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.

More difficult math courses, for example, will inevitably give future math teachers stronger knowledge of mathematics; but it seems doubtful that studying multivariable calculus, abstract algebra, or complex analysis (all of which are common required courses for undergraduate math majors) is an ideal use of a future middle-school math teacher’s time. It’s true that, when time can be spared from the state curriculum, some of the more obscure advanced topics in math can provide interesting enrichment material—I know private school teachers who do wonderful high-school level elective courses in non-Euclidean geometry, topology, and the like—but it doesn’t make sense for these topics to be required material for future sixth grade public-school math teachers. There is other mathematical content that is much more relevant to their work.

As I argued in my last post, the kind of mathematical knowledge needed by a math teacher is significantly different from (but no less rigorous than) that needed by, say, an engineer. A couple examples will give a better idea of what I mean. To my mind, a well-prepared eighth grade math teacher ought to be able to solve problems like these:

1. Consider the second degree equation below:
   
   
   
   4x² – 6x = (x + 2)²
   
   
   Part A:   Come up with an application word problem that one might reasonably solve using this equation. Do not use the words “add,” “subtract,” “multiply,” “times,” “minus,” “plus,” “product,” “sum,” “difference,” or “variable.”
   
   
   (See a sample solution)
   
   
   Part B:   Create a sequence of three simpler problems, based on your word problem from part A. The three new problems should gradually build in difficulty, so that the first is as simple as possible and the second and third build towards the difficulty of the problem from part A.
   
   
   (See a sample solution)
   
   
   
2. A house is twice as tall as a signpost which stands 12 feet in front of it. At 10am one day, the sun is directly opposite the house from the signpost, so that the shadow of the house falls directly towards the signpost. At this moment, the shadow of the house on the signpost falls 3 feet from the top of the post, and the shadow of the post on the ground extends 2 feet beyond the shadow of the house on the ground.
   
   
   
   Part A:   How tall is the house? Show all work and explain all steps.
   
   
   (See a sample solution)
   
   
   Part B:   Derive a formula to solve problems like problem 2, regardless of the distances given and the ratio of the height of the house to the height of the signpost. Clearly specify all variable names.

In terms of mathematical content knowledge, problem 1 requires nothing beyond basic algebra concepts and the most fundamental understanding of rectangle area. Problem 2 requires nothing beyond the ability to use similar triangles, to assign variables, and to solve ratios. There are no mathematical techniques required to solve these problems that are not part of the basic middle-school math curriculum, but these problems are much harder than what we currently expect most eighth graders, or most adults, or most teachers, to solve. You wouldn’t learn how to solve these in an upper-level undergraduate math course, either. It would require a special kind of class to learn this material, one that focused on the specific type of content knowledge that teachers need to know.

I speak with less authority about other subject-areas, but I suspect that, in every subject, you could find content knowledge that’s highly relevant to teachers but not necessarily taught in university courses. A science teacher, for example, must know the various laws, forces, processes, and concepts through which we understand scientific phenomena, but she need not have memorized the massive arrays of facts that take up much of college-level chemistry and biology nor the difficult mathematical derivations that take up most of college-level physics courses. On the other hand, she ought to have far broader knowledge than, say, a doctor or an engineer, regarding the development and history of science, its connections to other subject areas, its applications in the world around us, and its relevance to daily and civic life.

A history teacher need not have done the hundreds hours of primary-source research that characterize graduate-level study in history, nor need she have a strong grounding in the critical theories that modern historians use to problematize the reconceive their discipline. What she needs, instead, is a thorough and broad knowledge of historical phenomena: not only the major political events, but their connection to technological development, intellectual history, literature, and so on. She will also benefit greatly from a detailed knowledge of the many captivating narratives, dramatic moments, and vivifying details scattered throughout history: how Genghis Khan’s warriors, in order to travel quickly without stopping for supplies, would make yogurt from the milk of the mares they rode, mixed with blood from the same mare’s ankles; how J. Robert Oppenheimer, speaking in 1965 about his thoughts upon hearing that the bomb had dropped on Nagasaki, quoted what is thought to be his own translation of a passage from the Bhagavad Gita: “Now I am become death, destroyer of worlds,” and how sad he looked while he said it.

All of this applies with only slight modification to the training of primary-school teachers. Currently, content knowledge requirements for primary-school teachers are minimal: after all, everyone knows math, grammar, history, and science up through the fifth grade level—or if they don’t, they can pick it up quickly enough on the fly. The truth is, though, that primary school teachers lay the foundation for the study of each academic subject in the later years, and their content knowledge ought to go deeper than the curriculum they’re teaching. A fourth grade teacher with a weak understanding of math may know enough about long-multiplication to teach students how to perform it, but he is unlikely to instill in them any love of the subject; he will not know the many ways that multiplication can be understood in application problems or related to algebraic concepts, and he will not lay a strong foundation for the study of math in middle- and high-school.

Of course, primary school teachers typically have to teach math, history, reading, and writing, and often science as well, so they cannot be expected to study each subject as deeply as secondary-school teachers study their specialty, but they will be more effective teachers and better respected adults if they have a thorough working knowledge of each of the main academics. Currently, training for primary school teachers focuses heavily on developmental psychology and puts little emphasis on academic content, but it seems to me that primary-school teachers, because of the breadth of material that they teach, need just as much preparation in academic content as do secondary-school teachers.

Sample solution for problem 1:
(back to text)

Part A:

John and Pete both have square gardens. The length of John’s garden is two feet more than half the length of Pete’s garden. One day, Pete cuts three feet off one side of his garden, in order to extend his house. Now John and Pete’s gardens have the same area. What is the area of each garden?

Part B:

1. John has a rectangular garden and Pete has a square one, but both gardens have the same area. Pete’s garden is six feet on a side. John’s garden is nine feet long. How wide is John’s garden?
   
   
   
2. John and Pete both have square gardens. John’s garden is two feet wider than Pete’s garden. If the area of John’s garden is 40 square feet greater than that of Pete’s garden, what is the area of each garden.
   
   
   
3. John has a rectangular garden and Pete has a square one. Pete’s garden is one foot wider than John’s garden. John’s garden is ten feet long. One day, Pete cuts three feet off the side of his garden in order to extend his house. Now the two gardens have the same area. What are the dimensions of each garden? (Find all possible answers.)

(back to text)

Sample solution for problem 2 (part A only):
(back to text)

*** thanks to one of my readers for pointing out a careless error in this solution. It is now corrected. ***

We have three similar triangles in this problem, one formed by the wall of the house and its shadow, one formed by the signpost and its shadow, and one (which we can imagine) formed between the signpost, the shadow of the house on the ground, and the diagonal where the shadow of the house cuts through the signpost:

We know they’re all similar, because we assume that the rays of the sun are parallel (even though, in reality, they’re not quite parallel). Thus, we have

From the first pair, we get:

We plug this in for the y terms in the first and third ratios, and we get: 

Simplifying, we get:

The signpost is 18 feet tall. The wall is 36 feet tall.