In the KOM project we have identified eight such competencies, which may well be seen as forming two clusters, each containing four competencies.
The ability to ask and answer questions with and in mathematics: (Judging(Thinking/Logic and Feeling/Ethics)
Mathematical thinking competency(Introverted Thinking/Logic) - mastering mathematical modes of thought includes
- understanding and dealing with the roots, scopes and limitations of given concepts;
- abstracting concepts generalising results;
- distuingishing between different types of mathematical statements concerning single objects and particular cases;
- possessing awareness of the kinds of questions that are typical of maths, and insight into the kinds of answers to be expected;
- possessing and ability to pose such questions.
Problem handling competency(Extraverted Thinking/Logic) - formulating and solving mathematical problems, includes
- detecting, formulating, delimitating, and specifying mathematical problems, pure or applied, open or closed;
- possessing an ability to solve problems, posed by oneself or by others, if desireable in different ways.
Modelling competency(Introverted Feeling/Ethics) - being able to analyse and build mathematical models concerning other areas includes
- analysing the foundations and properties of existing models, and assessing their range and validity;
- performing active modelling in given contexts i.e. structuring and mathematising situations, handling the resulting model, drawing mathematical conclusions from it, validating the model, analysing it critically, communicating about it, monitoring and controlling the entire process.
Reasoning competency(Extroverted Feeling/Ethics) - being able to reason mathematically, includes
- following and assessing others' mathematical reasoning;
- understanding what a proof is (not) and how it differs from other kinds of reasoning;
- understanding the logic behind a counter example;
- uncovering the main ideas in a proof;
- devising and carrying out informal and formal arguments, including transforming heuristic reasoning to valid proof.
The competencies in the second cluster focus on
The ability to deal with mathematical language and tools: (Perceiving(Sensing/Intuition)
Representation competency(Introverted Sensing) - being able to handle different representations of mathematical entities, includes
- understanding (decode, interpret, distinguish) and utilising different kinds of representations of mathematical entities;
- understanding the relations between different representations of the same entity;
- choosing, making use of, and switching between different representations.
Symbols and formalism competency(Extroverted Sensing) - being able to handle symbolic language and formal mathematical systems, includes
- decoding symbolic and formal language;
- translating back and forth between symbolic language and natural language;
- handling and utilising symbolic statements and expressions, including formulae
- understanding the nature of formal mathematical systems.
Communication competency(Extraverted Intuition) - being able to communicate in, with and about mathematics, includes
- understanding, examining, and interpreting different kinds of written, oral or visual mathematical expressions or texts;
- expressing oneself in different ways, and at different levels of precision, on mathematical matters to different sorts of audiences.
Tools and aids competency(Introverted Intuition) - being able to make use of and relate to the tools and aids of mathematics, includes
- having knowledge of the existence and properties of different relevant tools and aids for mathematical activity (e.g. rulers compasses, protractors, tables, centicubes, abaci, calculators, computers, the internet);
- having insight into the possibilities and limitations of such tools;
- reflectively using tools and aids.