*This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.*

Lampert covers a great deal of territory in this chapter, but for a subset of it she focuses her attention on a single lesson. Students are placing a number of Fraction Bars — shaded rectangles representing different fractions — in order from least to greatest. In the discussion following the task, one student, “Saundra”, asserts that five-sixths and five-twelfths go in the same place on the number line. Lampert’s thought process as she responds is fascinating:

I need Saundra to learn the correct placement of fractions on the number line. I need her to understand why five-sixths is larger than five-twelfths. And at the same time, I need her to learn to think of herself as a person who can study and explain her mathematical reasoning and that she can do it in school, where her peers are watching everything she is doing (305).

Lampert returns several times to these major goals. I identify them as:

- Students learning content
- Student learning that math makes sense and they are capable of reasoning logically about it
- Students learning the courage to take intellectual risks in a public setting

Lampert goes on to describe the tensions inherent in working with these challenges:

To build Saundra’s mathematical competence, and at the same time maintain her academic self-confidence, I must now work like an air traffic controller, keeping all of the planes that are trying to come in for a landing from crashing into one another (307).

At the same time, she has decisions to make about which goals to prioritize over the course of the discussion. Her first decision is interesting: she helps Saundra to visualize the fractions on a number line at the blackboard, using that simple tool as a support for Saundra to make sense of the problem. Saundra makes a simple mistake — she counts the tick mark at 0 as one-twelfth, and counts up from there, meaning each fraction is off by one-twelfth. Lampert chooses to make that correction herself, rather than opening it up to the class or prompting Saundra to notice it, for the sake of expediency and to focus the class’s attention on the larger questions that she sees more potential for learning in.

As the class begins to discuss the relative placement of five-sixths and five-twelfths, Lampert makes another deliberate decision in who she calls on:

I first called on Charlotte because I had observed a pattern of respectful consideration in the way she responded to other students in the class when she disagreed (308).

Here she prioritizes teaching Saundra to take intellectual risks, by choosing a student to respond who will be likely to be respectful and productive in building up Saundra’s belief in herself as a mathematician.

The class then moves into a broader conversation, with a number of students contributing, and many disagreeing with Saundra’s assertion about five-sixths and five-twelfths. There are a number of teacher moves worth discussing in these exchanges, still working at the tensions between Lampert’s goals for the discussion. Interestingly, in the final exchange, Saundra states, “So I mean, I guess really it could be either way”, stating that either five-sixths and five-twelfths could be equivalent, as she states, or five-sixths could be larger, as several other students articulated. Lampert, in summarizing the discussion, writes: “She performed her understanding publicly and maintained her dignity” — Saundra was willing to take several risks, defended her reasoning, and did not come out of the discussion having been hurt or unwilling to contribute to a future discussion. At the same time, Lampert notes the challenge: “Reasoning mathematically, one would have to conclude that there is a contradiction between these two approaches, but this did not trouble Saundra (324)”. In managing the tensions between her goals, she may have been successful in instilling intellectual risk-taking in Saundra, and provided a new perspective on fractions, but was unsuccessful in teaching Saundra to believe that mathematics makes sense and that her reasoning should be consistent.

This is not a reflection on Lampert’s teaching; rather, it reflects the ongoing challenge of teaching students to be people who study in school, and the fact that this is a challenging, multi-year project, happening in fits and starts, and often without clear victories along the way.

I want to end this post with a nod to Joe Schwartz. He wrote one of my recent favorites posts over at his blog, and offers an equally fascinating case study of a student who is struggling, both with content knowledge and with his beliefs about mathematics and about himself. These are hard challenges, and I see in Joe’s post another issue — that it is often hard even to notice when a student’s beliefs and dispositions about math are counterproductive unless we ask the right questions and take the time to hear out student answers. That’s a larger challenge for another post, but more food for thought about what is actually happening in the minds of my students while they are in my class every day.

Michael Paul GoldenbergIf we assume that the goal of every lesson and/or discussion is to eliminate all misunderstandings of all topics up to and including the one for that day in the minds of all students, then clearly this was a failed lesson. The student apparently ends the day unclear as to whether 5/12 and 5/6 are equivalent or different. But from another perspective, she grew, in that she had possibly moved from her belief that they absolutely were equivalent to entertaining different (though contradictory) viewpoints.

Knowing the worlds of mathematics and mathematics education (often at odds with some of each other’s fundamental goals and assumptions), I would anticipate both mathematicians and teachers reacting to my claim of growth with revulsion: how can being “wrong” indicate growth? And therein, I believe, lies one of the most basic problems in discussing the teaching of mathematics through examining actual practice in real classrooms (rather than through purely theoretical conversations about what “should” happen in idealized settings): those who believe that the single goal of mathematics is “getting the right answer” will never accept the notion that process is important and that at times it is at least as important as answer-getting. Focusing on the student’s process here, we see a teacher possibly moving that student along a continuum from being fixed on a wrong notion towards seeing the mathematical truth (at least about these two rational numbers and their relative positions on the number line). At the end of the lesson, she has moved from the first extreme to a middle ground that could allow her to eventually come to a truth. Will she get there? It seems likely, but of course we do not yet know (based solely on the information at hand). She might be stuck indefinitely. Why not just tell her she’s wrong, explain why, and move on?

I’ve read enough discussions about mathematics teaching to know that in the minds of many, there is no real justification for not taking the direct path: correct the error in student thinking. GIve the reasons the student is wrong, or simply emphasize the right way to reason about comparing two positive rational numbers (particularly with the same numerators and different denominators). If the student remains confused, repeat the explanation – louder and slower. If the student is still confused, she fails. The teacher has done her job; it’s the student who has come up short and will be found wanting. Time to move on to the next topic, example, etc.

It is probably obvious that I have no sympathy for that sort of thinking. I believe Lampert’s approach, while not the only possible path to more effective teaching/learning, is a good one, and clearly superior to the more typical alternative. But I’m going to see what this column and my comment engenders before fleshing out the rest of my argument.

dkane47Post authorI’m definitely curious to hear more — I’m not sure that taking an either-or approach is the best way to move this discourse forward. I do think that both sides you refer to have more in common than they think, but the rhetoric tends to mask all that.

One quote from this chapter that struck me was this one:

“Now I have at least three pressing agendas. I need to bring the class to a close because it is time for recess. Shahin is very anxious to say something to me and the class and when he is anxious he gets silly and distracts other students. And I am still hoping Saundra will revise (314)”.

It highlights a challenge that I didn’t bring up above — the institutional challenge of teaching in a large classroom with pressures of time and space that force us to move from ideal teaching moves to reality. I think that very many teachers who believe what you wrote above make much more traditional teacher moves on an everyday basis because of these structural challenges — and they provide an interesting perspective on when these different moves may be appropriate or inappropriate. I need to flesh this out a bit as well, but I’m pessimistic on the value of an “all or nothing” approach that tries to mimic Lampert or others who might be seen as progressive educators, and I much prefer thinking about when their ideas make the most sense and using that to further more conversation.

All that said, the traditionalists who I’d love to have this conversation with don’t frequent my blog, so I don’t know if that dialogue will happen.