Profound knowledge of “elementary” mathematics (such as may be covered in secondary education).

Ability to construct and develop logical mathematical arguments with clear identification of assumptions and conclusions.

Facility with abstraction including the logical development of formal theories and the relationships between them.

Ability to model mathematically a situation from the real world and to transfer mathematical expertise to non mathematical contexts.

Readiness to address new problems from new areas.

Capacity for quantitative thinking.

Ability to extract qualitative information from quantitative data.

Ability to comprehend problems and abstract their essentials.

Ability to formulate problems mathematically and in symbolic form so as to facilitate their analysis and solution.

Ability to design experimental and observational studies and analyse data resulting from them.

Ability to formulate complex problems of optimisation and decision making and to interpret the solutions in the original contexts of the problems.

Ability to use computational tools as an aid to mathematical processes and for acquiring further information.

Knowledge of specific programming languages or software.

Ability to present mathematical arguments and the conclusions from them with clarity and accuracy and in forms that are suitable for the audiences being addressed, both orally and in writing.

Knowledge of the teaching and learning processes of mathematics.