Mastery – as problem solving

My last non-fiction blog looked at thinking and asked ‘should thinking be on the curriculum? Yes, it definitely should. I ended that last blog by relating thinking to problem solving. So let’s explore thinking and problem-solving as a further dimension within my mastery theme.

This week it is half term and my Year 3 (age 7 to 8) grand-daughter has been doing mental maths with me. The first thing I noticed was her confusion with certain words, for example: What is the sum of…. and ….? She did not recognise ‘sum’ as another word for ‘add’. It occurs to me that, even at Year 3, children have many different words for adding and subtracting, that include: sum, total, difference, altogether – or questions such as, ‘how much more/less/smaller/greater is ….. than …..? Consider this last problem in two ways. How much smaller is 14 than 25? How much greater is 40 than 23? Both sums are easy. But the subtractions are not obvious. Notice also that the numbers can be in a different order, causing confusion.

The second source of confusion for my grand-daughter lay in the wording of some questions: In half an hour it will be 11.00. What time is it now? We need to think backwards and subtract half an hour – half past ten. But again, this is not obvious. The first part of the problem uses the future tense: the second part uses the present.

The third puzzle lay in number lines that showed the calculation backwards. Most children can jump forwards using number lines as visual cues. Jumping backwards from the highest to the lowest number is a potential problem.

A fourth area of confusion was equations with missing numbers, because the thinking behind these puzzles involves inverse. Consider: 26 – …. = 19 and ….. + 7 = 13. What to subtract from what is not straightforward, I realised. The first subtraction is 26 – 19 to get 7. The second is 13 – 7 to get 6. Juggling with the order of numbers and (for the second) recognising the inverse, caused confusion.

Once she had grasped these nuances of language, this child was able to work out the numerical problems without difficulty. It strikes me that many children are basically okay with numeracy, and can do ‘sums’ at the expected levels, but when those sums are enveloped within word problems, it is a different matter. Confusion often reigns.

What do children have to think about when faced with a word based problem? Firstly, is the problem single or multi step? Is it add, subtract, multiply or divide – or a mixture of two or more? If multi step, in which order are the calculations? Many problems at Key Stage 2 involve a range of calculations, and at Key Stage 3, these are even more complex.

Yet, exam calculations are mainly disguised as problems, so young people must eventually master maths that is wrapped around words. So my message here – is for schools and parents to focus on helping children to master problems. The sums are the fillings. But they are wrapped in language.

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