The Danish Ministry of Education introduced competence thinking to put an end to syllabus thinking, e.g. long lists of concepts and routines to be learned.
There are several reasons:
• The traditional syllabus thinking causes too low a level of ambition. Reproduction from syllabus is acceptable, however not at all sufficient, and the curriculum is not in agreement with it any more, be it in primary school, secondary school or at educational colleges.
• Growth of knowledge brings a packed syllabus to schools and educations and thus a risk of superficial or skewed choices.
• Assessing student qualifications only based on the knowledge of type and amount of subject matter does not supply sufficient evidence of their comprehension and uptake of the eight mathematical competences as prerequisites to optimal mathematical progression between education levels.
Competence thinking is in accordance with an international tendency for ways to represent the teaching proficiency.
Competence-descriptions of mathematics present interesting possibilities.
First, concerning two kinds of descriptions:
• Normatively, when weight and degree of mastering every competence define a curriculum. This is now done in Denmark.
• Descriptively, when competences are used to describe and analyze educations including mathematical issues, as well at a curriculum level as in daily education.
This is partly done by the PISA testing of 15-year-old students in many countries.
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Second, the competence descriptions offer a meta-cognitive support in daily use – both descriptively and normatively – by simply being used as foci between students and teachers about:
• what the math teaching is intended for, and
• what is actually happening – or ought to happen, both concerning math teaching and learning.
A matrix structure
In the math curriculum at every level in schools and teacher education a set of mathematical competences and a set of subject areas can be said to span the mathematical content in a matrix structure (Table 1):
Subject area →→→→ Competence
Numbers Arithmetic Algebra Geometry ...
1. Mathematical thinking 2. Problem tackling 3. Modeling 4. Reasoning
5. Symbol and formalism 6. Representing
7. Communicative 8. Aids and tools
Table 1 Mathematical competences vs subject areas [4]
The way to think of the link between subject area and competences is to fill out the big blank space. Ideally all subject areas can be linked to all eight competences, but of course some links seems more natural than others, like functions and modeling, or arithmetic and representing, but it is in no way exclusive. It is also important to realize that we don’t need to start with a subject area and then add on a competence. It is equally valid to choose a competence to focus on, and then pick a suitable subject area to work on the given competence. This is what we have done in the following example taken from a developmental work carried out in Denmark in a 5th grade class focusing on reasoning competence [5]. It was a 90 minutes lesson with a small brake, and the pupils were expected to work on an open-ended problem:
House 1
Here we have a 3 room house seen from above. The red dot is Casper, a small ghost who can pass through walls. He is thinking if it is possible to pass through all the walls
Mathematical competences 35 once and come back to where he started. It might work in house 1 but how about the following house 2 (make yourself some more houses):
House 2
It is important that when presenting the task the teacher makes sure that the pupils understand it is the leaning of reasoning competence that is in focus. It is not only to get the results of “he can” he can’t” for each of the houses, but to find ways of arguing why he can or can’t. So the pupils need to find and express rules to judge by the look of a house if it is possible or not for Casper to go through all the walls once and come back.
The focus of the reasoning competence is maintained by the teacher asking questions like: Why is it so?, Is it so any time?, How can you be sure?, etc. We found it very important for the pupils to recognize results from the previous tasks with houses in order to accumulate experiences sufficient to build an argument. Pupils who just went from one house to another answering “can” or “can’t” (as if it was 20 multiplication exercises) did not get very fare in the reasoning. It is also important to leave time (as much as 20 minutes) in the end to have a class discussion to keep focus on the arguments, and letting the pupils listen to the arguments of the others.
How about the gifted pupils?
By keeping the perspective of the competences we can as teachers choose to change the focus from the reasoning competence to the representation competence. The gifted pupils can gain a lot in shifting the perspective from walls and houses to places and routes:
House 1 as places and routes
The walls are now the routes (lines) between the rooms (red ovals), the sizes of the rooms are now neglected and finally the prezentation of the space around the house as
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a “room” (the big oval) as the rest of the rooms is now possible. Now we can look at the problem in a way similar to the problem of the seven bridges of Königsberg [6] by Euler and we can link it to the more general topic of topology.
So one way to support the gifted pupils is a shift in focus with respect to the competences in order for them to develop a better representation of the problem, and thereby use this problem to introduce a new mathematical topic. Here it will be possible to expand the pupil’s degree of coverage within the representation competence and by shifting back to the reasoning competence expand the degree of coverage there as well.
Another way to use this task is to keep the focus of the reasoning competence but ask for reasoning at a higher technical level. This could mean bigger houses, houses of more than one floor, where Casper can pass between the floors or not allowing Casper to use the outer walls. Another way of increasing the technical level is to ask the pupils to write up the arguments in a structured way, with a focus on the logical structure of the argument, the definitions, the premises, and so on. This way of organizing the teaching sets high demands for the teacher. Next we will look at ways to describe the competent teacher.