kitchen table math, the sequel: on improving traditional teaching

Friday, January 26, 2007

on improving traditional teaching

Here's a question I've been struggling with for some time now:

Under the traditional curriculum why didn't mastery learning become the norm?

With the exception of the haphazard presentation of material is some of today's constructivist texts, traditional texts typically present a lesson, provide some practice, and then move on to the next topic in the sequence. Many students learned the material using this approach, but there was no effort to get the student to master the material and firmly place the material into the student's long term memory where it is somewhat protected against the ravages of forgetfulness. The inevitable result is that the student partially or fully forgets much of the material once the class moves on, unless the skills taught are used in subsequent lessons (like in elementary math). It was rare that we ever got a cumulative exam at the end of the year which was probably intentional because most of the students had forgotten the material taught in the first half of the year. This practice was mitigated to an extent by the fact that much of the material was retaught year after year--a precursor to today's spiral curriculum.

Nonetheless, this seems to be a horribly inefficient way of teaching to me. Yet it seems the have developed as the dominant form of (pre-constructivist) instruction by the latter half of the 20th century.

The question is why did it develop this way? Why not mastery learning?

Before you answer take a look at this blurb from Engelman's new book (pp. 30-31):

Mastery is essential for lower performers. Unless the practice children receive occurs over several lessons, lower performers will not retain information the way children from affluent backgrounds do. Prevailing misconceptions were (and are) that children benefit from instruction that exposes them to ideas without assuring that children actually learn what is being taught. If you present something new to advantaged children and they respond correctly on about 80 percent of the tasks or questions you present, their performance will almost always be above 80 percent at the beginning of the next session. In contrast, if you bring lower performers to an 80 percent level of mastery, they will almost always perform lower than 80 percent at the beginning of the next session.

The reason for this difference is that higher performers are able to remember what you told them and showed them. The material is less familiar to the lower performers, which means they can’t retain the details with the fidelity needed to successfully rehearse it. After at-risk children have had a lot of practice with the learning game, they become far more facile at remembering the details of what you showed them. When they reach this stage, they no longer need to be brought to such a rigid criterion of mastery. At first, however, their learning will be greatly retarded if they are not taught to a high level of mastery.

This trend was obvious in the teaching of formal operations. At first, the low- and high-performing groups were close in learning rate. Later, there were huge differences. Group 2 was able to learn at a much higher rate, largely because it was not necessary to bring them to a high level of mastery. On several occasions, I purposely taught the children in Group 2 to a low level of mastery (around 60 percent). I closed the work on the topic with one model of doing it the right way, and I assured the children that this was very difficult material. At the beginning of the next lesson, almost all of them had perfect mastery.

So, I think the answer to my question as to why mastery learning didn't become the norm is simply that it wasn't needed. Why go through all the effort of mastery learning when the higher-performers really didn't need it to learn? If the teacher is basing their performance on the feedback they are receiving from the successful students (only 60% mastery is needed), it's easy to see how one could reach the false conclusion that that's all the teaching a student needs to learn. And human nature being what it is, why teach more when less will do.

Nonetheless, I think we now know enough about how the brain works to know that retention of learned material is greatly enhanced when the learner engages in distributed practice after the initial mass practice. All students would benefit from distributed practice. So why haven't traditional educators changed their ways to offer more distributed practice?

I understand there is a philosophical objection to distributed practice (i.e., drill and kill)at the elementary school level. But what about at the secondary and post-secondary level where traditional education is still the norm? At this level, distributed practice just means giving a a couple of additional independent work problems that keeps previously taught material alive for the student until the material is better retained in long term memory. So why are classes at these levels still taught like the need for distributed practice doesn't exist?

Moreover, if the goal is to eradicate the worst practices of constructivist teaching, wouldn't it be beneficial to improve traditional teaching methods to incorporate techniques that will improve student performance? One of the reasons why constructivism has gained the foothold it has is due to the underperformance of the traditional curriculum, especially among lower-performers.



SteveH said...



"One of the reasons why constructivism has gained the foothold it has is due to the underperformance of the traditional curriculum, especially among lower-performers."

I think you give them way too much credit. They just wanted to do what they wanted to do. Philosophy and assumptions first, justification second.

Catherine Johnson said...

haven't read yet and I MUST get some writing done, but I've had the same question myself: why didn't mastery learning take hold?

It's not that nobody ever thought of it; I took two courses using the Keller Method in college, one at Wellesley & one at Dartmouth.

That was mastery learning; it worked beautifully.

People wrote "programmed instruction" books; my first job out of college was writing these books.

Then it all went away.

Catherine Johnson said...

I've actually gone through Amazon pulling all the programmed instruction books I can find. I found a couple for vocabulary & bought them at once.

Catherine Johnson said...

I'll have to finally post the link to the law professor in Australia who put together a law course using the Keller method.

It worked, but the other professors were so hostile that he abandoned the effort.

Catherine Johnson said...

On several occasions, I purposely taught the children in Group 2 to a low level of mastery (around 60 percent). I closed the work on the topic with one model of doing it the right way, and I assured the children that this was very difficult material. At the beginning of the next lesson, almost all of them had perfect mastery.



Catherine Johnson said...

I believe things are getting much, much worse.

Virtually all of our teachers here now are in their 20s. They don't have kids; they haven't spent many years teaching.

They have ZERO interest in distributed practice.

The ONLY thing any of them ever talks about is "comprehension."

If a student does badly on a math test, that's because he didn't understand.


"If students need distributed practice, parents can find worksheets online." Irvington math chair

I talked to a friend whose son is now drowning in 6th grade accelerated math taught by another new young female teacher who grades the kids on binder organization and whose stated goal is "I want the kids to take ownership of their learning."

Her son is down to Ds and Fs on tests.

Teacher says he doesn't understand the material.

The tutor went over both of his tests with him & says he absolutely does understand the material (ratios & proportions, fairly easy topics to understand first time around).

The problem is "careless error."

I hate that phrase!

These aren't careless errors; they are what you get when you understand something but have never done it.

Our math department now fully conflates undertanding with doing.

Declarative & procedural knowledge are, to them, one and the same.

The teacher who just retired used to tell all the kids to do as many math problems as possible. She'd tell them, "The secret to math is DO MORE PROBLEMS."

She would also Xerox copies of the teacher's manual, so the kids could check their answers.

A friend just asked the 8th grade teacher for copies of the teacher's manual (impossible to find online). He checked with the math chair; she said 'no.'

He himself - he's the best teacher in the middle school - tells kids and parents that the retired teacher went overboard; there's "no need" to do so many problems.

That teacher was legendary for getting math inside students' heads.

Now she's gone and the belief system is:

no need for "all that practice"

no need to re-do the problem if you got it wrong

You can see why these teachers aren't interested in checking homework.

They aren't interested in procedure, period.

They believe that after they've put something on the board everyone's got it.

If they don't have it they should come in for "extra help," which means the teacher will explain it again.

Constructivism leads directly to sage on the stage teaching unsupported by drill.

Barry Garelick said...

My experience in high school math was not as you describe. Our algebra book was written so that mastery of one skill was needed to proceed to the next chapter. Equations got progressively more complex and if you didn't have the rules down for multiplying or dividing terms containing exponents, etc., then you couldn't do subsequent problems. Factoring and rational expressions were mastered before proceeding to quadratic equations, which then enabled students to understand completing the square and derivation of the quadratic formula.

(Take a look at Dolciani's algebra books. They are very sequential and require mastery of material.)

In geometry, we used a very structured text in which material built upon itself. One needed the earlier theorems to progress to more complex areas, as it should be.

Pre-calc was primarily trig (which of course builds), analytic geometry, polar coordinates, parametric equations, conic sections (which builds upon the principles set forth in analytic geometry to derive the various formulae for the curves), and then the beginnings of calculus.

I would agree that the "traditional" texts today (Larson, Boswell Stiff's "Algebra" for example) get away from anything logical, and have a minimum of distributed practice. Of course, the book is touted as meeting NCTM standards, so in a sense, even the traditional books that are written today are inherently informed by the trappings of constructivism. Exposure = understanding as Catherine characterizes it.

Catherine: We must talk about programmed learning. I had a grammar book that was programmed (English 2600) and have a calculus text that is programmed, written in 1968. I have reservations about its effectiveness for math because one needs drill and you don't get much of it.

TurbineGuy said...

I don't know if the military is considered the "real world", but we are one of the largest employers (if not the largest) in the country.

We expect/demand mastery. Our technical schools are broken into blocks that are scheduled in a set logical sequence. The content of every block is expected to be mastered before moving on.

One of the most amazing things that I find in the education system is how disconnected its practices are from the actual job market.

Can you imagine an auditor that hadn't mastered his trade. What would happen if a computer programmer had a 30 - 40% bug rate in their software (answer: Microsoft. JK).

It's amazing that our production oriented capitalist society that values not only hard work, but quality produced such an arbitrary education system that thinks that people will mircaculously become competent without any practice.

KDeRosa said...

Barry, let me clarify.

My high school math instruction was also the same as you describe. Very linear with mastery of one skill needed to progress. The rrsult was that many skills were subsumed into the new skills. So for these skills there was a degree of distributed practice. (We also moved at a quick clip and there was no time for playing games.) However, Many skills were not partially or fully subsumed into the new skills and once we were done with those skills, you didn't look back except to study for the periodic tests.

There was some distributed practice available in this scheme but I don't think it was sufficient. We might have mastered a certain skill in the sense that we learned how to do it quickly and accurately, but once we moved on that was it as far as distributed practice went. Once we moved on, we stopped doing problems related to the previously learned material. The inevitable result was by the end of the year I had forgotten how to do a fair bit of the mateial we learned in the first half of the year. It was true that I could have easily relearned it and/or was retaught it in later courses, but the fact remains I had forgiotten the material because the instruction had let the some of the material we learned lie fallow as we moved on to bigger and better things. Ultimately, my point is that if the object of learning is to make a permanent change to long term memory, this didn't always happen, not because we hadn't learned the material, but because we were not provided with enough distributed practice to make it stick.

Everybody, think back to your high school days what were the two most dreaded words (instructionally speaking)?

Cumulative test.

(P.S.: I'm almost done)

Barry Garelick said...

Again, my experience doesn't match yours. What we learned previously we continually used. Cumulative tests were common. I think you're a bit younger than I, so you probably had a smorgasboard of different concepts. When I was taking math, it wasn't yet a mile wide, and was very focused. What we learned, was continually applied, so was very efficient.

I tutor students in pre-calculus occasionally so I see what you are talking about. I would have a tough time now if faced with the large menu of topics they must learn. For example, we did not cover De Moivre's theorem for all complex and real solutions to an equation.

Unrelated topic: How're ya comin' with that letter??

Instructivist said...

One reason mastery is disfavored has to do with educationist insistence on wildly heterogeneous classrooms with respect to ability. Under such circumstances, mastery for all is an elusive goal. High and medium ability students would be bored stiff and teaching would be stuck in place. So when advocating mastery one needs to decide for what portion of the class when ability ranges widely.


Query: Some time ago (KTM I), Catherine, I believe, mentioned a book that makes probability easy (if that's possible). I wanted to buy it way back then, didn't, and now the title has slipped my mind.
Please let me know, if you remember the title.

I don't get along well with probability. It gave me the heebie-jeebies in college. I am ashamed to admit that I don't have quick recall of prob formulas for simple problems. One such problem for which I need the formula has to do with the number of different numbers that can be formed with a given number of digits that may not be used twice, e.g. 5, 3, 7, 9, 2.

SteveH said...

"One reason mastery is disfavored has to do with educationist insistence on wildly heterogeneous classrooms with respect to ability."

I agree with instructivist. It may be a chicken and egg thing, but it seems that the desire for treating all kids as equals comes first on the philosophical pecking order. They know that facts and skills are important because they teach them. But after the mixed ability assumption, they are stuck. They can't separate and they can't accelerate. They can only dream or try to say that enrichment is more important than acceleration. Facts and skills are just not that important. They can differentiate "understanding" and "critical thinking" in a mixed classroom, but they can't differentiate content and skills; especially mastery. Enrichment, understanding, and critical thinking are vague enough to give them great pedagogical cover. They placate with generalities, but then do what they want.

SteveH said...

"I don't get along well with probability."

You're not alone. I made this same sort of comment before. I forgot the name of the book that was recommended, and there might be more than one. I bought the one that looks like cartoons for my brother-in-law, but I don't remember the name. Don't ask me why I didn't buy a copy for myself. (Some people have a life list of places to go. I have a life list of things to learn, but probability is not near the top. It's on the list, however, because I need to simplify it all for my son who is getting some of these problems already.)

The book looked good (before I sent it out), but there are a lot of topics besides combinations and permutations in the book. What I have problems with is not statistics like standard deviation, but all of the problem variations of combinations and permutations.

The basics aren't too bad. The equations for combinations and permutations are really very simple. For simple problems, it's easy to decide which formula to use and what numbers to use for the variables.

However, it can be very difficult to translate many word problems into these simple formulas. Then, there are variations to the fomulas and the problems of conditional probability and Bayes' rule. This is definitely a case where the the math is easy, but translating the word problem is difficult.

I'm looking at my copy of Schaum's Outline Series for Probabability and Statistics. You can see the simple formula for permutations is

nPr = n!/(n-r)!

Factorials (!) are easy and this formula is easy. How to apply it takes a lot of practice.

Then, there is the variation

nPn1,n2,n3,...,nk = n!/(n1!n2!n3!...nk!)

K-8 math loves to give combination and permutation problems to kids to have them discover the answer. This perhaps can be done with very simple problems, but it cannot be done (except by very few people) with more complex problems like this:

1.34 Five red marbles, two white marbles, and three blue marbles are arranged in a row. If all the marbles of the same color are not distinguishable from each other, how many different arrangements are possible?

It would be nice to have the ability to "discover" the solution, but is that reasonably possible for any combination/permutation problem? This problem from Schaum's has an interesting explanation, but I think it would take a lot of PRACTICE and a good knowledge of the basics to discover the solution to any problem.

The difficulty with combination and permutation problems is that many of the problem statements look so different. Practice (not discovery) is the only way to master the subject. But I would like to see an explanation that talks about how to translate word problem statments to the appropriate formula.

Do you ever get the feeling that something you are studying just HAS to be easier to understand; you just haven't found the right explanation? This is how I feel about permutations and combinations.

I know how many kids feel about math. They can do problem A, problem B, problem C, but they have absolutely no confidence that they can understand, let alone do, problem D. I feel that way with these problems, even though I completely understand simple combination and permutation problems.

Instructivist said...

"Facts and skills are just not that important. They can differentiate "understanding" and "critical thinking" in a mixed classroom, but they can't differentiate content and skills; especially mastery. Enrichment, understanding, and critical thinking are vague enough to give them great pedagogical cover. They placate with generalities, but then do what they want."

I just want to say that I greatly enjoy and benefit from your keen insights into the educationist mind.

I see this egalitarianism that seems to be a sacred dogma in edland as one of the MAJOR obstacles to achieving quality education. Things are steadily getting worse with the trend toward full inclusion.

If grouping by ability is not palatable in the upper elementary grades (HS is less of a problem), then at least there MUST be behavioral grouping. It is unconscionable to allow the behavior-disordered to prevent those willing to learn from learning.

Instructivist said...

Thank you for the comments on probability. You are right about the translation of problems into the formulas.

"K-8 math loves to give combination and permutation problems to kids to have them discover the answer."

These problems are part of the state standard/test here in Illinois and probably elesewhere, too.

Students are not expected to solve them using formulas. It's a guess-and-check game like solving the heads, two legs, four legs barnyard problem without algebra.

I see this as part of the fuzzy math desire to move away from traditional math and plaster the curriculum with estimates, visuals, pseudo statisitcs and the like in an effort to make math (with the math taken out) "accessible" to previously excluded populations in NCTM'S view.

KDeRosa said...

Barry, my parenthetical PS was too cryptic, but that was my update.