Why Left-Brainers Don't Get High Grades, Part II: Formative Assessment

Back when most schools used the Iowa and Stanford achievement tests, children could demonstrate skills that exceeded their current grade levels. The watered down, standards-based tests that more and more states are using under No Child Left Behind only rate children on whether they meet standards that, in fact, are low relative to grade level.

Further obscuring grade-level math skills in particular, schools are giving up traditional assessments like end-of-unit tests in favor of what's called "formative" or "authentic" assessment. As described, for example, by the standards-based report card guidelines of the Christiana School District in Newark Delaware, this involves:

• Observations of students at work (observing, predicting, testing, recording, computing, analyzing)

• Oral responses (explaining, reporting, questioning, justifying)

• Journal entries

• Samples of student work.

While these guidelines allow testing, they warn that it “should not be the primary instrument used to determine a report card grade.”

As a means of informing teachers about student learning, formative assessment has its virtues (for a recent discussion, see kitchentablemath).

But as a basis for grading, formative assessment severely shortchanges certain types of math buffs.

Worst off are those to whom math comes so easily--particularly in its watered-down, standards-based incarnation--that they can ace the actual math without doing the extra work. These students not only see no virtue in the journals, the group assignments, the discussions, and much of the homework, but are so bored by these activities that they often opt out or tune out.

But, as we will see in my next post, even those left-brain math-buffs who play by the new rules suffer lower grades than many of their less mathematically capable classmates.

Why Left-Brainers Don't Get High Grades, Part I: "Grading to the Standard"

In an accidental bipartisan collaboration, the liberal brainchild known as the Standards of the National Council of Teachers of Mathematics and the conservative brainchild known as No Child Left Behind have drastically diluted and diminished the math standards against which our children are graded.

NCTM Standards tell educators that communicating problem-solving strategies is key to learning math. NCLB tells states to devise tests to hold schools accountable to student achievement. The resulting state tests, following the standards, only award full credit to explained answers, and, so that not too many schools will "fail," dumb down the actual content.

The risk of failure has also pressured school districts to make teachers both teach to the test and “grade to the standard.”

Accordingly, the guidelines for the standards-based report cards now used by the Montgomery County District in Maryland say that grades must reflect “what students know in relation to the standards.” Or, in the words of the Christina School District in Newark, Delaware, “their progress towards meeting the state standards.”

All this discourages teachers from assessing skills that extend above and beyond what are now the official benchmarks for particular grades. Indeed, the Montgomery guidelines explicitly rule out “extra credit.”

The upshot: children who once would have been assessed for, and rated with, above grade-level math skills--e.g., a 3rd grader who rates at a 6th grade level--now merely “meet expectations.”

And many of these children, as we’ll see in my next post, end up receiving lower grades than their less mathematically apt peers.

Right-brain biases against school boys

In an Op-Ed in today's Philadephia Inquirer, veteran education reporter Richard Whitmire calls on presidential candidates to take on the under-appreciated plight of today's school boys. 

Boys, Whitmire reminds us, have long been earning lower grades than girls, and graduating from high school and college at lower rates. He cites the 23-campus, 450,000-student California State University system as typical: 2/3 of its graduates are female.

Whitmire doesn't speculate on causes, but others haven't hesitated. Typical culprits: reduced recess and the resulting restlessness of high-energy children (purportedly disproportionately male); video game addiction (ditto); learning disabilities and attention deficit disorders (ditto). 

Less obvious culprits are the various contemporary practices that shortchange left-brainers (again, purportedly disproportionately male). In The War Against Boys, Christina Hoff Sommers cites one: the shift from competitive classrooms to cooperative ones. Boys in particular, she argues, perform better when competing with classmates than when required instead to cooperate with them.

Another problem that may afflict more boys than girls, I propose, is the intrusion of language arts into math. Many boys I know, especially in the early grades, dislike writing (often struggling with spelling and penmanship) and far prefer doing math problems to communicating about them.

Finally, as I'll discuss in later posts, more and more schools are reserving their top grades for those with the kind of attentive, diligent, eager-beaver attitude that many boys fail to display--particularly in the growing number of classrooms that discourage competition and "mere calculation."

Math problems of the week: Reform Math vs. other math

One way to see how Reform Math stacks up with other math programs (whether traditional American math, or math as it's still taught in other countries) is to compare specific assignments.  Once a week, OILF will do just that.

We'll rotate between different Reform programs, including Everyday Math, Investigations in Number Data and Space, MathLand, and Trailblazers, as well some of the secondary school programs.

We'll pair up a specific assignment drawn from this set with a specific assignment drawn either from a traditional series like McGraw-Hill, or from the foreign series most popular in America: Singapore Math.  

I'll try to pick assignments that take place (assuming the curriculum in question is used chronologically) at approximately the same point in the school year.  For example, I might choose two assignments from the first few weeks of first grade, or from the last few weeks of second grade, or from approximately 2/3 of the way into third grade.

I've picked Wednesdays for this weekly feature because it's the day my daughter's math homework is due.  And so it seems appropriate to kick off this series with examples from the two first grade programs she's been using, one at school and the other at home.

Tonight's homework:

1. Investigations:  Investigation 1, Sessions 4-5, Number Games and Story, Student Sheet 7.

Assignment:  "Total of Ten"

Materials:  Deck of Number Cards [each card displays a number from 0 to 10.]

Object:  Find combinations of cards that total 10.

How to Play:

1. Lay out 20 cards faceup [sic] in four rows of five. 

2. Players take turns.  On your turn, look for a combination of cards that totals ten. Remove those cards and put them aside.

3. The game is over when no more combinations of 10 can be made.

4. List all the combinations of 10 you made.

From a similar point in the Singapore Math curriculum:

2. Singapore Math:  Primary Mathematics 1B Workbook. Exercise 61, p. 142-143:

Add:

45 + 10 + 3  = 24 + 10 + 2  =

45 + 13 = 24 + 12 =

37 + 10 + 3  = 76 + 10 + 4  =

37 + 13 = 76 + 14 =

25 + 10 + 17 = 48 + 10 + 6  =

25 + 17   = 48 + 16 =

42 + 30 + 6  = 35 + 40 + 2  =

42 + 36 = 35 + 42 =

55 + 20 + 5  = 28 + 60 + 2  =

55 + 25 = 28 + 62 =

37 + 30 + 8  = 65 + 20 + 9  =

37 + 38 = 65 + 29 =

In short, a group activity (Investigations) vs. a solo assignment (Singapore Math). And a haphazard exercise in finding multiple ways to sum single-digit numbers to 10 (Investigations), vs. a structured exercise in adding two-digit numbers by breaking up one number into tens and ones (Singapore Math).

Math vs. language arts: an increasingly blurry line

"Can you help me understand why your son had so much trouble with this math problem?" a veteran first grade teacher asked my friend E at a recent parent-teacher conference.

The problem: "explain what this graph shows." The graph: a representation of the latest class survey results. Her son's answer: a sentence about one thing it showed.

E was happy to see that her son had actually written a whole sentence. To her, the source of his "trouble" was obvious: he hates writing. The only thing that surprised her was that a teacher was interpreting a six-year-old's verbal brevity as "trouble with math."

Such entanglement of grade school math with language arts dates back to the 1989 NCTM Standards, with its emphasis on communicating about math. NCTM is particularly zealous about this when the topic is representations (e.g., graphs).

So zealous that an intelligent, veteran teacher mistakes writing blocks for math deficiency. And assigns a report card grade of 2 (basic) on a 1-4 (4=advanced) scale to a child who is actually quite good at math--at least as measured by the ease with which he solves the non-writing-intensive math problems that E gives him at home.

Reform an even greater threat to high school math?

An anonymous poster sent me a link to this article in the Ledger-Enquirer, which reveals the latest front in the secondary school math wars: the state of Georgia. 

Georgia's Department of Education is now calling on schools to switch from the traditional Algebra I-Geometry-Algebra II-Trig sequence to a program in which, in the words of "instructional specialist" Suzanne Evans, "It's all going to be integrated."

Pondering this--my Reform Math-afflicted children haven't yet hit middle school--I suspect that there's even more at stake here than in elementary school math.

At least, however superficial, haphazard, and sloppy the pedagogy, Reform elementary students still come across most of the topics that traditional students do. They get at least a passing exposure to most of the standard algorithms, and those that have been chucked--e.g., long division or inverting and multiplying--are still relatively accessible to them and to their parents. When all else fails, most parents can teach most children these topics at home. Especially if they are left-brained math buffs.

But mixing up and reconstituting algebra, geometry, trigonometry, and calculous into a novel, "integrated" stew opens up all sorts of opportunities for Reformists to ditch whole topics. Especially those that don't integrate nicely with those things they consider important: calculators and real world applications. Rotated parabolas? Polar coordinates? Proving properties of triangles from the axioms of geometry? 

These topics won't be ones that most parents have at their finger tips, however eager their children are to learn about them. 

Georgia's motivation for integrating math? Below average math scores. Its education experts may be thinking of continental Europe, whose secondary schools use an integrated curriculum. But one look this curriculum--I've used it myself--shows it to be much more rigorously mathematical than anything these American Reformers have in mind.

As with all our other reforms, we lament how poorly our students do compared with counterparts in Europe and Asia, but can't be bothered to look at how these countries educate their children.

American Exceptionalism--it rears its pompous head in the most surprising of places.

Front page accolades for right-brained classrooms

Breathless articles on right-brained classrooms are a staple of today's education reporting. Consider a front page article in today's Philadelphia Inquirer about the University Park Campus School in Worcester, Mass. This public high school receives student teachers, mentors, and training for school staff from nearby Clark University, and "international recognition and numerous accolades for its ability to take low-performing students and turn nearly all of them into first-generation college-bound teens."

The Inquirer article opens with three 9th-graders huddling around an algebra problem. Which summer job would be more lucrative for the son of Mr. Knittle, their teacher? The assignment stipulates:

Explain your reasoning.  Show all work that supports your effort.  Write a note to Mr. Knittle, telling him which is the better offer, and convince him to take that job. Make sure you COMPLETELY explain how to change this into a mathematical exercise that everyone can understand.

Tellingly, the article makes no mention of the mathematics in question. Presumably it involves equations for hourly rates, hours per week, and overtime. It's hard to see how it could include much more than simple, two-variable, linear equations--for instance, the factoring of polynomials, solving of quadratic equations, and graphing of conic sections that once defined 9th grade college-track math.  

Instead we have the usual modern trappings: "real life" problems, group work, and an emphasis on non-mathematical communication--persuasive letter writing, language that "everyone can understand"--that bores math buffs and stymies the many on the autistic spectrum who excel in math but struggle to put words together and understand the perspectives of others.

Mr. Knittle's classroom reflects these increasingly fashionable priories. Students sit in clusters "to foster interaction." "Make sure you're an expert, talk to each other," he tells them.

University Park's English classes are similarly right-brained--at least as described by the Inquirer. In one 11th grade English class "students spent the morning looking through a portfolio of work they have written since seventh grade and reflecting on their growth." Self-esteem over self-education.

And personal inspiration over literary analysis. One student's Adrienne Rich presentation seems to center on how the poet learned "how to live" from books, and on how she herself understands Rich's feelings: "I feel like books really do have a life, and they gave me life as well." 

I agree. And I'm all for students appreciating literature. In previous generations, too many high school lit classes turned off too many students with too serious an approach to too many gloomy works. (Death in Venice and Ethan Frome come to mind; yes, I attended such classes.)

But reducing literature to self-help, inspiration, and life lessons, and reducing literary analysis to personal feelings about a text's "messages," raises a generation not of literature readers, but of advice manual consumers. And it shortchanges any budding wordsmiths--and other left-brainers--who might benefit from close attention to alliteration, simile, and other techniques.

 

As it turns out, University Park has had to modify its curriculum somewhat. In 2003, the five graduates who enrolled at Clark University all quit. The school now requires high school students to audit a Clark course and meet with Clark professors, and 12th graders to take high school classes "structured as college courses, with syllabi, lectures, and lots of independent work."

As growing numbers of college courses relinquish such structure for hands-on, group centered activities, such 11th hour adjustments may no longer be necessary. For many professors are as enthusiastic about right-brained teaching methods as journalists are.

New data on learning style differences: gender and ethnicity

In support of such right-brained Constructivist practices as hands-on, group-centered learning, American education experts are quick to enlist multiculturalism. Claiming that traditional instruction favors white, Western males, they happily hold forth on how much more relational, holistic, multi-modal, and/or social girls and nonwhites are.

Much of this, of course, is sexist, racist bunk.  But look closely, and you'll find some actual respected research that suggests where some of this bunk is coming from.

Recent studies by Simon Baron Cohen, for example, suggest that girls tend to be more right-brained (empathetic, social, holistic) and boys more left-brained (systematic, unsocial, analytical).  Baron-Cohen stresses that these are not absolutes, but tendencies--a caveat too subtle for many of our education experts.

As for race/ethnicity: since the turn of the millennium, several studies are reputable enough to have made it into the science section of the NYTimes.  

One article, back in 2000, reports on two studies comparing U.S. students with counterparts in Japan and Korea.  Asked about an animated underwater sequence in which a large fish swims among smaller ones, the Japanese students focused more on the general scene; the Americans more on the big fish.  Asked about someone who was forced to write an essay endorsing a particular position, the Korean students were more likely than the Americans to recognize how external pressure may have influenced the writer's conclusions.

A second article in this week's NYTimes reports on a study comparing how Japanese and Western students judged the emotions of a smiling child flanked by four others who smiled in one picture and frowned in the other.  Only the Japanese students gave the central child different happiness ratings depending on his companions, assessing him as less happy when they were frowning.

All three studies suggest that East Asians may be more sensitive to certain contexts than Americans and Westerners are.  

But:

1. The studies only compared Americans and other Westerners to a tiny subset of non-Westerners: Japanese and Koreans.  (And, as if these studies were typical, they only examined a particular subpopulation of each culture known as "psychology majors").

2. Their conclusions are limited to the relative weights that such subjects assign to certain types of background information.

3. Whatever these conclusions might suggest about holistic thinking, Japan and Korea use a much more rigorously analytical math and science curriculum than we do, and their students consistently outperform ours in these especially left-brained of subjects.

In short, to conclude from these three studies that members of all non-white, non-Western cultures are unequivocally more right-brained than white Westerners are would be a huge leap of Constructivist faith.

Chess in the classroom: left-brained, right-brained, or hare-brained?

The state of Idaho, today's NYTimes reports, will universalize coverage for First Move, a chess curriculum that currently serves 100 grade school classrooms. Next year's education budget guarantees funding for all 40,000 Idaho 2nd and 3rd graders. The estimated cost, $200,000-$250,000 a year, includes DVDs, DVD players, training sessions for teachers, and take-home chess sets for students.

What exactly does First Move accomplish? No studies show chess actually benefitting children. State Superintendent Tom Luna cites anecdotes.

One First Move-trained teacher, for example, observes how video games, iPods, and TV isolate today's students, whereas with chess “they learn give and take." She adds, "There are courtesies that you follow. It has been really beneficial for them.”

“One of the things that we hear is that too much of what we do is based on rote memorization,” says Superintendent Luna. “The part I really like about this program is that kids are thinking ahead.”

Chess strikes me as at least as analytical and left-brained (chains of likely logical outcomes) as it is holistic and right-brained (whole-board configurations). Indeed, for skills both left-brained and right, it may be the perfect fusion.

But classroom chess, as characterized by these Idahoans, sounds like yet another unequivocally right-brained Constructivist move: a time-consuming and empirically unfounded substitute for rote learning; a vehicle for vaguely defined "higher level thinking" and social development rather than academic achievement.

Furthermore, we must ask:

--Does First Move pair up students with similar chess skills? Or, as with so many other contemporary classroom practices, does it favor mixed ability groupings that stress good sportsmanship over cognitive challenge?

--What are they giving up during the hour per week of classroom chess? Rote facts like where Afghanistan is and who sits on the Supreme Court?  Math and science problems in which, as much as in chess, they learn to think ahead?

Reform Math and nonverbal learning styles

kitchentablemath has been hosting a lively discussion on Reform Math's take on children with different learning styles. Reviewing the National Council of Teachers of Mathematics' January, 2008 paper, "Equity in Mathematics Education," SteveH observes the NCTM making such lofty pronouncements as:  

The school community acknowledges and embraces all experiences, beliefs, and ways of knowing mathematics.

and:

All students have access to and engage in challenging, rigorous, and meaningful mathematical experiences.

In fact, Reform Math proponents display much more interest in celebrating supposed cultural differences in math appreciation than in respecting documented cognitive differences:

Such practices empower all students to build a relationship with mathematics that is positive and grounded in their own cultural roots and history.

The practices in question include a group-centered, hands-on pedagogy that favors a concrete, holistic, social, altogether right-brained learning style.  In furthermore requiring all children to explain their answers verbally, these practices downgrade those who solve the typically easy (as compared with traditional math) problems in their heads without thinking them through in words.  

Here's an excerpt from an email exchange I had with our school's math consultant, an outspoken Reform Math proponent who holds a Ph.D. in Educational Leadership from one of the top education programs in the country.  This exchange took place in January, 2008--the same time as NCTM published its Equity paper.

Lefty:
If I'm remembering correctly, when we were discussing mathematically-gifted but language-impaired kids, you told me that the curriculum allows strategies to be demonstrated in words, numbers, OR, pictures. Is that right? I'm now wondering about problems that use the word "explain", as in "explain [how you got] your answer" rather than "show" ("show how you got your answer"). Is it still the case that the child could answer in numbers (e.g. a series of number sentences) or pictures (e.g. a geometric representation of fractions?)? I'd like to get a better handle on just how well the curriculum accommodates, in particular, the mathematically-gifted but language impaired children that comprise so many of the children on the autistic spectrum, who often literally see the answer, pictographically or numerically, with no accompanying words in their heads.

Math Consultant:
I don't have a simple answer to your question, as it depends somewhat on the problem, concept, grade level & teacher. Most problems do ask the students to show how they got an answer using pictures, numbers, and/or words. There is no set rule as to what is meant by "explain" but the idea would be that someone else should be able to look at the work and know exactly what the student did--often a combination of pictures, numbers and words is necessary to communicate clearly in mathematics. As far as the curriculum is concerned, a central goal is for students to learn to express mathematical thinking through drawing, writing and talking. A teacher would therefore work to develop students' skills in all three areas. Also keep in mind that instruction is also driven by state tests, and on the PSSA students need to explain their thinking in writing on some of the open-ended questions. Therefore teachers need to have students practice this skill throughout the year...

L:

So, to follow up, if the idea is for the teacher to know exactly what the student did, what about a child gets the answer automatically in her head, and simply doesn't know what she herself did in her head (i.e., the answer just "came to her")? In this case, could the student simply write "mental math" as the explanation? A Narberth school allows the mathematically-gifted, non-autistic son of a friend of mine to do this, even on his PSSAs. (Though I don't know whether PSSA graders mark him off for this!)

MC:
Well no, the idea is not just for the teacher to know exactly what the student did, but rather for the student to learn to communicate his or her thinking in a clear way. So "mental math" would not be an adequate explanation. I would certainly think the child would be marked off on the PSSA for that response, given the guidelines that are put out (unless there is some special accommodation in place.) Communication has been a goal of reform mathematics programs since the publication of the standards in 1989.

L:
So if a child doesn't know what his/her thinking was in solving the problem (because it was all subconscious and/or nonverbal thinking), how could this child possibly explain his/her thinking in words? Except to say "I solved it pictographically with the following pictures in my head..." or "it just came to me."
The only alternative I can think of is that such a child would have to imagine how a more verbal person would have solved the problem, and then explain how this hypothetical person would have solved it.

MC:
I can certainly appreciate the fact that it would be much more of a challenge for a mathematically gifted but language impaired child, but your question is beyond the realm of my expertise. Perhaps your research will shed some light in this area? I hope I have answered your question about the goals of the curriculum.

L:
Yes, you have. 
Unfortunately, these goals, with their narrow notion of what math is about, shortchange the mathematically gifted (ALL those who see the answers nonverbally, including many mathematicians I know). This is, as you know, one big problem I have with the curriculum.
It is too bad that those who have chosen these goals and this curriculum don't seem to know much (or care much?) about how nonverbal children and mathematically inclined curriculum solve math problems.

Too bad, in particular, for the increasing number of children on the autistic spectrum who are being mainstreamed into regular classroom and whom the education establishment purports to be embracing.

And too bad for the mathematically gifted, whose contributions our mathematically impaired nation must be doing all it can to nurture rather than discourage. 

Presidential platforms on math, science, and schools

OILF studiously avoids war, environmental regulation, healthcare, and economic policy. Its focus on left-brainers and right-brain biases should give you no inkling of my opinions on these other issues--however important they are in general, and in the presidential election.

Here the political issue that matters most is math and science education. As Alan Leshner, chief executive officer of the American Association for the Advancement of Science asks in the opinion page of today's Philadelphia Inquirer, what do Clinton, McCain, and Obama propose to do about the abysmal rankings of American 15-year-olds with respect to their peers in 29 other wealthy nations: 17th in science and 24th in math?

Clinton

The AAAS website has Clinton advocating new NSF fellowships for math and science professionals interested in public school teaching. The education page of Clinton's own website makes no mention of math and science education; as for education in general, her only proposal is that thousands more "outstanding" teachers and principals be recruited.

McCain

The AAAs website has McCain advocating improving school performance through accountability, standards-based assessments, and competition for students. The education page of McCain's own website has him highlighting school choice as key. Nowhere does he mention math and science in particular.

Obama

The AAAS website has Obama calling for an increase in the number of students pursuing degrees in math, science, and technology, and a Teacher Service Scholarship program to recruit such graduates. He also wants to expand access in the public schools to computers and broadband connections. Finally, he wants to invest in science education r&d to determine which curricula and instruction work best. The education page of Obama's own website specifically lists math and science education as a national priority.

OILF's assessment

Computers and broadband are icing on the cake.  Science education r&d postpones improvements that must occur yesterday (and, if it involves curriculum consultants and education professors, may not lead anywhere good). The key questions are how to recruit good teachers, and how to hold schools accountable, now.

Recruiting good teachers

Here, McCain makes no proposals, and Obama and Clinton make just one: fellowships/scholarships for math and science teachers. 

As this blog discusses earlier, however, international comparisons question whether money makes the difference. Far more effective may be freeing applicants from education coursework and allowing them greater autonomy in the classroom.  But no candidate espouses a position this radical. 

Holding schools accountable 

Topping Clinton's education proposals is scrapping No Child Left Behind. She offers no alternatives for school accountability.  

Obama's education proposals include funding NCLB, improving how it assesses student performance, and making it support rather than punish failing schools. He remains vague on how he would alter assessments, and on how he would support failing schools in ways that prompt improvement. Whether NCLB has improved classroom instruction is highly uncertain; no less uncertain are the consequences of Obama's proposed tweaks.

Much more radical, and promising, are McCain's proposals for competition and school choice. Failing schools that hemorrhage students are much more motivated to improve--or shut down and make room for ones that can--than those that continue to receive our support. Parents who can choose their public schools are less likely to abandon them--as many politicians do, Democrats and Republicans alike.

Once public schools compete like private schools, they're more likely to behave as such, hiring teachers more for their talents than for their paper credentials and seniority, showing them respect, and deferring to their judgment in picking their materials, planning their lessons, and teaching their students. All this, as we've noted, is essential to attracting and retaining the best teachers, including the most elusive ones:  those qualified in math and science.

Unfortunately, only McCain can propose this most promising of reform strategies. However Clinton and Obama may privately feel about school choice, publicly supporting it is political suicide. For, as Democrats, they are accountable to one of the largest blocks of reliably Democratic voters: the education establishment, whose most powerful opinion-makers oppose school choice more resolutely than almost any other education reform.

Social norms vs. the alternatives

"Indeed, it has been said that democracy is the worst form of government except all those other forms that have been tried from time to time," observed Winston Churchill in a speech to the House of Commons back in 1947.

I feel the same way about free market capitalism.

So, reading in today's NYTimes book review of Predictably Irrational that author Dan Ariely prefers "life with fewer market norms and more social norms," I shuddered. What kind of norms are we talking about here?

Does the MIT-trained economist mean those that emerge freely and democratically from the unenlightened masses?  Such norms, of course, are the very worst kind there are.  

Except for those other kinds "that have been tried from time to time:" norms handed down from on high by those who've managed to convince themselves that they know better than the rest of us do what constitutes decent social behavior. 

I'm thinking, of course, of the education establishment.  

Consider what its norms have done to our public school report cards.  Children today are rated not just for academic achievement and classroom behavior, but for things like "socially appropriate behavior," "working cooperatively," and "participating in large groups." Check out these two grade 3 report cards from Michigan and New York, links to which were passed on to me by concernedCTparent.

These sorts of ratings end up penalizing certain children--the shy or otherwise unsocial, the child on the autistic spectrum, the child with Asperger's Syndrome--for core aspects of their personalities over which they, their parents, and their teachers have scant control. 

Today's right-brained classrooms aside, sociability isn't an academic qualification and has no place in report cards. Many of our most accomplished left-brainers--mathematicians, scientists, engineers, computer programmers--don't work cooperatively, participate in large groups, or display what others would consider "socially appropriate behavior."

True, unsociability does disqualify people from certain jobs: most obviously, those that involve dictating social norms from on high. But I'm guessing that most unsocial people have little desire to make that kind of contribution to society.

False dichotomies in math and social skills

Math

The Washington Post's report on the National Mathematics Advisory Panel repeats the tired dichotomy of concepts-oriented Reformists vs. memorization-oriented Traditionalists. It ignores that:

1. No memorization means no knowledge means no foundation for concepts.

2. Memorizing arithmetic tables frees up short term memory for higher level math concepts.

3. Traditionalists teach concepts.  Some of the deepest conceptual instruction appears in long out-of-print math texts (one of which I've used with my son) that are nothing if not traditional.

Social skills

Then there's David Brooks' social skills dichotomy in today's NY Times Op-Ed.  Speaking of Eliot Spitzer and like-minded alpha male strivers, he pits those people (presumably most of us, including Brooks) capable of genuine intimacy towards friends and lovers against their "emotionally avoidant" counterparts: those governed by ambition, false intimacy, schmoozing, deference to bosses, narcissism, and... Asperger's Syndrome!  Brooks ignores that:

1. Social climbing requires social skills that Aspies fundamentally lack.

2. Aspies are among the least socially manipulative and narcissistic of all of us.

3. A number of factors can block intimacy:  some cognitive (e.g., difficulties with perspective-taking or Theory of Mind); others emotional (e.g., selfishness, social insecurity, self-absorption, emotional neediness, and narcissism).  Some are more controllable, and thus more reprehensible, than others. 

The moral 

The right-brain view isn't always the right view:

1. Not all concepts depend on right-brained re-conceptions of math.

2. Not all social sins stem from left-brained deficits.

Earning high grades in Reform Math

A's in math is one of our family's left-brain traditions.  Many of us are mathematicians, scientists, economists, and programmers, and even those with learning disabilities or non-quantitative careers have maintained the streak of top grades in grade school math--at least through trigonometry.  Then, a year and a half ago, my daughter entered grade school.

Because her oldest brother is old enough to have escaped Reform Math, and her middle brother has autism and an IEP that lets him do algebra independently, she is the first in our family to be fully immersed in the Reform Math curriculum that has recently permeated almost every school here in Philadelphia.  And instead of receiving the top grade of 4 ("advanced"), she consistently gets 3s ("proficient").

Perhaps our elementary schools are rare exceptions to two rules.  

Rule #1 is the tendency of all assessments--letter grades, grades out of 100, teaching evaluations, employee evaluations, surveys, and informal expressions of preference--to divide into four gradations that translate, roughly, into "excellent," "good", "fair," and "poor."

Rule #2 is grade inflation, rampant throughout high schools, colleges, graduate schools, and even the compliments we bestow on our children and friends--for faint praise, invariably, is damning. Could it be that our elementary schools have bucked this trend for grade deflation?

Apply rule #2 to rule #1, and the top grade means not just "excellent," but "good," while the next highest shifts down to "average."

But if elementary students rarely get 4s, I'm not worried that continued 3s in math will compromise my daughter's options for one of the few magnet high schools that offer Philadelphia students a decent secondary school education.  

Otherwise, I not only worry, but question. What exactly are my daughter's 3s based on? One thing is certain: they don't reflect her math skills.

I know this because, in the Singapore Math she does at home, she is doing problems that are significantly more advanced than those she's doing in Reform Math.  For example, she's currently solving, accurately and without help, Singapore problems like:

--4 children share 12 crackers equally.  How many crackers does each child get?

--20 less than 98 is ___ 

Meanwhile, in her Reform "Today's Math" book, she's currently (accurately and without help) solving problems like:

--How many black triangles are there in Pattern A?

--There are 9 counters in all.  Four of them are next to the cup.  How many are under the cup?

Most kids in her class, so far as I know, aren't doing simple division and two-digit subtraction on the side.  And there are no opportunities to demonstrate such skills in the classroom.

Indeed, this is the crux of the problem with the new math grading system.  The actual math skills it assesses are so basic that most students are at ceiling.  The only way to distinguish them is through non-mathematical aspects of their performance.  (For similar observations, see kitchentablemath).

The Pennsylvania Math Standards on which the Philadelphia public schools base their grades, in fact, include numerous non-mathematical factors: explaining in words, drawing pictures, manipulating objects. Perhaps my daughter's explanations and drawings aren't as elaborate as some of her peers'. Perhaps she doesn't complete hands-on tasks as quickly as others do. And perhaps her shyness and passivity keep her from making oral contributions that "demonstrate superior understanding of concepts, skills and strategies" and from "independently explor[ing] ideas and topics:" two of her report card's benchmarks for grades of 4.

It's of course way too early to say just how strong my daughter's mathematical talents are.  But I can't help wondering how many math buffs are being lost in the new system, and what this means for both their future, and that of this country.

Rethinking traditional academics

One good thing about the Constructivist revolution is that it provokes fresh thought about the grade school traditions we all take for granted.  Why, for example, should we bother teaching English grammar to native English speakers?  

Why, indeed, I thought last night as I watched my daughter mechanically underline the verbs (or "action words") in a vocabulary list for her language arts homework.  What was she learning that she didn't know already?

Modern linguistics tells us that we master the grammar of our native languages automatically, without explicit teaching.  This, indeed, is one of the first lessons of linguistics 101.  But there's grammar, and then there's grammar.  

On the one hand, there's basic, intuitive grammar. Even if we native English speakers don't know the labels "subject," "verb," and "object," we know to put the subject first, then the verb, and then the object.  This is the focus of linguistics 101.

Then there's grammar as style, or what distinguishes good syntax from bad.  This is--or should be--the focus of writing 101. For it's where we native speakers make mistakes--all the time. Dangling our modifiers, we say "returning home, the front door was ajar." Failing to keep conjoined phrases parallel, we say "a time not for thinking about it but acting on it."  Many of us don't master good syntactic style automatically, but depend on explicit teaching.

To see a dangling modifier as such, you need to identify the subject of the main clause (e.g., the front door), infer the tacit subject of the modifier (e.g., whoever is returning home), and notice that they differ. To see a lack of parallelism as such, you need to group the sentence into phrases, notice which phrases are conjoined (e.g., "for thinking about it" and "acting on it") and determine whether they have the same basic syntax (e.g., do they both begin with prepositions?).

Here it helps to visualize sentences as trees.  Traditional grammarians use the diagrammed sentence, but I prefer the more transparent syntax tree of modern linguistics. To construct such trees, you need to be able to identify the different parts of speech, just as my daughter was starting to do last night.

Recognizing good style, in other words, means becoming consciously aware of the grammar we all know intuitively.

What about long division?  This is one of the traditional algorithms of arithmetic that our right-brained Reform Math has marginalized. Ever fewer students can automatically, effortlessly, work their way through its many steps.  Does it matter?

Just like fractions, long division resurfaces in algebra.  It's often the easiest way to turn improper rational expressions into sums of proper ones.

So later last night, watching my son attempt to long-divide (x + 2) into x to the 4th, I asked myself whether it helped that he already knew how to long-divide 12 into 10,000. As soon as he started to flounder, the answer became obvious.  Just as I'd reminded him last week how common denominators work, I was able to reboot him on long division--in both cases by revisiting what we'd already helped him master back in arithmetic.

The assault on reason is bipartisan

In her NY Times review of Susan Jacoby's "The Age of American Unreason," Michiko Kakutani singles out conservatives as the main political force against science and reason.  They include the Bush Administration, religious fundamentalists, and those who insist on local control and funding of public schools. The other culprits, Jacoby and Kakutani agree, are psycho-cultural: the rise of video, the internet, and a "culture of distraction"; the decline of print and attention spans; the triumph over self-education of self-improvement and self-esteem.

But within education, religious home schoolers and Creationist school boards aside, the biggest assault on science and reason comes from the political left--from the public education establishment that is one of the largest blocs of reliably Democratic voters in this country. 

It is, in part, the power-brokers within this monolith--curriculum consultants, education professors, and the National Council of Teachers of Mathematics--whom we can thank for the facts that, quoting Kakutani, "two-thirds of Americans cannot name a single Supreme Court justice," and "American 15-year-olds rank 24th out of those of 29 countries in mathematics literacy."  

With near unanimity, our education experts speak out against the "rote learning of facts" that underlies knowledge, and the "mindless algorithms" and "mere calculation" that underly math. What Kakutani says about conservatives--"conservatives have turned 'intellectual' into a dirty word in politics"--applies as well to the arbiters of our curricula.

Jacoby's book bemoans the absence of national education standards.  In fact we have them, especially for math:  the National Standards of the National Council of Teachers of Mathematics; the state tests, mandated by No Child Left Behind, that measure students against these standards; and the curriculum packages that enact them, designed and funded by the Directorate for Education and Human Resources of the National Science Foundation.

Jacoby laments America's insistence on local control.  But most of our public schools lie within large municipalities that have consolidated their control over education. No longer can such schools choose their textbooks and curricula; instead they must defer to superintendents and boards of education, which, in turn, defer to education experts who preach their party line.  

Compared to other countries, we in America enjoy the worst of both worlds. Localized funding, with all the self-reinforcing economic disparities it brings, and centralized control by unelected non-experts in math, science, and knowledge, accountable neither to voters nor to reason.

Social pressure, self-esteem, and cooperative learning

During a segment on today's NPR Morning Edition on WHYY in Philadelphia, family therapist and Philadelphia Inquirer columnist Dr. Dan Gottlieb discusses a study in the Archives of Pediatrics and Adolescent Medicine that links popularity to weight gain.  Adolescent girls who perceived themselves as less popular were more likely to gain excess weight over the study's two-year period.

The culprit, Gottlieb believes, are our competitive schools.  Competition raises pressure, and, "since there's no self-worth in doing better than others," only lowers self-esteem.  Low self-esteem makes girls form cliques and alienate, scapegoat, and bully one another.

The solutions, Gottlieb proposes, are reducing school competition and "encourag[ing] our daughters to be with peer groups who are socially accepting and not as competitive."

By now, Gottlieb's concerns and solutions are old hat.  Our schools have long striven to raise self-esteem by lowering academic competition.  They've reduced academic tracking and academic awards, and increased social promotion and "cooperative learning."

But this has heightened social competition.  Cooperative learning requires all students to work in groups with peers--even those who prefer working alone.  Many classrooms don't offer each student a "socially accepting" peer group.  In general, the more time in groups, the more occasions for alienating, scapegoating, and bullying. 

All this grows particularly nasty during junior high and high school, not just because adolescence is adolescence, but because much of the group work is homework.  It occurs not in classrooms, where teachers might supervise students, but in dens and living rooms from which parents tend to keep their distance.

Cooperative learning thus lowers the social self-esteem of the unsocial child, who once could avoid all but the most accepting of peer groups. And it reduces his or her chances for academic esteem.  However much we'd like to believe otherwise, there is self-worth in doing better than others--lots of it.  Reducing academic honors while increasing social competition simply favors the socially savvy over the analytically talented.

Science and science prodigies

Science now turns off so many people that Philadelphia's nearly 200-year-old Franklin Institute Science Museum has shortened its name to The Franklin--treating science like the grease that, accordingly to urban legend, KFC has tried to hide behind its own recent name abbreviation.  

As today's Philadelphia Inquirer notes, with 3D movies about U2, a crew of storm troopers that hawks the current Star Wars exhibit, and recent exhibits on King Tut and the Titanic, The Franklin's box office goals have it focusing less and less on science.  To today's visitors, it seems, almost any other subject is preferable.

So it's refreshing to see an article in today's New York Times profiling a high school "science prodigy," and observing that more and more schools across the region are trying to cultivate star science students.  Key to these students' success, reporter Joseph Berger observes, are teachers who recognize their talents.  

And key to recognizing scientific talent, I must add, is looking beyond the science fair poster to the science that hides behind it.  Many budding scientists lag in their organizational, communication, and graphic-arts skills, and today's interdisciplinary project-oriented priorities too often let these factors trump meticulous analysis and rigorous experimental design.

Teacher pay and teacher quality

An article in today's New York Times profiles a new NYC charter school, The Equity Project, that plans to pay its teachers salaries of $125 k plus bonuses. TEP will fund these salaries with large class sizes (30 students), reduced support staff (only 2 social workers), and extended teacher responsibilities and work hours. To qualify, applicants need scores in the 90th percentile on the GRE, LSAT or GMAT, and in their subject area tests, and "excellent" grades in their subject area courses. 

Because TEP's salaries are so high relative to what other schools pay-- twice the average for NYC teachers and 2 1/2 times the national average--you can bet it will attract at least 28 highly qualified applicants for its 28 teaching slots.

But, like so many other worthy models, this one won't replicate on a large scale. Whether it's higher than average salaries, smaller than average class sizes, better than average students, greater than average control over the curriculum, or some combination thereof, what attracts the best teachers to model schools are the perks they offer relative to other schools.

To attract the best teachers to all schools, we must change the demographics of the entire applicant pool--a pool which currently scores on average in the bottom third on their SATs. This is where the connection between teacher pay and teacher quality gets murky.

Internationally, there is huge variation in the academic credentials of those who apply to teaching jobs, with teachers in Japan and Singapore coming from the top third of high school graduates. But a recent (October, 2007) article in the Economist shows no correlation between teacher pay and teacher quality. The determining factor instead appears to be teacher status.

But that only begs the question.  Surely how much status we assign to different professions depends largely on how impressed we are by the people who follow them. 

So what is it, besides pay, that will get the ball rolling, attracting better applicants to our schools, increasing the status of the teaching profession, attracting even better applicants, and so on?

Even now, many highly academically qualified college graduates would love to teach in public schools. What turns them off are the hoops they must pass through first. The most annoying of these--and I speak, in part, from first-hand experience--are the teacher training courses. Typically dull, intellectually disengaging, divorced from the specific subject matter (language arts, math, science, etc.), and geared more towards indoctrinating students in the dominant, jargon-infested teaching philosophy (right-brained Constructivism) than in educating them about practical teaching strategies, these courses specifically deter the smartest, most intellectually curious, most pedagogically talented, and most left-brained (essential for math and science instruction) of potential applicants.

To improve the overall applicant pool, therefore, we must begin by revamping these courses. Or, as The Equity Project appears to have done (its website mentions absolutely no teaching training coursework requirements), scrap them altogether.  Indeed, TEP's success may depend as much on allowing applicants to forego teacher training courses as on paying them high salaries.

Politics and the Math Wars

Recent developments will probably only aggravate our already overly politicized math wars. An article in yesterday's Wall Street Journal reports that the Bush-appointed National Mathematics Advisory Panel "is expected to urge the nation's teachers to promote 'quick and effortless' recall of arithmetic facts in early grades, mastery of fractions in middle school, and rigorous algebra in high school or even earlier."

That this panel was appointed by the Bush administration will be enough for some people to peremptorily dismiss its recommendations.  Many in the education establishment have long branded as "reactionary" any reforms that smack of drills, rote memorization, and back-to-basics--along with anyone who promotes such reforms (e.g., E.D. Hirsch, Diane Ravitch, Charles Sykes).

Politics, of course, has no more business in math than in evolutionary biology. Our grade school math curricula should be devised and chosen by the following professionals and no one else:  

(1) Mathematics professors, who know better than anyone else which concepts students must master in order to do college-level math and science.

(2) Cognitive scientists specializing in math acquisition, who know better than anyone else how children learn math and which math skills people use in everyday life.  

(3) Our most successful grade school math teachers, who know better than anyone else which classroom teaching strategies work best.  

That's it.  

No curriculum consultants, no education professors, no math textbook publicists--unless they also happen to be mathematics professors, cognitive scientists (real ones who publish in cognitive science journals; not education journals), or former grade school math teachers with years of hands-on experience and outstanding success in the classroom.  

And, finally, no high-falutin' education philosophy pitting the right-brained "relational understanding," "real-life" problems, and "communicating about math" in peer groups against the abstract manipulation of symbols--all that left-brained stuff dismissed all too often as "mere calculation."

Why teach fractions?

So asks Dennis DeTurck, an award winning math professor and dean of the college of arts and sciences at the University of Pennsylvania.  Children don't understand fractions; fractions are as obsolete as slide rules.  Calculators and decimals let us breeze where once we slogged, baffled by expressions like 1/2, by finding common denominators, and by inverting and multiplying.

Of course, many children--left-brainers and math buffs--start grasping fractions as early as first grade.  And surely all children must master them in order to advance through algebra?

Pondering this, I once asked Professor Deturck about rational expressions like 1/x and y/z. How do you express these as decimals and manipulate them with calculators?

Maybe we should wait until algebra before introducing fractions, he replied.  Or (see above link) until calculus.

Those of us who teach children know well the pedagogical nightmare that arises when we introduce two tough concepts at once-- e.g., fractions and variables, or fractions and derivatives.  

A second reason for fractions before algebra struck me last night.  I was helping my son through an algebra problem, and amid all the messy denominators--e.g., (a-b)*(x-a)--he'd lost sight of how to find common denominators.  It was only when I gave him an analogous problem with numbers in place of variables that he rediscovered the algorithm and why it works.  Had fractions with numbers not been something familiar we could return to, he might have continued to flounder.

Girls as justification for social classrooms

In the cover story of yesterday's New York Times Magazine, which asks whether we should teach girls and boys separately, we see yet another education professional repeating one of the most commonly cited justifications for social classrooms:  the purported needs of girls.  David Chadwell, the coordinator of Single Gender Initiatives at the South Carolina Department of Education, says that girl-only classrooms should focus on "the connections girls have (a) with the content, (b) with each other and (c) with the teacher."  He recommends "a lot of meeting in circles, where every girl can share something from her own life that relates to the content in class."

Such unstructured, personal-sharing sessions serve left-brain girls no better than left-brain boys. Left-brainers of both genders tend to find other people's personal connections to classroom lesson content far less interesting than the content itself, and the shyer, more private ones are loathe to offer up their own personal connections.

If we are going to assign classrooms based on Chadwell's considerations, it might make more sense to partition children not by gender, but by how right- or left-brained they are.

Except that Chadwell's recommendations for boys are also right-brained--in a different way. Citing boys' energy levels, he advocates a kind of  hands-on learning that often bores and disengages the abstract, analytical thinker--for example "do[ing] physical representations of number lines."  Left-brain math buffs quickly grasp the symbolic number line representation offered by traditional textbooks. The more energetic of them--boys and girls alike--would much rather spend the time it takes to get in line and act out a number line in the classroom instead running around for a few extra minutes at recess.

Good writing requires both right- and left-brain thinking

But in today's published fiction, right-brain inspiration too often trumps left-brain precision. B.R. Myers observed this seven years ago in his provocative Atlantic Monthly piece, "A Reader's Manifesto," in which he critiques the sloppy, pretentious prose of many of our most esteemed contemporary novelists.  More recently, Ian MacKenzie's letter in today's New York Times Book Review takes reviewer Liesl Schillinger to task for praising a sentence in Charles Bock's Beautiful Children that depicts the tattooing of a character named Ponyboy:

"Electricity lit up Ponyboy's skeletal structure as if it were a pinball machine on a multi-ball extravaganza, and the mingling odors of brimstone and sulfur and sweat and burning skin filled Ponyboy's nostrils."

In its original simile and flamboyant imagery, this sentence is nothing if not inspired.  In the implausibility of the simile and (as MacKenzie points out) the redundancy of "brimstone" and "sulfur," it is also imprecise, sloppy, and unrevealing.  Connecting body and pinball machine might work, but it cries out for some left-brain editing.