What Teachers Should Know Before they Start Teaching

Mark J. Perry posted an essay on Carpe Diem two days ago about grade inflation in university education departments (see the image to the right). Aside from offering me, as the holder of a BA in education history and policy, some personal embarrassment, the post gives strong evidence for the lack of rigor in teacher training that I discussed in my last post. These soft standards have a double effect: they lower the public perception of teachers, and they leave teachers worse prepared to transmit knowledge.

I want to argue, however—and this follows pretty directly from the discussion of expertise in my last post—that upping rigor is not a sufficient solution to the problem of weak teacher preparation; indeed, low-rigor is more symptomatic of our teacher-training problems than causal. The more important question is, what exactly are we trying to teach teachers? We want to up the rigor, yes, but the rigor of what?

There are, generally speaking, two types of material in a teacher training program: subject-area content and pedagogical technique. I talked briefly about the issues surrounding pedagogical technique in my last post and at considerable length in my post on how to improve teacher training. Upping the rigor on the psychology and pedagogical theory courses that dominate traditional training programs will not make teachers more effective in the classroom; what we need is a different kind of pedagogical training entirely, one that occurs in actual grade schools, under the mentorship of master teachers.

What I want to talk about today are the issues surrounding subject-area knowledge. I touched a bit on this in my last post, but I want to go into more detail, because this is something I don’t hear anyone talking about. No matter how it’s done, more rigorous subject-area classes for secondary-school teachers are probably a good thing, but it’s worth thinking carefully about exactly what type of rigor we want. The word rigor gets tossed around a lot in education discussions, and I’m not the first to point out that it’s meaning has gotten a little vague: rigorous has become more or less synonymous with difficult. But there are a lot of ways to make classes harder.

The Sweet Spot: Balancing Conceptual and Procedural Instruction in Grade-School Mathematics

The purely procedural approach to long division.
I stole these colorful illustrations from this website.

A few years ago, I was working in a charter school where the sixth-grade math teacher was having a hard time teaching long division. (I myself, let it be noted, was having a hard time teaching anything at all.) I sat in on some classes, and it was easy to spot the problem. The teacher had reduced the process of long division to a series of five steps: divide, multiply, subtract, bring down, repeat, with a mnemonic, which I forget, to help remember them. This must have seemed like an easy-to-follow script when they were planning the lesson, but it’s deceptively complicated: you have to know which numbers to divide, multiply, subtract, and bring down and what to do with all those quotients, products, and differences once you get them. (See Appendix A, for an idea of just how complex this gets.) The students didn’t understand why they were doing any of these steps, so they found all that information extremely difficult to keep track of. Long division, taught this way, became a dull and intricate labyrinth, riddled with small procedural booby-traps to derail the unsuspecting scholar. Yet, pure procedural methods like this one seem to be prevalent in contemporary instruction—try Googling “long division” and see what comes up.

More recently, my girlfriend asked me to teach her long division, a skill she’d never gotten her brain around back in elementary school. Eager to show her that mathematics is logical and comprehensible, not arbitrary and byzantine, I dove into a thorough explanation of the inner-workings of the long-division algorithm, complete with diagrams and concrete examples. After ten minutes, she was frustrated

Concepts vs. Procedures – the Great Math Debate

In last week's post—read it before you read this one—I presented evidence of deep math concept deficits in four 5^th graders in inner-city Brooklyn. Math is America's weakest subject, according to the results of international exams,^[1] and it's traditionally the most feared and hated among students. There's a lot of debate as to why that is, and as with so many other issues, that debate tends to polarize around two camps. For years, I have found myself torn between those camps, but I believe the evidence that I presented last week offers a glimpse of an elusive middle-ground and a more nuanced approach to math education.

The Confusion Beneath Confusion: Brief Glimpses of Number and Quantity

For the past few months, I’ve been spending a couple hours each week teaching remedial math to four fifth-graders in inner-city Brooklyn. My assignment: help my students to develop a conceptual grasp of numbers and elementary mathematics.

Over the course of my first week or two with my charges, I gauged my students’ knowledge and understanding—or so I thought. In fact, I had discerned only the superficial: their addition tables were shaky, but they could perform two-digit column-addition; they could carry but were prone to mistakes; three of them could borrow, but again with errors; they could name the place values of two- and in some cases three-digit numbers; they could quickly add multiples of ten in their heads.

It took me several weeks to realize what should have been obvious: that that was a procedural assessment, not a conceptual one. Beneath those weak basic skills were fundamental gaps in my students’ understanding of number and quantity.