1. From the final page of the 5th grade Math Trailblazer's Student Guide:

## Thursday, July 31, 2008

### Math problems of the week: Grade 5 Trailblazers vs. Singapore Math

## Wednesday, July 30, 2008

### Gender differences in today's math tests

I doubt that my brother, an accomplished mathematician with a Ph.D. from the University of Chicago, would have cooperated at all: he would then have gotten only half credit for each problem, and, I’m guessing, would have scored way below average.

## Tuesday, July 29, 2008

### The end of linear, analytical reading?

A third education article in this past weekend's New York Times addresses the "new kind of reading" done by the growing number of "Internet readers."

For example, the article describes how one child conducts Internet research on 19th-century Chief Justice Roger B. Taney:

First there's Dana Gioia of the National Endowment for the Arts: “What we are losing in this country and presumably around the world is the sustained, focused, linear attention developed by reading.”

Then there's Nicholas Carr, author of “Is Google Making Us Stupid?” in the current issue of the Atlantic, who speculates that the Internet “is chipping away my capacity for concentration and contemplation,” making it difficult for him to read long books.

Then there are scientists. Noting neurological studies that "show that learning to read changes the brain’s circuitry," the Times reports scientists as speculating "that reading on the Internet may also affect the brain’s hard wiring in a way that is different from book reading." Some scientists, it notes, "worry that the fractured experience typical of the Internet could rob developing readers of crucial skills."

## Sunday, July 27, 2008

### Right-brained science: explaining your answer in pictures

This weekend's New York Times EducationLife section reports on:

...a continuing collaborative project called Picturing to Learn, supported by a $500,000 National Science Foundation grant and also involving Duke University and Roxbury Community College in Boston. The project is an effort to improve basic science education.

1. From a student’s perspective: undergraduate students can clarify their own understanding of scientific concepts and processes by creating drawings that explain these concepts to non-experts.

2. From a teacher’s perspective: drawings can be useful as:

• assessment tools, allowing instructors to identify students' scientific understanding and pinpoint their misconceptions

• educational tools, to help inform instructors’ lecture preparation.

Donald R. Sadoway, who teaches introductory chemistry at M.I.T., [and] collaborates with Ms. Frankel. He assigned his 600 students to answer a question about the boiling points of calcium oxide and calcium sulfide by drawing a picture for a high school student. The crux was to see if they understood which forces holding molecules together are stronger. A typical answer showed atoms holding hands while others tugged at them.

“M.I.T. students are usually good at math,” he says, “but sometimes you discover they’ve memorized the equations and use the right buzzwords. You don’t know if they’re just not a good writer or if they’ve bungled the whole concept. If you make them do a picture, you can zero in on things that words might conceal.”

## Saturday, July 26, 2008

### The charter school alternative: how much of an alternative, II

Today's New York Times EducationLife reports on particular trend in charter school education: the environmentally-themed charter:

...emphasizes the environmental costs of big cars and big houses, and how cities like New Haven can be sustainable communities.

## Thursday, July 24, 2008

### Math problems of the week: grade 2 Trailblazers vs. Singapore Math

1. From the fractions unit in the grade 2 Math Trailblazers Student Guide Book Two:

## Wednesday, July 23, 2008

### ...and how much they haven't

Now most head teachers are chosen because they possess a number of fine qualities. They understand children and they have the children's best interets at heart. They are sympathetic. They are fair and they are deeply interested in education. Miss Trunchbull possessed none of these qualities and how she ever got her present job was a mystery.

"There is a little girl in my class called Matilda Wormwood..." Miss Honey began....

"...Now what is it you want, Miss Honey? Why are you wasting my time?

"I came to talk to you about Matilda, Headmistress. I have extraordinary things to report about the child. May I please tell you what happened in class just now?"

"I suppose she set fire to your skirt and scorched your knickers!" Miss Trunchbull snorted.

...

...Miss Honey was determined to have her say and she now began to describe some of the amazing things Matilda had done with arithmetic.

"So she's learnt a few tables by heart has she?" Miss Trunchbull barked. "My dear woman, that doesn't make her a genuis! It makes her a parrot!"

...

"It is my opinion," Miss Honey said "that Matilda should be taken out of my form and placed immediately in the top form with the eleven-year-olds."

"Ha!" snorted Miss Trunchbull. "So you want to get rid of her, do you?..."

"No, no!" cried Miss Honey. "That is not my reason at all!"

"Oh, yes it is!" shouted Miss Trunchbull. "I can see right through your little plot, madam! And my answer is no! Matilda stays where she is and it is up to you to see that she behaves herself."

"But Headmistress, please."

"Not another word!" shouted Miss Trunchbull. "And in any case, I have a rule in this school that all children remain in their own age groups regardless of ability. Great Scott, I'm not having a little five-year-old brigand sitting with the senior girls and boys in the top form. Whoever heard of such a thing!"From Roald Dahl's Matilda.

## Tuesday, July 22, 2008

### How things have changed

"You say you don't find it difficult to multiply one number by another," Miss Honey said. "Could you try to explain that a little bit?"

"Oh dear," [Five-year-old] Matilda said. "I'm not really sure."

"For instance," Miss Honey said, "if I asked you to multiply fourteen by nineteen...No, that's too difficult..."

"It's two hundred and sixty-six," Matilda said softly.

Miss Honey stared at her. Then she picked up a piece of paper. "What did you say it was?" she said, looking up.

"Two hundred and sixty-six," Matilda said.

Miss Honey put down her pencil and removed her spectacles and began to polish the lenses with a piece of tissue. The class remained quiet, watching her and waiting for what was coming next. Matilda was still standing up beside her desk.

"Now tell me, Matilda," Miss Honey said, still polishing, "try to tell me exactly what goes on inside your head when you get a multiplication like that to do. You obviously have to work it out in some way, but you seem able to arrive at the answer almost instantly. Take the one you've just done, multiplied by nineteen."

"I..I...I simply but the fourteen down in my head and multiply it by nineteen," Matilda said. "I'm afraid I don't know how else to explain it. I've always said to myself that if a little pocket calculator can do it why shouldn't I?"

"Why not indeed," Miss Honey said. "The human brain is an amazing thing."

"I think it's a lot better than a lump of metal," Matilda said. "That's all a calculator is."

"How right you are," Miss Honey said. "Pocket calculators are not allowed in this school anyway." Miss Honey was feeling quite shivering. There was no doubt in her mind that she had met a truly extraordinary mathematical brain...From Matilda, by Roald Dahl, published in 1988.

Today's schools, far from banishing calculators, have made them an integral part of their curricula.

## Monday, July 21, 2008

### The best of hands-on, real-world, interdisciplinary, multi-media learning

In true, left-brained, analytical spirit:

Over his 38 years at Yale, Bennett carried out research in diverse fields ranging from atomic physics to computer science and acoustics...

Many of the approaches Bennett used to collect data for his projects provided much amusement to his students and colleagues. For one project, he rented a truck and filled it with equipment and a mattress and, together with his wife and dog, set out to measure the "Fifth Force" at a site where a large body of water changed height rapidly. The site he chose was the locks on the Snake River in Washington, which gave him special dispensation to camp there with his truck for the summer.

He was also frequently seen at various sites around the Yale campus collecting data for his popular course on "The Computer as a Research Tool." For this course he was named one of the 10 best professors at Yale for many years in a row. His lectures in that course were multi-media events and included demonstrations of firestorms, removal of warts by laser, calculations of how long it would take monkeys sitting at the typewriter to produce phrases recognized from great works of literature, and comparisons of the sound waveforms of the French horn and the garden hose.

One time the professor was spotted dressed in scuba gear and pushing scales and other gadgets at the bottom of the Yale swimming pool, measuring drag coefficients.

...He used his expertise in physics and sound to make calculations on how to decrease the noise levels in the Yale dining halls and used those successfully to improve the ability to converse and to enjoy chamber music concerts there. He also measured magnetic fields around campus and around New Haven. With the magnetic field data, he showed that it was improbable that those fields could cause cancer.

## Friday, July 18, 2008

### The charter school alternative: how much of an alternative?

When a city mandates that all its public schools use a Reform Math curriculum, and when its only math and science magnet stresses leadership skills and cooperative, project-based learning, you'd think that a demand would arise for some charter schools that teach traditional math.

What teaching methods will be used? How will this pedagogy enhance student learning?

The essential experiences we provide will broaden their world beyond the classroom and the neighborhood and will offer them the opportunity and the challenge to develop the critical skills necessary to make the difficult decisions as they grow to become truly productive and contributing citizen [sic] of the world.

## Thursday, July 17, 2008

### Math problem of the week: 4th grade Trailblazers vs. Singapore Math

1. From the final word problems in the Math Trailblazers Discovery Assignment Book:

## Tuesday, July 15, 2008

### Autism diaries

Catherine's post today at Kitchen Table Math about "permissive parenting" and the latest escapades with her autistic son inspire me to share some recent tales about my own.

## Monday, July 14, 2008

### How to teach subtraction without linguistic barriers

Too often, Reform Math lets language get in the way, whether in its convoluted, poorly written directions, its convoluted, poorly written word problems, or in its relentless demands that children explain their answers.

## Friday, July 11, 2008

### In autism, as elsewhere: it's all about education

When we speak of fixing our schools, we too easily point to everything but education itself. We tinker with school size, architecture, scheduling, community involvement, and technology. Too few of us pay any attention what's actually being taught.

"The circuits are there but you have to give it an extra push," said Dr. Gary Goldstein of the Kennedy Krieger Institute in Baltimore, which wasn't involved in the gene hunt but is well-known for its autism behavioral therapy.

The genetics suggest that "what we're doing makes sense when we work with these little kids — and work and work and work — and suddenly get through," he said.

## Wednesday, July 9, 2008

### Math problems of the week: grade 3 Investigations vs. French Math

1. From Landmarks in the Hundreds, a student activity public for grade 3 Investigations (TERC):

## Tuesday, July 8, 2008

### How most language therapies fail to teach grammar to children with autism

We've already seen how many children with autism need systematic, rule-based grammar instruction.

## Sunday, July 6, 2008

### Please visit an actual classroom before you make recommendations, II

Here we go again:

Andrew Hodges, a fellow at Oxford and the author of the lively new book “One to Nine,” would have been horrified, but not surprised. My cousin, in his view, is a victim of the pedagogical tradition that presents math as an eternally fixed array of computations, to be memorized and repeatedly executed without motivation or explanation. The result, he writes, is a “legacy of fear and anxiety generated by schools, which leaves most of their victims with a lifetime of mumbling apologetically about ‘my worst subject.’”

[The opening paragraphs of Jordan Ellenberg's review of Andrew Hodges' One to Nine: the Inner Life of Numbers, which appears in today's New York Times Book Review.]

Both writers--each of them math professors-- also characterize classroom math as "abstract and remote." To this, Hodges' book, in Ellenberg's words, is an antidote: "offer[ing] a different model for teaching math."

Hodges hails from Britain, which hasn't yet gone whole hog for Reform Math, but if either professor had visited any number of American elementary or middle school math classrooms, they would see that:

1. in place of an "eternally fixed array of computations to be memorized and repeatedly executed," we have math as a mess of multiple, ad hoc solutions that students are required to explain and motivate ad nauseam.

2. far from "abstract and remote," today's math marginalizes pen and paper calculations and relentlessly requires students to learn through hands-on activities and real-life applications.

3. all this has so watered down the material and so slowed the pace of learning that, objectively speaking, math is the "worst subject" of more students than ever, however much their fear and anxiety levels about so-called "math" (as currently defined) have diminished.

4. too many math classes are as conceptually disorganized (organized more by "topic" than by concept) and as fleeting and superficial in their coverage as Ellenberg reports Hodges' own "discursive rather than linear" prose as being:

...The book is composed of nine chapters, each focused — very, very softly focused — on one of the first nine natural numbers. Chapter Four, for instance, starts out with the observation that four is a perfect square, and from there skips along to the construction of Latin squares, the irrationality of the square root of two, the definition of the logarithm (whose relation to “four” never comes entirely clear), complex numbers, and the even more exotic quaternions (a number system in which “numbers” are actually strings of four integers, and the product of two numbers depends on the order in which you multiply them!), the theory of four-dimensional spacetime and Einstein’s equation E=mc2 (squares again) before finishing with a short and speculative account of the theory of twistors, one of many competing candidates for the universe’s underlying geometry.

As Ellenberg notes:

The overall effect is like that of a lecture by the type of professor who paces back and forth in front of the blackboard, with insistent voice and waving arms, and has trouble adhering to the ostensible syllabus for any extended period. Being this type of professor myself, I can attest that the style is popular with students. But it requires discipline to convey real information as well as enthusiasm.

Indeed.## Saturday, July 5, 2008

### Why many autistic children need explicit grammar instruction

Full immersion.

It's just that when it comes to first languages, we take full immersion for granted. Our native language, by definition, is the one in which we are born, and immersed within our homes and communities, throughout those crucial first years of language development.

Immersed, that is, so long as we tune into the people around us, share attention with them, and appreciate them as intentional agents that communicate--with one another as well as directly with us.

A child on the autistic spectrum who rarely attends to what other people are attending to or communicating is like a visitor to a foreign country who takes out her powerful earplugs only occasionally, for fleeting moments that encompass fragments of disconnected discourse. Such a child might eventually acquire a basic vocabulary, and even put words together that express intelligent thoughts, but is likely to remain mired in an ungrammatical pidgin unless, before those critical years are over, his social attentiveness increases substantially.

Fully capable of mastering grammar, but immune to the full immersion that could take him there, this child desperately needs an alternate route.

For him or her--this systematic, rule-seeking child with autism--what better route to grammar--the systematic, rule-based structure that underlies every language....

...than that of systematic, rule-based grammar instruction?

## Wednesday, July 2, 2008

### Math problems of the week: grade 3 Trailblazers vs. Singapore Math

1. From the middle of the grade 3 Math Trailblazers textbook, p. 101:

## Tuesday, July 1, 2008

### Why grammar matters for autism, part I

I intend this as Part I of a three part series, collectively addressing the question of why grammar is something that no comprehensive autism therapy should overlook.