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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
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posted 2 Jan 2016, 10:34 by Unknown user
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posted 29 Nov 2015, 04:03 by Unknown user
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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. 
