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The Mathematics Education of Teachers: One Example of an Evolving Partnership Between Mathematicians and Mathematics Educators

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Five year program: Degree Internship. Capstone course- part of university requirement ... Honors college student asked : does this mean 3 .12199 or 3 x .12199? ... – PowerPoint PPT presentation

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Title: The Mathematics Education of Teachers: One Example of an Evolving Partnership Between Mathematicians and Mathematics Educators


1
The Mathematics Education of Teachers One
Example of an Evolving Partnership Between
Mathematicians and Mathematics Educators
  • Gail Burrill (burrill_at_msu.edu)
  • Michigan State University

2
Given m/n where m and n are relatively prime and
m lt n, what can you say about the decimal
representation?
Usiskin et al., 2003
3
Theorems
  • Terminate after t digits if n 2r.5s,
  • tgt max (r,s)
  • Simple repeating if can be written in form
    m/(10p -1), p is number of digits repeated
  • if 2 or 5 is not a factor of n
  • Delayed repeating if can be written in form
    m/(10t(10p-1)), t is number of digits before
    repeat, p is the repeat

Usiskin et at, 2003
4
The Mathematical Education of Teachers
  • Support the design, development and offering of a
    capstone course for teachers in which conceptual
    difficulties, fundamental ideas, and techniques
    of high school mathematics are examined from an
    advanced standpoint. (CBMS, 2001)

5
Related factors
  • Teachers for a New Era
  • Strong push from math educators
  • Interest on part of some mathematicians
  • Required capstone course for math majors

6
Background
  • Senior mathematics majors
  • Intending secondary math teachers (grade point
    requirement to be admitted to TE)
  • Five year program Degree Internship
  • Capstone course- part of university requirement
  • Concurrent with course in TE related to
    interfacing in classrooms

7
Capstone Course
  • Initially (2003) taught by Sharon Senk
    (mathematics educator in math department) and
    Richard Hill (mathematician)
  • Taught in 2004 by Gail Burrill (Division of
    Science and Math Education) and Richard Hill

8
Broad Goals of the Course
  • 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.

9
Resources
  • Mathematics for High School Teachers An Advanced
    Perspective (Usiskin, Peressini, Marchisotto,
    Stanley 2003)
  • Visual Geometry Project (Key Curriculum Press,
    1991)
  • Exploring Regression (Landwehr, Burrill, and
    Burrill 1997).

10
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

Usiskin et al, 2003
11
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.

Usiskin et al, 2003
12
Topics
  • Real and Complex Numbers
  • Functions
  • Equations
  • Polynomials
  • Trigonometry
  • Congruence Transformations
  • Regression
  • Platonic Solids

Usiskin et al, 2003
13
Shared Teaching
  • Assumed responsibility for certain topics
  • Interactive presentations
  • Play to each others strengths- knowledge of the
    core junior level mathematics courses, linear
    algebra, algebra and analysis and knowledge of
    high school mathematics and pedagogy

14
Mathematician
  • Clear links back to both junior core mathematics
    and to remedial courses that seniors worked in as
    TAs
  • Mathematical way of thinking (back to definition-
    is this an isometry?)
  • P(x) anxn a n-1 x n-1 ao. What are the
    restrictions on n, a?

15
Mathematics Educator
  • Engage students in activities
  • Links to classroom, curriculum, and pedagogy
  • Questioning
  • Reflection on learning
  • Fundamental Theorem of Algebra

16
Grading
  • Homework- alternated grading selected problems
    for each half of the alphabet
  • Tests- each graded half of test
  • Projects - each graded all papers on given topic
  • Final Grades- consultation

17
Grades
  • Grading-three hour-long tests, two
    papers/projects, a comprehensive final exam, and
    homework problems.
  • Test 1 100 points
  • Test 2 100 points
  • Project 1 50 points
  • Test 3 100 points
  • Project 2 100 points
  • Homework Problems 50 points
  • Final Exam 200 points

18
Concept analysis of topic not been discussed in
any detail in this class
  • Ellipse, Logarithm, Matrix, Slope
  • Trace the origins and applications
  • Look at the different ways in which the concept
    appears both within and outside of mathematics,
  • Examine various representations and definitions
    used to describe the concept and their
    consequences.
  • Address connections between the concept in high
  • school mathematics and in college mathematics.

19
Fragile Knowledge
  • Write 3.12199 as p/q where p and q are integers.
    Honors college student asked does this mean
    3.12199 or 3 x .12199?

20
Poor feeling for convergence
  • Find q(x) and r(x) guaranteed by the Division
    Algorithm so that
  • P(x) ( x33x24x -12)/(x24)
  • 2. Find the equation of the asymptote
  • 3 Sketch a plausible graph of P(x), along
    with the graph of the (labeled) asymptote.
    (Note You may assume that p(x) has only one real
    zero, namely x -1.)

21
Surprises
  • I never did believe that .9999.. 1.
  • I didnt bring my calculator.
  • Missed the connection between Pascals Triangle
    and Binomial Theorem

22
Surprises
  • Find possible roots of
  • x4 -3x22x-60

23
Issues
  • Credit for teaching as a team
  • Amount of planning and coordination
  • Relation to TE
  • Strengthening connections to earlier math courses

24
Text
  • Not enough history that is interesting and useful
    in high school content
  • Text is flat- theorems seem to have equal
    weight
  • Key areas not covered extension of lines in
    plane to space data and modeling
  • Underlying mathematical habits of mind not
    explicit

25
Text
  • Little discussion of reasoning and proof
  • No discussion of some key concepts such as why
    v-4 v-9 is not 6, parametrics.
  • Organization of topics - ie how to position
    trigonometry in relation to complex numbers
  • Links algebra and geometry could be stronger

26
Text
  • Interesting connections and approaches
  • Opportunities for making links back to analysis,
    linear algebra, abstract algebra
  • Some excellent problems
  • Good basis for beginning to think about the
    mathematics- and does start from the mathematics
    that teachers will need to know

27
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)

28
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)

29
References
  • Conference Board on Mathematical Sciences.(2001).
    The Mathematical Education of Teachers.
    Washington DC Mathematical Association of
    America
  • Landwehr, J., Burrill, G., and Burrill, J.
    (1997). Exploring Regression. Palo Alto CA
    Dale Seymour Publications, Inc.
  • Polya, G. (1965). Mathematical discovery On
    understanding, learning, and teaching problem
    solving. (Vol. II). New York John Wiley and
    Sons
  • Usiskin Z. , Peressini, A., Marchisotto, E., and
    Stanley. R. (2003) Mathematics for high school
    teachers An advanced perspective. Upper Saddle
    River, NJ Prentice Hall
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