

(Algebra 1, Algebra 2, Advanced Mathematics)

-- CatherineJohnson - 21 Nov 2006