Making Mathematics Conceptual

Knowledge, skills and understanding. This is the way that we educators breakdown the learning which we are to achieve in students. We work from detailed curricula, schemes of work, instructional plans, syllabi - whatever it might be called - and it is usually stated in detailed lists of things to know about and how "to do" things.
Recently, the move to more "conceptual" approaches, where content is not directly specified, has been discussed. Even assessment orientated organisations such as the International Baccalaureate have been discussing having a more conceptual curriculum for the IB diploma subjects. In order to be clearer about the teaching and learning of this curriculum, they are considering ATTL - specifying Approaches to Teaching and Learning (an extension of the ATL aspect in the lower IB programmes). This might include:

• A recommended pedagogy
• constructivist learning
• subject specific conceptual learning
• contextualised authentic learning
• differentiated learning
• inquiry and critical thinking
• independent, lifelong learning
• stimulating learning environments
• study skills common to all subjects
• e-learning/technology component

(from Andy Atkinson's presentation at the IB Heads World Conference - he is the new Curriculum Director for the IB)

This is quite interesting territory, needing all the usual caveats regarding the value of the IB diploma curriculum to the next stage of education and thus its currency for university entrance. However, it could herald in a new age of learning and teaching at this normally dry and traditional level.

To see how this might work in mathematics, and at an extreme of the concept, I recalled Conrad Wolfram's presentation from October 2010 at TED. Here he presents his arguments for making math more practical and more conceptual, using real world problems for the calculations ("real problems look knotty, they have hair on them"). As Conrad put it:

MATHEMATICS  ≠ CALCULATIONS
MATHEMATICS > CALCULATIONS

He stated that it is not the case that computers will dumb down mathematics but that we have (choose to have) dumbed down problems in mathematics (to be able to deal with the calculations that we can handle).

Here is the original presentation: