“It is time to reverse this prejudice against conscious effort and to see the powers we gain through practice and discipline as eminently inspiring and even miraculous.”
This is the second of a three part series. If you wish to read the first part, click here. Otherwise, read on…
John Mason – The Mystery of Mastery
Mention the names ‘Mason’ and ‘Watson’ in Mathematics teaching circles and a reverential air suddenly eminates. John Mason and his wife Anne Watson have worked for years on researching the best ways to engage and develop student thinking whilst studying Maths, particularly through questioning and the creation of tasks that push students into their individual zones of proximal development (hello Vygotsky). In fact if I had to choose two texts that teachers starting out in our rarified field should read, then Mason and Watson’s Questions and Prompts for Mathematical Thinking and Mason (et al)’s Thinking Mathematically would probably be my top candidates.
You can probably understand my excitement, therefore, when John Mason took the stage. His opening remarks betrayed principles that correlate with mine own: everything he proposed is merely conjecture (very Socratic!) and that professional development should be fundamentally phenomenological, i.e. a ‘lived experience’.
He started out his thoughts on mastery by answering two questions: a) what is a concept? and b) what is a procedure? John stated that
- Concepts provide access to relevant actions
- Procedures are a sequence of actions, organised by underlying concepts
In other words, concepts and procedures are interdependent, and mastery is the journey of layering concepts on procedures on concepts on procedures and so on. Most importantly, this journey never has a final end point: mastery is asymptotic (hello Mr Fitzpatrick!).
John then proceeded to punch through we delegates’ (assumed) mastery of ideas such as ordering decimals, counting, functions, and area/perimeter. He did this via a similar approach to Tony Gardiner’s earlier examples: start from simple principles, vary the initial conditions, run the exercise again, constantly challenging the human brain’s need to find pattern and run with it (to paraphrase John’s words), all the while paying attention to how these problems are seen.
John then moved into discussing his belief in the need for ‘didactic tactics’ – when students have completed a task, they shouldn’t put it down, but think how it can be extended? By varying conditions of a problem testing a skill, we are eventually diverting students attention away from the fact they are carrying out the skill, to the point where it becomes second nature – ergo, mastery. Mastery is focus on the goal, rather than on the process (which is automatic).
John closed by stating there are strategies for deepening appreciation and comprehension of concepts by:
- Enriching example spaces and methods of example construction;
- Refining personal narratives;
- Extending connections between pervasive mathematics themes (doing and undoing)
Ultimately, we were told (and in full agreement) that there are many ways to gain procedural fluency, moving along the path from cognition, to affect, and on to awareness.
I have to be honest I could have sat for hours listening to John and trying out his activities. The measure, honesty, practicality and rationalism that his ideas and methods demonstrate are quite someting to behold. There is no sell, there is no overconfidence – as he says, it is all conjecture, and if it works, it works. There is a great deal of thought that he and his wife Anne put into how we can test the true mastery of concepts and I want to know more. So I will be adding more of their writings to my reading list in due course, and my department have a lot to look forward to in terms of trying some of the ideas out in meetings, and hopefully their classrooms, in the not too distant future.
Robert Wilne – What does ‘mastery’ mean?
After Tony Gardiner and John Mason’s academic (but very important) viewpoints on mastery, it was now the turn of Robert Wilne to take on the baton, framing his findings very much in terms of what is going on in classrooms as we speak.
Robert started out by defining mastery in very simple terms: masterry is achieved if students
- demonstrate solid conceptual understanding
- are fluent in their methods
- can apply their mathematics to a range of applications
all whilst making connections to other facts and ideas. If at this point you’ve read the Cockcroft Report and thought ‘knowledge, skills and understanding’ then you will not be alone!
Robert stated that mastery does not refer to what the teacher does but what the pupil gets out of it. The teacher should work to ensure that all students are confident, secure, flexible and connected in their mathematical understanding. Robert was anxious to emphasise the word ‘all’: all students can perform well mathematically, given the opportunity; “don’t reify ability into character… only talk about attainment hithero”.
Robert has been heavily involved in the NCETM Shanghai project, and shared his findings, which were extremely relevant to the mastery concept. Lessons in Shanghai:
- have less chopping and changing of concepts;
- work through conceptual steps slowly;
- rapidly move towards increasing abstraction;
- have more questioning and focus on intelligent practice;
- provide interventions quickly to close gaps (same or next day);
- provide more time for pupils to discuss and improve on concepts.
All students made progress at broadly the same pace, and teaching, whilst not brilliant, was consistently good as a result of building lessons around these principles. Study was done through problems that had a richness and sophistication that did not require differentiation.
Teaching for mastery, Robert therefore conjectured, was based on four things:
- provision of good models and representations of concepts and procedures;
- offering procedural and conceptual variation in teaching and practice;
- in other words, predicting likely misconceptions by raising and resolving them;
- provision of intelligent practice through increase creativity.
Example thereof Robert shared in great detail, and it would be remiss of me to try and summarise these at this stage, but I am sure that Robert and his colleagues at the NCETM would be happy to share them.
Importantly, (developing on an analogy on kite-flying that the Shanghai teachers proffered) as learning progress, find out what the students have seen, and then establish if they’ve seen the right things. Make connections through intelligent practice, which is practicing the thinking process through more creative means.
Robert then moved on to how we can manifest these ideas in our own classrooms, based on what the Shanghai teachers suggested:
- Start with the topic, establish your objectives and how they’ll be achieved through the specific focus of your lesson;
- Have a quality textbook that tests procedural and conceptual variation;
- Use department meetings to develop ways of implementing intelligent practice;
- Use the knowledge of colleagues both in ‘real life’ and social media;
- Try things out and discuss the impact!
Now some people reading this will say, well duh, we knew this already, and did it need trips back and forth from Shanghai in order to confirm our thoughts? I have to say I have been skeptical about the whole Shanghai affair, however I do feel there are subtleties that have been shown in much of the teaching and learning that goes on which I do not think in this country we appreciate: particularly how conceptual and procedural variation play out in the classroom. I was glad to hear from Steve McCormack that work continues apace in transmitting and testing the impact of practice taken from the Shanghai project and I do hope that the subtleties Robert and his team have found do show up in teaching in this country, particularly in our quest for students working towards true mastery.
If Tony Gardiner and John Mason focused mainly on the ‘coalface’ actions of enabling the journey towards mastery then it is clear that Robert, along with Bruno Reddy, laid out how mastery can be facilitated for in terms of environment, culture, teacher development and planning. Both approaches are two sides of the same coin, and as you will see in the next part of my notes, it was down to Jeremy Hodgen to further tie these viewpoints together. Don’t change that dial…