Avoid ickiness at all costs

This is my last guest post before Joanne returns. (Update: In addition, one of my satirical pieces appeared on The Cronk of Higher Education today.) I have enjoyed this guest-blogging stint and am grateful that Michael Lopez has been on such a roll.

If I were to give advice to a new teacher, it would be twofold: (a) avoid ickiness at all costs; and (b) do not be afraid of repetition.

Teachers often get told what to do and how to do it, but intelligent administrators realize that they won’t (and shouldn’t) follow directives to the letter. When deciding what to follow, what to adapt, and what to ignore, a teacher can safely put icky things in the last two piles. One can take an icky thing and make it less icky, or one can avoid it altogether. No teacher should have to descend into anything tacky or dumb. (Of course, what’s icky for one teacher may not be so for another.)

For me, “turn-and-talk” activities are often icky. I do recognize the exceptions. When learning a language, students can benefit from talking to the person next to them; the more practice they get, the better. When learning a poem, students working in pairs can take turns reciting. But the usual “turn-and-talk”—where the teacher poses a question and then has students talk to each other, right away—goes against my grain, because it makes for a room of chatter, and it often interferes with the thinking. Why turn and talk? Why not pause and think?

I have been in many situations where I had to turn and talk to my partner, and I found it silly. I would have been better off with a bit of space and quiet, and then a full-forum discussion later. Of course, not everyone shares my preferences—but teachers should have some room to be true to their own. There is usually a reason for them.

Yet there are teachers for whom such activities are not icky in the least. They conduct “turn-and-talk” activities as though they were drinking water. It is possible, also, that a teacher might find it offputting at the outset and then warm up to it, or vice versa. In any case, teachers should have some room to listen to their gut, especially when it is squirming and turning.

The second piece of advice seems unrelated, but actually it comes from the same principle. Do not be afraid of repetition, especially when it is repetition of something good. There are many kinds of repetition in the classroom: daily repetition of routines and of subject matter; the act of returning to ideas and works over time; and even drill. I find that my students get a lot out of reading a passage several times. The first time, it hasn’t yet sunk in; the second time, they are starting to find their way through it; and the third time, they can start to notice some of its subtleties. Return to it later, or even in a subsequent year, and they find even more.

Far from being boring, repetition can actually be exciting, as you start to anticipate things: a turn in the poem, a favorite phrase in a passage, or a difficult cluster of syllables. Young children enjoy hearing stories over and over; so, often, do older children and adults. I enjoy rereading books more than I enjoy reading them the first time. (That’s why I have difficulty reading large numbers of books, or part of the reason.)

What do the two pieces of advice have to do with each other? Both come from the principle that you can have exciting lessons (or thoughts) when you allow for a bit of calm—that there’s room for interesting things when you aren’t constantly pursuing novelty and change. Teachers often feel pressure to keep things exciting and active (and to be “innovative“), but this may crowd out some of the greater excitement (which by nature cannot be there all the time). By contrast, if you turn something around and around, day after day, you start to see its textures and patterns. I don’t mean that instruction should be entirely repetitive; of course it has to move in a direction. But the repetition helps it do so.

For whom is this advice intended? For me and for anyone who finds that it makes sense.

More on the Russian math textbook

The other day I described the math textbook that was used in the Soviet Union when I attended school there.  My question was: how could this textbook teach so much in just 220 pages? My tentative hypothesis was that it eliminated needless “scaffolding” (yes, I recognize that the term is problematic) and went right to the matter. I realized later that this was only a partial explanation.

An acquaintance suggested to me the other day that Russians (and Soviets) had a particular way of presenting mathematics that made it clear to the reader and student. As I started to think about my textbook in this light, I saw that its material was presented not only lucidly, but also artfully. There is a well-considered and well-timed movement from specific problems to larger topics and back. 

The textbook’s editor and lead author is the eminent mathematician Andrei Kolmogorov (1903–1987). I suspect that he shaped a good deal of the book and didn’t simply attach his name to it. But since I don’t know that for certain, I’ll leave that aside and look at what’s actually there.

I’ll begin by describing the contents of the first four pages of the book. Then I’ll examine their structure.

The book begins with a discussion of induction. It presents a series of odd numbers (1, 3, 5, 7, … 2n – 1) and posits that for every natural number n, the sum of the first n elements of the series is n2. That is,

1 = 1 = 12;

1 + 3 = 4 = 22;

1 + 3 + 5 = 9 = 32;

1 + 3 + 5 + 7 = 16 = 42;

1 + 3 + 5 + 7 + 9 = 25 = 52;

and now the hypothesis: “For all natural numbers n, this equation holds true: 1 + 3 + 5 + … + (2n – 1) = n2.”

All that is on the first page (or half-page) of the text. The book leaves the reader with this concept: having observed a number of examples, you make a hypothesis about the general rule that unites them.

Then, on the next page, the book examines a few more problems in order to shed light on hypothesis-making. It shows, first, how a single counterexample is enough to disprove a hypothesis. For instance, if you have the formula P (x) = x2; + x + 41, and considered the natural numbers 1 through 5 as values of x, you might conclude that P (x) is always a prime number. However, an obvious counterexample is P (41).

After this, it gives two more problems where you can easily verify the hypothesis.

Now it goes on to explain the principle of mathematical induction. It returns to the original problem and restates it in this manner: If we know that the equation is true for a certain value of n, and if we can also demonstrate that where it is true for n, it is also true for n + 1, then we know that it is true for every natural number n

It then puts this in simple notation. The letter A represents the function. A (5)  is true, we know, because 1 + 3 + 5 + 7 + 9 = 52. We can also see that the truth of A (6) follows from that of A (5). (I wont’ go into that; it’s straightforward.) We represent this as follows: A (5) => A (6).

So our goal is to prove that A (k) => A (k + 1). That is, we want to prove that from the equation

1 + 3 + 5 + … + (2k – 1) = k2

this equation follows:

1 + 3 + 5 + … + (2+ 1) = (k + 1)2

In fact, this is easy to demonstrate, since

1 + 3 + 5 + … + (2k – 1) + (2k + 1) =

= (1 + 3 + 5 + … + (2k – 1)) + (2k + 1) =

= k2 + 2k + 1 = (k + 1)2.

(In the penultimate equation, just substitute k2 for everything contined within the outer parentheses. You can do that because we’ve already established that for a cerain value of k, the sum of the first k elements is k2)

From here, the textbook goes on to more complex examples.

How is this structured? First we are given a series with a hypothesis about the underlying pattern. The possibility of proving this pattern is dangled before us. We learn a little more about hypotheses themselves and how they might be proved or disproved. We see a few examples. We go on from there to the concept of induction. At this point we return to the original problem and prove, through induction, that it holds for all values of n, where n is a natural number. What in particular makes this presentation compelling, easy to understand, and at the same time challenging?

For one thing, there’s the lure of something to come. It’s a real lure. We know that a problem has been left hanging and that there’s more to it. Also, each concept is developed and at the same time reduced to the simplest possible notation. The student gets both the concrete examples and the abstractions. The careful reader understands the meaning of A (k) => A (k + 1). And then the proof of the original problem may jog the mind slightly at first, but then it’s obvious.

Now, some readers may be saying, “I have no idea what you’re talking about.” But that’s the thing. When I first read these pages, I could maybe make a glimmer of a tail’s feather out of them, but that was it. But then I sat down and puzzled over them. Then, when they came clear, they were straightforward and elegant to boot. This applies not only to the first four pages. I found that where this held true through page n, it also held true through page (n + 1).

Maybe that’s part of the “secret” here, too, if there is one: the understanding that the student must take time with the material, at home, without distraction.

Are quieter students considered less intelligent?

Today’s issue of Education Week has an article by Sarah D. Sparks about quiet, shy, and introverted students in the classroom. It’s gist is that current pedagogy (and teachers themselves) favor the extraverted child. Teachers commonly perceive quiet children as less intelligent than talkative ones, according to studies cited here. The article distinguishes (up to a point) between introversion and shyness.

A 2011 study found teachers from across K-12 rated hypothetical quiet children as having the lowest academic abilities and the least intelligence, compared with hypothetical children who were talkative or typical in behavior.

Interestingly, teachers who were identified as and who rated themselves as shy agreed that quiet students would do less well academically, but did not rate them as less intelligent.

As many as half of Americans are introverts, according to the Center for Applications of Psychological Type, located in Gainesville, Fla.

There’s a distinction between shyness—generally associated with fear or anxiety around social contact—and introversion, which is related to a person’s comfort with various levels of stimulation.

A shy student, once he or she overcomes the fear, may turn out to be an extrovert, invigorated by being the center of attention.

By contrast, an introverted child may be perfectly comfortable speaking in class or socializing with a few friends, but “recharges her batteries” by being alone and is most energized when working or learning in an environment with less stimulation, social or otherwise, according to Mr. Coplan and Ms. Cain.

I was interviewed for the article, but some of my points didn’t make it in. I find the denominations “introvert,” “shy” and even “quiet” limiting. There are students who speak very little in class on the whole but liven up when particularly interested in a topic. There are students who speak a lot but are not necessarily “extraverts”; they enjoy the exchange of ideas in the classroom. Many students who might classify as “introvert” do not desire “lower” levels of stimulation; rather, they find certain intellectual activities highly stimulating. And, of course, there are students who seem quiet in class but are social ringleaders outside.

What’s important is to stay alert to the students and to do what will bring out the subject matter. Most subjects require a good deal of focus and quiet thought. Even a class discussion can set the tone for that. Ideally, all students would learn to both speak and listen, to grapple with problems out loud and in quiet. But for this to have meaning, there must be things worth thinking and talking about.

As to whether teachers consider quieter students less intelligent, my experience says no, but this may be because my trachers, especially in high school, recognized the importance of thinking about the subject and not rushing to speak.

Teaching students to ask questions

What would education be like if students knew how to pose, prioritize, and use their own questions? Vastly better than it is now, argue Dan Rothstein and Luz Santana, authors of Make Just One Change: Teach Students to Ask Their Own Questions (Harvard Education Press, 2011). If students learned how to formulate good questions, according to the authors, they’d be that much closer to becoming “independent thinkers and self-directed learners”  and practitioners of “democratic deliberation.”

On the face of it, the idea sounds terrific. The ability to ask good questions can enhance both individual lives and common culture. Many people need special instruction in this skill; most of us have room for improvement. I am not convinced, though, that any of this requires the elaborate group processes that Rothstein and Santana describe.

The research started when the authors were working in a dropout prevention program. They heard from parents that they wouldn’t come to meetings at school because they “didn’t even know what to ask.” Rothstein and Santana began by giving them questions but then realized that this was only increasing their dependency—that they needed to know “how to generate and use their own questions.” Over time, the authors developed a technique for teaching just that. They and others founded the Right Question Project, now known as the Right Question Institute, which teaches the technique to people around the country and abroad.

The book explains the Question Formulation Technique, which consists of six components: (a) a Question Focus; (b) a process for producing questions; (c) an exercise for working on closed and open-ended questions; (d) student selection of priority questions; (e) a plan for the next steps; and (f) a reflection activity. The authors provide numerous case studies to show how these components have played out.

Before starting the process, students are introduced to the four rules: “(1) Ask as many questions as you can; (2) Do not stop to discuss, judge, or answer any of the questions; (3) Write down every question exactly as it was stated; and (4) Change any statements into questions.” Students are supposed to reflect on these rules before proceeding. The authors explain:

The rules ask for a change in behavior, officially discouraging discussion in order to encourage the rapid production of questions. Students thus need to think about how they usually work individually and in groups. They name their usual practices and become aware of how they generally come up with ideas. They then must distinguish their present learning habits from what the rules require of them.

After receiving their Question Focus from the teacher, the students begin producing questions in groups. They are reminded to ask lots of questions and to refrain from judging, answering, or editing them. The teacher is not supposed to give examples of questions, even if the students are having difficulty.

From here, the students work on improving the questions. [Read more…]

American Educator: Content matters

The new American Educator includes Jeffrey Mirel on Bridging the “Widest Street in the World,” (pdf), the divide between education school professors and their liberal arts colleagues.

Instead of continuing to debate the relative merits of pedagogy versus content, professors on both sides should realize that prospective teachers need to know not only their subject matter, but also how to teach it so students will understand.

Lauren McArthur Harris and Robert B. Bain write on Pedagogical Content Knowledge for World History Teachers (pdf).

Deborah Loewenberg Ball and Francesca M. Forzani write on Building a Common Core for Learning to Teach (pdf). They see the Common Core State Standards as an opportunity to establish “a common core of professional knowledge and skills for prospective teachers.”