‘Old maths’ vs ‘new maths’ – the balancing act

    Published: 08 June 2021

    You may or may not have come across a video appearing on social media feeds in late 2020 comparing ‘old maths’ to ‘new maths’. The video in question, and the comments in reaction to it, may or may not have caused you annoyance. They certainly did for me. Not because I don’t have a sense of humour (I do – I promise!) but because it goes against what I believe in terms of encouraging children (and adults) to really think about the numbers in any given calculation before choosing the most efficient strategy. I also think that it does little to overcome potential maths anxiety, especially for adults who weren’t exposed to multiple strategies and given opportunities to consider the value of having them.

    Let’s take a calculation similar to the one in the video… 27 + 46.

    We could of course use formal written addition to solve this, as they do in the video.

     

    Equation

     

    So what’s the problem?

    Well, there are a number of processes that I had to consider as I was doing this. I needed to…

    • write the calculation into the correct format
    • consider the bonds to 10
    • regroup the 7 ones and 6 ones for 1 ten and 3 ones
    • write the regrouped 1 ten in the correct place
    • remember to include this regrouped ten when adding 2 tens, 4 tens and 1 ten to find 7 tens

    How many of your KS2 pupils (and possibly Year 2 children) would resort to using the formal method to solve this? They might have the answer right, but arguably this calculation could be solved with an easier calculation and more efficient strategy by rebalancing and using equal sum. And no matter which year group you teach, it is never too late to introduce rebalancing and encourage it as a strategy and I can say this from my experience of introducing it with my Year 6 class.

    Equal sum – let’s rebalance

    My go to manipulative for exploring equal sum with any class is the beadstring. Imagine (or even better, go and grab a beadstring) that you have 16 beads between your fingers on your beadstring. Here, I have split my 16 beads into two parts (addends), 9 and 7.

    9 beads and 7 beads totals 16 beads.

     

    Beads

     

    No matter how I split the beads between the 2 groups, I will always have 16 beads. Addends of 12 and 4, addends of 1 and 15, addends of 5 and 11. The total amount of beads between my fingers, even if I move them around, will never be anything other than 16.

    It is important that children understand this concept before we consider why this is a useful mental strategy. What we are then looking for is how we can rebalance the calculation 9 + 7 to make it easier to solve. If I slide 1 bead from the 6 ones to join the 9 ones, I now have the calculation 10 + 6. And this is arguably easier to solve using place value knowledge and number facts – 10 and 6 more.

     

    Beads

     

    Now let’s go back to the calculation 27 + 46. Picture (or make) that beadstring! Where are the beads going to slide to help make this calculation easier to solve? We could use this speaking frame to support our thinking:

     

    Speaking frame

     

    Here are two examples:

    1) Slide 3 beads from the 46 onto the 27.

    “27 + 46 can be rebalanced and renamed 30 + 43.”

     

    Equations

     

    2) Slide 4 beads from the 27 onto the 46.

    “27 + 46 can be rebalanced and renamed 23 + 50.”

     

    Equations

     

    When written down like this, it can appear more complicated than the formal written strategy shown earlier. However, these jottings simply demonstrate one stage of the learning process.

    Once this strategy is understood and can be carried out mentally, I would need to:

    • consider the bonds to 10
    • take some from one number and add it to the other
    • use place value understanding to add

    Compared to the formal written strategy where I needed to:

    • write the calculation into the correct format
    • consider the bonds to 10
    • regroup the 7 ones and 6 ones for 1 ten and 3 ones
    • write the regrouped 1 ten in the correct place
    • remember to include this regrouped ten when adding 2 tens, 4 tens and 1 ten to find 7 tens.

    The relative ease of using equal sum becomes even more evident when children begin working with larger or decimal numbers.

    See Rachel Rayner’s blog for more detailed exploration of this strategy.

    Equal difference

    Can we use rebalancing for subtraction? Absolutely.

    Is it the same as equal sum? No.

    There are similarities but also a key difference, and this can be confusing to children. So for exploring equal difference, I would suggest a starting with a different manipulative – cubes (or counters).

    Let’s start with the calculation 8 – 5.

     

    Counters

     

    Here we have 8 cubes and 5 cubes with the arrow representing the difference between 8 and 5. The difference between 8 and 5 is 3.

    What would happen if we add 1 cube to each of the numbers? What would the difference between 9 and 6 be?

     

    Counters

     

    We can see that, again, the difference is 3!

    What if we subtract 1 from each number? 7 – 4 is still a difference of 3. What if we add 2 to both numbers? Is it still a difference of 3 if we subtract 4 from both numbers?

    I love letting children continue to explore this and discovering the ‘rule’ for themselves.

    It’s only useful if it makes it easier

    Again, as with equal sum, these strategies are only useful if we are manipulating and rebalancing the calculation to make it easier. Children may well use other strategies such as known facts to subtract 3 from 8 as these may be more efficient.

    However, by exploring simple calculations, generalisations can be made to be applied to more complex ones. To keep the difference equal, we must operate on each number in the same way. This is different from equal sum where if we subtract from one number, we must add to the other.

    Once the children can ‘see’ the strategy, calculations can then be explored considering how the rebalance makes the calculation more or less simple.

    Take the calculation 28 – 13 and the following 2 examples of how this calculation could be rebalanced.

     

     

    Equations

     

    Although pupils may use known facts to halve 30, subtracting ‘1 ten’ from ‘2 tens and 5 ones’ is perhaps a more manageable calculation.

    How would you rebalance 56 – 29?

    Would it be more manageable to rebalance to include the number 50? Or would the number 30 be better? What would you do?

     

    Equations

     

    A toolkit of strategies

    Let’s think back to the video I mentioned earlier.

    The point of the clip seemed to be about efficiency and not over-complicating strategies by just having 1 choice but is that what we really want for our children? Or do we want them to have a toolkit of strategies that they can then choose from with some real thought going into the numbers that they are dealing with?

    A really effective rehearsal opportunity is to use multiple strategy mats.

    The children are given one calculation and they note different ways to solve it. A further adaptation could be that both the calculation and answer are provided, really encouraging focus on the strategies rather than the answer. These could also be built collaboratively on the working wall and could include pictures and diagrams or manipulatives if children are also building the calculations. 

     

    Equations

     

    Equations

     

    Once children have filled their mats, rich discussions can then take place.

    Which is the most efficient method?

    Which is the least efficient?

    Which is the most creative?

    Who has used a method that nobody else has considered?

    Who can see a method that they would like to try in the future?

    A further adaptation can be seen in the mat below which was shared recently with Year 5 colleagues during our Summer Success in Maths Project webinar. Here, children are tasked with taking a strategy and creating 3 calculations where this strategy would be useful, and one where it would not be. If children can create their own calculations to match a strategy, this really does show depth of understanding.

     

    Equations

     

    Tracking back to ‘see’ the maths – it’s never too late!

    So what if you are teaching in UKS2 and your pupils have never been exposed to the rebalancing methods? Is it too late to teach them this? From my own experience of introducing equal sum and equal difference to a Year 6 class, I don’t think that it is!

    Ensure simple calculations (such as 9 + 7 and 8 - 5) are explored and modelled first using manipulatives (you could use beadstrings and cubes as suggested above) before providing rehearsal which gradually tracks back up to age-related pitch. The exploration is so important to allow children to ‘see’ how the strategy works and form generalisations for each:

    • to keep the difference equal, we must add or subtract the same to each number.
    • to keep the sum equal – if we subtract from one number, we must add to the other and vice versa.

    A possible progression for equal sum could include the following calculations:

    29 + 4

    57 + 8

    6 + 98

    187 + 14

    2.998 + 23

    11,599 + 202

    780,577 + 1,996

    Revisit, revisit, revisit

    In order for a strategy, or any learning, to become secure, it will need to be revisited throughout the year. Strategy walls can be a useful addition to a working wall. Different to working walls, which are updated regularly throughout a learning sequence (see Nicola Adams’ blog about making your working wall work for you), strategy walls are maintained over a longer period of time.

    As the name suggests, a strategy wall is a visual collection of all the differing calculation strategies (both mental and formal) which the children have been exposed to and are useful aide memoirs for both children and adults. With them displayed in one place, discussions could include:

    • what’s the same and what’s different about the strategies?
    • which strategy might be best for a particular calculation?
    • which strategy would not be efficient?
    • how to create calculations for the wall that are suitable for X strategy but not another

     

    Tree wall display

     

    Old vs new

    So back to that film again. It is up to us as classroom teachers to find those balances between when and how to introduce new strategies to our classes. A balance between rehearsal of formal written methods and informal methods. A balance between exploring simpler examples and building confidence before moving to appropriate pitch. And then comes the balance of which strategies, when and how these strategies are shared with parents and carers – but really that’s a whole other blog!

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