Am I getting warmer?

In The Teaching Gap, Stigler and Hiebert examine the details of typical lesson designs found in German, Japanese, and U.S. math classes from the Third International Mathematics and Science Study (TIMSS). A table found on pages 30 and 31 describes three specific lessons and notes whether or not each lesson's features represent the overall teaching in that country. The U.S column looks something like this:

• Teacher asks students short-answer review questions. (Typical to begin with "warm-up" activity.)
• Teacher checks homework by calling on students for answers. (A common way to check homework.)
• Teacher distributes worksheet with similar problems. Students work independently.
• Teacher monitors students' work, notices some confusion on particular problems, and demonstrates how to solve these. (Typical for teacher to intervene at first sign of confusion or struggle.)
• Teacher reviews another worksheet and demonstrates a method for solving the most challenging problem.
• Teacher conducts a quick oral review of problems like those worked earlier.
• Teacher asks students to finish worksheets. (Unusual to not assign homework.)

My work with teachers brings me into a lot of secondary math classrooms, and I can say that not much has changed in the typical math lesson since the printing of The Teaching Gap in 1999. However, I want to focus on just one aspect of the U.S. lesson, the warm-up.

Because of my own interest in engagement, teachers often ask me to focus on when students are engaged (or on-task) during a lesson and what to do when they are not. Probably the most off-task period during many lessons is the first part of the lesson, the warm-up.  This might be for any number of reasons but I have a few ideas.

First, the students find little purpose in this activity. It is rarely collected, usually graded only as a consequence for causing a disruption, and often disconnected from the overall lesson. Without some purpose, it is difficult for students to engage in a task.

Next, there are times when the task is an extension of current work rather than a refresher. It would be like running a race without warming up. Consequently, some learners do not see themselves as having the potential to be successful with this initial task - a tough way to begin a lesson.

Finally, by sixth grade students are familiar with the design of most math lessons. They know that someone will at some point provide the answers to the warm-up. So some students do not work on the warm-up (off-task) but copy down the answers when they are shared (on-task). Others, work on the warm-up (on-task) but pay little attention to the answers because they already have them (off-task). Either way, this seems like an inefficient way to start just about every lesson.

What do you think? Have I accurately portrayed what is going on during the warm-up in many math classes or is this a straw man? If you agree that the warm-up is in need of a makeover, then perhaps in a future post we can discuss the alternatives. Otherwise, I do not want to waste your time any further - going over something that is not really a problem has no purpose and does not make any sense to me.