Inquiry and mastery
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 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.

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}
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.
Five Myths of Mastery in Mathematics (December 2015) published by the National Association of Mathematics Advisers quotes this post in its analysis. 
The differences between investigations and inquiries When running workshops for experienced maths teachers, I hear the claim that Inquiry Maths is just another name for investigations. On one occasion, a teacher appeared exasperated as she accused me of "reinventing the wheel" and declared that "we've come full circle in maths teaching." Evidently, I had failed to distinguish between investigations and inquiries, but, more importantly, I had also failed to understand that the colleague remembered a time when the investigation classroom was very different to what we know of it today. The reason the investigation survived at all during this period, it could be argued, was its inclusion in the GCSE specification as coursework. However, coursework investigations became so structured by the requirements of the exam board's mark scheme that they fell into disrepute. No longer was a piece of coursework a reflection of a student’s independent thinking, but rather a reflection of how well the teacher knew the mark scheme. Coursework was scrapped in 2007. Investigation classrooms were not always like this. During my PGCE year in the early 1990s, I visited a maths department that taught the whole curriculum through investigations. It was one of the last, if not the very last, school in the country to do so. It was like no other department I have visited or worked in since. Students were allowed to investigate or not, depending on whether the teacher’s questions about a starting point had aroused their curiosity. They investigated individually, occasionally having discussions with the teacher, until they discovered the mathematical concept for which the starting point had been designed. There was no wholeclass instruction, which seems unbelievable today. This is how I imagine the classroom Marion Bird describes in her 1983 booklet on generating mathematical activity. Marion refers to the activities she uses as 'inquiries' and I would class some of them as inquiries in the sense I use the word today. Splitting decominoes, for example, starts with a diagram and, even though Marion sets an initial question, she allows the activity to develop into multiple pathways that encompass different forms of mathematical reasoning. However, another one of her activities  The greatest number of intersections  has become a classic investigation in which students have to draw more diagrams, tabulate results, identify a pattern and discover the generalisation. This is the inductive method of a science experiment in which more results confirm the hypothesis or lead to its revision. The exclusively inductive approach is not consistent with the combination of induction and deduction that characterises mathematics, and certainly not with the deductive nature of mathematical proof. Polya explains in How to Solve It and Mathematics and Plausible Reasoning how "deduction completes induction." While the mathematician finds an interesting result through plausible, experimental, and provisional reasoning, the result of this creative work is established definitively with a rigorous proof. It is the certitude given by a proof that makes further results unnecessary. (I have discussed the limitations of the inductive approach in an article from 2008.) The table below summarises the differences between investigations and inquiries as I see them.
Andrew Blair August 2013 (updated June 2016) 
Shanghai Maths: teacher led and student centred? 
Postscript: It was argued on social media that this post makes general comments about Shanghai Maths based on the observation of only one lesson. Even if this were true, another post about a different Shanghai lesson in southwest London shows that the lesson I observed is not atypical. The lessons were similar and, in some respects, identical. The criticism on social media might have originated in the fact that the analysis above ignores the hubs' preferred analytical framework. The hubs encouraged observers to study lessons in terms of ideas, such as concept/nonconcept and intelligent practice, that are said to underpin Shanghai teaching. However, the concepts of childcentred and social and sociomathematical norms have broader relevance and greater validity when analysing the introduction of a model of teaching into a new environment. 
Levels of Inquiry Maths
* The profiles of classes are not related to ability or prior attainment. In my experience, 'top' sets can show less propensity to inquire than 'bottom' sets. Indeed, students who have previously achieved in maths by successfully completing teacherset exercises can become anxious and even dismissive when faced with the challenges of open inquiry. They require structure or guidance just as much as bottom sets. The following articles have been influential in drawing up the level descriptors:
The work of Galina Zuckerman, particularly her concept of the "breakthrough group", has been important when profiling classes:
Andrew Blair January 2015 
Mathematics by Inquiry The Australian government has announced funding of $7.4 million for the ‘Mathematics by Inquiry’ project in an attempt to improve the teaching of maths from early years to year 10. The project, which runs from November 2015 to June 2018, will prepare and disseminate inquiry resources for use in classrooms across the country. Alongside the resources, the project will provide teacher training related to assessing higher order thinking and supporting inquiry in STEM contexts. The project represents a huge opportunity to develop ideas about mathematical inquiry and create a model of inquiry that can have a systemwide impact. However, in the spirit of critical inquiry, we should examine the project proposal more carefully. The term ‘inquirybased pedagogy’ is problematic for mathematics education and terms such as a problem solving approach or an investigative approach are more commonly used. The pedagogy of inquirybased learning is founded on the principle that students should be actively and socially engaged in the process of learning, constructing new concepts based on their current knowledge and understanding. Inquirybased learning, as described in the research literature, often refers to highly student driven approaches where the student decides the questions to ask, the research methods to use, and different learning occurs for different students. This very open studentled interpretation of inquirybased pedagogy has only a very small place in mathematics.  Instead the best investigative pedagogies for mathematics use ‘well engineered’ mathematical problems, where engagement in the problem solving process individually and with others and supported by the teacher will assist in the development of targeted concepts, or strategic skills, or the ability to transfer knowledge. (p. 11)
