The Book Gods?

Something that always blows me away is how teachers will follow books blindly in the face of what should be big warnings from what they know about their own students. All too few teachers are immune to book worship, having been led to believe by their own experience as kids and education students: the math textbook (and its magical authors) knows more than any regular old K-6 teacher.

Two cases in point: I was coaching upper elementary teachers in math at a low-performing K-6 school in a district near Detroit a few years ago. They were using the Everyday Math program for the first time. I was asked to guest teach some lessons on fractions in a couple of the 4th and 5th grade classrooms. I noticed that in one lesson, involving pattern blocks, there were three problems for classroom discussion. The first one was clearly needed to establish the relationships amongst the smaller shapes (triangles, parallelograms, trapezoids, that could be fit together to make a hexagon (if you have the standard pattern blocks, the relationships were that two triangles formed a parallelogram; three triangles made a trapezoid; six triangles made a hexagon, which in the first problem was the "whole" or "1." Also, you could make the trapezoid from a triangle and a parallelogram; and you could make the hexagon from two trapezoids or from three parallelograms, etc.



In the first problem, therefore, students establish that the triangle is 1/6 of the unit hexagon (which is an outline drawing in the book that you cover with these various combinations); the parallelogram is 1/3 of the unit hexagon; and the trapezoid is 1/2 of the unit hexagon.

While some students had difficulty with this, most did pretty well as I had figured. But when I saw the second problem, I sensed danger (interestingly, the third problem was much easier, and relatively few students struggled with it). #2 was an outline drawing of TWO of the hexagons with an adjacent edge, but the line where the edges met was erased, so you had a double hexagon as the new unit. The idea was that students would cover this new figure and see it as a new "unit"; it would take, for instance, 12 triangles to cover it, and so the triangle would now represent 1/12 of this unit, and so on. Each figure would represent a fraction half its previous size, since the new unit was double the old unit.

I would like to say that I brilliantly smelled a rat, but I didn't. Or that is to say, I smelled it, but I didn't trust my reactions sufficiently. Then again, I can plead lamely that I was not experienced with this book (though the mathematics wasn't a problem for me) or teaching kids this age. I was there because I had some previous experience as a coach, and because I knew the math well, and because I am very quick to adapt to new ideas and approaches).

In any event, I went ahead with the first group and had them do the problems in the order given by the book. As I had sensed but failed to act upon, the students struggled mightily with the second problem. They couldn't wrap their minds around the shift in the unit. The picture, I suspect, was perceived by most of them not as a new "unit" but simply as "two"; "obviously" it WAS equal to two of the OLD units connected to one another. They likely were visually filling in the removed boundary line where the adjacent edges met. It was very difficult to get a lot of them to make the leap to seeing this as a new "one." Even when they did the individual tasks ("cover this new shape with the triangles. How many triangles does it take?") and got the correct numbers, they were not going to budge from the notion that if a little green triangle was 1/6th in the first problem, then it was still 1/6th in the second problem. When I asked about why it now took 12 triangles, not 6, to cover the figure, they just said in essence, "Well, sure: there are two hexagons there."

I hope I have made this clear enough without the exact problem drawings that would likely made it more transparent.

So, when I guest-taught the same lesson in another class, I changed the order of the problems. I got much better results, overall. And I warned them that the problem just described, now their last problem, was challenging and might prove upsetting until they played with it a while. Interestingly, some of the metaphors I tried that bombed the previous time worked well here (e.g., a quarter is a half of a half-dollar, but it's only 1/4th of a whole dollar, etc.). Was it really that simple? Just change the order, build their confidence, warn them of quicksand, and things would go better? I'm not sure.

But the teacher in the second class was truly SHOCKED that I changed the order, and after class, when we debriefed, it took a lot to convince her that I had made an informed choice based on the previous class and my own previously ignored intuition that there was too great a leap in that second problem to go to it directly from the first one and without offering some warning bells. Her feeling was that the textbooks authors knew more than she did and must have a good reason for the order of the problems.

I assured her that SHE was the expert on her students in her class, and no author would dare usurp that position. I was making an educated guess, and had the experience of the previous class to back me up. (I know one of the main authors of EM personally, I told the teacher I would be e-mailing him with a summary of the experiences I had that day. I told her that I had no doubt at all he would understand my decision. May he might suggest a change in the next edition as a result. This was all amazing to her. And she was not a rookie teacher.

In another example from the same school and book, I chose to omit entirely a couple of problems that introduced mixed numbers into a situation where they were NOT the main point of the lesson. I anticipated (and this time trusted my instinct) that throwing mixed numbers into the fray was an error that would distract students from the real mathematical residue I wanted them to take away. Again, the classroom teacher (not the same one), was surprised by my choice and not confident that she was allowed to make a decision to remove problems, temporarily or permanently (I made clear in class to the students that we would come back to them another day).

I think this is tragic. And I think it extends even to home schooling parents, even though they know their own child(ren). They can make pedagogical choices based on that knowledge, as well as other factors. There's no textbook that can anticipate the needs of each kid, and a good teacher must intervene to make the best choices s/he can given the limitations of either whole-class or individual instruction. Clearly, teaching one child or only a few is a huge advantage and allows great flexibility. Not being under the thumb of a district or even the state or NCLB makes things even better. But the god of the book is intimidating. What if you're wrong?

But I must add this caveat: if math isn't "your thing," you need to think carefully about your choices and you are going to make some definite mistakes. Few of them will be fatal (I'm talking about the sorts of pedagogical choices already described, not mathematical errors, which are a separate issue entirely). There is what is called "pedagogical content knowledge": an understanding of both the subject and effective ways to teach it. If you're weak in content, it's hard to be strong in this area, but if you are strong in content, you may not be strong in it anyway. (And college math departments prove this daily throughout the land). It takes a lot of thinking, before, during, and after teaching lessons to be effective. It takes a willingness to really anticipate how a reasonable-seeming problem could be a mine-field for your student(s), to think about where your student(s) may go astray in the task and/or topic at hand, based on their strengths and weaknesses AND your knowledge of the mathematics (obvious example: kids learning fractions will be prone to do addition by adding the numerators and adding the denominators. It's a good idea to be prepared with an example like 2/3 + 1/5 so that you can ask the student(s) if it makes sense that the answer would be 3/8, since 2/3 is already greater than 1/2, but 3/8 is less than 1/2).

So I don't advocate just tossing things out because they look unappealing or you don't like the topic or something like that. Or because you read somewhere that a particular method isn't good. You want to think it through and in terms of the student(s). Then, you do what you can, being prepared to change course if necessary. No shame in that. Indeed, it's a wise teacher who can admit error, doubt, and change course. Too bad some people in Washington, DC seem to lack that wisdom.