In my last post I discussed the importance of focusing on good teaching instead of just trying to identify good teachers. One of the cardinal principles of good teaching is to provide not only positive examples of a new concept, but also negative examples as well. Learners who are trying to categorize information according to a criterion or set of criteria new to them need to know not only what the new thing is, but what it isn’t as well.

It is with that thought in mind that I will devote the next couple of posts to the topic of bad teaching. I hope that is is clear from my earlier posts (and my general disposition, for those who know me) that my purpose is not to embarrass or make fun of the teachers responsible for the bad teaching. As I have mentioned, I have done bad teaching myself, and I will include this in the current discussion. Rather, the goal of this discussion is to use examples of bad teaching to flesh out a description of good teaching and suggest some important ways to evaluate the quality of teaching, whether one’s own or that of another.

One of the worst examples of teaching that I’ve seen involved a homework assignment for spelling practice required in a 6th grade classroom. The teacher instructed the students to rewrite each of their spelling words in a code, with a tall rectangle used to represent any “tall” letters like “l” and “d”, a short square to represent “short” letters like “c” and “v”, and a tall rectangle below the base line to represent “basement” letters like “g” and “p.”

Now, even in this very bad assignment there is a kernel of a good element, namely, an attempt to help students analyze the content they are learning. As I shall argue over and over, helping students analyze what they are studying is a critical component of good teaching.

The problem with this particular assignment, of course, is that besides being tedious and requiring a significant amount of time to complete, it really has nothing at all to do with spelling. The kind of analysis required of the students to perform this activity involves absolutely no attention to the underlying patterns and relationships of our writing system. These fundamental patterns and relationships include how individual letters and groups of letters represent individual speech sounds, how certain functional units such as “un” and “ed” and “ion”, known as morphemes, are used over and over in the formation of words, spelling patterns such as when to double a consonant when adding a suffix to a base word, how both the pronunciation and spelling of words can change as various morphemes are added to base, and how knowledge of a word’s origin can often provide insight into why it is spelled the way it is. In other words, there is plenty of appropriate analysis that students can do to help them learn to spell words correctly. Moreover, this productive analysis helps students recognize fundamental patterns that reveal the architecture (and, dare say it, the beauty) of our writing system.

A similar sort of problem bedevils a technique used in several classrooms I know of to teach 6th graders how to divide fractions. This practice entails having students memorize the phrase “Keep-Change-Flip” to help them perform a calculation such as 1/2 ÷ 3/4. What it means is that the students are to rewrite the the equation by modifying it in the following way. They are to **keep** the first number (unchanged), **change** the operation from division to multiplication, and then **flip,** meaning, write the reciprocal of, the second number, thus yielding the expression: 1/2 x 4/3.

There are several problems with this approach. First of all, it shares a problem that is very common in American mathematics instruction, namely, it focuses student attention initially on a procedure for calculation without adequately developing an understanding of the underlying patterns and numerical relationships that make the procedure work. Now, this technique is effective if the goal is to get a student to do a set of problems exclusively involving division of fractions as quickly as possible. Only a few minutes of instruction is required to accomplish this objective. But that objective is very useful. Students will rarely ever confront a series of similar division problems all in a row. It is much more likely that they will have to perform an unpredictable mixture of various arithmetic calculations, in which case, they have to remember which calculation procedure goes with which operation. And there is nothing in “Keep-Change-Flip” that designates it as the procedure for dividing fractions.

This approach makes the procedure more difficult to memorize for another reason as well. It is presented as an isolated fact about arithmetic, rather than as something firmly embedded within a set of fundamental patterns and relationships. To see this limitation clearly, consider an alternative approach, namely, having students learn that “dividing by a number is the same as multiplying by its reciprocal.” This statement, like “Keep-Change-Flip” provides all of the information students need to perform the procedure for dividing fractions correctly. But unlike “Keep-Change-Flip,” it clearly identifies this procedure as something involved in the division (and multiplication!) of numbers. It also includes another key concept, the idea of reciprocal. And it places the procedure within the context of a fundamental pattern of arithmetic which the student can (and should) explore. Indeed, it is very easy to give the student a set of investigations in which the concept of reciprocal is examined and then this fundamental relationship between division and multiplication is made apparent. When taught correctly, in fact, the students use the experiences and analysis they have done in these preliminary investigations to state the procedure themselves. All the teacher does is to give them the precise wording for a relationship the students already understand, as well as to provide sufficient practice to perform the calculation correctly. (I will have lots to say in the future about both of these points, namely, providing vocabulary AFTER students have the relevant experience and hence know what the vocabulary refers to, and the characteristics of efficient practice.)

Now some readers may be thinking, “But the presentation of ‘Keep-Change-Flip’ is something like a mnemonic, that is, a device that helps us memorize things. Are you saying that mnemonics are bad?” No, not at all. Mnemonics have a very important place in good teachingand I will talk a lot about such tools when I devote some blog posts to the topic of memory and helping students memorize.

But the purpose of mnemonics lies primarily in helping people learn things that are related in an arbitrary way, such as the digits of pi, or the order of the planets in the solar system, or the order of the colors in the rainbow. In contrast, the procedure for dividing fractions (as well as the skills involved in learning how to spell words correctly, in the first example in today’s blog post) do not involve isolated and arbitrary information. On the contrary, both involve concepts that are connected to other material according to a set of underlying relationships. And when we want students to learn this type of material, our job is to direct their attention first to these patterns and principles.

In short, good teaching helps students make connections. The corollary is also true. If some teaching does not involve helping students make connections, then there is almost always a way to improve it!