The Mathematics Education of Teachers: One Example of an Evolving Partnership Between Mathematicians and Mathematics Educators Gail Burrill 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= 2 r.5 s, t> max (r,s) Simple repeating if can be written in form m/(10 p -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/(10 t (10 p -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) = a n x n + a n-1 x n-1 + …+ a o. 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 # 1100 points Test # 2100 points Project # 1 50 points Test # 3100 points Project # points Homework Problems50 points Final Exam200 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 as p/q where p and q are integers. Honors college student asked : does this mean 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+3x 2 +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 = 1.” “I didn’t bring my calculator.” Missed the connection between Pascal’s Triangle and Binomial Theorem

Surprises Find possible roots of x 4 -3x 2 +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