Tuesday, July 31, 2012

Is algebra necessary?

So asks political science professor Andrew Hacker in an Op-Ed Piece in this past weekend's New York Times.

To answer him, yes, it is:
1. For STEM jobs
2. For not burning bridges for kids who might pursue STEM careers

It doesn't necessarily follow, though, that algebra is ncecessary for everyone. In particular, it's not clear that algebra should be used as an absolute requirement for college. Only admitting students who take algebra (and get decent grades in it) might screen out some who can't hack algebra but who might thrive in college-level humanities and go on to write brilliant books or screenplays. Or provocative New York Times Op-Ed pieces.

But the fact that someone can't hack algebra doesn't mean that they are constitutionally incapable of hacking it. Much more likely is that their entire pre-algebra math background consisted of Reform Math taught by teachers who scored in the bottom third of their math SATs: true of a growing proportion of young Americans.

Hacker's piece has gone viral and inspired an enormous critical response. This is presumably precisely what the Times was looking for. So I'm hoping it's part of a series:

1. A political science professor asks whether algebra is necessary.
2. A math professor asks whether civics is necessary.
3. A computer science professor asks whether health class is necessary.
4. A philosophy professor asks whether spelling tests are necessary.
5. A critical theory professor asks whether chemistry is necessary.
6. A religious studies professor asks whether gym is necessary.

Sunday, July 29, 2012

Systematizing in different dimensions: the linear, left-brained thinker

The standard tests of creativity, I've just argued, are skewed towards visual creativity. What about tests of what some might consider to be creativity's polar opposite--i.e., systematicity? Here the best-known test is Simon Baron Cohen's Systemizing Quotient test, which solicits self-ratings of systematic thinking and preference for systematic topics. This test involves a similar sort of skewing.

Of the 60-questions, 13 are focused on mechanics and technology. Three are about car, machine or airplane mechanics; four are about how things are put together; two are about transportation systems; and four are about computers, Wi-Fi, stereo equipment, and new technologies.

There are two questions about interest in grammar, two about geology, and two about botany, but the remaining topics--math, music, painting, religion, current events, zoology, cooking, science/nature, meteorology, and culture--get only one question each.

Baron-Cohen's website has so far tallied over 85,500 completed tests, with the average male test score about 8.5 (out of 60) points higher than the average female score. Echoing what he has said about autism and Asperger's syndrome, Baron-Cohen concludes that Systemizing, is a "male brained" phenomenon.

But what would his results look like if he'd posed fewer questions about mechanics and technology, and more about social rules, philosophy, plotlines, computer programming, and invented systems (as in Middle Earth or made-up languages)? Perhaps the gender differences would still persist. But more linear left-brainers like myself, who aren't particularly obsessed with transportation and technology and whose heads spin when we try to reason in 3 dimensions or keep track of moving parts, but who nonetheless will happily systematize till the cows come home, would get the SQ scores we systematically believe we deserve.

Friday, July 27, 2012

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

I. The rounding numbers assignment of the 4th Grade TERC/Investigations Student Activities Book (Unit 5, "Landmarks and Large Numbers," p. 43):


II. The rounding numbers assignment of the 4th grade Singapore Math Primary Mathematics 4B Workbook (Unit 1, "Whole Numbers," pp. 19-20):


Wednesday, July 25, 2012

Invisible creativity

One of the most commonly cited 21st century skills is creativity. Supposedly today's jobs require more of it than ever before. But no solid research shows how to teach it, whether it even can be taught, what it is, or whether it even is a particular "what"--i.e., a single, well-defined concept.

Plenty of people in education ignore these uncertainties. Many seem to think of creativity in fuzzy, "right-brain" ways: i.e., primarily in terms of the visual arts and open-ended activities and multiple solutions.

Encouraging this perception are the sorts of questions one finds on the standard creativity tests, for example the Torrance Test of Creative Thinking. Many of them are visual in nature, soliciting drawings or explanations based on open-ended visual prompts, as in 1 and 2 below:

1.


2. 
 

Even the more verbal creativity questions often tap into the visual (as well as favoring open-ended prompts and multiple solutions): for example, those of the "imagine as many uses as possible for a paper clip" variety.

What about the creativity that goes into composing music? Proving mathematical theorems? Coding new software? Engineering a new type of building material? Figuring out how to survive a wild fire? These varieties are much harder to measure on general creativity tests. Far less visible (literally!) than stereotypical creativity, they're also much harder for lay people to notice, comprehend, and appreciate.

And so it goes with the creativity of our left-brained professionals.

Monday, July 23, 2012

Autism Diaries XXXVI: Non-savant skills

J is not a savant. But he's related to one. His deaf and autistic first cousin twice removed (my mother's first cousin) can tell you which day of the week you were born on a split second after you tell him your birth date.

Temple Grandin, herself autistic and highly intelligent, has argued that savant skills in autism may arise when language deficits allow specialized subconscious processes to prevail that would normally be overridden by a more generalized (linguistic) consciousness. J's first cousin twice removed is verbal, but not very. He cannot, for example, explain how he calculates birthdays.

As for J, aside from his memory for the specs, locations, and rotational statuses of the hundreds of ceiling fans he's observed on thousands of specific dates and times (down to the specific resting positions of specific stationary fans at specific times of day), the one savant-like symptom I've witnessed was early on in his language development. Early on is his sign language development, I should say, for this was back when sign language was our deaf, autistic child's primary mode of communication.

He was toddling around his grandparents' living room, flicking out the fingers of his right hand into different configurations, and it slowly dawned on me that what he was doing was signing the sign language alphabet--smoothly, steadily, and backwards. (People who think this is easy should try saying the alphabet backwards--smoothly and steadily, without pausing between letter names.) When I asked him to do this years later, he couldn't.

"Maybe J is a savant," J's older brother casually proposed at a recent family reunion dinner.

"I don't think so," I replied. I explained to the extended family why I thought so, contrasting J with his first cousin twice removed (who wasn't in attendance).

"Have you tested him?" asked one family member.

"J," called out another, "I was born on March 6th, 1970. Can you tell me what day of week I was born on?"

J turned his back to the table, and about 30 seconds later, announced "Friday."

"He's right."

And around the big round table we went. Birth dates from 2007 down to 1938 were tossed out; birthdays were calculated after a 15-60 second pause. Not one leap year was missed; he was right every time. But unlike the split second duration of J's first cousin twice removed's calculations, J's seemed to correlate with how far in the past the dates were (and whether he could short-cut from one date to the next). It seemed quite clear that he was deliberately deploying straightforward arithmetic algorithms rather than deferring to the sort of subconscious, rapid lookup process that autism experts have proposed is the basis for split-second calendrics.

There's something magical about savant skills--maybe too magical. Or maybe this is only the impression of a biased mother who also has a special predilection for that which can be called to consciousness and expressed in words. But I actually find J's 15-60 second deliberate calculations more impressive than the automated ones of a split-second savant.

If language interferes with savant skills, perhaps it does so specifically by involving the consciousness in mental processes that would otherwise be purely subconscious. Generally the deliberated processes of the conscious mind are considerably slower than the automated processes of the subconscious. But, because they involve the person as a conscious being, they're in some sense more meaningful--both to the person him or herself, and to all his extended family.

Saturday, July 21, 2012

Correlating test scores with self-reports

In a back-page commentary in this week's Education Week, Yang Zhao, associate dean for global education in the college of education at the University of Oregon, argues that testing well on standardized tests correlates with weak entrepreneurial qualities skills. His evidence?

Out of the innovation-driven economies, Singapore, South Korea, Taiwan, and Japan are among the best PISA [Program for International Student Assessment] performers, but their scores on the measure of perceived capabilities or confidence in one's ability to start a new business are the lowest. The correlation coefficients between scores on the 2009 PISA in math, reading, and science and 2011 GEM in "perceived entrepreneurial capability" in the 23 developed countries are all statistically significant. (By the way, these countries have also traditionally dominated the top spots on the other influential international test, the Trends in International Math and Science Study, or TIMSS.)
Note that Zhao's evidence is based on perceived capabilities or confidence. Using such data, one might also conclude that testing well on standardized math tests correlates with weak math skills. After all, as Jay Matthews reports in the Washington Post:
According to the Washington think tank's annual Brown Center report on education, 6 percent of Korean eighth-graders surveyed expressed confidence in their math skills, compared with 39 percent of U.S. eighth-graders. But a respected international math assessment showed Koreans scoring far ahead of their peers in the United States, raising questions about the importance of self-esteem.
Several countries in Asia and some in Europe tend to beat the United States in math scores, even though their students show less satisfaction with performance.

Thursday, July 19, 2012

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

I. The first multiplcation homework assignment in the 4th grade TERC/Investigations Student Activity Book, pp. 21-22:



II. The first multiplication problem set in the 4th grade Singapore Math Primary Mathematics 4A Workbook, p. 51:
III. Extra Credit
At this rate, which cohort is more likely to be ready for algebra (real algebra) in middle school?

Tuesday, July 17, 2012

Autism Diaries XXXV: Crossing things off the "bother list"

According to J, only one person remains on his "bother list": his younger sister. Everyone else has been "crossed off." Her reactions to his provocations, more reliably intense and robustly satisfying than anyone else's, have been the hardest for him to give up. It's like eliminating all desserts except chocolate.

And it is, of course, a real drag for younger sister. Try not to scream when he calls you "baby," we tell her, while withholding the usual privileges from J (not answering any questions about ceiling fans until he plays 10 more games of chess) and encouraging them into separate rooms when chaperoning isn't possible.

In comparison to J's earlier, extreme, relentless and indiscriminate mischief, occasionally calling his sister "baby" (and grinning derisively and pretending that he was actually saying "maybe") is a huge improvement.

Not that all behavior problems have suddenly vanished. As a recent 10-day camping trip has reminded us, transitions, wait times, and neurotypical conversations all inspire boredom and restlessness--and Wanderlust. No longer is he locking us out of the car, rading the dessert bags before supper time, or puncturing all air mattresses except his, but an urge to run across bridges (they might collapse!) and down inclines ("because of gravity") without waiting for human obstacles to get out of his way means frequent barreling through groups of hikers on narrow trails. His greatest obsession of all has him rushing ahead of us into stores and restaurants and filming the ceiling fans with his digital camera--occasionally standing up on chairs to pull chains and adjust speeds while customers look on hushed surprise.

Indeed, more than anything else, it's that fan fandom that's now J's biggest liability in public. And on Wikipedia, where he no longer gets banned for deleting the Delete Key entry, but for repeatedly adding "Some restaurants have ceiling fans" to the Restaurant entry. Not to mention in his social interactions, where (besides hypothetical mischief and the Number 2) fans have long been a topic of extreme perseveration.

"I saw a fan; what speed do you think it was on?" he asks me for the upteenth time during some recent downtown errands, having finally earned back (via 10 games of chess at the Apple Store, which has no ceiling fans to get on film), the privilege of me answering his fan questions.

"I'm sick and tired of that question," I decide to say to him, playfully parotting back his latest response to questions like "How much longer are you going to be on the computer?"

"It's not same question!" he said. "I never asked you about this fan before."

Half an hour later we're almost home.

"I saw a fan; what speed do you think it was on?"

"I'm sick and tired of that question." This time it isn't a different fan.

"You forgot about the 4th dimension!" he shouts. Yes, it's a fan he's already asked me about many times. But never before at this particular moment in time.

Sunday, July 15, 2012

The guide on the side: obsessed with assessment

One of today's truisms is that schools are over-assessing kids at the expense of educating them. People tend mostly to blame No Child Left Behind. But educational ideologues--generally not fans of NCLB-- are equally to blame.

While the kinds of "authentic assessments" favored by ideologues may look to some like a big improvement over standardized, multiple choice testing favored by politicians, such alternative assessments, "authentic" and "formative," and "holistic" as they are, extend over longer periods of time and are arguably at least as intrusive. Those promoted by curriculum developers at Pearson and embraced by the visionaries running the Montgomery County MD Public School, for example, involve assessing kids every 9 weeks during a 3 week period, via rubrics, checklists, and student self-ratings. Teachers walk around and observe kids during classroom activities, asking them how they are doing, what they are doing, and why, presumably taking copious notes on clipboards.

The odd thing is that the point of such assessments doesn't appear to be to  provide feedback for curriculum and pedagogy, but, in the words of one Pearson developer, "tracking progress to predict success in post-secondary education.”

Why all this assessment merely to make predictions?

Answering this question means considering the broader Constructivist framework, which views teachers as "guides on the side" rather than the "sages on stages." If teachers are no longer front and center directing lessons, then they need other tasks to occupy their class time. How about facilitating, managing, and... assessing? And if teachers aren't teaching, they need some other raison d'etre. How about predicting which of their students will succeed in post-secondary education?

Friday, July 13, 2012

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

Culminating division problem sets:

I. The final division problem set in the 4th grade TERC/Investigations Student Activity Book, p. 72:


II. The final division problem set in the 4th grade Singapore Math Primary Mathematics 4B Workbook, p. 73:


III. Extra Credit

Relate these problem sets to this recent report.

Wednesday, July 11, 2012

Against inclusion

Many parents and teachers quietly complain that the drive to include special needs children in regular classrooms impairs the education of non-special needs children. From what I know about autistic students, I'd add that inclusion also impairs the education of certain types of special needs children. I'm thinking, in particular, of high functioning autistic children like J; autistic enough to be reading several years below grade level in English class and autistic enough to get little out of sitting in classrooms of typical peers, but high functioning enough to be ahead of most typical peers in math and computer programming.

Socially, he's more likely to interact with peers like himself than with neurotypical peers, whom he finds baffling and who find him insufferable. But there aren't any peers like himself at his school. Instead, the handful of high schoolers at his particular level of functioning are dispersed throughout the city and beyind (many special needs parents taking flight for better services in the subbarbs).

So I like to fantasize about a magnet program for kids like J, in which a 9th grade curriculum would include pre-calculus, advanced programming, and an English class consisting of nonfiction texts written (depending on the specific subject) at a 6th to 9th grade level. (Let's stop pretending that one can get language-impaired kids to appreciate Shakespeare by feeding them simple paraphrases, explanations, and visuals--about as satisfying as "explaining" a joke.)

After school activities in my autism magnet school would include math team, chess team, computer science club, and sports (here, other kids could join in). And cafeteria talk could focus on ceiling fans, transportation systems, baseball statistics, and other topics of popular interest.

Would I be depriving my son of valuable social skills by segregating him from his typical peers? People forget that Zone of Proximal Development applies as much to social skills as it does to academics. Spending his high school years immersed among high functioning autistic peers with whom he shares common interests and abilities taps into his Proximal Development; immersion with typical peers doesn't--yet. If only he weren't stuck in this "least restrictive environment" now, he might be ready for it eventually.

Monday, July 9, 2012

Dazzled by "giftedness" Or desparate for learning?

Some school systems seem to have reconciled their distaste for ability-based grouping with pressure to offer something in the way of gifted programming by limiting advanced academics to a tiny select group (often chosen based in part on subjective teacher recommendations and questionable definitions of giftedness). For everyone else, one size fits all.

Under these circumstances, parents of the many under-challenged kids often have no recourse other than to seek out the gifted label. Given the scarce spots for gifted kids, this often ends up pitting parent against parent. The dividing and conquering powers that be, meanwhile, start accusing parents of hypercompetitiveness and of being "dazzled by the label"--in other words, of seeking gifted labeling simply for the sake of having their kids called "gifted." Our society as a whole, convinced as it is that parental competition is at an all-time high, is all too ready to believe this.

But for the parents in question, desperate as they are to help kids who come home bored and disengaged, sometimes to the point of tears or of dropping out of school, this is nothing but an extremely red herring  that only adds insult to injury.

Saturday, July 7, 2012

Publicly funded vs. public

In a recent primary battle in my neighborhood, a neophyte politician attempted, unsuccessfully, to upset a long-standing incumbent by citing his opposition to school choice and vouchers. The incumbent’s establishment supporters argued that this would only further impoverish the already bankrupt Philadelphia public schools. The issue of vouchers is complicated, but the notion of a straightforward dichotomy between public and private schools, at least in big cities like Philadelphia, is totally out of date.

1. “Public” schools are harder for parents to visit (let alone choose among) than private schools.

2. “Public” school parents have no input in what matters most: who’s hired; what curricula are used.

3. "Public" school principals often stonewall when parents volunteer to run after-school activities at the school, particularly if those activities are academic in nature and/or pitched at higher-achieving children (e.g., math team, computer science club).

4. Many of the nation’s biggest “public” school systems have appointed rather than elected boards and superintendents who answer to cronies and corporate partners (textbook and software companies and other suppliers) rather than to citizens.

5. While some religious schools fail to teach certain key topics in science (evolution of species; pre-4004 BC geological and cosmological history), many public schools (via Reform Math) fail to teach even more key topics in math (arithmetic fluency; fluency in multi-step manipulations of algebraic expressions; rigorous multistep proofs)--math being the subject that underpins all of college-level science.

One indication that our schools aren’t truly public is extreme disconnect between supply and demand--something that transcends financial constraints. In particular, there’s the disconnect between the curricula and pedagogy used by our public schools and the curricula and pedagogy desired by public school parents. I’m thinking not just of Reform Math vs. traditional or Singapore Math, phonics versus Whole Language, or mixed-ability groupings vs. ability-based groupings; I’m thinking also of the extremely long waitlists for the tiny numbers of Montessori, bilingual, and KIPP schools our biggest school districts have to offer.

However complicated the vouchers debate, calling our publicly-funded schools “public” is becoming more and more of a red herring.

Thursday, July 5, 2012

Math problems of the week: traditional vs. IMP trigonometry

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.407-408 [click to enlarge]:



II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 19-20 [click to enlarge]:


Tuesday, July 3, 2012

Computerized math enrichment

In preparation for our after-school math enrichment program, I tracked down some websites offering free math practice: mostly straight up, fluency-fostering practice with basic arithmetic, with corrective feedback but no distracting bells and whistles (e.g., Free Rice). When I attempted to link to these sites on the school’s computers, however, I discovered that they had been blocked by Philadelphia School District’s Internet Firewall, which doesn’t recognize them as educational games. The kids were quick to tell me about a site that isn’t blocked: CoolMath. They all knew about this site, and their eagerness to play CoolMath immediately raised my suspicions.

CoolMath certainly looks like an educational site. It’s got lessons and drills, including basic arithmetic, fractions, pre-algebra, algebra, and even pre-calculus. But it also has “puzzles” and “strategy games” like the monster in the maze and the obstacle course and the road race, and it was naturally these games, and not the math lessons, that all the students, without exception, consistently chose to do. When I asked them where the math was, they insisted that these were puzzles and strategy games. After all, math (especially Reform Math’s version of it) is all about puzzles and strategies, and not about “mere calculation.”

When we revamp this program in the fall, CoolMath will be gone. Not only did the students learn no math during the 15 minutes at the end of the afternoon when we let them access the site; many of them so pre-occupied with how much time was left until they could get on the computer and do CoolMath that they had trouble focusing on the actual math lessons that were the centerpiece of our program.

I’m not sure how much time students in general are spending on sites like CoolMath during their school days, but these students in particular (many of whom don’t have computers at home) were suspiciously familiar with, and suspiciously addicted to, this particular unbanned “educational” Internet site.

Sunday, July 1, 2012

Two ways to ensure learning

When students get the right answer, how can you be sure that they did so via the skill that they're supposed to be learning? This is particularly problematic in automated learning situations, where there is no third party monitering the student. When I click on the correct picture in a Rosetta Stone Spanish exercise, how can one be sure that I didn't get the right answer simply because I recognized a single key word in a sentence, as opposed to having correctly parsed out the syntactic structure being taught?

When it comes to language learning, the best way to ensure that students are exercising the intended skill is to make displaying it an inherent part of getting the right answer. If you're teaching grammar, for example, answering correctly should involve actually producing a sentence with the grammatical structure being taught, not just clicking on a picture.

Ensuring that students are applying the intended skill is one of the justifications for today's obsession with having them explain their answers to math problems. But that's the easy way out, and it frustrates students and decelerates their progress. An alternative would be construct math problems whose solutions are unlikely to be found by any means other than by using the skill in question: arithmetic problems that involve more than single digit or "friendly number" solutions; algebra problems that don't lend themselves to arithmetical, guess & check solutions. Then there are problems complicated enough that there's actual work to show; work that, if written out systematically, helps the students at least as much as the teacher.

Constructing such problems isn't easy--they are studiously avoided by the various Reform Math curricula, which consistently place the burden on the students to explain their answers. But, when it comes to ensuring that the relevant skills are being exercised, shouldn't the burden be on the deep-pocketed textbook companies and their teams of professionals rather than on the students themselves?