Numeracy

The vertical axis of the course begins with the following topics:

• Sorting and classifying
• Counting with whole numbers and using them to measure and draw
• Adding with whole numbers
• Subtracting with whole numbers
• Multiplying with whole numbers
• Dividing with whole numbers
• Understanding about parts of whole numbers
• The four operations for money
• The four operations for decimals
• The four operations for fractions

The axis of the spiral of progress is regarded as a continuum, rather than sets of skills to be acquired separately. The pupils’ own characteristic approaches are encouraged. For example, a pupil is encouraged to view a division such as 24 ÷ 4 as:

• A reverse multiplication, giving 6, because 6 × 4 = 24
• A repeated subtraction down to zero:
  
• Repeated subtraction in chunks:
  
• Repeated additions up to the right answer:
  

Note that special attention should be given to the number facts for single‐digit addition, subtraction, multiplication and division, since knowing these facts, or having quick and reliable strategies with which to work them out, reduces the load on working and short‐term memories during calculations. This knowledge helps not only computation but also understanding of numbers.

General mathematical topics

Topics such as perimeter, area, equations, angle sums and graphs are introduced only when an appropriate level of numeracy has been reached. By using core numbers, this can be achieved relatively soon.

The mathematical variety needs to be as wide as possible as early as possible in order to maintain motivation and extend experience. It is particularly useful to introduce algebra very early, in the form of simple formulae at the conclusions of pieces of fully understood work. Algebra is a useful vehicle for developing the generalisation of skills, concepts and knowledge. For example, when working out a perimeter has been grasped for numbers, it is not a large conceptual leap for a pupil to understand that a + b + c is the formula or generalisation for the perimeter of a triangle with sides a, b and c.

Besides maintaining the students’ interest, dealing with a wide variety of topics has an even greater value in helping build foundations that are wide, so that difficult areas can be spanned just as missing bricks can be spanned in a wall (see Figure 18.1). Furthermore, relationships between mathematical topics are revealed and alternative paths are explored and developed. In this way, it becomes possible to put into practice the earlier claim to ‘build upon pupils’ strengths and extend what they do know and understand.

Using and applying mathematics

Investigational and practical work are sometimes viewed as ‘bolt‐ons’ to the curriculum when the student is struggling with the basics, but they are integral to the pupils’ understanding of the reasons for studying the subject and how it all works collectively. These activities help pupils with the organisation of their work and offer experiences with which to strengthen their concepts. Pupils usually respond with strong motivation for this alternative kind of work.

The use of patterns

The authors have long advocated the use of patterns in mathematics, as part of its structure. Patterns act as the mortar that holds the bricks of the subject together (Chinn and Ashcroft, 2004). We consider that patterns and their recognition can help in the following ways:

• Streamline and secure the learning of related facts
• Add interest and motivation (as puzzles and games can)
• Help with conceptual problems, by providing another way of looking at things
• Rationalise idiosyncrasies
• Provide structure
• Encourage generalisation skills
• Help long term memory

Mental arithmetic

Throughout the course, pupils should be encouraged and shown how to develop methods of calculating answers mentally, something most of us need to do in everyday situations. They should not be expected to create these methods by themselves, but any methods they have already adopted should be welcomed and discussed. Whether the expectation is for a correct answer or merely an estimate, pupils should be encouraged to use mental calculations as their first resort and then as their last resort when verifying or checking an answer.

However, the problems of weak short‐term and working memories are likely to make mental arithmetic very challenging. The pupil is likely to need appropriate differentiation if he is not to be seriously demotivated by this activity. One example of differentiation is to show the problem, for example, written on a white board, so that the pupil can at least focus on the computation.

Spread of ability

A teacher’s organisation and preparation should enable both faster and slower pupils to make progress. If he is devoting time to slower pupils, then faster pupils should have selected extra work to cover. Pupils can often learn well from each other, so faster pupils can sometimes be given the opportunity to develop their ‘communication level of understanding’ (Sharma, 1988), by helping slower pupils.

Pupils’ mathematical cognitive styles

Chapter 3 explains the two extremes of thinking style. Pupils will lie somewhere on this spectrum. At one extreme is the pupil described as an inchworm learner, who works part‐to‐whole, and at the other extreme is the pupil described as a grasshopper learner, who works whole‐to‐part. The inchworms follow a rigid, step‐by‐step, formula/algorithm‐based style when tackling mathematics. This is usually the way this type of learner is best taught. Conversely, grasshoppers work more intuitively and are very answer‐oriented. They may have been stifled and demotivated by being taught in the first style, to which many mathematics teachers arguably belong. A good mathematician needs to be flexible and make appropriate use of a mixture of the styles, so pupils need to be encouraged to consider more than one method when appropriate.

Pupils can be helped to find their own best way of working if the teacher:

• Begins each lesson with an overall picture of its contents, using both oral and visual stimuli
• Thoroughly explains the logic behind each method (as overview and then detail)
• Offers alternative methods
• Puts the work into a familiar context, or relates it to the pupils’ own experiences and existing knowledge
• Reviews the outcomes of the lesson

Lesson management

To be taught in a whole class by an empathetic teacher offers some advantages over the individual help some dyslexic pupils receive. The members of the class should work together, share problems and accept mistakes, safe in the knowledge that few learners are perfect. They are encouraged to lose their fear of being wrong and thereby gain confidence. There should be a risk‐taking ethos in the classroom.

As pupils improve their mathematics at different paces, the differences between them will grow and become more evident. Each will have his own expectations and require these expectations to be met. Allowing and guiding them to work individually for some of the time encourages them to fulfil more of their potential. Class lessons may remain the main learning mode, but individual routes can be provided, where pupils choose from the following:

• Help or further practice with troublesome current work
• Revision of recent or seemingly mastered topics
• Extension work at higher levels

Published schemes, texts and carefully chosen work from the Internet are becoming increasingly valuable to support this way of working.

Teaching materials

A relevant teaching philosophy can be summarised simplistically as, ‘Mathematics is easy, only writing it down is hard’. Sometimes we explain by copying Figure 18.2 onto the board and ask pupils what they think it is.

Pupils will inevitably guess that it is a bike or a moped or a motorcycle or a scooter, and then become annoyed when this is refuted. It is possible to tease them even further by saying, ‘If it’s a bike, ride it down to the shop’. Eventually they are realise that we are showing something that it is not a bike, but a picture of a bike. In many ways, the mathematics we study is not real life, but only a written representation of a real problem. Sharing money between people is a real problem, which we can all do, while the usual written version of, for example, £12.48 ÷ 4 is just a picture, a theoretical version. If the written version can be shown to mimic the actual processes of dividing up the money, then more pupils will understand. This philosophy suggests the following sequence of steps:

• New topics are introduced through practical work, demonstrations, investigations, discussions and physical experiences. The use of a variety of these will facilitate the development and understanding of each concept. It can put them in a context.
• The next step is to try and translate the concept into a written form, linking the concrete experiences directly to the symbolic representation.
• This will lead into the use of worksheets or textbooks. The worksheets written and used by the authors start with worked examples, which are related to the earlier experiences. Then questions provide practice and revision. The worksheets are thus the third stage of the procedure and not the sole source from which the pupils are expected to learn.

Worksheets can follow a structured course and can be designed to enable the following:

• Present an advance overview of the section of work to be followed
• Eliminate the need for taking down notes, with its inherent risks of slow progress, mistakes, lack of clarity and readability
• Start at the most appropriate point
• Cover only a single concept, so that any point of difficulty can be readily identified, isolated and dealt with (that is, be diagnostic)
• Contain no more information than can be comfortably digested in one bite
• Present work clearly
• Use the fewest possible and simplest words, yet introduce the necessary technical terms carefully
• Incorporate exercises
• Provide a practical revision aid (used as ready‐made and organised notes)
• Carefully relate to other sheets
• Be designed to be criterion referenced, at least with some of the questions

As stated earlier, later on there will be a point when the differences between individual pupils begin to outweigh their similarities. Short worksheets, answered together at the same rate, are no longer ideal. At this point, it is also important for pupils to begin to use mainstream materials, as they must eventually face public examinations. An appropriate textbook or a series of textbooks, which becomes familiar and trusted, can provide help and reassurance, especially at the time of examinations. Textbooks provide different viewpoints and a variety in language, to which pupils must acclimatise when they are ready. The presentation of work in textbooks has become more and more appealing, with good graphics, uncluttered pages and well‐structured sets of examples. Textbooks must be chosen with great care, so as not to risk a reversal of the changes in attitude previously achieved. New approaches, as used in Singapore for example, are now being introduced.

Writing paper

If the responses of younger, newer pupils need more space than a worksheet allows, there are considerable benefits in using squared paper, either in loose form, to file with their worksheets, or in exercise book form. If centimetre squares are too large for many written answers, half‐centimetre (5 mm) squares offer invaluable help with:

• Lining up calculations vertically and horizontally
• Setting out tables and charts
• Doing measurements and diagrams, especially those in centimetres and/or using right angles
• Working out area problems

It is our view that dyslexic and dyscalculic pupils should be coached specifically through the transition from squared paper to lined (vertical lines can help with the alignment of calculations) and blank paper. The transition can take place when the value of the squared paper becomes outweighed by the need to prepare for public examinations, or at a time of the pupil’s choosing.

Calculators

With a course that initially has numeracy as its axis, the use of a calculator would be counterproductive. Unless there is another purpose to the work and it involves repetitive calculations, calculators should be discouraged at first.

Mental arithmetic is necessary for everyday life and for checking answers that are worked out using a calculator. Premature over‐reliance on a calculator could well delay the acquisition of these mental arithmetic skills, particularly estimation skills.

Furthermore, many of the later mathematical processes, such as (a + x)(b + y), are based on the early numerical processes, for example 35 × 24, which therefore need to be thoroughly understood.

There will be a point when pupils have learned all they can about manual calculations. Powerful calculators can be bought cheaply, but some have too many functions that will never be used and that make the machine difficult to operate. A simple solar‐powered calculator can be sufficient, with scientific functions, including fractions, percentages, and degrees/minutes/seconds, and which does not resort to scientific form in unnecessary cases.

A calculator is an ideal aid for the short‐term memory and can help compensate for a pupil’s remaining computational difficulties (which might only be a matter of speed). Logic and keying errors can be filtered out using checking methods (see Chapter 17). Specific calculator functions should be introduced on the basis of need, as with the trigonometric functions, for example. Specific time can be set aside for the exploration of other functions, such as n! (n factorial), which are of investigational interest.

Monitoring progress

Standardised tests are called so because they have been given to a standard sample or population, before publication. The performance of any child can then be compared with this standard population. The results can be expressed in various ways.

Some tests produce a ‘mathematical age’, which can be compared with a child’s chronological age to give an idea of how far behind his attainment is in years and months. The results of subsequent repeat tests will show how much improvement has been made. This can be related to the time that has elapsed to give a ‘value‐added’ factor. For example, 18 months’ progress in a year could be defined as a value‐added factor of 1.5.

Some tests produce a mathematical ‘quotient’, which resembles IQ, and relates performance to an average figure of 100. For example, 96 is just below average, but within an average band.

Other tests produce a ‘centile’ or ‘percentile’ figure, which shows a child’s position in the standard population as a percentage. For example, a percentile figure of 20 would indicate that 20% of the population would be expected to perform at or below the child’s level.

Ideally, standardised tests should be repeated at similar times in the year for comparison purposes. Test results should be considered in conjunction with progress in class, and any other changes that may have occurred, to obtain a real picture of progress. Single test results should be treated with caution, as leaps in progress may not coincide with test dates. There is always the possibility that the test may have occurred on a bad day. Many pupils are labile in their performances.

Diagnosis of difficulties

Some tests are designed to assess mathematical sub‐skills separately, so that particular problem areas can be identified. The results can be used to direct subsequent teaching towards the areas of weakness. In an ideal classroom teaching and diagnosis should go hand‐in‐hand.

Pupils’ mathematical styles

The Test of Cognitive Style in Mathematics (Chinn, 2017a) can be used to determine mathematical learning/cognitive style. It distinguishes between the step‐by‐step ‘inchworm’ and the intuitive, holistic ‘grasshopper’. This test is for use with individual pupils and provides a measure of the pupil’s position on the continuum of styles from extreme inchworm to extreme grasshopper (see Chapter 3). It encourages discussion as to ‘How’.