# Combined HL/SL mathematics course at FKG

## Mathematics: the science of structures and patterns

In the history of mankind, mathematical ways of thinking and solving problems typically evolved when societies became more complex. Number systems and geometric figures were used to understand and explain phenomena in nature and human interaction. Examples are the advanced civilizations of the ancient Babylonians, Egyptians and Greeks. Quite early, international cooperation developed (e.g. Greek mathematicians working in Alexandria) and has been intensified ever since (in most high ranking universities of the world a multicultural team of mathematicians can be found). This makes sense since the mathematical approach of solving problems is rather independent from the spoken language although a cultural touch sometimes can be noticed. In fact, mathematicians have built an international symbol system which can be seen as a language of its own and which is understood worldwide by insiders. This point of view will often be emphasized in the course. Historical references in the lessons support this approach. Though English will be the language of instruction, technical term comparisons to other languages, especially German, might be helpful to enhance the international perspective of our students.

The course is aimed for the students to develop logical, critical and creative thinking. They are trained to be patient and persistent in solving problems in a variety of contexts. Recognizing links between different areas of mathematics as well as stressing connections to other fields of science should be realized as useful concepts by the course members. Additionally, the students should be able to present their results and to communicate clearly with other people dealing with the same or similar matter. Knowing and using correct mathematical notation and terminology belongs to the objectives of the class as well as applying appropriate technological devices as mathematical tools (e.g. graphic display calculators [GDC]). The course covers all the seven core topics of the IB guide and (for HL students) one optional topic (currently sets, relations and groups). Working on portfolio tasks - a kind of extended homework paper initialized and guided by the teacher - gives the students the opportunity to practice mathematical investigation and modelling as well as proper mathematical writing and thoughtful reflection. These pieces of work make up the internal assessment part accounting for 20 % of the final grade (the remaining 80 % of the final grade are determined by the results of the externally accessed examination papers). At the end of the two-year course students should have completed two of these portfolio papers, successfully. Finally, the course members ought to appreciate historical and multicultural aspects of mathematics. Short histories of mathematical pioneers and events are going to be part of the lessons.

Students who decide for the SL level possess knowledge of basic mathematical concepts and are equipped with the skills needed to apply simple mathematical techniques correctly. This is usually expressed by the German grade 4 or better (if the common grade used to be 4- or worse, a student might think twice before enrolling for the class). The SL level gives a sound mathematical base for studying university subjects such as chemistry, economics, psychology and business administration. The course focuses on introducing important mathematical concepts through the development of mathematical techniques. This can be achieved best by many illustrative examples which are typically given as problems (consequently, the students must be willing to do their assigned homework regularly and diligently). The teaching time averages 3 lessons per week. The HL students usually take part in these lessons. Due to fact that a combined class is given, the number of lessons can vary from week to week whenever the teaching process requires this.

For students choosing the HL level the course is designed to prepare them to study university fields which contain mathematics as a key element such as physics, engineering or mathematics as a major. Besides finding correct results, students learn to justify the solutions they have created using techniques of mathematical reasoning (proof). These goals can be achieved best by treating interesting and complex problems and exercises. Open-minded classroom discussions support the students in the outlined way of thinking. Intensively and persistently working on given problems either in groups or on their own also belongs to the preferred processes proposed for this course. The students should have a good background in mathematics which is usually expressed by the German grade 2 or 1. Even more important than the grade is a strong interest in the field which includes a touch of enthusiasm about certain mathematical phenomena without being frustrated if hard work is required to solve a given problem. The teaching time averages 5 lessons per week including 3 lessons in combination with the SL students.