How Do I Understand Addition? Let me Count the Ways
How would you teach a Year 1 class to add 5 to 27? How much time would you devote to teaching it? If you knew that some of the children were already capable of getting the correct answer to that sum, how would you extend these children mathematically?
These were the questions that teachers were prompted to think about when observing Shanghai teacher Du Juan, teaching a Y1 class on the ‘return leg’ of this year’s England-China teacher exchange. More than 60 teachers from the South Yorkshire Maths Hub area observed Du teach a Y1 class in the school hall at Hayfield Lane Primary School in Doncaster.
Du poses the addition question to the class in the form of a problem:
There are 27 eggs packed into boxes of ten. There are five eggs laid by the hen in the henhouse. How many eggs are there altogether?
* helpfully, eggs are sold in boxes of ten in China!
“27 + 5”, the calculation required to solve this problem, is what Du spends the main part of the lesson addressing.
It’s not that she doesn’t realise that some of the children can do this calculation before the lesson even starts. She understands that being able to do the calculation is only the beginning, particularly as it rapidly becomes clear that most children use a ‘counting on’ method. The egg box picture lends itself well to encouraging children to see the value of partitioning the five eggs into three and two, but she also demonstrates, and encourages children to try it with multilink cubes. She shows how a model that illustrates counting-on:
Can be shown in a way that illustrates partitioning:
As well as the concrete cubes demonstration and the pictorial illustration using the egg-box, Du uses a couple of different ways of writing the calculation to show the partition - see these animated PowerPoint slides.
Teachers express their surprise, in the discussion afterwards, that Du even addresses the possibility of partitioning the bigger number (in this example, 27 into 25 and 2).
What we see in Du’s lesson is not the sort of polished perfection that a teacher from this country might hope to put on for an observation. The Y1 children, in the high-pressure situation are stage-struck and fidgety. They find the teacher’s accent and use of language a barrier, and the expectation to sit still in rows of desks for 40 minutes means that they lose concentration at times. But it seems that the local teachers observing have moved beyond expecting to see a lesson they can copy in their own classrooms. Instead they are successful in pulling out what is impressive from Du’s lesson – the meticulous planning of progression, the relentless focus on a single calculation and the many different ways that it can be looked at to help expose the mathematical structure of adding over the 10s barrier.
After each showcase lesson, Maths Hub staff have organised a discussion amongst visiting teachers. Teachers in Doncaster were asked what they would take away from the lesson they saw. One was particularly impressed by how the children were encouraged to ‘play’ with the numbers, rather than simply being shown a method and repeatedly asked to practice it. Another seemed to have had the idea that “mastery’ is ‘the extension work that the top kids get to’ challenged – she left saying that she liked the idea of starting with a problem that all children could be encouraged to solve. Others were pleased to see the ‘ping-pong’ lesson style (children working in short bursts interspersed by frequent teacher input) in action and resolved to try more of this in their classrooms.
Background to the England-China Teacher Exchange
2016/17 is the third year in which English and Chinese teachers have taken part in a classroom exchange. First, English teachers have spent a fortnight in Shanghai schools learning about Chinese maths teaching pedagogy. Shortly after, Chinese teachers have been welcomed into their exchange partners’ schools to teach a class of English children for a fortnight. The Maths Hubs programme website has more about the 2014 England-China Exchange (involving primary schools) and the 2015 Exchange (involving secondary schools).
One of the purposes of the return leg of the exchange has been to provide first-hand experience of Chinese teaching techniques for many more English teachers. Classrooms have been opened up to small groups of teachers, for observation and lessons have been analysed afterwards. Each Maths Hub has also been holding two large showcase events where a lesson is staged in the school hall, observed by up to 100 teachers.
For a flavour of a Y6 lesson given by a different Chinese teacher (in a Surrey Plus Maths Hub school), and of what most impressed teachers about it, listen to teachers talking in this six-minute video.
The bravery of the Chinese visitors, some of whom have never left Shanghai before, has been striking to all the local teachers. Teachers returning from Shanghai suggest this is a result of a very different culture of lesson observation in China. Teachers are regularly observed by colleagues, as a basis for ongoing collaborative lesson development. Observations are weekly and formative, rather than less often only to be graded. (Indeed many Shanghai schools have classrooms with one-way viewing windows so that groups of teachers can observe lessons without influencing them). Implementing more formative peer observation is something that many schools and Maths Hubs in England are now experimenting with, but many English teachers remain shy about opening their doors. One head teacher, visiting a showcase event in the London South East Maths Hub area commented:
‘I was so impressed when Lilly (the Chinese teacher) stood up at the end of her lesson and asked the hall full of teachers, for comments. She said “This was not perfect – tell me how I can do better”. This is not something you would see teachers in this country do – it’s not part of our professional culture.’
No teacher can adopt teaching for mastery from seeing one showcase lesson from one Chinese teacher teaching a class of English children she doesn’t know. But the events have given Maths Hubs the opportunity to raise awareness and interest in other opportunities to learn about teaching for mastery – opportunities to see local teachers who have adapted this approach successfully in their own classrooms, and opportunities to engage in professional development where teachers support and develop one another rather than being talked at by an expert. These opportunities are available from Maths Hubs all over the country. To find out what is happening in your region – contact your local hub.