Math problems of the week: 5th grade Investigations vs. Singapore Math

Polygons: recognizing terminology and properties vs. deducing area I. A 5th grade Investigations worksheet (click to enlarge):

II. The first 5th grade Singapore Math assignment involving polygons (click to enlarge):

III. Extra Credit

One of these problems builds on a previous unit on fractions; the other one might charitably be interpreted as confusing labels with concepts. Which is which?

The race to remediation

I'm just finishing up a college visiting trip with my oldest, now a high school junior. For all the pressures we hear about kids his age being under today (in what some have dubbed a Race to Nowhere), he seems surprisingly laid back, perhaps more than he should be. The odds of getting into your favorite college are seemingly worse now than ever, and one hears dozens of stories of kids with straight A's, double 800s, and tons of extra curricular leadership activities (my son is not among them), who get rejected from all their top choices.

What I don't understand about all this is the disconnect between our cries about how much more selective colleges have become, and our cries about how ill-prepared our students are, even at the top colleges. They can't write, they can't think, they can't do college-level math, they don't know how to do research, they have no study habits, and they cheat. Related to this is a second disconnect: our cries of how much academic pressure our hard-working high school students are experiencing, and our growing realization of the dwindling amount of time they spend attending classes and doing homework once in college.

Perhaps students are just crashing once admissions pressures end? But what about their increasing academic ill-preparedness? What about all those remedial math and writing classes that more and more colleges are offering? What about all those complaints by professors, even at top colleges, about the inability of so many of their students to read critically, think analytically, construct coherent sentences, and synthesize material into research papers? What about the fact that even highly selective schools like Penn now require all their undergradutes to take what is essentially a remedial writing class?

The overall ill-preparedness of college-bound kids, of course, is readily explained by current trends in k12 education. But given the sheer numbers of students in those record-sized applicant pools, why aren't colleges able to select against those who can't write or think analytically and have no work ethic? Surely there are enough diligent, articulate, numerate, literate, critical thinkers out there to fill the dorms at least of our top colleges?

Here's where I worry about my son. Well, let's forget about my son; since I'm his mother, how can my judgments about him be anything but highly inflated and subjective? Instead let's suppose, for argument's sake, that you have a child who writes well, tests well, and does well in math and science classes, but is a bit of an underachiever, and lacks a resume stuffed with leadership roles. Let's suppose that a top college is deciding between your child and several others who do have resumes stuffed with leadership roles, as well as higher GPAs, equally high test scores, and equally well-written college essays? How is a college to know which of these applicants can really write, analyze, solve problems, and read critically? How do they know who may have had a homework tutor, parents who micromanaged the completion and handing in of homework assignments, intensive coaching on standardized tests, extra time on standardized tests because of ADHD or processing speed disorders, and/or a ghost writer for their college essay?

In other words, while top colleges surely have plenty of highly qualified college applicants to choose from, is there a way for them to identify who these applicants are?

Home schooling: week 4

Highlights of the week include identifying and constructing independent and dependent clauses, completing the multiplying and dividing fractions unit of Singapore Math 5A, understanding more and more of the first 12 lessons of French in Action ("It's starting to sound like English"), and riding the train up to the grandparents with big brother for two intensive days of museuming and piano and violin practice with her veteran piano & violin teaching grandmother. 

Isn't it great when you can take off to your grandparents' for several days and not miss any school?

The potential of charter schools: still largely theoretical

My review of Diane Ravitch's "The Death and Life of the Great American School System" is now up at the Nonpartisan Education Review.  In it, I note that Ravitch doesn't fully appreciate the potential of charter schools:

Ravitch rightly describes charter schools as falling short on both measures, and as not, on average, providing a better alternative to public schools. What she doesn't admit, however, is that these shortcomings aren't inherent to charter schools per se, but largely result from the obstacles placed in their way by state governments and the education establishment. Since most states require that most charter school teachers be certified, it's hard for charters to avoid hiring teachers who haven't been indoctrinated by education schools. The lengthy and highly technical application that would-be charter school founders must submit can often be properly filled out only with substantial help from establishment insiders. Applicants must demonstrate in detail how the charter's curriculum will line up with those infamous, NCLB-inspired state standards. Many school districts limit the number of new charters they will license, and their highly political application process tends to favor insiders. School districts also limit enrollment and/or prevent expansion. Both of these factors force some charters to accept only a fraction of their applicants. Beyond all this, charters face many of the same regulatory burdens as existing schools, as well as, simultaneously, all of the challenges and startup costs that come with starting a new school without the logistical and initial financial support of the local school district.

It's hard to appreciate just how constraining all this can be unless you've actually attempted to start your own charter school. In particular, imagine the possibilities if cities couldn't limit the number and size of charter schools, and if charters (just like private schools) were free to hire the best teachers they can attract, regardless of whether they have official certifications from ed schools?

Yes, charter schools, as a collective whole, haven't yet offered nearly enough of an alternative to public schools, but perhaps that's partly because their gatekeepers don't want them to. Some of us hope that will change.

Math problems of the week: factoring polynomials in the 21st century vs. the 1900's

I. From the Chapter Review of the "Factoring" chapter of the University of Chicago School Mathematics Project Algebra text (second edition), (Chapter 12, pp. 769-771):

SPUR stands for Skills, Properties, Uses, and Representations. The Chapter Review questions are grouped according to the SPUR objectives for this chapter.

SKILLS DEAL WITH PROCEDURES USED TO GET ANSWERS

Objective C: Factor quadratic expressions

18. d² - 8d - 20

...

20. 11a² - 26a + 21 =

(a) (11a - 7)(a - 3)

(b) (11a + 7)(a - 3)

(c) (11a - 7)(a + 3)

(c) (11a + 7)(a + 3)

...

24. -3 - 2k + 8k2

...

Write the difference of squares as the product of two binomials:

30. 25t² - 25

Objective D: Solve quadratic equations by factoring

31. x² - 2x = 0

...

40. 0 = 16m² - 8m + 1

PROPERTIES DEAL WITH THE PRINCIPLES BEHIND THE MATHEMATICS

Objective G: Determine whether a quadratic polynomial can be factored over the integers.

53. Multiple choice. Which polynomial can be factored over the integers?

(a) x² - 11

(b) x² + 11

(c) x² - 121

(d) x² + 112

54. Suppose m, n, and p are integers. When wil the quadratic expression mx2 + nx + p be factorable over the integers?

Use the discriminant to determine whether the expression can be factored over the integers.

...

57. 3r² + 2r - 21.

58. In attempting to factor x² - 16x + 20, Rachel made a list of pairs of factors of 20 and checked the sum of each pair.

Factors of 20: -1, -20 -2, -10 -4, -5

Sums of factors: -21 -12 -9

From this list she deduced that x2 - 16x + 20 was not factorable over the integers. Determine whether she was right or wrong. Explain your answer.

59. Find two integer factors whose sum in 10. What does this tell you about x² + 10x + 24?

USES DEAL WITH APPLICATIONS OF MATHEMATICS IN REAL SITUATIONS

When a old ball is hit with an upward velocity of 80 feet per second, an equation that gives its height (in feet) above the ground after t seconds is h = 80t - 16t².

72. How long with the golf ball be in the air?

REPRESENTATIONS DEAL WITH PICTURES, GRAPHS, OR OBJECTS THAT ILLUSTRATE CONCEPTS

Objective J: Represent quadratic expressions and their factorizations with areas.

78. Show that x² + 7x + 6 can be factored by arranging tiles representing the polynomial in a rectangle. Sketch your argument.

79. A square has an area of 9a² + 30ab + 25b2. What is the length of a side of the square?

=============================================================

II. From the chapter review of the "Factors" chapter of Wentworth's New School Algebra (published in 898) (Chapter 7, pp. 105-107):

1. a² - 9a

...

11. a² - (m + n)2

...

21. x² - 7x + 6

...

31. (x - y)2 - b2

...

41. 3x⁴ - 6x³ + 9x2

...

51. 16x⁴ - 81

...

61. a³ - b³ + a - b

...

71. a² + a + 3b - 9b2

...

81. a² - 2ab + b² + 12xy - 4x² - 9y2

...

91. x⁴ + 8x² - 9

...

101. x³ - y³ - 3xy(x - y)

...

111. 25a² - 4x² + 4x - 10a

...

121. x⁴ - 2abx² - a4 - a2b² - b⁴

=============================================================

III. Extra Credit

Which problem set presents a greater variety of factoring problems?

Which is more likely: that someone who can complete the first set of factoring problems can also complete the second set of factoring problems, or vice versa?

Left brainers as sacrificial lambs for "diversity"

In their attempt to make gifted programs, selective high schools, and top colleges more diverse, more and more influential people are promoting policies that discriminate against left-brain students.

I've already talked (here and here) about Robert Sternberg, provost and senior vice president of Oklahoma State University and the author of College Admissions for the 21st Century. Sternberg has designed both an alternative to the IQ test used for deciding admissions to gifted programs, and an alternative to the SAT test.  Sternberg's tests replace analytical questions with open ended ones of the sort that totally stymie many left-brainers, such as:

Number 7 and Number 4 are playing at school, but then they get in a fight. Why aren't 7 and 4 getting along?

and:

Use one of the following topics to create a short story: 

a. The Spam Filter

b. Seventeen Minutes Ago 

c. Two by Two 

d. Facebook 

e. Now There's the Rub 

f. No Whip Half-Caf Latte 

g. The Eleventh Commandment

One of the virtues of such tests, Sternberg argues, is that they "virtually eliminated the admissions edge enjoyed by some ethnic groups."

As for selective high schools, consider Thomas Jefferson High School for Science and Technology in Fairfax County, Virginia, perhaps the most selective high school in America. As Joanne Jacobs recounts in a recent post, Jefferson no longer seeks out top math students. Rather, as Washington Post education columnist Jay Mathews, cited by Joanne Jacobs, reports in his latest piece on the school:

On the first page of Jefferson’s letter to teachers writing recommendations, in boldface type, was the school board’s new focus: It wanted to prepare “future leaders in mathematics, science, and technology to address future complex societal and ethical issues.” It sought diversity, “broadly defined to include a wide variety of factors, such as race, ethnicity, gender, English for speakers of other languages (ESOL), geography, poverty, prior school and cultural experiences, and other unique skills and experiences.” The same language was on the last page of the application.

...

Recommenders are required to assess three qualities: intellectual ability, commitment to STEM [science, technology, engineering, math] and whether the applicant’s background, skills and past experiences “contribute to the diversity of TJHSST’s community of learners.”

Particularly upset by this new policy is Vern Williams, one of the top math teachers in the country, the recipient of two national awards from The Mathematics Association of America, and a presidential appointee to the National Mathematics Advisory Panel, who, for 25 years, has prepared middle-schoolers for admission to Jefferson: 

Last year, he said, Jefferson rejected one of only two eighth-graders in Virginia who qualified to take the Junior USA Math Olympiad test, six scary problems to be done in nine hours. At the same time, “students who had very little interest [or] motivation in math and science were admitted,” he said. “Some admitted students had even struggled with math while in middle school.”

Why is it left-brainers in particular who as the sacrificial lambs for diversity? Is it all part of a vast, right-brain world conspiracy? Or perhaps it's simply that left-brain skills tend to be quantitative skills that are less easily manipulated for the sake of desired outcomes than are soft skills like "creativity" and "leadership."

Either way, Vern Williams, who is "familiar with the failings of math education for low-income minorities," and who himself is African American, "doesn’t think rejecting top math students is the best way to make the school more diverse." Rather:

The solution, he said, is to “get rid of all warm and fuzzy math programs at the elementary school level and teach real academic content to all students.”

Precisely.

Home schooling: week 3

Highlights include making maps of the neighborhood, tracing and filling in maps of the Ancient World, biking to the zoo, Blue Planet, learning how to divide fractions, and not having to take the PSSAs!

An unexpected benefit of home schooling has emerged: now that there's no longer any friction between them during the school day, she's getting along better with her best friend from school. This is one of several ways in which home schooling has actually improved her social life.

Bliss vs. longevity

This past week, Philadelphia public radio's resident pyschologist Dan Gottlieb discussed the implications of "new research" that purportedly shows that "self-compassion" (silencing your inner critic) improves weight loss, health and general well-being. 

The research itself may be new, but not its conclusions. For decades we've been hearing bromides about the virtues of promoting self-esteem, and of slowing down, de-stressing, smelling the roses, and thinking good thoughts. We have seen this attitude permeate our schools, reaching even our math classes, whose top priorities now include the prevention of math anxiety. Don't worry about hard problems: there's no one right answer or one best way to do math; as long as you try, you'll get partial credit; and we'll make sure there aren't any truly hard problems anyway. And don't concern yourself with math nerds: they might look like they're good at math, but what they are good at, if anything, is mere calculation. 

How refreshing, then, to find, just one day after Philadephia public radio's "Self-Compassion" segment, an article in the Philadelphia Inquirer entitled "Longevity: Revenge of the female nerds." In it, reporter Julia Baird discusses the new book The Longevity Project (by psychologists Stanford Friedman and Leslie Martin), which analyzes the results on a study that tracked, in detail, the lives of 1,528 Californians since 1921.

As Baird puts it:

The results are startling because so many of the platitudes we use to soothe and inspire each other are terrible advice if we want to live a long life: Don't worry, don't work so hard, live a little, think sunny thoughts. Friedman and Martin find that cheerful people are likely to die younger, that hard work, stress, and worry are good for you (though you must manage them), and that eating raw vegetables, going to the gym, and attending church are good practices - but will not seriously affect the number of years you will live.

Those who live long and prosper are decidedly Spock-like:

The prudent, dependable, persistent, thrifty, and organized will die well after their peers. They are less likely to be depressed, anxious, smoke, have diabetes or tuberculosis, or suffer strokes. They also tend to find better marriages and jobs.

Female Spocks, of course, live especially long, whence this article's title.

Naturally, some might argue that, these other virtues aside, longevity isn't everything. Here's H. L. Menckin, cited at the beginning of this article:

Men have a better time of it than women. For one thing, they marry later. For another thing, they die earlier.

Math problems of the week: 6th grade Connected Math vs. Singapore Math

I. The first "Investigation" in 6th grade Connected Math Bits and Pieces II Using Rational Numbers booklet ("Using Percents," pp. 9-10): [click to enlarge]

II. The first word problems in the 6th grade Singapore Math Primary Mathematics 6A Workbook, "Pecentage" chapter (pp. 48-51): [click to enlarge]

III. Extra Credit Which math curriculum is more "inclusive" of children with autism and/or language

and/or reading delays?

Ignorance and blissful cooperation

Over the years, New York Times Op-Ed columnist David Brooks has written a number of pieces extolling the virtues of group cooperation over solo intellectual pursuits, and, naturally, I've criticized various of his conclusions, most recently here.

Brooks now has a new book out entitled--why did I not this one coming?--The Social Animal. In it, he sings the virtues of social connectedness and argues that the emotional, intuitive subconscious is much more important than the rational conscious in determining how we feel and what we do.  A review of The Social Animal by philosopher Thomas Nagel appeared in this past weekend's New York Times Book Review, and  Nagel discusses two fallacies that underlie Brooks' conclusions. First:

Brooks has a terminological problem here. He describes the contents of the unconscious mind as “emotions, intuitions, biases, longings, genetic predispositions, character traits and social norms,” and later he includes “sensations, perceptions, drives and needs.” A majority of the things on this list are “conscious,” in the usual sense of the word, since they are parts of conscious experience. The sense in which they are unconscious, which is what Brooks has in mind, is that they are not under direct conscious control. I may consciously choose from a menu, but I do not consciously choose what foods to like.

Second:

There is moral and intellectual laziness in his sentimental devaluation of conscious reasoning, which is what we have to rely on when our emotions or our inherited norms give unclear or poorly grounded instructions.

Along these lines, it's interesting to reconcile Brooks' reverence for social connections and for the intelligence of cooperative groups with a recent Economist article subtitled "A group's 'intelligence' depends in part on its members' ignorance." Discussing Iain Couzin's computer models of fish shoals, the article observes:

If the models are anything to go by, the best outcome for the group—in this case, not being eaten—seems to depend on most members’ being blissfully unaware of the world outside the shoal and simply taking their cue from others. This phenomenon, Dr Couzin argues, applies to all manner of organisms, from individual cells in a tissue to (rather worryingly) voters in the democratic process. His team has already begun probing the question of voting patterns. But is ignorance really political bliss? Dr Couzin’s models do not yet capture what happens when the leaders themselves turn out to be sharks.

Hmm. Is ignorance bliss? Is the slavish adherence to your subconscious urges, and the slavish following of leaders via mimicry of those closest to you and blindness to the larger world around you, the best route to happiness?

Yes, absolutely--assuming you are a sardine.

Home schooling: week 2

Highlights of last week included the final installments of David Attenborough's Life of Mammals series; the Puma chapter of North American Wildlife; Philip Larkin's "The Trees";  Icarus and Daedalus; the Story of Joseph; Sumer and the Babylonians; the rest of Singapore Math 5A fractions; analysis and imitation of sentences with sentential modifiers and appositives (from Story Grammar for Elementary School); the first 9 lessons in French in Action; listening to Corelli; recorder lessons on Youtube; and the Philadelphia Flower Show.

It's going very well, and I feel very lucky to be able to do this. I feel lucky, as well, to have had such supportive feedback, including your many comments.  

Basic computer programming: not a 21st century skill?

In a couple of earlier posts (here and here), I lamented how difficult it is to do basic computer programming on today's computers. Commenting on one of these posts, Vlorbik adds:

in the DOS environment, one had BASIC
right there, ready to go, 
on every box on every desk.
you'd open it up, find a program that runs,
and start banging on to see what happens.

Several other commenters responded with links to free downloads of platforms for basic programming--particularly SCHEME (from MIT Open Courseware) and SciLab. While grateful for these suggestions, I haven't been able to get any of these to run on my Mac (OS 10.4.11). I haven't yet tried them out on my Windows (Vista), but since this computer is the one I use for most of my work, I like to keep the mischievous child to whom I'm trying to teach basic programming (J, of course) entirely off this computer.

Ultimately, therefore, I resorted to using with J the same built-in programming language I myself use for coding GrammarTrainer: Javascript. But I discovered last week (after a long hiatus in programming with J) that my Mac has managed (apparently while I wasn't actively supervising it) to "upgrade" itself in ways such that it's no longer possible to trick it into letting you program in Javascript. 

Earlier, all I had to do was this:  copy an html page, "view" its source code, copy this code into a text file, edit it, and save it as an html file.  But now such a file, whether opened up in Firefox or in Safari, no longer behaves like an html file, but instead gives a static display of the earlier text file with no functionality whatsoever. Only by downloading TextWrangler--suggested to me by an extremely helpful friend--was I able to bypass this problem. But why does an upgrade to my computer result in my having to download stuff that I didn't need earlier?

My new PC has also made basic programming more difficult (as well as making buttons like "save" and "refresh" less transparent for us non-visual, verbally-dependent left-brainers), but so far a special download has not been necessary. Here, it used to be possible to edit source directly from the window that pops up when you "view source." No longer. But luckily the trick that used to work on my Mac still works on my PC--for now.

The assumption by those who create operating upgrades seems to be that no one (or no one else?), at least on this side of the Pacific, wants (or needs) to do any basic coding any more. And so we're raising a generation of American STEM students that not only is increasingly dependent on things like Dreamweaver and Macromedia, but, for those few who might be interested in more basic programming, has fewer and fewer convenient venues for it.

In an earlier comment Seth, who has taught intro programming at Penn, observes one of the possible consequences of this:

It was universal that the students who came from the best high schools and had the most experience had never been exposed (consciously) to problems of recursion. 

And as Joel Solsky observes in his blog post The Perils of JavaSchools, even in college, where Java has edged out more basic programming languages like Scheme, students are no longer learning basic programming concepts like pointers and recursion.

Personally, I love recursion. Ever since my great-grandfather had me thought-experimenting about cereal boxes whose front sides contain pictures of front-facing copies of themselves, and even more so since reading Gödel, Escher, Bach, but most especially since taking LISP in college, I've been fascinated by it.

Why is it so darn hard to teach it to my son?

Math problems of the week: 1st grade Investigations vs. Singapore Math

Button problems, from about 1/3 of the way through the first grade curricula: I. Investigations:

II. Singapore Math:

III. Extra Credit:

How long will it be before boys start trailing girls in American math classes as much as they do in language arts classes?

Why am I home schooling my daughter?

Thanks to all of you who commented with words of support on my first home schooling post. It has been a big decision and it's nice to feel validation--especially when the school I'm pulling her out of is so coveted relative to other schools in the city that real estate prices in its catchment area have soared and parents who want to secure spots in its limited-enrollment kindergarten classes now start lining up outside the front doors by midnight on the school's mid-winter enrollment day.

As I noted in my previous post, I've been deliberating about this for years. And over the years more and more reasons for home schooling have piled up. None of them have anything to do with my daughter's teachers, all of whom have, in their different ways, been very good to her. Instead, the most compelling issues are:

1. She was increasingly bored and miserable at school, tearful on Monday mornings, elated on Friday afternoons, and plagued with preoccupations and nightmares about unfriendly classmates.

2. She spaced out frequently during class and missed out on a lot of material and activities.

3. Her grades suffered because of this. Recently, she also lost substantial points on her science fair project for not making sufficient eye contact and not speaking loudly and clearly enough during her presentation.

4. She hates Investigations Math, personal reflections, dioramas, projects, and the daily homework sheets almost as much as I do. And these things plague all of the otherwise decent area schools, public and private alike.

5. To get ready for the one potential alternative, what I've taken to calling the True School of the Future (more on that later), she needs to start learning French as soon as possible.

In addition to these issues, there were a few others pertaining to the idiosyncrasies of our school's parent-staff relationships, though none of these were decisive:

6. I'm under the impression that the principal despises me. 

7. In comparison with many other parents, I'm an outsider. I no longer volunteer at the school, and I am not friends on Facebook with any of my daughter's teachers, nor do I (or my daughter) hang out with any of the teachers socially after hours.

8. Some of the insider parents, when they volunteer in the classroom, are tasked with grading homework assignments including those of my daughter.

I have no evidence that any of these latter factors have had, or later would have had, any ill effects on my daughter's educational experience, but how can I be certain?

Home schooling: week 1

After years of deliberation and weeks of planning, I've started home schooling my fourth grade daughter. Here's a sampling of what she did last week.

Literature: classical tales (The Gift of Athena, Adam and Eve, Cain and Abel, Noah's Ark, Abraham and Sarah, the Lion and the Mouse, the Ants and the Grasshopper, the Father and his Sons, the Bat and the Weasels); poetry ("In the beginning was the three pointed star..."); social fiction (Sheila the Great, chapter 1); nighttime reading (several chapters of "Little Town on the Prairie").

Writing and grammar: reading summaries; exercises in Story Grammar.

Math: 6 problem sets from the Singapore Math 5A fractions chapter.

Social Studies: first 3 chapters of Story of the World, volume 1 (nomads through the early ancient Egyptians).

Science: chapter 1 of North American Wildlife; David Attenborough's Life of Mammals (CD 3, episodes 1 and 2).

Problem solving: Analogies Challenges; Venn Perplexors; math contest problems (misc).

French: First seven lessons of French in Action.

Field trips: Academy of Natural Sciences; Penn Museum/Secrets of the Silk Road.

Miscellaneous: exercises from Drawing with Children; Renaissance music; music composition; ping pong; roller skating; map reading (the local neighborhood; the Silk Road quadrant of the globe)

After school: pottery; piano; violin; bike riding.

Social: Saturday playdate with E.

True neurodiversity?

In a recent post, I discussed three popular models for autism: The Normal Child Locked Inside model, the Brain Injury model, and the Quirky Personality model. Different types of people, I noted, favor different models—partly depending on what sort of children they know well and/or work with. Another factor is professional: medical people tend to focus on the Brain Injury model; miracle cure peddlers favor the Normal Child Locked Inside model.

Among advocates of neurodiversity and autism self-advocates, two of these models are popular: the Quirky Personality model, which implies that autism is simply one kind of human identity and that it’s society that needs to be more accepting and accommodating; and a version of the Locked Inside model that assumes that assistive (augmentative or alternative) communication devices (and, perhaps, devices that help with organization or with sensory issues) are largely all that autistic individuals need to get along on society. In the neurodiversity framework, autism is like a physical handicap or deafness: what holds autistic people back are sensory differences (especially in perceiving facial expressions), obstacles to oral communication, and (in some cases) organizational difficulties, as well as lack of full acceptance by the rest of society. Neither intensive remedial therapies and nor parental grief and anxiety are appropriate responses; simply accept, accommodate, and assist, and everyone is happy.

It’s particularly easy to limit your view of autism to the neurodiversity view if you are either:

1. the parent of a child with only mild autism or autistic-like symptoms that are largely attributable to apraxia or sensory processing difficulties.

2. associated with a clinic, school or summer camp that cherry-picks its clients as such: the easiest clients to work with are those with the mildest autistic symptoms; and the easiest ones to “cure” are those who emerge as relatively socially normal once supplied with assistive technology.

3. someone who him or herself has mild autism or autistic-like symptoms that are largely attributable to apraxia or sensory processing difficulties.

In this third category is Ari Ne’eman, the founder of the Autistic Self-Advocacy Network and the first-ever autistic individual to be appointed to the National Council on Disability. Ne’eman is a big proponent of acceptance and accommodation, and a big critic of those who focus on brain injuries, fixing autistic brains, and preventing autism.

Ne’eman is a great spokesperson for people like himself, but, as critics point out, he represents only those at the milder end of the autistic spectrum. In the words of Jonathan Shestack, a co-founder of Autism Speaks (a group focused on cures and the Brain Injury model of autism):

He doesn’t seem to represent, understand or have great sympathy for all the people who are truly, deeply affected in a way that he isn’t.

And here we hit on some of the central absurdities/paradoxes that arise when individuals with autism serve as advisors, informants, and representatives of the autistic community as a whole:

1. The more able such individuals are to serve as leaders, the milder their autism; the milder their autism, the less representative they are of the autistic spectrum as a whole, and the less able they are to speak for others on the spectrum (particularly to the extent that they themselves are afflicted with the impairments in empathy and perspective-taking that remain defining impairments of autism even for those at the mildest end of the spectrum).

2. The more able such individuals are to introspect about their autism and describe it to others, the milder their autism and the less applicable their self-reflections and descriptions are to the general condition of autism.

3. And, finally, my pet peeve: the more those “autism-focused” clinics, schools, and summer camps cherry pick their applicants, the less applicable their interventions are to autism in general, and the easier it is for them to pretend either that they can cure autism, or that their teachers, counselors, and non-autistic enrollees are “fully accepting” of those on the spectrum and that autism is a “tiny disability.”

Math problems of the week: 4th grade Everyday Math vs. Singapore Math

I. Last 6 problems in the Multiplication and Division chapter of the 4th grade Everyday Mathematics Student Math Journal Volume 1, pp 77-78:

[Math Boxes]

Complete the multiplication facts.

3.

a. 4 × 7 = __________

b. 7 × 7 = __________

c.  5 × 7 = __________

d. = _________ = 3 × 6 

e. = _________ = 6 × 5 

f. =__________ = 9 × 6

4. Tell whether each number sentence is true or false.

a. (7 × 4) - 2 = 80 - (9 × 6) __________

b. 8 × 11 = (8 × 5) + (8 × 6) __________

e. 34 < (3 × 5) + (63/9) __________

d. 12 = (6 × 7) / (19 - 7) __________

5. Make a true sentence by inserting parenthesis.

a. 7 × 4 - 4 = 0

b. 45 / 9 + 10 = 15

c. 8 × 7 - 6 = 8

d. 24 / 3 + 5 = 13

[Time to Reflect]

1. Why are multiplication facts called "turn-around facts"?

2. In this unit, your teacher encouraged you to use "A Guide for Solving Number Stories" when solving problems. Do you think the steps and suggestions in the guide are useful? Why or why not?

3. Is the World Tour Project a good way to learn about numbers? Is it helping you? Why or why not?

II. Last six problems in The Four Operations of Whole Numbers chapter of the Singapore Math Primary Mathematics 4A workbook, p. 62:

[Multiply and use the answers to complete the cross-number puzzle]

DOWN

A. 

895

×31

B.

676

×62

E

346

×28

F

406

×53

G

119

×29

I

135

×65

III. Extra Credit

Are Singapore Math students less reflective than Everyday Math students?

Models for Autism

1. The Normal Child Locked Inside Model

Originating with Bruno Bettelheim (The Empty Fortress), this model holds that locked inside every autistic child is a normal child and that sufficiently drastic changes to his or her environment, including sufficiently intensive therapy, can unlock this child. 

In Bettelheim's day, the favorite cause of autism was "refrigerator mothers" who had willed their child's non-existence, and what would unlock the normal child was Freud-inspired psychotherapy, along with removal of the child from his or her parents.

Later, blame was lifted, somewhat, from mothers, and the intensive therapeutic remedies morphed into some combination of play based therapy (e.g., Floor Time) and behaviorist therapy (e.g., ABA). While professionals no longer claim that these therapies can unlock a fully normal child, bestselling miracle cure books by parents (Let Me Hear Your Voice; The Boy Who Loved Windows) claim otherwise.  Some locked-in children may also be partially liberated by assistive communication devices like keyboards or a Dynavox.

More controversial remedies for locked-in children include Facilitated Communication (The Mind Tree), whereby previous mute children, via a keyboard and a supportive hand from a highly trusted companion, apparently express complex emotions, empathy, and subtle (if quirky) humor, perhaps in sophisticated prose and/or poetry.

The locked-in child version of autism best describes those whose autistic-like symptoms are caused either by some sort of withdrawal-inducing sensory overload, or by a serious impairment in fine motor control (e.g., apraxia) that makes speaking, gesturing, and controlling eye gaze difficult or impossible. Help the child regulate his intake and processing of sensory information, or supply him with assistive communicative devices, and a socially normal personality may start to emerge.  

2. The Brain Injury Model

This model holds that some kind of brain damage/abnormal brain development is responsible for autism--usually pre- or perinatal, for example, German Measles--that may be deep enough that there is no longer a normal child inside and that restoring full or partial normality can only be achieved, or restored, via drastic measures. Specific recent theories about causes include the dysfunctional mitochondria theory (causing deficiencies in the fueling of neurons), theories about problematic in utero environments (the parental age theory; the close birth spacing theory) or insufficient time spent in an utero environment (the premature birth theory). 

More popular with lay people is the discredited vaccine injury theory, which first indicted the MMR vaccine for its live measles virus (which, it is imagined, induces a "leaky gut" that then allows "opioid-like" gluten molecules to enter the brain and stupefy it), and then vaccines in general for their mercury (Unraveling the Mystery of Autism and Pervasive Developmental Disorder; Evidence of Harm).

No responsible medical professional has proposed cures for autism-related brain damage; pet remedies of the vaccine injury proponents include the Gluten-Free Diet and Chelation (the highly dangerous and ineffective process of removing heavy metals from the body). Much safer, effective remediations, again, are tried and true therapies like ABA. Specific remedy aside, the idea here isn't to unlock a normal child, but to reverse the damage, bypass damaged circuitry, and/or or teach the child ways to compensate, "habilitate," or pass as normal.

The brain injury model best fits those whose autism includes not just social impairments, but cognitive impairments, linguistic impairments that are unrelated to social impairment, and/or impairments in sensory processing and self-regulation.

3. The Quirky Personality (or Extreme Maleness) model

Popularized by Simon Baron-Cohen (The Essential Difference), this model holds that autism is caused by excess exposure to testosterone in the womb, and/or by "assortative mating" in which highly unsocial, analytical people are more likely to find one other in today's interconnected world, to intermarry, and to produce even more extremely unsocial, analytical offspring (concentrated in areas like Silicon Valley).

This model best explains those children whose primary deficits are social, and who excel in things like engineering and computer programming: the Temple Grandins of the world.  Some of those who embrace this type of autism have argued against intensive treatments (beyond helping the child with basic social skills). In this view, AS is just another personality type, and why should the burden be primarily on the child, rather than on the rest of society, to make the necessary adjustments?

Reconciling the models?

Different specialists don't seem to spend a lot of time reconciling their particular autism model with others. Depending on who you read, or what kind of child you have, either a more normal child is waiting in the wings; or a once potentially normal child was neurologically damaged; or there never was a normal child in the first place, but why worry?

Looking at my son, I wonder if all three might somehow be true.