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One of the many pleasures in the life of a teacher is observing and learning from other teachers. We’ve just returned from a lovely morning at Thorndown Primary School, St Ives, where we were fortunate enough to observe two teachers taking part in another round of the NCETM England-China teacher exchange.
Whilst definitions of and arguments about mastery abound, we thought it was worth sharing some of what we observed in our brief visit. We saw two very interesting lessons, one looking at learning the four times table with year two and the other a year five class learning to subtract fractions with the same denominator. The pupils were wonderful and coped incredibly well with 50+ adults sitting behind them in the school hall observing their every move and sound. The two teachers, Mandy Zhu and Noah Chen were consummate professionals who delivered fascinating lessons in their non-native tongue.
Language and vocabulary
One of the biggest things that struck me about both lessons was the precision with which the teachers and pupils used language. This was not just in using technical terms in normal conversation, such as product, factor, numerator and denominator, but the way in which solutions were constructed and explained.
For example, when discussing how to build the four times table pupils drew on previous knowledge in trying to work out 3 x 4 (three groups of four). The link was made to 4 x 3 (four groups of three) and pupils realised that the two calculations gave the same answer because of the commutativity of multiplication. Yet every time the teacher was talking about the four times table she insisted on groups of 4, hence it was always 5 x 4 or 6 x 4. A link could be made to other tables but with a clear distinction that this was the four times table. This careful construction came into its own when pupils began considering contextualised problems such as, ‘A car needs four wheels, how many wheels will 5 cars need’? The calculation involved groups of 4 and you needed 5 of them hence you were trying to calculate 5 x 4, five groups of four. This structure enabled pupils to make more sense between the concrete cars and their wheels and the abstract calculation 5 x 4. They saw a reason and were able to explain what each part represented.
Similarly, during the fractions lesson it was imperative that the pupils talked about not just one but one whole. At one point the teacher talked about 7 one ninths subtract 4 one ninths equals 3 one ninths, the importance of the units in which the problem was set (ninths) being imperative. The concept of units is something that is coming up again and again in our curriculum development work – see here for more discussion of this.
The progression of learning was carefully designed to ensure future learning was not hampered and was in fact supported. For example, during the fractions lesson the following question was asked:
Linda has a book. On the first day she read 2/7 of the book. On the second day she reads 4/7 of the book. How much of the book does she have left to read?
One pupil suggested that the solution was 7/7 – 2/7 – 4/7 = 1/7
Now I imagine most of us, myself included, would be very happy with this calculation. However, the teacher picked up on the 7/7. Where had this come from? Why did the calculation start with this? Where in the problem was the 7/7?
It transpired that pupils were expected to write every line of their calculation down carefully to begin. The question included the information that Linda had one book. Hence the first line was then 1 – 2/7 – 4/7. A conversation then took place about changing the one whole into sevenths. Why sevenths? Why not fifths? Ninths? The idea that it had to be sevenths to allow the subtraction of sevenths emerged – this seems obvious to us but it really exemplified how the teachers expected pupils to understand each and every step of the process.
So the calculation became….
1 – 2/7 – 4/7 = 7/7 – 2/7 – 4/7 = (7-2-4)/7 = 1/7
This small change wasn’t something that I think I would have picked up on, but considering it in depth and the conversation that followed highlighted its importance.
These lessons were obviously carefully designed and prepared: thought for language, continuity in learning and setting the foundations of understanding were key. The two lessons demonstrated that no matter what stage in your teaching career, no matter the phase in which we teach, you can always learn from observing exceptional practitioners at work.