Slovenia: Description of the Advanced Mathematics Programs and Curriculum
In curricular documents for teachers and students, mathematics is presented as one of the basic subjects of general gymnasia in which students learn mathematics concepts and structures, critical thinking, and reasoning; develop creativity, formal knowledge and skills; recognize the practical usefulness of mathematics; gain mathematics knowledge and competencies needed for future mathematics studies as well as learning in other subjects and everyday life. The gymnasium mathematics course is compulsory and the same for all future university students, regardless of their area of study. The national curriculum for advanced mathematics is available in the form of printed and e-books containing general goals, contents and topics, expected student outcomes, and recommendations for teaching, including the incorporation of ICT, homework, and assessments into mathematics courses. In addition to the curriculum that is written for teachers’ use, the expected standards, list of topics and examples of questions for basic and advanced level of the mathematics matura examination exist in printed and e-documents for students.
Contents and topics are given in the general order of teaching the advanced mathematics course through four years. For each topic, expected goals for students are followed by list of specified topics to be taught, expected hours of lessons needed for the content, and didactical recommendations about use of ICT. Included also are suggestions and guidelines for connecting the topics with material from other academic areas and how the topics could be presented and taught in these contexts. There are some topics classified as optional or as left to the teacher’s discretion based on the teacher’s expectations for students’ achievement. The prescribed topics in each compulsory and elective content are listed below.
| Content Area | Topics |
|---|---|
| Sets and Logic | Basics of logic; sets |
| Numbers | Number sets with whole, rational, real, and complex numbers (mathematical induction and the polar form of complex numbers are optional topics) |
| Algebraic Expressions | Equations and inequalities and their methods of solution (parametric equations are optional); powers and roots |
| Geometry | Lines, angles, circles and triangles in a plane and in space; sines and cosines; the areas of 2-D geometric shapes and the volumes of 3-D shapes and sections; Cartesian coordinate systems; vectors in a plane and in space, scalar product (vector product is optional) |
| Functions | Limits, continuity, inverse and composite functions; linear functions; solving systems of linear equations; quadratic, exponential, rational, logarithmic and trigonometric functions; conic sections |
| Sequences and Series | |
| Calculus | Differential calculations; integrals; applications of integrals |
| Probability and Statistics | Combinatorics |
Expected outcomes are given by main topics as a list of content and procedural knowledge, provided in the table below. Procedural knowledge outcomes include general skills and processes linked to mathematical knowledge but transferable also to other areas.
| Knowledge | Expected Outcomes |
|---|---|
| Content Knowledge | Calculate with numbers
Use properties of sets Use logic in proofs Understand linear, power, root, quadratic, exponential, logarithmic, rational and trigonometric functions and calculate with them Draw graphs and use them in modeling Use Euclidean geometry and trigonometric functions in the context of Euclidean geometry; link Euclid geometry and vectors Use conic sections in problems Know and use arithmetic and geometric sequences and series, and apply them in financial mathematics and natural growth context Understand and use derivatives and determine tangents and simple extrema problems Know the meanings of indefinite and definite integrals; find indefinite integrals in simple situations, and use definite integrals for calculations of the area of a surface of revolution and volume of a solid of revolution Understand and use the fundamental principle of counting and other principles of combinatorics Know the classic definition of probability and calculate the probability of compound events Know statistical concepts, use them in other subject areas, and provide statistical analysis for a given problem |
| Procedural Knowledge | Abstract thinking
Understanding of formal mathematical reasoning Analytical problem solving with different strategies Use of mathematics in everyday life (geometry, measurement, estimations, data analysis, interest expenses) Developing effective reading strategies for future learning Communicating mathematics in oral, written and other forms in the mother tongue and in one foreign language Designing and carrying out a research study and critically reporting findings Formulating research questions and hypotheses Thinking about necessary and sufficient conditions Using ICT and the Internet responsibly Making decisions and giving estimates of risks |
Cross-curricular connections are provided as examples of activities that can link together knowledge from different subjects and mathematics.
Didactic recommendations describe the compulsory use of ICT in as many possible forms and activities as possible:
- To develop skills
- To reach new knowledge
- To help students with disabilities
- To help with calculations, statistics and in communication
All available digital devices (computers, tablets, graphic classroom boards, advanced calculators) and specialized software for learning mathematics (geometry simulations, symbolic calculations, drawing) are encouraged to be used for learning mathematics.
Homework is presented as the basic form of self-motivated learning and primary source for discussions in a class. It is said to help student attain better knowledge and may indirectly influence students’ grades. Students should be assessed by at least four written tests and one oral examination in class per year. Other forms are also suggested (projects, research, group work) with the recommendation that students be giving enough opportunities to demonstrate their knowledge in different situations and are encouraged to develop responsibility for their own learning.
For the matura, students can decide whether to take the basic level or advanced level of the compulsory mathematics exam. The curriculum contents for both are the same, but required standards differ. The written test for the advanced level, in addition to compulsory items for the basic level, contains additional advanced level items. For oral examinations, the expected knowledge for the advanced level is specified in the matura standards (i.e. theoretical explanation of the definition of a limit versus the calculation of the limit only). Students receive grades from 1 to 5 for the matura exam at the basic level and from 1 to 8 at the advanced level. The sum of grades from all five matura subjects is used as a criterion for entrance to tertiary-level education programs with a limit on the number of new students.