by Margaret Langfield
October 2000
Many preservice mathematics teachers in Canada and the United States do not know all the mathematics needed to teach in the International Baccalaureate (IB) Diploma Program. All four mathematics options (mathematics higher level, advanced mathematics standard level, mathematical studies, and mathematical methods) in the program require a broad range of mathematical topics—some core and some teacher chosen. Common to all are number, algebra, functions, geometry, trigonometry, probability, and statistics.
The two syllabi chosen by the majority of students include such other topics as sets, logic, twodimensional vector geometry, financial mathematics, and some calculus. An IB teacher must have a ready knowledge of these topics; in addition, for the optional topics, the teacher must know more calculus, some matrices and graph theory, and statistical methods. The two upper levels of the IB program (mathematics higher level and advanced mathematics standard level) require threedimensional vector geometry, complex numbers, analysis, and yet more calculus, probability, and statistics. Three of the syllabi require that students use a graphing calculator. From a mathematical standpoint, most prospective secondary teachers simply have not had this much mathematics in the undergraduate programs required for teacher certification.
In addition to mathematical knowledge, prospective teachers of IB mathematics need to "be able to represent mathematics as a coherent and connected enterprise" (NCTM 2000, p. 17). The externalassessment component of an IB program poses creative questions requiring both teachers and students to see connections and draw on their knowledge in many areas; the IB mathematics classroom affords many opportunities for this type of assessment to be practiced. Too many undergraduates regard mathematics as a static subject with a set of algorithms to be applied unthinkingly to known problems. The intent of the IB program is that "students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge" (NCTM 2000, p. 20). Prospective IB teachers must be capable of developing and fostering classrooms where students can use their imagination, skills, and knowledge to explore new situations with confidence and with the expectation of success. From their mathematical experiences, students should understand the importance of rigor and communication, and they should appreciate the power of mathematics and mathematical reasoning. Few teacherpreparation programs educate prospective teachers in this manner (NCTM 2000, p.17).
For the internalassessment component of the IB mathematics program, which is either a project or portfoliotype assignments, a student must spend time on extended problems. The students have opportunities to explore new topics, apply knowledge, make conjectures, construct mathematical models, pursue particular interests, use technology, and write both well and meaningfully about mathematics. An IB teacher must have had experience in these types of things, as well as experience in using criterionreferenced assessment to evaluate these activities.
Typical prospective IB teachers have energy, stamina, perseverance, the courage to move away from standard textbooks, and the ability to draw on many resources while constructing the needed activities for their IB students. In their undergraduate mathematics preparation for teaching, they should have developed confidence in a coherent, unified body of knowledge and should be able to share this with students. Until teacherpreparation programs change, few firstyear teachers are prepared for IB teaching. They need more mathematics and the training offered by the International Baccalaureate Organisation.
References
International Baccalaureate Organisation. Subject Guides for Mathematical Methods Standard Level, Mathematical Studies Standard Level, Mathematics Higher Level, Further Mathematics Standard Level. Available at www.ibo.org.
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.
Margaret Langfield, mlangfie@pegasus.cc.ucf.edu, is a visiting instructor in the mathematics department at the University of Central Florida. She is an examiner, an internalassessment moderator, and a Teacher Training Workshop leader for the International Baccalaureate Organisation, and she previously taught in IB programs and high schools in Florida, Belgium, and Great Britain. 
