Mathematical habits of mind and ways of thinking for prospective teachers

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Mathematical habits of mind and ways of thinking
for prospective teachers

• Gail Burrill
• Michigan State University

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Background

• Association of Mathematics Teacher Educators
  (AMTE)
• Preliminary Teacher Education Development Study
  (PTEDS) (NCES)
• Teacher Education Development Study (TEDS) (NCES)
  
• PROM/SE Michigan State Mathematics Science
  Partnership (NSF)
• Knowledge of Algebra for Teaching (KAT) (NSF)
• Teachers for a New Era (Carnegie Grant)
• Capstone Courses (MSU)

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Association of Mathematics Teacher Educators
(AMTE)

• Goals for the Preservice Education of Prospective
  Secondary Teachers
• 1. Analyze and purposefully transform their
  beliefs and dispositions about what mathematics
  is and what it means to learn, do and teach
  mathematics

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PTEDS
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(No Transcript)
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Doing mathematics

• In general, a mathematical approach involves
  defining a problem through conjecturing in an
  established mathematical area. Conjecturing may
  be supported by technology, by compelling ideas
  based on past work, computation, or pattern
  exploration.
• (Teachers for a New Era, 2003)

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Doing mathematics

• Mathematics involves representing a mathematical
  concept concretely a single concept can have
  multiple representations. One of the beauties of
  mathematics as a whole is the interplay between
  various areas of the subject. A particular way of
  representing a problem may lead to an especially
  efficient or enlightening result.
• (Teachers for a New Era, 2003)

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Capstone Course - MSU Senior Math Majors

• Deepen understanding of the mathematics needed
  for teaching in secondary schools.
• Prepare students to
• 1. describe connections in
• mathematics2. figure things out on their own.

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High school math from an advanced perspective

• Analyses of alternative definitions, language and
  approaches to mathematical ideas
• Extensions and generalizations of familiar
  theorems
• Discussions of historical contexts in which
  concepts arose and evolved
• Applications of the mathematics in a variety of
  settings

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High school math from an advanced perspective

• Demonstrations of alternate ways of approaching
  problems, with and without technology
• Discussions of relations between topics studied
  in this course and contemporary high school
  curricula.

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Habits of mind Mathematics should make sense

• Know that all mathematics is not equal
• Understand that mathematics can be done different
  ways
• Be willing to work hard on a challenging problem
  for a long time
• Recognize that mathematics is about how
  mathematical results are obtained not how they
  are presented

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Mathematically confident enough to take risks

• Using technology
• Asking for explanations
• Doing problems with students
• Trying something new
• Celebrating mistakes

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A habit of mind
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Polyas Ten Commandments

• Read faces of students
• Give students know how, attitudes of mind,
  habit of methodical work
• Let students guess before you tell them
• Suggest it do not force it down their throats
  (Polya, 1965, p. 116)

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Polyas Ten Commandments

• Be interested in the subject
• Know the subject
• Know about ways of learning
• Let students learn guessing
• Let students learn proving
• Look at features of problems that suggest
  solution methods (Polya, 1965,p. 116)