COURSE AIMS AND OBJECTIVES: The course aims to introduce students  prospective mathematics teachers to the basic concepts of mathematics teaching and learning (that is, to the mathematical didactics) at primary and secondary school level. Particular attention will be paid to various forms and methods of mathematical thinking and reasoning and to formulation, proving and implementation theorems, as prerequisites for understanding processes of teaching and learning mathematics.
COURSE DESCRIPTION AND SYLLABUS: The course contains lectures, tutorials and seminars. Lectures provide theoretical foundation of teaching and learning mathematics. In tutorials, acquired theoretical knowledge will be applied to selected examples  concrete topics from the school mathematics, through various forms of instruction and working methods (individual study, hands  on activities, pair work, group work, team  collaborative work, project work). Seminars consist of students' group or individual oral presentations of assigned topics from school mathematics, followed up by group discussions.
The headlines of the course are:
1. Introduction. Mathematics as a science and a subject taught in primary and secondary education  their definitions and relations. Position of mathematics in the Croatian national curriculum and a comparison to selected European countries.
2. Aims and learning objectives of mathematics education. General and specific aims. Three crucial components of mathematics education: mathematical concepts, strategies (problem solving, theorem proving, model building etc.) and algorithms. Learning objectives snd strands (contents) for each education level (according to ISCED 1997). Educational standards (in Croatia and elsewhere).
3. Didactics of mathematics. The notion of the field (mathematical) didactics  interdisciplinary scientific discipline and a theoretical and practical guidance (the knowhow) for successfull mathematics teaching and learning.
4. Forms of mathematical thinking and reasoning. The language of mathematics (development, use, symbols). Basics of logic (mathematical ideas, assumptions and concept development). Mathematical notions. Example and counterexample construction. Interpretation and implementation of the definitions of mathematical concepts. Formulation, proving and implementation of theorems in mathematics education. Inductive reasoning  complete and incomplete induction. Deductive reasoning.
5. Methods of mathematical reasoning. Analysis and synthesis. Variation. Analogy. Generalization and specialization. Abstraction and concretization. Distinguishing cases. Superposition of particular cases. Descartes' method. Experiment. The method of undetermined coefficients. Substitution. The method of recurrence relations.
