Sunday, October 31, 2010

Does making children work in groups foster cooperation and kindness?

I've devoted several blog posts to critiquing the ideas of education expert and frequent New York Times op-ed contributor Susan Engel (see here, here, here, and here), including a recent post in which I object to her ideas about using group work as a way to prevent bullying. Back in August I had the privilege of being interviewed with Engel on this very topic on Rae Pica's BAM! radio. That interview has just gone live:


Friday, October 29, 2010

Math problem of the week: traditional algebra vs. CPM

I. The final page of problems in the second chapter on quadratics ("Simultaneous Quadratics") in Wentworth's New School Algebra, published in 1898:

II. The final page of problems in the second chapter on quadratics ("The Rocket Show: More about Quadratic Equations") in College Preparatory Mathematics 1 (Algebra 1), published in 2002:

3. Extra Credit:

Relate the road from Wentworth's Algebra to CPM to the new report from the National Academies about the deteriorating prognosis for future American competitiveness in science and technology.

Wednesday, October 27, 2010

Preconceived notions about place value

One thing that struck me about the math talks given at this weekend's New England Conference on the Gifted and Talented was the emphasis on manipulatives and the concerns about whether children understand place value. Are these the most appropriate things to be focusing on when it comes to students who are gifted in math? The mathematically gifted kids I know grasp place value and other aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade. But the education establishment seems bent on convincing itself that children--however gifted--don't understand place value. 

Why would you want to convince yourself of this? Because it gives you an excuse not to teach the standard algorithms of arithmetic. If children don't understand place value, then they can't understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though mathematicians disagree), using them will permanently harm their mathematical development.

So, given how nice it would be not to feel any pressure to teach the standard algorithms (because, let's admit it, they are rather a pain to teach), wouldn't it nice to convince ourselves that our elementary school students, however gifted in math, don't understand place value?

But how do you convince yourself of this? As that ground-breaking math education theorist Constance Kamii  has shown, it's child's play. All you have to do is ask a child the right sort of ill-formed question. Here's how it works:

1. Show the child a number like this:
2. Place your finger on the left-most digit and ask the child what number it is.

3. When the child answers "two" rather than "twenty," immediately conclude that he or she doesn't understand place value.

4. Banish from your mind any suspicion that a child who can read "27" as "twenty seven" might simultaneously (a) know that the "2" in "27" is what contributes to twenty seven the value of twenty and (b) be assuming that you were asking about "2" as a number rather than about "2" as a digit. 

Monday, October 25, 2010

More on conferencing about the gifted and talented

It was fun being a fish out of water at this weekend's New England Conference on the Gifted and Talented. Most of the talks centered on social and emotional approaches to social and emotional problems, or looked for emotional causes of academic under-performance, while my own mood-centered talk focused on intellectual approaches to problems that may look social and emotional but really stem from boredom and lack of intellectual peers.  Those talks that addressed math focused on math games, manipulatives, and "real life" math, while my math talk argued instead for the opposite: more hard core, abstract math--especially for the mathematically gifted.

I also got into a friendly argument with Ms. Math about whether numbers are adjectives and whether it's OK to say "one hundred and forty five" instead of "one hundred forty-five." And I won one of her $1.00 pig die while playing B-A-C-O-N.

But my most significant takeaway was a comment made by one of the participants at my "How about a Friendly Game of Chess?" talk.  We were discussing how teachers often miss the subtle teasing that goes on during group activities, and this participant pointed out that it isn't just that the teasing is often subtle and sub-audible, but that, even when heard, it may not sound even like subtle teasing to an outsider observer. When the quirky kid is the target, it's often quirky things that bother him or her. His/her classmates are as quick to figure out what pushes his/her buttons as they are with more typical victims, but the teacher may have no idea that the idiosyncratic taunts they've come up with are serving this function. In the worst case, the tauntee then acts out in frustration, thus becoming a target of one more person: his/her teacher (and would-be, should-be, protector). 

Saturday, October 23, 2010

How about a friendly game of chess?

From a talk I just gave at the New England Conference on the Gifted and Talented:

Who is to say that an intellectual, or collaborative, or competitive exchange is any less meaningful than the more emotional ways in which more typical people connect? (Raising a Left-Brain Child)

Thursday, October 21, 2010

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

I. From the 3rd grade Investigations Brain Challenges, Brain Puzzlers #3671:

II. From the final 3rd Grade Singapore Math money word problems, in Primary Mathematics 3B, p. 88:

II. Extra Credit:

Compare the challenges posed by the two sets of problems.

Tuesday, October 19, 2010

Autism diaries XXI: the fading pleasures of petty theft

When it comes to cell phones, wallets, and keys, being in our house is like being out in public. Leave any of these items unattended for a few minutes (as our guests, too, have learned) and they're apt to be snatched up and made off with. The difference is that here in our house it's not so much a matter of theft (we keep a running tab of how much money J is supposed to have, for example, and he knows it) as of mischief. You know, the sheer delight in disappearing people's valuables and then watching them react.

But even this pleasure, however intense it once was, can fade. This weekend after we realized that my husband's wallet had disappeared (I had accidentally left it out after paying the babysitter: silly me!), I asked J where he'd put it. He was in the middle of doing something on the computer at the time, and sighed, got up, wandered into one room, realized he'd misremembered, and then wandered back to where he'd been, pulled a chair over to the wall, and reached up to the ceiling molding. Retrieving it, he handed it over to my husband and returned to the computer. Nary a glint in his eye nor a smile on his face. 

But here's the rub: will such fading delights inspire an end to the mischief, or new innovations?

Saturday, October 16, 2010

Free riding off of smart self-starters

Some kids have what it takes to master core academic subjects, become well-informed citizens, and land satisfying jobs without instruction from k12 teachers, and, indeed, in spite of whatever goes on in their k12 classes.  These are, most typically, the smart, resourceful, self-starters; those who thrive in the absence of structure and are endowed with a strong drive to learn and create, and ability to learn and create independently--whether by devouring books, tinkering with machinery, improvising on a stage, or putting pen to paper, brush to canvass, or fingers to keyboard; or otherwise exploring the world around them. It helps, of course, if their parents have the resources to facilitate these activities, or to supplement them with things like private lessons, field trips, and inspiring conversation.

And if you could convince enough parents of such children that, despite all this, it would be a good idea enroll their children at your new k12 school, and if you fill your school with books, art supplies, science equipment, plenty of space, and a certain number of adult "facilitators", and then give the children the freedom to do what they want, the results would be quite impressive.  And people all around would credit, first and foremost, the school, its pedagogy, and its teachers and principal.

At least, that's my prediction for the various curriculum-free "free schools" that (dating back to 1921 with the Summerhill School in England) have been popping up around the country, one of which, the Philadelphia Free School, was recently profiled in the Philadelphia Inquirer:
The Free School, which plans to launch a pilot program in January in South Philadelphia for students ages 4 to 18, follows a democratic model of education, meaning no tests, no curriculum, no bells every 45 minutes, no separation into grades, and no teachers. The adults at the school will be called "staff" and be elected by the students each year. The students will also vote on the school's budget and serve on a judicial committee that deliberates on misbehaving peers.
The school isn't literally free; it plans to charge between $9,000 and $10,000 in tuition. Nor does it fully renounce actual instruction; it just farms out this instruction to others:
[Founder Robert Loucas] said all students enrolled in the program would receive additional schooling each weekday from a separate certified education program. The students will be homeschooled, take online courses, or be enrolled part-time at a public school so they meet their legal requirements.
How does such a school ensure that their students are mostly smart, self-starters with well-to-do parents? To some extent, this will happen automatically through selection bias. First there's the tuition. Then there's the sort of parents most likely to think this school a good match for their children. The parent of the unmotivated child, or the child with learning disabilities, or the structure-craving left-brainer, is not going to be banging on the doors trying to get in.  

But the school itself can do some strategic handpicking--assuming that it generates enough buzz (e.g. through front page articles in the Philadelphia Inquirer) to have more applicants than spots. Besides weeding out the weaker-looking students, you might do what schools like the Science and Leadership Academy (not a free school, but a free-ish school) does. Interview the applicants; ask for portfolios of their independent work and have them present it; require them to be the ones who initiate certain key parts of the application process. Shy away from the shy, the inarticulate and uncharismatic, and those who under-emote about their interests.

Is there a problem with schools like these? If parents of intelligent, driven, self-starters want to shell out this kind of money for a school that may not be adding much more value than that which comes from surrounding their children with like-minded peers, more power to them. The problem is, rather, when people look at these schools and see how happy and productive their students are, and how well they do after graduation, and conclude that this model (or certain aspects of this model) is the ideal way to educate everyone.

Friday, October 15, 2010

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

I. From the second-to-last assignment in the 6th grade Everyday Math Student Math Journal Volume 1, p. 205:

In this unit, you had the opportunity to learn a second way to construct circle graphs. Which geometric tool do you find easier to use, the percent circle or the protractor?

Which activity in this unit do you believe is an example of your best work?

Which activity in this unit did you find the most challenging? Explain why.

What is something new you learned about geometry in this unit?

II. From the second-to-last assignment in the 6th grade Singapore Math Primary Mathematics 6A workbook, p. 103:

Ali had $120. He had $45 more than his brother. After his brother spent some money on a toy, Ali had twice as much money as his brother. Find the cost of the toy.

There were 20% more boys than girls in a swimming club. After 50 girls left, there were twice as many boys as girls in the club. How many boys were there in the club?

III. Extra Credit

The student who used my copy of the Everyday Math Student Math Journal (before I acquired it through Amazon used books) had the following answer to the third question:
The part when you have to use a protractor I think because I don't understand how to use it.
Is this the kind of answer that the authors of the Everyday Math program are shooting for?

Wednesday, October 13, 2010

Teaching multi-digit multiplication to my 4th grader

(click on picture for a closer view)

Saturday, October 9, 2010

Even more artsy science and science appreciation

This year another festival has emerged to rival Briane Green's World Science Fair in its right-brained take on science... the USA Science and Engineering Festival. As described by the festival's website:

The Inaugural USA Science & Engineering Festival, hosted by Lockheed Martin, is the country’s first national science festival and descends on the Washington, D.C. area in October 2010. Opening on 10/10/10 with a concert of amazing science songs performed by over 200 children and adults at the University of Maryland, the Festival promises to be the ultimate multi-cultural, multi-generational and multi-disciplinary celebration of science in the United States. The culmination of the Festival will be a free, two-day Expo on the National Mall and surrounding areas on October 23 and 24 that will feature over 1500 fun, hands-on science activities and over 75 stage shows and performances on four stages. In addition, several exhibitors will be hosting talks and performances in their exhibit areas. The Festival is a grassroots collaboration of over 500 of the nation’s leading science organizations. The Festival has a bipartisan Honorary Congressional Host Committee with over 100 Members supporting its efforts.
The top ten exhibits listed on the website involve images of the sun and the night sky; a virtual journey across mars; a virtual helecopter ride through the DC skies; a simulation of a fighter plane ride; iphone apps for facial recognition, biometrics, and geopositioning; chatting with an interactive robot; solving a mystery using the latest Microsoft and Apple technology; meeting astronauts; meeting two actors from the hit TV show NCIS as they "pose for photos, sign autographs, and MC the Lockheed Martin Engineering Primetime demonstration"; and various interactive virtual reality demonstrations. 

Spanning two weeks and activities across the country, culminating in two-day expo on the Washington Mall, the festival is the brainchild of entrepreneur Larry Bock. As quoted in Edweek, Bok notes that:
The premise that I'm operating by is society gets what it celebrates. We celebrate Lindsay Lohan and Britney Spears and we get a lot of young people who want to be like them.
and that the Festival is:
Not a competition, it's more a celebration of science and engineering. There will be lots of hands-on activities, but also theater, art, and music, all celebrating science and engineering.
Yes indeed, Mr. Bok, society gets what it celebrates. And so the effectiveness of festivals such as these depends on what our goals are. Do we want to cultivate a generation of technology appreciators and consumers, and of people who view science as performance? Or do we want to show people what science really is--its logical and experimental rigor; its narrow focus; the strong knowledge base and hard work it requires--and give those who relish these things the educational foundation they need to become scientists, computer programmers and engineers?

If we keep avoiding the harder, less showy route, the objects of our technology appreciation and consumption will be engineered increasingly by those who are educated in other countries, while our own students will be increasingly ill-prepared, both scholastically and psychologically, for what science really is.

Thursday, October 7, 2010

Math problems of the week: two ways to calculate angles

I. From the "Measurements in a Right Triangle" chapter of Weeks & Atkins A Course in Geometry (1961), p. 401:

In the triangle XYZ, Z = 90o, x = .24 in., z = 12.0 in. Find angle Y.

In the triangle PQR, R = 90o, Q = 73o10', r = 12.50 ft. Find the remaining parts of the triangle.

A triangle has sides 10.0 in, 10.0 in., 8.25 in. long. Find the size of the smallest angle. 

II. From the "Triangles and Trigonometric Ratios" chapter of Contemporary Mathematics in Context Course 2, part B (2003), p. 403:

Suppose you know sin A = 4/5 = .8 Use the sin-1 function of your calculator to compute the angle whose sine is 0.8. (Make certain your calculator is set in degree mode.)

Use your calculator to find the measure of the angle in each of the following cases.

tan B = 1.84
sin A = 0.852
cos B = 0.213

III. Extra Credit

Discuss the possible connection between spelling out that 4/5 = .8 and telling students to use their calculators.

Tuesday, October 5, 2010

Extra time on standardized tests: pitting underachievers against slow processors

As my older son gets ready to take the PSATs in two weeks, I was shocked, shocked to find out that a significant portion of his classmates, perhaps even more than half of them, have qualified for time and a half on standardized tests.

For the most part, these are kids who've been making their through a highly demanding K12 private school without special accommodations, until such time as those high-stakes college tests start looming, at which point their well-to-do parents shop around until they find a psychotherapeutic professional willing to identify a processing speed disorder, for which a mere discrepancy between performance on timed vs. untimed activities can be sufficient justification. Competitive schools, eager to boost their college admissions statistics, can also be complicit, encouraging parents who might otherwise hesitatate to pursue this route.

While time and a half certainly aids these children--as it would aid countless others who remain undiagnosed but for whom time is a factor, including yours truly--it disadvantages those whose parents, whether out of financial considerations, ethical considerations, or mere lack of awareness, don't pursue extra time. And it also disadvantages kids with rapid processing speeds who test well, especially if they are underachievers for whom unusually high standardized test scores can make up for mediocre grades.

Time and a half may also be one reason why college admissions committees at many of the top schools have been de-emphasizing the SATs--another development that has been disadvantaging underachievers.

Is there any way to make things more equitable? I have a couple of thoughts. The most obvious strategy, of course, is to make it harder for parents of cognitively typical children to secure time and a half. But wealthy, determined parents will always find a way, and all it takes is one willing psychologist. 

What about creating accommodations that are less crude than mere time, and tailored to where actual distortions in measured vs. actual ability lie? For kids with language deficits, like my autistic son, this would mean accommodations specific to the mathematics subtests, tailored to ensure that the language of the questions involved is not a barrier to answering them. For kids with fine motor or other manual deficits, this would mean accommodations (a keyboard being the most obvious) that ensure that handwriting does not interfere with performance on the essay writing section. For kids with visual processing disorders that interfere with the ability to rapidly decode strings of letters, this might mean having the critical reading subtests read out loud. 

What about processing speed itself? Arguably it is both an integral component of intelligence and a predictor of intellectual performance (in everything from the science lab, to the tenure track, to the law office). But if we honestly don't care about processing speed as far as college admissions goes, then we should eliminate this factor for all students, and turn the PSATs, SATs, ACTs, and all and sundry achievement tests into untimed tests for everyone.

Sunday, October 3, 2010

How to make Singapore Math sound like Reform Math

From Friday's front page New York Times article:

By the time they get to kindergarten, children in this well-to-do suburb already know their numbers, so their teachers worried that a new math program was too easy when it covered just 1 and 2 — for a whole week.

...The slower pace is a cornerstone of the district’s new approach to teaching math, which is based on the national math system of Singapore and aims to emulate that country’s success by promoting a deeper understanding of numbers and math concepts. Students in Singapore have repeatedly ranked at or near the top on international math exams since the mid-1990s.
(Don't mention that, while this slower pace may characterize kindergarten-level Singapore Math, the curriculum is already significantly ahead of U.S. math by the end of first grade--especially U.S. Reform Math programs, infamous for their slow progress through actual mathematics). 
Singapore math may well be a fad, too, but supporters say it seems to address one of the difficulties in teaching math: all children learn differently. In contrast to the most common math programs in the United States, Singapore math devotes more time to fewer topics, to ensure that children master the material through detailed instruction, questions, problem solving, and visual and hands-on aids like blocks, cards and bar charts. 
(Don't mention that Singapore Math's hands-on aids cease after kindergarten, or shortly thereafter, while they persist further into Reform Math than any other math program used in the U.S. or elsewhere. And don't mention that it's the Reform Math programs, not Singapore Math, whose curricula are informed by empirically unfounded "learning styles" fads.) 
Franklin Lakes, about 30 miles northwest of Manhattan, is one of dozens of districts, from Scarsdale, N.Y., to Lexington, Ky., that in recent years have adopted Singapore math, as it is called, amid growing concerns that too many American students lack the higher-order math skills called for in a global economy.
(Don't mention that it's the Reform Math programs, with their emphasis on calculators and other "technology," and on "real world" problems and "data analysis," that harp the most on "higher-order math skills called for in a global economy.")
Bill Jackson, one of Scarsdale’s new math coaches, scribbled notes the other day as he watched a fourth-grade math class. For nearly an hour, the students pored over a single number: 82,566 (the seats in New Meadowlands Stadium, where the Giants and Jets play football). They built it with chips on a laminated mat, diagramed it on a smart board and, finally, solved written questions.

Mr. Jackson said that students moved through a three-step learning process: concrete, pictorial, abstract. American math programs, he said, typically skip the middle step and lose students when making the jump from concrete (chips) to abstract (questions).
(Don't mention that the Singapore Math materials cover content, but not specific lesson plans, and that activities like the 4th grade investigation of  the number 82,566, with its slow pace, hands-on materials, arts and crafts, and use of a smart board, are much more likely to occur in Reform Math than in Singapore Math classrooms--especially in Singapore itself.)
Singapore math’s added appeal is that it has largely skirted the math wars of recent decades over whether to teach traditional math or reform math. Indeed, Singapore math has often been described by educators and parents as a more balanced approach between the two, melding old-fashioned algorithms with visual representations and critical thinking.
Reform Math devotees don't embrace Singapore Math's visual representations and critical thinking; Reform Math critics find Reform Math's versions of visual representations and "critical thinking" seriously wanting and hardly comparable to the Singaporean approach.

It's nice to see reporters writing articles about Singapore Math  The next step is for them to look at the actual curriculum.

Friday, October 1, 2010

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

I. An assignment from the 4th grade Investigations curriculum, given in the third week of school:

II. An assignment from a similar point in the 4th grade Singapore Math curriculum:

III. Extra Credit:

Today's front page New York Times article on the rise of Singapore Math consults a Scarsdale math coach on what he views as the chief advantage of Singapore Math over American math:
Mr. Jackson said that [Singapore Math] students moved through a three-step learning process: concrete, pictorial, abstract. American math programs, he said, typically skip the middle step and lose students when making the jump from concrete (chips) to abstract (questions).
Does Mr Jackson have it right?