Beauty in mathematics is seeing the truth without effort
George Polya
Long gone (or I hope) are the days where good maths lessons are judged upon whether or not children proudly present pages upon pages or nice, neat, correct calculations, as their end products. This shift away from such a perception of what good mathematical learning and thinking looks like has made space within our maths lessons for those ever so important discovery-learning experiences. This is where our children collectively and creatively unpick mathematical concepts. We know these experiences are vital in helping children appreciate the structure of what later becomes embedded within the calculations we hope they will then be able to solve, both independently and accurately.
In order to ensure children can lay down this final piece of their mathematical learning journey’s jigsaw puzzles, we have to carefully consider which methods or tools they require to succeed. The national curriculum directs us to a range. When looking particularly within the context of addition and subtraction you can see that this starts early on with the acknowledgment of using concrete and pictorial methods, preparing children for when they move into KS2 where the word ‘formal’ begins to regularly rear its head. It is here where children are then increasingly expected to be able to use mental methods, estimation, even rounding to be able to QA their own calculations. Interestingly, in Year 4 and 5 they are expected to also be able to decide which operations and methods to use and why. A huge ask, as there was once was a time, where even as their class teacher, I am not 100% certain I thoroughly understood or gave enough thought to the ‘what method and why’ argument. I was once very guilty as charged for railroading into and solely relying on formal written methods.
Mason, Stephens and Watson (2009) discuss the importance of mathematical methods being greater than simply discovering and relying upon ‘clever moves’. This for me does to an extent resonate with how the formal written columnar methods often are deployed within children’s mathematical tool kits. Their repeated and somewhat simple, repetitive steps often provide fast, foolproof steps (or clever moves so to speak) for children to solve increasingly more difficult calculations. What we know in reality is how un-fool-proof these are when children fail to have basic number facts readily and accurately available.
Ofsted (2021) appears to have a somewhat alternative opinion regarding the place of formal written methods. They claim that accurate calculations and careful presentation provide children with the ability to spot important and interesting patterns with numbers, as well as errors that need to be corrected. And don’t worry, they then insist that some children naturally develop this ‘neatness’ and subsequently accuracy. I won’t dwell on that idea. It all seems a little too good to be true, or at least something I am yet to fully experience within my classroom!
Referring back to Mason, Stephens and Watson (2009), they discuss how actually we should really be inviting children to find ‘quick ways’ to do calculations. This for me brings us back nicely to why it is so important to make sure we have that all-important sound understanding of which method and why? The problem for me lies with our (adult) perception of ‘quick’. As competent mathematicians, formal methods tend to be rapid and mostly accurate. So they should be, we have rehearsed them for 100s for years…
Therefore, we sometimes cannot help ourselves but naturally and instinctively assume that the formal methods are the ‘quickest’ most efficient ways also for our children, who are less well-practiced within this art. This is also probably because like I have previously eluded to, formal written methods are here there and everywhere within our national curriculum, whereas terms such as compensation and equivalence are nowhere to be seen…
In simple terms, compensation and equivalence is the process of reformulating a calculation to make it more easily computed mentally. It has its own spine within the NCETM mastery support materials, which I highly recommend you cast your eyes over…
However, in the meantime, here is a quick story to share, of something I witnessed first-hand whilst working with one very wonderful mathematician…
The task was simple. I gave them a sum, told them they could solve it however they wanted, as long as they could tell me why!
The calculation was 1732 -999
Below was their outcome:
It was a neat and accurate answer. So why was I complaining? The problem was, their reason for using this method was pinned really on what they thought would impress me! They thought that was what I wanted to see…
When we looked closely at that calculation together lots of discussions emerged. We studied their steps, and how they had undergone 7 (if not more!) steps to get to this answer- which was 7 chances to trip up.
What would have happened if they’d have used compensation and equivalence and seen this calculation as:
1733 – 1000?
By balancing and compensating the calculation, in this case by adding 1 to 999 and then 1732, we made really easy numbers to work with. The child looked at me like I was some mathematical magician. They not only saw a now quick, efficient, mental method but discovered something new within the structure of addition. They could now do that calculation and similar follow-up examples very efficiently, without even putting pencil to paper. How many clever moves? Now, a couple of quick, yet comfortable ones.
This is one of the many millions of reasons why I love all things ‘teaching and learning’ within primary mathematics. There is so much to be discovered. No stones can ever be left unturned.
So I return to my initial question, do methods really matter? I’ll leave that answer up to you.