Subtraction Strategies: The Problem of Mixing Practice and Challenge

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 decided to do the Week 3 MTBoS Challenge, and worked on Challenge 6 from Collaborative Maths.

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:

pieces of eight - part 1

And my solution to part 2 (find ALL the ways of doing so) is here:

pieces of eight - part 2

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.

balancing method 1 balancing method 2 borrowing method 1 borrowing method 2 number line method 1 number line method 3 number line method slow and steady

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:)


Filed under Interesting or Fun

7 responses to “Subtraction Strategies: The Problem of Mixing Practice and Challenge

  1. I’ve used a blend of b, c, and d. My approach to extension is one of depth not acceleration, and so students who already meet the core objectives get extended through problems and challenges – which is far more fun and develops a far greater love of learning than does being given a different worksheet, or acting as a permanent TA. Using low threshold high ceiling tasks that ‘trick’ students into lots of practice as part of a bigger challenge is the dream! Either way I find you need c to build up sufficient volume of practice. Approach d is really just a reworking of approach b. Understanding a variety of methods is just deepening your understanding of the underlying concept, and its the concept, not the method, that we’re trying to make stick.

    On the calculator issue you mention a few times, have you seen QAMA calculators? They’re my new best friends.

    • QAMA – no, but I keep hearing about them and will have to look now.

      I agree with what you say re: depth v acceleration (we have massively slowed down the KS3 curriculum – painful to make so many cuts, but we think it’s working). My concern is at the stage where about 8 pupils need to spend 2-4 lessons of learning and practising, and most have done about all the exploring and conceptualising they possibly can by the end of lesson 1 (lesson 2 at a push). It’s at that point I get really stuck – if it’s new content, I’m pushing them ahead and will still be trying to catch the others up, if it’s more and more puzzles I’m soon not letting them think or do anything new. Such a bind! Thankfully it’s mostly restricted to written calculations.

      What have decided to do less of in order to privilege depth over acceleration?

    • I found the QAMA calculator ad on line last year but could not wrap my head around the funding needs for a classroom set. Should I revisit my budget? Looked fascinating for our middle school kids.

  2. Very interesting topic! This reminds me of a seminar I attended at ICTM last year. The speaker asked us to solve a simple addition and subtraction problem in our heads. When he polled how many of us computed it the “traditional” way we were taught in school, very few hands were raised. As we mature in math, we learn to manipulate the numbers in the best way possible for that situation as opposed to performing a set of rules. While this can be very challenging to teach, it is the ultimate goal. We want students to be able to understand the relationships between the numbers, not just how to solve specific problems. This never dawned on me until my first year of teaching and I am learning ways to help my students understand this reality each and every day!

    • That’s interesting – I was still totally stuck in using column algorithms in my head (basically acting as my head was a whiteboard that I could watch – NOT EFFECTIVE!) for +,-,x until I started teacher training. I had no idea that ‘counting on’ could be used for so many calculations, and is so useful for mental maths.

  3. What a terrific post – loved the first video. How much time did it take you to work through that solution? How much time might it take a class of students? I wrestle with posing problems that are (a) maybe too easy for my best students AND (b) probably too complex for my least students. It is hard for me to find a way to feel good about an activity that will take a great deal of time and may result – simultaneously – in some students feeling unchallenged and others feeling overchallenged. I know that there are smart ways to do this, I just have not internalized them effectively yet.

    • Hi,
      Thanks for the comment. I have the exact same concerns; so much so that I did time myself, and it took me about 9 minutes to do the whole thing, including deciding how to set it out/show my ideas. My rule of thumb is that, for the strongest kids, it’ll take them about 50% longer (so it’ll take them about 12-15 minutes) and that the weakest take about 3 times longer (although that may still include support/scaffolding from me).
      I totally empathise with the problem of ‘spreading challenge’ for the range of ability in a class. In particular, if there are too many challenges/activities/elements, I get spread too thin, become confused about who’s doing what, and it’s too messy to wrap up learning at the end of the lesson. I’m always on the lookout for ‘low floor/high ceiling’ tasks that combine the full spread. Nrich are quite good at designing these, and I find that generally reading around (e.g. the blog links of other teachers) can really help. I wish there was some sort of handbook of ‘safe starting points’ for all topics though, so that you knew a lesson could at least be solid while you contemplate going off piste!

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