I’ve learned much of what I know about teaching from colleagues, but when I started teaching high school, there is one book that I found extremely helpful: Every Minute Counts, by David R. Johnson (1982, Dale Seymour Publications, with great illustrations by cartoonist John Johnson). In the 37 years since the book’s publication, society has changed, math education has changed, and I’ve changed. I decided to take another look at it, to see how well it’s holding up.
Early on, Johnson shares his goals as a teacher. Here are three good ones:
- I will give no options for nonparticipation.
- I will allow students to make mistakes without fear of failure or embarrassment.
- I will encourage student interaction during a portion of each class period.
After listing his eight goals, he adds: “I don’t remember a single day when I can honestly say that I have achieved all of them…” That’s evidence that the author is an actual teacher, not someone promoting the latest fad. In fact, the whole book confirms that: it consists of a mix of big picture ideas and specific implementation ideas about everything from homework, to the daily routine, to assessment, all of it stemming from one teacher’s experience.
The most useful part of the book, both in the 1980’s and now, is the chapter on “The Art of Questioning”, in which he presents some great ideas about how to run a class discussion. He starts out by challenging what he calls the “one-on-one questioning method” (you ask a question, get the correct answer from a student, and move on, without knowing whether anyone else in the class understands what you’re talking about.) He then offers a list of 20 “try-to” principles of questioning. (Yes, twenty!) Here are three of the best:
- Try to avoid yes/no questions.
- Try to follow up student answers with “why?”
- Try to ask for reactions to a student answer.
Johnson’s overriding philosophy of class discussion is two-fold: on the one hand, the questions should get all students to engage, and on the other, student answers should reveal to the teacher the extent of student understanding. He discourages us from asking “Does everyone get this?” or similar questions, which rarely yield accurate feedback about what is really going on. He believes it is the teacher’s responsibility to figure out who gets it. This leads to his key insight: he needs answers from every single student. To achieve that, he has the whole class answer questions using the “paper-and-pencil method.” He asks a question, has all students write down their answers, and he walks around to see what they wrote. This way he can proceed with the lesson with full knowledge of who’s on board, whether he needs to backtrack, and so on. (Of course, there are other ways to get this sort of whole-class feedback, which I should probably discuss in a future post.)
To make this method more efficient, he lays out the student desks in a half-circle, with the teacher at the center — what he calls “a U-shape arrangement”. This means that no one is sitting in back, and the teacher can quickly see all the students’ work. Unfortunately, this is not optimal for student collaboration, so this is not a suggestion I would take up. Still, even in a collaborative classroom, teacher-led discussions play an important role. David R. Johnson’s techniques for genuine student engagement in class discussions are mostly excellent, and far superior to the standard pseudo-interactive lecture format, where a small number of vocal students finish the teacher’s sentences.
In fact, his ideas on questioning would make a great topic for a math department meeting, and for targeted visits from colleagues to your classroom. It appears the book is still in print, or is back in print, so you may want to get a copy.
Oh, one more thing. Johnson ends the book with 15 discussion starters, most of which address common student misconceptions. Here are three of them:
- Which is greater, x or -x?
- When is 1/x greater than x?
- Are some numbers greater than their squares?