Designing a deep and durable primary mathematics curriculum
By KEVIN HUBBARD
Following a report from the Organisation for Economic Co-operation and Development (OECD, 2014) into the findings of the 2012 PISA tests, it was suggested that 15-year-old students from education systems such as Shanghai and Singapore are, on average, up to three years ahead of 15-year-old students in England. Following this, the NCETM (National Centre for Excellence in the Teaching of Mathematics) created a national network of Maths Hubs, and has invested in supporting schools to develop a maths mastery approach – a methodology commonly used in high-performing jurisdictions in Asia that involves building conceptual understanding, language and communication and mathematical thinking to problem-solve effectively (Drury, 2014).
So how do schools design a whole-school curriculum to enable the key elements of such an approach to be implemented effectively? This article provides key questions that any teacher – whether they teach mathematics or not – might want to consider.
What are the big ideas underpinning the curriculum that I am going to be teaching?
Not all maths concepts are equal: some are far more important than others. Teachers need to be clear about which are the crucial concepts – or big ideas – and ensure that these have an appropriate amount of exposure across lessons each year. Mccrea (Mccrea, 2017) recommends building in sufficient retrieval practice to develop ‘automacy’ for the most indispensable concepts, enabling children to recall them rapidly and effortlessly.
Am I providing opportunities for students to explore and make mathematical connections?
The NCETM (NCETM, 2016) advocates that ‘the structure and connections within mathematics are emphasised, so that pupils develop deep learning that can be sustained’.
Using a range of well-selected concrete and pictorial representations is an effective way of encouraging students to make connections to an abstract representation, as they expose the structure of a concept. For example, using individual paper straws alongside bundles of tens and hundreds to represent three-digit column subtraction in Year 3 can be particularly useful in helping students to understand the concept. Firstly, they get to see where the number being subtracted actually comes from (which is from the whole amount that you begin with). Secondly, they can see what is actually happening during ‘exchanging’, as well as why it might be required in the first place.
How do I know where to start from?
It has been well-documented that the transition from Year 6 to Year 7 can be challenging for teachers and students alike (Ofsted, 2015) (EEF, 2017). Assessment can be a useful tool to find the correct starting point. Open lines of communication between primary and secondary schools, mutual respect and a willingness to learn from each other can make a big difference. Maths leads need to engage with each other, share development opportunities, visit each other in their schools, and talk to students about their learning and their progress.
Am I going slow enough?
The NCETM (2016) promote the notion that ‘significant time is spent developing deep knowledge of the key ideas that are needed to underpin future learning’. Once the ‘big ideas’ are established, it is important to explore them in depth. The NRICH website is a fantastic source of problems and activities that can slow students down to ensure that they are thinking deeply (https://nrich.maths.org).
For those students who are ready to go deeper, simply asking them to find more than one solution or to find all possibilities (and then prove that they have done so) will significantly slow them down and encourage thinking at greater depth. A mantra that I have advocated in this respect is: ‘Don’t practise until you get it right; practise until you can’t get it wrong.’
Where are the opportunities to use and apply the skills I have taught?
One way of deepening a concept that has been taught is by applying it to different contexts. This could take the form of problem-solving or cross-curricular links. However, cross-curricular links have to be relevant and accessible, so they should only be made when using a maths skill that the children are already secure with – the purpose is to reinforce the learning.
To summarise, here are the key points any teacher should consider when designing a deep and durable primary mathematics curriculum:
- Know what concepts are most important
- Make connections
- Find out what the children already know
- Slow down
- Use retrieval strategies in problem-solving and cross-curricular activities.
Mathematics in and across the curriculum
By CAROLINE LOCKE
Mathematics has long been considered a core subject. It was one of three subjects offered to school children in Roman Britain – alongside reading and rhetoric (Orme, 2006) – and has continued to be a core part of the curriculum in the UK and internationally for over 1,500 years. Every student in the UK studies mathematics until the age of 16, and the UK government are considering whether to make it completely compulsory up to age 18. It is therefore timely to ask: why do we teach mathematics? What purpose does it serve and is our curriculum working towards that purpose?. The value of studying mathematics for its own sake is well-documented, but it also serves a wider purpose within and beyond the school curriculum.
Mathematics and technology
With the increasing dependence on technology across various industries, Malyn-Smith and Smith (Malyn-Smith and Smith, 2013) conclude that new skills are required for young people to compete in the workplace. They argue that ‘computational thinking’ and the ability to ‘model’ and ‘think critically’ about results from computer calculations are essential for success in the digital age and that these skills derive from a strong mathematics education (Malyn-Smith and Smith, 2013). Mathematics education can equip students with the skills to interpret data and calculations, to create models of situations, and to be able to critique and refine existing models.
The government has specified that ‘The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.’ (DfE, 2016) Mathematics Education Innovation (MEI), a charity and curriculum development body, has produced guidance on integrating technology into schemes of work (http://mei.org.uk/integrating-technology), and highlights that students will need access to either a graphical calculator and/or a graphing tool such as GeoGebra or Desmos, both mathematics web applications.
It is important not to focus too heavily on developing interpretation and problem-solving skills. A 2017 review of the USA mathematics curriculum found that this approach had reduced memorisation of facts to the point that students struggled with numeracy, and cognitive science suggests that there must be fluency in the basic facts to reduce cognitive load sufficiently for problem-solving (Nelson, 2017). The systematic and regular use of retrieval practice to build fluency can help to lay the foundations upon which to build problem-solving and interpretive thinking.
Mathematics also has strong links to other curriculum subjects. Rich et al. (Rich et al., 2013) conducted a study on the ‘convergent cognition’ of mathematics and computer science and, as might be expected, found an extremely strong link between the two subjects, such that developing mathematics skills has a concrete correlation with computer science, above and beyond the general positive correlation found with most subjects. Less predictably, Huckstep (Huckstep, 2007) discusses the mathematical foundations of art and music, including the use of perspective in art and the sequences of numbers that reveal the structure and mood of a poem.
Mathematics may, then, inform and aid understanding of other areas of the curriculum. To fully capitalise on mathematics’ rich links to other subjects, it is important for mathematics teachers to have opportunities to jointly plan with teachers of other subjects (Yuan, 2015).