# Mathematics Didactics PPUMAT15

## Course Description

##### Introduction

Mathematics originated in the study of numbers and simple geometric figures, and many people still associate the subject with formulas and calculations. In addition, geometry may be considered an essential element of mathematics. Some mathematicians claim that all mathematical theory is based upon geometry.

Another view of mathematics dates back to Plato, who believed that mathematics deals with ideas that have a separate reality. In a sense, we manipulate ‘representations’ of these ideas, when we express ourselves in mathematics. The symbol ‘3’ represents the number three. The number 3 is not a physical object, but an abstract idea, which we may represent with many different symbolic expressions, such as ‘3’, ‘III’ or ‘…’. It is convenient to work with these representations on paper, but in our thoughts they represent more abstract ideas. From this perspective, it is easy to agree that mathematics involves more than symbols, as the mathematical symbols also represent abstract objects.

In working with mathematical relationships, we follow certain ‘rules’ when manipulating mathematical symbols. Mathematics may also be expressed as a formal system, a logical construction. We seldom think about what the symbols represent, but manipulate them according to fixed rules, detached from outer references in the ‘real’ world.

However, the subject does not merely concern this system, but also the *activity* that we carry out when we work within the formal system. The concept of ‘mathematics as a human activity’ has also been proposed. This implies that mathematics is something we *do*. Mathematics is the activity involved in finding, for example, the largest common denominator; it is not the answer or the symbols, but the actual *activity* itself.

If we delve more deeply into the question of what mathematics *really* is, we will simply be confronted with a new series of questions. What are mathematical objects? What does a mathematician work with? Are mathematical theorems discovered, or are they artificial, man-made constructions? The answers to such questions are dealt with in mathematical philosophy. However, there are no unambiguous answers to these and related questions, just alternative approaches to the subject of mathematics.

The various views on how people learn mathematics are related to the differences over what constitutes mathematical knowledge. Different theories about learning may provide alternative approaches to teaching. In schools, the subject of mathematics has alternately been regarded as a gauge of academic refinement and culture, and a practical tool for achieving specific objectives. More recent curricula reflect a more complex view of the subject. The creative and curious student who creates his own mathematical concepts is more highly regarded than was previously the case. Discussions about mathematics are also more common. In addition, hand-held calculators and information technology tools have also provided rich opportunities for new approaches to the subject.

This post-graduate course in mathematics will prepare students for teaching in a rapidly evolving society where the abilities to evaluate, review and develop will be essential.

The subject curriculum in *Mathematics Didactics* builds on the *Curriculum and Regulations for the Post Graduate Certificate in Education *(Ministry of Education, 1999). The course qualifies students who have teaching skills in mathematics, gained at degree level, to teach mathematics in primary and secondary schools. Students will study topics which will, in combination with their background in the subject, provide a basis for planning and executing reflected and varied instruction based on the current curricula in use in primary and secondary school education.

Fundamental questions will be addressed, such as: what is meant by mathematical skills; how do people learn mathematics; and why do people need mathematical skills? Similarly, various groups of students’ approaches to, and knowledge of, the subject will be explored.

##### Aims and target areas

Students will acquire didactic skills for teaching mathematics which will enable them to evaluate its role in the school and contribute to the development of mathematics as a school subject.

Students will develop:

- Knowledge within mathematics didactics theory and practice.
- The ability to plan, execute and evaluate mathematical instruction.
- An understanding of the factors that influence how much pupils benefit from the instruction they receive.

The course consists of the following target areas:

*Mathematics as a Field of Knowledge *(3 ECTS)

*Learning Mathematics* (6 ECTS)

*Teaching Mathematics *(6 ECTS)

These three target areas overlap with one another. Consequently, we will generally work within more than one target area at a time.

###### The scope and content of the target areas

*Mathematics as a Field of Knowledge *includes approaches to the subject that will prove useful to the teachers when they plan their teaching. The curriculum presupposes that students will familiarise themselves with the history and role of mathematics as a subject in Norwegian schools.

In addition, students will learn to interpret and evaluate the current curricula in mathematics, and be able to use them as a basis for their own teaching efforts. Students will develop insight into the use of mathematical models, and be capable of evaluating the role of the subject in society and in the field of science. In addition, students will also learn to describe mathematics as a dynamic subject, in which creativity and imagination play an important role in developing understanding. They will also be able to explain the inductive and deductive nature of the subject.

Different learning theories may support varying didactic approaches. Within the target area *Learning Mathematics*, students will encounter several alternative theories on learning, and develop the ability to reflect on the various didactic approaches that these learning theories support. In addition, students will learn to describe different views and methodical approaches to learning, and to evaluate them. Students will learn to create adapted learning environments in which pupils are stimulated to use their imaginations, language and mathematical skills and develop insight into their own learning. Students will acquire an understanding of common misinterpretations and specific learning disabilities in order to satisfy the educational needs of various pupil groups.

The following approaches will be included in the target area *Teaching Mathematics*. Using the current curricula as a foundation, students will learn to formulate goals for their teaching initiatives, and be able to motivate, guide and evaluate pupils. Students will also learn to evaluate various teaching and learning methods in mathematics, and encourage active pupil participation in the planning and organisation of adapted mathematics instruction in which the ideas, skills and knowledge of the pupils are treated as resources. Knowledge and skills in the use of information and communications technology (ICT) are a prerequisite for using ICT-based teaching aids as an integral part of the teaching, as described in the curriculum. Students will learn to describe how the teaching of mathematics in upper secondary schools may be adapted to specific occupational educations. Students will learn to assess and use various forms of assessment and discuss criteria for determining pupils’ skills in mathematics, based on the current assessment criteria and examination requirements.

## Learning Methods

The course includes lectures, group work and individual assignments. Each student will prepare a didactics assignment in mathematics related to his or her period of teaching practice, which will take place during the course of the study period. Emphasis is placed on students using teaching and learning methods that they will be able to employ in their future careers as teachers.

Students who take 30 ECTS in *Mathematics Didactics* will, in addition to the above-mentioned assignment, write an independent project consisting of a report and a study of syllabus texts related to the theme of the project. The size of the project report should be in proportion to the weight of the 9 ECTS to be awarded for it, while the study of the syllabus texts related to the project field will count for 6 ECTS.

## Assessment Methods

**Students taking 15 ECTS**

An individual, 5-hour written examination.

**Students taking 30 ECTS**

An individual, 5-hour written examination that corresponds to the written examination taken by 15 ECTS students. The written examination counts for 60% of the final grade.

The project report will be assessed. The grade for the project report counts for 20% of the final grade. In addition, candidates will be given an oral examination on their project report. Candidates will also be examined on the syllabus texts associated with the project. The oral examination grade counts for 20% of the final grade.

A single grade will be entered on the diploma, ranging from A to F, where A is the highest grade and E is the lowest passing grade. Each part of the assessment must receive a passing grade in order to achieve a final passing grade for the course.

Please refer to Telemark University College’s examination regulations for further information.

*Minor adjustments may occur during the academic year, subject to the decision of the Dean*

Publisert av / forfatter Frode Evenstad <Frode.EvenstadSPAMFILTER@hit.no>, last modified Jorunn Halvorsen - 25/08/2011