Friday, August 26, 2011

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.”

    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.

  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.

    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.)

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.


  1. You come close, in this post, to convincing me that the training teachers really need is in content not pedagogy. Some posts back, there was a comment to the effect that you can’t teach teaching, much as it’s often said that you can’t teach writing. Yet if a non-writer has an experience of a certain kind, he may teach himself to write simply out of an inner need to tell it. Similarly, a person filled with knowledge may find the means to convey this (teach it) because the knowledge and its importance presses against him with an inner urgency. Frankly, it seems much more logical to teach people who know things how to impart them than to train someone in the teaching skill, then hope to fill them up with something they can use it on.

  2. Back in the 70s I took one lesson with a great musician who had incredible knowledge - in addition to being a virtuoso on his main instrument, he played at least a half a dozen others in different families at a professional level. After I played for him, he said, "There's nothing wrong with your playing that 20 years experience won't fix." Well, all well and good, but that was no use to me at all, plus, I was a novice, my playing was rudimentary (ie shit). He was a great musician but a poor teacher.

    The point here (and I do have one) is that to be a good teacher you need both - deep knowledge of both content and pedagogy (ie knowledge of how to teach). Either one by itself is not enough. Some people say you can't teach composing music - well, you can't teach someone how to be brilliant and original and inspired, but there's a hell of a lot of just nuts and bolts stuff that you can't do much with your so-called 'inspiration' without. Anyone who's been taught by a great teacher knows that. (I'm sure writing is the same)

    It's hard to get past the problem here that the skills of teaching, (and the art of teaching) are just not valued enough by our current culture (for lack of a better word) to invest properly in them. We don't invest in them, and then we blame the teachers for not magically acquiring these very difficult skills. ie we treat them like artists, which they are in their own way.

  3. As someone who has come to teaching as a second career and completed a 2 year Masters of Teaching at The University of Sydney, which required 20 weeks of practicum inclusive of an in school research project; I find the relationship between content knowledge and pedagogical technique to be a sliding scale along a continuum dependant on the age and ability of the class you teach.

    To try and quantify this is a dark art that every teacher has to attempt to perfect on a daily basis by knowing your subject broadly, your curriculum closely, your students and their learning styles rigorously and be prepared to be wrong and fail,learn from the experience and go back and do it again.

    Universities cannot fully prepare teacher trainees for the reality of attempting to teach 30 14 year old boys last period on a Friday, only time and experience will do that. I have found that although as a History and Social Sciences teacher I will always find my content knowledge lacking, my ability to adapt and enjoy learning along with my students improves and grows on a daily basis.

  4. Re: The examples you give of problems an eighth grade teacher ought to be able to solve, and which should be part of teacher training.

    These are problems that test the ability not to solve math problems, but to "think in math", to understand the intellectual nuances and implications of a given math problem, to be comfortable with the conceptual elements to the point where these become material the teacher can enjoy playing with, can enjoy testing its range and limits, can stretch, fold, imagine in new contexts, how to make new intellectual things out of them. You are talking about the art of teaching creative teachers (sorry for using such cliched expressions but in this case they are apt in a literal way), and (to borrow from Olson) I take play – the ability to play with math ideas - to be the central fact of creative teaching and learning.

    Obviously, not only should math teachers be taught to do this, but math students at every level should be taught the same approach. If future math teachers were taught that way as kids, it would be part of their thinking/understanding before they even got to teacher training.

    I would also venture that acting classes should be part of teacher training. This sounds frivolous, but it isn't. A good actor communicates effectively and compellingly with an audience. A teacher, to be good, must be able to do the same.

  5. I would really like someone to quantify how many education undergraduates actually become teachers.

    Most people I know who become teachers do so after receiving a BA, most of them in either a 5th year program or a grad school.

    And I don't know where teachers are being taught content at all. Nor should they be. Content is a standard that must be determined by the state, through state tests.

    So why even mention content as part of teacher training?

  6. To Anonymous & Fujio:

    Thanks for your comments, both of you. I definitely agree that good teaching requires deep knowledge of both content and pedagogy. That deep content knowledge, or to be precise deep passion for subject matter, is the only important prerequisite for good teaching is a position that you encounter in some Romantic educators, but it's not one I agree with and I never intended to argue for it.

    Beyond the problems of knowledge-transmission that Fujio raises, there's also the problem of classroom management, which becomes a major factor in public-schools where class-sizes hover around 30, and students sometimes arrive at school with little will to learn.

  7. Fujio's anecdote provides a particularly interesting example, because it's likely that, in addition to his music teachers' pedagogical shortcomings (and apparent lack of will to teach), he may have had the wrong kind of knowledge about music. I know little about musicianship, but it seems to me that some virtuosic artists, artisans, athletes, etc. come to their craft so naturally that they actually don't have a very good understanding of how they do what they do. This can make teaching difficult and (a smattering of anecdotal evidence leads me to suspect) frustrating and unpleasant as well.

  8. To Jackie Charles:

    Thanks for your comments. I agree with a lot of what you're saying here. The relationship between content knowledge and pedagogical knowledge is indeed murky, and it's true that the amount (and kind) of content knowledge needed depends on both the age and ability of the class being taught. What I'm trying to argue, though, is that all teachers should know a lot of content: as you say, there's a sliding scale, but the bottom of that scale is still quite high.

    I absolutely agree with you that university training can't prepare someone for the realities of teaching. That's why I believe that teacher training programs should be modeled on medical residencies, with most pedagogical training occurring in actual grade-school classrooms, under the mentorship of master teachers.

    There are two main reasons why I think such training (as opposed to just throwing teachers into classrooms to sink or swim) is necessary. First, we need to reduce burn-out. For almost all teachers, the first year in an inner-city classroom is traumatic, and too many teachers simply give up before they become competent. A strong mentorship program could ease teachers in, with greater support and fewer responsibilities during the first year or two teaching.

    Second, the sink-or-swim model sacrifices students' educations to the training of teachers. The medical profession understands the delicacy of this trade-off: you have to train surgeons but you also have to save lives; you just throw a novice into an operating room and give her a scalpel. For every teacher who swims, there are several more who sink, and even the swimmers take a couple years to hit their stroke. That all adds up to many classes of kids having ineffective teachers for many years. Clearly, that's to be avoided.

  9. To Leora:

    Interesting idea about acting classes as part of teacher training. I like the idea, but you have to be specific. I don't think method-acting or scene-building classes would be a good use of teachers' time. A class in speaking, story-telling, character, and body language, on the other hand, might make an interesting component of a teacher training program. It would have to be an acting class designed for teachers. I'm pretty sure that such courses exist in some education schools.

    I agree that solving the sample problems in my post requires real mathematical thinking, which necessarily entails some creativity, but it's not just about creativity. It's also about a very high degree of familiarity with the concepts and processes of algebra, how they're used, what they describe, etc. Expertise requires not only the ability to use ones knowledge creatively, but also simply a lot of relevant knowledge.

  10. Cal,

    You're right, of course, that grades in education departments are not the same as grades in teacher training programs-- this did not escape me. Accusations of low academic rigor in education departments is widespread, however, and grade inflation at the undergraduate level offers further evidence of that problem. Research comparing grade distributions in MAT programs to those in other professional credentialing programs would provide more direct evidence on this point, but I don't have it.

    Regarding your second point, are you arguing that someone who intends to be a math teacher doesn't need to know any math? I'm baffled.

  11. No, I'm saying that ed schools should stay out of the job of teaching subject matter.

    A middle school teacher doesn't need any additional training--or shouldn't--as the knowledge required to pass the tests is usually at the (genuine) 10th grade level.

    High school teachers either can learn on their own (as I did for two of my three credentials) or learn through college through a BA in English, Math, whatever.

    States set the standard through their subject matter tests. Ed school should stay out of it.

    By the way, I think your suggested content instruction is more suited for a curriculum degree. While many teachers (including me) design their own curriculum, they do it because they like to, not because it's part of the job. That's why God created textbooks.

    . Accusations of low academic rigor in education departments is widespread, however, and grade inflation at the undergraduate level offers further evidence of that problem.

    As to the first, so what? They're wrong. The knowledge level of teachers has NOTHING to do with ed schools. Teachers either pass the subject matter test or they don't, and that's set at the state level. All the stats bandied about regarding the supposed knowledge level of teachers--which is actually about education majors, who comprise an unknown (and probably small) number of teachers.