Autism diaries XVIII: of Mangos and Mad Men

The latest things in our house to magically disappear are the champagne mangos and the latest Nextflix Mad Men CD.

The mangoes, originally intended for Boy I because he's fond of eating them, turned up in the bedroom of Boy II with wires protruding from them and connecting them to various bits of Snap Circuitry. Apparently Boy II had read that high acid mangos make better batteries than potatoes

A few days later, the Mad Men CD turned up, undamaged, in the master bathroom wastebasket. As Boy II explained with great glee, he'd decided to "be a Mad Man about the Mad Men CD." Since he's almost 14 and rather nutty, I believed his explanation.

Math problem of the week: Core-Plus Math vs. traditional math on triangles

1. The first assignment on proving triangle similarity in Core Plus Mathematics, Contemporary Math in Context (a unified high school "core curriculum appropriate for all students") Course 2 (year 2), Part B, p. 395:

Imagine that you and a classmate each draw a triangle with three angles of one triangle congruent to three angles of the other triangle Do you think the two triangles will be similar? Make a conjecture.

a. Now conduct the following experiment. Have each member of your group draw a sement (no two with the same langth). Use a protractor to draw a 50^o angle at one end of the segment. Then draw a 60^o angle at the other end of the segment to form a triangle.

-What should be the measure of the third triangle? Check you answer.
-Are these two triangles similar to one another? What evidence can you give to support your view?

b. Repeat Part a with angles measuring 40^o and 120^o. Are these triangles similar? Give evidence to support your claim.

2. The first assignment on proving triangle similarity in Weeks & Atkins A Course in Geometry (a high school geometry text first published in 1961), p. 233:

P, Q are points on the sides AB, AC, respectively of the Δ ABC such that AP = 6 in. and AQ = 8 in. If AB = 16 in. and AC = 12 in., prove that the Δ APQ and ACB are similar. Is PQ parallel to BC?
[No accompanying diagram]

3. Extra Credit:

Discuss the perils of having today's students draw solo conclusions based only on rigid axioms and rigid logic rather than through empirical hands-on investigations in groups.

More grammar denialism

I've just written about how Applied Behavioral Analysis (ABA), one of the standard therapies for autism, neglects grammar instruction. Since then I've been reviewing for an online class I'm designing the language curriculum of ABA's main competitor, the D.I.R (Floor Time) approach of the late Stanley Greenspan. Greenspan's curriculum is called the Affect Based Language Curriculum, available on Amazon for $55, and it, too, neglects grammar.

Part and parcel of giving short shrift to grammar is dismissing the work of that seminal linguistic grammarian, Noam Chomsky. Chomsky's name is anathema among ABA devotees; Greenspan and his accolytes are merely dismissive. In the Affect Based Language Curriculum, for example, Greenspan writes that Chomsky was wrong to claim that language is innate to the human brain since learning language depends on the social environment.

Anyone familiar with Chomsky will recognize this as a straw man caricature. Of course learning language depends on the social environment. But this does not rule out an innate component, and Greenspan does not address any of the evidence or arguments adduced by Chomsky and others to argue that this innate component exists. Instead, Greenspan goes further, arguing that developing "purposeful affect", or socially-grounded motivation, is all that grammar acquisition depends on:

We create purposeful affect by being a pain in the neck. [E.g. by standing in front of the door the child wants to open.]... We go away a step and come back saying “go, go?”… Once children have that purposeful affect and begin connecting it to the word “go,” guess what they do? They begin using words meaningfully and even using that verb and noun properly.

A nonlinguist, Greenspan has no idea how complex grammar is. Neither verb conjugation nor noun phrase formation are trivial, and nowhere in the Affect-Based Language is there any discussion of how to teach this. Instead, as with ABA's Teach Me Language, the entire curriculum is organized around non-grammatical categories: requesting, using words to protest, talking about the past, adding to what the speaker just said.

Meanwhile, I and others have established that there are many children on the autism spectrum who can do all these things with words, but can't speak grammatically. In neglecting grammar, we neglect these children, many of whom are quite capable of learning grammar when it is taught explicitly and systematically.

Why are therapists all across the autism treatment spectrum so dismissive of grammar? Part of it is that their ranks (and the ranks of speech/language therapists more generally) don't include linguists. Here, though, the causality goes in both directions: we linguists haven't exactly been welcomed into the ranks of autism and speech therapists. But their grammar denialism, I believe, extends into the broader population.

Most people think that grammar amounts to learning the labels for word categories (parts of speech), grammar-related spelling distinctions (e.g., "their" vs. "they're"), punctuation (as in run-on sentences), and stylistic rules about dangling modifiers and parallel structure. They have little appreciation for the more basic but complex rules and structure that underlie these rules of style: since the latter rules are innate, we don't need to learn them explicitly, and therefore most of us aren't even aware of them.

Unless, that is, we learn a foreign language grammar whose rules are significantly different from the rules of English grammar. And here is where Americans in particular fall short. If we learn a language at all, it tends to be one whose grammar isn't particularly exotic compared to English, or difficult for English speakers to learn--i.e., French, Spanish, or Chinese. Languages that Americans used to learn, like German, Russian, Latin, and Greek, presented us with far greater grammatical challenges than these languages do--and perhaps with a greater appreciation for grammar than we currently have.

Furthermore, deep appreciation of the grammar of a foreign language depends on in-depth study of that grammar. Most Americans never get to that point. Many abandon their foreign language after a year or two (as in the "world languages" language-surveying model that has become popular in U.S. schools). And, as I discuss in my book, more and more foreign languages classes have reduced grammar instruction to make room for what's called "communicative competence" (which includes conversing in classroom groups, putting on skits, and designing menus, travel brochures, and tissue boxes).

That leaves it up to the linguists of the world to somehow get across to others what grammar really is, and how best to teach it to those who need explicit instruction--be they foreign language learners or children with autism.

Neurologists argue for expanding classroom group work

An article in this month's Harvard Education Letter, just referenced by Catherine Johnson at kitchentablemath, argues for further expanding classroom group work. The article opens with the group-centered classroom of Rachel Otty, a 10th grade social studies teacher in Boston.

“I remember history in high school as just lecturing, and I didn’t enjoy it much,” recalls Otty, who has garnered a reputation at the school for her skillful use of group work. “In graduate school, we learned about differentiated instruction, and group work is a way to do that. It’s tough work and it requires a lot of brain power.”

True to its genre as an education article, the article then moves on to what "research shows":

Research by educators, psychologists, and, increasingly, neuroscientists supports Otty’s personal experience. Done right, group work can harness the natural propensity of humans to interact, and—most important—make learning for a wide variety of students more engaging, memorable, and equitable. While it is more difficult to do than traditional lecturing, teachers say, most of the hard work is in the preparation, and the payoffs make the time invested well worth it.

The article proceeds to invoke neurology, citing in particular a neurologist names Chris Frith:

Recent research by neuroscientists points to the existence of a “social brain” that enables humans to interact with each other. Summarizing the evidence gathered so far, Chris Frith, professor in neuropsychology at the Wellcome Trust Centre for NeuroImaging at University College London, has identified four regions of the brain associated with functions that allow humans to “read” others’ states of mind and predict what they will do (see sidebar “The Brain and Collaboration”).

Chris Frith is the husband of renowned autism expert Uta Frith, who has conducted experiments showing that the "social brain" is highly impaired in autism. So I'm wondering if Chris has consulted Uta on how expanding group learning might affect autistic spectrum students who are mainstreamed into regular classrooms.  Here's Frith's take on neurologists and group learning:

...Frith said there was “consensus that work on the social brain does argue for expanding the group learning and cooperative learning projects” in schools. “I certainly believe that the special feature of humans is that we can work together to achieve more than the sum of the individuals in the group. This is because we can share experiences,” he wrote in an e-mail, adding, “But we do have to learn how best to do this.”

Well, yes, if a group consensus is so powerful that it can substitute for scientific evidence, than certainly we should worship the power of the group wherever possible.

But what about the many humans in whom this "special feature" is impaired--and whom Frith's wife has so thoughtfully written about?

Not to mention the many others who are hesitant about groups and their consensuses?

I suggest that Chris Frith get into a group with his wife and have a conversation with her about this.

The article moves on to another neurologist--one who's now become a classroom teacher who devotes 50% of her class time to group work:

Group work can also work against factors known to inhibit learning, such as the fear of making mistakes or becoming discouraged, says Judy Willis, a neurologist who has taught elementary and middle school in Santa Barbara for 10 years and often writes and speaks about how findings about the brain can inform teaching.

...

Group work can also increase engagement because individuals can be assigned roles that allow them to be “experts in something,” so that they can be challenged at a level appropriate to their understanding, she says. To discuss and present various theories for why the Jamestown settlement failed or why the dinosaurs became extinct, for example, more advanced students may be “producers” charged with stopping their group periodically to summarize what is being said; those with attention deficits might be assigned to be “prop directors” to keep track of supplies needed to make a chart for the final presentation. 

And the shy, socially awkward child could play the part of the quiet guy who keeps to himself and is shunned by the rest of the community.

And the child with Asperger's could play the role of "little professor", who spends the entire time lecturing his group mates. Unless that's somehow contrary to group-centered discovery learning...

GrammarTrainer and Verbal Behavior

I've just discovered that Grammar Trainer has been listed as a resource on the Verbal Behavior website. This is good news, not just for me, but also for addressing what I've always considered a weak point in Applied Behavioral Analysis (ABA), one of the standard therapies for autism, which includes Verbal Behavior as its approach to language teaching. 

In some ways, GrammarTrainer and ABA have always been very much on the same page.  Both teach by breaking things down into small pieces and teaching those pieces one at a time, sequentially and incrementally.  Both include initial modeling, gradually fading of prompts, and repetitive practice throughout.

What's different is in how the two therapeutic approaches conceive of grammar.  ABA/Verbal Behavior follows a behaviorist model of grammar that dates back to B.F. Skinner. GrammarTrainer follows the modern, structural approach that dates back to Skinner's rival, Noam Chomsky. The problem with the behaviorist model is that it fails to capture the structural complexity of grammar.  Verbal Behavior, traditionally, is all about categories, and not about hierarchical structure and transformations from deep structures to surface structures.   Questions, for example, are categorized into different types (who, what, where, etc.), but they aren't analyzed in terms of the grammar that underlies all questions: do-insertion and inversion of auxiliary verbs (as in Does he want a sandwich?). This is what makes question-formation particularly grammatically challenging, and where many children with autism go astray.

GrammarTrainer captures the grammar of question formation incrementally, first teaching auxiliary verbs and do-insertion in negative declarative sentences ("he doesn't want a sandwhich"), and later teaching do-insertion and auxiliary inversion in questions.  This incremental strategy is totally compatible with ABA's overall pedagogy. The fact that question formation isn't currently taught this way within ABA is most likely a reflection of the fact that few, if any, modern grammarian linguists have been involved in developing ABA's language curriculum. Perhaps times are changing, and I am happy to do my part to contribute to that change.

Math problems of the week: 3rd grade Everyday Math vs. Singapore Math

I. Final division problems in the first half of the 3rd grade Everyday Math curriculum, Student Math Journal, Volume 1, pp. 154-5 (final set of Math Boxes):

3 ÷ 6 = _____

12 ÷ 4 = _____

20 ÷ 5 = _____

_____ = 14 ÷ 7

_____ = 9 ÷ 3

Write 4 division facts you need to practice.

____________

____________

____________

____________

Complete the Fact Triangle.  [Triangle with 45 in the top corner, 5 in the lower left corner, blank line in the lower right corner, and × and ÷ in the middle.] Write the fact family.

____________

____________

____________

____________

II. Final division problems in the first half of the 3rd grade Singapore Math curriculum, Primary Mathematics 3B (Standards Edition), p. 180-183 (Review 5):

440 ÷ 9 = 

The product of two numbers is 72. If one number is 9, what is the other number?

6 children bought a present for $42. They shared the cost equally.  How much did each child pay?

A fruit seller packed 108 apples into bags of 5. How many bags were there? How many apples were left over?

The total weight of a textbook and 8 notebooks is 800 g. The weight of the textbook is 200 g. If the notebooks are of the same weight, find the weight of each notebook.

III. Extra Credit

Do Singapore Math students miss out on opportunities to develop meta-cognitive awareness by not being asked to write down what they need to practice?

Cultural variation in litigiousness: a legalistic, left-brained account

I'd always assumed that the extraordinary litigiousness of American society said something deep about our culture.  It must be that we expect to have more control over the chance events in our lives than people from other cultures do; that we accept less readily that sometimes bad things simply happen and no one's at fault; that we're always looking for easy ways to make big bucks--if not through the state lottery, then through the lottery of the civil court.

But a German national who's spent time in the U.S. and knows something about this sort of business professionally proposes to me an alternative explanation. In Germany (and throughout Continental Europe) governments limit lawyers' fees and don't allow class action suits. This both reduces large awards, and prevents such rewards from bringing big bucks to those who make them happen.  German lawyers, in other words, have much less incentive to chase ambulances, or to represent people who want to sue for huge demands.

Were German lawyers to have the same incentives as their American counterparts, my German contact assures me, they'd have no trouble finding large numbers of injured clients eager to act as litigiously as we do.

Autism Diaries XVII: Magical Thinking

J's drive for mischief is tempered by his drive to construct new things with Snap Circuits. But mischief means longer waits for the next electronic part--provided you get caught. So J has figured out what all too many neurotypical individuals eventually figure out as well: to lie effectively, first convince yourself.  We see this in J's increasingly passionate, indignant denials of responsibility for mishaps we're 99% sure he caused.

It's now no longer possible to tell him he has to wait another five days for a new capacitor without provoking angry bursts of righteous indignation that may last all day or longer.  And, worse, may inspire him to punish us in return, sucking us all into a downward spiral of anti-social behavior and withheld privileges.

Time to prevaricate back.

"Bad things are happening in this house," I tell him one evening after making several 11th hour discoveries.  "My toothbrush magically few off the counter and into the bathroom wastebasket."

"The drying rack magically knocked itself over and snapped three of its dowels in two."

"Your sister's homework magically crumpled itself up, flew over to the toy closet, turned the handle, went inside, closed the door, and threw itself on the floor."

"A comb magically opened the medicine cabinet from the inside, flew out, hit the toilet bowl handle, and, just as the toilet started flushing, flew into the vortex and lodged itself into the pipe below."

Lamely, he starts to offer explanations.  "Maybe K was combing his hair and he accidentally dropped the comb into the toilet."

"Maybe Daddy thought the homework was trash."

But he knows, and (the beauty of his blossoming theory of mind skills) he knows that I know, and he knows that I know that he knows, etc., how lame these excuses are. He peters off, and I deliver the punch line.

"And, magically, your two electric fans disappeared from the basement."

He's speechless.

"Bad things are happening in this house," I repeat, "and I don't think your fans will come back until we go a week without any other bad things happening."

By now, of course, he knows that I'm lying. But each time he accuses me I can feign a shocked "Why would I lie to you about your fans?" without giving him any room for the slightest bit of righteous indignation.

Math problems of the week: 7th grade Connected Math vs. 1920s Math

I. From Hamilton's Essentials of Arithmetic, Higher Grades (published in 1919), final problem set (for 7th grade) involving fractions, p. 154:

Make out sales slips for the following purchases in this form:

[A model sales slip including fields for Quantity, Article, Amount, Total, and Change Due]

1. Edith Mason bought 2 ¼ yd. lace @ 32 ¢; 2 5/8 yd. edging at 7 ¢; 1 ¼ yd. chiffon @ $2.50. Amount received, $5.00.

2. Cora Wise bought ½ doz. lienen towels @ $6.75; ¾ doz. linen napkins @ $5.94; 1 ½ doz. kitchen towels @ $2.95. Amount received, $13.

3. Ruth Wade bought 5 ¼ yd. serge @ $2.35; 4 7/8 yd. wool Jersey cloth @ $2.75. Amount received, $30.

4. Alice Miner bought 1 chiffon blouse @ $5.95; 5 5/8 yd. ribbon @ 29 ¢; 2 ¼ yd. braid @ 19 ¢. Amount received, $10.

5. Clara Hold bought ½ doz pencils @ $.3; 1 box writing paper @ $.59; ¼ gross pens @ $1.0; ½ doz pads @ $1.15; 1 founatin pen for $2.75. Amount received, $5.

6. Emma Stone bought 3/8 yd. damask @ 225; ¼ doz. napkins @ $5.90. Amount received, $3.

7. Maud Smith bought 13 /8 yd. cretonne @ 45 ¢; 12 3/8 pd. ribbon @ 12 ½ ¢; 6 5/8 yd. braid @ 19 ¢; 7 ¼ yd. lace @ 17 ¢; 2 5/8 yd. edging @ 9 ¢. Amount received, $15.


II. From Bits and Pieces 2, Using Fractions Operations booklet, final problem set, p. 64:

Unit Review

Use Your Understanding: Fraction Operations

1. The Scoop Shop sells many types of nuts. Jayne asks for this mix:

½ pound peanuts
½ pound almonds
1/6 pound hazelnuts
¾ pound cashews
¼ pound pecans

a. Nuts cost $5.00 per pound. What is Jayne’s bill?

b. What fraction of the mix does each kind of nut represent?

c. Diego does not like cashews, so he asks for Jayne’s mix without the cashews. What is his bill?

d. Kali is making small bowls of nuts for a party. Each bowl uses ¼ cup of nuts. Kalli has 3 3/8 cup of nuts. How many bowls can she make?

2. Shaquille likes dried fruit. He wants a mix of peaches cherries, pineapple chunks, an apple rings. The following chart shows how much The Scoop Shop has of each fruit and how much of each fruit Shaquille orders.

The Scoop Shop’s Stock: 1 ¼ pounds dried peaches.
Shaquille’s Order: ½ of the stock
The Scoop Shop’s Stock: 4/5 pound dried cherries
Shaquille’s Order: ½ of the stock
The Scoop Shop’s Stock: ¾ pound dired pineaaple chunks
Shaquille’s Order: 2/3 of the stock
The Scoop Shop’s Stock: 2 ¼ pounds dried apple rings
Shaquille’s Order: 3/5 of the stock

a. How many pounds of dried fruit does Shazuille order?

b. Dried fruit costs $5.00 per pound. What is Shaquille’s bill?

Explain Your Reasoning

When you use mathematical calculations to solve a problem or make a decision, it is important to be able to support each step in your reasoning.

3. What operations did you use to find the cost of the nuts in Jayne’s mix?

4. How did you find the fraction of the mix for each kind of nut?

5. 4 ÷ 1/3 = 12 and 4 ÷ 2/3 = 6. Why is the second answer half of the first?

6. Use the following problems to show the steps involved in algorithms for adding, subtracting, multiplying, and dividing fractions. Be prepared to explain your reasoning.
a. 5/6 + ¼
b. ¾ - 2/3
c. 2/5 × 3/8
d. 3/8 ÷ 3/4

III. Extra Credit

What is Diego's bill?

A Chinese "Marshall Plan" for U.S. Math Instruction?

An article in last week's New York Times describes a teacher exchange program in which teachers from China work "for up to three years in American schools, with their salaries subsidized by the Chinese government."  As the Times explains, "China wants to teach the world its language and culture."

And, in this age of Chinese ascendency, there's certainly a great demand among American parents and educators alike for Chinese instruction, ideally by native Chinese speakers.

But there's another way in which these same Chinese exchange teachers can help us, and that relates to the growing demand for qualified math teachers.  While there's been, for decades, a dearth of k12 teachers who are sufficiently good at k12 math to teach it well, Reform Math is only making matters worse: educating (or miseducating) a new generation of students who lack even the most basic mathematical literacy and fluency.  In not too many years, some of those same students will become math teachers.

Those who have become concerned about what's going on have found a solution in one of the highly successful math curricula used in East Asia: the only one written in English, namely, Singapore Math. Many parents, for example, are home-schooling or after-schooling their children using this curriculum.  Some schools have even tried adopting it. And those who use it have generally found it highly effective.  Where Singapore Math has failed in this country, and where schools have abandoned it, appears to be in cases where teachers lacked the math background necessary to understand the curriculum. For Singapore Math is a much more challenging curriculum than Reform Math, and requires a deeper understanding of math than perhaps the majority of American elementary school math teachers have.

Here's where China can once again come to our aid.  Chinese k12 math teachers are much better trained in math than their American counterparts are.  The curriculum used in Mainland China (as with East Asian math programs generally) is quite similar to Singapore Math.  If we could encourage large numbers of exchange teachers from China to not only teach Chinese, but also teach Singapore Math, we might be able to reverse the alarming course we've set for ourselves.

It's unclear, however, whether the Chinese government would be as willing to subsidize math instruction as it is to subsidize Chinese instruction.

The demise of a right-brained approach to autism

Reading about the death of Dr. Stanley Greenspan in last week's New York Times, I wondered about the future of Floor Time.  This was the approach that Greenspan, a well-known child psychologist, championed as the most effective therapeutic approach to autism--an approach I've critiqued first here and later in Raising a Left-Brain Child.

Some of Floor Time seems reasonable but obvious:  a wheel re-invented time and again by those who work with autistic children--parents and professionals alike--whether or not they've ever heard of Greenspan. What do you do with a child who pays you no attention; who remains immersed in a world of his or her own? Jump into this world and follow the child's lead.

More problematic are Floor Time's not-so-obvious recommendations: avoiding teacher- or therapist-centered instruction; avoiding formal structure; using a social- and emotion-based mode of interaction and language acquisition and concept-development that flies in the face of the specific strengths and weaknesses of children with autism--depending, for example, on an ability to read facial expressions and tone of voice.

The curious thing about Floor Time is that, in both its child-centered approach, and in its emphasis on social and emotion-based learning, it strongly resembles the right-brained Constructivist classrooms in which many children (neurotypical as well as autistic) are languishing.  

Floor Time's biggest competitor is a behaviorist approach called ABA.  It has its own problems (which I've also critiqued first here and later in Raising a Left-Brain Child), but it at least has been subjected to fairly rigorous empirical studies, has something of a proven track record, and provides the structure and direct instruction that children with autism depend on.

Who is winning this competition?  In light of current trends in education, you'd think it would be Floor Time. But autism is a pretty powerful condition, and the reality of autism tends to favor ABA.  

Just yesterday I finally watched Temple Grandin, the movie, which concludes with a scene at the 1984 Autism Society of America Conference.  Here a aged, bearded psychotherapist stands at a podium, holding forth on an outdated emotional attachment theory of autism (the philosophical father of Floor Time).  The Temple Grandin character, sitting in the audience with her mother, rises and starts talking about her own experience:  all about structure, drills, being pushed by her mother and others out of her world and into the worlds of science and engineering.  All heads turn towards her, and she ends up literally upstaging the bearded sage on the stage.

I thought of Stanley Greenspan then, and then I read his obituary.

Yet more media praise for hands-on science

An enthusiastic article in last week's Philadelphia Inquirer reports on a visit by Bill Gates to the Science and Leadership academy, a partnership high school between the School District of Philadelphia and The Franklin Institute. During his visit, Gates listened to students present their science projects and then asked them specific questions.

"It's great to see people doing hands-on science," said Bill Gates after the presentations were over. "Science is fun."

...

"Hopefully, many of you will go into these fields which are so cool, so interesting," Gates said.

According to the article, Gates also discussed philanthropy, how he taught computer programming to his fellow students in high school, about being an overconfident math student at Harvard, and how the SLA students should learn as much as they could about science.

The article reports that, as with SLA's principal:

Frederic Bertley, a Vice President at the Franklin Institute, was also thrilled.

The museum takes the partnership with SLA very seriously, Bertley said. Watching the students - who were not prepped for the session - engage in such high-level dialogue with Gates was a thrill for him.

"I couldn't have asked for a better litmus test for the project," said Bertley. "This is why we do what we do."

An alternative litmus test would be one that measures how much scientific knowledge SLA students have acquired. As I've noted in a recent post, (1) there's reason to believe that scientific knowledge acquisition should be a higher priority in k12 science education than hands-on learning, and (2) on this measure, SLA has not tested well. According to the Philadelphia Inquirer's own Report Card on the Schools, which came out just two weeks ago, 59% of SLA's 11th graders weigh in at "below basic."

This makes Gates' most specific advice (as reported) especially urgent:

Gates implored the students to take advantage of the wealth of knowledge available to them for free. For example, Gates wanted a refresher course on physics, so he followed an MIT course online available for anyone to take advantage at no cost.

If k12 schools won't provide this for free, perhaps our universities will.

And perhaps Gates could give the same advice to the students at his School for the Future.

Math problems of the week: middle school 1900's math vs. Core Plus Math

I. From Hamilton's Essentials of Arithmetic, Volume II (published in 1919), first four word problems involving rate (intended for students in "year 7"), p. 253:

How much will 30 dozen eggs cost when 70 dozen cost 42?

When Irish potatoes are 5 lb for 17 cets, how much should a bushel of 60 lb. cost?

If 25 men can build a bridge in 18 days, how long, at the same rate, will it take 15 men to build it. (Notice that fewer men require more time. 15: 25 = 18: ?).

It is estimated that 90 men are necessary to grade a certain street in 45 days. If only 81 men are hired to do the work, how long will it take them?

II. From the Core-Plus Mathematics ("a four-year curriculum that replaces the traditional Algebra-Geometry-Advanced Algebra/Trigonometry-Precalculus sequence") Course 1 Student Study Guide, first four word problems involving rate, p. 30:

Which is the better deal? $18 for 3 bracelets; $30 for 5 bracelets

Which has the fewer calories per serving? 120 calories in 2 servings; 360 calories in 6 servings

Which is the better pay? 4 hours worked for $12; 7 hours worked for $28

Which is the better deal? 15 blank CDs for $5; 45 blank CDs for $15

III. Extra Credit
Comment on the following:

Numerical calculations in the pre-calculator age.

Everyday math problems before "Everyday Math."

Rote patterns in word problems before and after the age of aversion to rote learning.

No Whole Child Left Behind?

An article in last week's Education Week discusses how U.S. lawmakers are moving to revamp No Child Left Behind so that it focuses on "the whole child," and not just on academics:

As Congress gears up for renewal of the Elementary and Secondary Education Act, lawmakers and the Obama administration are seeking to address a perennial complaint: that the current version of the law, the No Child Left Behind Act, places too much emphasis on students’ test scores and pays little attention to their health and other needs.

And at a hearing of the Senate Health, Education, Labor, and Pensions Committee last week, lawmakers agreed that the idea of educating “the whole child” encompasses a wide range of support services, which advocates are hoping could be reflected in the rewrite of the ESEA.

Those include dental and mental health, as well as programs aimed at providing prekindergarten and library services, summer and after-school enrichment, mentoring, college counseling, and increased parent and community involvement. The whole-child concept can also refer to making sure schools attend to students’ nonacademic interests, through programs such as the arts and physical education.

Some of this sounds innocuous enough, but to anyone familiar with current trends in education, the term "the whole child" carries a lot of baggage. Concerned about some of this baggage, I posted the following comment:

I'm concerned that the emphasis on "the whole child" will move beyond measures to ensure basic physical and psychological welfare to include measures that require students to spend even more of their time working in groups; writing about their personal feelings; making dioramas for language arts, posters for science, and illustrations for math problems; and doing large-scale, open-ended, interdisciplinary/multi-media projects.

There's altogether too much of this at those k12 schools that are considered models for other schools, and it alienates large numbers of what I call "left-brainers," who languish with groups and arts & crafts requirements as much as they thrive with independent work and academic challenge.

Also, nearly all American students need more rigorous math and science than they are currently getting (thanks in part to the bar-lowering that No Child Left Behind has inspired, as well as to Reform Math). "The whole child" should include these priorities; unfortunately, the analytical needs of children are not the first things that come to mind when we ponder terms like "the whole child."

Autistic students: looking beyond accommodations to academic alternatives

As the mother of a mainstreamed middle school child on the autistic spectrum, and as the designer of an online course on high functioning autistic students in mainstreaming environments, I spend much of my waking hours thinking about how best to accommodate students with autism in regular ed classrooms. So when an article featuring an Asperger/autism inclusion middle school teacher by the name of Cherie Fowler appeared in this week's Teacher Magazine, it caught my eye immediately. 

According to the article, Ms. Fowler's goals are to teach her students to express themselves better so they are successful academically in general education classes in middle school and beyond. The article credits Ms. Fowler with five specific strategies:

1. Allowing autistic students to type assignments others would have to write by hand. 

2. Allowing them to use other assistive devices.

3. Shortening some of the assignments. 
4. Allowing them one class period that is designed just for them.
5. Educating each general-education class about what Asperger's/autism is.

Laudable goals, and very much in line with what the eminently practical Asperger's expert Tony Attwood recommends in his Complete Guide to Asperger's Syndrome.

However, this list left me wishing for more.  As I commented online on Teacher Magazine:

These are good measures, but, I'm hoping, not the only things that Fowler is doing for her autistic spectrum students. When it comes to reading assignments, AS students often read at a lower level when the content is primarily social, emotional, or grounded in popular culture, and at a higher level when the content is more scientific, technical, and linearly organized, and/or removed from everyday culture (as are fantasy fiction and science fiction).

In writing, AS students often languish when asked to write about their personal lives and personal feelings, or to produce realistic fiction. When the topic is science or fantasy, on the other hand, they are often much more inspired and have much more to say.

In math, AS students often do complicated problems in their heads and aren't able to explain their answers verbally. They should be exempted from having to give such explanations, and should receive full credit for correct answers that lack verbal explanations.

When it comes to large, interdisciplinary/multimedia/ open-ended projects, AS students are often so overwhelmed by the breadth of material that they don't even know where to begin. In lieu of such projects, they should be given a larger number shorter, more structured assignments that offer the same degree of academic challenge.

AS students also flounder when required to work in groups. While group activities specifically targeted at improving their social skills, run by an expert in AS, are fruitful, group activities centering on learning tasks should be replaced by independent learning opportunities.

Finally, AS students are often way ahead of their peers in certain subjects and need to be allowed to progress at their own rates.

What AS children, in other words, are not just supports for, and modifications of existing assignments, but a wholesale replacement of many of these assignments by alternative assignments and learning opportunities that are specifically tailored to their strengths and weaknesses.