Today, I had the horror of reading the speech made by Elizabeth Truss to the North England Education Conference on Thursday. This speech advocates, amongst other things, the almost forced return to the teaching of algorithm calculation methods.
I initially felt very worried after reading this, and this afternoon decided to pen some form of response to Elizabeth Truss, which you can find below I’ve decided to blog this, an ‘open letter’ if you like, as I would be very interested in what you think :)
A bit of background here- my Mathematics Specialist Teacher second year project is on developing a conceptual understanding of calculations with a focus on division, and I have been meaning to (and still will) blog more about this. I also feel I need to put a reminder that these views are my own and not necessarily those of my employer etc etc…
Anyway, my thoughts are below…
I read with quite considerable alarm the speech you made to the North England Education Conference on the 17th January 2013. As a committed teacher, who wants their children to not just learn facts but develop understanding, this speech made me feel very un-easy.
In your speech you state that children must learn to speak the ‘language of maths’- something which I fully agree with- we need to develop a conceptual understanding of maths within our children, so that they are able to effectively use and apply their mathematics skills, and are able to problem solve beyond the dreaded and much over taught ‘word problems’. I would like to address just some of these issues from your speech.
Let’s first tackle your accretion that those ‘new’ methods which are based on conceptual understanding do not allow for progression- are you able to provide evidence of this? By their very nature the methods taught in many schools now are rooted in the structure of our number system- and therefore can easily be extended to any number. Children frequently do this themselves- for example, when taught how to subtract by finding the difference on the number line for 2 digit positive numbers, they are easily able to extend this, often independently, to subtracting larger numbers, decimal numbers, negative numbers and indeed time intervals. The same applies for division by grouping/chunking, and multiplication using the grid method (which, in reality takes multiplication back to what it is- an array)
The ‘algorithm’ methods bode little resemblance to what is actually happening with the numbers. Can you explain why long division works? There is no logic to these methods- they work by simply following n a set of instructions, which, to many children at least, ‘magically’ end up with the correct answer- and I believe this means children will struggle to extend these methods. After all, maths is not ‘magic’.
You correctly state that no-one can predict when a child will experience the ‘light bulb’ moment- but by removing the conceptual understanding behind calculation, you will be virtually eliminating the chance of this happening- as a child will not truly be able to say ‘ahh…I understand’- all they will be able to say is ‘I can follow the method…’
There is a considerable bank of research evidence showing that the ‘new’ methods which foster a conceptual and relational understanding of number and calculations lead to far greater accuracy and confidence than the ‘traditional’ algorithm based methods.
In fact, you don’t have to go far to find evidence of this- for 3 pieces of leading research in your very own county of Norfolk[i] have consistently shown that methods which foster conceptual understanding ‘win’ when it comes to children being able to calculate accurately. Surely it is by using methods based on a conceptual understanding that children are more likely to spot their errors- for they actually ‘see’ what they are doing to a number, rather than just robotically following a set of instructions ‘just because it works’. And further research clearly states the importance of developing conceptual understanding.[ii] Can you provide research to prove that these methods lead to greater errors as implied in your speech?
Of course, this is not the first time you have heard this. Many groups and organisations have tried to explain this before, and many of us responded in a similar vein to the (absolutely appalling in parts) draft curriculum published in March 2012. Indeed, the joint response[iii] to the proposed curriculum by the Association of Teachers of Mathematics (ATM) and the Mathematical Association (MA), which together represent some of the leading thinkers when it comes to mathematics education, clearly states and provides evidence for the need to develop a conceptual understanding and not rush to formal algorithms too quickly. Could I suggest that you re-read this response and ask if the new version of your curriculum addresses the many issues raised within it before it is released in a few weeks time?
The government has spent money on, and now continues to support the Mathematics Specialist Teacher Program, of which I am now in the second year, and this very program, designed to ensure that each primary school has a mathematics specialist and therefore raise the standard of mathematics teaching, advocates conceptual and relational understanding and the calculation methods which are based on these.
Indeed, these ‘new’ methods are, in fact, nothing ‘new’- leading academics, backed up by research, have consistently proven the case for relational understanding (Skemp, 1976[iv]). Without methods based on this understanding you constantly find in classrooms that children who struggle with maths find the algorithm methods incredibly confusing and are unable to use them.
I also take great issue with your statement that these methods are not efficient. These methods may not be efficient for you- I’m guessing because you are not comfortable with the method, nor do you probably have the conceptual understanding of number which underpins these methods- this is demonstrated by your rather lax explanation of these methods in your speech. This is of course, not surprising as they are different to the way you were taught.
But for the children who are taught these methods they are efficient. They, used alongside robust teaching of mental strategies, which themselves are rooted in a conceptual understanding of our number system allow for efficient and accurate calculations in all four operations (yes, whilst they would work for any calculation, even the most simplest, it’s never been advocated to conceptual based written methods ‘blindly’ used for any calculation), . Surely a method is only efficient if it means the child is actually able to get the correct answer?
East Asia, and specifically Singapore, is, yet again, used as an example by you as a country where things are done ‘right’. So are you proposing we import all of this teaching, if so, let’s first look at Singapore’s mathematics teaching, which includes a very refreshing approach to the purpose of mathematics teaching (for the solving of problems) , which encourages mathematical thinking and includes the Singapore bar methods for calculation and solving problems- based on a conceptual understanding of number and can be which can be seen to be similar in its thinking to the conceptual methods taught by teachers in the UK.
You state that parents are not able to understand these methods. Yes- these methods are different to the way in which most parents were taught themselves at school, and it is the job of the school to ensure that parents are aware of the different methods taught in school- something that many schools do through successful calculation events, in which many parents realise for the first time themselves what division, subtraction, multiplication or addition actually means. Once aware of the new methods and the reason behind them, parents are normally, in my experience anyway, very supportive. But despite this, are you therefore creating a maths curriculum for parents? I would assume that a curriculum is meant to be written for the benefit of the pupils not their parents? Children understand and are able to easily explain the conceptual methods they are taught, they are accurate when using them and can use them ‘quickly’- this is what matters to me as a teacher. (It is also worthwhile pointing out that in only a few years time our new parents will, themselves, have been taught the conceptual based methods- so we could end up with the same problem in reverse!)
I am also not advocating the complete abolishment of the use of algorithm methods, however ideal that would be, as I appreciate this would never happen- but do believe it is about using the right methods, backed by conceptual understanding, at the right stage for the child. For many children, the forced introduction of algorithm based methods in primary school when they do not have this conceptual understanding is too early, and will ruin their chances of actually understanding maths and increase their chances of becoming one more of adults who say ‘I can’t do maths’ (when they are normally really saying ‘I don’t get maths’) that you refer to in your speech (I agree by the way, that is an awful attitude to have!). [Of course, the converse is true, for a few children, they will be 'ready' and be able to extend their conceptual understanding for the traditional algorithms by the time they leave primary school- surely we should be reactive to the child?]
There is one glimmer of hope in your speech- that the new curriculum will state that children must be taught efficient methods- and I could argue that by teaching the methods based on conceptual understanding, that these methods are the most efficient for the children I teach.
But, then you state that, from 2016 children will achieve a ‘method’ mark only if they have used the ‘prescribed’ algorithm methods. [By the way, what will this mark be awarded for? as by the very nature of algorithm methods there is little margin of error.]
This means you are effectively forcing schools to take a calculated ‘gamble’- do they solely teach the algorithm methods, which they know (as backed up by research as stated earlier) will lead to a far greater error rate but which will pick up a ‘method’ mark, or do they stick to the research proven methods, which they know children will generally stand a higher chance of getting the correct answer, and therefore two marks, but with the risk that if they do not arrive at the correct answer they will be penalised for using a method which they do not understand?
All this strikes me as change for changes sake- a dangerous change which stands a good chance of reducing our children’s ability to calculate and solve problems, but which will please many traditional conservatives and elements of the press who do not understand the real issues in mathematics education, and a change which, yet again, erodes the professional judgement and expertise of teachers. Please do not think, as I am sure someone will be quick to suggest, that teachers are simply resisting change. I am not teacher who resists change and am more than happy for change, and embrace it, after all teaching has to be a profession where we are constantly moving forward. But change has to be positive change that will have a positive impact on the education and lives of the children which I teach- and I, as many teachers do, believe this change certainly will not.
Of course, there are many more things in your speech which were incredibly worrying, but I do not have the time to get started on, for example, why rote learning times tables up to 12 times is unnecessary (the structure of our number system means that tables up to 9x would be sufficient) or why children’s ability to think mathematically can be increased by calculator use (but not by dependence on calculators . What your speech simply proves, is yet again, this government is not listening to what teachers and leading academics are saying- but why am I surprised?
[iv] R Skemp: ‘Relational understanding and instrumental understanding’, Mathematics Teaching, 81.