Knowing and Learning Mathematics for Teaching Proceedings of a Workshop (2001) / Chapter Skim
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Investigating Teaching Practice: What Mathematical Knowledge, Skills, and Sensibilities Does It Take?
Investigating Teaching Practice: What Mathematical Knowledge, Skills, and Sensibilities Does It Take?
Pages 39-64
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From page 39... ...
Torres, Tucson, Arizona Public Schools Analyzing Student Work Michaele ~ Chappell, University of South Florida Managing Class Discussion Buck Smith, University of Rlinois-Chicago
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From page 40... ...
Revisited" anti discussed a set of focus questions in both small and large groups. During this part of the (liscussion, the conversation was designed to allow participants to share and (1ebate their ideas about the mathematical thinking of the students (lescribe in the print case.
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From page 41... ...
How could I be asking them to clivicle 2 candy bars among 3 kids if we already "knew/' that we couldn't do it. From Developing Mathematical Ideas, Number and Operations, part 2: Making Meaning for Operations Casebook, by Schifter, Bastable, & Russell.
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From page 42... ...
) RAYMOND: So, if each kid was going to get equal shares, they would have to cut the 5 candy bars into little equal pieces.
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She is displaying some confidence ancl some clear mathematical thinking that I have not seen before. ANTHONY: I think the same thing, that each person will get one piece from the first candy bar and one piece from the second and then from each one aher that and will end up with 5 little pieces, so 5/39.
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From page 44... ...
DISCUSSION SUMMARY In the first part of this discussion, participants consi(lere(1 the mathematical thinking of the students represented in the print case. The conversation had several dimensions: · clarifying exactly what was said in the print case; · making hypotheses about what aspects of mathematics the students understood; · making hypotheses concerning the source of the students' confusion; · locating the mathematical logic in the student's thinking.
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From page 45... ...
The list provides a description of the mathematical issues considered by the class described in the print case. For a particular mathematical situation, this list illustrates the range of mathematics and knowledge about the nature of mathematics and mathematical reasoning that a teacher must have to negotiate this kind of classroom conversation.
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From page 46... ...
This leads to the following implications for teacher preparation: Mathematical content should not be separated into individual topics; the ways preservice teachers encounter mathematics as learners is significant; and understanding how an idea develops over time (and across grade levels) is necessary.
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From page 47... ...
For example, in our discussion we talked about mental images of the operations; we drew diagrams to express mathematical ideas; and we expressed ideas through language. This lea(ls to the following implications for teacher preparation: When preservice teachers encounter mathematics content, they need to have experience with a variety of forms not only symbolic representations; mathemetics class shoul(1 be a forum for communicating mathematical ideas in both writing and speech.
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From page 48... ...
This is especially true if we commit ourselves to providing opportunities for preserves teachers to explore mathematical ideas, develop as mathematical reasoners, and deeply engage with a subset of mathematical ideas to determine the ways children make sense of those ideas over time. Even while ~ expect that once such ideas about mathematics and its nature are understood, teachers would be in a position to continue to learn mathematics through the process of teaching, ~ am still left asking, 'what is the set of i(leas around which the undergraduate experience should be built?
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From page 49... ...
What mathematical knowledge does a teacher need to remodel a mathematical task to make the task simpler, to broaden student access, to add another dimension to the task to move students forward in their thinking, or to use the same concept in a slightly differentform to provide another experience with the idea? OVERVIEW The chief activity of the workshop was a variation on a riddle, "Coins in the Purse.
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The activity helped to make it apparent that anyone involved in teaching mathematics at the elementary level needs to confront the complexity of the apparently simple mathematical ideas these students need to acquire and develop. A constant series of questions was RECONSIDERING THE MATHEMATICS
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Throughout this exercise, ~ was conscious of how important it is to know about key number concepts, such as inclusion, 1:1 correspondence, and conservation of number, and of how these concepts contribute to acquiring mathematical understandings. At the elementary level, it is critical to know these key concepts, which help provi(le the foun(lation for developing number and operation sense.
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From page 52... ...
One must be informed about human development, knowledgeable of mathematics content, curious about how mathematical knowledge is constructed, patient with the knowledge that understanding is developed over time, and possess a disposition towards learning and teaching that capitalizes on the dynamics of the learning environment the people, their ideas and experiences, and the physical materials.
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From page 53... ...
The session focused on these questions: What mathematical knowledge does a teacher need to analyze student work? What level of mathematical knowledge does a teacher need to respond to varying strategies in ways that will move the entire class forward in their thinking?
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From page 54... ...
in the task Participants identified several mathematics content areas embedded in the "Mixing Juice" task. Major topics included the following: · un(lerstan(ling proportional reasoning; · comparison of (lifferent ratios, percentages, an(l/or fractions; · part-to-whole relationships; scaling (e.g., how many "recipes" are nee(le(1 to make the total amounts; conversions to (lifferent units for comparison (e.g., reducing numerator to l)
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had ~—n~ \~r~s'E~ ANALYZING STUDENT WORK ~\~> ~ 2~ 1~o ~ ~ ~1e ~z=~Zj33_ ~= ~°1^ ~ t)
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Student work is used with permission from Traverse City Public Schools via the Connected Mathematics Project.
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From page 57... ...
ANALYZING STUDENT WORK Examining and analyzing the stuclents' responses In(livi(lually, in pairs, or in trios, the participants examined the 12 responses provi(le(1 by the student teams (Figure 21. Much of this discussion concerned the mathematics involved and the overall focus question, although comments were made either about the students' work or about the analysis process itself.
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From page 58... ...
In addition, because problem solvers all have their own individual methods, teachers must be prepared to think differently in order to understand the various methods students may use in solving a problem of this nature. Iclentifying what matters This discussion pertained to the overall focus questions for the session.
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From page 59... ...
Other factors that matter: · what the teacher does with all of the information obtained from analyzing students' work; · how the teacher plans for the next day once the information is obtained; how to use students' work to generate teacher research questions, such as asking what this child really understan(ls about"x"; the role of the National Council of Teachers of Mathematics (NCrM) Standards (NCrM 1989, 2000)
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From page 60... ...
Thus, from Brown's perspective, the mathematical meaning a teacher constructs is in relationship to "these things," that is, the mathematical worI(1 of students. If we take this contextual view of mathematical meaning seriously, it naturally lea(ls one to wonder how the mathematics of teaching is relate(1 to the content of typical mathematics courses.
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From page 61... ...
MANAGING CLASS DISCUSSION DISCUSSION SUMMARY After some discussion of these issues, the sessions ended on the more general issue: Now step out of the role as teacher in the moment to be the reflective teacher. What kinds of knowledge and understandings did you call upon to make the decisions you made as the teacher of the moment?
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From page 62... ...
+ 3 + 3 + 3 + 3 =6 4 4 4 4 From Developing Mathematical Ideas, Number and Operations, part 2: Making Meaning for Operations Casebook, by Schifter, Bastable, & Russell.
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From page 63... ...
We can make lists of general mathematical topics important to teaching, but understanding how the knowledge of those topics becomes integrated into the actual act of teaching is MANAGING CLASS DISCUSSION more difficult. A growing awareness of these issues has led to the development of curricular materials to support teacher learning of mathematics that attempt to connect that learning to the contexts of classrooms by embedding the mathematics into classroom contexts, students' work on mathematics, and teacher interactions about their own classroom mathematizations.
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