Mathematical competence is a complex concept. The idea of using it all the way from primary school to university was introduced, developed and exemplified by the so-called KOM-report [1]. This report was produced in 2002 by a Danish math commission in response to 10 questions from the Danish Ministry of Education.
The report is a comprehensive paper (212 pages) prepared over a period of two years by a group comprising 12 people (mathematicians, researchers of mathematics didactics, mathematics teachers from all levels and people applying mathematics in other fields).
As we find mathematical competence a promising concept for description and prescription of mathematics teaching and learning, also with the focus on the gifted students, this chapter is intended to clarify the concept through some examples taken partly from the report and partly from developmental – and research work in Denmark.
In the KOM–report possesing mathematical competence is described as follows:
Mathematical competence comprises having knowledge of, understanding, doing, using and having an opinion about mathematics and mathematical activity in a variety of contexts where mathematics plays or can play a role. This obviously implies the presence of a variety of factual and procedural knowledge and concrete skills within the mathematical field, but these prerequisites are not sufficient in themselves to account for mathematical competence [1; p. 49].
The report identifies eight mathematical competences that are described as:
… a well-informed readiness to act appropriately in situations involving a certain type of mathematical challenge [1; p. 49].
The eight mathematical competences are:
1. Mathematical thinking competence – mastering mathematical modes of thought i.e. being able to pose questions which are typical for mathematics. This is in focus in Chapters 3 and 4 (on competitions and clubs).
2. Problem tackling competence – formulating and solving mathematical problems, i.e. being able to detect, formulate, demarcate and define mathematical problems, and eventually solve them.
3. Modeling competence – being able to analyze and build mathematical models concerning other areas,
i.e. being able to analyze the basis and characteristics of a given model and to build mathematical models of situations outside mathematics itself.
This is in focus in Chapters 3, 4 and 5 (on competitions, clubs and IT).
Chapter 2 24
4 Reasoning competence – being able to reason mathematically,
i.e. being able to follow and evaluate a mathematical argumentation as well as being able to think of and carry out one’s own informal and formal reasoning.
This is in focus in Chapter 3 (on competitions).
5 Representing competence – being able to handle different representations of mathematical entities,
i.e. being able to choose among and interpret various forms of representation.
6 Symbol and formalism competence – being able to handle symbols and formal mathematical language,
i.e. being able to deal with and make use of symbol-containing expressions.
7 Communicative competence – being able to communicate in, with and about mathematics,
i.e. being able to study and interpret written, oral or visual statements or “texts” as well as being able to express oneself about matters involving math issues.
This is in focus in Chapter 4 (on team competitions).
8 Aids and tools competence – to be able to make use of – and relate to the aids and tools of mathematics,
i.e. having knowledge of possibilities and limitations of relevant tools for mathematical activity as e.g. concrete materials, calculators and ICT-programs.
Fig. 1. The competence flower [1]
In addition to these eight competences the KOM-report mentions three kinds of active insights in the nature and role of mathematics in the world, which do not describe desired actions. Namely an overview and critical judgment on:
Mathematical competences 25
• the actual application of mathematics in other fields and practices
• the development of mathematics in time and space, in culture and society
• the character of mathematics as a well-structured subject area.
In the following we will present an example on how to use the competence thinking in describing the mathematics of a situation in another way than just stating the need for a specific knowledge and some given skills.
How much money to use on an apartment?
Let us say that you want to purchase a place to live, and for that you need to take on a loan/morgage of 400.000 euros. The interest is 0,5% per month for 20 years (240 months). Usually you find the formula for annuity loans:
I Mp I
A
n
+ −
⋅ −
= 1 (1 )
where A is the amount loaned, Mp denotes the monthly payments, I – the interest, and n – the number of payments. In this example the equallity works out as follows:
1 (1 0,005) 240
400 000 2851
0, 005
− + −
= ⋅
How do we look at that situation from a perspective of the eight mathematical competences:
• Thinking competence: recognizing the mathematical questions, e.g. Why divide by the interest I?; How to calculate the monthly payments Mp in the formula above?, or What happens if you need only 300 000 Euros? Whereas a question like: Why is your interest so big? should be recognized as a non-mathematical question. It is more likely for an economist to answer a question like that.
• Problem tackling competence: to formulate and solve problems similarly to what we are doing now. Finding the right formula, gathering the right information, and using them to answer the questions we are asking. Here we see the link to the Thinking competence.
• Modeling competence: to understand the decisions behind this model, viz. that the model is based on the monthly payments being made at the same time as the calculation of interest. If that is not the case, then the formula cannot be used as it is. Another point is to see what is not included in this model, e.g. fees to the bank.
In this case the model is already made, but a part of the modeling competence is to make models or improve models, e.g. to adjust this model by including fees to the bank.
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• Reasoning competence: to like to work on the proof of the formula – or just to understand a presentation of the proof. To argue that for a constant interest (I) and a constant number of months of payment (n), there is a linear correlation between the size of the loan (A) and the monthly payment (Mp).
• Representing competence: the ability to read and write different representations of the same mathematical object, e.g. the interest of 0,5% = 0,005, also remembering that the numbers represent money in Euros, or for different sizes of loan to represent the relationship to the Monthly payment in a graph to see the linear correlation. Here we see the connection to the reasoning competence. With a well-chosen representation arguments can be made easier.
• Symbol and formalism competence: here we use the symbols I, Mp, A and n, and (1+I) is raised to the power of –n. Again we see a link to the reasoning- and the representing competence. The formalism and the use of symbols make the reasoning easier if you understand the symbols. For instance, to argue that there is a linear relation between the size of the loan A and the monthly payments Mp one can argue for a substitution of the fraction with another symbol since it is a constant for constant I and n.
• Communicating competence: to use the right way to communicate your own mathematical ideas and to perceive the ideas of others. Here again the use of the above mentioned substitution of the fraction with another symbol to argue for the linear relation between A and Mp has a communication value as well, and shows the importance of a communicative focus in handling mathematical situations.
• Aids and tools competence: how to choose and use the right tools and aids to your mathematical activity. Similarly to above to work out a table on all the payments by using a spreadsheet or a calculator just to calculate the monthly payments. Or to use a table containing the relation between the interests and the number of payments as it was done in the good old days. It is important to know the use of the different tools and their limitations. One should not just copy a result with 18 decimals of the calculator when we are talking money. We have to round the result to an appropriate number of decimals that makes sense in the situation.
The three dimmensions of a competence
It goes for all eight competences that they carry within them a duality of both a productive side and a perceptive side. On one hand a part of a given competence is to produce a valid mathematically action to a given situation. The other part of the competence is to perceive and judge the actions of others in relation to a given situation. This is called the coverage of the competence.
In all we talk about three dimensions of a competence, in which coverage is one of them. The other two are the degree of action, and the technical level of a given competence.
Mathematical competences 27 The degree of action relates to the number of areas within the subject of mathematics or in more general the number of different situations one can act out a given competence. For instance not all have the same level of symbol-formalism competence within higher algebra as within arithmetic.
The technical level of a competence refers to the level of mathematical complexity of a given situation one can handle. We would say it is easier to handle the representation of a situation of linearity in relation to buying potatoes 75 eurocents a kilo than the representation of the linearity of the situation presented above.
Below we will present a few of the eight competences in more details, to exemplify how the competence thinking can be used to challenge the more gifted students within the frame of the ordinary class work.