Schools are expected to begin implementing the national curriculum this term. We are planning a series of articles that examine the curriculum and particularly the areas of mathematics where there have been significant changes. To start us off, in this article we explore the aims that you will find in the introduction to the national curriculum for mathematics. Many will be familiar with these aims which are to ensure that all pupils:
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
can solve problems by applying their mathematics to a variety of routine and non- routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
The first aim – mathematical fluency
The first aim stresses the importance of children developing fluency in and a conceptual understanding of the fundamentals of mathematics. Let’s unpick what is meant by fluency:
Children need to know their number pairs for all numbers to 10 and then 20 and 100. They need to know their multiplication facts up to 12 x 12. Not only do they need to know these facts, they also need to be able to use them to generate other facts. If they know, for example, that 63 + 37 = 100, they should be able to use this information to work out, quickly, calculations such as 163 + 137 = 300, 300 – 137 = 163, 300 – 163 = 137. If they know that 8 x 7 = 56, they should be able to work out quickly that 8 x 70 = 560, 8 x 35 = 280, 0.8 x 35 = 140 and so on by using strategies such as multiplying and dividing by 10 and doubling and halving.
The children need to develop a bank of mental calculation strategies, which is something that the National Numeracy Strategies (NNS) first bought to many teachers’ attention in 1999. It is important that the children are able to use efficient strategies such as: counting on to find a small difference, rounding and adjusting when adding and subtracting near multiples of, for example, 10 and 100, using near doubles. It is a requirement that children are taught the formal written methods for the four operations. It is also required that they develop the ability to choose an appropriate and efficient strategy to calculate. This might be using a written method, or, if more efficient, a mental calculation strategy. More about this in the next article!
Conceptual understanding is crucial if children are to succeed in mathematics. Many schools have begun to use manipulatives (such as Dienes, Numicon and place value counters) to enable the children to gain a conceptual understanding of the procedures that they are taught. In good schools this has replaced rule based learning where children were simply taught procedures without necessarily understanding the structure of the maths they were learning. Procedural fluency and conceptual understanding should go hand in hand at all times.
The second aim – reasoning mathematically
The second aim talks about reasoning mathematically. Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a child’s success in mathematics. It is therefore crucial that we provide our children with opportunities to develop the ability to give reasons for their thinking and actions, for them to draw inferences and make deductions, to use precise language to explain what they think and to make informed judgements and decisions. You might find it helpful to look at the section of the NCETM’s National Curriculum microsite entitled ‘Developing opportunities and ensuring progression in the development of reasoning skills’ which gives suggestions on how to embed reasoning skills within all the mathematical curriculum requirements.
The third aim – problem solving
The third aim is that children will be able to solve problems. One of the main reasons that we learn about mathematics is to solve problems. Therefore it is important that we provide plenty of opportunities for the children to apply the skills, knowledge and understanding that they have acquired to solve a variety of problems. The paragraph after the aims states that: “Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects. “ We therefore need to help the children to see the ‘big picture’ of what they are learning. We should be helping them to make connections within all areas of mathematics and also look for opportunities to apply mathematics to the other areas of the curriculum, for example, science, geography, art and music. You might be interested in reading about an NCETM project undertaken by seven primary schools, who each took one area of the curriculum and developed a project that linked their area with mathematics. This was a very successful project and enriched the learning of the children involved. You will find this in the microsite Maximising opportunities for mathematical learning across the primary curriculum.
Another important paragraph under the aims states that:
The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace.
However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on. There is no time to consider this now but it is a challenging paragraph and we will be looking closely at what this means and how we can achieve it in a future article. We hope you have found this helpful. The aims of the curriculum are very sound and what we should want for all the children that we teach. So keep these in mind as you begin to plan and teach from the national curriculum.
Many thanks to our guest blogger Caroline Clissold
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