Developing Mathematics Syllabus Documents

I attended one of the Dean’s Lecture Series seminars, at the University of Melbourne’s Graduate School of Education, last week. It was titled ‘Competencies and the Fighting of Syllabusitis’ and was given by Associate Professor Tomas Hojgaard from Denmark.

He has recently been working with the Mathematics Faculty of the Graduate School of Education. He was also one of the people behind ‘Mathematical Competencies and the Learning of Mathematics’ by Mogens Niss (2002), a paper that has guided the re-writing of syllabus documentation for mathematics in various European countries.

He started with saying “Curriculum only works if it works in the classroom”

He also stated that many curriculum documents he has seen focus on content and pay little attention to the purposes of teaching the ‘what’ that is to be learnt. Guidelines for assessment also seem to refer more to the ‘what’ and ‘how’  (ie the format of the assessments. For example: the number of tasks, the timing of these tasks, the structure of these tasks, consequences for late submission etc) rather than a purposeful focus on what it means to be working mathematically.

The analogy was made that creating a syllabus is like creating a house. If you were to present the builders with a list of the content required to build a house (the number of bricks, amount of cement, number of pieces of wood etc) this would not be of much use to them. It wouldn’t lead to the construction of something with a purposeful structure nor inform one on how the various elements link together to create the house.

His contention is that the, sadly, majority of syllabus documents that are merely a listing of content should be replaced with a series of competencies. By his definition, a competence is something that is being enacted, the use of knowledge to show understanding. His formal definition is “someone’s insightful readiness to act in response to the challenges of a given situation”. ‘Insightful readiness’ is a way of encapsulating the difference between merely doing a skill and being capable of understanding and knowing how to go about approaching a problem set in an unfamiliar context.

He believes that the answer to “What does it mean to be mathematically competent?” is addressed in the following 8 competencies:

  • Mathematical thinking competency: carry out and have a critical attitude towards mathematical thinking
  • Problem tackling competency: formulate and solve both pure and applied mathematical problems and have a critical attitude twards such activities
  • Modelling competency: carry out and have a critical attitude towards all part of a mathematical modelling process
  • Reasoning competency: carry out and have a critical attitude towards mathematical reasoning, comprising mathematical proofs
  • Representing competency: use and have a critical attitude towards different representations of mathematical objects, phenomena, problems or situations
  • Symbol and formulation competency: use and have a critical attitude towards mathematical symbols and formal systems
  • Communicating competency: communicate about mathematical matters and have a critical attitude towards such activities
  • Aids and tools competency: use relevant aids and tools as part of mathematical activity and have a critical attitude towards the possibilities and limitations of such use

Not only would instruction be geared towards these, but assessment as well. He thinks that focussing on competencies as ‘bands of strength’ would mean a more positive approach to education instead of the can/cannot, either/or dichotomy he believes currently exists. These competencies underly the essence of the discipline of mathematics.

Prof. Hojgaard sees two key foci in an authentic education:

  • The need for student directed processes and
  • The need for maintaining educational focus (the essence and rigour of the discipline)

To enact this curriculum in the classrooms, teachers need to engage with thinking about the competencies and how they can be achieved in their learning plans…the need to develop teacher-directed autonomy. (As an aside, it is interesting to consider this in the light of the proposed national curriculum. Will the way the curriculum is designed allow for, or indeed compel, teachers to control their learning plans by thinking about how to teach so that a coherent set of linked ideas create a mathematical structure in their students’ minds? Or will it try to take the control away…and thus, in my opinion, treat teachers more as transmitters of knowledge instead of designers of their students’ creation of knowledge?)

So the question then becomes: “How can a syllabus document be structured so that these competencies can be integrated across a content-rich curriculum so that they are a focus and not able to be ignored by teachers?” There needs to be a way that teachers are compelled to address the competencies and not just teach the same way as done before. Teachers have a difficult job – to develop and target a learning focus but, at the same time, encourage their students to engage in autonomous learning. It is imperative, if students are to internalise the learning, that they take more responsibility for the teaching of what it is teachers want them to learn.

Prof. Hojgaard’s suggestion for the design of syllabus documents is as below:


Number Algebra Probability & Statistics Geometry Measurement
Mathematical thinking competency   The body of the document would have sample questions, examples, suggested
Problem tackling competency   activities for learning etc.      
Modelling competency              
Reasoning competency              
Representing competency              
Symbol and formulation competency            
Communicating competency              
Aids and tools competency              


An interesting talk…and it is very interesting to consider this idea in concert with the Understanding by Design curriculum framework that I am currently toying with.

At the conclusion of the talk, I had the opportunity to chat with Tomas for a while and he agreed it was imperative to compel teachers to engage in thinking more about the purposes for specific content and then delivering a learning plan that addresses those purposes. Too many teachers, he fears, see themselves as deliverers of a curriculum and a methodology that is determined elsewhere and by others. (And he laughed when I asked whether Denmark’s educators were as sick and tired of having Finland given as the ideal example of education, as we in Australia were!!)


About Linda

I have been involved in secondary mathematics education in Victoria, Australia for over 25 years.
This entry was posted in Ideas for teaching & learning, Systems, Vision. Bookmark the permalink.

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