Knowing and Learning Mathematics for Teaching Proceedings of a Workshop (2001) / Chapter Skim
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Investigating Alternative Approaches to Helping Teachers Learn Mathematics
Investigating Alternative Approaches to Helping Teachers Learn Mathematics
Pages 77-104
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From page 77... ...
rat ~r.r~ ~ kY`~ .r~ Alibi _ ~ ~ ~ rev Participants attended two of four concurrent sessions in which they engaged in activities to help them understand how opportunities might be designed to enable preservice teachers to learn the mathematics they wall need to teach well. SESSIONS Curriculum Materials Shin-ying Lee, Michigan State University Marco Ramirez, Tucson Public Schools Case Materials Carne Barnett, WestEd Programs and Practices Virginia Bastahie, SummerMath for Teachers [ill Lester, Mount Holyoke College Deborah Schifter, Education Development Center Video as a Delivery Mechanism Bradford Findell, Mathematical Sciences Education Board Deborah Ball, University of Michigan
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From page 78... ...
Participants ha(1 opportunities to think of new ways to improve teachers' mathematical content knowle(lge anti pedagogical content knowledge at the preservice and inservice level. One possible avenue is to examine the teacher training practices of countries that demonstrate effective instruction in mathematics.
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From page 79... ...
Japanese elementary school teachers can acquire a clear idea of the structure of elementary mathematics through the scope and sequence chart in the teachers' manuals. The scope and sequence Systematic Charts included at the back of every teacher's manual lay out the sequence of all the mathematics concepts students should learn in the four mathematics content areas over the elementary school years.
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From page 80... ...
Figure 1. Sequence Chart for Numbers and Operations~perations First Grade 5 Addition (1 )
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From page 81... ...
Fourth Grade 1 Multiplication · (3 digit) x (3 digit)
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From page 82... ...
In addition, the key points of the discussion address the particular features of the mathematics being taught in the lesson, the (leveloping nature of the mathematical thinking of the students, and the critical mathematical ideas students should learn from the activity. The conceptual basis is illustrated, along with a goo(1 lesson script, for every lesson in the curriculum.
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From page 83... ...
An example of one Japanese scope and sequence chart and its English translation were shown to illustrate the sequential and systematic approach to the mathematical concepts throughout the elementary school curIAPAN E SE TEAC H E RS' MAN UALS riculum (see Figure I)
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From page 84... ...
For example, it is necessary to assist teachers who have students who speak primarily in a language other than English. Some participants chose to focus the discussion on issues embedded in the American system or their own particular institutions, for example, the lack of good mathematics textbooks at the college level.
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From page 85... ...
The college structure of two separate departments mathematics and mathematics education, both of which are involved in the training of prospective elementary school teachers may not be the best way to enhance the mathematical knowledge of future teachers. Professors in the mathematics department are not likely to teach content necessary for elementary mathematics nor to understand mathematics in the setting of elementary classroom practices.
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From page 86... ...
The curriculum is presented through a series of teacher books. Each book provides lesson plans; description of the mathematical emphasis of the lesson; the materials preparation needed; reproducible student sheets for activities and games; a family letter; homework suggestions; opportunities for skill and practice; assessment activities; notes to the teacher including vocabulary, notation, and a discussion of the mathe
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From page 87... ...
The participants identified strengths of Investigations as: teachers' notes, mathematical emphasis, classroom routines, and translations. Participants considered the degree of support necessary for teachers anti spent some time commenting on the reality of teaching from a school perspective, raising issues of teacher preparation in mathematics, credentialing and licensing procedures, time for planning lessons, (liverse nature of students' mathematical and cultural backgrounds, home environment and support, and language.
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From page 88... ...
Figure ~ . Fourth-Grade Lesson , ~ ~ ~ / / / / / / / W0~ ~ / ^0 / / 1 / ..
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From page 89... ...
REFERENCE Battista, M T., & Clements, D
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From page 90... ...
One goal of the presenter was to illustrate how a deliberately facilitated case discussion can help teachers acquire an advanced and flexihie knowledge of the mathematics content they teach. Anothergoal was to demonstrate how deep analysis of the mathematics- in the context of a classroom situation can prepare teachers to make informed and strategic teaching decisions.
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From page 91... ...
Next, participants were invited to take part in a discussion of the case titled "Six-Tenths or Four-Fifths of a Dollar." The presenter modeled a step-by-step process, developed by the Mathematics Case Methods Project, which provides a familiar routine for discussing cases and for focusing the discussion on productive issues. After reading the case but before the discussion, participants worked on a "Starter Problem" related to the case, designed to help them think about what might be difficult or confusing for a child.
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From page 92... ...
by this approach Many participants agreed that cases offer a site for discussing complex mathematical concepts. A mathematician in the group pointe(1 out that mathematics courses are specifically designed for architects and engineers, using problems faced by architects and engineers in their work.
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From page 93... ...
The facilitator must focus the discussion and coordinate the ideas offered by the participants to make the discussion useful. As case discussion participants gain experience, they learn the importance of carefully analyzing one idea before moving on to the next.
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From page 94... ...
illustrating student thinking as described by their teachers. In addition to case discussions, the curriculum offers teachers opportunities to explore mathematics in lessons led by facilitators; share and discuss the work of their own students; plan, conduct, and analyze a mathematics interview of one of their own students; view anti (liscuss videotapes of mathematics classrooms anti mathematics interviews; write their own classroom episode; analyze lessons taken from innovative elementary mathematics curricula; and rea(1 overviews of relate(1 research.
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From page 95... ...
Especially challenging for teachers is the work on operations with fractions. WORKSHOP ACTIVITIES: THE BREAK-OUT SESSION In the teacher preparation workshop, we (new mainly on material from Building a System of Tens, showing examples of videotape, print cases, mathematical explorations, anti seminar participants' writing assignments to illustrate the flow of ideas anti (1emonstrate how the various strategies provi(le (lifferent win(lows on the same or relate(1 pleas.
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From page 96... ...
By taking apart Thomas's solution methods anti i(lentifying what is solid in his thinking it would be an effective strategy for adding 17 + 30 anti where his logic has gone awry, and comparing elements of strategies to those of his classmates, teachers can practice important pedagogical skills while working out for themselves concepts of multidigit multiplication. This case illustrates how teachers can transform student error into a learning opportunity for their classes.
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From page 97... ...
. PROGRAMS AN D PRACTIC When inquiry is the style of the class, participants' thinking is validated, appreciated, and valued (as opposed to settings in which all ideas come from the instructor)
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From page 98... ...
The intent was to think about characteristics of a video that make it productive for engaging in discussions of the mathematical prohiems of teaching A VIDEO CASE: CINDY The first video came from the Vi(leoCases for Mathematics Professional Development project, (lirecte(1 by Nanette Seago.i In their work, they foun(1 that it was important to ask participants to engage in the mathematics before viewing the video. To that end, the participants were aske(1 to solve the problem in Figure I, trying to anticipate also how a mi(l(lleschoo} student might approach the problem.
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From page 99... ...
The emphasis in the discussion was that different approaches yield solutions that look different but are algebraically equivalent. Some participants wondered whether the solutions would have been different if they had been asked to think about how preservice or inservice teachers would approach the task.
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From page 100... ...
A VIDEO LECTURE: HERB GROSS The second video excerpt was a portion of "Gateways to Arithmetic, Lecture #7: Subtracting Whole Numbers" from a 51lecture collection on mathematics content. In each 30-minute videotape, Herb Gross lectures for a video audience, sometimes speaking directly to the camera.
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· The vicleo can be use to explore missed opportunities proviclecl by student statements. Students · The vicleo provides a window into student thinking Using the Video · The vicleo cannot be used without a facilitator.
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From page 102... ...
DISCUSSION SUMMARY Again, the participants identified opportunities and pitfalls (see Table 2) , several of which provoked extended discussion.
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From page 103... ...
Although the video provides a useful synthesis of some mathematical ideas, some conceptual difficulties are omitted. Some participants suggested that the lecture might be used as a source of mathematical questions for deeper consideration anti reflection.
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