How Do We Do It? Teaching Mathematics to U.S. Teachers Jeremy Kilpatrick University of Georgia

How Do We Do It? Teaching Mathematics to U.S. Teachers Jeremy Kilpatrick University of Georgia

My experience • Graduated in mathematics (and science), Chaffey College, Spring 1954 • Graduated in mathematics, U.C. Berkeley, Spring 1956 • Student teaching in mathematics, Richmond High, Spring 1957 • Mathematics (and science) teaching, Garfield Jr. High, Fall 1957 • No internship • More mathematics at Stanford Richmond High King Middle (Garfield Jr. High) Berkeley High

Outline: • Who teaches? • What mathematics? • To whom? • And how?

Who teaches?

Teacher developers: those who teach mathematics to prospective or practicing teachers • Mathematicians at two-year colleges, four-year colleges, and universities • College and university mathematics educators • Mathematics supervisors in schools, district offices, and state education departments • Commercial providers of professional development in mathematics • Teacher leaders and coaches

Mathematics for prospective teachers • Taught by faculty in 4-year undergraduate programs, usually in large public institutions • Two-year college faculty teach roughly 45% of all undergraduates • Increasing numbers of extended (e.g., 5- or 6-year) and alternative programs • In 1998, 28% in teacher education programs began at the postbaccalaureate level • Newer teachers more likely to hold degrees in an academic field rather than education

Mathematics for elementary school teachers • Students likely to major in education but often take courses in the mathematics department Coursework in mathematics or mathematics education Grades K–5 Source: Horizon Research. (2002). 2000 National Survey.

Mathematics for middle school teachers • Students likely to major in education Source: Horizon Research. (2002). 2000 National Survey.

Mathematics for middle school teachers • Most have equivalent of at least a minor in mathematics Coursework in mathematics or mathematics education Grades 6–8 Source: Horizon Research. (2002). 2000 National Survey.

Mathematics for high school teachers • Students likely to major in mathematics or mathematics education Source: Horizon Research. (2002). 2000 National Survey.

Mathematics for practicing teachers Source: Horizon Research. (2002). 2000 National Survey.

Teacher developers in mathematics departments • What preparation in mathematics? • What preparation in mathematics education? • What criteria for promotion and tenure? • How to keep up with developments in mathematics? • How to keep up with developments in mathematics education?

What mathematics?

The Mathematics Learning Study • K–8 teachers need to learn more mathematics • More mathematics courses • Not more standard mathematics courses

Elementary school teachers • CBMS The Mathematical Education of Teachers (2001) calls for at least 9 semester hours of coursework on fundamental ideas of elementary school mathematics for teachers of grades 1–4 • 2000 Survey found majority of teachers in grades K–5 have at most 7 semester-long courses, and a quarter have fewer than 4 courses

Mathematics for elementary school teachers Source: Horizon Research. (2002). 2000 National Survey.

Middle school teachers • CBMS The Mathematical Education of Teachers (2001) calls for 21 semester hours of mathematics, including at least 12 on fundamental ideas of school mathematics for teachers of grades 5–8 • According to 2000 Survey, NCTM recommends coursework in abstract algebra, geometry, calculus, probability and statistics, applications/problem solving, and history of mathematics • 2000 Survey found nearly two-thirds of teachers in grades 6–8 have taken 8 or more semester-long mathematics courses

Mathematics for middle school teachers Source: Horizon Research. (2002). 2000 National Survey.

High school teachers • CBMS The Mathematical Education of Teachers (2001) calls for the equivalent of an undergraduate major in mathematics (9–12) • But “future high school teachers need to know more and somewhat different mathematics than mathematics departments have previously provided to teachers” • Recommends a 6-hour capstone course to connect college with high school mathematics

Mathematics for high school teachers Source: Horizon Research. (2002). 2000 National Survey.

Mathematical knowledge for teaching • An application of mathematics to the practice of teaching • The mathematics that is imperative/useful/ important for teachers to know • Just as the school mathematics curriculum is a selection from all that could be taught, so is the curriculum of mathematics for teaching

To whom?

Elementary school teachers • 93% of K–5 teachers are female; 90% are White; 58% are over 40; 29% have taught more than 20 years; 42% have a master’s degree • In mathematics, 54% of K–5 teachers consider themselves very well qualified, 45% adequately qualified, and only 1% not well qualified Source: Horizon Research. (2002). 2000 National Survey.

Middle school teachers • 72% of grades 6–8 teachers are female; 85% are White; 52% are over 40; 29% have taught more than 20 years; 44% have a master’s degree Perceived qualifications to teach subjects (in %) Source: Horizon Research. (2002). 2000 National Survey.

High school teachers • 55% of grades 9–12 teachers are female; 91% are White; 59% are over 40; 34% have taught more than 20 years; 51% have a master’s degree Perceived qualifications to teach subjects (in %) Source: Horizon Research. (2002). 2000 National Survey.

High school teachers • The more undergraduate mathematics that high school teachers have studied, the better the performance of their students (effect is small and may decrease beyond 5 courses; larger for teaching advanced than remedial courses) • “Whether a degree in mathematics is better than a degree in mathematics education … remains disputable” • Students of teachers certified in mathematics do better than students of uncertified teachers Sources: Floden & Meniketti; Wilson & Youngs. (2005). In Cochran-Smith & Zeichner (Eds.), Studying Teacher Education. AERA Report

Ingersoll, R. M. (2003, September). Out-of-field teaching and the limits of teacher policy. Center for the Study of Teaching and Policy.

And how?

Pólya’s principles of teaching • Active learning: The ideas should be born in the students’ mind and the teacher should act only as midwife • Best motivation: Pay attention to the choice, formulation, and presentation of a worthwhile task • Consecutive phases: Learning begins with action and perception, proceeds to words and concepts, and ends with ideas

Pólya on teaching • Mathematics consists of information and know-how • “Nobody can give away what he [or she] has not got.” • Teachers need “experience in independent (‘creative’) work on the appropriate level in the form of a problem-solving seminar or in any other suitable form.”

Teaching teachers mathematics • In a genetic approach, the learner rediscovers, retracing the major steps in the path followed by the original discoverers • Otto Toeplitz (1963/2007) The calculus: A genetic approach(MAA & U. Chicago Press) • Prospective teachers need a synoptic view of mathematics • Just as prospective high school teachers need a capstone mathematics sequence, so do teachers of other grades

Felix Klein(18491925) Elementary Mathematics From an Advanced Standpoint • Vol. 1: Arithmetic Algebra Analysis (Dover, 1932/2004) • Vol. 2: Geometry (Dover, 1939/2004)

Felix Klein Purpose: “to take into account, in university instruction, the needs of the school teacher” “My task will always be to show you the mutual connection between problems in the various fields, a thing which is not brought out sufficiently in the usual lecture course, and more especially to emphasize the relation of these problems to those of school mathematics” Real goal of your academic study: “to draw (in ample measure) from the great body of knowledge there put before you a living stimulus for your teaching”

Jens HøyrupHistorian of mathematics and philosopher of scienceRoskilde University Denmark

Extended episodes from history: Scribal computation in Mesopotamia Axiomatics in Greece Merged traditions during Latin Middle Ages Relations between • Development of mathematics • Character of the mathematical discourse • Institutional setting of mathematics teaching (mainly adults) Jens Høyrup In Measure, Number, and Weight SUNY Press, 1994

Mathematics is a reasoned discourse It is the product of communication by argument Not only is teaching the vehicle by which mathematical knowledge and skill are transmitted to the next generation, but also Mathematics is constituted through teaching

Hans FreudenthalMathematician and mathematics educatorUtrecht University The Netherlands(19051990)

Freudenthal on mathematics • Mathematics starting and staying in reality • UCSMP International Conference on Mathematics Education, Chicago, March 1985 • Mathematics starting and staying within common sense • Revisiting Mathematics Education, Kluwer, 1991

Mathematics starting and staying in teaching • Starting, as noted by Høyrup • Staying, because teaching preserves mathematics • Therefore, those who teach mathematics are keeping it alive • And those who teach teachers mathematics are keeping it alive for future generations

How Do We Do It? Teaching Mathematics to U.S. Teachers Jeremy Kilpatrick University of Georgia