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 m/(10p -1), p is number of digits

repeatedif 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 repeatUsiskin 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 EraStrong 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 to1. 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

GradesGrading-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 KnowledgeWrite 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

P(x) =( x3+3x2+4x -12)/(x2+4)

2. Find the equation of the asymptote3 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

x4 -3x2+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 studentsGive students “know how”, attitudes of mind, habit of methodical workLet students guess before you tell themSuggest it; do not force it down their throats (Polya, 1965, p. 116)

Polya’s Ten Commandments

Be interested in the subjectKnow the subjectKnow about ways of learningLet students learn guessingLet students learn provingLook 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