OCR’s Level 2 award specification in Thinking and Reasoning Skills states:
The term Thinking and Reasoning Skills is used to denote not only the well established critical thinking skills of analysis, evaluation and synthesis but also a far wider and more extensive range of thinking skills including problem solving, information processing and creative thinking. These thinking skills are classed together as Thinking and Reasoning Skills because together they are essential to mature, developed thinking, whether in the classroom or in the laboratory or in the public place of work.
Developing numerical reasoning skills in the National Literacy and Numeracy Framework (LNF) reflects all of these skills in the process of applying and using procedural (mathematical) skills in a variety of contexts.
In the classroom, a teacher’s questions are central to the development of learners’ reasoning. They prompt learners to analyse, justify and evaluate their problem-solving strategies. For example, learners can be asked to revisit data in a systematic way so that conjectures about patterns and relationships in the data are more focused. Prompting learners can strengthen the validity of their justifications by checking whether they have tested all possible cases, or found all the possibilities that meet particular conditions.
Several different questions can be useful in probing learners’ thinking.
- Why do you think that . . . ?
- Can you explain why that is right?
- How do you know?
- How did you reach that conclusion?
- What might explain that . . . ?
- How is that possible?
- Can you show me . . . ?
- Is there another way . . . ?
- What explanation do you think is best . . . ?
- Have you tried all the possible cases?
- Does it always work? Why?
- What do you notice when . . . ?
Once learners’ thinking is secure, then ‘What if . . . ?’ questions can be used to promote new ideas and to extend the scope or context of the problem (Mathematical Reasoning).
It is essential that learners are provided with opportunities to practise and extend their skills in these areas and to gain confidence and competence in their use. These opportunities should be age appropriate and in context, relevant to the subject area.
The impact of developing numerical reasoning skills on later achievement
A longitudinal research report involving some 4,000 learners in England entitled Development of Mathematics Capabilities and Confidence in Primary School (Nunes et al, 2009) (external link) looked at the development of competence in different aspects of maths and the effect of this on young people’s key stage results. Key research findings included the following.
- Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics in Key Stages 2 and 3.
- Spatial skills are important for later attainment in mathematics, but not as important as mathematical reasoning or arithmetic (although the teaching and testing of geometry and spatial competence gains in importance beyond Key Stage 2).
- Children’s attention and memory plays a small but consistent part in their mathematical achievement.
- 4. Children from high socio-economic status backgrounds are generally better at mathematical reasoning than their peers.
- 5. Streaming, or ability grouping, in Primary school improves the mathematical reasoning of children in the top ability group, but the effect is small. It hinders the progress of children in the other groups.
- 6. Children’s self confidence in maths is predicted most strongly by their own
competence but also by gender (girls are less confident than boys) and by the ability group in which the child is placed. Children’s attainment, although largely determined by cognitive and social factors, is also influenced by their self-confidence.
(Nunes et al, 2009, p.1)
Read the introduction on pages 1–5 of the Development of Mathematics Capabilities and Confidence in Primary School report (external link).
- How are mathematical reasoning and arithmetic defined by the researchers?
- What key implications for practice do the researchers suggest result from their findings?
- Follow up on one of the research findings within the report to identify and understand more fully why the researchers have drawn that particular conclusion.
Numerical reasoning within the LNF and Essential Skills Wales (ESW)
Numerical reasoning is also needed to solve real-world problems. According to the Welsh Government (2012):
Through reasoning, people are able to recognise how to use numbers to tackle a real-life situation, and planning a strategy to solve it. In a variety of situations, the ability to reason using numbers enables people to access and understand information such as sports statistics, reading maps, building an object to scale, managing money and others. Without this skill, people will not be able to use their mathematics ability in the real world.
Numerical reasoning is a key strand within the LNF for Wales. The elements within this strand refer to the skills learners need to identify what processes are needed to solve a real-world problem, how to express that approach in their workings, and how to draw conclusions by reviewing their own processes and answer for reasonableness. The elements are:
- identify processes and connections
- represent and communicate
Look in more detail at these elements within the Developing numerical reasoning strand in the LNF. How do these process skills compare and/or contrast with:
(a) the PISA study notion of ‘mathematical literacy’ as concerned with the ability of learners to analyse, reason, and communicate ideas effectively as they pose, formulate, solve and interpret solutions to mathematical problems in a variety of situations
(b) the Essential Skills Wales ‘Application of Number’ standards which are concerned with developing and recognising learners’ ability to select and apply numerical, graphical and related mathematical skills to tackle a task, activity or problem by collecting and interpreting information, carrying out calculations, interpreting results and presenting findings in ways appropriate to their particular context
(c) the ‘using and applying’ strand (Attainment Target Ma1) of the GCSE mathematics curriculum that is concerned with learners being able to tackle substantial tasks, analyse complex situations, interpret mathematical information and communicate findings.
- Within your own teaching context, do learners have to use any numerical reasoning skills? Which? When? Why?
- Are these numerical reasoning skills being used when your learners are applying number, measuring, or data handling skills?
- How do you recognise, value and develop these numerical reasoning skills explicitly with learners?
- How do you evidence this in your planning for learning, in your resource development, in your lesson delivery and in your assessment?
PISA tries to examine how well prepared young people are in terms of their mathematical literacy and problem solving skills to meet the challenge of living in the twenty-first century. In PISA, the fundamental process that students use to solve real-life problems is referred to as mathematisation. According to the OECD (2009, p.105) there are five stages involved in mathematisation.
- Start with a problem situated in reality.
- Organise it according to mathematical concepts and identify the relevant mathematics involved.
- Gradually trim away the reality through processes such as making assumptions, generalising and formalising. These processes promote the mathematical features of the situation and transform the real-world problem into a mathematical problem that faithfully represents the situation.
- Solve the mathematical problem.
- Make sense of the mathematical solution in terms of the real situation, including identifying the limitations.
The OECD (2009, pp.106–107) also identifies eight characteristic competences required to mathematize.
- Thinking and reasoning.
- Problem solving and posing.
- Using symbolic, formal and technical language and operations.
- Use of aids and tools.
These competences are targeted within PISA assessments and may be useful in identifying ways we might expect to see learners working mathematically or using numerical reasoning skills.
Look at some exemplar questions from PISA (2009, pp.85–104). As you read the commentaries, try to identify which of the eight competencies are being used in each of the stages of mathematisation.
As outlined in Topic 1 (Numeracy and society), there is convincing evidence (Smith Inquiry, 2004) to suggest that many post-14 learners are unable to transfer their mathematical skills to different situations or use them to solve problems. Wake (2005) suggests this is in part due to many learners’ experiences of classroom mathematics as:
- application of tools and techniques in a limited range of ‘classroom maths’ problems – doing without sense making
- mainly ‘pure’ maths with little opportunity to model in authentic contexts – hard to transfer to new or different situations
- not involving real data – no ‘messy’ numbers or measures from ‘genuine’ real world situations/scenarios (e.g. pie charts representing 725 people)
- not being required to communicate using mathematics in meaningful or purposeful ways.
Habits of mind
De Lange (2001) suggests that numerical reasoning involves ‘habits of mind’ but that compared with traditional school mathematics, mathematical literacy is:
. . . less formal and more intuitive, less abstract and more contextual, less symbolic and more concrete . . . (It) focuses more attention and emphasis on reasoning, thinking and interpreting as well as on other very mathematical competences.
Wake (2005) argues that such ‘habits of mind’ allow learners to quickly make sense of new situations in ways that would not otherwise be possible. He suggests learners need to learn how to:
- make sense of
- seek structures
- use quantitative arguments.
Implications for practice
From their research and classroom experimentation with primary and secondary learners in the USA, Pearse and Walton (2011, p.7) identify nine critical habits ‘to ignite numerate thinking’. They suggest teachers need to encourage learners to develop their numerical reasoning in everyday decisions through:
- monitoring and repairing understanding, developing schema and activating their background knowledge
- identifying similarities and differences, recognising patterns, organising and categorising ideas, investigating analogies and metaphors
- representing mathematics non-linguistically
- predicting, inferring, recognising trends, using patterns, generating and testing hypotheses
- questioning for understanding
- summarising, determining importance, synthesising
- developing vocabulary
- collaborating to learn.
- Choose a couple of these ‘critical habits’ related to numerical reasoning and generate some practical strategies and examples from your own work with learners to show how these could be developed within your subject area.
- Identify how and where these would fit within:
- your subject specifications
- the LNF
- the Essential Skills Wales Application of Number specifications
- the ‘habits of mind’ and OECD competency characteristics outlined above.
Nunes at al (2009, p.3) differentiate between arithmetic as ‘learning how to do sums and using this knowledge to solve problems’, and mathematical reasoning as ‘learning to reason about the underlying relations in mathematical problems they have to solve’.
‘Monitoring and repairing understanding, developing schema and activating their background knowledge’ might refer to predicting, estimating and questioning if the answer is reasonable. It might require slowing down, rereading of the question, looking again, identifying ‘sticking points’ (e.g. word or phrase), summarising progress so far, seeking help, connecting to background knowledge, visualising, raising new questions or using a different strategy.
De Lange (2001) Mathematics for Literacy. In: Steen, L.A. (Ed.) Mathematics and Democracy: The Case for Quantitative Literacy, pp75–89. Princeton, NJ: National Council on Education and the Disciplines. (online) (accessed 29 September 2013).
Pearse, M. Amd Walton, K.M. (2011) Teaching numeracy – 9 Critical Habits to Ignite Mathematical Thinking. California, Corwin, Sage.
Wake, G. (2005) Functional Mathematics: More than “Back to Basics”. Nuffield Review of 14-19 Education and Training, Aims, Learning and Curriculum Series, Dicussion paper 17 (online) (accessed 29 September 2013).