Inquiry and Shanghai maths
Last week I attended the launch of one of England's 34 maths hubs, which are part of a government attempt to improve maths teaching. Each hub is based in a school and aims to develop a regional network of schools that embrace hub initiatives. The initiative that maths teachers are hearing most about is the England-Shanghai exchange. Despite regular assurances from hub speakers that this is an exchange of ideas, the flow has all been one way to date with Shanghai primary teachers currently giving model lessons in UK schools. I suppose this is to be expected as the UK finds out more about the methods used in PISA-topping maths classrooms, but an exchange of ideas implies we keep an open, critical and inquiring mind in our engagement.
Unfortunately, the lead organisation that has benefited from the government investment seems to have lost its critical faculties with its on-message response to the Shanghai teachers. The Chinese teachers themselves, however, seem to have exactly the inquiring approach that has the potential to develop both systems. At the hub launch, for example, we were told by a UK teacher who had visited Shanghai that teachers and researchers she had met there were interested in developing inquiry-based learning. Furthermore, the Chinese teacher who also spoke at the hub told us of China's concern that it has "very talented young mathematicians, but no Nobel Prize winners" and, as an aside, of his regard for the mini-whiteboards he had seen in UK lessons.
So how have Chinese teachers developed an inquiring and critical approach? Shanghai maths is an "evolving research-based system" centred on the universities, which have close links with teachers and schools. Teachers give
two 35-minute lessons a day (three lessons on a "bad day") and spend a part of the remainder of their day in discussions with their colleagues about how to teach concepts to particular classes. They also engage with current theory by carrying out research projects. This key professional development aspect to Shanghai maths is unlikely to be implemented in the UK because of the shortage of maths teachers and the government's fixation on isolated features of the Shanghai model, such as textbooks. Here, it is worth noting that the Chinese notion of a textbook is different to that in the UK - textbooks are "just as much a professional resource for teachers as they are a learning resource for pupils" (from the NCETM's mid-exchange report).

At the hub launch, we were shown a demonstration lower primary lesson. It was tightly structured and broken down into "deliberate, small steps". The same 
concept - repeating patterns, in this case - was presented in different contexts. In typical fashion, the lesson started with a common everyday example. The teacher presented paintings of scenes from the four seasons (summer, autumn, winter, and spring), discussing how the sequence continued, its order and what the pattern would be with different starting points. He then moved on to coloured balloons in cycles of three and five balloons: How would the sequence continue? What would be the colour of the 33rd balloon in a cycle of five? Why is it the same colour as the third balloon? What is the colour of the 105th balloon? The lesson ended with a word problem that again related the concept to a familiar context: If on Thursday it was ten days before I returned to China, on what day of the week would I leave? The audience was reminded that the demonstration was accelerated from a real lesson. Nevertheless, it took 10 minutes to talk through the slides, so we can imagine that, in the classroom, this is a rapid process of quick-fire questions. 
I was impressed by the meticulous planning and the depth of thought that had gone in to mapping out the development of a concept in the minds of young children. The focus on cognitive processes involved in learning mathematics is an important lesson for all teachers. 
However, from the viewpoint of inquiry, there were a number of serious omissions. Firstly, pupil questioning and conjecturing, through which mathematical thinking develops, were totally absent. Secondly, pupils are not expected to think at a metacognitive level at all. So, there was no reflection on what had been learned. (A teacher described to me how in a Shanghai lesson he had observed in a UK primary school, it was only at the end that a pupil exclaimed, "Oh, so that is what the lesson was about!" That individual realisation seemed almost accidental and was not common to all pupils.) Nor was there an opportunity for students to regulate their own learning by deciding on which direction to take. So reflection and self-regulation, both of which have emerged from research as key characteristics of successful learners, are absent from Shanghai maths.
Finally, at the hub launch, we were informed that a popular saying in Shanghai maths classrooms is "the answer is just the beginning." The implication, for me, is that what follows from the answer is a structural analysis of mathematical methods. This might be an advance on mathematics lessons in the UK, but it is not an advance on inquiry classrooms. While the cognitive approaches evident in Shanghai classrooms offer important tools for focussed teaching episodes when students request instruction during inquiry, the absence of an intertwined metacognitive level restricts students' learning. Thus, the Shanghai model as a whole, in lacking opportunities for reflection and self-regulation, is deficient.

Andrew Blair
November 2014