Comprehension Across the Curriculum: Perspectives and Practices K-12
This collection of 13 essays is divided into three major parts: theoretical perspectives, classroom and school applications, and historical perspectives. The applications section will prove to be the most helpful to teachers. Among the chapters in this section are: “Promoting Comprehension in Middle and High School: Tapping into Out-of School Literacies of our adolescents,” “Thiniking and Comprehending in the Mathematics Classroom,” “Comprehension in Social Studies,” and “Comprehension Connections in Science.”
Douglas Fisher, Nancy Frey, and Heather Anderson, the authors of he mathematics chapter, include selections ostensibly from a student’s journal about his experience in a ninth grade algebra class. Here is some of what “Doug” wrote.
“The day starts like every other day so far this semester. As we enter the room, our teacher calls off odd numbers. By now, we know that when we’re called on like this, we have to solve the assigned homework problem on the board. Our teacher watches the group of students assigned to complete the problems and offers periodic criticisms and compliments. When everyone is finished and the answers are correctly posted on the board, we check our homework and then pass it forward to the teacher (of course, we’ve all checked our homework on the bus to make sure we’ve all got the same answers because homework counts for 25% of the grade).”
At the next juncture in class, the teacher lectures. “Our task during this time is to take notes exactly as he presents them. Our notebooks must have specific page numbers that match his and are worth 25% of our grade…. We do not summarize our notes or organize them in any systematic way; we copy them exactly as they are presented in class.”
“When he finishes the lecture and note-taking component of the class, we have the remainder of the period available to start our odd-numbered problem set for the day. If we do not finish the problems in class, we are to take them home and complete the rest. If we talk during class, our teacher will call out an even-numbered problem for which there is no answer in the back of the book. As punishment for talking, we have to go to the board and attempt to demonstrate our prowess in front of our peers.”
Frankly, I doubted the authenticity of these journal entries, but even if they are fictitious, they are instructive. I had to wince as I read them; they were painfully familiar. Even now, a half century out of high school, I can still recall the leaden opacity and oppressive atmosphere that enveloped my algebra and geometry classes.
But, this chapter is not just a broadside against so much of our schools’ mathematics instruction. The authors credit the teacher with worthy motives. They write: “…it is clear that the teacher values practice—he provides his students with lots of opportunities to engage in independent learning. He also values students having correct information and right answers. He wants their notes to be exact, and he wants students to practice testing formats. Unfortunately, this teacher has no way of understanding his students’ thinking and he types of errors they make. Although he explains information, he doesn’t let his students in on his thinking—the thinking of an expert.”
These are generous minded folk. I think they are trying hard to find good things to say about this teacher. He appears to me to be someone who has done exactly the same thing, day after day, year after year, throughout a long, long career.
Even so, the good news here is that the authors observe: “With a few adjustments to the structure of the classroom, students would likely develop a deeper understanding of mathematics and begin to see the relevance of this content in their lives.” I’m not sure about the second half of that assessment, but I certainly agree with the first half. The authors devote the bulk of their essay to showing how this teacher might improve his instruction.
The first suggestion is for him to model his thinking for his students, so they can see how he approaches problems. For example, in order to provide or remind them of background knowledge, me might say, “When I see a triangle, I remember that the angles add up to 180 degrees.” Or, in order to show students how he is setting up a problem, he might say, “The first thing I will do is…because….”
The authors also recommend that the teacher work methodically to develop students’ math vocabularies. Some words, “per” and “quotient,” for example, automatically signal division. Others: “difference,” “diminished by,” and “minus” signal subtraction. These may seem obvious, but what is obvious to some is opaque and seemingly unknowable to others—until they are shown.
All in all, teachers will find the subject specific sections of this book to be welcome and helpful.