posted 29 Jan 2017, 12:27 by Andrew Blair
[
updated 29 Jan 2017, 12:28
]
posted 8 Jan 2017, 12:46 by Andrew Blair
[
updated 9 Jan 2017, 13:07
]
posted 11 Dec 2016, 07:39 by Andrew Blair
[
updated 11 Dec 2016, 22:17
]
posted 6 Nov 2016, 11:19 by Andrew Blair
[
updated 25 Nov 2016, 08:28
]
posted 27 Jul 2016, 14:40 by Andrew Blair
[
updated 27 Jul 2016, 14:41
]
New on the website: Inquiry and curriculum As Inquiry Maths becomes more widely known, teachers are asking how they can incorporate the prompts on the website into their schemes of learning. More broadly, they are asking whether inquiry classrooms that promote curiosity and student agency are compatible with covering the content of a mathematics curriculum. In response, we have created a new page called Inquiry and Curriculum that can be reached through the link on the menu bar or by clicking here.
 
posted 2 Jul 2016, 02:09 by Andrew Blair
[
updated 8 Jan 2017, 07:33
]
posted 2 Jan 2016, 10:34 by Unknown user
[
updated 16 Feb 2017, 13:05 by Andrew Blair
]
posted 29 Nov 2015, 04:03 by Unknown user
[
updated 13 Jul 2016, 13:15 by Andrew Blair
]
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 ‘childcentred’ to mean that teachers adjust their teaching by taking account of students’ levels of understanding, then at no stage did the lesson I observed appear to be childcentred. There was neither assessment for learning, nor
The lesson The 50minute lesson I observed was about multiplying fractions and involved a year 8 class (set 2). The teacher based the four phases of the lesson on twelve expertlycrafted questions, with each one increasing the level of complexity or introducing a new concept or procedure. The first phase was typical in combining a short episode of wholeclass interaction followed by practice. It started with ^{4}/_{5} x ^{2}/_{3}, which, after a very brief discussion, one student answered correctly. The teacher then showed a pictorial representation of the calculation on a 3 by 5 square grid, confirming the answer as ^{8}/_{15}. (In discussion with me afterwards, the teacher said he might have extended this phase in Shanghai by asking students to create their own representations and showed me the blank squares of paper he had prepared for this.) Students then practised by finding the answers to four more questions. This phase ended with the teacher presenting the general equation: There was no discussion of why q ≠ 0 and n ≠ 0. The next stage of the lesson was designed to teach students to cancel down before completing the calculation. The second of two questions used to show the value of this procedure (^{5}/_{48} x ^{24}/_{15}) led to one of the two extended wholeclass episodes (‘extended’, that is, in the context of this fastpaced lesson) because students did not identify 24 as a factor of 24 and 48. One student eventually arrived at 24 after the teacher had refused to accept other common factors. The third phase involved just one question: ^{3}/_{4} x 7. The students invariably gave the answer as ^{3}/_{4} because ^{3}/_{4} x 7 =^{ 21}/_{28} = ^{3}/_{4}. The teacher explained how this could not be true by using repeated addition. The second ‘extended’ wholeclass episode then started when students struggled to recognise that 7 could be written as ^{7}/_{1}, which would have allowed them to see multiplying a fraction by a whole number as a special case of the general equation. Each of the students’ suggestions was written on the board, but was not discussed, and was rubbed off when superseded by the next suggestion. With time running out, the teacher tried to initiate the fourth stage by introducing ^{4}/_{9} x 13^{1}/_{2}, but the bell went signalling the end of the lesson. In discussion, he told me that he had designed one more stage involving the multiplication of a proper fraction, a mixed number and a whole number. 
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 ‘childcentred’, 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 what Professor Paul Cobb has called social and sociomathematical norms about how to discuss and explain. 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 childcentred. 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 in inquiry lessons. 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
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. 
posted 14 Nov 2015, 08:21 by Unknown user
[
updated 2 Sep 2016, 10:55 by Andrew Blair
]
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. Firstly, we should heed the experience of two previous largescale European projects to promote inquirybased learning in maths. The PRIMAS and Fibonacci projects ran between 2010 and 2013 in 12 European countries. The €9 million spent on the projects went to universities, with nearly €1 million going to three universities in the UK. One problem with the projects was their aim to promote inquiry in mathematics and science. This led to confusion, particularly in the PRIMAS theoretical documents, about inquiry in the two subjects. The inquiry processes, which are different in maths and science, were conflated into one set of generic stages (see this article for an extended discussion). Another problem has been the low impact of the projects on classrooms. In the UK, the universities promoted materials they had already developed in a handful of oneday conferences with highlypaid marquee speakers and restricted audiences. The teaching community is left with some disparate webbased collections of resources and training materials that hardly anyone knows exist. Secondly, we should analyse the statements from the two organisations chosen to manage the project in order to understand their conceptions of mathematical inquiry. The Australian Academy of Science (AAS) and the Australian Association of Mathematics Teachers (AAMT) were invited to submit 'desktop reviews' of the current state of maths teaching in Australia before being confirmed as the managers of the new project. Their answers to question 2 are of most interest to us. The question was: What is the role of inquirybased pedagogy in the teaching of mathematics? The AAS used its answer as an opportunity to raise objections to inquiry as a legitimate pedagogy in maths: 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) This response is disappointing. Yes, inquiry teachers structure and guide learning (see levels of inquiry), but their ultimate aim is to develop independent inquirers who leave school able and enthusiastic to engage in open inquiry. The AAS define inquiry as problem solving in which the teacher 'engineers' the process from start to finish. Even more concerning is the AAMT document, which repeats exactly the same mistake made in the European projects; the document states that "there are clear parallels between science inquirybased approaches and contemporary thinking about pedagogy in mathematics" (p. 6). It then lists six generic inquiry 'principles': articulating goals, making connections, fostering engagement, differentiating challenges, structuring lessons, and promoting fluency and transfer. According to the AAMT, mathematics is "a practical vehicle for implementing these principles" (p. 7). Where are the processes associated with mathematical inquiry? Questioning, noticing, conjecturing, generalising, deducing and proving get hardly a mention in the AAMT’s document or no mention at all. The most interesting aspect of the project is its aim to base the inquiry resources on realworld contexts. I have written here about concerns at using 'real life' in mathematics classrooms, particularly at secondary school level, because it can inhibit the development of abstract reasoning. However, the Australian project has the opportunity to evaluate the role of realworld contexts in the inquiry learning of mathematics across phases of education. The Mathematics by Inquiry project is a great opportunity. It should seek to develop a distinctive approach to inquiry that is appropriate to the discipline of mathematics. Furthermore, the project managers should utilise the expertise that already exists in Australian schools, rather than, as in the case of the European projects, allocate the resources solely to higher academic institutions. In this way they will not only have greater credibility, but they will also have a greater impact.
Andrew Blair November 2015
Postscript Kath Murdoch (author of The Power of Inquiry and a primary teacher, international speaker and consultant based in Australia) responded to this article on twitter by saying that, "The focus should be on inquiry specific to the discipline and on generic, shared inquiry skills and processes." This comment opens up new avenues for inquiry. How are the generic and specifically mathematical inquiry processes merged in classrooms? Is the balance the same in different phases of education? You can follow Kath on twitter @kjinquiry.

posted 20 Jun 2015, 08:12 by Unknown user
[
updated 11 Feb 2016, 05:56
]
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 welldesigned 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 ‘overpitched’ 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 coconstructed 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

