# HANDBOOK OF MATHEMATICS PDF

So in the work at hand, the classical areas of Engineering Mathematics The \ Handbook of Mathematics" by the mathematician, I. N. Bronshtein and the. This guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an Download book PDF. It is in this spirit that AFRL offers The. Handbook of Essential Mathematics, a compendium of mathematical formulas and other useful technical information that .

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This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated . This Mathematical Formaulae handbook has been prepared in response to a request from the Physics Consultative. Committee, with the hope that it will be. PDF | Authors Foreword Main Notation DEFINITIONS, FORMULAS, METHODS, AND THEOREMS Arithmetic and Elementary Algebra.

For the 5th edition, the chapters "Computer Algebra Systems" and "Dynamical Systems and Chaos" were fundamentally revised, updated and expanded. Skip to main content Skip to table of contents.

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Advertisement Hide. Handbook of Mathematics. Pages Linear Algebra. Algebra and Discrete Mathematics. Infinite Series. Integral Calculus. Differential Equations. Calculus of Variations. Linear Integral Equations. Functional Analysis. Vector Analysis and Vector Fields.

Pupils learn from more knowledgeable others peers, teachers, parents by enquiry, and at a pace and time that suits them. Interest and motivation is self-initiated so pupils do not become bored or disengaged in the work. They also experience a feeling of involvement in the community in which they are a member, whether they are weaving colourful patterns, finding out about local art or history, or assisting others in an extended project.

The twofold nature of maths The next section uses two key learning paradigms—the work of Bruner and that of Gardner—to explain the twofold approach to mathematics teaching and learning. It should be noted, however, that other research in this area also exists and the reader is encouraged to review a range of perspectives. McLaughlin and Shepard make the case for eschewing the behaviourist model in favour of the development of higher-order skills: Teaching for understanding 23 Much of the current instructional practice is based on the behaviourallearning theory from the early part of this [twentieth] century.

According to this theory, learning occurs by reinforcement of low level skills that become the building blocks for more complex understandings…. In more recent decades, learning researchers have demonstrated that memorising the facts first does not lead automatically to an ability to analyse and apply what has been learned. Learning requires thinking. However, little attention is paid to the well-wrought psychological theory available to guide their decision-making processes.

The first stresses the acquisition of prepositional knowledge by learning facts and algorithms for future regurgitation in tests. Knowledge put into the head is taken as cumulative, with later knowledge building upon priorly existing knowledge.

Active interpretation or construal does not enter the picture. It is blankly one-way: Children arrive at school for the first time with a mathematical model albeit a poorly developed one of how numbers are manipulated. By working alongside more knowledgeable others, these naive ideas are replaced by more accurate, mathematical concepts. Teachers are advised to offer pupils opportunities to use the discipline in question as a mode of enquiry, through access to discovery learning, but this should not exclude the learning of facts and algorithms—mathematics as a body of knowledge.

It is familiarity with the basic mathematical concepts and skills that underpin the ability to solve problems. It is these intuitive theories that often persist long after formal education has been completed. Clearly, teachers have a responsibility to address these misconceptions, challenge them, demonstrate their inadequacy and then show the pupils the correct way of representing the concept using facts or algorithms as necessary: Indeed, unless such confrontations take place, it is likely that the intuitive theories will continue to exist, potentially re-emerge and dominate once the expert theories are no longer supported by the trappings of school.

Gardner, , p.

## Handbook of Mathematics for Classes XI & XII

The third type of learner, the disciplinary learner, has a true understanding of the subject matter, being able to apply the knowledge and skills in novel and unfamiliar contexts.

These learners can see beyond the disjointed facts and algorithms to the general principles underlying the concepts. Gardner and Boix-Mansilla , p. Gardner declares that the testing process is partly to blame for the lack of awareness of the hidden misconceptions. The current assessment and grading of children fails to ensure that an understanding of the taught topic s has been achieved and that the internalisation of the knowledge and skills has occurred so that it can be utilised in other contexts or newly posed problems.

The successful student is one who learns how to use research materials, libraries, notecards and computer files, as well as knowledgeable parents, teachers, older students and classmates—in order to master those tasks of schools that are not transparently clear. Each of these activities involves the use of higher order thinking.

## Soil-structure interaction : the real behaviour of structures

Open-ended questions for which there is no single correct answer encourage responses that involve the development of problem-solving skills. Pupils benefit from the switch from drill and practice to explorations. The latter equips pupils with the ability to solve problems, communicate, reason and interpret, refine ideas and apply them in creative ways.

Teachers should avoid, where possible, imposing modes of mathematical thinking on pupils; rather, mathematical representations should, ideally, be actively constructed from within. When children start school they possess a considerably developed, informal body of mathematical knowledge, augmented by thinking skills from everyday experiences Bruner, ; Gardner, By encouraging the active participation of students in problem-solving and exposing them to situations requiring the use of reasoning and communication, qualities such as self-regulated learning and thinking may be developed.

Children, like adults, are seen as constructing a model of the world to aid them in construing their experience. Pedagogy is to help the child understand better, more powerfully, less one-sidely. Understanding is fostered through discussion and collaboration, with the child encouraged to express her own views better to achieve some meeting of minds with others who may have other views. Hunt states: Thinking beings solve problems by manipulating mental models of the environment, rather than trying out responses until they find one that works.

They build these models by combining their conceptualisation of the problem with personal information about the world, abstracted from previous experience. Cited in Kulm, , p. Pupils must be convinced of the need to reject old naive and intuitive models and adopt new ones. Lesh highlights the constructivist view of learning as: Teaching for understanding 27 Schema theory Schema theory models the mind as a graph consisting of a number of nodes that may or may not be connected to other nodes.

Typically, nodes will occur for the connections between fractions, decimals and percentages initially, with nodes for proportionality, enlargements and probability appearing at a later stage. Each schema consists of well-connected facts, features, algorithms, skills and strategies. These schema can be viewed as problemsolving vehicles, since they allow pupils access to strategies used in solving similar problems already encountered, the methods used to simplify particular procedures, and the routines for establishing targets.

Marshall focuses on four types of knowledge needed for schema development: Modern pedagogy is moving increasingly to the view that the child should be aware of her own thought processes, and it is crucial for the pedagogical theorist and teacher alike to help her to become more metacognitive—to be as aware of how she goes about her learning and thinking as she is about the subject matter she is studying.

Achieving skills and accumulating knowledge are not enough. The learner can be helped to achieve full mastery by reflecting as well upon how she is going about her job and how her approach can be improved. Lester and Kroll suggest that higher order skills such as planning, evaluating, monitoring and regulating allow the pupils to marshal and allocate cognitive resources. Again, notice the links between these skills and the four Polya stages. The level of control and regulation metacognition distinguishes experts from novice, successful from unsuccessful problem-solvers.

Poor control limits the effectiveness of the problem-solving Kroll, By mentoring others, pupils are able to monitor themselves, and by discussing alternative solutions with other pupils, internal dialogue or thought Vygotsky, is encouraged Wertsch, Teaching mathematics 28 Mathematics as a body of knowledge Convincing arguments have been made in the previous sections for the inclusion of problem-solving in the process of teaching and learning—in particular, its value in distinguishing between what has been covered in class by the teacher and what the pupils actually understand.

The content is structured hierarchically so that skills and concepts can be revisited and extended as the pupils progress up the eight-level scale.

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Continuity and progression across Key Stages can also be assured, particularly during the transition from primary to post-primary education. Chapter 12 will look at the post mathematics provision in terms of Key Skills. Quite simply, it is teaching that centres on student thinking, that focuses on powerful scientific and mathematical ideas, and that offers equitable opportunities for learning. What is its purpose? What are the aims?

Does it change the way maths teachers teach? What are the implications for the future? This chapter offers an overview of the place of mathematics in the National Curriculum for primary and secondary pupils in the UK. It introduces the terminology and policy requirements used in teaching mathematics and endeavours to offer a backdrop and context for the chapters that follow. Learning outcomes By the end of this chapter the reader should be able to: Mathematics is the central tenet of all things scientific.

Without mathematics, how would we have measured distance and time, navigated across the oceans, explained planetary motion, created some of the great wonders of the world, which are still visited today due to their sheer size and architectural beauty?

In more recent times, mathematics has played an important role in technological advancements in engineering, the computer, in communications and data security such as encryption, to name but a few. Initially, it was skills of bartering, sharing, exchanging, trading and taxing people. Nowadays, importing and exporting goods, stock market trading, dealing with other nations, taxing people and businesses, and distributing financial resources around government agencies are the advancements on the traditional skills already mentioned.

Consequently, the diversity of applications of mathematics around us illustrates the need to ensure that the future population are well prepared for the increasingly complex, numerical lifestyle in which money can be transferred and invested online, computers can assist in engineering—the design and manufacture of materials, buildings and cars—and people can communicate from anywhere across the globe at any time using mobile phones.

Teaching mathematics 30 For these and many other reasons, mathematics as a subject is recognised as a core skill across the world. As a result, simple activities such as counting and becoming familiar with the number system are introduced in pre-school classes, while basic numeracy skills such as addition and subtraction are dealt with in the early years of primary school before developing into the more complex representations of number in the form of algebra and geometry later in the school career.

So how has mathematics education changed over recent years? The Cockcroft Report Mathematics Counts highlighted the concerns revealed from extensive research into mathematics education and the skills of young people entering employment. A number of recommendations were suggested and much of the current emphasis being addressed via the National Numeracy Strategy indicates that there have been few improvements over the past two decades.

With the new examination came the changes resulting in the National Curriculum. Changes in the National Curriculum in mathematics from The National Curriculum aimed to offer pupils a curriculum that: These twelve years of schooling were subdivided into four Key Stages: Key Stage 1 spanning ages 5—7 years, Key Stage 2 addressing the needs of 8—year-olds, Key Stage 3 focusing on the first three years of post-primary education age 12—14 years and Key Stage 4 covering the content for General Certificate in Secondary Education GCSE examinations spanning ages 15 and By using this structure, the National Curriculum became a coherent and continuous learning process where ideas were introduced in their simplest form during Key Stage 1 and then revisited in subsequent Key Stages with increasing levels of depth and sophistication.

While at Key Stage 3, they should be using and understanding the conventional notation of algebra, i. In addition, pupils should be able to formulate, interpret and evaluate algebraic expressions; manipulate simple expressions, simplify, remove brackets and factorise as appropriate; use these techniques with a range of more complex expressions; and use the rules of indices and fractional values.

It is clear that progression within and between Key Stages is easily documented and a greater emphasis is placed on the continuity of subject coverage compared to the past. National curriculum for mathematics 31 Although separate Programmes of Study PoS have always been offered for each Key Stage, prior to these were general and quite vague in terms of content. Since the Dearing Review, these Programmes of Study have undergone considerable changes, including an increased emphasis on the content and organisation of the information.

In the case of mathematics, there are four attainment targets: Basic statistical techniques are also evident in terms of collecting and recording data, and the use of the correct terminology mean and range during the analysis.

Task 3. Consult the relevant documentation to identify the progression across the Programmes of Study of Key Stages you will be teaching. The transition from primary to post-primary education is particularly important as pupils may be coming from a diverse range of primary schools.

Attainment Targets, levels and Statements of Attainment From its inception, all National Curriculum subjects were subdivided for assessment purposes into a small number of Attainment Targets ATs set out in eight pseudohierarchical levels. Each AT had a number of performance criteria or Statements of Attainment SoA describing the type and range of performance that pupils working at that level should characteristically demonstrate.

Therefore, a pupil whose achievement in Shape and Space or in Measures could broadly be characterised by these SoAs was deemed to have mastered level four in Shape and Space or in Measures. Box 3. Use data drawn from a range of meaningful situations, for example, those arising in other subjects. Collect, classify, record, represent and interpret discrete numerical data, using graphs, tables and diagrams including Venn Decision tree and Carroll diagrams pictograms Teaching mathematics 32 block graphs, bar-line graphs and line graphs with the axes starting at zero initially with given intervals ; explain their work orally or through writing and draw conclusions.

Design an observation sheet and use it to record a set of data leading to a frequency table; collate and analyse the results; progress to designing and using a data collec-tion sheet, interpreting the results. Enter information in a database and interrogate it, using at least two criteria; use an appropriate computer package to produce a variety of graphical representations of the data. Understand, calculate and use the mean and range of a set of discrete data, for example, calculating the mean score of two teams that have played different numbers of games in order to compare their performance.

Introduction to Probability Pupils should have opportunities to: Become familiar with and use the language of probability, including certain, uncertain, likely, unlikely, impossible and fair, by participating in games and other practical activities.

DENI, a Box 3. Measures 4 Pupils should: There were no difficult numbers or twists in the wording of the questions to confuse the students. To determine if a pupil had mastered a particular level in any AT, teachers were required to compile evidence. Normally, this took the form of simple pen-and-paper tests composed of items related to each of the SoAs. Many schools adopted the practice of recording mastery of an individual SoA by ticking the appropriate box in an elaborate pupil record sheet, as shown in Figure 3.

If a pupil had been tested, then a diagonal line was drawn upwards from left to right; if the pupil had been assessed and had attained the level, then an additional line was included from left to right in a downwards direction to complete the X-shape.

Teaching mathematics 34 Figure 3. It is worth noting that the averagely successful pupil reaching level 7 during his or her eleven years at school would have been assessed by classroom teachers against some SoA. A classroom teacher at Key Stage 1 with 35 pupils in his or her class who assesses all the class against all the SoA would make some 8, judgements.

They understand and use language associated with line and angle. They know the eight points of the compass and understand the terms clockwise and anticlockwise. They use co-ordinates to plot points and draw shapes in the first quadrant.

They understand the relationship between metric units. They find perimeters of simple shapes, find areas by counting squares and find volumes by counting cubes. They begin to make sensible estimates using standard units in relation to everyday situations. They understand and use the twelve and twentyfour hour clocks. For instance, the level description for level 4 in Shape, Space and Measures now reads as shown in Box 3. Comparing the original SoAs for Shape and Space and Measures at level 4, it is clear that little of the content and terminology has changed, and it is therefore hardly surprising to find that teachers are breaking these levels descriptions into their component parts for the purpose of assessment.

The latest versions of the National Curriculum for mathematics can be freely downloaded from the QCA website at http: In which AT is the topic assessed?

Can you draw a line of development from the lower levels to the upper levels which address your chosen topic? National Curriculum assessment: Should discrepancies occur between the Teaching mathematics 36 level awarded by the teacher and by the test, the majority of parents will favour the test as the more reliable measure judgement hence undermining the role of the teacher in the assessment process.

A further discussion on assessment issues will follow in Chapter Current changes in education The White Paper, Excellence in Schools DfEE, a 2 extended the current goals of the National Curriculum to include an additional set of teaching priorities facilitating flexibility, breadth and balance across the subject areas.

It advocated the incorporation of literacy, numeracy and key skills within each subject and encouraged the development of interest in citizenship; personal, social and health education; and in spiritual, moral, social and cultural education. The three main issues are stated below: The first task of the education service is to ensure that every child is taught to read, write and add up.

But mastery of the basics is only a foundation. Literacy and numeracy matter so much because they open the door to success across all the other school subjects and beyond. It offers opportunities to gain insight into the best that has been thought and said and done. There are wider goals of education which are also important. Schools, along with families, have a responsibility to ensure that children and young people learn respect for others and for themselves.

They need to appreciate and understand the moral code on which civilised society is based and to appreciate the culture and background of others.

They need to develop the strength of character and attitudes to life and work, such as responsibility, determination, care and generosity, which will enable them to become citizens of a successful democratic society. DfEE, a The new basics? Literacy, numeracy and ICT in teaching and learning In response to the requirements of improving literacy skills in the primary school, the government approved the introduction of the National Literacy Strategy DfEE, b which outlined the scheme of work for 5—year-olds, to be completed during the Literacy Hour—a daily period of structured Literacy development lasting one hour per day.

Five main areas are addressed: Numbers and the number system called Counting and recognising numbers in Reception , Calculations called Adding and subtracting in Reception , Solving problems, Measures, shape and space, and Handling data for Year 3 upwards only. Few teachers may have the confidence to abandon the Framework to assist their pupils in overcoming the problems for fear of becoming seriously out of step with the guidelines.

The National Numeracy Strategy will be discussed further in Chapter 4. A third recent addition to the teaching repertoire has been the incorporation of information technology into all curriculum areas see Connecting the Learning Society, DfEEc, 3. In the case of mathematics, this innovation was welcomed due to the wide range of skills-based educational software currently available for both primary and secondary pupils. Games with built-in levels of difficulty ensure that differentiation according to ability is offered to the less able and also the gifted members of the class without changing the program.

The role of ICT in the mathematics classroom will be a recurring theme through-out this book and a chapter is dedicated to the pedagogical issues associated with this mode of teaching and learning in Part II. These six cross-curricular themes include information technology, economic awareness, health education, careers education more recently renamed as careers education and guidance , education for mutual understanding EMU , and cultural heritage currently part of education for citizenship.

Ideally, all six themes should be incorporated into subject teaching; however, some themes are more applicable to certain subjects than others. For example, health education and science are interrelated, information technology and mathematics overlap, EMU, citizenship and cultural heritage have distinct links to history and geography, and so on.

The two cross-curricular dimensions are environmental education and the European dimension. Again, these issues should be discussed in a wide range of subject contexts. Teaching mathematics 38 These new changes are exemplified in the new curricula in England and Wales, and in the proposed Northern Ireland Curriculum Review.

More recently, these ideas have developed into the aims of the new curriculum and the development of citizenship across Key Stages 3 and 4. The modular nature of these courses facilitates the award of the AS qualification after one year has been successfully completed.

To maintain a breadth of learning, Dearing recommended pursuing a science or technology-related subject, a modern foreign language, an element from Arts and Humanities and a subject illustrating how the community works such as business studies, economics, politics, law, sociology. Ideally, this mix of subjects would facilitate the development and assessment of a range of Key Skills such as the three assessed skills of: Application of Number, Communication and Information Technology; and the three non-assessed skills: National curriculum for mathematics 39 The new AS and A2 level specifications for mathematics have been rewritten to account for the changes in curricular policy and can be downloaded from the relevant exam body website e.

From September , all Advanced-level courses AS and A2 include opportunities for pupils to gain Key Skills qualifications at levels 2—3. For the less able pupils, the increased flexibility resulting from the revised National Curriculum granted pupils over the age of 14 the opportunity to follow Foundation and Intermediate level GNVQ courses and to take some NVQ units through school which include work-related placements.

Identify the section s devoted to Key Skills. Are all six Key Skills addressed? Teaching mathematics 40 4 The National Strategies for Literacy, Numeracy and Key Stage 3 Introduction This chapter deals with the three main initiatives impacting on the role of a teacher in primary and post-primary schools today.

Why were the National Literacy and Numeracy Strategies introduced? A number of reports highlighted the current weaknesses in literacy and numeracy skills among the adult population.

Evidence of low levels of basic skills and the resultant impact is will have on everyday life includes: Startling headlines in newspapers have also highlighted the major problems being experienced by the public as a result of professionals making basic numerical errors.

The current need for emphasis on the basics is supported by findings from recent research. Literacy and numeracy are viewed as vital skills for the twenty-first century. Pupils must learn properly from an early age otherwise they will struggle in later life. The second is an ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance graphs, charts or tables or by reference to percentage increase or decrease.

Mathematics Counts, , p. It includes an ability and inclination to solve numerical problems, including those involving money and measures. It also demands familiarity with ways in Teaching mathematics 42 which numerical information is gathered by counting and measuring, and is presented in graphs, charts and tables. National Numeracy Project, p. Numeracy at Key Stages 1 and 2 is a proficiency that involves a confidence and competence with numbers and measures.

It requires an understanding of the number system, a repertoire of computational skills and an inclination and ability to solve number problems in a variety of contexts. Numeracy also demands practical understanding of the ways in which information is gathered by counting and measuring, and is presented in graphs, diagrams, charts and tables. DfEE, d All three definitions of numeracy focus on the ability to work comfortably with numbers and to understand a variety of representations of numerical data.

Each daily maths lesson has three constituent parts: Depending on the age of the pupils, the length of the lesson varied: Teachers are free to adapt the strategy to suit the needs of their pupils. Additional support was available for schools with special circumstances. Additional support was available for schools where there were pupils with English as an additional language EAL. Advice was also offered regarding the use of calculators, namely that they should not be used as a prop for basic arithmetic and an introduction to them is not recommended before the upper years of Key Stage 2 DfES, a.

Pupils would also experience continuity in their numeracy education within and across schools as teachers would be working towards the same requirements regardless of school location. The structure of the numeracy lesson The first phase, mental maths and oral work, is teacher-led and lasts around 5—10 minutes.

Pupils are given the opportunity to practise and rehearse a variety of strategies to assist them in sharpening their non-written skills. Figure 4. During this time, the teacher can offer focused attention on the work of an individual group so that over the period of a week, each group will receive approximately 30 minutes of uninterrupted teacher attention. During this time, teachers can uncover misconceptions that can be dealt with as a whole class or individualised problems that require one-to-one attention.

The final 5—10 minutes are devoted to a plenary session in which the groups of pupils feed back to the whole class. This stage is facilitated by the teacher asking questions to each group, who share their work with the rest of the class. Questioning is differentiated by ability with the more able pupils being asked to explain why they made certain decisions or to offer reasons why a particular problem-solving strategy worked in that case.

The teacher then offers a summary of the key points of the session, making it clear to the pupils what they need to remember for the next day. Homework is set, offering further development of the class activity. Pupils must learn to work autonomously during the middle phase of the lesson as this time is devoted to the teacher working closely with each group in turn during the week.

By improving their skills at communicating, collaborating and listening to others, the pupils will value this time for independent activity. Numeracy is viewed as the outcome of being taught mathematics well, so that the skills can be applied in other situations.

The Framework encourages teachers to use the strategies and methods recommended to promote numeracy, in their teaching of National Curriculum maths. The main mathematical topics are also specified with a brief overview of their content domain. The main mathematical topics covered in the Framework for Reception are: Years 1—3 address skills in the areas of: At this stage the pupils have progressed from structured play to discovery learning and a basic introduction to the written format of maths.

Much of the work in phase 2 of the Numeracy Hour is completed using resource-based activities including the use of computers when appropriate. Pupils are encouraged to develop their mental maths strategies further.

For Years 4—6, the mathematics is categorised as: Over these three years a more formal approach to the learning of maths occurs with increased attention on mathematical reporting and showing their method of calculation. Despite the increased importance of written maths, the pupils still use a wide variety of resources in their learning. The effective use of calculators is also introduced to this age group.

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The NNS advocates an approach to teaching which sets clear learning outcomes and targets, promotes mental maths skills, ensures progress and continuity in learning, uses pupil-centred, active learning strategies and strengthens links within and across local schools.

However, they did not recommend the use of calculators until the upper end of Key Stage 2 as it was felt that younger pupils had to learn, practise and reinforce their mental maths methods in order that they did not become reliant on a calculator for basic arithmetic.

In post-primary schools, it is often assumed that pupils can use their calculators properly, so it is the primary school teacher who must focus on the technical skills of how to complete multi-step calculations when teaching his or her class. Using a calculator pupils can see that 3. DfEE, All these activities and associated software are freely available through the DfEE support and guidance materials. The pack addresses the requirements for the NNS and contains a CD of materials for use with classes and includes prepared activities, and photocopiable resources are also available for pupil use.

ICT will not replace a teacher, but it can be used as a tool to assist the teacher in offering differentiated activities to the class and to motivate reluctant or slow learners. The role of parents in the National Numeracy Strategy Parental involvement in the NNS encourages pupils to take an active interest in their own learning and progress. By seeing their parents are interested in their work in school, pupils also become more involved and keen to learn DES, a.

The pupils use and expand the skills they have learned in school by carrying out everyday activities such as shopping, baking, counting money, setting an alarm clock or VCR, or helping with measurements. Practical and useful applications of their knowledge are revealed at home and at school, making learning maths an important and valuable activity. The publication Money Counts FSA, offers a variety of classroom-based activities set in real-life contexts.

Talking about shapes and colours, counting objects and using basic knowledge of symmetry can also be addressed in the context of everyday activities. At the start of each academic year, teachers usually inform the parents of the techniques being taught in class for subtraction and other basic skills so that the parents can help or reinforce these methods at home.

Pupils should be well motivated and actively engaged in their work, whether it is whole-class, group- or individual work.

A shift in teaching style away from groupwork and towards whole-class teaching was envisaged using the allocated time-frame. Classroom organisation should allow teachers to spend increased amounts of time working alongside a group of children for a minute slot once a week.

By tailoring the questioning and answering to the ability of the small group, the teacher can ensure pupils are progressing during this stage in the lesson. Literacy in the mathematics classroom In the primary school classroom, a cross-curricular approach can be taken to the Literacy Hour so that pupils may work together on stories with a mathematical focus. A number of mathematical storybooks exist for the Key Stage 1 and 2 pupils, which are based on mathematical shapes and numbers, sorting, counting and grouping objects, odd and even numbers and concepts such as pi and exponential growth.

Some typical examples of these books are: In the upper primary school classroom Key Stage 2 , the more able children may be interested in working on simplified biographies of famous mathematicians, for instance. A timeline of great mathematicians or scientists could be constructed from the individual work of groups. Pupils can develop their research skills by finding and using reference books to obtain relevant information matching a writing frame.

Alternatively, the spelling of key mathematical terms and vocabulary can be reinforced during the Literacy Hour using calligrams or drawing and labelling the various two-dimensional and threedimensional shapes. Prefixes such as tri-, pent-, oct-, and dec- can be discussed in relation to everyday life with examples such as tricycle or tripod, the Pentagon building in the US or the pentathlon in sport, an octopus or an octogenarian and decimal numbers or a decade.

These techniques are often repeated in the early years of post-primary education to ensure that all pupils have the same standard of mathematical awareness. The mathematical meanings of words should also be discussed so that pupils understand the use of everyday words in the mathematical context as well as the real-life situation. For example, think of the difference between the everyday use and the mathematical meaning of the following words: The boy ate a Chinese take away for his dinner.

The young girl was taken to hospital with an acute pain in her side. The two girls were wearing similar dresses to the party. The Frenchman asked his son to translate the menu. The two friends asked for a regular coke with their pizza.

Perhaps now it is easier to see why some pupils are confused in the maths classroom! Task 4. Coursework and other oral and written pieces in mathematics also allow pupils to engage in communicating their mathematical ideas and findings to an audience within and outside the classroom. In many cases the most important form of literacy for pupils is being able to understand and interpret the question in the examinations. Like all aspects of education, the area receiving the most attention in the classroom is the one in which measurements can be made—namely, the assessed parts of the course.

Next, the pupil-centred activity occurs and finally the lesson ends with a whole-class plenary where the learning is reviewed. The four main principles of the Strategy are: DfES, b Teaching mathematics 52 Task 4. For example, your planning, preparation, assessment, monitoring and recording will assist pupils to progress in their learning.

Training materials to assist teachers in implementing the Key Stage 3 Strategy are available from the DfES for the following themes: In Year 7, the level 4 work is revised but most of the content is directed at level 5; in Year 8, level 5 work is consolidated and some work at level 6 begins; Year 9 constitutes a revision of level 5 work and continuation of level 6 activities. For the more able students, level 7 and some level 8 objectives may be addressed.

Examples of what the pupils should know and be able to do at the end of each academic year are included to illustrate the depth of content and coverage of material. The final section of the Framework contains a checklist of mathematical vocabulary spanning the content for the three years. A summary of the distinctive features of each area of maths at Key Stage 3 are included in the programme and teachers are encouraged to use a mixture of paper-based, practical and ICT approaches in each topic.

A three-part lesson similar to that of the NNS is advocated with the requirement that each lesson contains direct teaching and interaction with the pupils, and activities or exercises that the pupils will complete in class.

The development of thinking skills is highlighted in terms of the application of mathematics in novel contexts or investigations. The national strategies for literacy 53 The plenary A plenary is multi-functional.

It is usually a means of bringing the class together to round off and summarise the main learning points of the lesson DfES, a. Often it is used to re-focus the pupils on what they have achieved and to look forward to future work.

If pupils are offering their own feedback from a learning activity, then it is also the time when teachers can assess learning, identify misconceptions and plan accordingly. Although the majority of plenaries occur at the end of a lesson, many teachers find it useful to initiate some discussion midway through the lesson to consolidate understanding and move pupils to the next stage of the learning process.

Plenaries can range from two minutes during a sequence of related lessons to twenty minutes at the start or end of a topic. Some ideas for suitable pupil involvement in plenaries is offered in the DfES document The Plenary, which is available online from the Standards website www. The Springboard 7 materials issued in November DfES, c were designed as a two-term teaching programme to complement and not replicate the teaching materials created for summer numeracy schools.

High-ability Key Stage 3 students At the opposite end of the spectrum, the high-ability students need to be stretched and challenged in mathematics classes.

It encourages teachers to establish a classroom ethos that celebrates success for all pupils to reduce the social pressures that result in under achievement in this subgroup of students.

It also counsels the use of challenging questions at an appropriate level of difficulty with extension material of an open-ended nature.

The emphasis is on quality and not quantity: Years 7, 8 and 9 Teaching mathematics 54 offers suggestions for extension material for the more able students. Further sources of ideas, support and guidance can be found on the following websites: During this time pupils become bored and frustrated being exposed to the low-level work that they have already mastered at primary school.

Before the introduction of the Key Stage 3 Framework, the pupils also found the style of teaching in secondary schools too dry and didactic compared to the interactivity of the NNS lessons.

As a result, transition units have been introduced to complement the existing information available for each pupil. The transition units are intended to ensure that: This unit is then developed further in the Year 7 transition unit to sustain the same teaching style and also maintain the momentum in the learning process.

Year-on-year transition units are also available for use within Key Stage 3. The key topics addressed are the links between fractions, decimals and percentages, and thinking The national strategies for literacy 55 proportionally DfES, b; c.

Further exemplification and details of the transition units can be found on the DfES website. Why are some teachers more effective than others? Can you be an effective teacher all of the time and for all pupils? Or is your effectiveness more transparent to subgroups within the class? This chapter is composed of three main themes: What makes an effective teacher of mathematics? There are two basic requirements for teachers: The former ensures teachers have an up-to-date knowledge of the material they will be delivering and the latter ensures they can deliver it.

If teaching is solely the skill of transmission, then the teacher needs the material to transmit and an effective means of transmission. Stones describes the pedagogy of teaching as follows: Using analogies helps pupils to relate new knowledge to existing knowledge, so assisting pupils in remembering vast quantities of information or new rules for solving tasks. The simplest pedagogical skill to master is that of exposition or telling.

But learning is not a one-way street composed only of teacher talk. The pupils need to be engaged in the process—actively involved in the lesson and able to apply its outcomes to the real world. Negative effects lead to rejection of the subject matter, or low emotional feelings, so the content of the lesson is forgotten by the pupil.

It should be noted that reinforcement and feedback are two separate entities. Clearly, feedback is of particular importance in the teaching and learning process. Research completed by Askew et al. These teachers eschewed the transmitting of subject knowledge in favour of exploring the connections in maths with the children through discovery.

The teachers themselves were aware of these connections, particularly in the primary curriculum, thereby indicating solid subject know-ledge, so they felt that effective teaching was composed of the following four objectives: For further reading, consult Askew et al. Effective lesson planning Effectiveness in terms of a lesson can be summarised as: Stones advocates three phases in a lesson: In the preactive phase the teacher identifies the prior knowledge needed for the topic, plans the order of presentation of the material, gathers the necessary Being an effective mathematics teacher 59 resources for the lesson such as calculators, worksheets and rulers, and creates overhead transparencies or interactive whiteboard resources, and so on.

The teacher sets the pupils to task and then monitors their progress informally as he or she circulates around the room. The lesson is concluded by recapping on and discussing the learning outcomes of the lesson with the pupils in the plenary. The evaluative phase completes the teaching process.

The language used in the evaluative phase should mirror that used in the lesson plan. Task 5. Using bullet points, list the possible content of the first in the series of lessons in each topic.

Share and discuss your ideas with a friend or tutor. A pedagogical analysis A lesson plan always has aims and learning outcomes. The more specific and explicated intentions of the lesson are called the key objectives or learning outcomes. There are usually two categories of learning outcomes: Learning outcomes must be measurable, so they should be expressed using the appropriate vocabulary.

List, locate, define, create, know, explain, understand, state, write down, categorise, plan, think of, find, remember, identify, group, consider, bring, use, show, relate. Teaching mathematics 60 Stones avers that there are three types of cognitive skills in learning: Remembering, Identifying and Doing. See Table 5.

Table 5. The Doing skill is the high-level skill of applying the new knowledge in a novel context working out the length of a ladder resting against a wall. The breaking down of a concept in this way is called a pedagogical analysis of the concept. This matrix of skills is the optimal route for a teacher to take in order to attain the intended goal of teaching. Effective teachers complete a pedagogical analysis for every topic they teach.

As you become more experienced, this skill of breaking a concept into its pedagogical parts becomes natural and almost unconscious. Exemplars also illustrate the criterial attributes of the skill or concept being taught. Criterial attributes refer to the core elements of the concept or skill.

Non-exemplars have the converse meaning—namely, examples not possessing the criterial attributes of the concept or skill. For a more detailed explanation, consult Stones The criterial attributes for three-figure bearings are: So the following three diagrams are exemplars: Being an effective mathematics teacher 61 while these diagrams are non-exemplars: They also build in opportunities to test for misconceptions using this technique.

How would you integrate this into your teaching? First, think of the types of questions you will ask in the mental maths section. Regardless of the method the pupils use, they must still remember the place value of each Teaching mathematics 62 of the digits and ensure their addition is accurate. As part of the teacher-pupil interactions, a discussion of the types of approaches being used will ensue and from this, the pupils will be able to identify quick and effective methods compared to the slower, more tedious processes.

They will also be able to identify incorrect methods where the place value of the digits has been ignored. The main part of the lesson allows the pupils to practise their doing skill. Working in small groups or pairs, the pupils can use the strategies rehearsed in the first phase of the lesson to assist them with the calculations in the given tasks.

They will be applying their skills and understanding of mathematics in the new or novel context. Through peer discussion and collaboration, the pupils will be able to detect alternative approaches to the task through remembering and identifying similar examples covered in the earlier part of the lesson. The final stage—the plenary—will be similar to the last section of the mental maths phase where the pupils will be sharing and identifying strategies and methodologies for solving the task.

This discussion will encourage the pupils to remember the technique they used in completing the calculation as they explain it to other members of the class. They will be identifying reasons why their approach was effective or accurate, and finally the pupils will be re-doing the calculation in a step-by-step manner with the teacher or at the board themselves.

The three-stage heuristic advocated by Stones fits well within the recommendations of the NNS and Key Stage 3 Strategy due to the need for high levels of teacher-pupil interaction in lessons and the importance of addressing the key objectives for the lesson.

Planning problem-solving Stones , p. He also advocates the use of problem-solving to teach for understanding.

## Handbook of Mathematics

Closed problems have a single solution, so assessment is either correct or incorrect. For instance, consider the closed problem: A group of Year 4 children are going on a school outing to a zoo. How many coaches are required to transport the children if each coach has seats for a maximum of 35 children? For example, consider the case where Being an effective mathematics teacher 63 A golf ball lands 4 metres from a bunker and 3 metres from the hole.

How far apart are the bunker and the hole? Clearly, the direction in which the measurements are being made will influence the answer! Effective teachers are aware that it is only through exposure to problems of this latter form that pupils can begin to establish ways of thinking about the processes involved in problem-solving.

An essential aspect of successful problem-solving is being able to identify the problem. What are the key issues? What are the variables? What do we know and what do we need to know to obtain an answer? Means-end analysis requires the pupil to look at the difference between his or her current state of knowledge about the problem and the state of knowledge required for finding the solution to the problem.

Effective teachers often use language in the form of verbal cueing to assist and encourage the pupils in the right direction without domineering the problem-solving process.

He uses a four-stage process: Teaching skills embedded in the heuristics In both heuristics, the internalisation of the new concepts is a key learning outcome of the teaching process. So how does an effective teacher ensure that pupils have reached this stage? Teaching mathematics 64 The following list highlights the five steps inherent in all lessons where new concepts or principles are being uncovered either through direct teaching or discovery via problem-solving: In relation to the NNS, the questioning occurs in Stages 1 and 3—to engage the pupils in mental maths and also to review the learning in the final plenary.

If particular mental strategies have been discussed, there will be some Show and Tell from the pupils and perhaps a little Guiding and Cueing from the teacher. The second phase of the lesson the groupwork—addresses the Practice step with some Feedback from other members of the group. While the teacher is working alongside a group, there may be more Show and Tell from the pupils, Guiding and Cueing from the teacher, plus Feedback from both the pupils and the teacher.

Finally, the Plenary stage will focus predominantly on the Feedback with some Show and Tell as appropriate. Establishing a positive learning environment Effective teachers are aware of the importance of establishing a positive and open learning environment in their classroom.

Regardless of the age and ability of the pupils, these teachers ensure success in maths is achieved and celebrated by all students to foster self-esteem.All sections are contextualised in an educational setting.

But learning is not a one-way street composed only of teacher talk. Receptiveness is the willingness to learn. Perhaps now it is easier to see why some pupils are confused in the maths classroom! Skip to main content Skip to table of contents. After lunchtime, pupils tend to be restless or tired, which means that a more structured and controlled teaching style is more effective for afternoon classes.

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