kids need spiraled practice, not spiraled instruction

Hainish left a link to a terrific post by a math teacher:

11th grade here = 9th grade here. In fact, Algebra 2 was such a rehash of the district's Algebra 1 course that some teachers called it "Algebra T-o-o." And really, the same point could be made about math curriculum as a whole in the U.S., since most content for any given year is a review of content from previous years. (The Common Core State Standards may help change this, but I'll believe it when I see it.)

This approach, where we touch on lots of topics each year--rather than go deep with fewer topics--and then revisit them in subsequent years is often called spiraling. But what it is for many students is stifling. And this is as true for kids who've yet to master a skill as it is for those who nailed it right away. I first noticed this when I taught 9th grade Algebra classes where every student was performing at least two years below grade level.

"Meet them where they are," fellow math teachers advised me. Makes sense, I thought, since I couldn't imagine teaching Algebra to kids who didn't know basic arithmetic. But what I soon learned is that perception matters more to students than performance. For many kids, having seen something is akin to having learned something. "Man, we already know this," students said, as I presented lesson after lesson on fractions, decimals, and percents.

Other students, meanwhile, knew they didn't understand the material, but had given up hope of ever understanding it. The implication was therefore the same for all students: encore presentations on previous years' topics were pointless. And though I was able to engage a few students when I found new ways to present old topics, one group of students was always slighted: those who really did "already know this."

[snip]

The problem, of course, goes back to the disconnect between kids seeing something and actually learning--and retaining--it. But if it didn't sink in for them the first, second, or third time a teacher presented it, why should we present it again?

We shouldn't. At some point the focus needs to be on students practicing math rather than teachers presenting it.

[snip]

[W]e should provide students spiraled practice, not spiraled instruction. When I did this in 10th grade Geometry classes, students said they learned more Algebra than they had learned in their 9th grade Algebra course. And, as a result, they were ready for more advanced math--starting with Algebra T-w-o.
Spiraled Instruction, Stifled Learning
By David Ginsburg on March 5, 2012 8:35 PM

Wow!

yet another brilliant idea from the folks who brought you all those other brilliant ideas

Jen writes:

I was talking to a teacher friend the other day and he was lamenting that the curriculum is going to change yet again and that if they follow what they say they want to do, they'll move everything to a grade earlier.

Mind you, this is an urban district with not great scores. Mind you, they spent years with EM which is likely the worst way to teach kids who come in without number sense, without support for education at home, and without a parent who can figure out what's being asked and more importantly, what's being missed.

But, the new big idea (again, this is really part of the idea of spiraling) is that if kids aren't getting, 5th grade math in 5th grade it means you really need to teach those concepts in...4th grade! Brilliant! Bravo! Imagine how much better they'll do at it, not learning it at an earlier age!

Now kids coming into K and 1st grade without any number skills, 1-3 years behind other kids of middle class, well-educated parents, will be expected to be getting through 1-3 more years of math in their first few years of school, too. It's genius!

What teacher can't take 25-30 elementary students who are starting behind and teach them 2-6 years of math in a year? Whiner slacker teachers, that's who!

I say, Whiner, slacker teachers of the world, unite!

spiral curriculum in a charter school

When Courtney Sale Ross, the widow of Steve Ross, the former C.E.O. of Time Warner, and the founder of the Ross Global Academy, a charter school in the East Village, was told, last month, to expect a 9 A.M. phone call from the outgoing New York City schools chancellor, Joel Klein, she feared that it would not bring good news. Her academy, which was founded five years ago amid considerable fanfare—it promised a so-called spiral curriculum, encompassing the history of civilization across all cultures, and also offered instruction in eating organically and yoga—recently had the distinction of getting the worst progress report of any charter school in the city, with seventy-five per cent of its students failing English and seventy per cent of them failing math. The school, into which Ross and members of her board have poured eight million dollars of their own money, has had six principals, has occupied three locations, and lost three-quarters of its teachers last year; its charter is up for renewal this month. At eight-thirty on the appointed morning, Ross’s phone rang; on the line was Marc Sternberg, a deputy chancellor. “He said the most extraordinary thing,” Ross recalled last week. “He said, ‘I am informing you that the Department of Education is going to recommend a non-renewal.’
Spiralling
The New Yorker
by Rebecca Mead
January 17, 2011

If only someone would say the same to my district's "department of curriculum."

And see: The Trouble with Math by Ralph Abraham (pdf file)

And so much more: multiple intelligences, spirals, innovation, the whole child.... The Concept of Spiral Curriculum at Ross School



Educating the Whole Child for the Whole World: The Ross School Model and Education for the Global Era

Everyday Math author defends his program against Katharine Beals

In today's Philadelphia Inquirer Letters to the Editor, excerpted here:

Katharine Beals' article on the use of "reform math" with students with autism contains many misperceptions about Everyday Mathematics that, as the program's coauthor, I want to clarify ("The 'reform math' problem," last Monday).

Everyday Mathematics was designed for general education students, but it has been effective in special education, including with students with autism.

Beals' claim that students spend large chunks of time working in unsupervised groups is untrue. A teacher supervises student group work at all times. While some assignments are "open-ended and language-intensive," many are not. A balanced curriculum needs simple exercises to build basic skills, as well as more difficult problems.

Beals writes that students "lose points for failing to cooperate in groups, explain their answers, and comprehend language-intensive problems." While decisions about how to grade students are made at the local level, many people believe it's reasonable to require students to work cooperatively, explain their work, and understand word problems.

Everyday Mathematics is not just a "sequence of themes," but a carefully organized sequence of lessons resulting in mastery of a specific set of goals. Its approach is well supported by research, the authors' experience, and decades of classroom experience.

Naturally, accommodations for teaching children with autism must be made, and that's what professionals always do. As with any tool, Everyday Mathematics must be used with professional judgment.

Andy Isaacs

Chicago

Speaking of the spiral...

...my problems of the week this week show the spiraling vs. the linear approach within chapters called "Addition and Subtraction" in the 2nd grade Everyday Math vs. Singapore Math curricula.

Education Non-Myths

I couldn't resist sharing these maxims from a new blog :
www.incentiveseverywhere.com
whose author I know from a previous book he wrote entitled Power Teaching (it's in the list of books I recommended in a post a few months ago: http://kitchentablemath.blogspot.com/2009/02/recommended-reading-from-palisadesk.html



What follows is from the "Book of Right", the set of assumptions which will produce learning.

1. Although students come from different backgrounds, and some are much easier to teach than others, what education brings to the student is much more important than what the student brings to education.

2. All subjects are hierarchically arranged by logic and there is a sequence of instruction which must be followed by all but the most exceptional of high-performing students.

3. Reinforcement is a very powerful determinant of student achievement. The main reinforcer in education is the improvement the student sees in his skills. Ill-constructed curricula, the kind found in almost every government school, result in a steady diet of failure for most students.

4. Having a system of education which is not a civil servant bureaucracy is a necessary but not a sufficient condition for effective education. You can’t do it with such a bureaucracy, but just because you don’t have a bureaucracy doesn’t mean you can do it.

5. Higher order thinking skills are explicitly taught, not fondly hoped for.

6. Methods of teaching are determined by scientific research, not consensus based on experience and sincere belief.

7. Teachers use a curriculum and lesson plans which have been demonstrated to work best and are not expected to create their own.

8. Psychological assessments are used rarely, but assessment of student progress, which means assessment of the effectiveness of teaching, occurs at least daily.

9. Teachers are taught how to teach in detail rather than being expected to apply vague philosophical maundering.

10. Special education is rarely needed because students are taught well on the first go round.

11. If a student does not learn, the blame is not placed on neurological impairment, but on faulty teaching methods.

12. Self-esteem is not taught because it does not have to be.

13. Students are not given "projects" until component skills have been mastered and rarely thereafter.

14. No attention is paid to individual "learning styles" because these hypothetical entities have no effect on learning.

15. Academic success can be measured by reliable and valid standardized tests, although many of these tests are too simple.

16. Students are expected to perform correctly in spelling, writing, reading, and mathematics and it does not stifle creativity.

17. The precepts of Whole Language are not used to teach reading because these precepts are wrong.

18. Students are not expected to create their own reality because this leads to frustration and slow learning.

19. Students are not expected to learn when it is developmentally appropriate but when they are taught.

20. The concept of multiple intelligences is ignored because it has no positive effect on learning.

21. The teacher is a teacher and not a facilitator.

22. The spiral curriculum is not used because things are taught properly the first time.

23. The customer is the parent and the customer must have the economic power to move his child to another teaching situation when unsatisfied.

24. In private education, the cost of education is known. In public education, the cost can never be known because there is no motivation to tell the truth and every motivation not to.

25. The curriculum must be tested on children and provision must be made for mastery learning. Passage of time or exposure does not guarantee learning.

26. Students are not tortured by "creative problem solving" because this is just another crude IQ test and has no value aside from categorizing students yet again. http://incentiveseverywhere.com/2009/10/09/education-non-myths/


I'm not sure I agree that "special education will rarely be needed," because I have observed that students with certain exceptionalities (autism, some LDs, some language impairments) need the same effective instruction but can't benefit from it in an inclusive setting, at least not initially. However, I agree with the general case, that much "special education" is simply ineffective general education, watered down in in a smaller group. As Lloyd Dunne (I think) observed, "It's not special, and it's not education."

All students deserve better.

Sadly, Job Security for Me: Everyday Math Marches On

Job security as in academic remedation in the Palo Alto/Portola Valley/Woodside area -- all school districts having adopted Everyday Math.

Here's a snippet from Old Math, New Math : Everyday Math, aka Chicago Math by Roxane Dover at Silicon Valley Moms' Blog (who also blogs at Rox and Roll)

I had seen firsthand the effects of a problematic math curriculum on otherwise motivated kids. My daughter is a keen student who is interested in math and science to a surprising level. We want to encourage our daughter to stay her course – and hopefully, one day, she will be among those solving the energy crisis, curing cancer or healing the environment. Mastery of math is key to success in science. What I saw when my daughter experienced Everyday Math for three years in K-2 was a rejection of math out of frustration and a related inability to master basic mathematical concepts because of the Everyday Math approach of teaching every concept in too many different ways. This approach gives students a cursory understanding of several ways of addressing concepts at the expense of mastery.

Specifically, mastery falls victim to a concept called “spiraling.” Spiraling means that concepts are introduced but not necessarily mastered before new concepts are introduced, then the previously introduced concepts are revisited and built upon before something else new comes along, repeat. Mathematics learning, which should be progressive and built on a solid foundation, is replaced in this curriculum by a method of throwing a multitude of ideas at the kids without giving the kids time to properly internalize them to create that solid groundwork. It’s like cooking spaghetti and testing it by throwing a handful of noodles at the wall to see what sticks. Everyday Math doesn’t want it all to stick; it’s just concerned that some of it does. And that’s not good enough to build the solid mathematical groundwork that our children require.

From the San Jose Mercury News:

Critics, however, say the curriculum and its nontraditional algorithms are confusing. "Everyday Math" follows a "spiraling" method, where students move quickly through new concepts and may not necessarily learn them the first time around, but they revisit them over and over again in different formats or applications. They also say it encourages students to use calculators too often.

They never develop mastery, said R. James Milgram, a Stanford University math professor who sat on a state curriculum review committee in 2000 when the state rejected "Everyday Math."

"The mathematics these kids are seeing is hardly mathematics at all," Milgram said. "They learn a mush of things, most of which are just wrong."

Milgram said that while the program at its core makes sense, there are only 500 or 600 elementary teachers in the state with the expertise to teach it properly. He said he has seen enrollment in "Everyday Math" districts contract as parents pull out their students and send them to private schools, and said that could happen in Palo Alto.

One local preK-5 private school saw a 28% jump in admissions applications. This when local unemployment is at 11%.

unified theory

from Paul H:

It's instructive to put constructivism in context. I put it smack dab in the middle of; Spiraling Curricula, Curricula Bloat, ** Constructivism **, Inclusion/Immersion, and Grade Level Placement. I'll call these the five horsemen of the apocalypse.

Horse #1, Spiraling Curricula (the Trojan horse), lays the natural hierarchy of a subject on its side, preferring to teach everything in a strand as a set of increasingly complex parallel universes that never need to be mastered. We spend, on average, 6 years on concepts that other countries dispense with in 3.

Once you install horse #1 you observe that, without mastery, every stovepipe in the spiral is easier to achieve success in (because you don't really measure it anymore). This leads to the evolution of horse #2, Curricula Bloat, where you get to fill your newly invented extra time with bloated concept development (2-3 times more concepts per year than is common in the TIMMS countries that surpass us).

Horses #3 and #4, Inclusion/Immersion and Grade Level Placement, differ in motivation but produce the same noxious result, an enormous range of student capabilities in a single classroom, with an attendant reduction in the number and type of teachers that address them. And finally with horses 1,2,4, and 5 teamed up you're ready for the lead horse, horse #3,Constructivism.

With an exquisitely complex spiral, delivering an overwhelming number of concepts to a highly diverse population grouped by virtue of their hat size, there is no other choice but to have the kids teach each other. Horse #3 is inevitable.

Every argument you hear for constructivist philosophy is no more than rationalization, devised to make palatable, the misbegotten notion that kids can teach themselves the things that mankind learned over thousands of years, driven by the need to address the maelstrom created by the four horsemen that accompany it. Once you figure out where it comes from you're better able to appreciate why it's so popular. Unfortunately, it's very hard to change the direction of the lead horse when the rest of the team is not cooperating and these horses all tend to be discussed in isolation, where they can be made to sound plausible. Together though, they are undeniably toxic.

And yes I know there are only four horsemen but I'm invoking my 21st century skills to invent my own literary reality.

Schmidt on the elite kids

This system of ours has failed the elite kids, too. This is a little known fact because it wasn’t emphasized very much, but in the early TIMSS study there was a high school specialist exam for those kids that were the AP physics kids and those that were the AP calculus kids. Those kids were last among their counterparts in the rest of the world. That is, if you took the elite track in the French system that was leading to math and science, these [American] kids were at the bottom. So we’re failing those kids just as much as we’re failing the kids on the other spectrum.

William Schmidt, Michigan State University
U.S. TIMSS National Research Coordinator
Baltimore Curriculum Project/Leading Minds

sound bites

Does anyone have a succint way to explain the difference between Teaching to Mastery and Spiraling? I am eagerly searching for clarity/brevity of speech.

(I find myself giving too much information and sounding kindof like a conspiracy theorist when I talk to people about math education these days.)


update from Catherine

I would start with this "fact sheet" I put together for a PTSA Forum a couple of years back (scroll down).

The salient passages are these:

You know, talk about curriculum, if I put in front of you a fifth, sixth, seventh, and eighth grade textbook in math and opened up to page 200 and I jumbled them up, and said, “order them from fifth through eighth grade in order,” you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of “we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.” Instead of thinking, “let's teach the kids how to reduce fractions at a mastery level in fifth grade, maybe spend a little time reviewing it in sixth grade but let's move on to pre-algebra and let's move on to algebra then.” And that's been our take and so it's not that we have a different math curriculum as much as we have a different math strategy and a different math philosophy.

Interview with Mike Feinberg, Co-Founder Knowledge is Power Program

(oops - just found this sitting open on my desktop - more in a bit)

“the kiss of death”

During a meeting with my daughter’s teacher last week, she told me that a spiral curriculum (used in our school, of course) is “the kiss of death” for a child such as my daughter who requires a lot of practice and needs to master a lesson before moving on to the next one. What typically happens, this teacher said, is that when the class revisits the lesson 6-8 weeks after the first introduction, it’s “like she never learned it”.

These comments followed my explanation that C. had been achieving great success doing Kumon, which follows a logical cumulative sequence of topics, provides abundant practice and applies formative assessment to ensure mastery at each level. The classroom teacher agreeably observed, “Oh, that’s good” before she made the comment about spiraling. And the resource teacher, also present at this meeting and apparently trying to demonstrate how helpful he would be this year, told me he’d be happy to talk with the tutor anytime if it would help.

[The sound you don’t hear at this point is a suppressed scream from mom.]

This is the second year that I have explained to the school about the type of instruction that enables my daughter to excel. And, apparently, this will be the second year that they will inform me they will not change the way they teach my daughter. Worse yet, this year the school is implementing a new constructivist math program, complete with group discovery and spiraling that I expect will only impede my daughter’s learning more than ever.

But, here’s the kicker. My daughter has an IEP! Isn't that supposed to ensure that the school provides an individualized education plan that will meet the specific learning needs of the student????

[Another suppressed scream.]

I’m not sure what I must do to make the school teach my daughter in a way that works for her. I can be more forceful in explaining to the IEP committee that the school should use methods employed in direct instruction and precision teaching because they meet my daughter’s specific learning needs. Maybe I can try having them incorporate mastery at each step within the IEP goals. Really, they should just pay for her Kumon and be done with it.

To me, this crazy situation is just another example of how it’s frequently the parents taking up the slack when the illogical methods that pass for “quality education” let our children down.

same time, next year

I just noticed Ken's use of the phrase "how the spiral should work."

This is our problem.

A spiral curriculum is by definition a form of spaced repetition. You learn topic X in first grade; then you "revisit" topic X in second grade; then you take another return trip again in 3rd grade.

The notion that the space between repetitions will be 12 months' time is simply built in to the proposition. It's unexamined.

And, of course, no effort is made to ensure or assess whether students have reached mastery before the class peddles on.

MORE T/K


overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

"Overlearning" Overrated?

Science Daily had an interesting article out this week discussing the value of overlearning. Recent research has found that studying material beyond mastery may be a waste of time in the long run.

University of South Florida psychologist Doug Rohrer decided to explore this question scientifically. Working with Hal Pashler of the University of California, San Diego, he had two groups of students study new vocabulary in different ways. One group ran through the list five times; these students got a perfect score no more than once. The others kept drilling, for a total of ten trials; with this extra effort, the students had at least three perfect run-throughs. Then the psychologists tested all the students, some one week later and others four weeks later.

The results were interesting. For students who took the test a week later, those who had done the extra drilling performed better. But this benefit of overlearning completely disappeared by four weeks. In other words, if students are interested in learning that lasts, that extra effort is really a waste. They should instead spend this time looking at material from last week or last month or even last year.

Researchers concluded that once mastery was achieved it was better to leave that subject alone for a while and return to it later. They found that an optimal "study break" of about a month resulted in long-term learning-- something they refer to as the "spacing effect".

Is this "spacing effect" an argument for the spiral approach? Perhaps so, yet it does seem to be a well executed spiral in which the content is first studied to mastery and then revisted for reinforcement later. This is certainly not the haphazard "spiral" I've witnessed my children being subjected to with Everyday Math and seems to be more in keeping with Saxon or Singapore Math's idea of a spiral curriclum.

I hope they keep looking into this subject. Children have such precious little time to learn so many important things. Imagine all that could be accomplished if we started implementing teaching and study skills that were actually efficient.

Source: Back to School: Cramming Doesn't Work In The Long Term


ABSTRACT—Because people forget much of what they learn, students could benefit from learning strategies that yield long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions—the spacing effect—depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.

Increasing Retention Without Increasing Study Time


Catherine here, diving into Concerned's post.

What a find! I've just pulled the article; will read shortly.

In the meantime, here are the Willingham articles that discuss overlearning:

• Allocating Student Study Time: "Massed" versus "Distributed" Practice

• Practice Makes Perfect, But Only If You Practice Beyond the Point of Perfection

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary

from le radical galoisien - spiraling in Singapore

It is in part due to the flexible module system, isn't it? Sometimes the prerequisites aren't really cohesive ...

My current American high school generally has it to take one science subject per year, and thereby acquire the entire credit for that subject in one year. The next year, another science subject is taken, and so on. [Catherine: true of all high schools in NY state, I would imagine]

I assume this is standard for most American schools. At first, I thought it rather interesting compared to my old schools in Singapore, but now I am questioning their effectiveness. After all, I have not done any real biology for so long, none of the material is fresh in my head, even though I am very highly interested in oncology and so forth. The same applies to mathematics in terms of matrices, which were taught to me three years ago and were only re-introduced recently in calculus class following the AP exams.

It is this that makes me miss the Singaporean system and especially since I worry that I am falling behind my Singaporean friends.

Perhaps the whole problem is the credit-based system. Credit units should not be used until university. Having to accommodate classes to the nature of credits at a secondary level can make the interaction of knowledge between those classes ineffective.

Interesting.

I had never made the connection between credits and a "flexible module" system of curriculum, although I've skimmed a couple of accounts of the Carnegie unit & how it came to be.

So much of U.S. public education seems to be just a series of random "reforms that stuck"^*... some Committee of 10, or 9, or whatever number it happened to be at the time, and somehow managed to power its ideas into effect.

Of course, the Committee of 10 isn't the best example, seeing as how the Committee of 10 seems to be the last such committee to promote "traditional academic study."


a boy's will is the wind's will

^* Of course, random reforms that stuck may simply be what historians call contingencies, in which case the history of U.S. public schools is no different from the history of anything else... (One of these days I'll learn something about history, possibly after I learn high school math, earth science, chemistry & physics.)

spiraling in the Ukraine - from Exo

In Soviet schools the content with building-on spiraling was set across a number of years. Lets say, biology would start in grade 5 and it will be Botany (2 times a week), then Zoology (grades 6-7, twice a week), then Human Anatomy and Physiology (grade 8, twice a week), then General Biology (grades 9-10, twice a week). Of course, topics were studied in logical order according to evolution theory, and concepts of cells, tissues, organs, functions were coming up again and again. It was allowing for understanding of complexity and patterns in living things and processes.

Here I have to teach general bio (Living environment) to students who have no idea of plants stuctures, animals, or human's anatomy and functions; I have to mix it all together with emphasis on molecular biology (that requires knowlege of chemistry) and genetics, that heavily connects with evolution. Many of my honor students don't know that insects pollinate the plants!

When I was in school, by the time we had general bio, we knew evolution of plants, organization of animal kingdom, anatomy of humans, and were up to organic chemistry in chemistry class and nuclear physics in physics class that was well alligned with molecular bio and gene engineering. Oh boy... And when in vet school, we took the entrance exam in bio (covering everything in curriculum from Botany to General bio) and didn't have a general bio class ever again.

BTW, from my trip to Ukraine: the school curriculum is still in place the way I had it - still logically built, but spread over 12 years now. However, there is no school that would have students sitting 8 periods per day. The maximum number of periods students can have is 6 in grades 6-12, and 5 in grades 1-5. A period is 45 minutes. The school year starts September 1st, and ends May 25th, with final exams in higher grades scheduled from May 28-to June 12.

how to talk about spirals

I am just now getting to the Singapore/Saxon thread (and I STILL have not read the 888 lesson - which is why I haven't weighed in - SORRY! I'm frantic about my book, stuck on a question about stereotypies & environment & the brain, etc.)

We're going away for the night, so I may not get to it now, either, but I've begun reading the Comments thread.

Like instructivist, I appreciate this parsing of the difference between the Singapore spiral and the US spiral:

Singapore's Framework is an Additive Spiral that Builds Topic Content Grade-by-Grade; NCTM's Framework Is a Repetitive Spiral Approach that Covers Similar Topics Across Grade Bands

Singapore
Spiral approach by content strand (additive)
Specific for each grade (K-6)
Clear, specific topics
Mathematically logical sequence

NCTM
Spiral approach by content strand (repetitive)
Within broad grade bands (K-2, 3-5, 6-8 )
Imprecise topics
Absence of boundaries and inclusive

The fluency idea has raised a question in my mind.

What's good about spiraling-with-mastery is the fact that after studying the same material 3 years in a row people remember it for the rest of their lives:

Studies show that if material is studied for one semester or one year, it will be retained adequately for perhaps a year after the last practice (Semb, Ellis, & Araujo, 1993), but most of it will be forgotten by the end of three or four years in the absence of further practice. If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.*^{(see below *)} Exactly what knowledge will be retained over the long-term has not been examined in detail, but it is reasonable to suppose that it is the material that overlaps multiple courses of study: Students who study American history for four years will retain the facts and themes that came up again and again in their history courses.

Practice Makes Perfect -- But Only If You Practice Beyond the Point of Perfection

by Daniel Willingham
American Educator

(When I mentioned this finding to my sister-in-law, who is a federal prosecutor, she instantly said, "That must be why law school is 3 years.")

I'm wondering whether learning to the point of fluency, as opposed to mastery-defined-as-percent-correct, would alter this finding.

The article is about overlearning, which the fluency people suspect is the same thing as fluency. However, I don't know whether the subjects in the "50-year retention" studies studies had overlearned the same material in each of the three years they studied it.

shoot the moon

EM's spiral feels very flat as you watch the progression from grade to grade. On the other hand, and quite by random, things seem to drop from the moon, with no foundation knowledge taught.

This is called enrichment.

Lynn G

Research for The Spiral?

I'm trying to find the original research that spawned spiraling curriculum.

I can't find it. Admittedly, I haven't looked all that hard.

I've pulled out my copy of Everyday Math's Teacher Reference Manual. Surprisingly, spiral, spiraling, and spiral curriculum are not in the index or the table of contents. In fact, I can't find the word spiral anywhere in the book. It must be there, but I'm not going to read it cover to cover just to locate it.

Anyway, the introduction makes several vague statements that seem as close to promoting a spiral as I can find. For example, here's a representative quote:

Students using Everyday Mathematics are expected to master a variety of mathematical skills and concepts, but not the first time they are encountered. Mathematical content is taught in a repeated fashion, beginning with concrete experiences. It is a mistake to proceed too quicly from the concrete to the abstract or to isolate concepts and skills from one another or from problem contexts. Students also need to "double back," revisiting topics, concepts, and skills, and then relating them to each other in new and different ways. (p. 3)

Why is it a "mistake" to proceed quickly to the abstract? This is a reference manual for 4th through 6th grade teachers. This is not kindergarten where I can see abstraction might be counterproductive. But why not get to the abstract pretty quick once the foundation is set?

The problem as I see it, both on the concrete-abstract concept, and the whole spiral thing (repeated encounters?) is that EM isn't telling me how they reach these conclusions.

They toss these statements out there, with no reference, no support, no research. Where is it? They can't have imposed this spiraling drive-by exposure thing on our teachers and students with no research, could they?

The manual continues:

. [R]epeated exposures to key ideas presented in slightly different contexts are built into the EM program. (p. 5)

What makes them think that a key idea should be presented through repeated exposures?

I don't see this as intuitive and I'm more than a little annoyed that I can't find even a real kernel of research that might support the whole spiral thing.

In our experience over the past five years with EM, it is the spiral that is the most pernicious and destructive aspect of EM.

For example, the manual states that the square root symbol is introduced for the first time in 4th grade. Students need this exposure because of the symbol's "unfamiliarity and peculiar look."

By the time my daughter got to 5th grade, she had forgotten that she had ever known what the symbol meant. This is just one illustration of how the exposures are inefficient. There seems to be nothing to gain by introducing a square root symbol in the 4th grade when they won't be using it until the 5th grade. There was no time saving at the 5th grade level. A year is a long time to keep in the memory a concept you were exposed to, but never used or mastered.

So I'm left with my initial inquiry, is there some research somewhere that says kids learn better when they have repeated exposures to a concept before they are expected to master it?

the cost of spiraling

in Canada:

Reading and math are the two crucial elementary school subjects required for high school and life beyond, but British Columbia's elementary math curriculum is crippling learning, especially among disadvantaged students.

B.C. has used what is called a "spiral" curriculum since 1987, following a tradition of emulating U.S. educational practice.

A spiral curriculum runs a smorgasbord of math topics by students each year, the idea being that they pick up a little more of each with every pass. In reality, the spin leaves many students and teachers in the dust.

Ideally, the curriculum should cover fewer topics per year in more depth.

Presently, teachers face having Grade 4 classes who still cannot add 567 + 942 nor multiply 7 x 8 because the Grade 1, 2, and 3 teachers were forced to spend so much time on graphing, polygons and circles, estimating quantity and size, geometrical transformations, 2D and 3D geometry and other material not required to make the next step, which is 732 x 34.

And because elementary math fails to provide a solid foundation, many basically capable students simply give up when faced with the shock of high school algebra, which would be the doorway to advanced technical training at all levels. High school math teachers cannot make up Grades 1 to 7 while teaching Grade 8.

Alarm bells about the math curriculum have been ringing in B.C. since the United States, which used spiralling almost exclusively, registered a dismal performance on the Third International Mathematics and Science Study (TIMSS), a test that comparatively evaluated more than 500,000 students from 15,000 schools in 40 countries, first in 1995 and again in 1999 with the same results.

The B.C. ministry of education, to its credit, realized right away in 1995 that the U.S. performance on TIMSS suggested weaknesses in B.C.'s curriculum.

Also aware of some then-emerging data indicating that students in Quebec -- which had retained a sequential curriculum when B.C. went to the spiral -- were outperforming other Canadian students in math, Victoria commissioned researcher Helen Raptis, now a University of Victoria professor, to compare B.C. and Quebec test results and curricula.

In her report, submitted to the ministry in late 2000, Raptis showed that the average B.C. student was more than two years behind the average Quebec student in math by Grade 10, and explored the extent to which curriculum might be responsible.

source:
Things Don't Add Up in B.C. Math Classes
by Bill Hook and Karen Litzcke
NYC HOLD

My own district philosophically opposes acceleration for gifted students and does not discuss time costs of curriculum and pedagogy with parents.

I've been copied on emails from administrators stating that they do not feel it is important for students to "rush" through the curricula.

As a friend of mine said, "What is rush? There's rush as in race around frantically, and then there's rush as in you've got some place to go."

School defines rush.

Not parents.

Although I think this may begin to change.

KIPP & the spiral

from Making Schools Work School-by-School reform:

[Hedrick] Smith: Okay. Does KIPP have a fifth grade math curriculum?

[Mike] Feinberg: No, KIPP does not have a fifth grade math curriculum; it has a fifth grade math philosophy, it has a fifth grade math scope and sequence but not a curriculum. We realized early on that trying to view the solution as reinventing the wheel and creating a brand new curriculum didn't make a lot of sense. There're a lot of smart people in this country who've already spent a lot of time working on what is good curriculum at first grade, fifth grade and ninth grade. The issue is not that we don't have good curriculum; the issue is that we're not getting the kids to learn it.

Smith: But what's that all about then, getting the kids to learn?

Feinberg: Getting the kids to master the material.

Smith: No, I understand that but what's the key to that? If the curriculum is reasonably good, then what's the key?

Feinberg: Instructional delivery, being very good at teaching in front of the room, very good at using those resources. Being very good at assessing the students and where they are and re-teaching and whatever, doing whatever is necessary to get the kids to really, truly master the material.

You know, talk about curriculum, if I put in front of you a fifth, sixth, seventh, and eighth grade textbook in math and opened up to page 200 and I jumbled them up, and said, “order them from fifth through eighth grade in order,” you'd have a very tough time because they all look the same. That's because, unfortunately, we have this national strategy of “we're not really going to teach to master, we're going to teach to exposure and over lots and lots of years of kids seeing page 200 in the math book, eventually somehow they're going to learn it. We're going to teach them how to reduce fractions in fifth grade, in sixth grade, in seventh grade, in eighth grade, in ninth grade and continue until finally somehow magically they're going to get it.” Instead of thinking, “let's teach the kids how to reduce fractions at a mastery level in fifth grade, maybe spend a little time reviewing it in sixth grade but let's move on to pre-algebra and let's move on to algebra then.” And that's been our take and so it's not that we have a different math curriculum as much as we have a different math strategy and a different math philosophy.

the semiotics of PBS

This passage tells you why the media continues to carry articles on the need for progressive education to replace ineffective traditional practices:

[Feinberg]: The issue is not that we don't have good curriculum; the issue is that we're not getting the kids to learn it.

Smith: But what's that all about then, getting the kids to learn?

Feinberg: Getting the kids to master the material.

Smith: No, I understand that but what's the key to that? If the curriculum is reasonably good, then what's the key?

The interviewer, whose company produced this program, does not know that teaching to mastery is a practice unique to a handful of schools like KIPP.

Nor is he aware of the term "spiral curriculum" and its meaning.

The number of education writers in this country who know and understand these terms is very small. Jay Mathews, Andrew Wolf, Linda Seebach in Colorado, Debra Saunders in California....

What names am I missing?

Google is very confusing

Levin co-authored KIPP Math, a comprehensive fifth- through eighth-grade math curriculum that culminates in students completing a two-year high-school Algebra I course by the end of eighth grade.

So now I've heard 3 things about KIPP Math:

• they use Saxon Math
• they combine Saxon Math with Everyday Math
• they have a new constructivist teacher who invented her own curriculum in just one school year
• David Levin wrote "Kipp Math"

sigh

I'm thinking "KIPP Math" is probably a scope and sequence, not a curriculum.

Everything else is probably true.

The common thread is teaching math to mastery.