Sophie (not her real name) is a Year 6 student who does not find maths easy. Each week, she comes to see me and I always ask her what maths she is learning at school. One week, she described what I believed to be subtraction as addition in reverse, i.e. starting with the smaller number and working out how much more is needed to make the bigger number. What she wrote down wasn’t correct, but I was able to decipher it and got her computing differences correctly by writing them out as jumps along the number line. It helped her a lot and she was quite happy with that.
Thinking of subtraction as addition in reverse, and viewing these sorts of calculations on the number line, is a very worthwhile strategy. However, the following week, Sophie wrote down something which I can only guess was meant to be “Adding in Parts”. It was difficult to know. She drew a fork underneath one of the numbers but simply didn’t know what numbers to write.
In case you don’t know, Adding in Parts is one of the main additive strategies that over-complicates things for children. One number is split up so that the other number can be made up to (nominally) the next “tidy number”, then you add on whatever is left. For example, 27 + 5 = 27 + (3 + 2) = (27 + 3) + 2 = 30 + 2 = 32. Implicitly, this requires computation of the next tidy number (30), two subtractions (30 – 27, 5 – 3) and a new but hopefully easier addition (30 + 2). This type of (“part-whole”) thinking is highly regarded in the Numeracy Project, and is notably the glorious hoop that students must jump through to gain permission to learn/use column addition at school. Sadly some students never quite make it. It’s not a hoop, it’s a barrier. If you really want to understand how stupidly inefficient and cumbersome Adding in Parts is, try programming a computer to perform addition in this way.
I explained to Sophie what I thought needed to be done, but it didn’t ring any bells, nor did the task get any easier for her after numerous attempts. I admired her determination to master this strategy, but it upset me to see her so close to tears. All the while, I tried to reassure her that she really didn’t need this strategy! Sophie is perfectly capable of adding numbers using column addition and she executes the method well. There is nothing to be gained by her taking twice as long, struggling (or even failing) to add two numbers using this highly overrated strategy. In the end, I managed to persuade her to stick with column addition and she agreed that she felt better using this method.
With Sophie’s permission, I wrote to her mother and recommended that Sophie should not continue with Adding in Parts at school at this point in time. If it was deemed absolutely necessary for the purpose of assessment, then perhaps she could come back to it later, but it was harming her mathematical development and confidence to persist with it now.
Sophie’s mother arranged a meeting with Sophie’s teacher, and this is what happened:
“Our meeting went well. Sophie’s teacher listened and took on board everything we said. She sympathised with Sophie as we talked about her difficulties learning maths. I think the teacher is now going to use some of your strategies with the class. She is just so thrilled that we are working together and backing each other up.”
I am equally thrilled. Maybe the whole class will benefit. Hats off to Sophie’s teacher. I hope other teachers will follow suit.
Dr Audrey Tan, Mathmo Consulting