National Curriculum in Focus
National Curriculum in Focus is dedicated to unpicking the new curriculum and how to understand and develop the requirements of the new programmes of study for mathematics. You can find previous features in this series here
Assessing the Aims: Part One - Fluency
This is the first of three articles focused on assessment of the aims of the National Curriculum
The National Curriculum for mathematics aims 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”1
As stated in the NCETM Teaching for Mastery booklets:
“Progress in mathematics learning each year should be assessed according to the extent to which pupils are gaining a deep understanding of the content taught for that year, resulting in sustainable knowledge and skills. Key measures of this are the abilities to reason mathematically and to solve increasingly complex problems, doing so with fluency, as described in the aims of the National Curriculum.”2
As teachers and schools grapple with decisions about assessment it will be important for them to consider how the aims are reflected in their:
- assessment principles
- assessment criteria, and
- assessment practice.
The first of the aims, fluency, involves pupils making and justifying decisions and using what they know and understand to solve problems. Making decisions requires pausing before engaging with ‘doing’ the maths.
The need to pause before engaging with solving any problem is necessary in order to provide the opportunity to notice things. This is an important element of fluency which is not always appreciated by children. Noticing is crucial to good decision-making.
It is possible for children to get correct solutions without making good decisions; for example: in Year 2, calculating 40 + 8 by counting in ones, in Year 4 calculating 1003 – 998 using a formal written method; and in Year 6 calculating 41.79 + 25.3 + 25.7 – 41.79 by adding the first three numbers and then subtracting the fourth. Accuracy alone does not indicate fluency.
Assessment of fluency must, therefore, be more than assessment of children successfully ‘doing’ some maths and getting correct answers; it should include assessing whether children take a flexible approach and make appropriate decisions, based on what they notice and what they already know and understand.
Providing opportunities for children to focus on decision making can be achieved in a number of ways; the following suggestions are illustrated with examples from the Teaching for Mastery booklets:
- Making a direct connection between known facts and related calculations
With this example, the children could also be asked to write another set of calculations using the same known facts in order to further probe understanding.
- Comparing different methods and identifying which is most efficient
When comparing methods, it is useful to ask children to consider what has been noticed and why this is useful in each method, as well as whether there is a calculation they can think of when the method they have identified as more efficient would no longer be more efficient. It is important that they understand that no one method is going to best for all situations, hence the need to make decisions. For example, a child might suggest that rounding and adjusting is the best way to solve £163 - £28 but this might not be the best choice for £762 - £462.
- Considering how to use a given fact to work out related facts
This example reflects all three elements of fluency as described below:
“Fluency rests on a well-built mathematical foundation with three parts:
- an understanding of the meaning of the operations and their relationships to each other - for example, the inverse relationship between multiplication and division;
- the knowledge of a large repertoire of number relationships, including the addition and multiplication "facts" as well as other relationships, such as how 4 x 5 is related to 4 x 50;
- a thorough understanding of the base ten number system, how numbers are structured in this system, and how the place value system of numbers behaves in different operations – for example, that 24 + 10 = 34 or 24 x 10 = 240."3
- Making a situation easier to deal with
Seemingly difficult calculations can often be quite simple when they are viewed as a whole rather than a series of unconnected parts. This can start in KS1 where children should be expected to identify what makes calculations such as 7 + 15 – 7 easy.
Assessing fluency will, therefore, include assessing efficiency, accuracy and flexibility. Embedding this in assessment systems and practices is one of the challenges for schools at this time. Next month we will look at assessing reasoning.
- National Curriculum 2014
- Teaching for Mastery NCETM 2015
- Developing Computational Fluency with Whole Numbers in the Elementary Grades Russell 2000
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