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MEP learning pack

Masters in Educational Practice: Numeracy learning pack

3

Learning and teaching numeracy

3.3

Developing procedural skills

Procedural skills within the National Literacy and Numeracy Framework (LNF)

According to Welsh Government (2012), once numerical reasoning skills have been used to identify how to tackle a problem, a separate set of mathematical procedures are needed to reach a solution. Within the numeracy component of the LNF these procedures are broken down further into three additional strands.

  • Using number skills – the fundamental skills needed to be comfortable with using and manipulating numbers when carrying out procedures.
  • Using measuring skills – knowing what measurements to use in which context, what standard units to use and to what precision.
  • Using data skills – representing the results of tackling a problem that involves handling data in several ways.

Procedures and concepts

Nunes and Bryant (2012, p.5) state that:

Although it is important to make the distinction between concepts and procedures, learning numerical procedures does depend on children understanding related numerical concepts and principles.

As a result, learning procedures without understanding underlying numerical concepts will often result in children making mistakes in non-standard tasks that require them to adjust the ways they are used to tackling problems. Using non-standard tasks according to Nunes and Bryant (2012, p.5) helps children link concepts and procedures.

Teachers should balance the time given to procedural learning experiences and to conceptual development. An intertwining of procedural knowledge, conceptual knowledge, a can-do attitude and logical thinking are all necessary for numeracy proficiency.
(National Research Council, 2001).

Activity 3.18

1. Within your own teaching context, do learners have to use any number, measuring or data skills? Which? When? Why?

Look in more detail at the elements within one of the procedural skill strands of the LNF that is particularly relevant to your work with learners.

2. How do these elements compare and/or contrast with how learners use these procedural skills within your subject setting?

You may want to talk to colleagues to find out what sorts of ‘standard tasks’ are used with these elements in ‘mathematics’ lessons in different years and across phases.

Representation and context

Most numerical procedures that learners use are human inventions, e.g. counting systems, measuring systems, arithmetical procedures and algorithms for long multiplication and division. Providing some historical and global context to how and why such procedures have developed can help learners engage more meaningfully with these tools.

Concrete representations of number and use of practical equipment are common in early years’ classrooms but most learners are then quickly moved onto entirely symbolic representations of number in written form. In countries like Singapore, more extensive use of hands-on and visual representations are encouraged for longer and have helped learners work with diagrams to represent quantities and the relationship between quantities in numerical problems.

Activities to develop numeracy skills, concepts and principles with young children

A great deal of research is available to provide examples of classroom activities that support development of number, measure and data skills alongside numerical reasoning. These include teaching young children about:

  • correspondence and cardinal number
  • part-whole relations and additive reasoning
  • one-to-many correspondences and multiplicative reasoning
  • chance and probability
  • using measuring skills and thinking about shapes and space
  • using data skills.

Activity 3.19

Read Nunes and Bryant’s 2012 Report on the plans for a new National Numeracy Framework for schools in Wales.

  1. Find out how specialist mathematics and numeracy colleagues within your institution are using these approaches in their teaching.
  2. How could you develop and reinforce more hands-on and diagrammatic approaches within your own subject area when learners are using number, measuring or data skills?
  3. What sorts of contextualised realia, equipment, tools, and visual resources could you develop and use in your subject area?

Relational understanding

Skemp (1976) argues that learners should be encouraged to develop a relational understanding of number rather than simply an instrumental understanding (refer to Glossary). That is, an understanding of the properties of and the relationships between numbers and between operations, so that in solving new problems they can draw on this and their everyday life strategies.

Nunes and Bryant (2012, p.6) also highlight the importance of context in numerical classroom activities as familiar and interesting topics can promote effective problem solving, e.g. building on what learners know about money can help support use and understanding of base ten, negative numbers, percentages and proportion.

Welsh Government (2013, p.20) also stress that:

Numeracy is different to the mathematics subject in that it is the application of the skills learned in mathematics in a cross-curricular, real-world way, and not purely about the skills themselves.

Procedural fluency

Ginsburg et al (2006, p.28) identify procedural fluency as ‘an essential component in completing many numeracy tasks that require efficient and accurate calculation’. These may include pen and paper procedures, use of mental strategies, estimation techniques, and methods that use calculators, computers or other mobile and digital technologies. The choice of method will depend on the purpose, context and actual numbers involved within a task or problem scenario.  

Several researchers have observed that procedural fluency in using written methods to carry out calculations does not require a particular ‘textbook’ algorithm. In real life, a range of alternative strategies based on sound reasoning are often used flexibly and fluently to find answers successfully. These may be unique to a particular home or work setting, or learned in different countries (Scribner and Stevens, 1989; Schmitt, 2006).

Other studies (Dowker, 1992; Northcote and McIntosh, 1999) conclude that calculation is mostly estimation and mostly mental whether for mathematicians or adults in everyday life. Often such mental procedures involve greater insight and understanding of the relationships between numbers than traditional written methods Similarly, estimation often requires higher order reasoning and the integration of numerical reasoning and procedural knowledge. Even use of calculators and computers does not remove the need for understanding what the operations do and whether they are appropriate for a particular situation (Ginsburg et al, 2006).

Approaches

In their Maths4Life booklet on ‘Number’ (2007) (external link), Newmarch and Part provide several relational approaches to use with post-16 learners working from entry Level 2 to Level 1. These include:

  • linking the four rules of number
  • making connections with other mathematics topics and links with learner contexts
  • examples of questions learners frequently ask about numbers
  • developing alternative mental strategies for calculating with number
  • teaching points
  • the language of number
  • examples of sorting, matching, classifying and evaluating number activities
  • problem solving
  • estimation and checking strategies
  • links to other subject areas and everyday life
  • analysing misconceptions
  • suggestions for resources.

In their Maths4Life booklet ‘Topic-based teaching’ (2007) (external link) Ness and Bouch suggest that topic-based approaches can also help older learners make more sense of mathematical procedures because meaningful connections can be made with the knowledge and experience learners already have. Such approaches also allow different topics and skills to be integrated within a topic.

Activity 3.20

Choose one of the Maths4Life booklets discussed in this section.

  1. Read through the booklet carefully, making a note of any new or unfamiliar approaches.
  2. Choose one or two ideas that might be useful and discuss with a colleague how you might adapt these to use with your own learners to help develop their numeracy skills? To help support their ‘main’ learning programme?
  3. Reflect on the ‘do’s’ and ‘don’ts’ at the end of the booklet. Are there any implications here for your own practice?

Glossary

Instrumental understanding – learning a process off by heart, such as ‘area = length x breadth’ or using a formula A = L x W without understanding what ‘area’ is.

Relational understanding – understanding why length is multiplied by width to find area and understanding what area is.

References

Dowker, A. (1992) Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23(1), pp.45–55.

Ness, C and Bouch, D. (2007) Topic-based teaching. London, NRDC. (online) (accessed 3 October 2013).

Newmarch, B. and Part, T. (2007) Number. London, NRDC. (online) (accessed 3 October 2013).

Northcote, M, and McIntosh, M. (1999) What mathematics do adults really do in everyday life? Australian Primary Mathematics Classroom, 4(1) pp.19–21.

Nunes, T and Bryant, P. (2012) A report on the plans for a new National Numeracy Framework for schools in Wales. Cardiff, ISIS Innovation.

Nunes, T., Bryant, P., Sylva, K. and Barros, R. (2009) Development of maths capabilities and confidence in primary school. University of Oxford, DfE. (online) (accessed 29 September 2013).

Organisation for Economic Cooperation and Development (2009) PISA 2009 Assessment Framework Key competencies in reading, mathematics and science. Paris: OECD. (online) (accessed 29 September 2013).

Schmitt, M. J. (2006) The answer is still the same. It doesn’t matter how you got it. Snow Camp, NC, Peppercorn Press. (online) (accessed 3 Oct 2013).

Scribner, S. and Stevens, J. (1989) Studying working intelligence. In Rogoff. B. and Lave. J.(eds) Everyday cognition: Its development in social context (pp.9–40). Cambridge, MA, Harvard University Press.

Skemp, R. (1976) Relational Understanding and Instrumental Understanding, in Mathematics Teaching, 77 pp.20–26.

Welsh Government (2008) Mathematics in the National Curriculum for Wales. Cardiff, Welsh Assembly Government. (online) (accessed 3 October 2013).

Welsh Government (June 2012) Developing numerical reasoning from the numeracy component of the LNF (online) (accessed 29 September 2013).

Welsh Government (2013) National Literacy and Numeracy Framework: To support schools in introducing the National Literacy and Numeracy Framework (online) (accessed 3 October 2013).

National Research Council (2001). Developing Mathematical Proficiency (online) (accessed 6 October 2013).

Useful links

The Programmes of Study at Key Stages 2, 3 or 4 within the statutory Mathematics in the National Curriculum for Wales (2008) (external link)

The skills and competencies required at different levels within the standards for Essential Skills Wales Application of Number (2009) (external link)

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