kitchen table math, the sequel: 2/17/13 - 2/24/13

Friday, February 22, 2013

More slave math

Watching local news tonight.

Today's story: parents being interviewed about a new slave math scandal in school.

The mom who is head of School Council, trying to go easy on the teachers involved, said she thinks it's a very good idea to bring "social studies" into math.

Which reminded me of Histogeomegraph and the tragedy of content isolation. And of the middle school model. And of my own middle school, back when they were bringing in the middle school model here.

With the middle school model, we parents were told, wondrous connections amongst the disciplines would be made:
  • In math class, the teacher could point out that Pythagorus was a Greek 
  • In English class, the teacher could point out that the father in The Miracle Worker was a patriarchal male who is ashamed of his handicapped child
  • In science class, the teacher would take POINTS OFF!!!! for misspelling
The problem with "bringing social studies into math" is that a person who has spent a great deal of time and energy learning math has not spent a comparable amount of time and energy learning "social studies." So the social studies that gets 'brought in" is superficial (Pythagorus was a Greek), wrong ("ashamed of his handicapped child"), obnoxious (POINTS OFF!!!),* or, in the case of slave math, upsetting and weird.

From 1999, here's Nancy Granstrom:
During our eldest daughter's eighth-grade year in junior high school, I wrote a letter to the editors of the local newspaper praising the education in our school district. The teachers were highly knowledgeable, dedicated individuals, who imparted their expertise so well, and did an exceptional job of preparing students for high school. I was convinced that the Community Consolidated School District 21 in Wheeling, Illinois had cornered the market on how to truly deliver a quality education. Two years later, however, it was welcome to Jack London Middle School for our second daughter.

Our daughter had not been looking forward to all the spelling, vocabulary, history, pre-algebra, and reading of the classics she had seen her older sister do. It turned out that she did not need to worry because little of that took place at her middle school. The explanatory pamphlet from the district stated that middle school theory includes: "Integrated thematic instruction, cooperative learning, problem-based learning, multiple teaching styles, flexible block scheduling, and authentic assessment."

Under this system all teachers taught all subjects, even if they were not qualified to do so. The philosophy was and is that they should be able to teach anything. The reality is they have to, because the majority of the classes are extremely integrated.

A Parent Criticizes "Middle School Theory"
by Nancy L. Granstrom
Funny thing here in my district: we had two years of strife over the middle school model, then they brought in the middle school model, and then...nothing.

Nobody ever heard a word about it again, and except for some scheduling changes, things seemed to be pretty much business as usual.

Two or three years later, they brought in Lucy Calkins' reading workshop.


* I say "obnoxious" because the school was not teaching spelling at all. C.'s spelling was so bad that I had purchased a spelling curriculum I was using at home, and now, in the name of the middle school model, we were being told that the only help our kids would receive in the spelling department would come in the form of a cranky science teacher, nearing retirement, punishing the kids into better spelling by taking POINTS OFF!!! (That's what we always called it, back in the day: POINTS OFF!!!)

I wish now I'd kept a journal of those years. Ed attended that meeting....

Oh, wait! It's coming back to me.

Ed of course was strenuously lobbying against adoption of the "middle school model," and was playing a prominent and vocal role in the parent meeting convened to educate us in the nature of the coming changes. So when the science teacher said her contribution to Subject Integration was going to be taking POINTS OFF!!! Ed said forcefully & without a trace of irony, as if the POINTS OFF!!! plan explained everything (which it did), "That's very helpful to know. The science teacher will integrate English and science by docking points on science tests for spelling errors. That's the kind of information we need."

He told me later the science teacher glared at him.

Really, the best parents can hope for is to get these people's goat every once in a while.

Thursday, February 21, 2013

online courses: "decrements" in performance for all students

Using a dataset containing nearly 500,000 courses taken by over 40,000 community and technical college students in Washington State, this study examines how well students adapt to the online environment in terms of their ability to persist and earn strong grades in online courses relative to their ability to do so in face-to-face courses. While all types of students in the study suffered decrements in performance in online courses, some struggled more than others to adapt: males, younger students, Black students, and students with lower grade point averages. In particular, students struggled in subject areas such as English and social science, which was due in part to negative peer effects in these online courses.
Adaptability to Online Learning: Differences Across Types of Students and Academic Subject AreasDi Xu and Shanna Smith JaggarsFebruary 2013CCRC Working Paper No. 54

all MOOC all the time

The Times ran a terrific editorial on MOOCs yesterday:
The Trouble With Online College
Published: February 18, 2013

Stanford University ratcheted up interest in online education when a pair of celebrity professors attracted more than 150,000 students from around the world to a noncredit, open enrollment course on artificial intelligence. This development, though, says very little about what role online courses could have as part of standard college instruction....

...First, student attrition rates — around 90 percent for some huge online courses — appear to be a problem even in small-scale online courses when compared with traditional face-to-face classes. Second, courses delivered solely online may be fine for highly skilled, highly motivated people, but they are inappropriate for struggling students who make up a significant portion of college enrollment and who need close contact with instructors to succeed.

Online classes are already common in colleges, and, on the whole, the record is not encouraging. According to Columbia University’s Community College Research Center, for example, about seven million students — about a third of all those enrolled in college — are enrolled in what the center describes as traditional online courses. These typically have about 25 students and are run by professors who often have little interaction with students. Over all, the center has produced nine studies covering hundreds of thousands of classes in two states, Washington and Virginia. The picture the studies offer of the online revolution is distressing.

The research has shown over and over again that community college students who enroll in online courses are significantly more likely to fail or withdraw than those in traditional classes, which means that they spend hard-earned tuition dollars and get nothing in return. Worse still, low-performing students who may be just barely hanging on in traditional classes tend to fall even further behind in online courses.

A five-year study, issued in 2011, tracked 51,000 students enrolled in Washington State community and technical colleges. It found that those who took higher proportions of online courses were less likely to earn degrees or transfer to four-year colleges. The reasons for such failures are well known. Many students, for example, show up at college (or junior college) unprepared to learn, unable to manage time and having failed to master basics like math and English.....
O brave new world!

more MOOC

Letters to the Times

The President of Dickinson College has this to say:
MOOCs are separate from that highly desirable and precious residential degree. Online courses represent auxiliary income to support bricks and mortar and to increase brand recognition globally. New wine for an old bottle.

The high cost of a degree at M.I.T. — or most universities — is not lowered. That would be a revolution.

Devlin's Lament: the symbol barrier

(Cross-posted at Out In Left Field)

In an article in the most recent issue of American Scientist entitled "The Music of Math Games," Keith Devlin (head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University and NPR's "math guy") says that learning math should be like learning to play the piano. In doing so, he recalls (but does not credit) Paul Lockhart's Lament ("A piano student's lament: how music lessons cheat us out of our second most fascinating and imaginative art form"), which I blogged about here.

Though Devlin is no literary virtuoso, not all of what he writes here is mushy metaphor. He begins with a discussion of educational software, and here his points are clear and consistent with my own experience. Most "math games" and "math education" software programs I've seen don't make mathematics an organic part of the games or activities. Instead, math problems--mostly arithmetic problems of the "mere calculation" variety--are shoe-horned into non-mathematical situations. Here they serve simply as tasks you must complete before moving through the current non-mathematical activity or on to the next non-mathematical activity.

As Devlin writes:
To build an engaging game that also supports good mathematics learning requires... understanding, at a deep level, what mathematics is, how and why people learn and do mathematics, how to get and keep them engaged in their learning, and how to represent the mathematics on the platform on which the game will be played.
The same is true of language learning. Most linguistic software taps only superficial aspects of language, and, as I know from personal experience, it takes great effort to build a program that does more than that.

Where I begin to part ways with Mr. Devlin is in his discussion of traditional math and what he thinks is an excessive emphasis on symbols:
Many people have come to believe mathematics is the memorization of, and mastery at using, various formulas and symbolic procedures to solve encapsulated and essentially artificial problems. Such people typically have that impression of math because they have never been shown anything else...
By and large, the public identifies doing math with writing symbols, often obscure symbols. Why do they make that automatic identification? A large part of the explanation is that much of the time they spent in the school mathematics classroom was devoted to the development of correct symbolic manipulation skills, and symbol-filled books are the standard way to store and distribute mathematical knowledge. So we have gotten used to the fact that mathematics is presented to us by way of symbolic expressions.
This approach to math, Devlin suggests, is at odds with the resolutions of a "blue-ribbon panel of experts" serving on the National Research Council’s Mathematics Learning Study Committee ("Adding it Up: Helping Children Learn Mathematics," National Academies Press, 2001). In Devlin's words: these resolutions hold that math proficiency consists of:
the aggregate of mathematical knowledge, skills, developed abilities, habits of mind and attitudes that are essential ingredients for life in the 21st century. They break this aggregate down to what they describe as “five tightly interwoven” threads. The first is conceptual understanding, the comprehension of mathematical concepts, operations and relations. The second is procedural fluency, defined as skill in carrying out arithmetical procedures accurately, efficiently, flexibly and appropriately. Third is strategic competence, or the ability to formulate, represent and solve mathematical problems arising in real-world situations. Fourth is adaptive reasoning—the capacity for logical thought, reflection, explanation and justification. Finally there’s productive disposition, a habitual inclination to see mathematics as sensible, useful and worthwhile, combined with a confidence in one’s own ability to master the material.
Ah, "21st century skills," "habits of mind," "conceptual understanding," "real-world situations," "explanation," "disposition"...--all this makes me wonder about the ratio of mathematicians to math eduation "experts" on this blue-ribbon panel. (It should be noted that Devlin himself is not, strictly speaking, a mathematician; he holds a Ph.D. in logic from the University of Bristol, and, while affiliated with Stanford, is not a member of the Stanford math department.)

Standing in the way of these lofty goals is what Devlin calls the "symbol barrier":
For the entire history of organized mathematics instruction, where we had no alternative to using static, symbolic expressions on flat surfaces to store and distribute mathematical knowledge, that barrier has prevented millions of people from becoming proficient in a cognitive skill set of evident major importance in today’s world, on a par with the ability to read and write.
To the rescue comes... Devlin's math education software program:
With video games, we can circumvent the barrier. Because video games are dynamic, interactive and controlled by the user yet designed by the developer, they are the perfect medium for representing everyday mathematics, allowing direct access to the mathematics (bypassing the symbols) in the same direct way that a piano provides direct access to the music.
Devlin's notion that a well-designed math video game can help students meet the National Academy's goals for math education rests on two assumptions. One is that students can achieve a sufficient level of mastery in mathematics without symbols. The other is that playing such video games is to math what playing the piano is to music.

To address the first claim, Devlin elaborates the analogy to music:
Just how essential are those symbols? After all, until the invention of various kinds of recording devices, symbolic musical notation was the only way to store and distribute music, yet no one ever confuses music with a musical score.
Just as music is created and enjoyed within the mind, so too is mathematics created and carried out (and by many of us enjoyed) in the mind. At its heart, mathematics is a mental activity—a way of thinking—one that over several millennia of human history has proved to be highly beneficial to life and society.
But there's an important difference between math and music--and a reason why no one confuses music with a musical score. Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreduceable, qualitative impression--an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.

Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be be translated into different representational systems without losing their essence and beauty.

Devlin argues that while we might write down symbols in the course of doing real-life math, it is primarily a "thinking process," and that "at its heart, mathematics is a mental activity—a way of thinking." I agree. Indeed, math is much more appropriately compared with thoughts than with music. But this makes math symbols the mathematical equivalent of linguistic symbols. While thoughts, like math, can be expressed in a number of different symbol systems, you need some sort of symbol system in order to represent your own thoughts and to understand the thoughts of others.

This is especially true of abstract thoughts--and of abstract math. As Devlin himself admits, "the advanced mathematics used by scientists and engineers is intrinsically symbolic. "What isn't intrinsically symbolic, Devlin claims, is "everyday mathematics":
The kind of math important to ordinary people in their lives... is not, and it can be done in your head. Roughly speaking, everyday mathematics comprises counting, arithmetic, proportional reasoning, numerical estimation, elementary geometry and trigonometry, elementary algebra, basic probability and statistics, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. (Yes, even elementary algebra belongs in that list. The symbols are not essential.)
OK, but what does this mean for education? Are we going to decide before the end of middle school which students are going to become scientists, engineers, and mathematicians, and only help those students scale the "symbol barrier"? For a barrier it certainly is, as Devlin himself notes: "people can become highly skilled at doing mental math and yet be hopeless at its symbolic representations."

But Devlin is too busy appreciating the (well-studied) math skills of Brazilian street vendors, who do complex arithmetic calculations in their heads with 98% accuracy, and supposedly without the help of symbols (even mental ones?), to realize the educational implications of the fact that "when faced with what are (from a mathematical perspective) the very same problems, but presented in the traditional symbols, their performance drops to a mere 35 to 40 percent accuracy." No, not everyone is going to become an engineer. But not all non-engineers are going to become Brazilian street vendors.

It's ironic how deeply Devlin appreciates the difficulty that "ordinary people" have with the symbol barrier without appreciating what this says about their educational needs:
It simply is not the case that ordinary people cannot do everyday math. Rather, they cannot do symbolic everyday math. In fact, for most people, it’s not accurate to say that the problems they are presented in paper-and-pencil format are “the same as” the ones they solve fluently in a real life setting. When you read the transcripts of the ways they solve the problems in the two settings, you realize that they are doing completely different things. Only someone who has mastery of symbolic mathematics can recognize the problems encountered in the two contexts as being “the same.”
Instead of seeing this as a reason for exposing children to mathematical symbols early and often, Devlin sees this as reason to create computer games that somehow teach math non-symbolically.

He calls this "adaptive technology," a term that should raise red flags. In a recent blog post, I wrote about how assistive technology often becomes yet another excuse not to teach basic skills. Kids with dyslexia struggle mightily with the symbol system of written language; should they instead learn everything through text-to-speech and speech-to-text devices, and never learn how to read and write?

Devlin makes a few other strained comparisons to the piano:
The piano metaphor can be pursued further. There’s a widespread belief that you first have to master the basic skills to progress in mathematics. That’s total nonsense. It’s like saying you have to master musical notation and the performance of musical scales before you can start to try to play an instrument—a surefire way to put someone off music if ever there was one.
No it's not; it's like saying you have to master simple scales and exercises before you move on to Rachmaninoff.
The one difference between music and math is that whereas a single piano can be used to play almost any tune, a video game designed to play, say, addition of fractions, probably won’t be able to play multiplication of fractions. This means that the task facing the game designer is not to design one instrument but an entire orchestra.
Can one create a video game that functions "as an instrument on which a person can 'play' mathematics?"
Can this be done? Yes. I know this fact to be true because I spent almost five years working with talented and experienced game developers on a stealth project at a large video game company, trying to build such an orchestra.
What does Devlin's software do? The last two paragraphs of this article function as an extended but not very informative infomercial. Here's the most informative excerpt:
Available in early March, Wuzzit Trouble is a game where players must free the Wuzzits from the traps they’ve inadvertently wandered into inside a castle. Players must use puzzle-solving skills to gather keys that open the gearlike combination locks on the cages, while avoiding hazards.
Puzzle solving? As I argue in my last post on math games, existing games already offer some version of this, and it isn't math. This, indeed, is one of the other problems with so-called math education software.

Devlin suggests his software is different:
Unlike the majority of other casual games, it is built on top of sound mathematical principles, which means that anyone who plays it will be learning and practicing good mathematical thinking—much like a person playing a musical instrument for pleasure will at the same time learn about music.

Wuzzit Trouble might look and play like a simple arithmetic game, and indeed that is the point. But looks can be deceiving. The puzzles carry star ratings, and I have yet to achieve the maximum number of stars on some of the puzzles! (I never mastered Rachmaninov on the piano either.) The game is not designed to teach. The intention is to provide an “instrument” that, in addition to being fun to play, not only provides implicit learning but may also be used as a basis for formal learning in a scholastic setting.
If you say so. But I wonder how much it will cost schools (and society) to find out whether this latest incarnation of "math education" software helps prepare students to become mathematicians, scientists, engineers--or Brazilian street vendors.

Tuesday, February 19, 2013

David Boulton interview with Engelmann re: Follow Through and Head Start

Siegfried Engelmann: Then we became engaged in Project Follow Through. There were originally eighteen sponsors, about 500,000 kids, 180 communities, and pulled-comparison groups. It was supposed to be the definitive educational experiment. The idea was to work with kids in K-3 who had gone through Head Start.

Our students were first place in everything, but the reports were never really presented. We were first place in reading, first place in math, first in spelling, and first in language. And our kids had the most positive self image. Yet the report that APT Associates had developed, along with Stanford Research Institute, was just a summary of those reports.

David Boulton: The purpose of the study was to be able to see the effects of Head Start on education?

Siegfried Engelmann: Yeah.

David Boulton: Why was 'Follow Through' linked into Head Start in that way?

Siegfried Engelmann: Well, because it was originally an Office of Economic Opportunity project. And then it was taken over by the newly formed Federal Department of Education. It was designed to serve at-risk kids, disadvantaged kids, who had gone through Head Start. But that was just one of the requirements for the demography of the kids we were working with.

David Boulton: I see.

Siegfried Engelmann: A certain percentage of the students had to come from Head Start. Because Head Start was an obvious failure and they were concerned. It had no instructional component, and it was modeled after the middle-class preschool. While the middle-class preschool is probably okay for middle-class kids, the kids that we worked with were far behind in terms of language skills and...

David Boulton: So it was more concerned with creating parental freedom than it was in actually helping the children get ready for school.

Siegfried Engelmann: Right, yes. Anyhow, that made it a poor model for disadvantaged students. But fundamentally, Project Follow Through was designed to bail out Head Start. It was a horse race, the idea [of the APT reports] was to declare a winner or winners, those who produced the best results in K-3, to show that Head Start was not a total disaster.

David Boulton: How could it have done that unless it was also using a control group of kids that weren't in Head Start to show the advantages of Head Start?

Siegfried Engelmann: Well, they had that. They had a vast number of comparison groups. For each school that was involved, there was a comparison school. They weren't perfect, because the comparison schools tended to have higher socioeconomic ratings. They were not as disadvantaged. But, in addition to that, the data from all of the individual comparison schools were pooled. Then there was a certain non-disadvantaged mix as part of the formulated average school. So you had your non-disadvantaged population, and also (I can't remember the exact requirement) I think over 60 percent of the kids had to have gone through Head Start. But they had data on the Head Start kids and the non-Head Start kids. It was a very elaborate study. It cost, I don't know, hundreds of millions.

David Boulton: So 'Project Follow Through' was a prototype – a model that would later be followed in many ways by the National Institute of Child Health and Human Development.

Siegfried Engelmann: Right, right. The APT findings were suppressed largely for political reasons. In 1976 when Follow Through was being evaluated, Gene Glass, head of the Ford Foundation at the time, appealed to the National Institutes of Health with an incredible statement. He said something to the effect that, "The use of quantitative data is inappropriate and what we need is case studies. We need to document various aspects of the program so that informed consumers can make intelligent decisions."

And of course, it was total baloney. Wes Becker responded, with what I thought was an extremely succinct response, "As the problem with the disadvantaged is identified by data and scores; certainly the solutions to the problems would have to be manifested with data and scores.”

David Boulton: Certainly it all has to correlate somehow…

Siegfried Engelmann: [laughs] Yeah. They wanted to identify the problem qualitatively, and then solve it with methods that didn't generate any data. Becker also pointed out that if we're going to use case studies, how do we know we're using typical case studies unless we use some kind of intelligent sampling processes?

David Boulton: Yeah, and some common system of attributes that would allow you to scale through the data.

Siegfried Engelmann: Right. So, the net result was that the results of Follow Through were suppressed. The report that came out on Follow Through was that the project was a failure, which implies that all of the models were failures. And then they just rode off into the sunset with some kind of blazing saddles and that was that.

David Boulton: So buried in the dismissal of Project Follow Through as a whole, were the results it had gathered that showed the benefit of the work that you were doing, in contrast to the other systems or approaches that were compared.

Siegfried Engelmann: Yeah. I mean, it was the biggest part. But the suppression was intentional. It was contrived. It didn't just happen. The fact that the whole project failed, that the overall statements of the primary sponsors were true, did not necessarily mean that every one of them failed. That certainly was not the case.

David Boulton: And so the baby went out with the bath water there.

Siegfried Engelmann: Yes.

Head Start is based on a "whole child" model

From the Third Grade Follow-up to the Head Start Impact Study:
Since its beginning in 1965 as a part of the War on Poverty, Head Start’s goal has been to boost the school readiness of low-income children. Based on a “whole child” model, the program provides comprehensive services that include preschool education; medical, dental, and mental health care; nutrition services; and efforts to help parents foster their child’s development. Head Start services are designed to be responsive to each child’s and family’s ethnic, cultural, and linguistic heritage.


The Head Start Impact Study is a comprehensive, carefully designed study of a large-scale early childhood program that has existed for more than 40 years. It is designed to address the overall average impact of the Head Start program as it existed in 2002.
Russ Whitehurst summarizes the findings:
There is no measurable advantage to children in elementary school of having participated in Head Start. Further, children attending Head Start remain far behind academically once they are in elementary school. Head Start does not improve the school readiness of children from low-income families.
Ed told me that when he was in college a professor of his, applauding Head Start, said: "We have to get them away from their families."

That's the social model.

Siegfried Engelmann never passed judgment on the families. He taught disadvantaged children to read, write, and do arithmetic, and he didn't presume that he could replace a mother or a father. 

Siegfried Engelmann on Head Start
Siegfried Engelmann teaches fractions to disadvantaged 5-year olds
One Strongly-Confirmed Impact on Math
Third Grade Follow-up to the Head Start Impact Study
Is Head Start Working for American Students?
Can We Be Hard-Headed About Preschool? A Look at Head Start

from 2007: Siegfried Engelmann teaches fractions to disadvantaged 5-year olds

Excerpt from Siegfried Engelmann's War Against the Schools' Academic Child Abuse:
In the summer of 1966, the Anti-Defamation League expressed interest in making a film showing the achievements of the disadvantaged black preschoolers we had been working with at the University of Illinois. Two years earlier, these kids had been selected for the project as four-year olds on the basis that they came from homes that were judged particularly disadvantaged and nearly all of them had older siblings in classes for the mentally retarded. These kids came to our school half-days as four-year-olds and as five-year-olds.

The school, The Bereiter-Engelmann preschool, received a lot of bad press. It was called a pressure cooker. Sociolinguists took shots at it on the grounds that we ostensibly did not understand "black English," or even know the difference between "thinking and speaking."

Despite our alleged mental deficiencies, we managed to teach these kids more and make them smarter than anyboy else had done before or after. That was our goal, particularly with this first flight of kids--to set the limits to show what could be done. We felt that this demonstration was particularly important because Headstart was looming in the wings, and it was clearly moving in a direction of being nothing more than a front for public health, not a serious educational project. We saw this as a great contradiction because disadvantaged kids were behind their middle class peers in skills and knowledge.

We taught reading, language, and math to our preschoolers. And they learned these subjects. They also learned to learn well and therefore how to be smart. A film showing what these kids could do might moderate what seemed to be the inevitable mandate of the Office of Economic Opportunity to designate Headstart as a "social experience" based on the model of the middcle-class nursery school. It seemed obvious that the model would not work.

We rounded up seven of the kids who were in our top group. (We grouped kids for instruction according to their performance.) They were in the middle of summer vacation, and we didn't have an opportunity to work with them before the film to "refresh" or rehearse them. A professor at the University of Illinois found out about the filming and asked if she could bring her class to view it. Why not?

So seven little black kids came into the classroom, sat in their chairs in front of the chalkboard with big bright lights shining on them, with two big cameras on tripods staring at them, and with a class of university students in the background. And these kids did it. There were no out takes, no cut sequences, nothing but the kids responding to problems that I presented, the types of problems I had taught them to work. These were not necessarily the problem types that one would present preschoolers as part of a 12 grade sequence, but they were good problems to show that these kids could learn at a greatly accelerated rate.

On the film, the kids worked problems of addition, subtrction, multiplication, and fractions. They worked problems in which they found the area of rectangles and problems in which they found the length of an unknown side of the rectangle (given the number of squares in the rectangle and the length of one side). They worked column-addition problems that required carrying and problems that did not require carrying. They even worked problems involving factoring expressions like 6A + 3B + 9C. And they used the appropriate wording: "Three times the quantity, 2A, plus 1B, plus 3C."

The kids told me how to work a simple algebra problem: "The man at the store tells you that 1/4 of a pie costs 5 cents. You want to buy the whole pie. How much is the whole pie?"

After telling me how to work the problem by multiplying the reciprocal of 1/4, I wrote the answer as $20. The kids jumped up to correct my sign error, one boy observed, "Wow, you have to pay that much for a pie?"

And the kids did dimensional analysis involving the equation: A + B = C. They told me how to rewrite the equation so it told what A equals (A = C - B), what B equals (B = C - A), and what C equals (C = A + B).

The last problem type I presented on the film was the simultaneous-equation problem:

A + B = 14
A - B = 0

They had worked on similar problems in which A and B were the same size (inferred from A - B = 0) and they quickly told me that the numbers were 7 and 7. There was still time left so I presented them with a brand new problem type:

A + B = 14
A - B = 2

I pointed out that when you start with A and minus B, you end up with 2. So A is bigger than B. They frowned, they thought; and finally the little girl sitting on the end of the group -- who is now an engineer -- said in a wee voice, "8 and 6." These were kids who had not yet entered first grade.

The film made no difference in deterring Headstart from becoming a program that produced no real gains. Nor did it give notice that failure with disadvantaged kids was a failure in instructional practices. We had shown , however, that all the disadvantaged black kids we worked with could learn to read and perform basic arithmetic operations in the preschool and that the average IQ gain of these kids was 24 points.
pages 1-3
I tear up every time I read this.

The children of the poor don't need lessons in good character.

They need knowledge.  Head Start

Film of Engelmann teaching preschoolers 
Original post

Monday, February 18, 2013

"How Social Science Research Can Improve Teaching"

We marshal discoveries about human behavior and learning from social science research and show how they can be used to improve teaching and learning. The dis- coveries are easily stated as three social science generalizations: (1) social connections motivate, (2) teaching teaches the teacher, and (3) instant feedback improves learning. We show how to apply these generalizations via innovations in modern information technology inside, outside, and across university classrooms. We also give concrete examples of these ideas from innovations we have experimented with in our own teaching.
How Social Science Research Can Improve Teaching
Gary King and Maya Sen
Haven't read, just passing the link along.

The first paragraph is not promising, though:
Humans have theorized how to teach for thousands of years and update the substance of what we teach almost every year. Yet generations have passed without any major im- provements in the procedures and style of teaching in our classrooms. If your great-great- grandparents went to college, they probably sat in a classroom with all the other students facing forward, trying to look attentive, while the professor professed. If you’re professor at a university today, you probably lecture to the same sea of students, all still trying to look like they’re paying attention. To be sure, you may use some newer technologies (electricity, radio, TV, whiteboards, powerpoint slides, etc.), you may have added a few group activities, and you perhaps teach a seminar with lots of discussion. But if your ancestors were to walk into a classroom today, they’d know where to sit, what to do, and how to act. Our methods of teaching have changed very little.
In fact, a large body of research on teaching, learning, and memory exists, and a precision teaching classroom does not look like an ordinary classroom. Moreover, the difference between "group activities" and lecture is quite large and should not be shrugged off.

Papers like these remind me of the old saying in Hollywood: Everyone's a writer.

When it comes to the schools, everyone's an expert.

It's never a good sign when authors take as self-evidently true the blanket assertion that schools today are the same as they have always been.

people on boats

Earlier today I had reason to quote the "Hell is other people" line to a friend of mine, who pointed me to this:
Hell is other people. Hell is other people on a boat. What will it take before we accept this? 
Passengers ill-suited for loss of cruise control
By Monica Hesse, Published: February 15 | Washington Post

Sunday, February 17, 2013

Calculating Weighted GPAs

Like most schools, ours uses an online system for entering and tracking grades (Aspen). I really like being able to verify and track grades. Some teachers are really slow, but with other teachers, I know my son's grade before he does. Considering the importance of grades and GPA in high school, this is much better than his middle school's mysterious rubrics and work that disappeared into black hole portfolios never to be seen unless you set up an appointment with each teacher.

The reason for this post started when I couldn't verify a quarter grade in one subject. (I know that many teachers hate the system and that you have to set up weird rules to have it do what you want.) My calculations showed that he should have gotten a 93 instead of a 92 for a grade in one quarter. Then, I found out that fixing that number would have no effect on his semester (half-year) grade or his weighted GPA. What was going on?

Our high school calculates a weighted GPA to use for class rank. Regular college prep courses have a weight of 3, honors classes have a weight of 3.4, and AP classes have a weight of 3.7. (The AP arms race this causes is a separate issue.) Course grades (by semester - half year) are multiplied by the weight and a credit score; 1 for a full year course and .5 for a half year course. The score is normalized using the sum of the credits. If you get a 100 in a full-year honors class, this would give you a weighted and normalized score of 100 * 3.4 = 340. The weighted GPA is summed for all courses and then divided by the sum of the credits. This means that weighted GPAs for students typically end up between 250 and 325.

The first problem is that the semester grade for each class is rounded to the nearest whole number before it multiplied by the course weight. OK, but the problem is that they round grades at every step of the way. First, each assignment and test is rounded to the nearest whole number (that's fine), and then they are combined together into groups called things like formative and summative. The formative might be 30 percent of your grade and the summative might be 70 percent of your grade. Some teachers have five or more categories. Each one of these categories is calculated and the grade is rounded off to the nearest whole number. Then, these whole numbers are combined to form a quarter grade, which of course is rounded to the nearest whole number. This is done again for the second quarter. After the second quarter, students are given a midterm test. To form the semester (half-year) grade, they count each quarter as 40 percent and the midterm as 20 percent. (It's common for midterm grades to be far lower than either quarter grade, but that's another issue.) This semester grade is also rounded to the nearest whole number before multiplying it by the weight for the course. When the online system displays the weighted GPA, it's displayed using four decimal digits. The roudings are supposed to balance out? No. They are throwing away significant digits and hoping that statistics will recover accuracy. Interestingly, my son's grades have far more rounding downs than ups. Each assignment might be accurate to plus or minus one point, but there is no justification for rounding numbers at every other step of the process. That's why it doesn't matter whether my son got a 92 or 93 on his quarter grade.

Actually, it gets worse. Our high school just emailed us a list of the top ten seniors for this year. (My son is still a junior.). I guess many high schools set this ranking in stone after the first half of the senior year because they have to send something out on the transcrips to colleges. So how do they calculate this final GPA? They do something different at the half-year (semester) point. They take the grade for each course, multiply it by the course weight (3, 3.4, or 3.7), but then multiply it by the full year credit for he course. This gives the semester grade a weight equal to a full year course. I raised this issue with a guidance counselor who said that this is how they've done it for all of the 25+ years she has been there.

I'm sure they use all of those decimal digits of accuracy on the printout to rank kids. They really need to define a proper matematical bound on their accuracy and not just throw away accuracy by repeated rounding.