Arbitrary and Necessary

Our most recent LUMEN Event was led by Dr Dave Hewitt from the Mathematics Education Centre here at Loughborough University. Dave's focus was on the fascinating issue of 'When to tell and when not to tell', and the basis for this was his extremely helpful framework: Arbitrary and Necessary. For some of Dave's papers that go into much more detail about this, please click on the following links: Arbitrary and Necessary part 1, Arbitrary and Necessary part 2, Arbitrary and Necessary part 3 and Mathematical Fluency- The Nature of Practice and the Role of Subordination.

For Dave's Powerpoint slides from the Spring term LUMEN Event please see his slides‌.

For other materials from the session, please click on the following links: Do We Meet, Scenarios and Feedback - LUMEN and A Square.

'Arbitrary' mathematical knowledge are definitions, conventions or notations that students couldn't work out for themselves without being told. They are things that, from their point of view, could have been otherwise. An example is that there are 360 degrees in a full turn. This may not be arbitrary from a historical point of view (the reason is to do with Babylonians counting in 60s), but mathematically it could be otherwise, and in fact in some situations it is (a full turn might be 1 turn, 2pi radians, 100%, etc.). On the other hand, with 'Necessary' mathematical knowledge, you can give a reason why it must be that way. For example, once you decide that there are 360 degrees in a full turn, then it is necessarily the case that the total interior angle of a plane triangle is 180 degrees.

Dave suggested that the ‘arbitrary’ is in the realm of memory, whereas the ‘necessary’ is in the realm of awareness, and he talked about us needing to tell students arbitrary knowledge but not necessarily needing to tell students necessary knowledge. Memorising is hard work and costly for students, whereas helping students to develop awarenesses of necessary knowledge, rather than memorising it, is easier, more satisfying and longer-lasting. For more details, and lots of examples, please see Dave’s slides.

The LUMEN Events are not intended to be isolated standalone CPD sessions. So we began the day by sharing experiences of putting into practice the ideas Craig Barton presented at the first LUMEN Event, back in November. It was great to hear participants describe how they and their colleagues in school had implemented diagnostic questions and formative assessment and made various modifications to fit their different contexts. We ended the day thinking about concrete steps we might take to implement the ideas from Dave’s session in classroom practice, and I look forward to hearing how this is going when we meet again at Event 3 in the summer.

Please do sign up now, if you haven’t already, and continue to spread the word, as new colleagues are very welcome to join at this stage. Event 3 will be led by Tom Francome, from the Mathematics Education Centre here, and will focus on “Designing practice tasks to develop the mathematician as well as the mathematics”. As an additional bonus, all participants at Event 3 will receive a free copy of the book ‘Practising Mathematics’, written by Dave Hewitt and Tom Francome and published by the Association of Teachers of Mathematics.

See you in the summer!

Colin Foster

Director of LUMEN (Loughborough University Mathematics Education Network)

Reader in Mathematics Education, Mathematics Education Centre, Loughborough University