Introduction
One of the difficulties of the school year is that thoughts will occur without time available or easily found to work them out in detail. Then I find myself, when given some time on a break, with too many half-formed conceptions to work through; or having lost my original impetus and train of thought either dissatisfied with the direction of the argument as I put it down or else unable to reconstruct critical part needed to form a complete piece of work.
So here I am going in the opposite direction. What follows are summaries of some of my more current concerns, which is a purely relative description: in some cases these have been percolating for two or three years – or more. If I have, revive, or receive interest in particular items I may post more expanded thoughts on those topics later, but here the goal is brevity.
Knowing or Doing?
Take a simple mathematical idea, like calculating the area of a rectangle. In school, what does it mean to learn this? Often we talk about a progression of skills like this: you might know this idea, by which I mean describe the general case of how to perform this calculation; you could be able to do the calculation in a specific case; and then you should apply the idea – to calculate the cost of repainting a wall or any number of any specific uses. I have two reservations about this paradigm.
In the first place, the “knowing” step is today often considered unnecessary: memorization is downplayed in general, justified if at all by appeal to the amount of knowledge which can be looked up; and formulas or examples are expected to be given for tests and exams. Students, unfortunately, subjected to this weakening of expectations, come to depend on these supports and resent unexpected variations. A related evil, the inexplicable policy of many private schools, is to assign students only work they can “check” themselves in a key, accompanied by abandoning the teaching of methods of checking one’s own work. The results are similar.
In the second place, there is at least one more skill, which I will call understanding, which is in current American schools barely taught at all, at least in mathematics, outside the Geometry course. This skill is demonstrated by performing a proof or derivation, in which an abstract principle can be related to its foundations or – less fundamental but easier for most students to begin with – solution of a specific problem is fully justified to a degree appropriate to the level of the class. Short of re-writing our mathematics textbooks, I have no direct remedies, but strongly advocate at a minimum increased stress to be placed on students showing and checking their work: increased verbal explanations, both written and spoken, are probably also indicated.
Politics and the Teacher
I have found that in many private schools, teachers are supposed to keep their political – and depending on the school, also religious – views to themselves, even to the extent of refusing to answer or redirecting student questions. I believe this to be a serious mistake. The pragmatic reasoning seems to be that schools do not want to alarm students or families by revealing that some staff may not agree with each other, or even with all elements of the official emphases of the school as an institution – but this is rarely stated so baldly, no doubt because it would essentially admit to institutional cowardice.
What is more usually said is that these are important questions students should be left to form their own views on. But this is fundamentally dishonest. First, students are not going to be left to form their own views: what it means in practice is that the views of teachers of history, language, and – where applicable – religion are privileged, because these teachers must discuss controversial events and views and if any good at all as teachers must at least communicate a method of approaching such topics, which method inevitably contains its own assumptions. Second, when teachers avoid difficult topics, students are not in fact encouraged to form or challenge their own views, but to imitate their teachers and avoid these topics themselves. And third, children form their initial views in imitation of or reaction to their elders: if we do not trust an adult to be a role model in responsible admission and discussion of political views, why should we trust him to teach at all?
Teachers Ban Books
All schools periodically revise their curriculum. Generally speaking, in the study of literature, this means some books are taken off of reading lists and others are added. Over the years this has in many schools, led to authors such as Homer, Virgil, and Milton being removed and – despite so-called “multiculturalism” – not generally being replaced with other classics such as The Mahabharata or Journey to the West but with various “relevant” – meaning modern – reading. Some memory of the stories lingers, and the more ambitious students may eventually be introduced to selections in an honors or AP class, or the contrarians may seek them out when they learn of their existence, but as a shared text these works are removed – that is, banned just as effectively as whatever manual of sexual perversion national media has reported that your local “educators” are indignant that parents are upset about.
No school can own or teach every written work ever published: nor, in fact, do all that many books actually deserve the consideration implied by the phrase “being taught”. The question, “Should this book be taught, maintained in the library, ignored, or actively avoided?” is asked every year of hundreds of books in thousands of schools: this is just as much a question of a “ban” as any parent activism, and in this context we know the answer is never as simple as “ban = bad” but depends, obviously, on the work in question and the priorities of the school. That parents want to be involved in the process of educating their children is good; that their interference in the schools is often ham-fisted is not surprising; but since they are trying to do what curriculum committees do all the time they should not be simply vilified by any thinking person without regard to the details of each case.
Creativity in Mathematics
In language and art students create: poems, drawings, or music. In science, experiments are performed and results documented; in history, motivations are debated and papers written. But in today’s mathematics, only students who find satisfaction simply in solving a problem are easily satisfied by the subject itself. Some suggest overcoming this difficulty by “applying” mathematics with “projects” to give students something about which they can say, “I did that!” But the time taken mostly outweighs the mathematics actually learned, and older textbooks would show that traditional problems can apply mathematical principles quite as well as most projects.
Instead, I recommend working withing the discipline itself, but more often assigning something closer to an essay. I have asked students to document, then change the form, of proofs covered in their Geometry class; or prove and present theorems we have not proved already. Presentation and discussion are crucial not just to the geometry class. Students also ought in general to be asked to take more care with the work that they turn in than most math teachers expect. Certainly there are students who need to be encouraged over messy work: but in general we are far too lenient on this score.
We also ought to admit that the mathematics emphasized today does not lend itself to creativity. Why? Even a calculus class is still teaching – in linguistic analogy – grammar and vocabulary, although the difficulty may be that associated with, say, Sanskrit. Naturally, the work produced is not likely to be more satisfying than a three-paragraph essay on some set topic.
Suppose we pursue this thought: advanced mathematics as a sort of new language. (Of course fundamental skills with the arithmetic operations must be achieved first, as well as some decent conception of handling unknowns (algebra) and spatial reasoning (geometry), and their related calculations.) Then, while still encouraging the study of the advanced topics, we could more easily ask how widely they must be required, and for how long; and if not, whether the alternative is different mathematics, other more urgently required learning, or simply not demanding this portion of students’ time.
The College Conundrum
As late as the 1970s, Jacques Barzun could be found advocating for schools to hire gifted high school graduates as teachers with minimal additional training. One supposes he probably didn’t have them teaching high school immediately in mind, but then again maybe he did.
Or again, when Dorothy Sayers wrote her essay “The Lost Tools of Learning”, her primary motivation was to address the historical anomaly that it was implausible to imagine sixteen year olds regularly entering colleges. The classical schools who claim Sayers as an inspiration have, whatever their accomplishments, signally failed to address Sayers’ chief complaint. Many of them do at least produce graduates who could meet Barzun’s specification. But what state would let them try?
I’m increasingly alarmed by seeing most private high schools advertising themselves as “college preparatory”. Something over 50% of the population now acquires at least some college credit, where the percentages historically were well under 1%. Obviously, an institution – taking America’s higher education “system” collectively as one institution – that serves 50% of a population must to some degree be less rigorous than one that served only 1%. But is what is taught in high school itself so useless?
It would almost be the lesser evil to merely assume that the college degree has become a status symbol more than an accomplishment. Societies can deal with status symbols, while the alternative is that the education of our youth is such that, despite adult bodies and minds, the young citizen either cannot be trusted to behave as an adult, or worse still, could be but is not so trusted by our society.
Romance of Victory 