Here is what I like so far about Elizabeth Green’s Building a Better Teacher: It has a searching quality, as I mentioned yesterday.It has vivid descriptions of lessons in action. It discusses actual subject matter. It makes the important argument that teachers can improve their craft through deliberate study. It gives rich examples of such study. All of these qualities make the book a worthwhile read.
At the same time, I am puzzled by Green’s utter lack of skepticism over certain exemplars of pedagogy that she offers in the book. In saying this, I am not trying to disparage them. My point is only that they could use some critical questioning and examination–in the very spirit of the kind of lesson study that Green finds promising.
This is a preliminary review, with a focus on a particular passage (about a third-grade lesson) in the second chapter. I haven’t read the whole book yet (I read slowly and have been very busy), but I had so many thoughts about these few pages that I decided to start here.
The context: Deborah Loewenberg Ball, at the time a professor at Michigan State, a scholar of math pedagogy, and a teacher at the public school Spartan Village, was teaching her third-grade students about odd and even numbers. The lesson was one of many that she and her colleague Magdalene Lampert had filmed for close study and discussion. Just before this lesson, the fourth-graders had a conference with the third-graders in which they discussed their findings on the question: “Was zero even, odd, or, as some children argued, neither one?”
For this lesson, Ball intended to have the students move from conjectures to proofs about odd and even numbers. But something unexpected happens: a “tall boy named Sean” puts forth a surprising conjecture that six is both even and odd. His classmates then jump in to refute him. What follows is a lively but flawed discussion–flawed not because of the students’ insights, which are excellent, but because of the lack of attention to basic principles, such as the principle of identifying and building on one’s working definitions (or, in the absence of definitions, information leading up to them).
The problem throughout the entire passage is that we never learn whether the students have a working definition of odd numbers. This lack of information affects everything, as I will show. It seems that they have a working definition of even numbers–but at times they appear to confuse definitions with properties. Moreover, the working definition itself could be the cause of Sean’s confusion–but this possibility is not mentioned. More about all of this shortly.
Back to the conference: it is a brilliant idea to have fourth-graders present their findings to third-graders. This gives the fourth-graders a chance to teach others what they have learned, and it gives the third-graders a glimpse of knowledge and insights that lie ahead. In addition, a conference on zero is a great idea; there’s much to explore about zero. Yet I fail to see why the question of zero’s odd, even, or other status merits a conference (even a short one). If the students have a viable definition of odd and even numbers, they can immediately rule out the possibility that zero is odd. (If they do not have working definitions, then they have no way of discussing the question anyway.) Then, if the students have a viable definition of even numbers, they can see (without a great amount of trouble) that zero meets the criteria. One stumbling block might be the concept of dividing zero in two. Some students might think that can’t be done. So, that would be the meat of the discussion, but it’s easily digestible. There isn’t much gristle here.
The students themselves don’t seem to be clear about their working definitions, or whether or not they have them. After Sean has spoken, Cassandra goes up to the board to refute him. She says that six can’t be an odd number, because zero is even, one odd, two even, and so on up to six, which must be even.
Green comments on the reactions of the mathematician Hyman Bass as he watches the video.
Hy marveled as the video continued. These third-graders–not a gifted class, but average, public school third-graders from, Deborah said, a wide range of backgrounds and ability levels–were having a real mathematical debate. One of them had made a claim, and then the others were trying to prove him wrong. Cassandra’s proof followed a classic structure. First, she had invoked one definition of even and odd–the fact that integers alternate between the two types on a number line–to show that six could only be even. Then she had drawn out a counterargument. To be odd and still fit the alternating definition, she’d shown, zero would have to be odd too. But, she’d concluded with a flourish, they had just decided the other day that zero was even. QED: Sean’s conjecture was impossible.
The two descriptions of Cassandra’s words and actions don’t match–the second is much more sophisticated than the first–but that’s only a secondary problem. The bigger problem lies in the notion that “the fact that integers alternate between the two types on a number line” could be called a definition. To me, this appears as a property, not a definition. It makes sense that the students would be working from properties to definitions–but it’s essential to point out the difference.
The same confusion arises a couple of pages earlier, in a footnote regarding the evenness of zero: “Like all even numbers, zero can be divided evenly by 2, is surrounded on either side by odd numbers, and when it is subtracted from an even number, produces an even result.” Only the first of these qualifies as a definition, and it alone is necessary.
The discussion goes on.Apparently the students do have a definition of even numbers: one girl, Jeannie, reminds them that an even number is “one that you can split up evenly without having to split one in half.” If this is indeed the working definition, then it seems possible (though it never gets mentioned as a possibility) that Sean’s confusion arises directly from this wording, particularly the word “evenly.” (His own explanation of his reasoning seems to proceed from such a misunderstanding.) He may have taken this definition to mean that a number is even if it can be divided into even numbers–a circular definition, but one that “evenly” seems to invite. In that case, there’s more to say about Sean’s conjecture. More about that in a minute.
Now another student, Mei, makes a great argument: by Sean’s reasoning, it could turn out that all numbers were both odd and even, in which case “we wouldn’t be even having this discussion!”
What Mei suggests here–but no one brings out–is that they have been working with the premise that a number is odd or even, but not both. If that is indeed one of their working premises, then it should be on the table. If it isn’t, then I wonder how they conceive of odd numbers in the first place.
I admire Mei’s energy and logic, but I feel bad for the student who has been sitting there quietly–who gets odd and even numbers and yearns to move on. I also feel bad for the student who has no idea at this point what has been established and what hasn’t.
To draw something helpful–and fascinating–out of this discussion, the teacher only had to remind the students to go back to their working definitions (and distinguish them from properties). This is important mathematical practice. One has to return to working definitions continually. Sometimes they come up for questioning. Sometimes a definition may prove flawed, or it may need better phrasing. But one must be clear about what the definitions are.
If, as I suspect, Sean thought that a number was even if it was divisible into even numbers, then the teacher could have clarified the meaning of “evenly” (and “even” elicited a rewording of the definition).
Then, to take up Sean’s idea (which is actually very interesting), she could have asked: Which numbers are divisible into even numbers only (assuming one does not treat 1 or -1 as a factor)? Students would notice that the positive integers in this set were 2, 4, 8, 16, …. in other words (though they wouldn’t have the vocabulary for this yet) exponentiation of 2 to the powers 1, 2, 3, 4, etc.
Many interesting things happen in the lesson–but the confusion over definitions and properties prevents the discussion from moving forward. For this reason, I do not share Green’s amazement, though I am grateful to the lesson (and to Green’s description) for stirring up some thoughts.
Note: I made some minor edits to this piece after posting it. Also, on 8/26/2014 I added one parenthetical sentence.