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Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop (2001)

Chapter: Appendix H: Mathematics Case Methods Project

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Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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Appendix H: Mathematics Case Methods Project

Mathematics Case Methods Project

Teacher Preparation Content Workshop

National Academy of Sciences

Washington, DC

March 19-21, 1999

Introductory Packet

Carne Barnett

Director, Math Case Methods Project

WestEd

Oakland, CA

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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CHALLENGES TO OURSELVES AS PROFESSIONAL DEVELOPMENT DESIGNERS

How can we design learning professional development experiences that:

INCLUSIVE

Are inclusive and effective with teachers, regardless of their pedagogical views, knowledge, or skills.

PRODUCTIVE AND EFFICIENT

Have rapid, significant, and direct positive impact on the learning of teachers and their students—and on their ability to continue to learn from their own experiences.

MEASURABLE

Demonstrate a positive impact on students' understanding of mathematics which is measurable on a variety of instruments, including standardized tests.

OPEN TO CRITICAL EXAMINATION

Provide an environment that supports teachers questions themselves, each other's ideas, even the most highly regarded ideas of the mathematics teaching community.

TEACHER LEADERSHIP

Assumes that all teachers, regardless of their experience or pedagogical philosophy, should be given the opportunity to assume leadership roles.

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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WHY LEARN TO DISCUSS AND FACILITATE MATH CASES?

Mathematics

  • To deepen our own understanding of the mathematics

Children's Thinking about Mathematics

  • To experience the mathematics from our students' points of view

Mathematics Instruction

  • To make informed teaching decisions by examining the benefits and drawbacks of various strategies

Language

  • To understand the impact of oral, symbolic, visual, and written communication on the mathematics learning of all students

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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Case Discussion Roadmap Summary
Mathematics Case Methods Project
  • Opening inclusion activity

Purpose: to build trust and support open discussion; participants may pass.

  • Starter Problem

Purpose: to stimulate thinking about the mathematics and student thinking; participants work independently; solutions may be presented during the discussion, but are not shared prior to the discussion.

  • Read or quickly review the case

If possible, participants read the case before coming to the discussion and begin to think about issues to discuss.

  • Facts

Purpose: to build common background about the case; collected quickly popcorn style without comments.

  • Issues written in question format

Purpose: to generate questions that will stimulate deep discussion; focus on the mathematics, child's thinking, instruction and materials, or language; participants

generate issues in pairs.

  • Begin the discussion

The issues are read to the group; a volunteer chooses one of the issues and begins the discussion.

  • During the discussion

Participants build on and ask questions about each other's ideas; a few ideas are examined in depth; participants come to the board to make drawings or use materials to explain their ideas.

  • Closing inclusion activity

Purpose: to bring the group together for a brief reflection or appreciation of the experience; may be an oral or written reflection.

  • Process Check

Purpose: to allow participants to give feedback to each other on how well the group participated in the discussion; it is not an evaluation of the facilitator.

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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STARTER PROBLEM

How could you show which is greater, 6/10 or 4/5 of a dollar?

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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Six-Tenths or Four-Fifths of a Dollar?

The math experiences of the 30 students in my fourth-grade class vary widely. Esmarelda, a recent immigrant to the United States, is Limited English Proficient (LEP) and has never had formal schooling, Chris is a gifted student who enjoys calculations and problem solving. Michael participates in my class, then later in the day goes to the resource room for additional math help. There are also 3 children with special needs, 4 other LEP students, and 12 extremely economically disadvantaged students.

Often, it seems, I teach a lesson 4 or 5 times before I feel comfortable moving ahead Sometimes I worry that I'm beating a dead, horse. For those students who catch on quickly I try to plan enrichment activities or set up activity centers. This year—my third year of teaching—I've been trying to include more math journal writing before, during, and after lessons. I also try to use concrete materials before explaining a mathematical concept to the class.

I recently introduced the class to fractions. They have learned how to identify fractions using diagrams such as the following:

Each student made a fraction kit that allowed them to show and identify various fractions. The denominators of the fractions in the kits were 1, 2, 4, 8, and 16.

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×

My beginning lesson had focused on identifying and orally naming various fractions.

“Place your ‘whole' on the desk top,” I said. “Show me ¼ of a whole by placing ¼ on top of it.” The students responded by placing a “one-fourth” piece on the square representing 1 whole.

By the end of the lesson, they could successfully name and use the fraction kit to show unit fractions like ½ or 1/8 and nonunit fractions like ¾ or 5/8

The next lesson focused on equivalent fractions. I asked the students to figure out the answer to questions like how many eighths would be equal to ¼ or how many sixteenths would equal 3/8. They used their fraction-kit pieces to determine the answers. By the end of this lesson, students were very familiar with the relationship among the fraction pieces and could solve simple equivalency problems without using the pieces.

I was then ready to have students learn how to compare fractions that have different denominators and numerators. Prior to beginning, I asked them to write about fraction equivalency in their journals, so I could assess their understanding of previous lessons and know what information I needed to cover, I asked, them to answer this question:

Which would you rather have: 6/10 of a dollar or 4/5 of a dollar? Explain your reasons for choosing your answer.

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×

After reflecting, the students picked up their pencils and wrote.

Cindy's journal read: “If I had 6/10, I would have 2 more than 4/5. I would choose 6/10 so I could have more money.”

Chris wrote: “4/5 = 8/10, 8/10 is greater than 6/10. Of course, I'd take 4/5 of a dollar. Wouldn't you?” He included an illustration:

Esmarelda wrote: “No sé. 6/10 es mas grande.”

Nikki drew a picture to accompany her answer: “I want 6/10. It is bigger.”

Only 4 of the 30 students wrote 4/5. Their journals gave me some hints about how they were thinking about fractions, but I was not sure how to use this information to plan our future work. Since the students were familiar with the fraction kit, I wondered if they would have answered differently if the question had been, “Would you rattier have ¾ or 5/8 of a chocolate bar?”

Suggested Reading

Behr, M. J., T. R. Post and I. Wachsmuth. 1986. “Estimation and Children's Concept of Rational Number Size.” In Estimation and Mental Computation, edited by H. L. Schoen and M. J. Zweng, 103-111. Reston, VA: The National Council of Teachers of Mathematics.

Cuevas, G. 1990. “Increasing the Achievement and Participation of Language Minority Students in Mathematics Education.” In Teaching and Learning Mathematics in the 1990s, edited by T. J. Cooney and C. R. Hirsch, 159–165. Reston, VA: The National Council of Teachers of Mathematics.

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×

PROCESS CHECK

Mathematics Case Methods Project

Rating Scale: 1 2 3 4 5

Strongly Disagree Strongly Agree

Feedback to the group:

  1. ___ Group members gave different points of view respectful consideration even when there was disagreement or the ideas were unpopular.

  2. ___ I felt I had opportunities to comment, whether or not I contributed during the discussion.

  3. ___ Members built on and contributed to each other's ideas. Members asked questions about each other's ideas.

  4. ___ The group gave a high priority to discussing the mathematics and student thinking.

Feedback to the facilitator:

  1. What is one thing that worked well?

  2. What is one thing that might be tried next time?

Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 201
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 202
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 203
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 204
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 205
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 206
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 207
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
×
Page 208
Suggested Citation:"Appendix H: Mathematics Case Methods Project." National Research Council. 2001. Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop. Washington, DC: The National Academies Press. doi: 10.17226/10050.
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Page 209
Next: Appendix I: Biographical Information »
Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop Get This Book
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There are many questions about the mathematical preparation teachers need. Recent recommendations from a variety of sources state that reforming teacher preparation in postsecondary institutions is central in providing quality mathematics education to all students. The Mathematics Teacher Preparation Content Workshop examined this problem by considering two central questions:

  • What is the mathematical knowledge teachers need to know in order to teach well?
  • How can teachers develop the mathematical knowledge they need to teach well?

The Workshop activities focused on using actual acts of teaching such as examining student work, designing tasks, or posing questions, as a medium for teacher learning. The Workshop proceedings, Knowing and Learning Mathematics for Teaching, is a collection of the papers presented, the activities, and plenary sessions that took place.

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