Entries from DirectInstruction

You understand a piece of mathematics if you can do all of the following:

• Explain mathematical concepts and facts in terms of simpler concepts and facts.

• Easily make logical connections between different facts and concepts.

• Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.

• Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

Wonderful.

In terms of math and the math wars, I especially admire the first principle: you understand mathematics when you can explain mathematical concepts and facts in terms of simpler concepts and facts.

The NSF-funded curricula seem to have been trying for this idea. But they bungled it.

A person who understands something can explain it in different terms. But those terms don't have to be words - and, in the case of math, probably shouldn't be words, or at least not solely words. Alford's formulation is more sophisticated. Being able to explain something means being able to explain it in simpler concepts and facts.

I'm think I'm going to post these principles over Christopher's desk. They're universal. I've been using them for years, without having tried to sort them out or write them down. Alford has made them explicit for me. When I'm writing a book or an article, I know I'm succeeding when I can do these four things.

Seeing past the clutter — that's the big Kahuna.

Temple calls it "finding the basic principle."





Saxon Math

Saxon Math probably does a superb job of using these principles to teach math.

This year I learned from Saxon Math, for the first time in my life, that when we find areas we are always multiplying "two perpendicular dimensions." (Saxon 8/7 Lesson 82 Area of a Circle)

Of course, I sort-of knew that.....but I'd never made the connection between finding the area of a square and finding the area of a circle. (Have I mentioned my education in mathematics left a lot to be desired?)

That one observation, in Saxon 8/7, permanently changed my perception of area & volume, permanently increased my comprehension of area and volume, and permanently improved my ability either to remember area and volume formulas or to derive them when I don't remember them.

Saxon used seven sentences, illustrated by geometric figures, to make that observation. This passage embodies all four of Alford's principles:

We can find the areas of some polygons by multiplying two perpendicular dimensions.

• We find the area of a rectangle by multiplying the length by the width.
  
  A = lw  [illustration of square]
  
  

• We find the area of a parallelogram by multiplying the base by the height.
  
  A = bh  [illustration of parallogram]
  
  

• We find the area of a triangle by multiplying the base by the height (which gives us the area of a parallelogram) and then dividing by 2.
  
  A = bh/2 or A = 1/2bh  [illustration of triangle]
  
  

To find the area of a circle, we again beegin by multiplying two perpendicular dimensions. We multiply the radius by the radius. This gives us the area of a square built on the radius.  [illustration of circle]

-- CatherineJohnson - 18 Jun 2006

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MasteryLearningAndIq 26 Jun 2006 - 16:59 CatherineJohnson




Through sheer serendipity, I've stumbled across the book with the answers:

Standards and Mastery Learning: Aligning Teaching and Assessment So All Children Can Learn

by J. Ronald Gentile, James P. Lalley.

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....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. 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.

why do we have to learn all this stuff?

Finally, an answer:

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).

fast learners, slower learners, memory, IQ

Suppose, however, that we ask how IQ relates to all of this. We already know, for example, that IQ is moderately but significantly correlated with memory. But suppose we randomly assign half the students to have to achieve a preset standard, while the other half (within the same IQ range) are exposed to the same material but do not have to achieve the preset standard. What happens to the correlation between IQ and surprise delayedretention test scores?

A dissertation study on this very premise was completed recently, under the senior author’s direction, by Marianne Baker (1999).... for original learning, a short story was read aloud to fourth and fifth graders individually, immediately followed by a free-recall test on specific items of information as well as comprehension of ideas in the story. For the mastery group, this process was repeated until each student scored between 75% and 90% correct. The nonmastery group heard the story once and did the free-recall test. A week later, both groups were surprised with a written test of memory for the same items. Then students relearned under their respective conditions and finally were tested for retention again after 14 days and 28 days.

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.

I believe it.

I'll have more later. The preface and first chapter (pdf file)) are available online.

I'm ordering the book.



in a nutshell

• learning occurs in phases or episodes

• all initial learning results in forgetting

• learning to high mastery means immediate recall of 80% to 100%

• learning to "low" mastery means immediate recall of 60 to 80%

• below 60% you haven't learned; when you encounter the material again you'll be starting over again [ed.: hoo boy. we're looking at a big, honking Phase 4 math reteach-fest this summer.] [UPDATE 12-6-06: Actually, we're not. C. is now much faster at learning math; he's managing to absorb and, I think, hang onto some of the content in Phase 4 Grade 7.]

• IQ is moderately correlated with memory

• high-IQ allows one to recall more after one exposure: "higher-ability students tend to remember more, as society has come to expect"

• high mastery to a standard of 75% to 90% on original learning plus one relearning erases the correlation between IQ and memory
  

"What mastery to a high standard can do, in summary, is virtually bypass the effects of IQ for specified educational objectives."

MORE COMING ANON



-- CatherineJohnson - 24 Jun 2006

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JohnDeweyAtEdspresso 26 Jun 2006 - 20:01 CatherineJohnson




Kicking the Ed School Blues

The fan mail is rolling in and paparazzi are following me to work every day despite the great lengths to which I’ve gone to protect my identity.

[snip]

Of all the comments, two in particular stand out. One from a friend who asked if I thought I was making a difference with this little venture into blog space. The other asked whether I thought I’d be making a difference teaching in a system that prevents effective math teaching in a world infiltrated by NSF, NCTM/ed school dogma and math police.

I don’t know the answer to the first question. But I’m in ed school, where there are no wrong answers.

Good one.

Moving right along —

...what is the chance for change with only a few enlightened teachers battling the math police?

My answer to the second question is based on the fact that I’ve never had an original idea in my life. Being part of the baby boomer generation means that whatever so-called original idea is in my head is also in the heads of thousands of other people. Which means that many people getting ready to retire and who have science or math backgrounds may also be looking into teaching.

Whoa.

That's my experience exactly.

I'm a walking cliche. Always have been, always will be.

I keep telling Ed, who is not a walking cliche, "Look, if I'm obsessing over X that means five zillion other people are obsessing over X, too. Either that, or they're about to start."

It's true.



Unfortunately, when it comes to television I'm an outlier.


it's always worse than you think

In the class I just took, the professor one night espoused the ubiquitous ed school philosophy that one of the biggest hurdles to conquer in teaching math is students’ math anxiety.

[snip]

The ed school of thought holds that if you just relax and get over the anxiety, the greater truth will prevail. Not a word about how inadequate preparation may play a role. “At-risk” students are particularly vulnerable to math anxiety according to ed school wisdom. One instructor the professor knew was quite good with such students. He told how she gave each student a name having to do with a concept in algebra. One student was called “perfect square trinomial”, another was “binomial”, and so forth. (They may have had name tags). Their task was to learn how each of them “related” to one another, thus forcing them to learn what these terms meant.

Time to blow up the ed schools.


edspresso search: Dewey letters

John Dewey at edspresso, part 1
John Dewey at edspresso, part 2
John Dewey at edspresso, part 3
John Dewey has the stomach flu
John Dewey at edspresso Letter #5

John Dewey at ktm
John Dewey at ktm part 2
John Dewey experiences stomach flu
John Dewey writes again

johndewey

-- CatherineJohnson - 26 Jun 2006

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ShortStoryByVernWilliams 15 Jul 2006 - 19:19 CatherineJohnson




One night I walked into the 43/8 dimension and actually believed the following:

• We should write about math but never do math.
• Correcting students' papers using red ink is a threat to children's self esteem and that red pens should be banned from all public schools.
• Howard Gardner was right about his multiple intelligence theory (I think that he claims about nine at the moment) and that schools should value bodily-kinesthetic ability and the intelligence of self as much as mathematical and linguistic ability.
• The war on intellectual excellence is a great thing. It will make us all equal.
• Teachers Unions are actually concerned about students.
• Advanced courses and gifted programs should be banned because they are elitist and unfair. Since everyone is gifted in their own way (see Howard Gardner), why have special gifted programs?
• There are no bored students in US public schools.
• We can teach thinking even when there is no content to think about.
• We should treat members of politically protected minority groups as victims.
• We should never view our students as individuals but as members of racial and ethnic groups.
• We should buy into the latest educational fad even if it's based on political correctness and has nothing to do with learning or common sense.
• There is no money wasted on administration, specialists, and useless programs. In fact, we should have more of each.
• I should join the NCTM.
• I should join the NEA.
• I should feel guilty because I teach smart kids.
• I should feel really guilty because I enjoy teaching smart kids.

cont'd





National Mathematics Advisory Panel
(nationalmathematicsadvisorypanel)

-- CatherineJohnson - 15 Jul 2006

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ThreeStrikesRuleAgainstPureDiscovery 04 Aug 2006 - 19:26 CatherineJohnson




I'm hoping Ed can pull the full text of this article on guided discovery versus pure discovery:

Should There Be a Three-Strikes Rule Against Pure Discovery Learning?

The author's thesis is that there is sufficient research evidence to make any reasonable person skeptical about the benefits of discovery learning--practiced under the guise of cognitive constructivism or social constructivism--as a preferred instructional method. The author reviews research on discovery of problem-solving rules culminating in the 1960s, discovery of conservation strategies culminating in the 1970s, and discovery of LOGO programming strategies culminating in the 1980s. In each case, guided discovery was more effective than pure discovery in helping students learn and transfer. Overall, the constructivist view of learning may be best supported by methods of instruction that involve cognitive activity rather than behavioral activity, instructional guidance rather than pure discovery, and curricular focus rather than unstructured exploration. (PsycINFO Database Record (c) 2006 APA, all rights reserved)

Mayer, Richard E.

Mayer, Richard E.: University of California, Santa Barbara, Department of Psychology, Santa Barbara, CA, US

American Psychologist. 59(1), Jan 2004, 14-19.

Barry is well-versed - or on his way to becoming well-versed - in the distinction. I want to learn more.


-- CatherineJohnson - 03 Aug 2006

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LovelessOnEducationPhilanthropy 04 Aug 2006 - 20:27 CatherineJohnson

The title of my paper is how program officers at education philanthropies view education. It is inspired by a 1997 study of education professors done by the Public Agenda Foundation in New York City, and in that particular survey, what Public Agenda found was, and I quote from the study: "Professors of education have a distinct, perhaps even singular prescription for what good teachers should do, one that differs markedly from that of most parents and taxpayers."

[snip]

The study concluded with the following. Quote. "While the public's priorities are discipline, basic skills and good behavior in the classroom, teachers of teachers severely downplay such goals."

So what I decided to do was give the same survey to program officers at education foundations, and in a nutshell, what I found was that they too are far outside the mainstream, on some issues [program officers are] even farther outside the mainstream than education professors.

[snip]

You can see in these bottom... six different classroom activities. These are sort of mainstays of progressive education. These are thing that come under criticism by progressives over the last 100 years.

Take a look. Should kids be given--this is the percentage that responded that more of this would be a good thing. Should kids be given more homework assignments? Only 21 percent of the program officers felt that they should be given more homework.

41 percent. The education professors are tougher on homework. Penalties for students who break the rules. Only 19 percent of the program officers. 37 percent, again about twice as many of education professors. And in a minute I'm going to show you what the general thinks about these things.

The title of the public agenda report was Different Drummers. The program officers at the philanthropies appeared to be even more different than the different drummers, at least on issues of discipline and basic skills. Those are the two main differences.

Memorization, endorsed by only 11 percent of the program officers. Prizes to reward good behavior in the classroom. This is Alfie Cohn's [ph] big problem, he has a big problem with that. Only 11 percent. And then multiple choice exams, not popular at all.

source:
With the Best of Intentions: Lessons Learned in K-12 Education Philanthropy

character ed

Look at the question on schools fail to teach religious values. If you see that as a serious problem or not. Only 6 percent of program officers think that's a problem. Among traditional Christian parents, not surprisingly, 70 percent see as a problem.

But even in the general public, almost half, 47 percent, think it's a serious problem.

I'm surprised to discover that, if I had to choose, I would come down on the "serious problem" side of this issue.

I don't want public schools to teach religious faith, though I support vouchers for religious schools and I wish to heck our schools would teach the Bible as subject matter content.

Biblical literacy: a good thing.

On the other hand, I don't want public schools teaching values - namely narcissism and yay-me blather - that directly contradict my own religious values. At Main Street School (grades 4-5) the kids apparently recited some kind of self-respect affirmation each and every morning, after the Pledge of Allegiance. Christopher can't remember the words now, and neither can I, but he thinks he had to say something like, "I am an amazing person." Every single morning. I am an amazing person.

Then, at the 5th grade graduation, the superintendent read an "Alphabet of Values" - "A is for Achievement" - that kind of thing. Most of it was nice, but the entry for L was awful:

L: Love yourself first and always.

It may have been even worse; it may have been "Love yourself first and best."

blech

I was sitting there thinking, "So what happened to Love thy neighbor as thyself?" (Which leads me to think it probably was "Love yourself first and best.")

As far as I'm concerned, there are many, many occasions in life when loving yourself first is a very bad idea. Anyone who a) gets married and b) has kids is going to experience these occasions.

Probably anyone going into the teaching profession is going to experience them.

I don't want my kids being taught to love themselves first. I also don't want them spending a lot of time thinking, "I am an amazing person." As a matter of fact, I would go so far as to say that spending any at all time thinking the words I-am-an-amazing-person is a terrible idea, and I could probably support my view empirically if I had all day to scan the archives of the PERSONALITY AND SOCIAL PSYCHOLOGY BULLETIN. (I have no idea whether this particular study does or does not support the anti-I am amazing person viewpoint. I think it might.)

So, yes.

At this point I'm thinking a failure to teach "religious values" is a problem.



some good news

Loveless:

The program officers with teaching experience are closer to the mainstream public views.

This jibes with my feeling that teachers are more likely than their superiors to think radical constructivism is a crock. Hirsch has a nice observation to the effect that while no one has been able to defeat ed school ideology, no one's been able to defeat reality, either. Students learn the way they learn.

Teachers are in the trenches.



question from the audience

QUESTION: Hi. My question sort of is which side are you on, or actually, which side are the philanthropists on in regard to the education wars that are being fought across this country, often under the name of the reading wars or the math wars. You know, to what extent are they facilitating the reform agenda and to what extent are they facilitating maybe the opposite.

MR. LOVELESS: I don't mind taking a stab at that. For the most part, they're on the neoprogressive side, which in the--I edited a book, a couple years back, called The Great Curriculum Debate, and it's about the wars in both math and reading that have occurred over the last 15 years, whole language versus phonics, and in math, math reform or NCT and math reform versus more basic emphasis on arithmetic and other traditional mathematics.

And for the most part, the philanthropies have supported, financially, the neoprogressive side.

So the question is, given the fact that lots more money poured into the schools over the past decades hasn't improved them, can lots more money poured into the schools make schools worse?

My feeling is yes. It's possible to have too much money.

This reminds me of the various studies showing that rich people undergo all kinds of unnecessary surgery. My line on that used to be, "You go into Cedars-Sinai for a c-section, you come out with a nose job."

In context - i.e. living baja Beverly Hills - it was funny.



-- CatherineJohnson - 03 Aug 2006

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NationalMathematicsAdvisoryPanelUpdate 02 Sep 2006 - 15:31 CatherineJohnson





Robert Siegler


Ed can pull the article, so I'll take a look. Given my immersion in the world of non-math teaching, this line from the abstract to Siegler's article on children's learning is chalk on the blackboard for me (chalk on the electronic whiteboard, I meant to say):

Learning has many sources; one that is particularly promising for educational purposes is self-explanations.

I've come to feel that in math ed - though not in other subjects - this proposition is almost completely false.


From Seigler's homepage:

My colleagues and I have built computational models to illustrate how young children can make such intelligent decisions and also to show how the decisions improve as knowledge and skill improve.

He's not anti-content.




NCLB: one law, two philosophies

Meanwhile, Education Week has a terrific article on NCLB (registration required): What Works vs. Whatever Works: Inside the No Child Left Behind law’s internal contradictions by Mike Petrilli.

[T]he No Child Left Behind Act is the result of an uncomfortable truce between two groups of school reformers: the “what works” camp and the “whatever works” camp. The law is an amalgam of their ideas, and their ongoing competition will shape the contours of No Child Left Behind version 2.0.

First, let’s examine the what-works crowd. These reformers look across the education system and see its failings in terms of ignorance and ideology. They decry the pedagogical fads that sweep through our schools, bemoan educators’ resistance to scientifically proven reading-instruction methods, and abhor the quasi-religious nature of disliked educational “philosophies” such as constructivism and “multiple intelligences.” They seek to bring order to this chaos through the dispassionate eye of science. Using medicine as their model, they aim to employ rigorous research methods to determine what works, and then to use the force of law and regulation to ensure adoption of these methods throughout the land. After all, they say, we don’t allow doctors to wing it when they are practicing brain surgery; we expect them to use best practices in order to save lives.

The stamp of what-works advocates is clear throughout the No Child Left Behind legislation, but especially in two of its most controversial provisions: the Reading First program, and the “highly qualified teachers” mandate. The former requires schools to use funds for a narrowly defined type of reading instruction—namely, that which has been found by rigorous research to be effective. The U.S. Department of Education has dutifully implemented the program in this narrow, prescriptive way, leading to much gnashing of teeth and complaints of bullying. But for leaders of the what-works coalition, such as the former federal reading czar, G. Reid Lyon, anything else would amount to malpractice.

Then there’s the “highly qualified teachers” provision of the law, which demands that all teachers be able to demonstrate their subject-matter knowledge. This, too, is said to be based upon rigorous research, though even admirers of the mandate must admit that the evidence is somewhat flimsy. (Most studies linking subject-matter knowledge to teacher effectiveness have examined math or science at the secondary level; their applicability to elementary school, much less to subjects such as art, geography, or economics, is unknown.) But again, the what-works coalition borrows from the language of medicine, insisting that we wouldn’t allow doctors to practice if they didn’t have the relevant credentials. Surely we need teachers who are similarly well-qualified.

The whatever-works camp holds a very different worldview. These reformers look out across the education system and see its failings in terms of incentives, power, and politics. They decry the daily decisions made by school boards and district leaders that benefit adults instead of children (especially poor children). They abhor the red tape and bureaucratic inertia that keep educators from innovating. They don’t particularly care what happens inside the “black box” of classroom instruction; they just want children to be well-educated at the end of the day. They seek to right the system through the classic management model of “tight-loose”: Be tight about the results you expect, but loose as to the means. Put differently, the whatever-works camp combines accountability for student learning with flexibility around everything else. Using the entrepreneurial sector as its model, this camp aims to create a marketplace of schools, free to experiment, compete, and improve. After all, there’s a reason that America has the strongest economy in the world, they assert, and if we can empower educators with significant freedom (in return for getting results), they, too, will rise to the occasion.

The stamp of whatever-works advocates is also clear in the NCLB legislation. The very heart of the law is its accountability system, built into the Title I program, which was designed to create incentives for schools to boost student learning and close achievement gaps. Most important, the design of “adequate yearly progress,” with its disaggregation of test scores by racial groups, was meant to help local communities overcome the political barriers that keep resources and attention from flowing to needy children. In the spirit of “whatever works,” and in return for this increased accountability, the rules around the use of Title I funds were relaxed dramatically, and many more schools were given permission to use their federal dollars for “schoolwide” reforms. New “transferability” provisions were included in the law, too, allowing states and districts to move federal funds from one program to another. Just show us the results, Congress seemed to say, and we’ll leave you alone.

Is it any surprise, then, that educators feel whipsawed between competing demands? On the one hand, the federal government is saying to do whatever works to boost student learning, and on the other hand it’s saying to do things in a certain prescribed, preapproved way.

The result is frustration and anger. Imagine a poor, rural Title I school that is doing whatever works to get great results. In this case, it hires a former engineer from the local coal mine to teach 8th grade mathematics. She’s a natural, and her students’ test scores go through the roof. But because she didn’t major in math, she’s not considered “highly qualified.” How is that school’s principal going to feel when the feds come knocking, telling him to replace the teacher or risk being “out of compliance”? Or consider an elementary school whose reading results are soaring but which dares to use a reading program not on the state-approved Reading First list. Should the school’s principal be punished for insubordination, or celebrated for innovation and inventiveness?

What does all of this mean for the next version of the No Child Left Behind Act, due to be reauthorized in 2007? Surely both camps will try to consolidate their gains, further push their agenda, and avoid surrendering ground. The what-works camp will seek to expand its influence beyond reading to other areas, such as math. (This is especially true if the newly named National Mathematics Advisory Panel can identify rigorous evidence to support certain approaches to teaching math.)

[snip]

The whatever-works camp will try to expand on the transferability provision, perhaps allowing all of a state’s federal funds to flow through a simplified Title I formula. It will try to make it easier for schools to earn waivers from federal laws and regulations. And, perhaps most importantly, expect this camp to push for a larger role for charter schools, whose “accountability in return for flexibility” design epitomizes the whatever-works approach.

in a nutshell

what works

• Reid Lyon, reading czar

• Reading First

• qualified teachers mandate

• anti-constructivist, anti-multiple intelligence, anti neoprogressivism

• analogy to medical practice
  

whatever works

• analyze failure of schools in terms of "incentives, power, and politics"

• “tight-loose” management model

• marketplace of schools

• pro-charter

Coming off of Animals in Translation, I was a whatever works person.

I favored tight-loose because of McDonald's success reforming the meatpacking industry using Temple Grandin's 10-item animal welfare audit. Temple created simple metrics such as "No more than 3 in 100 animals can boo in distress." Just 10 items, all focused on the animals. Nothing about employee training, nothing about non-slip plant flooring, nothing about plant maintenance schedules. That was the "loose" part. A typical government regulator-type will audit at least 100 different aspects of a plant, and Temple has seen the results in countries where that's the case. The results aren't good.

When the audit went into effect, plants thought it was far too strict to pass. McDonald's told plants that if they didn't pass, they'd be off their supplier list. That was the "tight" part. Not only did plants figure out how to pass the audit, a lot of them ended up exceeding the standards. Most plants passed the audit using the same management, the same workers, even the same out-of-date equipment.

So I'm a believer in "tight-loose."

But I no longer believe that a tight-loose approach will reform the schools. Charters & vouchers might, over the long haul, put so much pressure on existing schools that they're forced to change. Might. But I'm not hopeful. As far as I can tell people don't give up core values to increase market share, and neoprogressive ideology has been a core value of U.S. ed schools for nearly 100 years.

They're not going down without a fight.


nationalmathematicsadvisorypanel



-- CatherineJohnson - 04 Aug 2006

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TwoSides 12 Aug 2006 - 14:53 CatherineJohnson




Shortly after posting a link to Michael J. Petrilli's What Works vs. Whatever Works: Inside the No Child Left Behind law’s internal contradictions (registration required) it struck me: there are only two sides, and neither of them is neoprogressive.

-- CatherineJohnson - 04 Aug 2006

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NixOnColumbiaTeachersCollege 07 Aug 2006 - 20:41 CatherineJohnson




new old schoolteacher is guestblogging for eduwonk. Until June she was, I believe, a graduate student at a school of education in NYC which I take to be Columbia Teachers College, although her blog doesn't currently identify the institution.

Columbia Teachers College, the employer of William Heard Kilpatrick, is Ground Zero for curricular & pedagogical awfulness. E.D. Hirsch argues that Kilpatrick, not John Dewey, is the real founder of progressive education.

Here is new old school teacher:

I am currently attending the KIPP Summit 2006 in New Orleans, attended by all the KIPP schools as well as other excellent charter schools and charter school networks (for example, Uncommon Schools and Achievement First). It is awesome. Compare some of the AERA presentations I talked about yesterday with these, presented over the last few days here in New Orleans:

--"Basics on Advising College-Bound Students"
--"Analyzing Test Scores"
--"Activities and Questioning with Bloom's Taxonomy"
--"Informal Assessment of Reading Difficulties"
--"Overview of Expository Writing, Parts I and II"
--"A Typical Day in Math 8"
--"Developing Number Sense"

These sound a little more practicable and useful than a session on Taiwanese mail-order brides, don't you think? Yesterday I learned a ton of great strategies for creating a safe, calm, effective learning environment and how to deal with students with difficult behavior. Today in 1.5 hours I learned exactly how to teach my kids to write summaries from fiction or non-fiction and how to highlight text effectively (college, hello!). THIS IS WHAT I ALWAYS WANTED from grad school and NEVER GOT. And was so sad about that I had to write an angry blog full of tirades. It's $30,000 worth of tragic.

That's $30,000 to not learn how to teach.


My favorite W.H. Kilpatrick saying:

activity leading to further activity without badness

That's the project method.





The Columbia School of Pragmatism
William Heard Kilpatrick UNESCO

-- CatherineJohnson - 07 Aug 2006

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EngelmannOnHowManyWordsADay 08 Aug 2006 - 19:03 CatherineJohnson




Ken left this passage from Engelmann in the Comments thread to 15 words a day:


Engelmann thinks that Hirsch's estimates are way too high. He think's it's more like 30,000 words total and 3 new words a day.

The numbers: Hirsch selects 60,000 as the number of word meanings for the top-of-class student. I did a very unscientific experiment that may be way out in left field, but I came up with a smaller number. I didn't have a top-of-class high-school student handy, but I had a top-of-class graduate student. I opened a college dictionary that had about 70,000 entries to four random pages. I read the words, spelled them, told her the part of speech for those she questioned, and asked her if she knew what they meant. On three of the pages, she did not know all the words. On one page, she did not know bourn, bourrée, bouse, boustrophedon, bouzouki, bowerbird, bow pen, bowsprit. She also didn't know a second meaning of bower (a bow anchor). She probably didn't know bovid, but I gave her half credit. "Could that be something related to a bovine?" "It is an adjective for bovine."

Also, I did not present most capitalized entries because I didn't think they were fair (Bournemouth, Bow bells, Bowditch, Bowen, Bowie State.) I did present Bowie and Bowling Green. I did not present six entries because they were either dialect, slight variations of the same word (two bowman entries for instance), obsolete, or spelling variations (bowlder for boulder). The page had 58 entries. Eleven were discards. Of the 47 remaining, she missed 9.5 (half credit for bovid). So her score on that page was 37.5/47 or 80%. Her performance on the other pages was 100%, 65%, 39%. The low-scoring page had lots of sodium words, which she could identify only as a substance composed of sodium. (She got sodium chloride, sodium fluoride, sodium glutamate, and Sodium Pentothal, but she was not able to identify the others.) Also, I threw out a lot of items on this page-variant spellings, obsolete words, capitalized words I didn't know and that seemed trivial, affixes, and obscure slang words. She also missed sociometry, socal, socman, sokeman, sodalite.

Indeed my decisions were less than operationally delineated, but if we assume that 15% of the entries are not fair and that the top-of-class person would get average 80% on the others, the total number would be something on the order of 48,000, which is quite a bit less than 60,000. Personally, I don't believe it's that high. Also of interest is that a very extensive analysis of morphology for spelling, conducted in the '70s, came up with a number of 30,000 words that seemed to be fairly exhaustive.

At least some cognitive scientists favor this range over the one that Hirsch suggests. Biemiller and Slonim (2001) concluded that the learning rate of new words for the top-of-class student is more on the order of about 3 words per day, not 8-18. So there seems to be far from perfect consensus on number of words. Also, Biemiller endorses explicit, direct instruction. So there isn't perfect consensus on methods for inducing vocabulary.

Three new words a day sounds much more likely to me, based on nothing more than instant reaction.

I just noticed that Ken is back on the job!

Can't wait to read.

Fischgrund on divorce and SAT scores
how much reading a day?
robust vocabulary instruction (400 words a year)
Vocabulary Workshop levels & grades
15 new words a day
Engelmann says it's 3 new words a day

-- CatherineJohnson - 08 Aug 2006

---

NctmReformsAgain 14 Sep 2006 - 16:52 CatherineJohnson




In today's Wall Street Journal ($):

Arithmetic Problem
New Report Urges Return to Basics In Teaching Math
Critics of 'Fuzzy' Methods Cheer Educators' Findings;
Drills Without Calculators Taking Cues From Singapore
By JOHN HECHINGER
September 12, 2006; Page A1

The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.

In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.

The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.

Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country.

The new approach puzzled many parents. For example, to solve a basic division problem, 120 divided by 40, students might cross off groups of circles to "discover" that the answer was three.

Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.

Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor.

Francis Fennell, the council's president, says the latest guidelines move closer to the curriculum of Asian countries such as Singapore, whose students tend to perform better on international tests.

So maybe it wasn't such a great idea after all for IUFSD to ban my Singapore Math course.



new timeline

According to their report, "Curriculum Focal Points," which is subtitled "A Quest for Coherence," students, by second grade, should "develop quick recall of basic addition facts and related subtraction facts." By fourth grade, the report says, students should be fluent with "multiplication and division facts" and should start working with decimals and fractions. By fifth, they should know the "standard algorithm" for division -- in other words, long division -- and should start adding and subtracting decimals and fractions. By sixth grade, students should be moving on to multiplication and division of fractions and decimals. By seventh and eighth grades, they should use algebra to solve linear equations.

Here's the Singapore sequence.




Lutherans turning into Catholics

A recent study by the Thomas B. Fordham Foundation, a Washington nonprofit group, found that only two dozen states specified that students needed to know the multiplication tables. Many allowed calculators in early grades.

Chester E. Finn Jr., the foundation's president and a former top official at the U.S. Department of Education, blamed the earlier math-council guidelines for state standards that neglect the basics. He described the new advice as a "sea change," saying that "it's a little bit like Lutherans deciding to become Catholics after the Reformation."

Understanding math, rather than parroting answers to poorly understood equations, was the goal of the council's controversial 1989 standards. Those guidelines called on teachers to promote estimation, rather than precise answers. For example, an elementary-school student tackling the problem 4,783 divided by 13 should instead divide 4,800 by 12 to arrive at "about 400," the 1989 report said. The council said this approach would enable children using calculators to "decide whether the correct keys were pressed and whether the calculator result is reasonable."

"The calculator renders obsolete much of the complex pencil-and-paper proficiency traditionally emphasized in mathematics courses," the council said then. In 2000, in another report, the council backed away somewhat from that position.

Still, in response to the earlier recommendations, many school systems required children to describe in writing the reasoning behind their answers. Some parents complained that students ended up writing about math, rather than doing it.

As the debate heated up, concern grew about U.S. students' math competence. In 2003, Trends in International Mathematics and Science Study, a test that compares student achievement in many countries, ranked U.S. students just 15th in eighth-grade math skills, behind both Australia and the Slovak Republic. Singapore ranked No. 1, followed by South Korea and Hong Kong. Fueling concern about the quality of elementary and high-school instruction: one in five U.S. college freshmen now need a remedial math course, according to the National Science Board.

low-income students

This is very exciting. The AIR report (pdf file) led me to believe that Singapore Math had been a flop in low-income schools because the student mobility is so high (and see Hirsch on this subject, too):

If school systems adopt the math council's new approach, their classes might resemble those at Garfield Elementary School in Revere, Mass., just north of Boston. Three-quarters of Garfield's students receive free and reduced lunches, and many are the children of recent immigrants from such countries as Brazil, Cambodia and El Salvador.

Three years ago, Garfield started using Singapore Math, a curriculum modeled on that country's official program and now used in about 300 school systems in the U.S. Many school systems and parents regard Singapore Math as an antidote for "reform math" programs that arose from the math council's earlier recommendations.

According to preliminary results, the percentage of Garfield students failing the math portion of the fourth-grade state achievement test last year fell to 7% from 23% in 2005. Those rated advanced or proficient rose to 43% from 40%.

Last week, a fourth-grade class at Garfield opened its lesson with Singapore's "mental math," a 10-minute warm-up requiring students to recall facts and solve computation questions without pencil and paper. "In your heads, take the denominator of the fraction three-quarters, take the next odd number that follows that number. Add to that number, the number of ounces in a cup. What is nine less than that number?" asked teacher Janis Halloran. A sea of hands shot up. (The answer: four.)

Ms. Halloran then moved on to simple pencil-and-paper algebra problems. "The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?" (The answers: 21 and 42.)

In this class, the students didn't use the lettered variables that are so prevalent in standard algebraic equations. Instead, they arrived at answers using Cuisenaire rods, sticks of varying colors and lengths that they manipulate into patterns on the tops of their desks. The children use the rods to learn about the relationship between multiplication and geometry. The goal: a visceral and deep understanding of math concepts.

"It just makes everything easier for you," says fifth-grader Jailene Paz, 10 years old.

Cuisinaire rods for bar models!

That's so cool!




TERC time

The Singapore Math curriculum differs sharply from reform math programs, which often ask students to "discover" on their own the way to perform multiplication and division and other operations, and have come to be known as "constructivist" math.

One reform math program, "Investigations in Number, Data and Space," is used in 800 school systems and has become a lightning rod for critics. TERC, a Cambridge, Mass., nonprofit organization, developed that program, and Pearson Scott Foresman, a unit of Pearson PLC, London, distributes it to schools.

parents don't get it part 1

Ken Mayer, a spokesman for TERC, says many parents have a "misconception" that Investigations doesn't value computation. He says many school systems, such as Boston's, have seen gains in test scores using the program. "Fluency with number facts is critical," he says.

parents don't get it part 2

Polle Zellweger and her husband, Jock Mackinlay, both computer scientists, moved to Bellevue, Wash., from Palo Alto, Calif., two years ago so their two children could attend its highly regarded public schools. She and her husband grew suspicious of the school's Investigations program. This summer, they had both children take a California grade-level achievement test, and both answered only about 70% of the questions correctly. Ms. Zellweger and her husband started tutoring their children an hour a day to catch up.

"It was a really weird feeling," says their daughter, Molly Mackinlay, 15. "I do really well in school. I am getting A-pluses in math classes. Then, I take a math test from a different state, and I'm not able to finish half the questions."

Eric McDowell, who oversees Bellevue's math curriculum, says parents misunderstand Investigations.

If it weren't for the parents, teaching would be a great job.




math wars and war wars

In the Alpine School District in Utah, parent Oak Norton, an accountant, has gathered petitions from 1,000 families to protest the use of Investigations. His complaints began more than two years ago, when he discovered at a parent conference that his oldest child, then in third grade, wasn't being taught the multiplication tables.

Barry Graff, a top Alpine school administrator, says the system has added more traditional computation exercises. Over the next year, Alpine plans to give each school a choice between Investigations or a more conventional approach. Mr. Graff, who says Alpine test scores tend to be at or above state averages, expects critics to keep up the attacks and welcomes the national math council's efforts to provide grade-by-grade guidance on what children should learn.

"Other than the war in Iraq, I don't think there's anything more controversial to bring up than math," he says. "The debate will drive us eventually to be in the right place."

wow

I bet things are hopping over at math-teach & math-learn.

[pause]

hmm

No action thus far.

Once Wayne Bishop posts this baby, we'll be in a shooting war.

update: Bishop's got it!

let the fun begin



what Singapore students can do at the end of 7th grade

-- CatherineJohnson - 12 Sep 2006

---

BattleLines 26 Sep 2006 - 19:57 CatherineJohnson




Midway through today's TIMES report on the state's fourth grade slump:

E. D. Hirsch Jr., the author of a recent book, “The Knowledge Deficit,” said students do not learn enough vocabulary and content knowledge at younger ages.

Daniel P. Keating, director of the Center for Human Growth and Development at the University of Michigan, said schools should prepare students earlier for the more abstract and sophisticated reasoning required in middle school.

“Perhaps the early preparation is not anticipating that shift to having those higher demands,” he said, adding that tests for younger children do not measure those skills. “All of a sudden we’re looking for the kinds of skills that just haven’t been assessed earlier.”

Note: one of these men is a developmental psychologist "whose research focuses on integrating knowledge about biodevelopmental processes, population patterns in developmental health, and social factors affecting individual and population development."

The other is a college professor who has spent a lifetime researching education, creating a superb core curriculum for K-6 students, and researching and writing new content in his own field of research.

The psychologist doesn't know anything about education or curriculum and is giving his best guess.

The curriculum specialist is stating the consensus view of cognitive scientists who've been researching this subject and publishing their results in refereed journals for many years.

Guess which man is winning the argument.


Irvington slump
NY scores slump
battle lines



-- CatherineJohnson - 22 Sep 2006

---

HowToCureMathAnxiety 23 Oct 2006 - 00:00 CatherineJohnson





So I mentioned that the Yonkers school system graduated Christian from high school with a 3rd grade level of math knowledge — and that last week I'd started him on the 4th grade Saxon Math book, Saxon 5/4.

It's working.

He breezed through the first lesson.

Then he breezed through the second lesson.

Then he breezed through the third.

Now he comes in, deals with the kids, and, after dinner, pulls out his Saxon Math books and does the lesson in 15 or 20 minutes while keeping an eye on Andrew & watching whatever Christopher has on TV.

This is the way you cure math anxiety.

I think.

You cure math anxiety the same way you remediate gaps, by starting a student back before the point where math got hard. That's the Siegfried Engelmann way; that's the KUMON way.

I don't know that Christian had a full-blown case of "math anxiety."

I don't even know what "math anxiety" is, other than the predictable result of years of lousy math instruction.

Christian was against math when he first started working with the kids.

Now he's neither for math nor against it, as far as I can tell.

He just comes in and does his math.

Long live Saxon.



Speaking of living long, I'm a bit concerned.

The Harcourt website seems to have pulled its online placement tests.

yikes

We need those.

I have copies myself, and I've found two other websites that have them posted (links below). Harcourt has an online test, which is good, but it doesn't give you the flexibility you need and reading online is difficult.

I hope they post those tests again.

Engelmann on student placement in curriculum

Rule 1: Hold the same standard for high performers and low performers. This rule is based on the fact that students of all performance levels exhibit the same learning patterns if they have the same foundation in information and skills. The false belief that characterizes the conventional wisdom about teaching is that lower performers learn in generically different ways from higher performers and should be held to a lower or looser standard. Evidence of this belief is that teachers frequently have different “expectations” for higher and lower performers. They expect higher performers to learn the material; they excuse lower performers from achieving the same standard of performance. Many teachers believe that lower performers are something like crippled children. They can walk the same route that the higher performers walk, but they need more help in walking.

These teachers often drag students through the lesson and provide a lot of additional prompting. They have to drag students because the students are making a very high percentage of first-time errors. In fact, the students make so many mistakes that it is very clear that they are not placed appropriately in the sequence and could not achieve mastery on the material in a reasonable amount of time. The teachers may correct the mistakes, and may even repeat some parts that had errors; however, at the end of the exercise, the students are clearly not near 100% firm on anything. Furthermore, the teacher most probably does not provide delayed tests to assess the extent to which these students have retained what had been presented earlier.

The information these teachers receive about low performers is that they do not retain information, that they need lots and lots of practice, and that they don’t seem to have strategies for learning new material. Ironically, however, all these outcomes are predictable for students who receive the kind of instruction these students have received. High performers receiving instruction of the same relative difficulty or unfamiliarity would perform the same way. Let’s say the lower performers typically have a first-time-correct percentage of 40%. If higher performers were placed in material that resulted in a 40% first-time-correct performance, their behavior would be like that of lower performers. They would fail to retain the material, rely on the teacher for help, not exhibit selfconfidence, and continue to make the same sorts of mistakes again.

If students are placed according to their first-time-correct percentages, they tend to learn and behave the same way, whether they are “lower performers” or “higher performers.” In Project Follow Through, we mapped the progress of students of different IQ ranges. The results showed that regardless of students’ entering IQ, the rate of progress was quite similar across all children and across different subjects. Lower performers learned as fast as higher performers. They simply started at a different place, with material that higher performers had long since mastered. Note that this conclusion may be somewhat biased because we paid particular attention to the instruction for the lower performers. They tended to have better teachers and their instruction tended to be monitored very closely. In any case, they learned at a very healthy rate, one that paralleled that of students with IQs 40 points higher.

The typical practices of placing and teaching students are completely opposed to appropriate placement and teaching procedures. At the University of Oregon, we place teaching-practice students in special-ed classrooms that use direct-instruction programs. During the years that we first offered these practica, we typically worked with teachers who were teaching DI but had not generally received much training. Before we arranged for a placement with a new supervising teacher, therefore, we made sure that the classroom was “appropriate” for our students, which means that the children the practicum students were to work with were placed appropriately and that the teacher was using and modeling appropriate practices. As part of the review of the new classrooms that were candidates for receiving practicum students, we checked the program placement of the students and changed their placement if necessary.

Our estimate is that in the first 40 or more classrooms we used, the children were moved back in DI reading programs an average of 100 lessons—sometimes 120 lessons. The children, in other words, were placed about 3/4 of a school year or more beyond the optimum first-time-correct percentages. Nearly all teachers had children that were seriously misplaced. Furthermore, I don’t recall a single classroom in which children’s percentages required us to move children ahead in the programs. Children were always “over their heads.”

source:
Student Program Alignment and Teaching to Mastery (pdf file)
by Siegfried Engelmann

Children were always "over their heads."

When you design instruction so that children are always in over their heads, you have poor instruction and bad student outcomes.

Period.




in a nutshell

• students of all performance levels exhibit the same learning patterns if they have the same foundation in information and skills

• when placed correctly in the curriculum - at a level determined by what they already know and can do - "low performing" or "learning disabled" students learn "at a very healthy rate, one that paralleled that of students with IQs 40 points higher"

• teachers must provide delayed tests to assess the extent to which these students have retained what has been presented earlier

• children who are struggling in school are almost always placed far ahead of themselves in the curriculum. They are struggling the same way a straight-A student whose never studied physics would struggle if you dropped him into the middle of a college physics course.
  

remediating math at the college level

We've had a number of discussions about how to remediate the math knowledge of college kids. Rudbeckia Hirta has written about this often, along with Carolyn &, I think, Steve H.

When Ed and I first discussed what to do about Christian's math, I said I didn't particularly feel like paying for Westchester Community College to teach Christian the math his mom already paid Yonkers to teach him back in grades 6-12.

I said I was thinking about just giving him the Saxon placement test & buying him the books & seeing how it went.

Ed thought that was a bad idea.

(It's fun to be right! It's even more fun to be right on your own blooki.)

Actually, "I told you so" isn't (really) the point. At the time I thought Ed was probably right. I thought handing Christian a couple of Saxon books and telling him to go teach himself math — or trying to teach him myself — was probably nuts. He doesn't (didn't) like math; he couldn't do math; he had no interest whatsoever in the whole question of math....none of this sounds like a recipe for success in self-teaching.

The fact that I went ahead and gave him the Saxon placement test anyway was probably a Bayesian moment. I've been teaching myself math long enough now, and teaching Christopher math, and writing ktm, and reading the Comments that I probably "knew" at the level of the cognitive unconscious that Saxon was a better idea than a $500 "developmental" math course at a community college.

The reason Saxon is probably a better idea (it's early days, I realize) is that the community college doesn't offer courses in 4th grade arithmetic.

They're trying to remediate years of missing math instruction in one or two semesters. I just don't see how it can work no matter how good their teachers are (and I bet their teachers are good). In fact, the effort to close years of math gaps in one or two semesters seems almost inevitably destined to produce more math anxiety than a student had going in.

I'm thinking that just as there is no royal road to geometry, there's no royal road to Algebra 1. People whose math education went completely off the rails back when they were eight have to go back to when they were eight.

At least, that's what I'm thinking at the moment.

We'll see how it goes.






Saxon middle grades online placement test
downloadable Saxon placement tests at Learning Things
downloadable Saxon placement tests at Sonlight Curriculum
Rainbow Resource (least expensive source of Saxon Homeschool texts I've found)
teachingtomastery teachtomastery programplacement
Christianlearnsmath

-- CatherineJohnson - 20 Oct 2006

---

AgainstChallenge 01 Nov 2006 - 19:53 CatherineJohnson




More from Engelmann on teaching to mastery:

Rule 3: Always place students appropriately for more rapid mastery progress. This fact contradicts the belief that students are placed appropriately in a sequence if they have to struggle— scratch their head, make false starts, sigh, frown, gut it out. According to one version of this belief, if there are no signs of hard work there is no evidence of learning. This belief does not place emphasis on the program and the teacher to make learning manageable but on the grit of the student to meet the “challenge.” In the traditional interpretation, much of the “homework” assigned to students (and their families) is motivated by this belief. The assumption seems to be that students will be strengthened if they are “challenged.”

This belief is flatly wrong. If students are placed appropriately, the work is relatively easy. Students tend to learn it without as much “struggle.” They tend to retain it better and they tend to apply it better, if they learn it with fewer mistakes.

source:
Student Program Alignment and Teaching to Mastery (pdf file)
by Siegfried Engelmann

This is a perfect description of Irvington educational philosophy as it pertains to the rapidly dwindling student "elite," i.e. the chosen few who are allowed entry into accelerated and Honors courses.

The kids have to gut it out.

If they can't gut it out, down they go.

They "don't belong."



As a direct result of my experience muscling our family's way through "accelerated instruction" here in Irvington, I have now stricken the word "challenge" from my list.

I don't want to hear "challenge" ever again.

I want to hear "teach;" I want to hear "learn;" I want to hear "assess." That's "assess" as in assessment for learning, not grading.

During our meeting with the principal — which was very good — we raised the question of Irvington's rationing of accelerated courses.

That's another story; I'll get to it later. Suffice it to say that the middle school plans to offer two sections of Regents earth science next year, out of 8 sections altogether. Compare this to the Pelham school district which, a few years back (and possibly still today) had everyone in 8th grade taking Regents science and earning high scores on the Regents earth science test. Irvington has 25% of 8th graders taking earth science and apparently our Regents scores for the kids who take earth science in high school aren't good, or so I'm told.

Spot the difference?

Anyway, Mr. Witazek said he'd already had parents in to discuss the issue with him (good); he also mentioned that he himself had taken earth science in the 8th grade because in his school everyone took earth science in the 8th grade.

He's going to have to spend some serious time figuring this out, because this year the SOP^* isn't operative, SOP being that parents wait until after their child has been rejected to complain. We're complaining now.

Since he couldn't promise anything at that moment, he offered the observation that, "All courses can be challenging." This was sincere; it sounds like spin, but it wasn't.

I swatted it back across the net anyway. Time is short. I can't spend any more of my child's middle years swapping edu-language with the administrators who are in charge of his education.

I said, "Everyone who wants to take earth science needs to take earth science. Just telling people that this course or that course is 'challenging' isn't enough. An earth science course is either a Regents course or it's not."

The thing I like about the new principal so far is that he has exactly zero problem with a statement like that. Although he uses a great deal of edu-blah-blah in his official communications with parents, he clearly seems to think that parents are within their rights not to want to talk edu-blah-blah face to face.

Then I delivered my stump speech:

I don't want to hear 'challenging' any more. We've had 'challenging' in Phase 4 math for a year now, and it's a nightmare. All it means is the parents teach the course. Anyone can challenge a child. Just pull something that's over their heads off the internet and tell them to go do it. Teaching a child is what's hard. We want our child to be taught, not challenged.'

He said, "I hear what you're saying," and he meant it.

Since that meeting we've seen tangible changes in the math instruction.

Solutions to the homework problems are given out; kids check their own homework in class. (To me this is preferable to having the teacher collect the homework and correct it at home. These kids are perfectly capable of correcting their own homework as long as the teacher gives them the answers, and this way they get much more timely feedback.)

Then Ms. K finds out who missed which problems and gives each child individual help on those problems. (I'm thinking that as matters improve she shouldn't have to do that as often, because the kids should be getting homework assignments pitched to their level of mastery.)

A couple of days ago she gave a quiz consisting of two problems of the type they'd had on the homework assignment.

This is fantastic progress; it's exactly what we all desperately need; and it's happening because of the new principal.

We are impressed and grateful.

I've mentioned in the comments section that we're thinking the principal's background is a help.

Previously he worked with a disadvantaged student population in Albany schools.

During our meeting he seemed almost stunned to be finding out that he's got parents reteaching courses at home. This widespread practice has become naturalized for us; when there's trouble in a class, people immediately begin to reteach the content themselves; if trouble continues the next step is to hire a tutor. Parents take both of these steps quickly. Unlike a school bureaucracy, we don't wait for "the meeting" to provide "services."

Many parents hire tutors from the same school their child is attending.

All of this seems normal to us.

It does not seem normal to a person who just got off the train from Albany.

So far, I'd say he's making progress in changing the tone and culture of the school. Here is a bellwether: some of you will remember my friend who was spending close to $200 a week on tutoring by a middle school teacher to keep her child from flunking 6th grade.

I talked to her a couple of days ago.

She is now spending half that and she's spending nothing at all keeping her other child afloat.

She may be heading towards dispensing with the tutor altogether.

All of this appears to be happening because of the new principal.

The district needs fundamental reform.

At a minimum, we need a laser focus on student outcome.

What student outcomes will result from the proposed program or pedagogical approach?

And then: what student outcomes did result from the proposed program or pedagogical approach?

And we need to focus always on creating and maintaining the conditions for success.

One of those conditions is to dump the idea that unless a child is being worked to the point of burn-out or swamped by homework problems he has no idea how to do, he's not learning because he's not being challenged.

Teaching to mastery means the child is not gutting it out.

^* standard operating procedure


-- CatherineJohnson - 20 Oct 2006

---

NoOneToWaste 26 Oct 2006 - 20:56 CatherineJohnson




Now that I've been involved in "remediating" at least two people (not counting a couple of the kids in Singapore Math &, I guess, not counting me, either) I've realized that: I am interested in the question of remediation.

It's entirely possible that we have some real research avilable to us in the area of remediation, as opposed to your standard edu-research, which is not research but log-rolling.

If so, this would jibe with my feeling that when it comes to teachers & principals, you want the teacher or principal who's taught special ed.

Christopher's ELA teacher, the one who gave them the SWB and so assignment, has taught special ed.

So has the new principal; special ed is his background, I believe.

And then there's Siegfried Engelmann.

Teachers and administrators who've worked in special ed have spent a great deal of time endeavoring to teach the kids who need teachers, a large point in their favor. They haven't been able to coast on the gifted kids who "breathe it in"; nor have they been able to "teach to the middle."

They've been teaching to the bottom, by definition. I don't mean this statement to hurt feelings; I know special ed kids can be brainy — I've got some brainy special ed types around here. But when a special ed teacher is teaching a high-end special ed kid with strong native intelligence, she's still teaching a kid who has many gaps in his knowledge and probably a roaring case of edu-shame and demoralization along with. Special ed teachers are working with the most challenging population, regardless of how naturally intelligent any given special ed kid may be.

Beyond this, special ed teachers have spent a great deal of time — countless hours of time, in fact, and I am in a position to know — sitting in face to face, legally mandated meetings with parents hashing out what will be taught in the upcoming year. These teachers experience strong external legal, moral, and emotional demands to get content inside their students' heads.

Obviously any good teacher feels internally pressured to get content inside students' heads. Depending upon the school, she may be feeling heat from NCLB & state standards, too.

But special ed teachers live with intense outside pressure from law and from parents.

They've been shaped by different forces.

And then there's the fact that people who choose to go into special ed are the salt of the earth.

I was drafted. They volunteered.

So I've come to feel optimistic the minute I hear that a teacher Christopher is going to have has a background in special ed. My reaction may be common; my neighbor said "Good" the minute she heard that the new middle school principal had a background in special ed.

Anyway, I realized this afternoon that this principle may also apply to education research. It's possible that research into remediation is more serious and less ideology-ridden than research into "literacy" and "mathematical reasoning" and the like.

So I'm going to start paying closer attention to the field.



no one to waste

I've been prospecting for remediation factoids for awhile now.

Here are a couple:

The study found that students who are successfully remediated become productively employed. Almost 16% become professionals; 53.7% obtain mid-level, white-collar or technical positions; 19.8% become high-skill, blue-collar workers; and only 9.2% remain in unskilled or in low-skill jobs.

source:
No One To Waste: A Report to Public Decision-Makers and Community College Leaders

Students who require remediation in reading are at a greater disadvantage than those with a math deficiency (McCabe, 2000).

source:
Remediation PostSec


...about half of the students entering college aren't prepared for credit-bearing college-level work and have to take remedial courses, according to the Department of Education (2001). Overall, more than one million entering college students take remedial courses, says R. H. McCabe in "No One To Waste" (2000). McCabe cites studies that show 20 percent of entering students are underprepared in reading, 25 percent in writing, and 34 percent in math. Students who take remedial courses are much less likely to graduate.

source:
ACTIVITY

It's always interesting seeing the same study characterized as optimistic ("students who are successfully remediated become productively employed") and pessimistic ("Students who take remedial courses are much less likely to graduate") both.

McCabe's study doesn't seem to be available online, but I've found an interview with him.

Remedial students range from those with a minor deficiency in only one area to people who are deeply deficient in all three. There are roughly one million students a year who begin in college and are assigned to remedial courses.

PC: Is that number for all institutions or just community colleges?

RM: In all of them. It’s 29 percent of all of the entering college students. In community colleges it’s about 40 percent, and I do not know of an urban community college where it is not at least half.

[snip]

...the average student takes a little over seven semester credits for remediation, approximately one-fourth of a college year.

PC: What is known about whether, or how well, remediation works? Your report talks about the perception, especially among some state legislators, that we simply are paying for failure one more time?

RM: There is that perception, but more often than not it’s based on faulty premises. First, some of these legislators see continuity between high school and college that doesn’t exist. They think that if students graduate from high school, they should be ready for standard college work. But the criteria for high school graduation do not match the competency requirements to begin standard college courses....

Second, our definitions of success may be too narrow. A variety of studies, including my own, show that between 40 and 50 percent of students who begin in community colleges are successfully remediated. But, the fact is that a high percentage is not going on to bachelor’s degrees, even though remedial programs are really geared for that purpose. Twice as many earn occupational associate degrees or certificates.

If you look at remedial students nine years later, 90 percent are employed in work above an entry-level type of job. Less than two percent are out of work. Remediation in the community colleges is much more occupational and vocational than we have thought, and measures of its success should recognize this.

For the seriously deficient, the program is basically a total failure. Only 20 percent of those students complete remediation, and very few go on to anything after that. Programs for the seriously deficient need to be redesigned altogether.

PC: How do you define “seriously deficient”? Out of the million students mentioned earlier, how many are there?

RM: Seriously deficient students are those who are deficient in reading, writing and math, and are required to take at least one course prior to the standard remedial courses. For example, they simply could not take one remedial math course and be ready—they would have to take a math course prior to that math course. As for numbers, I lack data for all institutions, but in the community colleges I looked at, about one-fourth were seriously deficient.

[snip]

Actually, remediation is the most productive education program we have. With just one percent of the budget, it salvages the lives of a half-million people, and enables them to become positive, con-tributing individuals in our society. It is a particularly good investment because of the country’s changing demography and economy. The part of our population that is growing the most is also the part that is least prepared for college and skilled jobs. We believe in opportunity and access, and we can’t have either without remedial education.

...you assert that remediation is essential for quality, not a detriment to it. Is this true?

RM: It is absolutely true. In regular courses, faculty must expect that students are prepared, and only remedial testing and placement can assure that this expectation is met. If it is not met, then quality will suffer because faculty will be forced to lower their expectations to meet the competencies of the unprepared, or too many will fail.

[ed.: this reminds me of my hypothesis that when you refuse to accelerate gifted children you also refuse to accelerate regular children]

...it is essential to America, morally, socially and economically, that a high percentage of young Americans have education or training beyond high school—as high as 80 percent, in my opinion.

[ed.: I think it's Robert Murnane (not sure) who has shown that the only reason it's essential for 80% of young Americans to have training beyond high school is that high school isn't high school any more. It's junior high. Will attempt to track down the source at some point.... yes, it's Murnane^*

[snip]

...teaching less-prepared students is hard and often frustrating work, for they need more personal attention and support. A belief that their job is helping human beings to develop, and that academic work is, in fact, an instrument to that end, is not typical of college faculty.

[snip]

One of the most disappointing things to me is that typically community colleges, where most academically deficient students enroll, do not seem to use the resources that they are given for remedial programs. In fact, they often use mostly part-time instructors without additional and necessary support. They give remediation low priority and run cheap programs.

[snip]

PC: If all of the K–12 improvement programs that the country has been working so hard on are successful, can’t we look to the day when remediation will not be around in higher education?

...I simply do not see reform wiping out the need for remediation in the fore-seeable future. We have to dig in for the long run.

[ed: amen]

[snip]

Testing should recognize a continuum; secondary school testing should be closely related to tests used to place people in college. With regard to under-prepared students, experience together with educational research has produced substantial knowledge of effective learning practices. Typically community colleges do not use that knowledge. They must.

[snip]

Most programs simply identify someone who is deficient in math or reading or writing, and then assign students to subject classes based on test results. That is a great waste of resources and student time; nothing is done to identify the differences in deficiencies within and across subject fields, or to relate deficiencies to the learning program.

We have the capacity to produce diag-nostic assessments and to align each student’s program to the results.

[snip]

A final word, if I may. I was surprised by our data on just how educationally far behind ethnic minorities are. [ed.: No kidding.] There is a monumental problem here that, in my opinion, needs to be addressed with mega effort and resources, not just by schools, but also by states and by communities.

what skills does one need to hold a middle-class job?

RICHARD MURNANE, Co-Author, "Teaching the New Basic Skills:" To qualify for a job that will pay a middle class wage as minimum the graduate needs to be able to read well enough to understand training manuals, basically ninth grade, able to do the mathematics that’s typically included in training manuals, fractions and decimals and line graphs, mastery of that, the ability to problem solve, to take a problem and find what will work, to shape it, to design a solution towards it, and two kinds of what we call soft skills, the ability to communicate effectively both orally and in writing, the ability to work productively with people from different backgrounds, and enough familiarity with computers to have the self-confidence and the knowledge to learn to use new software. You might say these skills are extremely modest, and they are in one sense, and there are lots of jobs that require, that pay good wages that require a lot more than these skills, but there are almost none, outside of professional sports, that do not require at least these new basic skills and also, remember, in terms of whether this is a challenge for schools to provide this, roughly half of American high school seniors are graduating without these new basic skills.

DAVID GERGEN: And if they graduate without them and never get them, they’re condemned to live in very low wages? RICHARD MURNANE: They can find work in most cases but these are jobs that pay six and seven dollars an hour, not enough to support children.

DAVID GERGEN: And if they can get the skills?

RICHARD MURNANE: If they can get the skills, they have a chance at acquiring middle-class jobs and have access to subsequent training when they need it. And these aren’t jobs that will be jobs for--that one holds for twenty-five or thirty years. To a large extent, those jobs have disappeared from the economy, but it will be the opportunity to move from job to job and to earn enough to support kids.

If you don't get these skills in high school, and black and Hispanic students routinely do not, you've got to get them somewhere else.

That somewhere else is going to be our community colleges.

Our community colleges & John Saxon.

^* Murnane and coauthor Levey do not appear to hold the view that public school decline has made it necessary for high school graduates to pick up what once were K-12 skills in college:

FRANK LEVY: That’s right. And that’s what makes the problem so hard to diagnose. Schools today are a little better than they were 15 years ago, but the job market skills have just escalated much faster than that. I mean, we have an example in the book, "Looking At a Modern Automobile Plant," and about half of today’s high school graduates couldn’t make the cut-off to be a production worker at a modern automobile plant.

DAVID GERGEN: Say it one more time. Half of the 17-year-olds--

FRANK LEVY: Half of 17-year-olds don’t have the skills necessary to make the cut-off of the production worker at a modern automobile plant today.

-- CatherineJohnson - 23 Oct 2006

---

OnNotTeachingToMastery 01 Nov 2006 - 16:43 CatherineJohnson

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.