The Mathematics Education of Teachers: One Example of an Evolving Partnership Between Mathematicians and Mathematics Educators : 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? : 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 : Theorems Terminate after t digits if n= 2r.5s,
t> 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
The Mathematical Education of Teachers : 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 : 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 : 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 : 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: : 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 : 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 : 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 : 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 : Topics Real and Complex Numbers
Functions
Equations
Polynomials
Trigonometry
Congruence Transformations
Regression
Platonic Solids Usiskin et al, 2003
Shared Teaching : 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 : 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 : Mathematics Educator Engage students in activities
Links to classroom, curriculum, and pedagogy
Questioning
Reflection on learning
Fundamental Theorem of Algebra
Grading : 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 : 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 : 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 : 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 : Poor feeling for convergence Find q(x) and r(x) guaranteed by the Division Algorithm so that
P(x) =( x3+3x2+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 : 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 : Surprises Find possible roots of
x4 -3x2+2x-6=0
Issues : Issues Credit for teaching as a team
Amount of planning and coordination
Relation to TE
Strengthening connections to earlier math courses
Text : 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 : 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 : 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 : 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 : 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 : 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