4.9.08

REAL ANALYSIS.

Charles Murray's Real Education promises, in its subtitle, Four Simple Truths for Bringing America's Schools Back to Reality. Book Review No. 38 will demonstrate simple inconsistencies in the book that mislead readers. I'll use Mr Murray's plan. His first "simple truth" is "Ability Varies." He goes on to note seven dimensions of ability (there may be more, but those suffice for me to get rolling): bodily-kinesthetic, musical, spatial, linguistic, logical-mathematical, interpersonal, and intrapersonal. He intends to lay out a framework of analysis that everybody can accept. He destroys that framework at page 21 (which, if the book were properly paginated, would be page 5).

Educational measures such as test scores and grades tend to make differences among schoolchildren look as though they are ones of degree when in reality some of them are on differences in kind. For example, a timed math test limited to problems of addition and subtraction, administered to a random cross-section of fourth-graders, yields scores that place children along a continuum distributed in a shape resembling a bell curve.
That "resembling" is a critical hedge. With a small sample a statistician will not be able to reject the hypothesis that the frequency count is a collection of random draws from a normal distribution. That failure to reject is the basis for grading on a curve (top 10% A and bottom 10% F and fill in the rest as you wish, a 20-40-20 pattern is easy to implement).

Those scores appropriately reflect differences in degree: Some fourth graders can add and subtract faster and more accurately than others, but they are all doing the same thing and almost all children can be taught to add and subtract to some degree.
The purpose of No Child Left Behind and its attendant testing is to evaluate schools' effectiveness at teaching addition and subtraction, and teaching methods that enable students to make use of their differing capabilities to do the operations helps. To a great extent, one gets good at simple calculations the same way one gets good at shooting free throws, by repetition.

The same is not true of calculus. If all children were put on a mathematics track that took them through calculus, and then were given a test of calculus problems, the resulting scores would not look like a bell curve. For a large proportion of children, the scores would not be merely low. They would be zero. Grasping calculus requires a certain level of logical-mathematical ability. Children below that level will never learn calculus, no matter how hard they study. It is a difference in kind. Not only that: The child withouth the logical-mathematical ability to learn calculus cannot do a wide variety of other things in mathematics that are open to the child with the requisite level of logical-mathematical ability.
We're in agreement on one thing. Calculus is different from algebra. The middle-school student will not encounter "The solution can be guessed at and then verified." That suggestion comes at about page two of any presentation of differential equations. There's also the "there is a delta such that f(x) - L is less than epsilon whenever ...", which says nothing about motion, accompanying language such as "x approaches ...". These things can be taught. Good teaching matters. To what audience? Mr Murray suggests a smaller audience. Pay particular attention, dear reader, to that "would not look like a bell curve." Perhaps we can transform the score frequency count to a bell curve by taking the logarithms of the scores, creating a log-normal distribution. (Perhaps some professors use this dodge to establish grading scales with 80 or better out of 100 as an A.) Alternatively, perhaps we're looking at a Pareto distribution, or power law. Almost all students learn to add and subtract. A relatively few of them get through calculus. The phenomenon appears elsewhere: here's a Newmark's Door opening to a Mark Perry analysis of scoring in professional basketball. One fifth of the players score four fifths of the total points. (Whether that generalizes to scoring with Basketball Annies is outside the scope of this post.)

I believe, however, that I have established that when it comes to point scoring, most professional basketball players are below average. Thus, Mr Murray cannot honestly argue, as he does in his second "simple truth", that Half of the Children are Below Average. Mr Murray seems a bit aware of this reality (he has worked with some halfway decent statisticians in his career) when he suggests that his readers received reinforcement to develop their cognitive and non-cognitive skills (statistically, that's self-selection; for policy purposes the value of developing the Habits of Effective People in all youngsters is critical) and when he observes that in gym and music and art (all endeavours strongly subject to the power law) the schools do not push all students to participate in a sport or a performance or an exhibition. He offers some suggestions for improving students' performances in the traditional subjects while not discouraging either the strongest or the weakest students.

Now comes the third "simple truth", Too Many People Are Going to College. Given my fulminations against access-assessment-remediation-retention, one might expect me to agree. Mr Murray's point is a bit different from my pet theme, however. I object to the mind-set that views the withdrawal of a disengaged student as lost revenue, rather than a Pareto improvement. Elsewhere in the book, Mr Murray notes that the college-eligible population by his criteria exceeds the capacity at the Ivies and the rest of the so-called top tier. You'll find people who would satisfy his criteria at your local college, and you'll often identify them by their distaste for the disengaged students whose presence is a drag on everyone else. His simple truth is more subtle. By his criteria, it's the deep thinkers who take joy in grappling with Darwin and Freud and Marx and Milton, preferably in the original language. OK, I exaggerated. But he's attempting to separate the value of exposure to the liberal arts, which he argues can begin in elementary school, from deep study of the liberal arts, which he argues is the purpose of college. That excludes baccalaureate degrees that are credentials: agriculture, business, education, engineering, health and safety. There's a path-dependence problem here, in that much of the higher education goes on in land-grants and mid-majors that were chartered by their states specifically to combine the practical arts with the liberal arts. When he argues that "Today's college system is implicated in the emergence of class-riven America" (p. 105) he forgets that Northern Illinois flunks more students in a year than Harvard admits. If the university system is a bar to social mobility, that might be the consequence of (take your pick) universities persisting in access-assessment-remediation-retention in the face of excess demand for high-reputation institution or government failure to live up to the social contract implicit in blending those liberal and practical arts.

Thus, "America's Future Depends on How We Educate the Academically Gifted." The chapter is more about instilling humility in people who have been told since kindergarten that they're talented (the late Randy Pausch called it Recovering Jerk, enough said) and about instilling the Habits of Effective People in everybody. (Here I go again, talking like an economist: where inequality is increasing, the incentives for people to acquire the highly rewarded skills are greater, thus the response to an "increasingly class-riven" society is to equip more people to do the things that get the rewards). Mr Murray's conclusion ultimately gets there, but not based on the four "truths" he asserts.

(Cross-posted to 50 Book Challenge.)

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