"Improving" education

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One approach to improving the achievement of the United States of America in the Olympics has been to introduce American sports into it, sports that are not widely played outside the USA.  (This would certainly improve the profits of the TV channels that carry the Olympics.)

And now for something completely different.

A recent article in the NY Times by two quite respectable mathematicians and educators suggest some changes to the mathematics curriculum in American schools.

For instance, [the authors ask,] how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a 'group of transformations' or a 'complex number'? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.

 The broad thrust of the article can be reduced to the following assumptions:

1. Every child doesn't need to be taught the same (science/engineering - type advanced) mathematics.
2. Concrete applications are more useful to students than abstract mathematics.
3. The useful mathematics (listed in the article) is not being taught now.

All this, of course, is being driven by poor results of American student samples on standard international tests.

The fact of the matter is that ALL THREE ASSUMPTIONS ARE WRONG.

Certainly, all children do not need to know the same mathematics (or the same social studies, or the same history, for that matter).  But there is no harm in teaching all students the same principles, while you allow for expected variations in interest.  It would be a big mistake to try to predict the precise sliver of mathematics a child is likely to need and teach him or her only that.  First of all, we can't predict needs that accurately, and people change careers so often that the prediction simply cannot be correct.  Finally, with the insistence that Education should cost society the very minimum it possibly can, (and that the majority of education dollars should be spent on building-beautification and athletics) we cannot afford to give a highly individualized education at lower levels.  But teachers can, and do know how to, adjust for individual interests in students, and the better teachers supplement their basic classwork with enrichment activities focused on student interest.  But, of course, that instruction cannot be easily tested in standardized tests.

Secondly, the concrete vs. abstract debate is so out of place here.  It is human to abstract; that is, to generalize.  Two chickens and two chickens is the same number of chickens--four--as two cows and two cows are the number of cows: four.  No one can possibly protest the most common abstraction of all, namely number.  Our parents tolerated quite well a large degree of abstraction (though, of course, it was a small elite that went on to college in their day).  In these times, we all recognize, we want the students to learn considerably more.  Why?  Because a lot of what our parents considered to be advanced knowledge has now been demoted to material that is accessible to kids, and which they must know.  A "complex number" might be regarded by Professor Sol Garfunkel as something mystical and specialized (though I know for certain that he is no stranger to them), but it is an easy enough concept for the typical student, and moreover, an easy way of teaching various topics in calculus.   [Easy for the teacher, but also easy for the student.]  Geometry is these days taught via the idea of transformation, where transformations was a topic reserved for mathematical black belts in the earlier part of the 20th century.  Abstraction is another way of killing several birds with the same stone, and hardly something to be deplored.

Finally, useful mathematics is being taught now.  As early as grade four, with the recommendations of the NCTM (National Council of Teachers of Mathematics) that were announced by a taskforce as early as 1980, and adopted by the full NCTM shortly thereafter, and widely adopted in schools across the country, basic descriptive statistics was to be taught to children of about the age of ten or eleven, and in classrooms across America you can find displays of various pieces of data in bar charts and pie charts.  These fellows should get out more.

The fact of the matter is that if we were to change the curriculum to be exactly what Garfunkel and Mumford recommend, the chances are that within a few years, this curriculum would be --in its essentials-- adopted by all foreign countries, or even taught in addition to their own, and foreign kids will be trouncing American kids once again.

Why is this?

In an insightful comment (Mike O'Shea?) says that a possible reason why academics in the US are weak for the majority of students is that excellence in academic subjects is valued less than excellence in athletics and sports.  He concludes with: "And if our teenagers aren't practicing sports, they're working at part-time jobs after school for pocket money for themselves. Competing against sports and money, academic subjects don't have a chance."

Mavis Tavis says, looking at the whole article:

Ah, yes, just what we need: a math for the masses and a math for their masters. This argument presupposes that the common people don't get it, don't need it, and don't want it. It echoes the argument that has gone on in the humanities and foreign languages for a generation now: why teach complex subjects and abstractions to the herd who don't need such instruction? Teach them what they need to know to become, at best, good Wal-Mart managers.

A time there was when math assumed not only a utilitarian function but also a theoretical function, teaching children not just what they need to know to get by, but also how to think logically--to make them better workers, neighbors, voters, parents, and citizens of a increasingly complex world.

Garfunkel and Mumfords's assertion suggests an unhealthy elitism. Worse, it smacks of classism.

Perhaps Garfunkel and Mumford were careless in their writing, and laid themselves open to severe criticism because of a number of poorly-reasoned, or poorly thought-out remarks.  It is true that weaker students destined to be highway-repair laborers or construction workers will probably not get motivated to study mathematics that are in the least abstract, even at the level of, say, elementary geometry.  But they are butting their heads against cultural principles that dictate that every child must be considered to be potentially a professional or an artisan of some sort.  To relegate a child to an easy curriculum based on an assessment of his or her ability may make life easier for him or her, and for his or her teacher, but it is a choice that we cannot ethically afford.  We cannot both take the high road about the Equality of Man, and take the easy way to education, and get high scores for our kids in standardized tests.  I'm not saying our curriculum (and we do not have a national standard curriculum, but rather a core curriculum that is a sort of "back-to-basics" nucleus that, as G & M claim, has been adopted by at least 40 states) is perfect; it has to be adjusted from time to time.  But good teachers can do a better job within this curriculum, and they are doing so.  But statistically, they are a minority.

The vast majority of teachers are poorly-paid and poorly prepared underachievers.  You cannot improve the quality of teachers in the USA by picking on the weaker members of that profession and making life miserable for them.  I suspect that the better ones among our young people do not go into the teaching profession precisely because it is a scapegoat for all that is bad in society.  Let's stop bullying our teachers, and concentrate on rewarding the best of them, and appreciating all of them.

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