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excerpt from: Standards and Mastery Learning
by J. Ronald Gentile, James P. Lalley

Memory by Fast and Slow Learners

One field of basic research that is relevant to mastery learning, perhaps even more than studies comparing mastery-based instruction with traditional forms of instruction, is that of memory by fast versus slow learners. The general problem was introduced in Chapter 1 in terms of learning/forgetting curves, as well as what happens to the relationship between IQ and memory when a mastery standard is required. Consider the question directly (e.g., Gentile, Voelkl, Mt. Pleasant, & Monaco, 1995):
If slow learners can be equated with fast learners in amount initially learned, then what will happen on an unannounced test of memory for that material a day or week later?
a. Fast learners will recall significantly more than slow learners.
b. Fast learners will recall significantly less than slow learners.
c. Fast learners will recall about the same as slow learners. (p 185)
The question has a long history because it is central fo understanding individual differences in learning and memory….
...c is correct....When fast and slow learners are equated on amount of learning, recall is about the same.
[snip]
…the fastest and slowest thirds of the distribution tended to be quite different indeed: The fast learners ranged from 3 to 7 trials (median = 5), whereas the slow learners ranged from 12 to 33 trials (median = 15). Nevertheless, in surprise retention tests days or weeks later, the fast and slow learners were not significantly in average number of words of the poem they recalled.
Gentile et al. (1995) also extended this research to ask what happens to fast and slow learners in relearning to the same 75% to 90% standard. They found, first, that there was tremendous savings in relearning. Fast learners, who on average requird 5.3 trials at original learning, relearned to criterion in an average of only 1.4 trials. The corresponding data for slow learners was 17.4 trials at original learning and, remarkably, only 2.1 at relearning. The 1:3 ratio (fast:slow) advantage at original learning was reduced to 1:1.5 at relearning. All students, the fastest and slowest, were ready to relearn quickly as a result of having initially learned to a high mastery standard. Such data refute the oft-heard argument that there is no time to make sure each student learns to a high level; the savings in relearning time suggests that in the long run, teachers will save time by bringing all students to a high standard at original learning. [emphasis in the original]
The data with regard to memory, however, were a bit more complicated. It happened hat despite the 75% to 90% criterion, fast and slow learners were not exactly equated…Nevertheless, as the learning/memory curves described in Chapter 1 predict, there was much less forgetting after relearning than after original learning.
…What seems clear, however, is that requiring mastery to a high standard at original learning accrues the following benefits: (a) Fast and slow learners will recall about the same amount and are therefore both ready to have their prior knowledge activated in the current lesson (whether it be a review lesson or new applications of the material), and (b) relearning will be faster for all, saving time in the long run.

Table of Contents

Preface (pdf file)
1. Understanding Mastery Learning (pdf file)
2. Examining the Standards: Math, Science, Social Studies, and Language Arts
3. Planning Standards-Based Lessons Using Mastery Learning
4. Implementing Standards and Mastery Learning in the Classroom
5. Professional Development and Mastery Learning
Appendix: What Does the Literature Tell Us?

outputs, not inputs

from the Preface: (pdf file)

The history of standards also reflects federal government versus states’ rights skirmishes, competition among professional organizations (for what will be required and what will be optional for graduation), religious versus secular issues, as well as numerous other political issues beyond the scope of this book.
What does seem relevant, however, is an apparent shift in the goals of standards from inputs to outputs, in Marzano and Kendall’s (1996) shorthand. Whereas previous standards emphasized what had to go into a course (specifics of the curriculum, how material was to be taught, how many credits or Carnegie units it was worth), current standards emphasize what comes out (what students know or can do and how to assure accountability). As Marzano and Kendall (1996) summarized, “The new, more efficient and accountable view of education is output-based; success is defined in terms of students learning specific standards” (p. 17).
Whether the “outputs” view is better or worse, we cannot be sure, though it certainly created a firestorm of rhetoric over the high-stakes testing, perhaps a too-narrow view of accountability that has become linked to the standards movement beginning in the 1980s. Interestingly, the input/output categories have rough parallels with fundamental variables in John Carroll’s (1963) model of school learning, which provides some of the theoretical underpinnings of mastery learning. Those variables are opportunity and perseverance. Opportunity, in rough parallel to input, is the time allowed or scheduled by the teacher to cover the material and induce students to learn. Perseverance, in rough analogy to output, is the time the students spend or are willing to spend to learn. Although both of these are important to and predictive of achievement, perseverance becomes the bottom line: Teachers have been successful to the extent that they induce students to do what they need to do to learn.
We bring this up here to point out that the mastery philosophy on which this book is based is potentially compatible with current views of standards, but it has developed methods for measuring and motivating students to persevere until they achieve those standards.
Sadly, many advocates of standards, not to mention those who are charged with implementing them, know little or nothing about implementing mastery learning: its philosophy, its evolution, what makes it work or undermines it, its varieties. [ed.: don't know of don't care to know?] Thus, although the movement toward higher academic standards carries with it assumptions about learning, development, and measurement that have traditionally been central to the theory and philosophy of mastery learning, the standards movement has neither embraced mastery learning nor shown evidence of having learned from its successful or unsuccessful practices.

learning & forgetting

from the first chapter: (pdf file)

Learning, in other words, occurs in phases or episodes, and this original learning phase includes (a) the readiness component (described above), (b) learning to initial mastery, and (c) forgetting. Although forgetting has not been mentioned up to now, it is clear that forgetting is the inevitable result of initial learning, even when a high mastery standard of, say, 80% to 100% correct is required. [ed.: which is why it makes so much sense to spend one day teaching dimensional analysis going into the state test, assign no homework problems, and never mention the topic again] When the degree of original learning is less than mastery, say, 60% to 80%, then forgetting is likely to occur more rapidly or be more complete. If it is less than 60%, it is questionable to speak of forgetting at all, because learning was inadequate in the first place.1
Students show that they understand this principle implicitly when they ask, “Why do we have to learn this stuff anyway? We’ll only forget it.” Our typical answers, “Because it will be on the test” or “Because I said so,” are not satisfactory. In fact, we have been able to find only one satisfactory answer to the question, and it was supplied in one of the first empirical studies of learning/forgetting (Ebbinghaus, 1885/1964). The answer is that relearning is faster—that is, there is a considerable savings of time in relearning compared with original learning. Furthermore, there is a positive relationship between amount of time saved in relearning and the degree of original learning, with essentially no savings when original learning is below some acceptable threshold (which we earlier argued was 60% or less).
...[C]onsider the experience we teachers have. The first year of teaching a unit, we have a lot to learn (even though many prerequisites have already been mastered in college and student teaching). The next year, when we reach that unit again, we find we’ve forgotten quite a bit. Fortunately, relearning is faster, and we find ourselves reorganizing the material, coming up with new examples, and so forth. The next year, forgetting has been less yet, and thus there is greater savings in relearning. By the 10th year, the material is almost totally recalled, with examples virtually falling off the tongue. The material seems so easy by this time that many teachers can now be heard complaining, “The students are getting dumber and dumber every year.” Sadly, this is one of the negative effects of becoming expert in something: We lose empathy for the novice. (Note the parallel to what happens once a nonconserver on Piaget’s tasks becomes a conserver: e.g., the 8-year-old who understands that the amount of water does not change when poured into a taller, thinner glass cannot recall that she expected more in the taller, thinner glass when she was 5). This is also what distinguishes a mere expert from a teacher: An expert can do it; a teacher can do it but also remembers what it takes to progress from novice to expert.

Beautifully put. I would add that this is what distinguishes an expert nonfiction writer from a nonexpert.

For the negative example of how these learning/memory concepts form the basis for mastery learning, consider some students whose original learning is 50% or less. Their forgetting in Phase 1 will be at least as rapid as it will for those who mastered the material and, because they learned so little, more complete. Furthermore, there will be little or no savings in relearning during Phase 2. Then, if this material is treated as a review (“This was covered last year”), less time will be spent on relearning (and after all, students who mastered originally will need less time). Thus the relearning episode for those who need it most will also be substandard, leading to relatively little residue in memory and therefore little or no savings for Phase 3 relearning. By Phase 3, the motivation to learn this material will also be eroding (“I was never very good at this”), an issue we shall explore in more detail in “Learned Helplessness.”

The bottom 3rd learners are probably routinely being put into states of learned helplessness; I would expect to see learned helplessness in a number of the middle 3rd, too. In the case of accelerated math here in Irvington, I believe that we have some children in the top 3rd who have been pushed into states of learned helplessness for doing and learning math.

That is what their parents report.

The tearful mother who asked, at the Parent Revolt meeing in winter 2005, "What are you going to do to repair my child's self esteem?" deserved an answer.

What she got instead was: "I don't understand why your children are so thin-skinned."

That was it.

schools creating "comas"

A summary of the long-term effects of the difference between mastery and nonmastery at original learning is provided in Figure 1.2. After four or five episodes, when learners in Part A say to those in Part B, “I forgot more than you ever learned,” the sad fact will be that they are telling the truth. A close look at the curve in Part B shows that after four or five episodes, those persons—they can hardly be called learners—have learning/forgetting curves that resemble the brain waves of comatose patients.

learned helplessness, mastery learning, and math

Let’s return for a moment to the students in Figure 1.2B. If their first experience in learning fractions, say, is unsuccessful, they will forget all or most of it. Next year, they may even claim that they “never had this stuff before,” and they will probably believe it. On the second time through, if they are still unsuccessful (which is likely, because teachers usually spend less time reviewing previous material than was spent on initial learning), they will demonstrate little savings and forget again. By the third and fourth exposures to the material, they at least remember they had it before, but may stop trying and explain their lack of motivation with statements such as “I was never very good at math” or “Why do we have to learn this stuff anyway? I’m never going to use it.” Learned helplessness has set in.

[ed.: I want to stress that "learned helplessness" is a well-established concept in psychology — and that it was first demonstrated in animals. These experiments, which I studied in college, are wrenching to read. The most upsetting, which couldn't be done today, involved rats who were, iirc, yoked together and shocked. One of the rats could turn off the shock, or escape it somehow. The other rat was helpless. He received the same level of shocks as the "actor" rat, but he couldn't turn them off.

After going through this experience, the rats were put in a tub of water with a platform somewhere in the middle. (again, working from memory - details may be wrong) The "actor rates" - the ones who'd been able to turn off the shocks, swam to the platform and scrambled out of the water. The helpless rats didn't even try to save themselves. They sank into the water and drowned.

I believe that today, in our schools, spiraling curricula are putting children into states of learned helplessness.

Until someone proves me wrong, I stand by that perception.]

Under laboratory conditions, learned helplessness is developed by first exposing animals or humans to a series of experiences in which failure is inevitable and beyond their control (e.g., Peterson, Maier, & Seligman, 1993; Seligman, 1975). Later, when success is now possible and personally controllable, the victim does not even try. On the emotional level, there is a heightened state of fear, which if prolonged, can easily turn into apathy or depression. On the behavioral/motivational level, there is no perseverance or willingness even for trial-and-error searches (because “Nothing I do ever satisfies these people”). On the cognitive level, there is no discovery of what works, no understanding or organization of an information base, and a long list of defensive excuses or causal attributions, such as “I was never very good at this” and “I could do it if I want to, but school sucks” (the former a primarily female attribution, the latter male; e.g., Dweck & Licht, 1980). It is also more self-protective to adopt a strategy of not trying—or pretending not to try—than to try and not succeed.
Math seems to be a field particularly vulnerable to learned helplessness, because new topics and courses seem to be quite different from previous ones (from multiplication of whole numbers to fractions, arithmetic to algebra to trigonometry, etc.). Even having great success at earlier levels does not immunize against having difficulty on a new topic. Thus even being a “good” student or having 100% success does not guarantee against learned helplessness later, particularly if what students have been good at is memorizing without understanding.
But those primarily at risk for learned helplessness are those who come to school and have not mastered fundamentals (as mentioned earlier). If we teachers cannot diagnose their problems correctly— and early—they are almost destined to fail to master the new tasks. Sadly, to continue the math example, many teachers are not skilled enough themselves to diagnose a child’s problems with addition and subtraction as being a deficit in rational counting or one-to-one correspondence. Thus these students comprise the population in Figure 1.2B.
Is there a cure? As in health, prevention is easier than cure.

Didn't Steve H say this?

I believe that the phenomenon of learned helplessness explains why it was important for Christopher and Christian both to teach themselves using a Saxon book. Christian is emerging from his state of learned helplessness vis a vis math so fast it's startling. Same thing for Christopher, except with Christopher we're talking about prevention. Even after all the mishegoss of last year, his slow but steady march through Saxon Algebra 1/2 almost certainly prevents him from developing a state of learned helplessness in the first place. His previous work with Saxon Math 6/5 probably innoculated him against developing learned helplessness last year.

I think I've told that story.

Christopher does not know that Ms. Kahl selected "Finds subject matter difficult" from the Comments Bank. When I mentioned it one day, thinking he'd read his report card, he said indignantly, "She did not put that on my report card!"

He doesn't have learned helplessness.

how to cure math anxiety, part 2

With his learned-helpless dogs, Seligman (1975) literally had to drag them across the barrier to escape electric shock, anywhere from 25 to 200 times, before they once again tried to explore and control their environment. With humans, whose patterns of thought (“I was never very good at this”) may reinforce the helpless-behavior patterns, dragging is more figurative than literal. In any case, the cure for helplessness is competence, and only when students are succeeding do feelings of self-efficacy, self-control, and self-esteem begin to follow (see also Bandura, 1977, 1986).

Saxon Math is the answer.

memory & IQ

Table 1.2 shows the remarkable results regarding intellectual traits and memory.5 Under nonmastery conditions—that is, a single exposure for original learning, recall after 7 days, a single relearning opportunity, and then recall after 14 and 28 days—the correlations between intellectual traits and recall are all positive and significant. That is, higher-ability students tend to remember more, as society has come to expect.
In stark contrast, imposing a mastery standard of 75% to 90% correct on original learning and then again at relearning renders those standardized intellectual measures nonpredictors of how much is recalled: The correlations hover around zero and are all nonsignificant.
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.

(wrong) conventional wisdom about mastery

Critics usually cite three points in opposition to mastery learning:
1. It helps slower students at the expense of the faster students.
2. It is too oriented toward basic knowledge and skills at the expense of creativity and higher levels of thinking.
3. It requires too much work of the teachers.

[snip]

That fast learners are bored to tears waiting for slow ones to catch up is far too true of many educational programs, including badly implemented mastery programs. It is not true at all of Keller’s (1968) individualized mastery plan, in which students complete course units at their own pace. For group-based mastery schemes, such as Bloom’s (1971), it would be true only if mastery were misconceived solely as passing minimum competency tests, with no incentive for students to use their new competencies for higher-level intellectual purposes.

[snip]

A grading system that earns a minimum passing grade—that is, a C or 70 for passing the initial mastery test with at least 75% to 80% correct. Under the concept of mastery as a beginning rather than as an end state of learning, even a test score of 100% is just the initial phase of learning-forgetting curves. Thus it should earn an entry-level grade. Not passing the mastery test should have a grade of zero or “incomplete” attached to it so that, like driver’s tests, initial mastery is conceived as an all-or-nothing affair: Either you get your license, or you do not.

A set of enrichment activities that use but go beyond the basic knowledge, skills, and principles required for mastery. This includes reports on how these principles are applied in real life, creative projects and experiments, further readings or advanced problems to be solved, cooperative investigations or debates, and—most important—tutoring others (we really learn something well when we teach it). Such activities, because they provide overlearning, distributed practice, organization and construction of knowledge, and the like, earn bonus points when adequately completed: Add 5 to 10 points for each project to the minimum pass of 70, or move from C to B for one advanced project and A for two or three such projects.

This is where I get off the boat.

This passage equates "mastery" (75% to 80% correct on a test) with "expertise." A child who has scored 80% correct on an initial test of mastery has not reached the level at which he is ready to start constructing knowledge, applying principles to real life, or solving "advanced problems."

What is happening here — and, again, until someone proves me wrong I'm going to stand by my opinion — is that constructivist teaching practices are being used to mask the fact that the fast kids are being slowed down to the slow kids' pace.

It's significant that these authors do not once reference Siegfried Engelmann's work.

There's a reason for that.

I think the reason is the Siegfried Engelmann directly confronts the question of maximum efficiency of learning for everyone, not just for the slowest third.

against enrichment

I'm fast reaching the point where I don't want to hear "challenging" and I don't want to hear "enrichment":

A commonly held belief regarding mastery is that for it to be successful, there must be a highly effective, if not innovative, remediation component. Although we do not underestiet the importance of remediation, it is our assertion that the strength of any mastery program is contingent on its enrichment component. The reasoning behind this assertion is at least twofold. The first issue is that a common criticism of mastery is that it focuses on those students who initially fail to master the objectives at the expense of moe capable students who often easily master those same objectives. One can certainly find examples of this. However, we suggest that poor planning, rather than mastery learning, is at fault. Teachers need to develop engaging enrichment assignments that are pertinent to their objectives and not seen as busy work by students…Contrarily, students in need of remediation will often need to simply revisit and revise previous assignments and not be in need of additional materials. However, what they likely will need is additional instruction from the teacher or classroom aide. These resources will only be available to these students if other students are engaged in appropriate enrichment activities....

...which, presumably, "other students" will be engaged in entirely on their own!

Because they've reached mastery!

So they can zip right through Bloom's Taxonomy without further ado!

Thus a comprehensive enrichment program is not only a critical component of a mastery learning system but also an effective classroom management tool. Furthermore, enrichment activities need not be narrowly defined as something that occurs following mastery. When scheduling permits, it is appropriate to allow students who have yet to master the critical objectives to engage in enrichment activities. This seems appropriate for at least two reasons: (a) The enrichment activities may assist the student in achieving the critical objectives, and (b) the enrichment activities may provide motivation for that student’s learning.

enrichment = classroom management

in conclusion....

....first of all, the thing to remember here is that no one is currently doing this in U.S. public schools.

No one is "teaching to mastery."

That's the rhetorical purpose of the book, to persuade educators to start teaching to mastery.

Why aren't schools teaching to mastery now?

Because everyone and his brother knows that if you teach to mastery in a mixed-ability class — and mixed-ability classes are taken as a given — you're going to be moving very, very slowly:

"The fast learners ranged from 3 to 7 trials (median = 5), whereas the slow learners ranged from 12 to 33 trials (median = 15)...[for a ratio of] 1:3 (fast:slow).

The fastest third of the class masters material 3 times as quickly as the slowest third. (We're not even talking gifted here. Just top third, middle third, bottom third — which is the way a real class "falls out" in the real world,^* leaving aside a tiny number of genuinely gifted children.)

So say you're studying two-digit multiplication, and the fastest third has got it after a median of 5 trials, which we can probably define as lessons-with-practice.

The slowest third still has another 10 trials (lessons-with-practice) to go while the fastest third does — what?

Investigates two-digit addition?

Solves real-world problems using addition?

Teaches two-digit addition to the slowest third?

Gentile & Lalley's answer to all of these questions is yes. The fastest third has got two-digit addition after a median of 5 trials, so now they spend the next 10 trials investigating, solving real-world problems, and tutoring their slower peers.

This is what Gentile & Lalley are proposing.

That is a huge drag on the wheel for the fast kids, one not likely to go unnoticed by parents who are going to come in loaded for bear.

Why do real-world teachers and schools not "teach to mastery," as teaching to mastery is defined by Gentile & Lalley?

Real-world teachers and schools do not "teach to mastery" for exactly the reason they give: teaching to mastery in a mixed-ability classroom, and the mixed-ability classroom is a given, would slow the entire class down to the pace of the slowest third of students.

Almost certainly real-world teachers and schools are splitting the difference by teaching to the middle.

Gentile & Lalley don't tell us how much faster the middle group learns than the slowest group, so let's split the difference ourselves. Let's assume they're literally in the middle; let's assume that the same material it takes the fastest learners a median of 5 trials to learn takes the middle learners a median of 10 trials and the slowest learners a median of 15.

If you have the fast-thirds, who've hit mastery after 5 trials, sit around enriching themselves while you teach another 5 trials to the middle, and then you move on, you might reach something like mastery with 2/3 of the kids, without inciting the fast-thirds (or their parents) to riot.

Meanwhile the spiraling philosophy apparently allows everyone to tell himself that the slow-thirds will see the material again next year so it's OK they didn't master it this time; the constructivist philosophy tells educators nobody should be committing anything to memory anyway; and Bloom's taxonomy tells us (or can be interpreted to tell us) that the important thing is for everyone to comprehend-apply-analyze-synthesize-evaluate....and the entire judicial system of the United States of America has spent the past god-only-knows how many years ruling that the school isn't responsible for getting content inside kids' heads, because "it could be something about the child" — and there you have it.

Nobody is "teaching to mastery," nobody apart from teachers who close their doors and try to do it anyway, in spite of the towering odds stacked against them.

Worse still, if people did start teaching to mastery, this is what it would look like.

"Teaching to mastery" would mean the fastest third learn at exactly the same pace as the slowest third, then spend 2/3 of their class & homework time being enriched.

Not "being" enriched; enriching themselves. The teacher and her aide, if she has an aide (questionable) is going to be busy with the slower-thirds while enrichment is taking place.

The fast-thirds are going to be spending 2/3 of their class & homework time enriching themselves, and the middle-thirds will spend 1/3 of class & homework time doing the same.

No wonder parents (and teachers) want smaller class size.

So this is where we get differentiated instruction, now the essence of an Irvington public school education and part of our strategic plan.

Our teachers are all to teach 3 different groups of kids in each and every class & in each and every subject.

That is their job.

We're not going to have formative assessment or teaching to mastery (except, again, from teachers who buck the odds, or try to).

We're just going to have differentiated instruction. Three different levels of classroom instruction delivered to three different groups of kids.

Plus a whopping big load of enrichment for the fast-thirds and the 1 or 2 truly-gifteds.

Parents have no say — none — in these decisions, which profoundly affect our children's lives.

The district is the decider.

0761946144.01._AA240_SCLZZZZZZZ_.jpg

Standards and Mastery Learning: ERIC abstract

^* Mrs. Panitz told Ed and me about this back when Christopher was in 4th grade. She said every classroom falls naturally into 3 groups; teachers expect this. She said that the Phase 3 classes all had 3 such groups; so did the Phase 4 classes. That was one of the main reasons why, as she thought about it a bit, she realized Christopher should move to Phase 4. He was at the top of the top 3rd of his Phase 3 class, which meant that at a bare minimum he ought to be able to hold his own in the bottom third of the Phase 4 class. We figured he could probably be in the middle of the pack with the extra work we gave him at home, and in fact he probably did quite a bit better than that when he moved to Mrs. Woeckener's class. I think he may have been in the top third of her Phase 4 class in 5th grade. I would guess he was towards the "bottom of the top," but I think he was in the top third.

on not teaching to mastery - Gentile & Lalley
Engelmann on diversity and teaching to mastery
IQ is a range, not a point

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-- CatherineJohnson - 25 Oct 2006

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