Pedagogical Knowledge for Mathematics: Consequences for Effective Professional Development

(c)svwWhat knowledge do mathematics teachers need in order to teach successfully?   In a first blog post I looked at the concept of pedagogical content knowledge of mathematics.  In the second I discussed research attempts to measure teachers’ knowledge and link it to students’ learning outcomes.  In this one, I write about the implications for teachers’ professional development.  In a next and final blog post, I will try to relate the first three blog posts to the South African context.

Traditional forms of professional development such as workshops and lectures, tend to be top-down, one-off activities, focused on transmitting ‘new’ ideas of teaching and learning. Research shows that such isolated and piecemeal models of intervention bring little significant change to teaching practices and student achievement (e.g. Borko, 2004; Cohen and Ball, 1999). Recent initiatives of teacher professional development follow a ‘socially and culturally situated process of knowledge construction.’  This implies more attention for collaboration, discourse, reflection, inquiry and application. Research indicates that effective professional development requires continuous interactive support over a substantial period of time, should focus on specific educational content under guidance of an expert adopting a hands-off role and revolve around artefacts that help fostering a sense of ownership with teachers (Borko, 2004; Shalem et al., 2013).  Communities of Practice form an attractive theoretical framework for this kind of activities.  This view aligns well with the situative vision on PCK, as ‘knowledge-in-action’ that cannot be separated from the classroom context.  Regular school-based professional development not only has the advantage that it limits teachers’ time away from their classes, but it also promotes involvement from the school management. New teachers are often asked to comply with established practices in the school, regardless of what they learnt and appropriated before (NORRAG, 2013).

Some illustrations of teachers’ training formats that incorporate these principles are lesson study (Sibbald, 2009), curriculum mapping (Shalem et al., 2013) and mentoring programmes (Nilssen, 2010).

Curriculum mapping is a collaborative activity during teachers seek to align what is taught in the classroom (‘enacted curriculum’) with what is expected in state or national standards (‘intended curriculum’) and assessments’ (‘examined curriculum’).  Shalem et al. (2013) report on a curriculum mapping project for basic education mathematics teachers in South Africa (DPIP).  The main objectives of this project were:

  • Improve use of (inter)national assessments by teachers
  • Enhance alignment between enacted and intended curriculum
  • Develop communities of practice based on a joint enterprise and artefact creation
  • Clarify teacher expectations about intended learning outcomes
  • Improve interpretation of an outcome-based curriculum (previously in place in South Africa)

Lesson study has Japanese roots and is based on joint lesson planning combined with observation lessons to refine teacher understanding of all details surrounding a particular lesson.  A detailed lesson plan is collaboratively constructed, tried out and discussed several times, with members of the lesson study group taking turns teaching the lesson.  Positive effects on both content and pedagogical knowledge have been published.  However, the method is time-consuming and requires a collaborative culture within the school.  Initiatives such as in Chile, where a law has been proposed to link test results of teachers’ content and pedagogical knowledge to their salaries, are likely to have an adverse effect, promoting competition among teachers.  Hattie (2009) identifies collaborative work by teachers in preparing and evaluating their teaching as one of the top factors affecting learning.

Ball et al. (2001) suggest that the ideal course would to witness an outstanding fifth grade mathematics class, complemented by later study to extend and make more explicit a global and overarching perspective on the lesson topic.  In fact, Seymour and Lehrer (2006) suggest that PCK for mathematics develops with experience, as a teacher supplants general heuristics with more concrete representations and ‘interanimated’, contextualized combinations of teacher and student discourses develop.

These findings pose challenges for teacher professional development in developing countries.  In countries where many donors are active and that lack a framework for in-service training, such as Cambodia, organizing such a coherent system of regular professional training is challenging. Donors may set different priorities, timeframes and implementation frameworks. In the best case, organisations can organize joint trainings and follow-up activities, as in the cooperation we had with the Stepsam2 project from the Japanese International Cooperation Agency (JICA).  In the worst case, teacher trainers and teachers are overwhelmed by a plethora of one-off workshops, each reducing valuable available time in school. A vision for teacher professional development and a framework wherein various initiatives can be fit is necessary to enhance the quality of professional development, improve alignment with educational goals and to find a balance between time for teaching and time for learning.

Selected references:

Ball, D.L., Lubienski, S.T. and Mewborn, D.S. (2001) ‘Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge’, 4th ed. In Richardson, V. (ed.), Handbook of research on teaching, Washington, DC, American Educational Research Association, pp. 433–456, Available here
Borko, H. (2004) ‘Professional development and teacher learning: Mapping the terrain’, Educational researcher, 33(8), pp. 3–15.  Available here
Shalem, Y., Sapire, I. and Huntley, B. (2013) ‘Mapping onto the mathematics curriculum – an opportunity for teachers to learn’, Pythagoras, 34(1), Available here

Seymour, J.R. and Lehrer, R. (2006) ‘Tracing the Evolution of Pedagogical Content Knowledge as the Development of Interanimated Discourses’, Journal of the Learning Sciences, 15(4), pp. 549–582.

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