### Overview

Every learner has the potential to learn and it is the information and structures present in the environment that create the conditions for learning. Teachers must acknowledge and respond to their learners' cultural, socio-economic, ethnic, linguistic and learning profiles. They must build on learners' prior knowledge, skills and understandings and respond to these individual learning needs with specific and targeted learning. High expectations, sufficient time, adequate support and focused learning provide the conditions for effective numeracy learning (Northern Territory Government, Australia) (external link).

The *Curriculum planning guidance* document (Welsh Government 2013) states that a review of the [school’s] current position will hone teachers’ thoughts as ‘to how classroom practice needs to be changed to ensure opportunities for developing literacy and numeracy skills’ (p.12). Teachers need to include rich numeracy experiences as part of their day-to-day learning and teaching programmes and focus learners on the importance and relevance of developing numeracy skills. This needs to be across the whole school and prove a consistent and cohesive approach to embedding numeracy in the curriculum. The choice of resources and context should be relevant to the learners’ interests, abilities and learning needs. For example:

- interventions should be available for those who need them, whether they are the lower achievers or those on the boundaries of the higher levels
- the use of the outdoor classroom
- the use of mobile technology to enhance learning
- work-related opportunities for developing numeracy skills
- strong home-school links to enable parents/carers to support learners in numeracy, including workshops to develop the numeracy skills of the parent/carer.

### The classroom ‘milieu’

Learning takes place within an interactive environment of social forces and classroom practices – what Brousseau (1997) calls a ‘milieu’. As well as the teacher, the learner, and the particular subject area, this ‘milieu’ also includes the learning and teaching environment.

- The classroom ethos and the characteristic ways of working.
- The degree to which learners and teachers are responsible for ensuring learning.
- The classroom organisation (social structure, resources, etc.).

Each of these will influence how the teacher, individual learners and the class as a whole respond to tasks and activities involving numeracy learning.

Mason and Johnston-Wilder (2006, p.35) identify ways in which some aspects of ‘milieu’ can interact to influence the effectiveness of teaching and learning numeracy.

- Learners’ intention and desire to learn mathematics and numeracy.
- Learners used to specialising and generalising are more likely to engage in numerical reasoning.
- Learners used to precise instructions are more likely to struggle with open-ended, creative tasks.
- Learners labelled as ‘low attainers’ in mathematics may have low expectations of themselves and may limit what they attempt or how hard they try.
- Learners labelled as ‘high attainers’ in mathematics may feel under pressure to perform or be over-confident.
- Learners who feel uncomfortable in a room will find it more difficult to focus their attention.
- If the resources are inadequate, then the task carried out will be different from the task intended.

### Collaborative and small group problem-solving

Restructuring the learning environment to include smaller, collaborative, problem-solving groups can support the development of numerical reasoning and communication. According to Ginsburg et al (2006, p.36):

Learning to be numerate and to function numerately outside the classroom is best done as a social activity, in which people brainstorm strategies, propose alternatives, articulate and defend their reasoning, and learn from one another.

In her research, Jo Boaler (1998, 2000) looked at groups of learners over a three year period in two different secondary schools in England. She found that young people learning mathematics in environments that used open-ended, project-based activities developed better conceptual understanding, reasoning and problem-solving skills. On assessments involving procedural skills, these students performed as well as similar learners using more traditional textbook-based approaches. However, they far outperformed on applied, realistic tasks that used the same mathematical content contained in the computational assessments.

Boaler (2000) argues that this was in part due to:

- learner awareness of the different constraints and affordances found in multiple situations
- learner engagement in practices similar to those in the real world
- learners recognising that ‘cues’ specific to a maths classroom, may be of little or no use in problem solving in other subject areas or outside the classroom.

Swain et al (2007) also suggest that group work and collaborative learning supports the development of procedural skills and numerical reasoning because learners can share knowledge, explore each others’ strategies and bounce ideas off each other.

They learn to make their learning visible and to listen to each other when they have to explain their reasoning, justify their strategies and answers. Thinking aloud consolidates their understanding and also involves learning to use appropriate language. In joint problem solving learners are more likely to estimate and develop checking strategies because they are discussing work.

(Swain et al, 2007, p.19)

# Activity 3.29

- What are the implications of Boaler’s research for how you organise your teaching and learning environment? Think about classroom furniture, learner groupings, resources, activities, displays, learning support, assessment.
- There is a difference between working
**in**a group and working jointly**as**a group to problem solve and develop numerical reasoning. What challenges might this sort of working pose for the teacher, the learners, the resources and the learning space?

### Resources and representation

Jerome Bruner’s (1966) three modes of representing our experiences are regarded as important to the development of children’s understanding in mathematics and numeracy.

- The enactive mode (taking some form of action, such as manipulating objects).
- The iconic mode (representing ideas using pictures or images).
- The symbolic mode (using language or symbols).

Children are able to solve numerical problems earlier if they use concrete materials for the ‘representation of ideas’ (Bruner 1966). This ‘enactive mode’ allows learners to ‘make predictions and to make a connection from a past experience to a new learning context (Drews 2007).

#### Further reading

For further information on Bruner’s constructivist theory refer to Learning Pack 2, 'Child and adolescent development 0–19').

#### Physical learning materials

Moyer (2001, p.176) defines ‘manipulatives’ as, ‘Objects designed to represent explicitly and concretely mathematical ideas that are abstract. They have both visual and tactile appeal and can be manipulated by learners through hands-on experiences’. Examples are: Diene’s material (multibase 10) and Cuisenaire Number Rods. These can be used to model the base 10 place value system. A more unstructured example would be ‘Multilink’ which is more flexible in its use (for example, to aid counting, to investigate patterns or shapes, etc.) (Drews, 2007).

# Activity 3.30

1. Consider which of the following statements on the use of physical learning materials to teach number in the early years you agree with.

- These are very important in helping to move children’s learning from the concrete to the abstract.
- These definitely help young learners to understand how we use numerical symbols.
- A number line is better (than concrete objects) for helping children to count because it helps them to count on and back.
- They can help children to solve problems, but they do not necessarily help them to learn how to solve problems ‘internally’.
- They might be used by learners in ways that the teacher did not intend – this might hinder, rather than support learning.
- Concrete materials should always be available to all children, irrespective of age.
- Using concrete materials is important because it makes it more fun for the learners.

2. Read the paper by Manches (2008) found in From research to design: Perspectives on early years and digital technologies (external link). In this paper he considers the role that physical learning materials plays in the learning of early number skills and the potential for digital technologies in supporting learning, through enhancing physical learning materials.

Now reconsider your responses to the statements in Activity 1. Use your professional journal to note any significant points of learning, aspects for further study or development of personal practice, for example:

- ways in which manipulatives might support early numerical development
- arguments against manipulatives
- conclusions about manipulatives and the arguments against them.

While the use of manipulative objects is often most commonly seen when teaching young children, Edwards (1988, p.18) argues that mathematical understanding is brought about for **all** children through the interconnection of the three modes of representation.

In other words, manipulation has a key part to play in how learners develop mental images and mental strategies. It is therefore important for the teacher to plan activities and learning conversations which take account of past experiences and make explicit links to the practical activities which learners have experienced while also encouraging them to develop ‘mental images’ (Delaney, 2001, p.128).

However, Anghileri (2000, p.10) guards against an over-reliance on the use of concrete materials, so that learners are able to develop mental imagery and work with imagined or real-life situations. She suggests that some learners might need a ‘transitional stage’ in which the materials or objects are hidden once they have been used in order to encourage learners to produce mental images of the procedures used (p.11). Teachers, therefore need to think carefully about the appropriateness of resources and how helpful (or unhelpful) they might be in supporting children to develop mental images. Cramer and Karnowski (1995) suggest that use of manipulatives should be followed by pictorial representations and then verbal and written representations. The latter are seen as critical if learners are to link informal mathematical knowledge to abstract representations and understanding.

#### Images

Harries and Spooner (2000, p.6) define three types of ‘image’ which are used by learners to scaffold their development.

**Sound images**– repeating words until they ‘sound right’ (e.g. counting).**Concrete images**– linking sound to either/both something they see and/or something they can touch.**Symbolic images**– mathematical symbols which are the representations of the sound, sight and touch images with which they are familiar.

An abacus or bead string, for example, can help children to link counting to movement. When beads are blocked through the use of colour, shape or in groups, these ‘concrete images’ can help children to develop a sense of number order and number pattern. They support children to bridge the gap between the quantity (real/concrete) and the representation of it (abstract). Progression is achieved through the use of pictorial images or shapes which are more closely aligned to the symbolic number system. For example, use of number tracks (including 100 squares), number lines, digit cards and place value cards (Drews, 2007, p.22).

As children’s understanding of the number system progresses, a more sophisticated abstract image is required. For example, a number frieze or track might be the first representation children see of the number sequence. This would then be followed by a calibrated number line (linear and curved) which allows for the continuous nature of number to be represented. Estyn (2013, p.3) identified that ‘Many pupils [in key stages 2 and 3] have difficulty working with decimals, fractions and percentages and do not understand the relationship between, for example, 2/5ths, 0.4 and 40%’. More advanced concepts such as fractions, decimals and estimation can also be explored using number lines as these help learners to visualise the relationships amongst whole numbers, fractions and decimals.

Activities which demonstrate and which allow for rational numbers to be positioned on a number line can support a greater understanding of place value, relationships between ‘numbers of a different kind’ and our overall number system: this is true for all children including the more able in mathematics.

(Drews, 2007, p.22).

A partly numbered line or empty number line can also provide the means by which mental computations can be modelled and recorded, with learners sharing their thinking strategies.

# Activity 3.31

Consider the information above. Do you think there could be an occasion where the use of number lines might hinder, rather than progress, a learner’s development in numeracy?

How might you support learners’ transition from using a number line to using mental images or visualisation?

What role does ‘recording’ play in developing learners’ mental calculations?

Reflect upon these questions within the context of sub-topic 3.4 ‘Planning for learning’.

#### Recording numeracy learning in pictures and symbols

Research by Hughes (1986) and Carruthers and Worthington (2006) has led to encouragement of children’s mathematical mark making in preschool settings. For example, young children might demonstrate the following graphical marks.

- Dynamic – lively and spontaneous.
- Pictographic – representing something they can see in front of them.
- Iconic – choosing marks based on one-to-one counting or ‘tallying’.
- Written – words or letter-like marks which are read as words and sentences.
- Symbolic – standard forms of numerals and abstract symbols (e.g. ‘+’).

(Carruthers and Worthington, 2006)

Research evidence suggests that creating an environment which gives children plenty of opportunities to explore graphical mark making reveals the development of mathematical language and cognition (Athey, 2006). Therefore, the role of the practitioner is key not only in providing stimuli and resources, but understanding and valuing children’s graphics within play and spontaneous learning contexts as part of the development of mathematical understanding.

# Activity 3.32 (relevant for practitioners in all age phases)

**‘When environments are mathematical, mathematics happens’** (Carruthers and Worthington, 2006, p.142).

Consider this statement. Could this be applied to ‘numeracy’? Now take an audit of your classroom environment. How does it support numeracy? Is there one area that you might develop? You might wish to consider opportunities for learner-initiated learning, displays, notice boards, labels and signs.

#### Diagrams

Nunes and Bryant (2012) note that children are often provided with the opportunity to work with concrete materials to solve problems, but are then expected to make a direct transition from an informal method to a conventional method. For example, from using number blocks to represent sweets in a problem (in which they ‘count on’ to find the number of sweets missing) to using subtraction. Nunes and Bryant (2012, p.18) suggest that by introducing a ‘step’ between the concrete representation (ie with blocks) and symbolic representation (i.e. with numbers and operations), learners are more easily able to make this transition to more formal methods. This transitional ‘step’ is to use diagrams to represent quantities in a problem (Greeno, 1989, cited in Nunes and Bryant, 2012).

Nunes and Bryant (2012, p.19) point out that countries that do well in international tests such as PISA (e.g. Singapore) tend to adopt a three-phase approach to representation.

Phase 1: Learners solve problems with the help of concrete materials.

Phase 2: Learners solve problems through creating and/using diagrams and graphs (‘model method’).

Phase 3: Learners solve problems using written numbers and symbols only.

The concrete-pictorial-abstract (C-P-A) approach in the development of mathematical concepts has been widely advocated by Singapore’s Ministry of Education (1990, 2000, 2006) (Wong et al, 2009).

# Activity 3.33

1. In this YouTube video Dr Yeap Ban Har explains the ‘concrete, pictorial, abstract’ approach (external link). Consider how this approach relates to Bruner’s theory of representation (1966).

2. Now view the following YouTube video on ‘Singapore Maths’ (external link) and read the text below, taken from paragraph 243 of the Cockcroft Report (1982, p.71) (Refer also to sub-topic 1.2):

243

Mathematics teaching at all levels should include opportunities for:

- exposition by the teacher
- discussion between teacher and pupils and between pupils themselves
- appropriate practical work
- consolidation and practise of fundamental skills and routines
- problem solving, including the application of mathematics to everyday situations
- investigational work.

Do you agree with the recommendations identified here? Are these recommendations made in 1982 relevant to numeracy teaching? Does your classroom environment reflect such opportunities?

3. This YouTube video presents an example of the ‘model method’ (external link) used in the teaching of mathematics in Singapore schools.

- How might this method support the development of ‘mental imagery’ and progression in mathematical understanding and skills?
- What issues does this approach raise for less able and more able learners?
- Can any connections be made between this approach and principles of progression in the National Literacy and Numeracy Framework (LNF)?

#### Further reading

Carpenter and Moser consider the appropriateness of the availability of concrete materials for use by all learners in:

Carpenter, T. and Moser, J. (1982) The development of addition and subtraction problem solving skills. In: Carpenter, T., Moser, J. and Romberg, T. (eds) Addition and subtraction: a cognitive perspective. Hillsdale, New Jersey: Lawrence Eribaum.

The following text provides useful information on the use of images in the development of mathematical understanding (Chapter 4):

Harries, T. and Spooner, M. (2000) Mathematics for the Numeracy Hour. London: David Fulton Publishers Ltd.

#### Everyday objects and materials

The use of everyday objects and materials is important for developing children’s understanding of numeracy in real-life contexts both in and out of school. The National Numeracy Programme (NNP) emphasises the need for learners to be provided with meaningful contexts in which to develop their numerical understanding and skills. Resources which help them to make connections from ‘mathematics’ lessons to everyday applications should be provided if a numeracy-rich learning environment is to be achieved. These resources could be used in whole-class teaching, small group activities, interactive classroom displays and role-play provision.

# Activity 3.34

- List the everyday objects and materials you currently have in your own classroom/setting which support numeracy learning and teaching. Examples might be: timetables, holiday brochures, utility bills, catalogues, calendars, diaries, clocks, receipts, recipes and consumer packaging. For practitioners in secondary schools, this might be subject-specific objects or materials.
- View this short video in which two teachers from Tŷ Sign Primary School discuss how they have created numeracy learning environments.
Unable to play video as javascript is disabled on this browser. Please enable javascript to play video.

- numeracy planning and provision
- role play areas and resources
- real-life or imagined learning contexts
- classroom display
- collaboration between teachers.

#### Making connections

Creating a successful numeracy learning environment depends upon a number of factors. While access to appropriate resources and the effective use of resources by both teacher and learner are extremely important, it could be argued that provision of the resources themselves will not improve numeracy:

. . . there is no mathematics actually in a resource’ (Delaney, 2001, p.124)

It is the teacher-to-learner and learner-to-learner interaction which develops learners’ mathematical thinking. This is most rich when learners are actively engaged in problem-solving tasks. One of the roles of the adult is to encourage learners to think about what they are doing when they encounter a problem and consider other ways to do it. Questioning and opportunities to articulate thinking before, during and after engagement in numeracy tasks also supports children to link **how** and **why** they used a particular resource as part of their problem-solving. This supports them to construct mathematical knowledge and understanding and to make the links between the concrete and the abstract (i.e. the symbolic representations).

Without some accompanying mental activity to reflect on the purpose and/or significance of the physical activity, concrete materials will not actually enable the child’s mathematical understanding to develop.

(MacLellan, 1997, p.34).

Haylock and Cockburn (2003) suggest that it is the ‘network of connections’ (p.8) between children’s concrete experiences, pictures, language and symbols which leads to the mathematical understanding (p.18). Montague-Smith and Price (2012, pp.15–16) take this further by including real world scripts as a fifth element which might be present in a learning experience of a mathematical concept. They suggest that one or more of the following five elements will be present in any learning experience of a mathematical concept and children will need to be able to make connections from one element to another if a sound understanding is to be achieved.

- Physical materials.
- Spoken language.
- Pictures.
- Written symbols.
- Real world scripts.

#### Further reading

Montague-Smith and Price explore the connections which learners must make between the elements of mathematical experience in:

Montague-Smith, A, and Price, A. J. (2012). Mathematics in Early Years Education. Oxon: Routledge.

#### Selection of resources

Research carried out by Moyer in 2001 revealed that teachers’ decisions on whether or not mathematical resources came from their own beliefs and attitudes. Why they teach mathematics and how it can be learnt effectively ultimately influenced the pedagogical approach they used.

Delaney (2001, p.24) poses the following questions.

- Do resources inevitably help in the learning and teaching of mathematics and where they do help, what can be said about the ways that they do?
- Are there social and political factors that affect the choice and use of resources?
- Are some resources better than others and is this affected by the context? (e.g. by the beliefs and/or attitudes of the teacher or learner)
- Do learners use the resources you provide? Are any learners unwilling to use resources even if available? Consider the issues raised in sub-topic 2.5 about learners’ beliefs and attitudes when responding to this question.
- How can teachers develop their use and understanding of resources?
- Do teachers recognise the strengths and weaknesses of particular resources and are they able to adapt them to different circumstances? (e.g. through demonstration or modelling by the teacher, or for use by individuals or groups)

# Activity 3.35

- Delaney encourages practitioners to consider these questions in relation to their own practice and beliefs. Consider these not just in relation to mathematics teaching, but in relation to your numeracy practice and beliefs.
- At the end of this topic, reflect upon key learning ‘moments’ in your professional journal.

You might wish to consider how your knowledge and thinking has developed. Will anything explored within this topic, cause you to make a ‘change’ in how a ‘numeracy’ environment is provided for your learners?

### Glossary

**real world scripts** – this term refers to the mathematics embedded in real world tasks (e.g. cooking, building) and the mathematics which is part of a ‘script’ in songs, rhymes or stories (Montague-Smith and Price, 2012).

### References

Anghileri, J. (Ed.) (2000). *Teaching number sense*. London: Continuum.

Boaler, J. (1998) Open and closed mathematics: Student experiences and understandings. *Journal for Research in Mathematical Education*, 29(1), pp.41–62.

Boaler, J. (2000) Exploring situated insights into research and learning. *Journal for Research in Mathematical Education*, 31(1), pp.113–119.

Brousseau, G. (1997). Theory of Didactical situations in mathematics 1970–1990, [Edited and translated Cooper. M., N. Balacheff. N., Sutherland. R. and Warfield. V.] Dordrecht: Kluwer Academic Publishers.

Bruner, J. S. (1966). *Toward a theory of instruction*. Cambridge: Belknap Press.

Cramer, K., and Karnowski, L. (1995). The importance of informal language in representing mathematical ideas. *Teaching Children Mathematics*. 1(6), pp. 332–6.

Carruthers, E. and Worthington, M. (2006). Children’s Mathematics: Making Marks, Making Meaning (2nd edition). London: SAGE Publications.

Drews, D. and Hansen, A. (Eds) (2007). *Using Resources to Support Mathematical Thinking, Primary and Early Years*. London: Learning Matters Ltd.

Delaney, K. (2001). Teaching mathematics resourcefully, in Gates, P. (Ed.) *Issues in mathematics teaching*. London: Routledge Falmer, pp.123–145.

Edwards, S. (1998). *Managing effective teaching of mathematics 3–8. London*: Paul Chapman.

Estyn (2013). *Numeracy in key stages 2 and 3: a baseline study*. Cardiff: Estyn.

Ginsberg, G., Manly, M. and Schmitt, M. J. (2006) *The Components of Numeracy*. NCSALL: Cambridge, MA.

Harries, T. and Spooner, M. (2000) *Mathematics for the Numeracy Hour*. London: David Fulton Publishers Ltd.

Haylock, D., and Cockburn, A.D. (2003) *Understanding Mathematics in the Lower Primary Years: A Guide for Teachers*. London: SAGE Publications Inc.

Hughes, M. (1986) *Children and number*. Oxford: Basil Blackwell.

Mason, J. and Johnston-Wilder S. (2006). *Designing and Using Mathematical Tasks* (2nd Edition). St Albans: Tarquin.

Montague-Smith, A., and Price, A. J. (2012). *Mathematics in Early Years Education*. Oxon: Routledge.

Moyer, P. (2001). *Are we having fun yet?* How teachers use manipulatives to teach mathematics. *Education Studies in Mathematics*, 47(2): 175–197.

Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper S. R., Johnston, C., Browning, C., Özgün-Koca, S. A., and Kersaint, G. (2009). Mathematics teacher TPACK standards and development model. *Contemporary Issues in Technology and Teacher Education* [Online serial], 9(1).

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

Swain, J. Newmarch, B. and Gormley, O. (2007) *Numeracy-Developing Adult Teaching and Learning: Practitioner Guides*. NIACE: Leicester.

Tangney, B., Weber, S., Knowles. D, Munnelly, J., Watson R., Salkham, A., and Jennings, K. MobiMaths: An approach to utilising smartphones in teaching mathematics. *MLearn*. Malta. January. 2010.

Welsh Government (2013). *Curriculum planning guidance*. Cardiff: Welsh Government.

Wong, K. Y., Lee, P. Y., Berinderjeet, K., Foong, P. Y. and Ng, S. F. (2009) (Eds). *Mathematics Education: The Singapore Journey*. Singapore: World Publishing Co. Pte. Ltd.