Inquiry and mastery  part 2 Since my post about Mathematics Mastery in January 2014, the UK’s National Centre for Excellence in the Teaching of Mathematics (NCETM) has converted wholesale to mastery or, at least, to a variant it claims to have observed in China. This swing to Shanghai is the inevitable outcome of having a national centre funded by governments with ideologicallyfuelled visions of education. Nevertheless, the NCETM’s definition of mastery brings two features, which were not addressed in the original post, into sharper focus: firstly, all students move through and ‘master’ the curriculum content at the same time; and, secondly, mastery is achieved by ‘intelligent practice’. Charlie Stripp, the NCETM’s director, feels his organisation is justified in promoting mastery because the new National Curriculum in England is “explicitly a mastery curriculum”^{1}. However, in requiring teachers to teach “a detailed, structured curriculum” in “a precise way”^{2}, the NCETM’s mastery allows students no space to explore and inquire. Moreover, in his characterisation of the new curriculum, Stripp ignores the mention of 'enquiry' in the second of its three aims. Ensuring students move through the curriculum at the same pace sounds like a good idea. Indeed, Stripp links the idea to the development of a growth mindset because students will no longer be labelled as ‘mathematically weak’^{3}. An individual who understands a concept ahead of her peers will spend time deepening her knowledge instead of moving on; and, just as happens in Shanghai, students who have difficulty understanding a concept will have their issues diagnosed and remedied on the same day to ensure they keep pace. Leaving aside the obvious practical problem of UK teachers having a vastly greater teaching load than their colleagues in Shanghai (see Inquiry and Shanghai maths), there is also a theoretical problem with the NCETM’s prescription. Mastery is driven by the teacher’s skills to pose “precise” (that word again) questions and choose tasks that are “sequenced carefully”^{2}. This level of ‘scientific’ exactitude denies students any agency. It is the teacher who defines the pace and level of challenge; there is no opportunity for students to set their own. Such passivity, in which students have no responsibility for their own learning, is the opposite of a growth mindset. To achieve mastery, asserts the NCETM, students need to be involved in ‘intelligent practice’. As if to foresee the danger in this term, the NCETM immediately warns against “repeating a mechanical activity” in favour of students being “taken down a path where the thinking process is practised with increasing creativity”^{4}. Unfortunately, when it comes to the students, mechanical repetition is exactly their experience of mastery (see Debra Kidd’s Mastery Overload).  The one example (right) that the NCETM provides of ‘intelligent practice’ is rather curious^{5}. The NCETM writes: “By working though the calculations in the example, a pupil has to carry out the procedural operation of multiplication, but through connected calculations, has the opportunity to think about key concepts involving multiplication and place value”^{4}. I gave the calculations to a year 7 (grade 6) class of mixed prior attainment. The twentytwo students all attempted to answer the calculations. In discussion, they remarked on the need for the same number of zeros in the answer as there is in the question. No doubt, my questioning was not ‘precise’ enough to overcome this misconception about multiplying by powers of ten, but then the exercise itself is designed to lead students towards a conceptual understanding. I gave the same class this inquiry prompt (right). The focus is not on procedurally finding the answer, but on understanding how the two sides are equal. The quality and depth of reasoning immediately improved as students debated the meaning of multiplying by multiples of 10, 100 and 1000. Some began spontaneously to make up their own examples. No longer were students being “taken down a path”; they were beginning to create their own paths. The aims of the new National Curriculum are balanced to incorporate different forms of mathematical thinking, which, in turn, imply different forms of teaching. Stripp can take one passage to support his mastery interpretation^{1}, but, in so doing, he ignores others. The second aim of the three states: Ensure all pupils “reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language”^{6}. The NCETM’s definition of mastery is opposed to this aim. Students are served up tightlystructured lessons characterised by teacherled episodes and the completion of exercises. This is far from the inquiry processes envisaged by those who framed the second aim of the new curriculum. Andrew Blair May 2015
1. Stripp, C. (April 2015) How can we meet the needs of all pupils without differentiation of lesson content? How can we record progress without levels? 5. A couple more examples are available deep into this presentation.
