Inquiry and mastery The most influential development in UK mathematics teaching over the last three years has been the appearance of the mastery curriculum, which is based on “the world leading curriculum and pedagogy of Singapore”.^{1} 'Mastery' sounds like memorising procedures that can be recalled and applied with fluency. However, it turns out that at the heart of the Mathematics Mastery programme, promoted by the chain of ARK Academies, is problem solving. If students know how to apply the knowledge they have learnt to an unfamiliar situation, ARK assert, then they demonstrate mastery.^{2} In order to solve problems, students must develop a conceptual understanding, learn how to think mathematically, and use language to scaffold thinking and communicate effectively. This leads to an integrated curriculum that goes beyond procedural routines and repetitive practice: The development of understanding, calculating and problem solving skills are not competing for time, but are developed simultaneously. The teaching of critical thinking and problem solving skills is embedded into the programme.^{1} The integration of these mathematical elements in a unified process is also the aim of the inquiry maths teacher. However, where inquiry maths departs from the mastery model is in the starting point of that process and in the nature of the process itself.
The inquiry maths teacher would welcome two of the three key features^{1} of a mastery classroom. The first – “more time on fewer topics” – is preferable to the predominant spiral curriculum in which topics are revisited regularly. In reality, the spiral often becomes cyclical as students revisit each concept at the same level, failing to develop a deep understanding in the narrow window of time given to them. In the mastery model, all students study the same content, broadening their knowledge rather than being accelerated onto higher levels. In similar ways to mastery, inquiry thrives on having an extended time to explore the links between concepts and involves students in developing a conceptual understanding from the same starting point. The second feature of mastery – “calculating with confidence and understanding why it works” – is incontrovertible for most teachers, inquiry or otherwise. Few teachers would argue against the aim of fluency combined with understanding. The third key feature, however, is problematic. In the mastery classroom, teachers are “always using objects and pictures before numbers and letters” (my italics) and employing a sequential “concrete, pictorial, abstract approach”.^{1} There are two problems here for an inquiry maths teacher, one related to the starting point of concept formation and the other related to the process of concept formation.
The starting point Superficially, the use of pictures to represent a concrete object sounds similar to the use of diagrams in V.V. Davydov’s curriculum developed in the Soviet Union (and popularised in the US by Jean Schmittau). However, for Davydov, the representation is introduced in the general form to children before they use it to solve problems. So students learn the wholepart schematic (left) before successfully solving concrete problems involving wholepart calculations. For instance, students using the schematic understand the requirements of a problem like the one in the box below. Their peers schooled on concrete examples look for numerical cues and regularly sum 7 and 11.^{3} Mary had 7 pencils. Joe gave her more pencils. Mary now has 11 pencils. How many pencils did Joe give to Mary? Thus, starting with an abstract schematic provides students with a tool to represent the specific relationship in a problem. According to Davydov, starting the process of concept formation with concrete objects leads to an impoverished understanding of maths. The concept becomes a hollow abstraction divorced from the richness of the contexts and activity in which it originated.^{4}The process The concretepictorialabstract process is not unique to the mastery programme. Ironically, it is precisely the one described by Bruner, a proponent of inquiry: learning sequences begin with concrete "instrumental operations", which "become represented in the form of particular images" before, finally, the learner
 "comes to grasp the formal or abstract properties" of the concrete.^{5} In short, pictorial representations of concrete situations are internalised to form the basis of abstractions. In a paper published in 1931, Vygotsky discussed the role of pictorial representations in the formation of abstract concepts. He reported on a characteristic feature of his psychological experiments: secondary school students could use concepts in visual situations, but they found it far more difficult to generalise by using the concept in other situations: At the time when [the] concept has not yet become detached from the concrete, visually perceived situation, it is able to guide the adolescent's thinking perfectly. The process of transferring the concept, i.e. applying it to the other, completely different things, proves to be a lot more problematic.... When the visual or specific situation changes, the application of a concept which has been worked out in a different situation can become extremely problematic.^{6} Even though Vygotsky conceded that the student learns to apply concepts consciously "in the end", visual thinking tied to a concrete situation can obstruct the formation of abstract concepts. Davydov overcomes the obstruction by reversing the movement from concrete to abstract that occurs in the mastery classroom. His research (and that of Schmittau) shows that students' conceptual thinking is far more sophisticated when learning ascends from the abstract to the concrete. Inquiry maths lessons follow a similar path to that of Davydov’s curriculum. Decontextualised inquiry prompts force students to apply their existing and new knowledge to 'fill' the prompt with meaning. As students develop their understanding of an abstract starting point, they make it concrete through their own reasoning and activity (including requests for instruction when required). In this way, students enrich their understanding by applying concepts in new contexts and linking them in new ways. Posing problems For the ARK schools, students gain mastery of mathematics by “creating problems” and “asking and investigating great problems”.^{2 }Yet, how can this happen with a prescribed sequential approach that is always employed? If a student’s question is not susceptible to pictorial representation, then presumably the question is not allowed. If the student wishes to investigate in an alternative way to the accepted sequence, then, again, presumably that is not permitted. These seemingly peripheral suppositions strike to the core of the mastery model. Students ask questions when they feel the questions will be treated seriously and acted upon. When they have no role in deciding how to answer the questions, as they do in inquiry classrooms, students will be discouraged from asking. Thus, an inconsistency exists in the ARK model between the learning sequence and students' role in problem creation. If posing problems is as much at the heart of the mastery curriculum as the authors assert, then the “concrete, pictorial, abstract approach” to learning cannot be the straitjacket it is presented as. Alternatively, problem solving is tacked onto the end of the mastery sequence in much the same way as it is in many conventional classrooms.^{7} Either way, the development of understanding and problem solving skills is not the “simultaneous” process that Mathematics Mastery claims. The inconsistency between the learning process and problem solving is resolved in the inquiry classroom through the mechanism of student regulation. Students pose questions, decide on the approach to solve them and determine when new concepts or procedures are required to make progress. The teacher suggests contexts and introduces new mathematical representations to develop the inquiry. The Mathematics Mastery programme, like the National Numeracy Strategy before it, is done to students; inquiry maths is coconstructed with students.
Andrew Blair January 2014 1.http://www.mathematicsmastery.org/wpcontent/uploads/2012/05/MathematicsMasteryInfoSheet.pdf 2. http://www.mathematicsmastery.org/secondary/depthofunderstanding/ (This link has now been deleted by Mathematics Mastery. The new page claims only that: "Problemsolving is integrated throughout every lesson." AB February 2015) 3. Schmittau, J. (2005). The development of algebraic thinking: A Vygotskian perspective. ZDM, 37(1), 1622. 4. Davydov, V. V. (1990). Types of Generalisation in Instruction: Logical and Psychological Problems in the Structuring of School Curricula. Reston, Virginia: NCTM. 5. Bruner, J. (1967). Toward a Theory of Instruction. Cambridge, Mass.: Harvard University Press. (p. 68) 6. Vygotsky, L. (1994). The Development of Thinking and Concept Formation in Adolescence. In Van der Veer, R., and Valsiner, J. (Eds.), The Vygotsky Reader. Oxford: Blackwell. (p. 253) 7. http://webarchive.nationalarchives.gov.uk/20141124154759/http://www.ofsted.gov.uk/resources/mathematicsmademeasure
