Reasoning Skills and Mental Calculation
A new academic year: a year filled with new pupils, new challenges, and, hopefully, a better understanding of what we have to do! I hope this new year has started smoothly for you and you’ve got your class figured out by now (don’t worry if not, it will come!).
In my role as a maths specialist, delivering maths lessons that challenge pupils, no matter their ability, and focussing on their reasoning skills, I gain a unique insight into the similar ways in which children approach maths.
I work with children from all primary year groups, across a range of schools in the Northwest and what I’d like to talk about today is both reasoning skills and mental calculation methods. For the first session, I have been using a magic square activity (remember them!?) with whole classes of year 6 pupils. For those that haven’t seen a magic square before - each column, row and diagonal add up to the same number. Magic!
The activity first asks children to identify what makes a magic square special by adding up the numbers in each column, row and diagonal (15 in the first example). And even in this I encountered problems in the wording of the question: at first I had written, ‘Every row/column adds up to…’ and when I first did the activity, children were writing 45! They clearly understood the activity and could add accurately, but they had missed out on the magic as, because of my wording, they had found the sum of all the rows and columns, effectively bypassing the understanding that each row, column and diagonal adds up to 15. A simple reword - changing every to each - has eliminated that initial problem (so far!).
The Magic Squares task, with additional questions on the reverse.
As the maths involved in this activity is at a year 2/3 level (adding three 1 and 2-digit numbers), I had planned that the focus of the activity would be talking about how we solve problems. And to some extent it was - we talked about why you would start in a place with the fewest possibilities. For example, if there was a row of three and the sum of the numbers was 21, with two numbers already known, there is only one possible answer for the missing number (5 is the only number that can be added to 9 and 7 to make 21). Whereas if only one number is known, out of a column of three then there are several, in fact infinite, possible answers. I kept asking children’ ‘why have you started there?’ and encouraged the vocabulary of, ‘because there is only one possibility’ or, ‘because there are more than 1 possible answer so I’ve left it until I’m certain.’
This skill of logical reasoning, of starting with what you know for certain, is an essential mathematical skill. Being able to find all possibilities and start where there are the least amount means that you can be more certain about being correct, and it means that you can eliminate possibilities as you work through a problem (and it should be noted that these thoughts only apply to problem solving!). After letting children have a go themselves, I stepped in and did a bit of teaching. They inevitably got the first magic square wrong, having the top row adding up to 18 and sometimes one column - but the rest just added up to random numbers. This is mainly because children don’t spot the diagonals, and they didn’t yet have the strategy of logical reasoning, and finding the one with the least possibilities.
After some teaching, the children were off and across all of the nineteen year 6 classes I have done it with, pupils were successful and enjoyed it. I’ll be using it again.
But there was something unexpected. The third magic square the children had to complete, had rows, columns and diagonals which added to 42, and after we had discussed the problem solving strategies, I noticed children doing reams of column addition and subtraction to do their working out. Things like: 42-26, 17+11, and even 23+10, as a column! I even witnessed a year 6 pupil write out the numbers 1-21 to do 21-7. she counted back from 21 and got the answer 15. I was quite taken aback by the lack of mental calculation strategies and reliance on written methods. And it happened in more than one of those 19 Y6 classes.
Interestingly, one of the schools that had this problem has asked me to deliver some CPD as well as the Reasoning sessions for year 6, which puts me in a fantastic situation in that I know exactly what that school needs before even arriving! They will have a day of CPD that focusses on mental calculation, starting in Y1 and progressing all the way up to Y6.
So, what CPD will I deliver? Well, lots of counting, using a hoop and a stick, and asking questions like, “How many more to make…?”; playing the 1 2 game; doing an activity called starting from (quick fire mental calculation questions all with the same root number), through which I will explicitly teach mental calculation strategies for adding/subtracting 10, 20, 100, and then 9, 19 and 99 etc; playing games like shut the box, pontoon (blackjack), darts, and dice/spinner games.
How strong are mental calculations strategies at your school and in your class? Do you overtly teach problem solving strategies? Do you use magic squares in your teaching? Do you write any of the resources you use in the classroom or is it all from a scheme? Please let me know on social media or in the comments below. If you would like more information on the work I do or the resources I use then you can browse the rest of the website or send me an email using the button below!
