The Trouble with Maths
A Practical Guide to Helping Learners with Numeracy Difficulties
- 192 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
About This Book
Now in its fourth edition, with updates to reflect developments in our understanding of learning difficulties in maths, this award-winning text provides vital, pragmatic insights into the often-confusing world of numeracy. By looking at learning difficulties in maths and dyscalculia from several perspectives, for example, the vocabulary and language of maths, cognitive style and the demands of individual procedures, this book provides a complete overview of the most frequently occurring problems associated with maths teaching and learning. Drawing on tried-and-tested methods based on research and Steve Chinn's decades of classroom experience, it provides an authoritative yet accessible one-stop classroom resource.
Combining advice, guidance and practical activities, this user-friendly guide will help you to:
- develop flexible cognitive styles
- use alternative strategies to replace an over-reliance on rote-learning for pupils trying to access basic facts
- understand the implications of underlying skills, such as working memory, on learning
- implement effective pre-emptive measures before demotivation sets in
- recognise the manifestations of maths anxiety and tackle affective domain problems
- find approaches to solve word problems
- select appropriate materials and visual images to enhance understanding
With useful features such as checklists for the evaluation of books and an overview of resources, this book will equip you with essential skills to help you tackle your pupils' maths difficulties and improve standards for all learners. This book will be useful for all teachers, classroom assistants, learning support assistants and parents.
Frequently asked questions
Information
1 Introduction
Mathematics learning difficulties and dyscalculia
- understand some of the reasons problems may arise in learning mathematics
- understand that many problems track back to the very early experiences and thus that these basics need to be addressed within any intervention
- pre-empt learning problems
- develop flexible cognitive skills and encourage metacognition
- circumvent problems in basic numeracy by developing understanding, rather than an over-dependency on rote learning
- address the difficulties pupils have with word problems
- teach alternative strategies for accessing basic facts
- recognise affective domain issues and suggest strategies for addressing maths anxiety, attributional style, self-efficacy and self-esteem problems
- stimulate the ability to create effective ideas for teaching maths to all pupils, but especially those who are facing difficulties with the subject
- select appropriate materials, manipulatives and visual images for teaching maths topics
- encourage students to develop an understanding of maths.
A few golden rules
- Donāt create anxiety (and thus demotivation).
- Experiencing success reduces anxiety (and increases motivation).
- Experiencing failure increases anxiety (and decreases motivation).
- Understand your pupils as individuals and accommodate their individuality.
- Be as consistent as possible. Address inconsistencies when they arise. Inconsistencies, in their many manifestations, confuse learners (but see the next two points).
- Teach to the individual in the group ā¦ also known as the āTeach more than one way to do thingsā rule, but ā¦.
- Make sure that teaching more than one way to do things does not confuse some learners.
- Remember where each topic leads mathematically and where its roots lie.
- Know what the pre-requisite skills are for that topic and check that your learners have these skills (and knowledge).
- Understanding is a more robust outcome than just recall and supports weak long-term mathematical memory.
- Try to understand errors ā¦ donāt just settle for wrong.
- Prevention is better than cure.
- When reviewing topics it is very likely that you will have to go back further than you may think.
- Be empathetic in the pace you set for your lessons.
- All the above rules have exceptions.
What do learners need to be good at mathematics?
- The ability for logical thought in the sphere of quantitative and spatial relationships, number and letter symbols; the ability to think in mathematical symbols.
- The ability for rapid and broad generalisations of mathematical objects, relations and operations.
- Flexibility of mental processes in mathematical activity (metacognition).
- Striving for clarity, simplicity, economy and rationality of solutions.
- The ability for rapid and free reconstruction of the direction of a mental process, switching from a direct to a reverse train of thought. (Reversing is a challenge that starts early in maths.)
- Mathematical memory. A generalised memory for mathematical relationships and for methods of problem solving.
- Problem solving. The process of applying previously acquired knowledge to new and unfamiliar situations. (Developing transferable skills.) Students should see alternative solutions to problems: they should experience problems with more than a single solution. (An effective question to ask learners is, āCan you think of another way of solving this problem?ā This is also about metacognition and flexible cognition.)
- Communicating mathematical ideas (receiving and presenting). Students should learn the language and notation (symbols) of maths. (Recent research by Habermann et al. has found that Arabic numeral knowledge (defined by numeral reading, writing and identification at 4 years of age) was the sole unique predictor of arithmetic at 6 years.5 Knowledge of the association between spoken and Arabic numerals is one critical foundation for the development of formal arithmetic.)
- Mathematical reasoning. Students should learn to make independent investigations of mathematical ideas. They should be able to identify and extend patterns and use experiences and observations to make conjectures. (This suggests to me that this should involve, where appropriate, the use of visual images and concrete materials. Their use should not be an age-specific approach to teaching. This classroom approach requires careful and continuous monitoring to avoid learners absorbing incorrect information and concepts.)
- Applying maths to everyday situations. Students should be encouraged to take everyday situations, translate them into mathematical representations (graphs, tables, diagrams or mathematical expressions), process the maths and interpret the results in light of the initial situation. (Maths in everyday life provides ample opportunities for estimations and thus to develop a flexible sense of numbers and their values.)
- Alertness to the reasonableness of results. In solving problems, students should question the reasonableness of a solution or conjecture in relation to the original problem. They must develop number sense. (This also links to estimation. As an example of everyday reasonableness, I saw a poster outside a travel agent in Bath (UK) recently. At this time the exchange rate for GBP (Ā£) to euros was 1.16 euros to the pound. The poster said, āĀ£876 = ā¬750ā. There was no alertness to the reasonableness of the result of their calculation.)
- Estimation. Students should be able to carry out rapid approximate calculations through the use of mental arithmetic (or perhaps via jottings on paper when working memory is weak) and a variety of computational estimation techniques and decide when a particular result is precise enough for the purpose in hand. (See Chapter 4 on cognitive style.)
- Appropriate computational skills. Students should gain facility in using addition, subtraction, multiplication and division with whole numbers and decimals. Today, long, complicated computations can be done with a calculator or computer. Knowledge of single digit number facts is essential. (Learning to access facts by using mathematical strategies can help in developing, for example, an understanding of the four operations and algebra. Estimation comes into play again in checking those calculator answers.)
- Algebraic thinking. Students should learn to use variables (letters) to represent mathematical quantities and expressions. They should understand and use correctly positive and negative numbers, order of operations, formulas, equations and inequalities. (Being able to generalise is a key skill here. Again, this tracks back to how earlier learning was absorbed and understood.)
- Measurement. Students should learn the fundamental concepts of measurement through concrete experiences. (This links to place values for 10n and 10-n.)
- Geometry. Students should understand the geometric concepts necessary to function effectively in the three-dimensional world. (This may be a problem for some students with Developmental Coordination Disorder dyspraxia.)
- Statistics. Students should plan and carry out the collection and organisation of data to answer questions in their everyday lives. Students should recognise the basic uses and misuses of statistical representation and inference. (And the abuses.)
- Probability. Students should understand the elementary notions of probability to determine the likelihood of future events. They should learn how probability applies to the decision-making process. (It is apparent that many of these components interlink.)
- Communication
- Recording knowledge
- The communication of new concepts
- Making multiple classification straightforward
- Explanations
- Making possible reflective activity
- Helping to show structure
- Making routine manipulations automatic
- Recovering information and understanding
- Creative mental activity
Table of contents
- Cover
- Half Title
- Endorsements
- Title Page
- Copyright Page
- Table of Contents
- List of illustrations
- 1. Introduction: mathematics learning difficulties and dyscalculia
- 2. Factors that affect learning
- 3. What the curriculum asks pupils to do and where difficulties may occur
- 4. Cognitive style in mathematics
- 5. Developmental perspectives: a pragmatic approach
- 6. The vocabulary and language of mathematics
- 7. Anxiety, attributions and communication
- 8. The inconsistencies of mathematics
- 9. Manipulatives, materials and visual images: multisensory learning
- 10. The nasties: long division and fractions
- Appendix 1. Further reading
- Appendix 2. Checklists
- Appendix 3. Resources
- References and notes
- Index