Thursday, 5 May 2011

How do the Chinese do it? 30 minutes in. Reciprocals in Calculation.

SDPQ 3,9
SDPQ 2,5
SDPQ 1,7
Each is a 5-10 minute starter.
Watch as you and your class discover and explore the fundamental structures of mathematical calculation.
One of the most surprising things which reliably emerged with my classes was their insight into reciprocals.

They're always looking for short cuts so they rapidly spot the links in the results for division:
4 and 1/4
3 and 1/3
2/5 and 5/2
7 and 1/7
They're reciprocal pairs.  Even though they only get half a mark for a second quotient calculated in this way, they still spot the pattern.  So my students have a strong familiarity with reciprocal pairs and an insight into their relevance for multiplication and division from early in year 7 (age 11) when I first teach them.  I'm sure that younger students could cope with this too.

It's so easy and natural but I've never seen this understanding in other classrooms. Usually in the UK reciprocals are taught as a disconnected entity at GCSE level or are found in algorithms, again without their connected context.  Yet a structural understanding of them is clearly there in the Chinese classrooms described by Liping Ma.  Chinese teachers use them fluently and flexibly for calculation.

Of course I'm not saying that the Chinese teachers taught this in the way I did.  I'm just trying to convince teachers that it is possible to replicate significant aspects of the Chinese teaching strategies with classes of students and teachers who have never thought structurally about calculations before.  Oh and I always taught in tough schools.  No docile classes for me.  And I found that students were more settled if I taught them in this way.  It suited them to focus on structures rather than to learn recipes.  They felt more in control of what they were doing and their education became more relevant to them.  They weren't good at 'learning recipes'.

There's more to come....


  1. Forgot to ask before but where did you get this idea from? SDPQ, very neat, very simple. My only worry is the lack of open-ness in the question. There is only one correct answer to each. How do we modify it for the brigther ones, the quicker ones, how do we help the weaker ones access it?

  2. Don't set pairs with trivial answers and the activity will easily last five minutes and seem more open than you expect. Please let me know if it doesn't turn out that way and we'll work out how to fix that.

    My classes were mixed ability and it lent itself easily to that so don't worry too much about differentiation in advance, just give it a go and adjust the difficulty as you see fit. Let us know how it goes!

    It was my original idea - just in response to students' relying on algorithms they didn't understand and easily forgot rather than structures which allowed them to puzzle things out more or less even if they'd forgotten exact methods.