Conventional Wisdom is Wrong: Anyone Cannot Teach and Teachers are not Born

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Conventional Wisdom is Wrong Anyone Cannot Teach
and Teachers are not Born
AACTE 2002

• Robert M. Capraro, Mary Margaret Capraro,
  Dawn Parker,
• Tammy Raulerson, Gerald Kulm
• Texas AM University
• Department of Teaching, Learning, and Culture
• College Station, Texas 77843-4232

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Presentation Outline

• Introduction
• Statement of Problem
• Participants
• Instruments
• Data Analysis
• Cases
• Discussion

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Introduction

• New vision for teaching and learning of
  mathematics (NCTM, 1989, 1991, 2000 National
  Research Council, 2001).
• Documents describe a different role for the
  mathematics teacher
• Preparation programs for prospective mathematics
  teachers need to examine their role in how these
  new teachers are being prepared
• Responsibility of institutions in providing
  access to appropriate mathematical preparation
  and creation of supportive learning environments

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Teaching Principle from the Principles and
Standards for School Mathematics (NCTM, 2000,
p.16)

• "Effective mathematics teaching requires
  understanding what students know and
• need to learn and then challenging and
  supporting them to learn it well.

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Profound Mathematical Understanding

• To be effective, teachers must have a profound
  understanding of mathematics (Ma, 1999).

vast
deep
thorough
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Teaching and learning mathematics with
understanding

• fundamental forms of mental activity
• (a) constructing relationships
• (b) extending and applying knowledge
• (c) reflecting about experiences
• (d) articulating what one knows
• (e) making knowledge ones own
  
  (Carpenter Lehrer, 1999).

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Effective Mathematics Teaching

• The National Commission on Teaching and America's
  Future (1996) stated that in order to teach
  mathematics effectively, one must combine a
  profound understanding of mathematics with a
  knowledge of students as learners, and to
  skillfully pick from and use a variety of
  pedagogical strategies.
• The Texas Statewide Systemic Initiative (TSSI,
  1998) confirmed that the teaching of mathematics
  not only requires knowledge of content and
  pedagogy, but also requires an understanding of
  the "relationship and interdependence between the
  two" (p. 6).

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Pedagogical Content Knowledge

• Schulman defined PCK as "a knowledge of subject
  matter for teaching which consists of an
  understanding of how to represent specific
  subject matter topics and issues appropriate to
  the diverse abilities and interest of learners
  (Shulman, 1988, p.9).

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Statement of Questions

• Are there indicators that can be predictors of
  performance on state tests and/or classroom
  teaching ability for prospective
  elementary/middle school teachers?
• What are ways to assess the effectiveness of the
  elementary/middle school teacher preparation
  programs in the areas of mathematics content and
  pedagogy?

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Participants

• Large state public university
• Spring semester of 2001 (Senior Methods Block)
• Subsequent two semesters
• 193 participants (n 193)
• Traditional teacher education students.
• 20 22 year-old females
• Predominately Caucasian.

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Instrumentation

• A 15 item multiple choice mathematics
  pedagogical instrument (Appendix A).This
  instrument was designed to mirror the pedagogical
  content questions contained on the ExCET test.
• A four item open-ended rubric-scored content and
  application instrument (Appendix B). This
  instrument was adapted from the PISA
  International Test and items were selected that
  covered the domains tested on the ECE 02
  mathematics portion of the ExCET test.

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ECE 02 Domains

• 020 Higher-order Thinking and Questioning
• 021 Problem-Solving Strategies
• 022 Mathematical Communication
• 023 Mathematics in Various Contexts
• 024 Number and Numeration Concepts
• 025 Patterns and Relationships
• 026 Mathematical Operations
• 027 Geometry and Spatial Sense
• 028 Measurement
• 029 Statistics and Probability
• 030 Recent Developments and Issues in
  Mathematics

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A pizzeria serves two round pizzas of the same
thickness in different sizes. The smaller one has
a diameter of 30 cm and costs 30 zeds. The larger
one has a diameter of 40 cm and costs 40 zeds.
Which pizza is better value for money? Show your
reasoning.(www.pisa.oecd.org), 2000
Pizza Problem
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Coin Problem

• You are asked to design a new set of coins.
  All coins will be circular and colored silver,
  but of different diameters. Researchers have
  found out that an ideal coin system meets the
  following requirements diameters of coins should
  not be smaller than 15 mm and not larger than 45
  mm. Given a coin, the diameter of the next coin
  must be at least 30 larger. The minting
  machinery can only produce coins with diameters
  of a whole number of mm (e.g., 17 mm is allowed,
  17.3 mm is not). You are asked to design a set of
  coins that satisfy the above requirements. You
  should start with a 15 mm coin and your set
  should contain as many coins as possible.
  
• (www.pisa.oecd.org), 2000.

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Sample Pedagogical Content Knowledge Question
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Sample Pedagogical Content Knowledge Question
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Summary of Regression Analysis
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Correlation Matrix
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• Because teaching and learning in increasingly
  diverse contexts are complex, prospective
  teachers cannot come to understand the dilemmas
  of teaching only through the presentation of
  techniques and methods
• (Harrington, 1995,
  pg. 203).

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Summary of Math Background
Table 3 Summary of Mathematics Background and
Scores on Instruments for Preservice Teacher
Cases  
  Note Mathematics Courses consistent among all
interns Math 365 and Math 366. Maximum score
on open-ended instrument was 30. Maximum score
on multiple choice instrument was 15.
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Mathematics Buddy

• 4 weeks
• 1 hour session
• Themes
• Computational Fluency, Problem Solving,
• Communication, Estimation
• Interns plan and teach lessons to 4th grade
  student
• Submit reflection of teaching and learning
  experiences

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Sally

• Math emphasis
• Loves mathematics
• Confident in ability to teach mathematics
• Consistently plans and teaches meaningful lessons
• Reflections show identification of strengths and
  weaknesses
• Makes connections to other content areas

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Jane

• Social Studies emphasis
• Conscientious and good planner
• Not confident in mathematics teaching ability
• Uses resources well to develop lessons
• Gains confidence in teaching of mathematics and
  science

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Molly

• Early Childhood
• emphasis
• Teaching experience prior to methods semester
• Sample reflection from lesson on equivalent
  fractions indicating conceptual misunderstanding

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Sample Reflection
After we had all the pieces out, I went back to
try and assess his understanding of equivalent
fractions. I asked him how many fourths make one
half. He struggled with this and looked at me
with a blank stare and then guessed four. I
asked him why he thought four, and he couldn't
give any explanation. So I had him show me one
half of a circle, and then I had him cover it
with fourths. He then realized that it only took
two fourths to make one half. So I said, are the
fractions 1/2 and 2/4 equivalent? He said yes
and explained because they take up the same
amount of area. I continued this with him for
fourths, eighths, and sixteenths. He was able to
do this by putting pieces on top of the
others.After I felt he had a good grasp of
equivalent fractions, I moved on to a game. The
game required him to turn over two fraction cards
and decide whether they were equivalent. When he
turned over the first two cards, I realized that
he did not have a good understanding of how to
simplify fractions. So, I didn't get to play the
game as planned. Instead, I decided to use the
fraction cards to decide if two fractions were
equal. The first two fractions he turned over
were 1/8 and 3/24. He didn't know where to
begin, so I asked him to show me 24 divided by
3.He already knew that eight divided by one was
equal to eight because any number divided by one
is the same number. I then explained that since
these numbers are the same the fractions must be
equal.
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New Questions

• With large numbers, how does a program create
  consistency in mathematics content courses taken
  prior to methods block?
• What are ways to provide interns with quality
  field experiences and effective role models for
  teaching mathematics?
• How do we know when preservice teachers are ready
  to teach children mathematics?