14 April 2022

Intro-ducing and outro-ducing methods

Welcome to my first blogpost as President of the Mathematical AssociationI am aiming to post to this blog every other Thursday during my year as President, and to address a variety of issues that will hopefully be of interest to MA Members and others across the whole range from early years up to university. That is a tall order, and means that I won't always know what I'm talking about, so please engage with the blog in the comments underneath, put me right when I'm talking nonsense, and make this a conversation. I will try to encourage this by being a bit provocative and controversial at times!

So, let’s get started. And thinking about how to introduce the first post of the blog got me thinking about how teachers introduce methods in mathematics - and also particularly the opposite: namely, how we help learners to move on from methods. And yes, I've invented the word ‘outro-ducing’, because my thesaurus couldn’t find me a word that really captured the opposite of intro-ducing!

I think teachers spend a lot of careful thought on how they will introduce methods to learners, but much less consideration is given to how particular methods might enjoy a dignified exit. You have probably taught many lessons where the main aim was 'to introduce X'. But how often has your main lesson aim been 'to outro-duce' something? “This is the last day on which you will do X - we won't be doing that any more after today." Is that something you would ever do?

It may sound negative to be thinking about removing methods from learners' toolboxes. Why would anyone want to do that? Surely the more methods learners have access to the better? But I don't think that's realistic. Picture having a cluttered toolbox, with new tools constantly pouring in at the top. Some tools are genuinely additive - they expand the range of things learners can do. But others really should displace older tools - and we ought to throw out those older tools that we no longer need. If we don't prune our toolbox, we make it harder for ourselves to find what we need. Do we ever talk about this kind of thing with learners (Note 1)?

Counting on

The problem is that learners at any level can get stuck on an inefficient method, which becomes comfortable for them through familiarity, and it can then be hard for them to 'move on' to more powerful approaches. I will take an example from primary/secondary school, but please substitute your own example that is relevant for the ages you work with.

It is important for young children to learn ‘counting on’ as a powerful strategy - far more powerful than 'counting all'. So, to work out $5 + 3$ they would begin with the larger number, 5, and say “6, 7, 8”, so the answer is 8. For young children, this is not trivial, and there are all kinds of pitfalls, such as starting counting the 1 on the 5, rather than the 6, and obtaining an answer of 7. The business of counting up to 3 while saying “6, 7, 8”, rather than “1, 2, 3”, is really quite complicated. This is all important to take time over and work on carefully. However, what do we do when we find older primary or even secondary-age learners who still seem wedded to counting on as their preferred method? Of course, they have been introduced to many other, more efficient, methods over the years. But they trust ‘counting on’ more than any of these. They are more comfortable with it, and believe that, for them, it is more reliable. This favoured method then becomes a barrier to other methods, and, the more they use it, the more alien other methods feel ("I don't do it that way; I prefer my method").

When I watch a child counting on to work out something like $14 + 14$ (“15, 16, 17, …), I can't help feeling that we are wasting their time. Suppose they reach the answer 29 - what does the teacher do? It is easy to feel sorry for the child and say something like, “Ooh, nearly. Try that again.” More wasted time as they repeat the process, and, even if they get it right second time, what do they learn? A systematic error, such as a fencepost error (Note 2), getting 27 because they ‘count the 14’, should be addressed explicitly, but if their working memory has simply been overwhelmed by the task, or they just made a slip, then what does repeating it achieve? Errors like this are a feature of the method, rather than the child, when used on numbers as large as this; if I had to do $14 + 14$ that way, I would also be slow and possibly inaccurate. Assuming that the learner just needs more practice simply traps them, and lots of mathematics lesson time can be consumed while apparently ‘low-attaining’ learners endlessly 'count on', while learning nothing except that they are apparently not good at mathematics. Until they can succeed with this method (by some measure), they are deemed not yet 'ready' to be urged onto a more sophisticated method.

Getting over the hump

But what is the teacher to do? The learner has been taught more powerful methods but claims not to understand them, or not to like them, or just to be more comfortable with their counting-on method. The problem is that if we simply allow learners to stay for as long as they wish with whichever method they feel most comfortable with, then they are very likely to get stuck on inefficient methods. Any new method is going to feel hard at first, simply because it's new and unfamiliar. Mastering a new method is bound to be challenging initially, even if, ultimately, it might feel far more comfortable than where you were beforehand. Transitioning to a new method is hard because it’s unfamiliar. You are stepping away from your comfort zone, so learners should expect to find the new method harder at first, as there is a hump to get over before you feel the benefits (Figure 1).

Figure 1. Getting over the hump when learning a new method. You cannot expect to experience the benefits immediately.

Learners need to understand that they can't judge whether they like a new method the first time they see it or try it - it is only when some fluency with it has been developed that they will be in a position to say what they think of it. "This may be your future favourite method, but you can't know that yet!" If we just introduce a new method and ask them what they prefer, they may be very likely to prefer the old method, simply because it’s familiar and it has served them well in the past, particularly if they have low confidence and a history of lack of success in mathematics. We may have to be a bit more pushy than just introducing new methods and hoping they will catch on. We can phrase this positively: “I know you’re really good at doing this by counting on. I’d like to see if you can do it using tens and ones, and I want you to try this method today."

If we do this, we need to be tolerant of the fact that learners trying a new method may initially be less reliable than they were with their old method, since they are not yet fluent with it. A new method may not give instant benefits. So, we might need to expect more errors (or perhaps different ones), at least at the start. So we need to praise the fact that they're trying the new method and not let them feel like they have failed because they are slower and less accurate than they were previously: “Great that you’re using tens and ones to do this. You'll get more accurate as you work at it.” Otherwise, if we (even subtly) reward speed and accuracy, they will want to revert to the old ways ("Counting on just suits me better"). Learning the new method is an investment that will most likely take time to pay off.

Fading out?

People often talk in terms of 'scaffolding and fading', but I think it is not enough just to introduce new methods and hope that the old ones will 'fade away' in the shadow of these new, more powerful methods. Often the old method will persist, and we need to help learners by actively 'outro-ducing' them. Letting some learners spend all lesson working out a handful of calculations like $34 + 27$ by counting on is not teaching them anything useful. It is not helping them withdraw from 'counting on' and transition to 'tens and ones' approaches - it is just reinforcing their dependency on something that is ultimately no longer helping them. It is just contributing to the problem, as they fall further and further behind their peers, who are accessing the more powerful methods. We need to be helping learners through a managed withdrawal from methods that have outlived their usefulness: “I would like to support you in moving from this method to this other method that I know will be harder at the start but in the end I think will really help you.” This doesn’t have to be flicking a switch overnight ‘banning counting on’. The word 'fading' suggests something gradual - which is helpful - but also perhaps something that happens naturally, without any intervention - which is, I think, less helpful. If you think of a toddler who has got hold of something they are not allowed to have, like a pair of scissors, then the ideal thing to do may be to distract them away from it with something bright and even shinier. But, while doing this with one hand, you might still need to use your other hand to gently prise their fingers away from the scissors. The attraction to the shinier object might do some of the work, but not all.

Planning the outros

None of this is saying that certain methods should never have been taught (see Foster & Ollerton, 2020). Outro-ducing a method your colleague painstakingly intro-duced years previously is no reflection on their judgment as a fellow professional. If you had been the teacher then, you would also have taught that method. Many methods are important - necessary even - for a time, and then the point comes when they need to be retired. The learner who is now wedded to 'counting on' was probably previously committed to 'counting all', and somehow made the shift from that. So, moving on from 'counting on' is certainly not saying that ‘counting on is bad’. But, when a particular method seems to have passed its use-by date for a particular learner, our role may be to help them say goodbye to it. All of this is obviously a matter of judgment for the teacher. We don’t want to accelerate learners prematurely onto formal methods like column addition that they don’t understand (Foster, 2019). But building understanding of place value needs to be actively worked on, and leaving learners counting on for years doesn't do this.

Two of the most important parts of a piece of music are the intro and the outro. The quality of the intro determines whether the listener will continue listening or click ‘next’; the quality of the outro has a strong influence on the listener’s memory and overall perception of the piece. I suspect that good outros are harder to write than good intros - look at how many popular songs end by looping a repeat of a couple of lines and fading down the volume. Ending things well can be difficult - personal relationships sometimes drift along because neither person knows quite how to end it. A good host at a party needs not just to be hospitable and welcome all the guests in, but also occasionally to boot out guests who’ve overstayed or had too much to drink! Waving goodbye to methods that learners have become accustomed to over years is hard and may feel like pulling teeth, but it is just as important as introducing them to new methods.

Questions to reflect on

1. Are there methods that your learners use that you wish they would move on from?

2. Do you recognise the challenge of 'the hump' (Figure 1) when learners encounter new methods?

3. How might you help your learners to let go of mathematical methods that seem to have outlived their usefulness for them?

Notes

1. Of course, one exception to throwing out old tools is when you happen to be a mathematics teacher. You still need to know how to do things in 'less sophisticated' ways, because you will have learners who are working in those ways. Saying, "Oh but I don't do it that way" doesn't work if you are a teacher!

2. A fencepost error is an "off-by-one error" caused by incorrectly including or excluding a boundary value (e.g., for 4 fence panels you need 5, not 4, fence posts).

References

Foster, C., & Ollerton, M. (2020). Mathematical white lies. Mathematics Teaching, 272, 24–25. https://www.foster77.co.uk/MT272Foster&OllertonMathematical_white_lies.pdf

Foster, C. (2019). Doing it with understanding. Mathematics Teaching, 267, 8–10. https://www.foster77.co.uk/MT26703.pdf

5 comments:

  1. These reflections apply to advanced mathematics too. If the ABC Conjecture is true then the proof of Fermat's Last Theorem condenses to a page or so.

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  2. Thank you, as a relatively new Year 4 teacher l found this article really useful.

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  3. This article got me thinking about the ‘procept’ - an amalgam of three components: a "process" which produces a mathematical "object" and a "symbol" which is used to represent either process or object.
    As well as the fact that more able mathematicians often do qualitatively different mathematics from the less able

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  4. I'm thinking about "fading" by discussing the relative efficiency of the new method. In the example of "counting on", if you use larger numbers it is easy to see that a tens and ones approach will be quicker and more accurate, but having that conversation is the important bit. 48 + 37 is more efficient using tens and ones but counting on probably works better for 48 + 7.
    Thanks Colin, I'm going to enjoy these.

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    Replies
    1. Thanks very much. Yes, I guess so, although, with 48 + 7, I think I'd eventually want to get to 'knowing' the '8+7' bit, or mentally partitioning the 7 into 2 and 5, rather than going '49, 50, 51, ...'.

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