Triggers of contingency in mathematics teaching

Abstract

Our research in the last decade has been into classroom situations that we perceive to make demands on mathematics teachers' disciplinary knowledge of content and pedagogy. Amongst the most visible of such situations are those that we describe as ‘contingent’, in which a teacher is faced with some unexpected event, and challenged to deviate from their planned agenda for the lesson. Our findings and the associated grounded theories have been open to enhancement and revision as new classroom data has been gathered. In this article, we propose a classification of the origins of contingent classroom episodes: namely the students; the teacher him/herself; and pedagogical tools and resources. This classification extends and elaborates our original conception of ‘contingent’, in response to more recent data.

Keywords:

• teacher knowledge
• Knowledge Quartet (KQ)
• contingency
• lesson planning
• technology

Notes

1. One can judge the significance of this episode for Bishop by the fact that he has returned to it several times. The pupil seems to have been a boy, ‘Jonathan’, in the original version (Bishop, 1976).

2. The mathematical potential of this scenario is examined at length in Rowland and Zazkis (2013)

3. This phrase, in connection with the KQ, is due to Michael Neubrand (personal communication, May 2005).

4. The second and third of these are implicit in the a priori possibilities raised by Schoenfeld (1998, pp. 3–4)

5. All names of teachers and pupils in this article are pseudonyms.

6. We are grateful to Lourdes Figueiras and Saínza Fernández, Universitat Autònoma de Barcelona, for sharing this incident.

7. The videotape of Máire's lesson was collected in the doctoral research of Dolores Corcoran.

8. There was nothing unusual about this: many teachers in service since the mid-1980s would recognise situations when the technology let them down, notwithstanding conscientious preparation and rehearsal.

9. This is corrected later to effect subtracting 20 rather than 10, but the children's grasp of the intended procedure, and its variations, is tenuous here and throughout the lesson.

10. The popular literature is much more forthcoming about such eventualities: see for example http://malbell.wordpress.com/2010/03/28/when-the-technology-lets-you-down/

11. Description of the self-evaluation process lies outside the scope of this article, but the trainees answer, in their own time, 21 items related to the mathematics curriculum for students aged 5 to 14 in England, of the type indicated in Goulding, Rowland, and Barber (2002). They then compare their responses with a ‘commentary’ on each of the items written by the course team. On the basis of this comparison, they rate their own response and submit a confidential report to their mathematics methods instructor.