kitchen table math, the sequel: 12/16/07 - 12/23/07

Saturday, December 22, 2007

a question most 8th graders miss

Maria has $5.00 more than Joseph. Together they have $37.50. Which of these equations would you use to find the amount of money Joseph has?

A. j + (5 x j) = $37.50

B. j + ( j ÷ 5) = $37.50

C. 5 x j = $37.50 + j

D. 2 x ( j + 5) = $37.50

E. j + j +5 = $37.50

This item ... is missed by the majority of eighth-grade students in the NWEA norm group.
The Proficiency Illusion
John Cronin, et al
October 2007

I'm going to guess C. would get this right.

We'll see.

Of course, this is a classic bar model problem in Singapore Math. Third grade level, I believe, although you don't use variables. Still, the bar model is as abstract as a letter variable, or close to.

today's factoid

How many teachers in the U.S.?

3,000,000 according to Gadfly

Direct Instruction post at Fordham

check yourself before you wreck yourself

Fordham draws the familiar analogy between scripting in medicine and scripting in education: do you want your physician to be inventing new concepts in hand-washing on the spot or do you want him to do it the way the Best Practices checklist says to do it?

What often goes unremarked in these discussions is the fact that master teachers and professors often write their own scripts. These scripts will be revised, edited, and polished over the years, but they are scripts nonetheless. Teachers of older students and professors who teach seminars write questions and create discussion structures they repeatedly use. Here, for instance, is a terrific set of videos of a professional development session in which the presenter explains how to write one's own DI questions. Again, these questions and structures are revised, edited, and polished from one semester or school year to the next, but the fact is that superb teachers and professors don't wing it. Nor do they reinvent each course from scratch each year.

Creative people use scripts.

Another point: creativity comes in many forms. There is no reason to assume that a 1st grade teacher following a script could be replaced by a robot following the same script. Here is Ken on the question of scripting: (you may have to hit refresh to bring the page up)

Inevitably, whenever Direct Instruction (DI) is discussed the subject of “scripting” is raised. One frequent objection is that the scripts stifle teacher creativity. Nothing could be further from the truth.

Before we jump in, let’s first look at some sample scripts. Here’s a sample script on Writing Fractions. Here’s one on Subtraction. And, here’s another. (pdf files)

In DI, teachers use pre-designed scripts when teaching. The scripts are based on extensive research regarding student retention, and every aspect of every script is based upon results that were demonstrated through research. The great advantage of this approach is that every teacher using the script becomes the beneficiary of that research and will probably teach much more effectively than if left to her own devices.

DI designers test the programs carefully before publishing them and each DI program is extensively revised based on specific student error data from the field test. Scripting the lessons allows sharing of these “polished stones” across teachers. Also scripting helps reduce the amount of teacher talk. Children learn by working through the sequence of tasks with carefully timed comments from the teacher. Children learn little from straight teacher talk. Too much teacher talk decreases pupil-motivation, draws out the lesson length unnecessarily, and often causes confusion by changing the focus of the tasks, disrupting the development of the larger generalization, of which a teacher the first time through is usually unaware.

Also, without guidance, teachers may use language that students do not understand or that distracts students’ attention from examples. Scripts also allow aides, parents, and other paraprofessionals to assume teaching responsibilities, resulting in increased quality instructional time.

Moreover, even though the DI programs are carefully tested and scripted, successful use of them requires training in the special techniques of delivery. Teachers must make many decisions in response to the children's performance. Some of the most important decisions involve placing each child appropriately and moving the children through the lessons at a pace that maximizes their learning potential.

Lastly, the scripted presentations do not comprise the whole lesson, and the lessons do not comprise the whole school day. There are opportunities for group and independent work. A good DI teacher also creates additional activities that allow students to make use of their learning in various situations. So, there is a great deal of teacher creativity involved in the interpretation and presentation of the script, in attending to the needs and progress of all students and in designing expansion activities.

why is scripting used in Direct Instruction?
big concepts in DI
Explicit Direct Instruction professional development videos (brief - very worthwhile)

Carol Gambill in a nutshell

seamless (w)holes

from instructivist:

[Carolyn once said that math was "a seamless whole" inside her head,...]

I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these concrete skills, math is more like a seamless black hole.

This became apparent to me again when teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.

The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!

So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.

This one's going in the Greatest Hits file. (on the sidebar)

wholes, not parts
top down teaching
whole math taught wholly

book club

We may need to form a book club for interested folks. I say this selfishly because I need people to help me brainstorm my way through Karen Pryor's Don't Shoot the Dog, especially now that I'm hyper-aware of how difficult it is to generalize newly-acquired knowledge.

Pryor's book is brilliant. Reading it I see that I've been barking up the wrong tree thinking cognitive science has the answers. (barking?) Cognitive science does have answers; it's a valuable and riveting field. But when it comes to education the behaviorists are way out in front. Direct Instruction, Precision Teaching, Don't Shoot the Dog -- they're so far out in front they're disappeared from view.

No wonder their work is no longer taught in ed schools. (scroll down)

life-altering factoid number 1

Chickens procrastinate.

Using either fixed or variable schedules, extremely long sequences of behavior can be trained. A baby chick can be induced to peck a button a hundred times or more for each grain of corn. For humans there are many examples of delayed gratification. One psychologist jokes that the longest schedule of unreinforced behavior in human existence is graduate school. [ed.: followed closely by middle school]


Another phenomenon occurs on very long schedules: slow starts. The chick pecks away at a steady rate once it gets started, because each peck brings it nearer to reinforcement, but researchers have noted that a chick tends to "put off" starting for longer periods as the schedule of reinforcement gets longer.

This is sometimes called delayed start of long-duration behavior, and it's a very familiar aspect of human life. On any long task, from doing the income taxes to cleaning out the garage, one can think of endless reasons for not starting now. Writing, even sometimes just the writing of a letter, is a long-duration behavior. Once it gets started, things usually roll along fairly well, but, oh! it's so hard to make oneself sit down and begin, James Thurber found it so difficult to start an article that he sometimes fooled his wife (who was understandably anxious for him to write articles since that was how the rent got paid) by lying on a couch in his study all morning reading a book in one hand while tapping the typewriter keys at random with the other.

Don't Shoot the Dog, Revised Edition, p. 25

palisdesk on change in ed school curriculum
The Misbehavior of Organisms
Marian Breland Bailey: many lives (pdf file)
book club
a heroine I didn't know I had

Friday, December 21, 2007

Rote Learning in Core Knowledge, An Example

Last week, my son came home from school with a study sheet for the last big test for the year. The test covered some basic US geography, including the names of the Great Lakes, some facts about the Mississippi River, some facts about the US Flag, and locating some major geographical features. In addition, the students were supposed to be able to name at least 25 of the states, given a list showing only the first letter of each state.

Over the next one-and-a-half weeks, A. and I studied for his test about 15 minutes each night. This studying was the exactly the sort of studying that ed. schools teach as being the most harmful: pure rote memorization.

Early in this process, A. objected to the continued practice of the entire list on the basis that, "I only have to know 25 states, not all the states." My sympathy was notably limited; we studied the whole list. 8-)

By a couple of days before the test, A. was pretty reliably naming all the states starting with a given letter when prompted with that letter. (BTW, "M" is particularly annoying.) By this point, he was starting to think that getting the entire list right was pretty cool.

The day before the test, the teacher announced to the class that any child who could name all the states would get extra credit and a small prize.

Of the 26 kids in the class, 8 named all the states on the test the next day. None of those kids needed to have his or her self-confidence artificially boosted after the test, and they all now have a much better understanding of the value of hard work.

Thursday, December 20, 2007

fight the power

Math is hard.

Didn't Barbie say that?

Mr. Percent

I'm having way too much fun.

how to pronounce ISEE

I - see

Now you know.

On the other hand, if you go around calling the ISEE the I-S-E-E, people do know what you're talking about.

personal narrative

from Tracy W:

Okay, I love my calculator. Sharp EL-5120. It's on my desk at the moment. It's not much to look at, but its functionality means that it rocks my world. In terms of calculator-adoration I am probably in the top 1% of the world's population. My calculator has literally travelled around the world with me (there's no way I'd trust it to any removal company). I'm not a poet, but if I was I would write love poems to my calculator. The only reason I do not sleep with my calculator is that I fear it will disappear down the end of the bed and I will never see it again. When it comes to using calculators, I strongly suspect I am not normal. However, despite my deep and undying affection for my calculator I am sometimes without it, and on those occasions it is useful to be able to do basic arithmetic such as long division with pencil and paper or in my head. This may not be normal, but why should we educate kids merely to be normal people anyway?


SAT problem

YouTube rules.

I love this guy

But I still need a percent chart.

help desk - percent chart

Is there an all-around mode of charting the values in percent problems similar to the charting taught for distance problems?

this one's fun

During a sale, a bookstore sold 1/2 of all its books in stock. On the following day, the bookstore sold 4,000 more books. Now, only 1/10 of the books in stock before the sale are remaining in the store. How many books were in stock before the sale?
SSAT & ISEE 2007 Edition Kaplan

p. 161

C. came pretty close to doing this one on his own.

Of course, close doesn't cut it on a standardized test.

I realize that.
This problem can't be done.


Not enough information

Or way too much information, as the case may be.

Two trains are loaded with equal amounts of rock salt and ball bearings. Train A leaves Frogboro at 10:00 A.M. carrying 62 passengers. Train B leaves Toadville at 11:30 A.M. carrying 104 passengers. If Train A is raveling at a speed of 5 mph and makes four stops, and Train B is traveling at an average speed of 86 mph and makes three stops, and the trains both arrive at Lizard Hollow at 4:30 P.M., what is the average weight of the passengers on Train B?

Kaplan SSAT & ISEE 2007 edition
p. 155

Tuesday, December 18, 2007

Center for Environmental Therapeutics

A psychiatrist friend of mine recommends this web site. She says it has a self-administered diagnostic test for seasonal affective disorder that tells you exactly when to use light therapy.

We have enormous, aging light boxes all over the basement, but we've never known when to use them. We got them because John (Ratey) told me he'd visited the NIMH and all the shrink researchers there had light boxes on their desks. That was enough for me. We used to train one on Jimmy every day.

Today's Times has an article. (sorry - this is a paid subscription link, I believe - can't get the Times link generator pages to open.)

In 2001, Dr. Thomas A. Wehr and Dr. Norman E. Rosenthal, psychiatrists at the National Institute of Mental Health, ran an intriguing experiment. They studied two patient groups for 24 hours in winter and summer, one group with seasonal depression and one without.

A major biological signal tracking seasonal sunlight changes is melatonin, a brain chemical turned on by darkness and off by light. Dr. Wehr and Dr. Rosenthal found that the patients with seasonal depression had a longer duration of nocturnal melatonin secretion in the winter than in the summer, just as with other mammals with seasonal behavior.

Why did the normal patients show no seasonal change in melatonin secretion? One possibility is exposure to industrial light, which can suppress melatonin. Perhaps by keeping artificial light constant during the year, we can suppress the “natural” variation in melatonin experienced by SAD patients.

There might have been a survival advantage, a few hundred thousand years back, to slowing down and conserving energy — sleeping and eating more — in winter. Could people with seasonal depression be the unlucky descendants of those well-adapted hominids?

Regardless, no one with SAD has to wait for spring and summer to feel better. “Bright light in the early morning is a powerful, fast and effective treatment for seasonal depression,” said Dr. Rosenthal, now a professor of clinical psychiatry at the Georgetown Medical School and author of “Winter Blues” (Guilford, 1998). “Light is a nutrient of sorts for these patients.”

The timing of phototherapy is critical. “To determine the best time for light therapy, you need to know about a person’s individual circadian rhythm,” said Michael Terman, director of the Center for Light Treatment and Biological Rhythms at the Columbia University Medical Center.

People are most responsive to light therapy early in the morning, just when melatonin secretion begins to wane, about eight to nine hours after the nighttime surge begins.

How can the average person figure that out without a blood test? By a simple questionnaire that assesses “morningness” or “eveningness” and that strongly correlates with plasma melatonin levels, according to Dr. Terman.

The nonprofit Center for Environmental Therapeutics has a questionnaire on its Web site (

Once you know the optimal time, the standard course is 30 minutes of fluorescent soft-white light at 10,000 lux a day.

It may sound suspiciously close to snake oil, but the newest promising therapy for SAD is negative air ionization. Dr. Terman found it serendipitously when he used a negative ion generator as a placebo control for bright light, only to discover that high-flow negative ions had positive effects on mood.

Now that is exciting. I've been interested in negative ions forever. Negatives ions probably explain why it's impossible to be depressed on the beach.

Santa may be bringing me a negative ionizer for Christmas.

Brought on by Darkness, Disorder Needs Light
December 18, 2007
Brought on by Darkness, Disorder Needs Light

Shadow Syndromes

Monday, December 17, 2007

Enhancing Academic Motivation

I've just ordered Enhancing Academic Motivation from Research Press thanks to a friend of mine whose child is seeing Dr. Brier. Every word out of Dr. Brier's mouth so far has rung true.

When I discovered that Dr. Brier had published with Research Press, I was sold. Research Press published the two books that shepherded Ed and me through our first years with Jimmy: Gerald Patterson's Living with Children and Wesley Becker's Parents Are Teachers. Both are classics.

Wes Becker worked with Engelmann on Project Follow-Through:

During the Project’s third year, we found out that Carl was leaving to go to Canada and become an investigator for the Ontario Institute for Studies and Education and a professor at the University of Toronto. He invited Valerie and me to join him. Valerie accepted; I tentatively declined.

Carl’s impending departure presented serious problems to the preschool project. The reason was that I was not qualified to head the project. The only degree I had was a BA in philosophy, and the position I held then was Senior Educational Specialist, which did not allow me to administer projects. Neither Jean nor Cookie could assume directorship of the project because they also lacked formal credentials.

The rumors were that the Institute for Research on Exceptional Children would take over the project and change it as soon as Carl left. I later found out that Jean and Carl met with Wesley Becker, a gifted professor in the Department of Psychology. Their goal was to seek his help in preserving the project. I had heard a lot about Wes Becker from my sister-in-law, Geraldine Piorkowski, who earned her PhD at the University of Illinois. Wes was her advisor; from her descriptions of him I assumed he could even run on water. Among other achievements, he had set the all-time track record at Stanford for attaining a PhD, entering as a freshman and taking only six years to earn a PhD in clinical psychology and statistics.

At the time he advised Geraldine, Wes was a cognitivist, but shortly after she received her PhD, he became an energetic exponent of Skinner’s behaviorism, which is based on evidence that behavior may be changed by manipulating positive or negative consequences that follow responses. Wes abandoned his earlier orientation because it lacked data of effectiveness, a signature characteristic of Wes. The professional articles that Wes wrote in the '60s show his change in orientation from 1961 to '67: “Measurement of Severity of Disorder in Schizophrenia by Means of the Holtzman Inkblot Test” (1961); “A Circumflex Model for Social Behavior in Children” (1964); “The Parent Attitude Research Instrument” (1965); “How We Encourage Cheating” (1966); and “The Contingent Use of Teacher Attention and Praise in Reducing Classroom Behavior Problems” (1967).

I had met Becker only once. He had presented to our project staff and graduate students. He summarized his current research, which involved working with teachers in failed classrooms and teaching them techniques for using positive reinforcement with their students. His data showed that even though most teachers had to be instructed in how to give praise, and even though the praise some of them issued sounded contrived and unnatural, it changed students’ behavior. The basic thrust of Wes’s training was, “Catch kids in the act of being good.” His studies were among the first applications of Skinner’s version of behaviorism to humans and school settings.

After the meeting I told him about some of the observations we had made in the preschools. He listened, then asked, “Where’s the data?”

I told him I didn’t have any formal data related to the observations. He smiled and shrugged. The message this gesture conveyed was that if I wanted to demonstrate the validity of my assertions, I needed data.

Jean and Carl had set up their meeting to ask Wes if he would assume the role of director of our project. They didn’t have a chance to ask him, however. As they entered his office, he greeted them, and said, “I know why you’re here, and the answer is yes.”

I count this as one of the more amazing commitments a person could make. The project was embroiled in controversy. The work was demanding. By saying “yes,” Wes made an official break with the fortress of higher learning and moved to the trenches, the gritty realities of working with teachers and kids.

Wes brought some of his graduate and undergraduate students with him. Thirty years later, I still work with three of them: Doug Carnine, a shy undergraduate who had already authored articles that appeared in professional journals; Linda McRoberts, an adventurous and outspoken graduate student who later would become Linda Carnine; and Susan Stearns (now Susan Hanner), only nineteen years old but very smart and industrious.


Wes devoted some of his “free time” to writing the book, Applied Psychology for Teachers, a very ambitious work that covered everything related to effective practices and background information—from behavioral principles to the theoretical underpinnings of effective instruction and how to interpret data on student performance. I believe that Wes considered this book his ultimate achievement, an opus that positioned effective teaching and the analysis of learning in a framework that could be comprehended by undergraduates and that would establish DI as at least a contender in the field of education. The work, positioned as a textbook for undergraduates, was published by SRA in 1986. It was a colossal work—472 pages, in 8 1/2” x 11” format, with 275 references (over 11 pages).

It was another false hope. The book did not sell, was not adopted by more
than a handful of the faithful, and after only a few years, was discontinued by SRA. No other publisher was interested in it. I recently bought a copy of it online. It was in very good condition and cost $4.50.


Goodbye to a Good Guy

Wes and Julie were divorced in1980. Wes continued in his role as associate
dean until 1992, when he became involved in political wars with the College of Education and quit the University. After retiring, he refused to talk about education. On three or four occasions, I tried to discuss the book we had started. I got the same response each time. He said that he would talk about golf or other sports and the stock market, but that was all. He declined to talk about the Association for Direct Instruction, or about anything else related to education. He told me, “That is something from a past life. It’s dead and I have no interest in it.”

In 1993, Wes sold his shares in Engelmann-Becker Corp. and moved from
Eugene to Sedona, Arizona. There was no going-away party or celebration
because he didn’t want one. Just before he left, I asked if there was anything I could do for him. He asked if I would give him a painting I had done of a lion. Yes.

I called him several times in Arizona to see how things were going. Not well. I called him once around noon and he sounded as if he’d been drinking. His leg had gone bad so he couldn’t play golf, and the stock market had not been kind to him. His son David lived with him for a while but left. Wes never remarried and lived alone. A couple of times I asked him when he was coming back to visit us in Eugene. He seemed to entertain the idea but it apparently didn’t make the seriousplanning list. I never saw Wes again after he moved to Arizona.


Wes’ death came as a great shock. I hadn’t been in touch with him for months.
I knew he was getting frequent tests, but I had no idea that he would die at 73. We felt we should do something to honor him and decided to hold a memorial service for him in Eugene. We put a notice in the paper, made many calls, and arranged to hold the service in the church that Wes had attended (the Unitarian Church). A lot of people showed up for the service, including Don Bushell, Wes’ daughter Jill (who is a professor of biopsychology at the University of Michigan), his son David, and his ex wife, Julie (who lived in Florida). We took turns telling Wes stories and feeling sad.

I said, “Those who worked with him were routinely amazed, not only by his
skill, but the speed with which he could do things. Perhaps his most impressive quality, however, was the strength of his will. In the face of terrible setbacks and impossible deadlines, Wes prevailed. If he promised to get something done by a particular time, it was not only done on schedule, but done very well.”

Several others echoed this observation. One researcher who studied under
Wes said something that I had observed many times, the amazing speed at which Wes could identify glitches in raw data or elaborate calculations. About the time I was looking at the first few numbers on a spreadsheet of data, Wes would point to a set of scores in the middle of the display and say something like, “It’s impossible for them to have a correlation of point 9 with these data. These scores account for no more than 5 percent of the variance.” Possibly a minute later, I would see what he meant, but if I’d figured it out on my own, it would probably have taken closer to an hour.

I pointed out that even with the incredible number of things he had to do,
Wes was a good dad (a lot better than I was during the Follow Through years). Wes’ daughter Jill expanded on this theme. She told about some of the nice things he had done and indicated that the only time he lied to her was a couple of months before he died. She had visited him in a hospital in California. The last thing she said before leaving was, “Now, you take care of yourself. I’ll be back in three months.”

He said, “I’ll be fine.”

The clinical causes of Wes’ death had to do with his liver, kidneys and blood
pressure. One of the contributing causes was that he probably drank too much. These may have been the measurable causes, but the psychological cause was that he killed himself. When the establishment rejected Wes and his beliefs in data, he rejected education. To do that, he had to reject a huge part of himself. The image of himself that he had to maintain afterwards was one with many amputated parts, the hollow core that could survive on what had been peripheral interests. When his physical health failed, he had nothing.

The sad part of this equation was that Wes had to reject himself not because
he did anything reprehensible but because the establishment made a mockery of his beliefs and accomplishments. Jill believes that someday he will be recognized for his singular contribution to Follow Through. I hope she’s right, but I can sympathize with Wes. It is not very comforting to know that you can help thousands of kids and teachers, but you lack credibility and have no access to these victims. It hurts to see your professional beliefs trampled by educators who cling desperately to myth and folklore.

A colleague recently showed me a picture from the ‘70s, taken at a “Zignic” (a
picnic at the Veneta property for all our trainers and friends). Six people, including Wes, Bob, and I, were wearing t-shirts with the motto, “Show me the data.” For Wes, it was a way of life.

How is Wes remembered? In 2003, the College of Education at the University of Oregon launched a fund-raising campaign to support construction of a mega-building to house the College. Part of what the planners did was to make up a price list for “dedications.” If you want an office named after somebody, donate $25,000, and the plaque goes up. For a decent-sized classroom, the ticket is about $100,000.

Shortly after the list came out, Doug called me about raising enough money to have a classroom named after Wes. I told him that we shouldn’t have to pay anything. My feeling was that the College should have dedicated an entire wing to Wes, with no donation required. The College didn’t see it that way. Doug is currently trying to negotiate the price of a plaque for both Wes and Bob at the entrance to the Clinical Services Building, which was one of Bob’s projects.

Siegfried Engelmann 2007

Parents Are Teachers table of contents
Living with Children table of contents

Rye Country Day


School says that since 1996, 97% of kids taking the Advanced Placement BC Calculus course received perfect scores on the AP exam.

How the Schools Stack Up (pdf file)

Sunday, December 16, 2007

still can't do fractions?

Tom Loveless is back:

The 2007 NAEP test results showed small but statistically significant gains in both math and reading. Mathematics scores at fourth and eighth grade continued the steady progress registered since the main NAEP test was first administered in 1990. Both grade levels notched 2 point gains in scale scores. Table 1-1 reports the magnitude of the math gains in scale score points and years of learning. Figure 1-1 illustrates the upward trajectory of the scores. The gains indicate that fourth and eighth graders in 2007 knew more than two additional years of mathematics compared to fourth and eighth graders in 1990. On the face of it, this is an amazing accomplishment. Previous Brown Center Reports have raised questions about such gains. The primary question concerns the content of the NAEP math tests. Students are clearly making progress, but at learning what kind of mathematics? Suffice it to say that students are making tremendous progress on the mathematics that NAEP assesses, in particular, problem solving with whole numbers, elementary data analysis and statistics, basic geometry, and recognizing patterns. NAEP pays scant attention to computation skills, knowledge and use of fractions, decimals, and percents, or algebra beyond the rudimentary topics that are found in the first chapter of a good algebra text. In sum, we know that students are getting better at some aspects of math. But we do not know how American students are doing on other critical topics, including topics that mathematicians and others believe lay the foundation for the study of advanced mathematics. Thus, the years of learning gain must be taken with a grain of salt.

The 2007 Brown Center Report on American Education:
How Well Are American Students Learning

plus ├ža change (scroll down)