Learning to Learn Mathematics Extracts from Digging Deeper Handbook 6- Numbers Count

Success in mathematics does not come from simply committing facts and procedures to memory, but from a growing understanding of how numbers work and of how they relate to each other, interact and make patterns. Constructing these understandings entails effort, risk, discussion and perseverance.

Each child will take their own journey in this learning and will require appropriate guidance and support to overcome the inevitable obstacles and difficulties and that they will meet along the way. For most children, the pace of class lessons and the general guidance given by their class teacher, other adults and their peers will be sufficient to enable them to learn about mathematics and to construct their own helpful and accurate picture of the mathematical world. For some children, this pace or content will be inappropriate and they may lose confidence and become disheartened. Research (e.g. Burhans and Dweck, 1995) has shown that even very young children are not immune to feeling and acting as if they are helpless when they have experienced failure.

Some young children also believe that success in mathematics is measured in terms of the neatness and quantity of their written work and the number of ‘ticks on the page’ they have collected, rather than in terms of their understanding of the concepts involved. This could be why they feel they are rewarded for copying from their peers rather than by engaging with the learning themselves. Turner et al. (2002) noted that children may pick up the message that ‘demonstrating ability and outperforming others are the reasons for engaging in academic behaviour.’ Where this is the case, children may perceive that ‘asking for help, trying hard, and approaching their work in novel ways is a threat to their self-worth and thus purposefully avoid the use of strategies that might enhance their understanding and achievement. However in classes where understanding and improvement are overtly prized and encouraged, children are much less likely to feel threatened and may not feel and need to avoid learning.

DEVELOPING MATHEMATICAL THINKING‘... harness the power of talk to engage children, stimulate and extend their thinking, and advance their learning and understanding.’ (Alexander, 2008: 37).

Many children come to the maths lesson having already developed the idea that mathematics is just about getting the “right answer”. This may have arisen from the emphasis placed by some teachers or teaching assistants on the “correct” completion of tasks rather than on the thought processes that underlie their mathematical understanding. One of the problems with this is that children stop thinking as soon as they are told what the right answer is. Teachers usually decline to tell a child whether she has produced a “right” or “wrong” answer: Teachers sometimes challenge a “right” answer to encourage a child to think and talk about their learning and to develop their confidence in their own judgement. Children must also unlearn that whenever teacher challenges their thinking it means that they have produced the wrong answer. Teachers sometimes challenge a “right” answer to encourage a child to think and talk about their learning and to develop their confidence in their own judgement.

Have you experienced/ observed any of these attitudes in your classroom?

How are these created?

How could they be overcome?

Children have often also learned that it is better to give no answer than a wrong one: if they wait long enough, then an adult or another child will eventually supply one for them. Long waiting times help children to understand that they are expected to collaborate with their teacher in solving mathematical problems and to talk about them.

One way that children learn what is important about mathematics is by attending carefully to the things that adults visibly value, and particularly to what they praise. So even when a child gives a ‘right answer’, their teacher will show what they are really interested in by focusing their praise on the child’s effort, process or explanation.

Haylock and Cockburn (2008) recommend that teachers also help children to build connections between four ways of understanding and representing mathematics: language, symbols, images, and concrete experience; the more connections they can make between them, the stronger their concepts will be.

Gattegno (1987) argued that ‘only awareness can be educated’, by which he meant that you can’t make a learner understand a concept by telling them about it, but you can help them to become aware of key features that will help them to construct the concept for themselves. One way to do this is by providing a child with models and images that are structured to show something about mathematics and to help the child think.

A range of teaching strategies to construct a better understanding:

Using a meaningful context Setting a problem that challenged the child to think Allowing the child to choose the resources she would use to solve the problems Assessing the child’s understanding Waiting for the child to answer Declining to tell a child whether the answer is right or wrong Asking the child to check an answer, even if it is right Praising the child for effective mathematical thinking and talking Helping the child to make connections Providing a range of suitable resources that embodied key concepts Drawing the child’s attention to key features of the mathematics she was doing Helping the child to achieve success Reinforcing key concepts Encouraging the child to reflect on her learning.

ReferencesAlexander, R. (2008). Towards Dialogic Teaching: rethinking classroom talk, 4th edn. York: Dialogos.Gattegno, C (1987). The Science of Education, part 1: theoretical considerations. New York:Educational Solutions.Haylock, D. And Cockburn, A. (2008). Understanding Mathematics for Young Children, 3 rd edn.London: Sage.Houssart, J. (2007). Investigating variability in classroom performance amongst children exhibiting difficulties with early arithmetic. Educational and Child Psychology, 24(2), 83-97.

What do you think about Gattegno’s argument?

How can you avoid these attitudes in the classroom?

Do you have children in yoir class who struggle with counting? How could these principles be used in supporting these children?

COUNTINGA child’s first experience of number is though counting, and learning to count can support understanding o the number system and become a major tool in the development of calculation strategies. However, teachers know that it is important that children do not rely too heavily on counting strategies but also develop more efficient mental calculation strategies.

OBJECT COUNTINGObject counting includes counting a range of objects that present different challenge to the child who is counting them, such as:

Objects that can be seen, touched and moved, e.g. toys Objects that can be seen and touched but not moved, e.g. toys in a picture Objects that can be seen but not touched, e.g. ceiling tiles Objects that can be seen but are moving, e.g. bubbles Objects that cannot be seen, e.g. sounds Abstract objects, e.g. the number of ‘happy words’ that a child can think of.

HOWTO

COUNT

The One-to-One Principle

Assigning a distinct counting word to each item, even if a child says ‘1, 6, 2’.

The Stable Order Principle

Knowing that even the list of words must be a consistent one, even if a child repeatedly counts ‘1, 3, 2, and 6’. We should not be surprised if children make their own lists of numbers up, given the inconsistencies in our counting system.

The Cardinal PrincipleCounting leads to a ‘product’ at the end. If a child has to count again in response to ‘How many ... ?’, he has not grasped the principle

WHATTO

COUNT

The Abstract PrincipleCounting collections of abstract, miscellaneous items, even if a child refers to them as ‘things’.

The Order-Irrelevance Principle

Knowing that the order of counting is irrelevant – when a child has grasped this, he knows what he is doing when counting.

ORAL COUNTING

Oral counting (recitation or rote counting) is the ability to correctly order a string of numbers. In early counting, this is committing the string to memory. In Numbers Count the early development of oral counting focuses on learning by rote the number word sequences both forwards and backwards, with reference to the five stages in the early development of the number word sequence outlines by Fuson (1988).

String level a continuous sound string

Unbreakable List level separate words but the sequence can’t be broken and always starts from 1

Breakable Chain level Start to count from any point

Numerable Chain level sequence, count and cardinality are merged so, if you are counting from 3, then 3 is the first number, 4 is the second number...

Bi-directional numbers can be said in either direction and start at any pointChain level

BEYOND COUNTING: DEVELOPING CONCEPTS OF NUMBERA wide range of counting skills would underpin children’s development of early calculation strategies. Some children who have difficulties with mathematics can become proficient at counting but find it hard to move on to calculating with numbers. Gray (2008) comments that children need to develop a broad range of concepts of numbers in order to become effective calculators. One obstacle to this can be when a child’s experience of numbers is largely restricted to object counting, which leads her to understand and use numbers as adjectives that describe sets of objects. In order for a child to fully develop the concept of a number and begin to develop calculation strategies, however, she needs to be able to see numbers as independent of the sets that they describe and to be able to talk about them as if they have a life of their own (Haylock & Cockburn, 2008). To be able to say, for example, ‘Three is less than five’ or ‘Three plus two equals five’ requires the child to make the transition from treating a number as an adjective, describing a set of objects, to a number as a noun that can be compared, made bigger, made smaller, manipulated, added, subtracted, multiplied or divided.

THINKING MATHEMATICALLY IN DIFFERENT

CONTEXTS End of year expectations

During a diagnostic assessment, a Teacher assesses a child’s mathematical thinking by considering a range of questions.

How can end of year expectations support you in teaching mathematics to children who struggle?

Solving problems

Do some of the tasks I give the child provide the opportunities for him to show me how he tackles problems, including those set in a real life context?

Representing

Have I provided opportunities for the child to choose and use apparatus to help with tasks? Does the child choose to use pictures, diagrams, lists, tables or calculations as thinking tools? Can he use scaffolded and unscaffolded recording? Does reorganising the information help him? If so, how? Can he check his answer? How does he do this?

Enquiring

Can the child explain his methods when I give open-ended sorting tasks? Can he interpret, use and draw lists, tables or diagrams to help to solve a problem?

Reasoning

Can the child copy, continue and create patterns with shapes, objects and numbers? Can he say what might come next and why? Is he systematic in organising data or beginning to find all possibilities? Can he offer his own generalisations?

Communicating

Does the child talk about the mathematics he is doing? Can he explain things to adults and other children? Can he use an appropriate range of mathematical vocabulary? Can he use appropriate language structures when talking about mathematics?

PLANNING OPPORTUNITIES TO THINK MATHEMATICALLY IN DIFFERENT CONTEXTS

In order to develop children’s thinking, reasoning and communication skills in mathematics, teachers give them regular opportunities to use their mathematical skills and knowledge in different contexts.

In some lessons, problem solving skills and strategies are the major focus for the current learning activity. For example, the learning may be tightly structured in order to teach the child a problem solving strategy such as how to create an ordered list to find all possibilities. The learning might be focused on solving problems of a particular type, providing an opportunity for the child to use and apply the calculation strategies he has learned.

In other lessons, opportunities to use and develop skills are integrated into the current learning activity, making links to real life or within mathematics.

THINKING MATHEMATICALLY IN REAL LIFE SCENARIOS

Children need to engage in ‘real-life’ problems where they can use their knowledge in contexts which are similar to ‘real-life’. However, teachers have to be careful about what they choose as ‘real-life’ contexts because ones that are ‘real’ for adults, such as VAT, shopping or petrol consumption, are rarely ‘real’ for children (Ollerton 2010).

Children should be presented with scenarios that they can relate to. When they are learning to read two-digit numbers, a contents page from an information book can be used to practise reading numbers in a real life context. Similarly, having learned to count in tens, children can then be given opportunities to apply this skill in different contexts such as oranges in nets of ten, crayons in boxex of ten or 10p coins. Other examples of real life contexts might include using money and measures. Imaginary and fantasy worlds are also a part of real life for children and offer teachers rich opportunities for developing mathematical thinking.

Small Worlds

Small world play allows children to create and bring mathematical meaning to their own scenarios/worlds and to explore mathematic in worlds that are meaningful and relevant to them. Small worlds can therefore be used to:

Meet the interests and motivations of the children

Set up mathematical problems as part of pretend play

Develop and explore mathematical ideas Use mathematical language and

vocabulary in a context.

Role Play

Engaging in role play such as shopping or setting the table can enabled children to make connections between mathematics in school and mathematics in the real world and provides opportunities to support and stimulate mathematical thinking.

Playing Games

Playing games gives children the opportunity to rehearse and refine mathematical skills. It supports them in talking about what they know and understand about different mathematical concepts.

THINKING MATHEMATICALLY WITHIN MATHEMATICS ITSELF

It is important that children are also given opportunities to use and apply their existing mathematical skills in other areas of mathematics and to begin to make links and connections within mathematics itself.

Interpreting simple bar graphs allows children to think and talk mathematically using different mathematical terms and vocabulary such as ‘more than’, ‘fewer than’, ‘same as’, ‘equal to’ and ‘the difference between’.

Having learned the bonds for ten, children can use this knowledge to solve ‘missing box’ calculations such as 6+ = 10 or 10 = 7 + . They can also be encouraged to use their knowledge of bonds for ten when adding three small numbers such as 7 + 5 + 3 = . Children can use their knowledge of known

Can a child in year 5 who is still working on year 1 objectives benefit from these foundation stage opportunities?

facts to derive new facts, such as using their knowledge of double 6 to work out a near double like 6 + 7 = .

Children also need to learn how and when to apply their existing skills, knowledge and understanding, which they may previously have learned, practised and recalled in isolation. For example, they need to learn how to use the skill of counting in multiplies of 5 to generate facts in the five times table. Similarly, when children are learning how to add two-digit numbers they need to know where and how to use and apply their existing skills of partitioning, recombining, adding units, multiples of ten, estimation and inverses.

All of the above examples give children opportunities to develop their mathematical knowledge, skills and understanding at the same time as developing their mathematical thinking.

QUESTIONING AND EXTENDING DIALOGUE

Effective questioning can also be used as a guide, support and stimulate children’s mathematical thinking. A well used question can lay the foundations for an extended dialogue that will stretch the child’s understanding and expose any misconceptions.

Questions can be used to develop a mathematical dialogue. This provides an opportunity for children to communicate their mathematical ideas and understanding. It also provides an opportunity to use mathematical language and vocabulary.

Children should be exposed to a range of questions, both open and close. The questions need to be carefully planned to ensure that they prompt, probe and promote the children’s thinking and understanding. However, not all questions can be planned: teachers need to be flexible and to ask appropriate questions in response to the learning they have observed. Asking questions which demand higher order thinking skills helps children to extend their thinking from the concrete and factual to the analytical and evaluative.

Teachers encourage children to ask questions and seek their own answers. Teachers expect children to provide extended answers to their questions. They seek

complete answers that employ the correct mathematical vocabulary.

When planning opportunities for mathematics in different contexts, teachers provide a variety of experiences over time. They aim for a balance between providing enough structure for children to lean and develop new strategies, skills and ways of recording, and allowing enough flexibility for them to choose appropriate strategies to solve a problem, to face and overcome difficulties and to follow their own lines of enquiry.

ReferencesOllerton, M. (2010). Using Problem-Solving Approaches to Learn Mathematics. In Thompson, I. (ed),Issues in Teaching Numeracy in Primary Schools. Berkshire: Open University Press.

Can you think of a few questions worth asking?