Inquiry and mastery - the final part
In looking at the relationship between inquiry and mastery, the previous two posts in this series have tried to define the idea of ‘mastery’. Over the 18 months of the posts, a convergence between the two main organisations promoting mastery in England – 
Mathematics Mastery and the NCETM – has occurred. Take, for example, these two passages from their latest blogs: 
"Our Mathematics Mastery curriculum encourages ‘intelligent practice’ to enable students to develop conceptual understanding alongside procedural fluency. We use multiple representations to support this understanding and to encourage students’ reasoning. We’re also solving problems from the very start of the curriculum journey, not seeing it as some ‘add on’ that can only be considered when all the facts are in place. The challenge is developing these skills and understanding concurrently."1
(Ian Davies, Mathematics Mastery Director of Curriculum)
"Carefully crafted lesson design and skilled questioning encourage deep mathematical thinking in all pupils, helping them to identify mathematical connections and steering them to develop mathematical reasoning and problem solving skills. Exercises and learning activities designed to provide intelligent practice enable pupils to develop conceptual understanding, at the same time as reinforcing their factual knowledge and procedural fluency."2
(Charlie Stripp, Director of the NCETM)
While Mathematics Mastery emphasises multiple representations (although it is noticeable that the concrete–pictorial–abstract cycle does not get a mention) and the NCETM focuses on careful planning, their view of mastery is, for all intents and purposes, the same. Unlike those in the mastery camp who advocate fluency as a precondition for problem solving and reasoning, Mathematics Mastery and the NCETM both stress that these aspects of mathematics should develop together. This, however, is not new. As Mike Ollerton says, the blogs simply describe “the best practice which has been around for 40 plus years.”
So where does all this leave the relationship between inquiry and mastery? In his response to my second post, Charlie Stripp claims that inquiry can be incorporated into a mastery model:
“I think many of the tasks on Andrew’s website Inquiry Maths, and the example he offered in his blog post, are excellent examples of tasks that engage pupils in mathematical reasoning and in developing their own conceptual understanding, whilst also reinforcing their factual knowledge and procedural fluency. These 
tasks could easily and successfully be used by a teacher committed to teaching for mastery. I believe that carefully designed collaborative learning activities and inquiry tasks can be used as intelligent practice within the well-designed lessons that are central to a teaching for mastery approach to mathematics.”
Thus, not only do we see harmony between the main purveyors of mastery, but we also have an argument that subsumes inquiry under the mastery umbrella – or, rather, inquiry becomes a ‘task’ that can be made to fit the mastery framework. 
To me, this argument glosses over the fundamental differences between the two pedagogical models. Ultimately, the test of their 
compatibility can only come in classroom practice. 
For a view of the mastery classroom, we are indebted to Chris T (a year 3 teacher) who writes a hugely significant blog on his efforts to implement the "Shanghai way". Chris, who has laid out his classroom in rows “against all the principles I previously believed I stood for," describes how he now presents a problem and its solution at the start of a lesson. The pupils spend the remainder of the time explaining how to get from one to the other. However, for Chris, “the ‘problem’ has been the biggest problem.” If he selects a problem that is too challenging for his pupils, Chris reports that he can feel the atmosphere in the classroom change. Many teachers will identify with his experience as they recollect ‘over-pitched’ lessons of their own.
Chris's account highlights the key difference between mastery and inquiry. In the mastery classroom, students solve problems and reason mathematically within the teacher’s design; in the inquiry classroom, student design interacts with teacher guidance to develop co-constructed learning. By encouraging students to question and to devise a pathway in response to a prompt, inquiry avoids the potential for dissonance between the teacher’s plan and the students’ reaction to that plan. 
Mastery is scripted by the teacher; inquiry promotes students’ critical agency. This makes them incompatible.
Andrew Blair
June 2015

1. Davies, I (June 2015) Mastery – what it is…and what it isn’t!
2. Stripp, C.
(June 2015) Teaching mathematics for mastery at secondary school