Knowing and Learning Mathematics for Teaching Proceedings of a Workshop (2001) / Chapter Skim
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Investigating Teaching Practice: Setting the Stage
Investigating Teaching Practice: Setting the Stage
Pages 25-38
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From page 25... ...
Deborah Loewenherg Ball, University of Michigan Hyman Bass, Columbia University Reaction to the Presentation by Deborah Ball and Hyman Bass James Lewis, University of Nebraska
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From page 26... ...
Those are all valuable, but we want to focus on what it means to look at children's mathematical reasoning as emergent mathematical justification. Whatlenses(lo elementary teachers need to understand that what they as teachers are involve(1 with is the (levelopment of chil(lren's capacity to construct proofs, to un(lerstan(1 anti follow mathematical proofs, to understand the need for justification, and to be able to distinguish valid justifications from invalid justifications?
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From page 27... ...
INTRODUCTORY COMMENTS BY BASS ~ came into the work as a mathematician, and my role is essentially the same as the assignment for the workshop, to look for the mathematics entailed in the core tasks of teaching, the kinds of problems a teacher has to solve, design of lessons, interpretations of student thinking and work, and how to assess whether to pursue an idea or return to the lesson plan. Many of these decisions entail mathematical knowledge and considerations, and the idea is to try to discern what that mathematics was, to place it in a larger mathematical context.
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From page 28... ...
If these were professional mathematicians, this base might consist of an axiom system for some mathematical structure like Euclidean geometry or group theory, simply a(lmitte as given, plus a bo(ly of previously (leveloped and publicly accepted mathematical knowledge derived from these axioms. In another setting, a university instructor who lists prerequisites for a course is defining part of the presumed common knowle(lge in the environment of that course.
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From page 29... ...
We're concerne(l, as well, with the transformation of mathematics embe(l(le(1 in experiential settings, (lescribe(1 informally with common language, into more formal mathematical expression susceptible to efficient mathematical manipulation. We emphasize that mathematical language is not simply an inert canon inherited anti learne(1 from a (listant past.
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From page 30... ...
The children typically first encounter whole numbers, and they may have some informally developed sense, say, of even and odd, and they probably could quickly decide correctly whether a given small whole number is even or odd. At first, they usually have no formalized definition of these notions and typically lack, for example, the unit-(ligit criteria for recognizing evenness or oddness.
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From page 31... ...
For example, the alternating definition extends in a very natural way to the negative numbers, but the notion of dividing something into equal parts or an equal number of pairs, or a whole into number of pairs with nothing left over, presents a different way of prolongation to the other domain. For example, when extending the notion of addition to negative numbers, you might want to preserve certain mathematical properties like the distributive law with respect to multiplication and addition, but arguments made about arithmetic with negative numbers, which invoke those properties, have tacitly made the presumption that the definition must be made in such a way as to preserve those properties, just as using exponential notation when the exponent is no longer an integer or a rational number W H AT I S E N TA I L E D I N T E A C H I N G ?
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From page 32... ...
At first, students simply wrote two elementary problems, such as six plus four equals 10 or five plus five equals 10. They were only doing addition and only using two elements.
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From page 33... ...
Adding many terms at once or adding two terms both larger than one is a more complex operation, not only for children but also mathematically. Thus, once Kip's dramatic formula was presented to the class, the culture of reasoning demanded that it be justified or proved in a manner appropriate to children at this level, and as a move toward creating the culture of reasoning, the teacher had the children publicly honor this mandate.
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From page 34... ...
When asked why, they would say, 'well, we keep coming up with the same answers over again," or, "I looked at so-and-so's list and she didn't have anything that ~ didn't have already." Essentially they were making an empirical argument, a kind of scientific or probabilistic claim, not a mathematical argument. It was the first encounter as a class with the challenge to try to mathematically prove a claim, "do you have a full set of possible solutions to this problem?
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From page 35... ...
" One obvious answer is that it takes quite a bit more mathematical knowledge, and this is certainly underscored by the videotapes that were shown as part of the presentation. Watching the tape of Deborah Ball teaching a class of third-grade students, it becomes clear that she is attempting to establish a classroom culture in which mathematical reasoning is calle(1 for.
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From page 36... ...
She clearly was establishing a classroom culture that calle(1 for mathematical reasoning, anti her students (lemonstrate(1 a sophistication regarding the nee(1 to prove statements rather than accept them on faith. As a mathematician teaching at the university level, ~ have very little knowle(lge of thir(l-gra(le RECONSIDERING THE MATHEMATICS
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From page 37... ...
classrooms, but ~ suspect that similar classrooms are in short supply. ~ am also convinced that the mathematical knowIedge required to teach in this manner is REACTION TO BALL AN D BASS far superior to the mathematical knowIedge that most of our students have when we certify them as ready to teach mathematics at the elementary school level.
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