NUMBER BONDS – Why don’t we just call them Sums?
WHAT? WHY? HOW?
Confession: I did not teach numeracy and maths well until recently. I followed schemes and learning outcomes and genuinely thought that by ‘ticking the boxes’ my pupils would become numerate. Some did. But many didn’t. I know this now.
It was only when I began studying for my Masters that I realised I was hindering my pupils growth and progress. Not intentionally, but because I didn’t really understand or know how the tiny building blocks could help a child become fact fluent – that is, getting an answer pretty quickly and with limited demands on 'working memory' and having a deep understanding of the how and the why. In fact, what we want for our children is that they move beyond fluency to automaticity. Let's unpack this...
WHAT are NUMBER BONDS?
From a pedagogical sense, number bonds are number relationships. They are representations of the composition (and decomposition) of numbers. They show how numbers join together, and how they break down into component parts. Number bonds foster relational understanding. In particular, they are moving towards a mental representation of the relationship between a given value.
Here are some compositions (how they are made up)
4 + 2 = 6 |
Put simply…
- A whole thing is made up of parts. If you know the parts, you can put them together (add) to find the whole.
- If you know the whole and one of the parts, you take away the part you know (subtract) to find the other part.
WHY are they Important? Why teach them?
On a basic level, they provide a solid foundation for understanding how numbers can be split and combined. But over the long term, they do a lot more — they give students an informed skillset for gaining mathematical fluency and navigating increasingly difficult concepts.
HOW are they currently being taught?
Learning and practising number bonds to 20 are a statutory requirement in England by the end of Y1 and by the end of First Level for Curriculum for Excellence in Scotland. Often in classrooms it is typical to see a Learning outcome such as ‘TO LEARN NUMBER BONDS TO 10’ and this is often covered in a single lesson or over a week or two.
This is contradictory to what we know about how children learn mathematics. We have known since the work of Hughes in the 1980s that the move from the more concrete to the abstract is not simple (Hughes 1986) and that it takes children a long time to acquire secure number sense (Munn 1996). Nunes and Bryant go on to say:
“Because the connections between quantities and numbers are many and varied, learning about these connections could take three to four years in primary school.”
Nunes and Bryant (2009, p4)
Yet much of what we do mathematically, certainly from the end of Y1/ P1, becomes quickly abstract.
Different frameworks/ schemes/ programmes / advise various methodologies for teaching number bonds but from my experience it usually involves:
- Teaching addition and subtraction to 5, to 10 and then to 20,
- A variety of models (PART - WHOLE DIAGRAMS are common such as cherry diagrams and bar models)
- Fact families are introduced – in the hope that the relationships shown among the facts can improve efficiency in memorising.
The whole process can become fragmented and lengthy and often looses focus!
HOW WOULD WE ENCOURAGE THEM TO BE TAUGH?
Have you ever wondered how other countries around the world lead students to deeper relationships that underlie the facts or to activate intellectual capacities other than memorising?
Knowing now what we know about memory, how can we improve this for children? Surely we can move children’s thinking from concrete representations to abstract and much quicker. How can learners become fluent with their number bonds and move to automaticity?
How else might NUMBER BONDS be taught?
Liping Ma (2011) gives us a brief, but comprehensive picture of how ‘thinking tools’ are needed and developed and how we can move children beyond a counting by 1/ counting strategy and being so reliant on using TOOLS (counter and fingers). In her paper Three approaches to one-place addition and subtraction: Counting strategies, memorized facts, and thinking tools
She outlines the structured and progressive approach to tackling number bonds. Liping Ma has undergone research to compare the differences on how mathematics is taught in China and the USA.
These are my 3 TAKE AWAYS from Liping Ma’s (2011) paper (and after implementing them into practice in the classroom and with my own children):
- The foundational importance of subitising – perceptual and conceptual subitising is so important in the development of number bonds. An important computational capability of children is their ability to mentally calculate small quantities. Children do not get enough experience of this. It’s been well researched since the 1960’s (Let that sink in!) This is not new.
SUBITISING helps develop a sense of number and quantity and is the foundation for the next stage – number bonds. It draws on the natural computational capacity that students already have. You can invest in our Subitising Grab & Go Bags which will help your child/ children build on their natural ability to subitise.
- STRUCTURE. Systematically partitioning quantities...
to 3,
then to 5,
then 6-10
and then to 20 (bridging 10)
Interchanging the addends (those are the bits that are added together) is encouraged at each stage. This is what we call the communative law (this means that if we change the order of addends the product doesn't change).
In this way, each later stage is based on the capacities developed at prior stages.
Controversially, Ma noted that addition and subtraction (operations) were introduced and taught along side each other at the same time. Subtraction is introduced as the inverse operation. There is lots of scope to Lead students to observe how to find a missing addend. These strategies are labelled ‘thinking tools’ which students are encouraged to use when they encounter obstacles.
Many concrete representations in the early years are objects such as bears. These are wonderful for getting to grips with counting. When it’s time for children to start conceptualising how maths works, a better concrete representation is needed. Five frames do just the job (with the help of double sided counters). When children are learning pairs of parts that make a whole of 5, arranging five frames clearly represents one of the first mathematical patterns that children come across. These representations are included in our RED BOX.
- Expressing the number bond using STANDARD NOTATION “+, –, =” (these are the symbols needed to connect numbers to make addition and subtraction equations). These mathematical expression accompany the concrete and pictorial representation very early on.
All of this is done to reduce the cognitive load, allowing students to focus on the new concept. This is in line with how we move children from concrete to abstract thinking and move them to automaticity quicker.
This report from the New South Wales Centre for Education Statistics and Evaluation is just about the most clear and concise summary of Cognitive Load Theory that I have seen.
Here are some links to other REFERENCES which may add value to your thinking and Teaching and Learning:
Clements, D. H. and Sarama, J. (2009) Learning and teaching math: the learning trajectories approach. London: Routledge
Erno Lehtinen Online Colloquium videos:
Douglas Clements and Julie Sarama, University of Denver From cognition to curriculum to scale: Learning trajectories for early math
Jon Star, Harvard University New directions in the study of (and assessment of) mathematical flexibility
Hughes, M. (1986). Children and number: Dficulties in learning mathematics. Oxford: Basil Blackwell.
Liping Ma - Three Approaches to One Digit Addition
Munn, P. (1996) 'Teaching and learning in the pre-school period' in M.Hughes (ed) Teaching and learning in changing times Oxford:Blackwell
Nunes, T., and Bryant, P. (2009) Key Understandings in Mathematics Learning. Paper 2: Understanding whole numbers. London: Nuffield Foundation