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) 
Inquiry and online learning  Inquiry as research

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)

Levels of Inquiry Maths As you gain experience in inquiry learning, you will often find yourself moving between the levels in the same lesson. You might give some students responsibility for developing their own lines of inquiry, while, at the same time, providing others with structure in a more directed inquiry.
* 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. ^{† }This phase is the principal feature of an Inquiry Maths lesson and is 'nonnegotiable'. For further descriptors of each level, see the Assessment Framework. 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 
Shanghai Maths: teacher led and student centred? In November 2015 the Shanghai Exchange reached the secondary school stage with maths teachers from China teaching year 7 and 8 classes in English schools. The maths hubs organised observations of the lessons, which were accompanied by an introduction to Shanghai Maths from teachers who had visited China in September and an analytical discussion after the lessons. The two teachers from the Sussex Maths Hub who ran the event I attended did an excellent job of reflecting on their experiences in Shanghai in an insightful way. Unlike the Schools Minister who writes that the exchange is about showing teachers Shanghai's “perfect formula for learning," they acknowledged the good practice that already exists in the UK. The two teachers from Shanghai also demonstrated a highly professional and critical approach when they analysed their lessons with the observers. It is indicative of the consistency of practice in Shanghai that the event did not teach me much more about the system than I had learned from attending a hub event during the primary exchange. In the post I wrote after that event, I characterised Shanghai Maths as teacherled and focused exclusively on mathematical concepts and procedures (as opposed to inquiry which can be studentled and involves reflection and regulation). However, since then, the organisers of the exchange have described Shanghai Maths as "teacher led but student centred." While this phrase is intriguing, it (or, rather, half of it) proves to be misplaced. If we take ‘studentcentred’ to mean that teachers adjust their teaching by taking account of childrens’ levels of understanding, then at no stage did the lesson I observed appear to be studentcentred. There was neither assessment for learning, nor assessment of learning. Questioning was focussed on getting the required answer and did not probe students’ understanding. Exercise books were treated as note books without any evidence of a teacherstudent dialogue. While it is easy to discuss a lesson in terms of what it did not contain, I have seen teachers in the UK threatened with competency proceedings for teaching lessons that did not include regular assessment from which to show progress. One fellow observer commented to me that a UK teacher might be in trouble if observed teaching the lesson. Interestingly, then, the Department for Education’s promotion of Shanghai methods might founder on the systemic demands it already makes of teachers through Ofsted and senior leaders. Of course, these considerations in no way count against Shanghai Maths, but they do remind us how difficult it will be to transplant the method into the English education system. The description of ‘studentcentred’, it transpires, relates more to the norms embedded in the culture of Shanghai classrooms. The exchange organisers herald such norms as “students commenting on each other’s work” and a “relentless insistence on pupils giving reasons." These were not evident in the lesson I observed. Indeed, the lesson was designed purely at a mathematical level with a focus on precise questions that aimed to develop understanding in small steps. There was no attempt to develop (and certainly not negotiate) what Professor Paul Cobb has called social and sociomathematical norms of classroom interaction.
Cobb and Yackel's examples of social and sociomathematical norms from the classrooms they researched.* It seems to me that if these norms are to become common in English classrooms, then some thought will have to be given to their development. Paradoxically, the Shanghai model might not be the best vehicle to do this because the social and sociomathematical seem to be taken for granted at the stage of designing the lesson. Rather, an inquiry model, in which students learn how to construct mathematical understanding, is better suited to achieving these norms. Even if students had been expected to give reasons in the lesson, Shanghai Maths still cannot be considered to be studentcentred. There is no acknowledgement of or adjustment for students’ different levels of prior knowledge, no alternative routes to understanding the concept, no encouragement of student questioning and certainly no opportunity for students to participate in the direction of the lesson as would occur during inquiry. Rather, Shanghai Maths is mathematicscentred or, it is more accurate to say, centred on a conception of the subject as a series of tiny increments in a logical progression. This definition is far from the idea of mathematics as a creative human construction that you would find in inquiry classrooms. The teacher has a script – an expertly designed script, but a script nonetheless. Shanghai Maths is, therefore, a teacherled, tightlycontrolled model of teaching. Andrew Blair November 2015 
Philosophical inquiry in maths classrooms
