## Rich Mathematics Rich CPD - Is there a connection?

The NRICH website is full of rich tasks and guidance. We want teachers to use what we have to offer having a real sense of what we mean by rich tasks and what that might imply about classroom practice.

We do not present lesson plans because this is contrary to the very nature of what we are about. If we wrote lesson plans it would imply there is only one way to use our resources and also that there is only one route through a problem. As many of our tasks have the potential for multiple starting points, multiple routes and multiple endpoints; any suggestion that there is a well-defined lesson that can be built around them is wrong. However, we realise that we need to help people get started, so we produce resources such as teachers' notes for each of the problems and mapping documents to help link NRICH resources to the bigger picture. These additional resources are professional tools which can aid teachers' development. In keeping with our aim to support teachers supporting themselves, there is also an online CPD (continuing professional development) package for Primary teachers all about rich tasks (it would be nice if there were one for secondary teachers too but we have not found the funding to implement this as yet). We also offer face to face CPD. Such development offerings are made in the context of NRICH's philosophy, so we hope we not only offer rich tasks but that we also offer rich CPD.

So, when I was approached by Pete Griffin (NCETM SW regional coordinator) to run a session on the CPD equivalent of rich tasks, our discussion of what that might look like led to the question:

"Can we make connections/is there an equivalence between
rich mathematics, rich tasks and rich CPD?"

Here is a list of characteristics of a rich task adapted from my article on the NRICH website. Not all problems on the website cover all these characteristics but I believe that any rich task will map to a number of these characteristics. The task:

• is set in contexts which draw the learner in
• is accessible to a wide range of learners, offering different levels of challenge
• is enjoyable and contains the potential for surprise
• can reveal patterns or lead to generalisations or unexpected results
• can reveal underlying principles or make connections between areas of mathematics
• can broaden learners' skills and/or deepen and broaden mathematical content knowledge

• challenges learners to think for themselves and make decisions
• allows for learners to pose their own questions
• allows for different methods and different responses (different starting points, different middles and different ends)
• encourages originality, invention and imaginative application of knowledge
• involves learners in testing, proving, explaining, reflecting and interpreting
• offers opportunities to identify elegant or efficient solutions
• encourages collaboration, communication and discussion

• develops critical thinkers
• develops confidence
• creates learners who can apply their knowledge beyond the original context.

You might notice some division of these descriptions and that is because I want to emphasise the point about rich tasks being more than a problem in isolation, it is also about:

• The learner activity
• The teacher activity
• The wider context, what the longer term benefits for learners might be from using rich tasks.
So, returning to rich CPD. John Mason [2003] talks about "The Discipline of Noticing". Identifying the possibility for change in what we do often comes from recognising (or noticing) a "phenomenon, situation or incident" which acts as the stimulus for change. To me this identification of potential for change might be thought of as the rich CPD equivalent of the rich task. A teacher notices something in their classrooms, for example a behaviour that they wish to encourage and become part of learners' everyday practice or something in learner behaviour that they would prefer to try to modify. An example of the latter might be a teacher who has become aware of the habit of learners putting their hands up immediately a problem is posed and before they have had time to think (lack of independence). Once this phenomenon has been identified, the action of trying to change one's practice in order to influence what happens in particular situations is the task. Rich CPD supports the teacher as they work on their task of, what John Mason calls, 'choosing to act freshly'. So, given this context, is it possible to use the list above and the potential for translating it into a description of rich CPD. As with the rich task, it is not the identification of the need that is enough, it is how the CPD influences and causes change to happen. Is it possible to say:

Rich CPD:
• is set in contexts which draw the learner in
• is accessible to a wide range of learners, offering different levels of challenge
• is enjoyable and contains the potential for surprise
• can reveal patterns or lead to generalisations or unexpected results
• can reveal underlying principles or make connections between areas of mathematics
• can broaden learners' skills and/or deepen and broaden mathematical content knowledge

• challenges learners to think for themselves and make decisions
• allows for learners to pose their own questions
• allows for different methods and different responses (different starting points, different middles and different ends)
• encourages originality, invention and imaginative application of knowledge
• involves learners in testing, proving, explaining, reflecting and interpreting
• offers opportunities to identify elegant or efficient solutions
• encourages collaboration, communication and discussion

• develops critical thinkers
• develops confidence
• creates learners who can apply their knowledge beyond the original context.

I am not looking for a one-size fits all model of CPD. Just like rich mathematics can happen in different ways in different classrooms, I am convinced that rich CPD is not about a particular model but an underpinning view of the role of the teacher as learner and the provider as teacher.

I also believe that, like rich mathematics, rich CPD has rich consequences, meaning that long term changes and benefits come from it. So, how could such changes be described? In my last school I was the Head of ICT and Learning Resources. A large title for a large job, which involved developing teachers' confidence in their use of ICT across the curriculum. I became interested in research around the adoption of technology and I think the models of adoption offered by research can throw light on what I mean by long term gains.

Finally, I am thinking about the relationship between the long term development offered by rich CPD and models of computer technology adoption within classrooms. I think what is said about the stages of adoption of ICT in these models can be said about the adoption of changes to classroom practice initiated by rich CPD.

Five stages of adoption (based on the work of Hooper and Reiber) which I think are useful and which we should be thinking about in terms of the long term success of rich CPD:
• Familiarisation - inset research - learning by doing' getting to know more about the issue /area you are interested in
• Utilisation trying things out in practice (once or twice) sometimes the change goes no further. You see then you do (recipe approach to teaching and learning)
• Integration - at this stage you find that what you have been doing is making you think more deeply about the classroom learning environment in fairly limited contexts but you might find yourself adapting the experiences and they become part of a bigger picture and not just bolt on.
• Reorientation the experience is considered in terms of enriching the learning experience more generally and makes you shift in the way you operate more generally (schemes of work are adapted and new ideas are introduced).
• Evolution you begin to grow and change as the needs of your learners and the learning context changes
It is not until you have "evolved" that you have truly developed.

References

Mason, J. (2003) "Practitioner Research as an Extension of Professional Development" in Holden, I (Ed.) Utvikling av Matematik kundervisning I Samspill mellom Praksis og Forskning, Skriftserie for Nasjonalt Senter for Matematik I Opplaeringen (1), Trondheim p181-192. (all quotations are from this paper)

Hooper, S., and Reiber, L. P. (1995). Teaching with Technology. Teaching: Theory into Practice. A Ornstein (ed.). Boston, Mass: Allyn and Bacon.