First off: thanks to the MTBoS team for inspiring me to have another go at blogging and not being scared to write and share even though I do feel uncharacteristically shy about it!
I really enjoyed working on the problem; my solutions to part 1 (find any way of making 1000 with unlimited 8s and addition signs) is here:
And my solution to part 2 (find ALL the ways of doing so) is here:
I decided to be brave and make a video, safe in the knowledge that most people in maths blogging/tweeting are delightful and hopefully can ‘hear past’ my odd accent (I went from ‘very English’ to ‘very American’ to ‘Irish Yorkshire mash’ in 3 attempts to make the video; I don’t know HOW my pupils understand me).
This brought me back to something I’ve been pondering this week: how can time for deliberate, sustained practice of a routine but important skill – like addition and subtraction – be reconciled with the fact that some pupils have clearly mastered the skill and are making no gains in those lessons? I know from baseline tests that almost all the classes have a considerable chunk of KS3 pupils who need to work on it and are making excruciating errors, often not even aware of how badly off they are.
Possible approaches and thoughts on them are thus:
a- Reteach it to everyone, often using the more able pupils to model the methods, then having the pupils who can do it supervise the ones who can’t (i.e. giving more individual attention to the ones who can’t do it yet). I eventually decided against this as it felt so unfair on the ones who could do it – they were unlikely to learn much, if anything, during that lesson, would be horribly bored, and weren’t necessarily going to be good ‘coaches’ for the strugglers. Pupils at my school are great, and would try hard to be helpful, but even I find it a challenge to be patient when a pupil is working through something slowly and deliberately. This also only afforded one lesson of practice to those who needed it, when this skill is definitely one honed through practice over time.
b- What I did: Use interesting puzzles, that rely on ability to add and subtract so that there was cognitive challenge for the ones who didn’t need practice, and there was ample practice for those who needed it. The puzzles included: 1089 puzzle (although a missing caveat is that the first and last digit should have a difference of at least 2), Eight 8s, “Reverse 1000” (as I dubbed it) and some other bits and bobs. The pupils really enjoyed this, as the puzzles allowed them to really think about place value and how to work within constraints, whilst it also freed me up to help those who needed it. However, it doesn’t create enough practice over time,
c- Considering doing: For pupils who need to practise written calculations, it will make up a small part of their homework every week for the next four weeks (ideally in the form of doing two questions an evening for the month, so that it is small regular bursts of practice, not a horrible task where they’re tempted to cheat and use a calculator). This is a bigger ask of pupils; the school – and I agree – that pupils and staff should have a ‘whatever it takes’ mentality, even if it means working harder or for a bit longer. Most of our pupils buy into this and want to succeed, but I am always wary of encroaching on their unstructured time (especially as many have several hours of Mosque per week also, which is additional structured time that makes big cognitive demands on them).
d- What I (also) did: Teach all pupils an alternative/additional approach that can be used, so they have a broader range of strategies for different types of calculations (see below). For example, column subtraction with ‘borrowing’ is not that easy when the lead number has a lot of zeroes in it (e.g. 2,000 – 324 becomes very messy very fast). Furthermore, for subtraction, some pupils persist in finding the ‘difference’ in each column, instead of regrouping, if they use column method. I find this interesting as all these pupils ‘get’ that it is incorrect, and can (sort of) explain why, but have a deeply embedded ‘bad habit’ that they can’t seem to break. In light of this, I insisted that they practise using a different method that could (at least) be used to check results.
Incidentally, the specific problem of some pupils’ failure to grasp and master subtraction was referenced in Hegarty Maths this week, where Colin suggests an alternative method that side steps some issues. I have my own reasons – and I’ll elaborate in the comments of Colin’s post – why I don’t plan to teach it to my pupils, but I’ve recorded the methods I taught to my own pupils so that you can share your own opinion on these. A second video on them is here.
Of course, in day-to-day I expect most of my pupils will use calculators, but I still want them to master written calculations as doing a calculation ‘by hand’ can reveal a lot of patterns within a problem and unlock understanding of place value really nicely.
From your own experience, what strategies work best to support pupils who are still struggling with written calculation in secondary school, and what subtraction strategy do you find most intuitive or elegant?
Thanks for reading – have a good Sunday:)