We’ve known for a long time that mathematics is a series of interconnected ideas. At its simplest, this means that different mathematical concepts are related to each other in different ways. For example, the concept of multiplication is related to the concept of addition, and the ideas of graphing and equations are related through a shared, deeper concept.
When we understand the ways in which different concepts are related, we plan more efficient lessons, reduce the amount of time that is needed to learn a given concept and help students become cognisant of the relationships between the concepts they study.
It is important for us to have a firm grasp of this interconnectedness because it can help us to teach our pupils more effectively, something that Askew and Wiliam outlined in their 1998 review of the research into mathematics teaching. When we understand the ways in which different concepts are related, we plan more efficient lessons, reduce the amount of time that is needed to learn a given concept and help pupils become cognisant of the relationships between the concepts they study. This can, and should, lead to a deeper understanding of mathematics for all pupils. A more cohesive learning experience allows pupils to see the wider picture of mathematics, how the pieces of the puzzle fit together and can be applied to different situations.
It is important for us to have a firm grasp of this interconnectedness because it can help us to teach our pupils more effectively, something that Askew and Wiliam outlined in their 1998 review of the research into mathematics teaching. When we understand the ways in which different concepts are related, we plan more efficient lessons, reduce the amount of time that is needed to learn a given concept and help pupils become cognisant of the relationships between the concepts they study. This can, and should, lead to a deeper understanding of mathematics for all pupils. A more cohesive learning experience allows pupils to see the wider picture of mathematics, how the pieces of the puzzle fit together and can be applied to different situations.
When we (as teachers) don’t take account of the interconnected nature of mathematics, our pupils see concepts as isolated islands and the story being told becomes disjointed, fractured or incoherent. If we take the example of equations and graphing, the equation y = mx + c is a perfect example of the intercept between the two, supposedly, distinct areas of mathematics. On its own, the equation can bamboozle pupils, but when they realise that each part describes a very real feature of a straight line, the seemingly abstract becomes a whole lot more tangible and manipulable.
As a result of the hierarchical nature of mathematics, the relationship between connections can be viewed sequentially, with early ideas essential for success with later ideas – the tools with which pupils can work on developing an understanding of new and seemingly abstract concepts.
But as with any tool it all comes down to how we use them. A spanner, in my hands, is not much more than a doorstop. Yet, in the hands of someone who knows how to use it, it gives the beholder the power to fix a car. And this highlights the fundamental thread at the heart of this metaphor. You have to have the tool (know the mathematics) and you have to know how to use it (see how it relates to other areas of mathematics).
Supporting Pupils in the Classroom
Teaching number bonds requires us to be systematic in our approach. This means that we should follow a clear and logical sequence of steps, starting from the simplest bonds and gradually moving to the more complex ones. For example, we can start with the bonds within 5, within 10, to 10, within 20 and eventually within and to 100. Outlining the journey from 1+1 all the way to 15 + 5 and beyond.
It also requires us to be explicit about which bonds the pupils do and do not know. This means that we should assess the pupils’ prior knowledge and identify their strengths and weaknesses. We should also monitor their progress and provide feedback and support as needed. For example, we can use quick quizzes, games, flashcards, or online tools to check the pupils’ recall of number bonds. We can also use error analysis, questioning, or scaffolding to help the pupils correct their mistakes and deepen their understanding.
Perhaps most importantly, it requires us to be consistent in our provision of opportunities for pupils to recall them. This means that we should not teach number bonds as a one-off topic, but rather as a recurring theme throughout the curriculum. We should also vary the contexts and formats in which we present number bonds, such as using stories, puzzles, patterns, or real-life situations. We should also encourage the pupils to practise number bonds regularly and frequently, both in class and at home. That little and often you may have heard me and Chris Such discuss before.
Finally, teaching number bonds must be driven by the evidence into the recall of key facts. This means that we should base our decisions on what is likely to work best for our pupils, not on our personal preferences or beliefs. We should also use reliable sources of information, such as peer-reviewed journals, books, or websites, to inform our practice and evaluate the impact of our teaching on the pupils’ learning outcomes, in terms of accuracy, speed and confidence.
If we consider all four of these pillars, we will have gone a long way to providing all our pupils with the best opportunity to learn and understand number bonds.
Systematic
Being systematic means that we have a clear and logical plan of how to introduce, practise, and review number bonds with our pupils. It also means that we have realistic and achievable goals for our pupils’ learning progress.
One way to be systematic is to map out the journey of learning number bonds from start to finish. We can use a simple formula to estimate how much time we need to cover all the number bonds within 10 and to 10. There are 20 unique bonds within 10, such as 1 + 1, 2 + 3, 3 + 5, and so on. There are also 5 unique bonds to 10, such as 1 + 9, 2 + 8, and so on. That makes a total of 25 number bonds to learn. Now, let’s assume that we have 38 weeks in a school year. If we divide 25 by 38, we get approximately 0.66. This means that we need to teach about two-thirds of one number bond per week. That sounds manageable, right? Of course, this is just an approximation, and we may need to adjust the pace depending on the pupils’ needs and abilities. But the point is that we have more than enough time to teach all the number bonds if we are systematic.
Being systematic helps us to avoid gaps or overlaps in our teaching. We can ensure that we cover all the number bonds without missing any or repeating any unnecessarily. We can also monitor the pupils’ progress and provide feedback and support as needed. By being systematic, we can make the most of our time and resources, and help our pupils achieve mastery of number bonds. It also means that no matter what year your pupils might be in, it is never too late to intervene.
Explicit
When teaching number bonds, it is important to find out what pupils already know and what they need to learn. For example, if most students know the number bonds to 10, but have difficulty with the number bonds to 20 or 100, we can design activities and tasks that target those specific number bonds.
I hear the phrase “they don’t know their number bonds” quite a lot in schools, and my response is “Which number bonds do they not know and what are we going to do about it?” If we know precisely which bonds they need to learn when they come to us, we can then track the changes and progress of the students over time. This can be done by using exit tickets, regular quizzes or flash cards.
Whatever we choose to do, we need to make sure we make life easy for ourselves, be realistic in our expectations for coverage or the rate of learning and consider how we can get the most information from our pupils with the least effort on our part. Not to be lazy, I might add but because our time is finite and it is entirely possible to dedicate just a few minutes of each day to the learning and tracking of number bonds, and although that might not seem like much time, it is exactly what is needed.
Consistent
As is true of learning anything – whether it be mathematics, Japanese, piano or chess – consistency is the key to success. 5 minutes every day for a month is infinitely more productive than two and a half hours of study on a single day. Sure, you might have crammed as much study into that one session but the benefits of repeated exposure to ideas and the science of memory – to which we will turn our attention soon – suggest that a little and often approach is a much better way to ensure learning takes place. If we have time for 15 minutes, then by all means, use the allotted time, but I contend that two minutes is all you need – especially given that you need only commit an additional one fact to memory every week if we spread out the journey effectively.
Of course, if we are supporting teenagers who have not yet learned their key bonds, then we may wish to expedite the process, but there’s an argument to be made about older, almost adults, having more resources to work with and are, as such, in possession of greater capacity for taking on new information. All that matters is that we take the context and our pupils into consideration when deciding our plan of action.
Evidence
When considering the evidence available to us there are three key areas to consider – retrieval strength and storage strength, as per the New Theory of Disuse, Retrieval Practice (in general), and Unison Chanting.
Much has been written about the impact of retrieval on memory with everything from making new, stronger memories when retrieving to the optimum spacing between retrieval events. Needless to say, little and often opportunities to try and retrieve number bonds can only serve to strengthen our pupils’ ability to recall them quickly when they need them most.
The research into Unison Chanting is, perhaps, less robust but there is enough consensus that it is useful for committing things to memory. There are some claims about how it contributes to understanding – but that’s not the point I’m making here as I’m not convinced at all. Our primary concern here is the automatic recall of number bonds, and (as I’ve said in the past) any curriculum worth its salt will have the “understanding” built in as standard.
There is a world where children discover key bonds and relationships for themselves, but it is a world in which the child receives a mathematically affluent upbringing full of games, conversations about number and opportunities to explore the world around them. Sadly, this is not the world that many of our children live in. If it were, then education and mathematics education would feel a whole lot different. We need to consider how our actions support pupils who are less fortunate in learning the key number bonds because it is the difference between an enjoyable, rewarding journey through mathematics in school and a hard uphill slog, during which you are continually hampered by missing pieces of kit in your bag.
At ALTA we have designed a resource which we know will be really useful for schools. It is based on a resource I made in my last group of schools and one that we used from nursery upwards to support pupils in learning number bonds before they needed to use them. We’re not trying to replace things like Numbots, Mastering Number, or Number Sense – we think they’re all truly wonderful – but we do think that we have something that can support teachers in whole-class and intervention settings when the order of the day is the automatic recall of number bonds. Something that will make life easier for teachers, save time, and provide our pupils with access to mathematics in all its glory.
If this has piqued your interest then why not check our resources out at www.alta-education.com/bond-builders or contact us at support@alta-education.com to find out more?
We hope you take advantage of Bond Builders but even if you don’t, our guiding principles outlined above are free for all to use.
Until next time, thanks for reading.