The Mathematics Education of
Teachers: One Example of an
Evolving Partnership Between
Mathematicians and
Mathematics Educators
Gail Burrill (burrill@msu.edu)
Michigan State University
Given m/n where m and n are
relatively prime and m < n, what
can you say about the decimal
representation?
Usiskin et al., 2003
Theorems
•Terminate after t digits if n= 2r.5s,
t> max (r,s)
•Simple repeating if can be written in form
p
m/(10 -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
t
p
m/(10 (10 -1)), t is number of digits before
repeat, p is the repeat
Usiskin et at, 2003
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)
Related factors
Teachers for a New Era
Strong push from math educators
Interest on part of some
mathematicians
Required capstone course for math
majors
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
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
Broad Goals of the Course:
• Deepen understanding of the
mathematics needed for teaching in
secondary schools.
• Prepare students to
1. describe connections in
mathematics;
2. figure things out on their own.
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).
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
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
Topics
• Real and Complex Numbers
• Functions
• Equations
• Polynomials
• Trigonometry
• Congruence Transformations
• Regression
• Platonic Solids
Usiskin et al, 2003
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
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?
Mathematics Educator
•Engage students in activities
•Links to classroom, curriculum, and
pedagogy
•Questioning
•Reflection on learning
–Fundamental Theorem of Algebra
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
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
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.
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?
Poor feeling for convergence
1. Find q(x) and r(x) guaranteed by the
Division Algorithm so that
2
P(x) =( x3+3x +4x -12)/(x2+4)
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.)
Surprises
“I never did believe that
.9999.. = 1.”
“I didn’t bring my calculator.”
Missed the connection between
Pascal’s Triangle and Binomial
Theorem
Surprises
Find possible roots of
4
2
x -3x +2x-6=0
Issues
•Credit for teaching as a team
•Amount of planning and
coordination
•Relation to TE
•Strengthening connections to
earlier math courses
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
Text
•Little discussion of reasoning and proof
•No discussion of some key concepts such
as why √-4 √-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
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
Polya’s 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)
Polya’s 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)
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