The Trouble with Maths

A Practical Guide to Helping Learners with Numeracy Difficulties

Steve Chinn

• Editorial: Routledge
• Año: 2020
• ISBN:
  9781000178555
• Edición: 4
• Categoría: Education
• Categoría: Education General

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.

1 Introduction

Mathematics learning difficulties and dyscalculia

This book was written to help teachers, classroom assistants, learning support assistants and parents who are dealing with pupils who are underachieving in mathematics. The level of underachievement might be significant enough to be recognised as a mathematical learning difficulty or severe enough to be considered as dyscalculia.

The book looks at the problems from several perspectives, from preventative measures to use in the classroom, to cognitive styles, to practical ideas for intervention. It works like a repair manual in some respects and like a care awareness manual (looking after your students) in other respects.

It is a book which can be accessed in different ways. It can provide an overview of where and how problems may arise. It offers insights for teachers into areas of potential difficulty for learners. It can focus on a specific problem and suggest approaches which can help the pupil overcome the problem.

It would be an impossible task to attempt to provide an answer for every problem for every child. One way of being as comprehensive as is practical within one book is to focus as much on prevention as on intervention. A pro-active awareness of learning issues can help in reducing their impact on learners.

In many respects then, the key purpose of this book is to provide a context with which to design and appraise any intervention, be it major or minor.

The Trouble with Maths can be used to:

• 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.

Sometimes you may find information repeated in different chapters of the book. This is deliberate, as some observations fit into more than one area. The new topic area should give a different perspective to that information. Mathematics is a multifaceted subject requiring a constellation of abilities.¹

Barriers to learning are often rooted in inappropriate and ineffective communication. This book encourages teachers and tutors to constantly reflect on what and how they are communicating. That’s a multifaceted task.

A few golden rules

• Don’t 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 … also known as the ‘Teach more than one way to do things’ rule, but ….
• 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 … don’t 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.

Although we might not be surprised to know that high-stake examinations, such as the national exams for 16-year-old English students, create stress as found in an AQA study from 2013, lower-stake tests can also create stress, as reported in a front page article in The Times of 30 December, 2002 headed ‘Exam stress strikes seven-year-olds’. The article reported that the Key Stage One tests caused symptoms of excessive anxiety including loss of appetite, insomnia, bed-wetting, forgetfulness and depression. These are our children! And now, in 2020, it would still be good if we were less obsessed with exams and tests and measuring performance, and maybe make less spurious conclusions and interpretations based on these activities. Maybe.

Furner and Gonzalez-DeHass (2011) offered wise advice²:

It may be favorable to the students if less of an emphasis is placed on test taking and competition, passing or failing, and winning or losing. Students with mastery goals are interested in learning new skills and improving their understanding and competence; they are engaged in the process, not focused on the product. They are taking responsibility for their learning and engage in activities that allow for self-regulation and self-direction. Their success is defined by individual improvement, they place value on effort, and their satisfaction is gained from working hard and learning something new. And they thrive in a classroom climate that helps students to feel they can take risks, make mistakes, and reveal their lack of understanding and seek help during their internal drive towards growth and personal mastery.

I like the interpretation of mastery as used in this context.

What do learners need to be good at mathematics?

Although this book is about ideas to redress underachievement in maths, it should also be valuable for considering what learners need to be good, to achieve in maths. I have two sources for this information. One is from the USSR and the other from the USA. In both cases I have added some comments in italics.

From the USSR, Krutetskii (1976) presents a broad outline of the underlying structure of mathematical abilities during school age.³ He specifies that, to be good at maths you need:

• 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.

He goes on to state that ‘These components are closely interrelated, influencing one another and forming in their aggregate a single integral syndrome of mathematical giftedness’.

Although Krutetskii makes these observations concerning giftedness in mathematics, they are equally appropriate for competence. It should be apparent as to where learning difficulties may create problems with these requirements. It is worth noting Krutetskii’s observation about the components being closely interrelated and influencing one another. So many of the issues around teaching and learning maths are multifaceted.

My second source is the National Council of Teachers of Mathematics in the USA,⁴ who listed and explained twelve essential components for learning maths. I have added some comments (in italics).

1. 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, ‘Can you think of another way of solving this problem?’ This is also about metacognition and flexible cognition.)
2. 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.)
3. 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.)
4. 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.)
5. 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 (£) to euros was 1.16 euros to the pound. The poster said, ‘£876 = €750’. There was no alertness to the reasonableness of the result of their calculation.)
6. 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.)
7. 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.)
8. 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.)
9. Measurement. Students should learn the fundamental concepts of measurement through concrete experiences. (This links to place values for 10ⁿ and 10^-n.)
10. 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.)
11. 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.)
12. 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.)

Picking up on Krutetskii’s first point and the NCTM’s second point concerning the use of symbols in maths, the British psychologist Skemp wrote:⁶

Among the functions of symbols, we can distinguish:

1. Communication
2. Recording knowledge
3. The communication of new concepts
4. Making multiple classification straightforward
5. Explanations
6. Making possible reflective activity
7. Helping to show structure
8. Making routine manipulations automatic
9. Recovering information and understanding
10. Creative mental activity

concluding that ‘The use of symbols appears to be indispensable to the use of reason’.

So, we have some characteristics for being good at mathematics. It seems reasonable to look at the converse situation and conclude that deficits in all or some of these skills can create difficulties in mathematics. I know from my own many years of experience of teaching students with difficulties in maths that there are many reasons why someone may underachieve in mathematics and that the picture is a complex one with no single root cause. My experience is that Krutetskii’s criteria are the most telling. When we look at the characteristics of dyscalculia and maths learning difficulties it is apparent that the deficits in skills and knowledge highlighted by research tally closely with the areas of strength listed by Krutetskii.

The term dysca...