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

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

Write a paragraph comparing two pieces of work in your portfolio that are alike in some way. For example, you can compare two labs or your solutions to two problems you solved. One piece should be new and one should be from the beginning of the year. Use these questions to help you write your paragraph:

* Which two pieces did you choose to compare?

* How are they alike? How are they different?

* Do you see any improvement in the newest piece of work as compared to the older work? Explain.

* If you could redo the older piece of work, how would you improve it?

* How could you improve the newer piece of work?         

Write about your favorite piece of work in your portfolio. Tell why you like it. Explain what you learned from it.

2. From the final page of the 5th grade Singapore Math Primary Mathematics Workbook 5B:

Roxanne types at a rate of 5 pages in 45 minutes.

(a) How long will she take to type 1 page?

(b) How many pages can she type in 1 minute?

(c) Let y = the minutes and x = the number of pages. Write an equation relating the number of pages typed to the minutes.

(d) Use the equation to find how long will she take to type 20 pages.

(e) Graph the equation.  Label the axes appropriately.

(f) From the graph, find the number of pages typed at 27 min.

(g) Use the graph to find the time taken to type 4 pages.

-----

Need we say more?

Gender differences in today's math tests

A recent study by University of Wisconsin psychologist Janet Hyde shows, quoting an article in Science Now:

"no gender difference" in [standardized math test] scores among children in grades two through 11.

...as compared with Hyde's previous study, nearly 20 years ago, which found:

a trivial gap in math test scores between boys and girls in elementary and middle school.

Hyde's study notes that:

Among students with the highest test scores... white boys outnumbered white girls by about two to one. Among Asians, however, that result was nearly reversed. Hyde says that suggests that cultural and social factors, not gender alone, influence how well students perform on tests.

The study's most disturbing finding, which hasn't received anything like the media attention that its gender comparison has garnered:

[N]either boys nor girls get many tough math questions on state tests now required to measure a school district's progress under the 2002 federal No Child Left Behind law. Using a four-level rating scale, with level one being easiest, the authors said that they found no challenging level-three or -four questions on most state tests. The authors worry that means that teachers may start dropping harder math from their curriculums, because "more teachers are gearing their instruction to the test."

Indeed.

Another factor that may distort today's test results is the new practice of not granting full credit to unexplained answers. The better you are at math, the more annoying it is to keep explaining how you solved easy questions.

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.

To the extent that these tests are used to determine admission to accelerated math programs, all the worse for the math buffs.

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."

Instead of the focusing on a linear progression of a single author's view point, from predetermined start to predetermined finish, as readers of traditional prose do, Internet readers can skip ahead via key-word searches and click from link to link, "quickly find[ing] different points of view on a subject and convers[ing] with others on line, and "compos[ing] their own beginnings, middles, and ends."

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

 ...[H]e typed Taney’s name into Google and scanned the Wikipedia entry and other biographical sites. Instead of reading an entire page, he would type in a search word like “college” to find Taney’s alma mater, assembling his information nugget by nugget.

In our education world, with its celebration of multiple solutions, multiple literacies, and child-centered, child-constructed learning, is it any surprise that every single person the article quotes in support of this new reading is an education professor?

First there's Rand Spiro, professor of educational psychology at Michigan State University, who argues that kids “aren’t as troubled as some of us older folks are by reading that doesn’t go in a line,” and adds, “That’s a good thing because the world doesn’t go in a line, and the world isn’t organized into separate compartments or chapters.” He also notes that “[i]t takes a long time to read a 400-page book,” and that the Internet lets people “cover a lot more of the topic from different points of view" in "a tenth of the time."

Then there's Donna Alvermann, professor of language and literacy at the University of Georgia: “Kids are using sound and images so they have a world of ideas to put together that aren’t necessarily language oriented.” She adds that “[b]ooks aren’t out of the picture, but they’re only one way of experiencing information in the world today.”

Finally we have Michael L. Kamil, a professor of education at Stanford who lobbied for an Internet component of the federal "nation's report card" tests, who says that today's children “are going to grow up having to be highly competent on the Internet.” 

Those who worry about the predominance of "the new reading," on the other hand, hail uniformly from outside the education establishment.

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."

In particular, Ken Pugh, a cognitive neuroscientist at Yale who, quoting the Times, "has studied brain scans of children reading," argues that:

Reading a book, and taking the time to ruminate and make inferences and engage the imaginational processing, is more cognitively enriching, without doubt, than the short little bits that you might get if you’re into the 30-second digital mode.

Where does all of this leave linear, "left-brained" thinkers, or the left-brain skills of the population at large?

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.

Picturing to Learn explains its core premises as follows:

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.

The article cites principal investigator Felice Frankel, "a science photographer who teaches at Harvard," as arguing that "having students draw... forces them to prove they understand the concepts."

As an example of this, the article cites:

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.”

As its $500,000 in public funding indicates, Picture to Learn  is jumping on an educational bandwagon that already favors the concrete and visual over the abstract and verbal--as a quick tour through Reform Math makes evident.

In the driver's seat is a sort of right-brained totalitarianism whose potential effects on left-brained science students are chilling:

What about people who are good writers and aren't good illustrators?

Why couldn't someone explain which forces are stronger--or any other number of scientific concepts--using words?

Why force all students to learn the material, and justify their answers, using a particular modality (whichever one happens to be most fashionable among education experts)?

What business does how you learned something, as opposed to whether your learned it, have in teacher assessments?

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:

The environmental theme is particularly popular among charters: it lends itself to the kind of interdisciplinary, project-based approaches to learning that they employ...

Profiling the 11-year-old Common Ground, a charter school in New Haven, CT, the Times describes a class called "Egg and Seed":

...a class combining biology, ecology and literature that reflects the educational philosophy called active, authentic learning. To make learning real and relevant, students aren’t just told how food is produced; they actually slaughter chickens for the lunch table.

Then there's the college prep curriculum, which:

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

For example:

In “Four Corners,” team-taught by a social studies teacher and an English teacher, students choose a neighborhood and document its stories, writing up what they learn online.

Interdisciplinary learning extends into English, foreign language, mathematics, social studies and science, where "environmental topics are often used within those classes to convey a lesson."

How many more years, or decades, must pass before the Powers that Be in education and the media stop treating interdisciplinary, active, hands-on, project-based, real-world, "authentic" learning as novel?

Given how widespread these practices are already, among charters and non-charters alike, what would be truly revolutionary would be an environmentally-themed school that gives students a solid, in-depth training in basic biology, ecology, and climatology--one course, one topic, and one rigorous problem set at a time.

Wouldn't it be nice if the abstract, linear, detail-focused, one-thing-at-a-time, "left-brained" student had somewhere to turn for a solid environmental education...

not to mention an education in general?

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:

Banana Split

After buying a banana at a local fruit store, Jess and Bess started squabbling about who would get more.

"Let's cut it in half," said the first to the other.

"We'll each have two pieces and none for our brother."

"Hold on one minute!" the second one cried.

"I think we'll get more if we further divide.

Halves means two pieces. In fourths, there are four.

If we split the fruit that way, we're bound to get more!"

"But if fourths are good, then eighths must be great!"

So, they both got four pieces which each of them ate.

Do Jess and Bess understand fractions? Explain. You may wish to draw a picture to help explain your thinking.

---

2. From the fractions unit in the grade 2 Singapore Math Primary Mathematics 2B:

Sam ate 5/8 of a pizza. How much was left over?

Mrs. Smith made 2 cakes that were exactly the same size.

She cut Cake A into fourths and Cake B into sixths.

(a) Which cake had more slices?

(b) Which cake had larger slices?

Paula had 3 pairs of shoes.

She has a pair of blue sneakers, a pair of brown dress shoes and a pair of brown sandals.

What fraction of her shoes are brown?

-----

Extra Credit:

For each set, estimate the ratio of language arts skills to math skills required to answer the questions.

Estimate the fraction of high functioning children with autism who are able to handle the Trailblazers problems vs. the Singapore problems without special help.

...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.

What has changed: today's education establishment would recast the teacher as a pushy "helicopter parent," deluded that her child is a genius, and the principal as the good guy, concerned about whether "the whole child" can handle such a grade skip.

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.

And today's Matildas, far from being appreciated for their extraordinary mathematical brains, are marked down for not satisfactorily explaining how they got their answers.

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.

From In Memoriam: William R. Bennett Jr., Laser Inventor and Collector of Data, in this past week's Yale Bulletin.

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.

But consider the case of Philadelphia.

As yesterday's Philadelphia Inquirer reports, the eight new proposed charter schools that have received Philadelphia funding include a school for immigrants that stresses foreign languages, a school for at-risk high school students, and the Democracy in Action Charter School, which will be "student-driven and project based." No mention of any traditional, hard core math offerings.

Nor does one find anything about traditional, hard core (rigorous, abstract, linear) math in the mission statements of any of the 63 existing Philadelphia charter schools (with one exception: the k-8 Philadelphia Montessori Charter School). Instead, what dominates are schools stressing culture (6), leadership (3), the arts (3), project-based learning (5), and social and emotional skills (6).  

As if this represents an alternative to what already pervades the regular public schools.

There are two possibilities:

Either these offerings represent what almost all parents in Philadelphia want for their children--which seems unlikely given the huge, multi-year waiting lists for the Philadelphia Montessori School.

Or the charter school application process--which notoriously lacks transparency--is skewed in favor of those who have satisfactory answers to application questions like: 

Briefly describe the core philosophy or underlying purpose of the proposed school.

Why is there a need for this type of school?

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

How enthusiastically would the Philadelphia School District, which decides which applications to accept, respond to answers like:

There is a need for this school because many parents desperately want a more rigorous alternative to the Reform Math currently mandated by the school district.

and

Our school will use a traditional, teacher-centered pedagogy based on explicit teaching and rote memorization of basic math facts--of the sort that has shown to be effective in studies like Project Follow Through.

Given the dominant paradigm that the powers that be have embraced, it would be hard for such declarations to compete with this mission statement, from the Philadelphia Academy Charter School:

The Philadelphia Academy believes that for children to perform at their highest levels, to become life-long lovers of learning, to live, work and grow with integrity, self-discipline, compassion and respect for themselves and others, they must learn and flourish in environments that honor their individuality and commonality.

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.

The upshot, in this fifth largest U.S. city, even with its 63 charter schools, is that there is almost no free access to traditional (rigorous, abstract, linear, mathematical) math.

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

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

Michael, Shannon, and Jessie decided to create a movie library for their neighborhood.  They asked parents and teachers to donate children's videos to create a library.  Neighborhood residents could take out a children's movie for free if they bought one in as a trade. In the first week of the drive to collect movies, Michael collected 17 boxes. Each box was filled with 22 movies. How many movies did Michael collect?

Shannon collected 11 boxes with 27 movies in each. How many movies did Shannon collect?

Jessie and her friends collected 36 boxes with 15 movies in each. How many movies did they collect?

2. From the final word problems in Singapore Math's Primary Mathematics, 4B:

Gene walked 5 times around a rectangular field measuring 45 yd by 20 yd. How many yards did he walk altogether?

Alan used 3/4 of his money to buy a watch which cost $45. How much money did he have left?

The perimeter of a rectangle is 30 in. The width of the rectangle is half its length. Find the area of the rectangle.

----

Extra credit:

Which problem set involves more rote repetition of a given algorithm?

Which problem set is more accessible to children with language impairments?

Estimate the percentage of American 4th grade math teachers who can do all three of the Singapore word problems.

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.

He really wants a Wii. He wants one so badly that we've made it conditional on extended good behavior. Six months of no hurting, bothering, breaking, messing up, and wasting. In particular, six months of no practical jokes. We actually don't want to get him a Wii, but if he can achieve this miracle, it's his.

He's starting to realize it won't happen, so he's turning to the next best outcome: getting invited over to lots of other houses that do have Wii's.  

Step 1: Every time he meets anyone new, or spots one of us talking to someone new, he butts in and asks: "Do you have a Wii?"

Step 2: He tries to gather enough information on each Wii owner so that, when I'm not looking, he can contact them for a Wii invitation. For school parents, a last name will do: he can look them up in the school directory, or, for uncommon names, there's the phone book or Google. 

Step 3: When we're not paying attention (early in the morning or late at night are usually the most promising times), he calls them up. "It's 9:00. Most people are not still sleeping now," I heard him say into the phone last Sunday morning, before I was able to catch up to him and grab the phone.

Step 4: For more elusive contact info, he seeks out the email addresses in my inbox. Once I realize what he's up to, I stop letting him look over my shoulder. But where there's a will...

Step 5: From the computer in his room, he brings up my Comcast account. He then clicks the "forgot password" button and up comes the security question. "What is your mother's maiden name?" No problem: he's been studying the family tree for the last month or so, mostly out of innocent interest. As it turns out, he can also hack into my brother's account.

Step 6: Having thus accessed my (and my brother's) account, he changes the password, and presto, he's in. He then copies all the email addresses he recognizes as Wii-owners into his account, and emails them about inviting him over.

Step 7: After my brother notices that his password doesn't work any more, quickly indentifies the culprit, and changes his security question, my son attempts bribery. Using the user name "Mail Delivery System," and referring to the download code for one of his favorite computer games, Cro-Mag Rally, he writes:

I will tell you a cro-mag rally code if you tell me your favorite beverage.

As of yet, no one has actually invited him over. But if at first you don't succeed...

Two things occur to me:

The default security question doesn't protect you from members of your own (extended) family. 

Perhaps my son's most promising job prospects are in spamming. (Should I embrace it?)

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.

But, for students whose math skills far exceed their language skills, even traditional math poses problems.  Consider the term "borrow," as in "borrow 1 from the 10's digit."  And consider the autistic spectrum child who understands neither the word "borrow," nor the underlying (socially-grounded) concept.

In the course of helping my autistic son realize his mathematical potential, I've thought long and hard about how to simplify and mathematize the accompanying language.

Now, in teaching regrouping to my daughter, I'm revisiting what I came up with for my son.

We start by exploiting a common counting error:

"Twenty-one, twenty-two, twenty-three, ....,twenty-eight, twenty-nine, twenty-ten, twenty-eleven, twenty-twelve, twenty-thirteen,...."

Then we look at a particular problem:

  31

-  8

----

I let my child notice how you can't subtract 8 from 1.

Then I remind him or her of the counting error, and discuss how thirty-one is the same as twenty-eleven.  Then I have him or her rewrite the problem accordingly:

2 11

-  8

----

First I apply this to the most straight forward problems (where the top number is between 30 and 99). 

Then I introduce the teens:  "onety-one, onety-two, onety-three,... onety-eight, onety-nine, onety-ten, onety-eleven, onety-twelve..."  (My daughter now regularly--tongue in check--refers to 11 as "onety-one", and 21 as "onety onety").

Then I introduce the ones:  "zeroty-one, zeroty-two, zeroty-three..., zeroty ten, "zeroty eleven." (And my daughter renames 11 as "zeroty onety").

Then I introduce, via 90, numbers over 100:  "ninety, tenty, eleventy, twelvety..."

Next I translate specific numbers in the hundreds:  705 is "six hundred and ninety fifteen;" 821 is "seven hundred and twelvety-one" or "seven hundred and eleventy eleven."

Lastly I introduce numbers over 1000, which don't sound so odd to our ears in translation: "ten hundred, eleven hundred, ..."

Finally, I have my child translate specific numbers in the thousands:  1111 is "eleven hundred and eleven" (for carrying from the thousands place to the hundreds place), "ten hundred and eleventy one" (for carrying from the hundreds place to the tens place), or "eleven hundred and zeroty eleven" (for carrying both from the thousands place to the hundreds place and from the tens place to the ones place."

Or, translating directly into numbers, one can write 1111 as:

1111    (eleven in the hundreds place, useful when subtracting 900)

10111 (eleven in the tens place, useful when subtracting 90)

11011  (eleven in the hundreds and ones place, useful when subtracting 909)

For my quirky kids, all of this has been surprisingly straightforward.

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 same goes for autism.  We obsess with diet, sensory regimens, and therapeutic philosophies (that eternal debate: child centered vs. adult led).  As for the details of an educational curriculum?  We can't be bothered.

But education, research suggests, is key.  Today the AP reports on a Harvard genetics study that suggests that autism results "in a brain that cannot properly form connections:"

The findings also may help explain why intense education programs do help some autistic children — because certain genes that respond to experience weren't missing, they were just stuck in the "off" position.

"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.

And exactly what we do in the way of work is key. Education. Social Stories. Social skills. Comprehensive language instruction. Facial expression reading exercises. Systematic rules about knowledge flow and belief formation (Theory of Mind).

Key as well:  teaching not just to weaknesses, but also to strengths. Math (not Reform Math); science (not Reform Science); computer programming (not "technology"); analytical essays (not journals and open-ended projects).

School teachers and education experts:  please take note.

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):

Calculator Skip Counting

Choose a number to count by. Pick one you think will land exactly on 300. 

Skip count by this number on your calculator.

Does it work?  If so, write how many of your number it takes to get to 300.

Numbers Did you land on If it worked:

we tried 300 exactly? How many in 300

--------------------------------------------------------------------------------------------------

______________ Yes No ________________

______________ Yes No ________________

______________ Yes No ________________

______________ Yes No ________________

______________ Yes No ________________

... (18 iterations in all)

--------------

2. From Cahier d'activités mathématiques, CE2 (3rd grade), translated from the French:

Goal:  Calculate in your head. 

Observe:  

32 x 4 = 32 x 2 x 2

32 x 4 = 64 x 2

32 x 4 = 128

32 x 40 = 1280

5 is half of 10

12 x 5 is half of 12 x 10

12 x 10 is 120

12 x 5 is 60

50 is half of 100

12 x 50 is half of 12 x 100

12 x 100 is 1200

12 x 50 is 600

1. Solve in your head:

43 x 2 = _____ 82 x 2 = _____ 16 x 10 = _____ 35 x 10 = _____

43 x 4 = _____ 82 x 4  = _____ 16 x 5 = _____ 35 x 5 =  _____

2. Solve in your head:

24 x 4 =  _____ 43 x 4 =  _____ 23 x 5  = _____ 74 x 5  = _____

24 x 40 = _____43 x 40 = _____23 x 50 = _____74 x 50 = _____

3.  Solve in your head:

33 x 4 =   _____ 13 x 40 = _____ 26 x 5 = _____ 14 x 50 = _____

120 x 4 = _____ 21 x 40 = _____ 31 x 5 = _____ 15 x 50 = _____

---------

Which is more "rote" and "mindless," mental math, or skip counting on a calculator?

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.

The problem is that most autism and language therapies--designed and implemented as they are by non-linguists with little appreciation for grammatical complexity--don't provide it.

One major approach, Floor Time (DIR), expects children to pick up all of language, grammar included, from the natural environment. As we've discussed, many autistic children simply don't.  In particular, they fail to master the Question Rule, saying things like: "The boy swimming?" and "Did the girl talked?" 

The competing ABA (Lovaas/Discrete Trials) approach, with its systematic step-by-step pedagogy, has the right framework for explicit grammar instruction. But it, too, falls far short of its purported goals. Its language curriculum, Teach Me Language, is based on an outdated Skinnerian theory that reduces grammar instruction to a series of stimulus-response sessions. As far as ABA is concerned, there's no such thing as a Question Rule, with its notions of auxiliary verb, inversion, and tense-marking. One simply teaches the child to categorize questions by type--yes/no; what, where, when, who and why--and to respond, passively, to each. At no point is the child prompted to formulate questions on his own and offered grammatical feedback.

This passive/receptive/minimal-feedback approach to language instruction also predominates in the world of educational language software.  

For a linguistically principled alternative: consider this.

Please visit an actual classroom before you make recommendations, II

Here we go again:

My cousin, bound for a top liberal arts college in the fall, was amused when I told her I was reviewing a book about big ideas in mathematics, from the classical to the contemporary. “Don’t they already know everything about math?” she asked. “You know, there’s algebra ... and then calculus ... and that’s it, right?”

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.

Mathematicians are in a unique position to hold Reform Math accountable for its shortcomings. By the same token, when they make recommendations that are as out of touch with current practices in math classrooms as physicist Brian Greene's are with current practices in science classrooms, they have the potential to do real harm.

You can bet that Reform Math proponents will start citing Hodges and Ellenberg as their allies, thus ascribing more credibility to practices that most mathematicians would be horrified by.

Why many autistic children need explicit grammar instruction

Full immersion.

Just as it's the only way to master a second language, so it is, as well, the only way to master a first.

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?

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

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

At the end of Tina's birthday party, she will give each of her guests a party bag. After she fills the bags with toys and candy, she will tie the bags with ribbon. She bought 45 inches of red ribbon and one yard of blue ribbon.

To help you answer the following questions, you should use a yardstick or rule, scissors, and ribbon or string. You may wish to use pictures, number sentences, or words. For each problem, explain how you found your answer.

1. If Tina cuts the red ribbon into pieces that are 7 inches long, how many pieces of red ribbon will she have?

2. She cut the blue ribbon into four equal pieces. How long is each piece?

3. Tina needs two more pieces of ribbon. She found one piece of green ribbon that is 7 inches long and cut it into two equal pieces. How long is each piece?

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2. From the middle of the grade 3 Singapore Math, workbook 3a, p. 138-9

A fruit seller sold 1 kg of grapes for $8.  He collected $96 altogether. How many kilograms of grapes did he sell?

Tim printed 900 pamphlets.  He packed them equally into 8 boxes. How many pamphlets were there in each box? How many pamphlets were left over?

8 Students sold 272 concert tickets at $3 each. Each student sold the same number of tickets. How much money did each student collect?

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Note the mandated use of concrete materials by Trailblazers, which make abstract calculations superfluous.

Note the large numbers in the Singapore problems, which make it impractical to bypass numerical calculations. Then there's the conceptual challenge of figuring out when to divide and when to multiply, and (in the third problem) of combining these into a multi-step solution.

Stone knives and bearskins vs. higher level thinking.

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.

Today:  Why grammar isn't trivial.

Later:  Why many autistic children need explicit instruction in grammar; What works and how most therapies fall short

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Why grammar isn't trivial:

Consider how we form questions in English:

1. The person I am thinking of is swimming --> Is the person I am thinking of swimming?

2. The person I am thinking of swims. --> Does the person I am thinking of swim?

3. The people I am thinking of swim. --> Do the people I am thinking of swim?

4. The person I am thinking of swam --> Did the person I am thinking of swim?

5. The person thinking of will swim --> Will the person I am thinking of swim?

6. The person I am thinking of might have been swimming --> Might the person I am thinking of have been swimming?

The Question Rule:  

Move the first auxiliary verb after the subject (here: the person I am thinking of), assuming there is an auxiliary verb (sentences 1, 5, 6), to the front of the sentence. If there is no auxiliary verb (sentences 2-4), take the verb do, change it to the same tense and number as the main verb (does, do, or did), and put it at the beginning of the sentence, while changing the main verb to its bare infinitive form (swims/swam --> swim).  

Key grammatical concepts: 

--inversion:  moving a verb to the front of a sentence 

--auxiliary verb: (including will, may, might, and various forms of the verb to be).

--do-support: (inserting do when there's no auxiliary verb).

--tense marking: (when to keep the tense on the main verb vs. move it to the verb do)

--subject: (can include a relative clause modifier, as in the people I am thinking of)

--main verb: (the verb that agrees with the subject)

Native English speakers who don't have language deficits acquire these concepts implicitly--as well as the complex Question Rule in which they figure.

If we didn't, we might make the following errors:

1. He is swimming? (failure to use inversion)

2. Swimming he is?  (not grasping non-auxiliary vs auxiliary verbs)

3. Swam he? (failure to use do-support)

4. Did he swam?/Do he swam?  (failure to mark tense on only the auxiliary verb)

5. Is the person I thinking of is swimming?  (failure to parse out and skip over the subject to the main verb)

The complexity of English question formation is just one example of how grammar--even that which most of us apply subconsciously--is anything but trivial.