Friday, September 30, 2011

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

I. A 4th grade (TERC) Investigations homework sheet, handed out in mid-November  (Unit 3, Section 4.1, p. 52) [click on picture to enlarge]:

II. From an earlier point in the 4th grade Singapore Math curriculum (Primary Mathematics 4A, p. 54) [click on picture to enlarge]:

III. Extra Credit

1. Compare the mathematical and logical demands of the two problem sets.

2. How do the different roles of logical reasoning vs. visual representation parallel what we see in Reform vs. traditional geometry?

Wednesday, September 28, 2011

Autism Diaries XXVIII: Deep thoughts

Answers to the week one survey questions at his new high school:

What is a big question that you wonder about a lot?
What is your password.

What do you know deep in your heart to be true?
It pumps blood.

What is something that you notice that other people don't?
Ceiling fans.

How do you feel about life's uncertainties and opportunities?
I feel fine.

What is your most favorite activity?
Making lots of money.

What is your least favorite activity?
Getting killed.

Monday, September 26, 2011

The mystery of high functioning autism

I used to think of Temple Grandin as someone who'd been severely autistic as a child but made extraordinary progress during adolescence and early adulthood. My impressions came largely from Grandin's Thinking in Pictures, in which she notes that at two and a half she had "no speech and no interest in people," and then mostly discusses her adolescent and adult years. It was only in the last month, when I finally got around to reading Grandin's earlier memoir, Emergence, that I realized how much of her extraordinary progress must have occurred in early childhood.

Here she is, for example, as a fourth grader:

One time when I was visiting my friend, Sue Hart, we were playing in her hayloft. From the loft we looked down on the garden of Mrs. McDonnell, our fourth grade teacher. Sue said, "Bet you can't throw the red jack ball into Mrs. McDonell's bird bath."
So I threw the ball from the loft and bounced it out off the bird bath. For some reason, I don't know why, there were about a hundred big brown empty whiskey bottles up in that hayloft. Sue said, "Why don't you throw a whiskey bottle out?"
So I threw the bottle and it smashed the bird bath... We proceeded to throw everyone one of those whiskey bottles out of the hayloft against the fourth grade teacher's chimney, her sidewalk, her porch, her rose bushes. There was broken glass all over her garden.
The next day in school Mrs. McDonnell told the class about the terrible damage that had been done to her garden. I wasn't about to get caught so at lunch time I sat down next to Mrs. McDonnell in the cafeteria. "Mrs. McDonnell, what a terrible thing to happen to your lovely garden," I said.
"Thank you, Temple, for caring." Mrs McDonnell smiled warmly at me.
For once, I looked her straight in the eye and told her that I had no idea who had ruined her garden. "But I was at my friend Sue's house," I said, "and we saw Robert Lewis and Burt Jenkins near your house yesterday."
"Thank you for telling me this, Temple. You're a nice little girl to care."
I didn't feel bad about getting [Robert and Burt] in trouble. They might have done it if they'd thought of it.
A few paragraphs later we find her proposing to her cousin that they did up his neighbor's yard after he complains that they are tattletales and says he'd "sure like to fix them."  When Peter worries that he'll get blamed, Temple giggles "Who's to blame? The dogs did it."

A number of things are remarkable here:

So I threw the ball from the loft and bounced it out off the bird bath: understanding linguistic idioms and social gambits ("Bet you can't...")

What a terrible thing to happen to your lovely garden: complex language, conversationally appropriate remarks, and sophisticated deception.

Mrs McDonnell smiled warmly at me: facial expression reading.

They might have done it if they'd thought of it: complex perspective taking.

Who's to blame? The dogs did it: sophisticated, non-literal subtext.

By Junior High her deception (and interest in others) becomes even more (self-sacrificingly) sophisticated:
When I think about it now, I realize that part of my mischief... was the thrill of wondering what would happen--the reaction of my peers--and if I'd get caught. A good example of this was gym class, where I'd wait until the other girls had gone into the gym and then hide their classroom clothes. When gym was over, I laughed and laughed to myself as I watched them run around trying to find their clothes... I always hid mine, too, so I wouldn't be a suspect.
If you watch and listen to Temple Grandin today, and if you know what to look for, you see immediately that she's the real thing. Her face, her gaze, her gestures, her tone of voice, her responses to spontaneous questions: all of these cry out Asperger's. But the social and linguistic skills of her early childhood--so seldom the focus of today's discussions and publications about her--cast all of this in a new light. Now the big mystery isn't how Grandin became such a high functioning adult, but how she went from "no speech and no interest in people" at two and a half to the level of social sophistication we see at nine or ten, and how, despite all this early sophistication, she still remains, after all these decades, so squarely on the spectrum.

Saturday, September 24, 2011

Autism diaries XXVIII: adventures to and from high school

As a special ed student, J qualifies for special transportation to and from his new high school. I'd like him to be able, eventually, to take public transportation just like his classmates, and so every morning I ride in with him, taking the green trolley to the orange train.

J has the logistics down pat and knows exactly where to get on and off. But every morning I see him do something that shows just how unready he is to travel on his own. He sneezes without covering his mouth; he lets go of his backpack; he can't keep his balance while standing while the trolley's in motion, leaning into nearby passengers; he pushes the turnstyle too hard. Most alarmingly, he can't resist the urge to charge past people, especially when there's one seat left on the trolley (as there so often is, way in the back, past all the standing passengers), or he sees a train a flight of stairs below him sitting on the tracks, its doors about to close.

So we have our morning transit riding tutorial, and then I let special transportation take over in the afternoon. But because apparently no other special ed kids are commuting between our part of town and where J attends school, this special transportation comes in the form of a taxicab.

J loves it. Especially because the driver readily follows his directions and drops him off wherever he asks.

So, if you have lunch money left over, why not get dropped at the Hoops Deli around the corner. You walk home from there grinning and brandishing a bottle of lemonade, and then, if your mother fails to catch on, tell her that "I stole a basketball and threw it through the hoop." Make sure she knows you are "speaking metaphorically."

Thursday, September 22, 2011

Math problems of the week: 1960's vs. 21st Century geometry

Second set of introductory problems:

I. From Weeks Adkins A Course in Geometry (1961), pp. 11-12 [click to enlarge]:

II. From Discovering Geometry: An Investigative Approach (2003), pp. 8-9 [click to enlarge]:

III. Extra Credit:

1. Which is more important in the 21st century: precise definitions or artistic designs?

2. Estimate the volume and mass of the Discovery and Akins texts. Hint: one of these texts has twice the volume and three times the mass of the other.

Tuesday, September 20, 2011

Grandiose theories about group selection

The book is out; the buzz is out. Here's National Public Radio, The New York Times, and Nature, all fawning over David Sloan Wilson and his The Neighborhood Project: Using Evolution to Improve My City, One Block at a Time.

Only a few weeks after all this do we get the critical review--and it is one of the most devastating New York Times Book Reviews I've seen, especially given the reviewer's credentials. He is Jerry Coyne, a professor of ecology and evolution at the University of Chicago.

First, Coyne argues, Wilson's basic premise is flawed:

Wilson is well known for his controversial work on evolution via “group selection.” While modern evolutionary theory emphasizes natural selection acting on genes and individuals, Wilson sees an important role for selection acting on entire social groups, particularly in the evolution of “prosociality”: that complex of behaviors, including altruism and compassion, that underlies human cooperation.


Group selection isn’t widely accepted by evolutionists for several reasons. First, it’s not an efficient way to select for traits, like altruistic behavior, that are supposed to be detrimental to the individual but good for the group. Groups divide to form other groups much less often than organisms reproduce to form other organisms, so group selection for altruism would be unlikely to override the tendency of each group to quickly lose its altruists through natural selection favoring cheaters. Further, we simply have little evidence that selection on groups has promoted the evolution of any trait. Finally, other, more plausible evolutionary forces, like direct selection on individuals for reciprocal support, could have made us prosocial.
Second, Wilson's data is flawed. It consists of  surveys of children in different neighborhoods that are supposed to measure (a) how "prosocial" they are and (b) how "supportive" their neighborhoods are:
Prosociality is determined by the level of agreement with statements like “I am trying to help solve social problems.” At the same time, support from their environment is measured by students’ agreement with questions like “I have a family that gives love and support” or “I have good neighbors that help me succeed.”
As Coyne points out, "both statistics derive... from self-report, so there is no independent evaluation of students’ environments." Nonetheless:
Wilson makes much of the correlation between prosociality and a supportive environment, arguing that prosocial nature comes from prosocial nurture.
This flawed conclusion then leads to a half-baked strategy that thus far has yielded no results:
After using the survey to identify neighborhoods with higher or lower degrees of prosociality, Wilson’s strategy is to improve the city by having them compete. This involves neighborhood contests to design city parks, studies of churches to determine why some are better able to recruit and retain members...


Five years into the Neighborhood Project, it has apparently yielded only one published paper, which gives the results of the prosociality survey. The Binghamton parks contest went belly-up, as people weren’t interested in competing according to Wilson’s schedule.
A flawed premise, flawed data; and no results: why, then, is anyone paying attention? Perhaps it's the way Wilson carries on about how many other areas his theory extends into, from economics to education, and because we love grandiose theories more than data and analysis. Or perhaps it's what Wilson has to say about schools in particular, and how this resonates with right-brained educational Romanticism:
Education, for instance, will be transformed by going back to the ways of our distant ancestors, who gave their children no formal instruction but let them learn from unstructured play and interaction with older kids.
Or perhaps what recent research has concluded about narcissistic leaders applies to narcisstic public figures in general:
Wilson’s enthusiasm has a way of shading into hubris, as when he proclaims: “Now that my intellectual life and my everyday life have been thrown together, I can almost feel the connections taking place inside my head. Like a Shakespearean play, the length and breadth of human nature are being enacted in front of me on a local stage.”
As an aside, I've often wondered how truly cooperative humans really are. As far as I can tell, the supposed mystery of our supposed cooperativity has been one of the main reasons why some evolutionary biologists (however much in the minority they are) have invoked group-level selection in the first place. But are we really as cooperative (and as altruistic) as we'd like to think? Sure, we're great at pretending we are; after all, we all depend on reciprocal support, and we zealously keep tabs on, and zealously gossip about, one another's behavior. But what about gossip's greatest thrill: revelations of all the backstabbing and cheating and pettiness that occur whenever people (especially those who try hardest to seem noble and cooperative and above it all) think they can get away with it.

Sunday, September 18, 2011

21st century Shakespeare

One thing that got lost in the discussion following the NY Times' recent exposé of a school district in Arizona that has spent $33 million on laptops, big interactive screens and so-called educational software and is asking for $46.3 million for more of the same even as its test scores are falling and its music, art, and phys ed classes are being cut, is what is happening to Shakespeare:

Amy Furman, a seventh-grade English teacher here, roams among 31 students sitting at their desks or in clumps on the floor. They’re studying Shakespeare’s “As You Like It” — but not in any traditional way.

In this technology-centric classroom, students are bent over laptops, some blogging or building Facebook pages from the perspective of Shakespeare’s characters. One student compiles a song list from the Internet, picking a tune by the rapper Kanye West to express the emotions of Shakespeare’s lovelorn Silvius.
This is a great way to ruin literature in general, but reducing Shakespeare in particular to the psycho/sociological perspectives of his characters is particularly ruinous.  Despite claims by some that Shakespeare invented the human, human psychology was not Shakespeare's strong suit. His characters are constantly falling in love--or tumbling into murderous jealousy--at the drop of a hat or handkerchief; switching love interests at the lifting of a disguise; and ridiculously prone to deception (c.f., e.g., the "transvestite comedies"), persuasion ( And be it moon, or sun, or what you please: An if you please to call it a rush-candle, Henceforth I vow it shall be so for me.) or (most infamously) forgiveness.

No, what Shakespeare excelled in was--of course--language, and the radiant gems of wisdom into which he worked it. Some of my favorites:

Uneasy lies the head that wears a crown.


The undiscovere'd country, from whose bourn
No traveller returns 


Be not afraid of greatness: some are born great, some achieve greatness, and some have greatness thrust upon them.  

What would these look like as status updates on Facebook?

Friday, September 16, 2011

Math problems of the week: Symmetry in Connected Math vs. Singapore Math

I. From the 8th grade Connected Mathematics "Kaleidoscopes, Hubcabs, and Mirrors" chapter, Investigations 2: Symmetry Transformations, end of unit Connections and Extension problems, pp. 38-39:

II. From the 4th grade Singapore Math Primary Mathematics 4B Workbook, "Congruent and Symmetric Figures" chapter, final exercises, pp. 100-101:

III. Extra Credit:

Singapore Math moves on from symmetry after 4th grade; 8th grade Connected Math spends 40 pages on it. What sort of real-world problems and careers will Connected Math students be better prepared for as a result?

Wednesday, September 14, 2011

Narcissistic Leadership

Personality, of course, is a multi-dimension affair, and there are as many personality spectra as there are personality dimensions. But one possible spectrum I've been wondering about goes something like this:

Autism___________Asperger's__________Normal Range_________________________Narcissism

At both ends you see varieties of extreme self-absorption. In autistic spectrum disorders, the source, and the essence, are cognitive: core to autism is severe difficulty in calculating the perspectives of others. In narcissism, the source and essence are emotional: self-love interferes both with gut-level empathy and with the desire to consider others' perspectives. Source and essence aside, perspective taking deficits obviously impair pro-social behavior, including, for example, the ability to lead others.

But because narcissists are better at hiding their disorder, it tends to be only those on the autistic side of the spectrum who strike us as bad candidates for leadership positions. Indeed, because narcissists also exude confidence and charm, many of us consider them leadership whizzes. Wouldn't it be nice if a study cited in last week's weekend Wall Street Journal could make us think twice:

When narcissists get assigned to leadership roles, they impress their charges with authority and confidence. They also underperform, a new study finds.

Workers assigned to narcissistic leaders tended to report that they were authoritative and effective. But, in fact, narcissists-led groups shared information less effectively than the others and picked the wrong candidate more often.

The study offers a cautionary tale to businesses, the authors said: Narcissists are equipped to ace job interviews, for example, but are unlikely to live up to expectations.
At some level, this explains a lot of what's wrong in the world. But why does it not surprise me that many people who ace job interviews are "unlikely to live up to expectations"?

Monday, September 12, 2011

Please visit an actual classroom before you make recommendations, VIII

The latest journalistic entities to carry on about how schools spend too much time on academics and drills are this past week's New York Times and Wall Street Journal. Both the Times' Tara Parker-Pope, in her Science Section's Well column, and the Journal's Jonah Lehrer, in his Weekend Edition Head Case column, make these assertions while discussing the importance of self-control, with those now-familiar longitudinal studies that conclude, in Mr Lehrer's words, that:

Children who could better regulate their impulses and attention were four times less likely to have a criminal record, three times less likely to be addicted to drugs and half as likely to become single parents.
 and that:
In many instances, the ability to utilize executive control was more predictive of adult outcomes than either IQ scores or socioeconomic status.
Both Lehrer and Parker-Pope argue that schools should spend more time teaching self-control instead of academics. Parker-Pope's piece, entitled School Curriculum Falls Short on Bigger Lessons, opens with the following:
Now that children are back in the classroom, are they really learning the lessons that will help them succeed?

Many child development experts worry that the answer may be no. They say the ever-growing emphasis on academic performance and test scores means many children aren’t developing life skills like self-control, motivation, focus and resilience, which are far better predictors of long-term success than high grades...

“What are we really trying to do when we think about raising kids?” asked Dr. Kenneth R. Ginsburg, an expert in adolescent medicine at Children’s Hospital of Philadelphia. “We’re trying to put in place the ingredients so the child is going to be a successful 35-year-old. It’s not really about getting an A in algebra.”
...In his new book “Letting Go With Love and Confidence: Raising Responsible, Resilient, Self-Sufficient Teens in the 21st Century”... Dr. Ginsburg draws a crucial distinction between hard work and simply getting an A or “being smart.
And Jonah Lehrer's piece, entitled Learning How to Focus on Focus, concludes with the following:
We worry about the periodic table instead of persistence, spelling instead of self-control.

That's almost certainly a mistake. Given the age in which we live, it makes no sense to obsess over the memorization of facts that can be looked up on a smartphone. It's not enough to drill kids in arithmetic and hope that they develop delayed gratification by accident. We need to teach the skills of executive function directly and creatively.
Underlying both pieces are two fallacies. One, of course, is the assumption that today's (dumbed down, No Child Left Behind) testing means that schools are emphasizing academic performance and fact memorization. The other is the false dichotomy between self-control and academic performance.  For most kids, learning the periodic table, or getting an A in algebra (the more so back when algebra was mathematically rigorous), involves a fair amount of "self-control, motivation, focus and resilience."

Mr. Lehrer implicitly admits this earlier in his piece when he includes Montessori education on his "surprisingly varied" list of activities that are "both engaging and challenging." What he should be arguing, therefore, isn't that learning factual information is "a mistake," but that schools should be offering the sort of "engaging and challenging" education that Montessori schools provide (and for which demand far outpaces supply), including reinvigorated math and science courses.

Saturday, September 10, 2011

Topics in Education

My interview on Ken Capron's Topics in Education is now available here.

Ken Capron is directing an American history bee project, which you can learn about here.
(Why do we have bees for spelling and math, I wonder, but not for history?)

Friday, September 9, 2011

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

Division problems:

I. A 4th grade TERC/Investigations sheet, assigned in early November [click to enlarge]:

II. From a similar point in the 4th grade Singapore Math curriculum [click to enlarge]:

III. Extra Credit:
a. Why do American math students need more clues than their Singaporean counterparts do?
b. Is 4th grade too early for the long division algorithm?

Wednesday, September 7, 2011

The Rectangle: an appreciation

Only the Triangle gets a course to itself, but, as a tool for teaching math concepts in general, the Rectangle (including the special case of the Square) is, as it were, unparalleled.  Rectangles are, of course, the basic constituent of Singaporean bar modeling, substituting for x and y in word problems that might otherwise require algebra:

Mr. Lim read 10 pages from his book on Monday. Mr. Smith read three times as many pages as Mr. Lim. Their friend, Mr. Samy, read 15 pages more than Mr. Smith. How many pages did Mr. Samy read from his book?

Rectangles are also a good way to visualize multiplication:
5 × 9:
74 × 368:

including the commutative law of multiplication:

And the distributive law:
 a (b  + c) = ab + bc:
And equivalent fractions:

And the factoring or multiplying out of quadratics:

And the area of triangles:
And the area under a curve:

Indeed, so versatile is the unassuming rectangle that I'm sure I've missed a number of other areas for which it is a powerful conceptual building block.

Monday, September 5, 2011

Differentiated instruction in the one room school house

Constraints sometimes force good practices. The presence of special needs students often leads to increased structure and direct instruction. Large enough classes can make group-centered instruction impractical (there's a limit to how many groups a teacher can supervise), forcing more teacher-centered and/or individualized instruction. Also forcing individualized instruction are multi-aged classrooms, as in the one-room schools of rural America.

My daughter and I just finished up the Little House series, and it has occurred to me that the one room school houses attended by Laura Ingalls & co do not exactly match our 19th century stereotypes. Sure, they featured daily recitations and occasional whippings, but the lecturing drillmaster shoving meaningless facts down students' throats is nowhere to be found. In his place sits a site-selected young adult (selected by local school boards populated by the [yes, male] heads of local families) who sends most of her time facilitating individualized instruction par excellence. And in place of meaningless facts and lists of spelling words we have high-quality literature and history, as students spend most of the day working their way, at their own rates, through the (today largely under-appreciated) McGuffey Readers.

Given a choice between this and what's happening in today's model classrooms, with their hours of group work, personal reflections assignments, dumbed-down, "relevant" math, and dumbed-down "relevant" reading (reading that is, in its own way, at least as preachy as McGuffey ever was), I'd have to choose the former. I wonder how many others would join me there?

Saturday, September 3, 2011

Special education and Everyday Math, revisited

Nearly two years ago, I wrote an Op-Ed in the Philadelphia Inquirer about how Everyday Math and other Reform Math programs disadvantage children on the autistic spectrum, even the many who have the potential to do quite well in math.  I discuss the need that AS children have for direct, structured, step-by-step instruction and well-defined tasks, and note that:

Reform math gives them the exact opposite. Instead of direct, structured instruction by teachers (for example, on how to add large numbers), it offers child-centered learning through incidental discovery (for example, of ad hoc ways to add particular numbers). Instead of a curriculum organized incrementally around math concepts (such as borrowing from the tens place), it favors a sequence of themes ("Sticker problems," "How many pockets?").
Naturally, I share these issues with my special education students, many of whom already work with AS students, in a session of my class that focuses on math and science learning. Invariably, some of my students share their own frustrations with Reform Math. Recently, one student wrote:
This session really hit home for me. I was thrown into a situation where I was teaching remedial math and was given Everyday Math textbooks. At the time, I knew very little about middle school math textbooks and curriculum However, it only took me about 3 weeks to realize that the Everyday Math textbooks were not working for my population, specifically a student with PDD-NOS [mild autism]. There was so much verbiage!! So, I found old Saxon Math texts and used those instead. It turns out that my student with PDD-NOS was not behind at all in his math abilities but his way of thinking was not compatible with Everyday Math.
The more I think about it, the more it seems to me that special ed teachers, perhaps especially those who teach academic subjects to autistic spectrum students, are some of our best teachers. They're especially dedicated to teaching, and at the same time are confronted with a population that is especially dependent on good teaching, and that, perhaps more than any other population does, challenges current teaching fads. And so our special ed teachers do what they can to modify their teaching methods, to sneak in better materials (often behind the backs of their principals), and to recognize talents that are too easy to dismiss in many of today's classrooms.

Would that we had more teachers like this--enough to teach regular ed as well!

Thursday, September 1, 2011

Math problems of the week: 1900's math vs. Interactive Math Program

I. The first chapter review in Wentworth's New School Algebra (published in 1898) [click to enlarge]:

II. The first chapter review in Interactive Math Program: Integrated High School Mathematics, Year 1 [click to enlarge]:

III. Extra Credit
Are writing cover letters, creating portfolios, and discussing personal growth examples of 21st century mathematical skills?