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Annex C
Initial Teacher Training National Curriculum for Primary Mathematics
Introduction
The initial teacher training (ITT) National Curriculum for primary mathematics specifies the essential core of knowledge, understanding and skills which all trainees, on all courses of primary ITT, must be taught and be able to use in relation to mathematics.
These requirements come into force for all courses from 1 September 1998.
The curriculum has been written for those who have a mathematical background, including providers of primary mathematics ITT in schools, higher education institutions and elsewhere. While every attempt has been made to avoid jargon, correct mathematical terminology has been used where appropriate.
The curriculum is in three sections:
Section A Knowledge and Understanding Required by Trainees to Secure Pupils' Progress in Mathematics
This section sets out the knowledge and understanding which, as part of all courses, trainees must be taught and be able to apply in order to secure pupils' progress in mathematics. It lists the mathematical knowledge, understanding and skills which trainees need to be able to describe, explain, manipulate and use when teaching mathematics at KS1 and KS2 and where applicable, to children in nursery and reception classes.
As part of all courses trainees must be taught how these mathematical ideas are connected and how they underpin pupils' understanding of and skills in mathematics.
Section B Effective Teaching and Assessment Methods
This section sets out the teaching and assessment methods which, as part of all courses of ITT, all trainees must be taught and be able to use. Only teaching and assessment methods specific to primary mathematics have been included in this curriculum. Standards which apply to more generic areas of teaching and assessment, and which all those to be awarded Qualified Teacher Status must meet, are set out in the Standards for the Award of Qualified Teacher Status (Annex A to this Circular).
Section C Trainees' Knowledge and Understanding of Mathematics
This section sets out the knowledge and understanding of mathematics which trainees need in order to underpin effective teaching of mathematics at primary level. All providers of ITT must audit trainees' knowledge and understanding of the mathematics contained in the National Curriculum programmes of study for mathematics at KS1 and KS2, and that specified in paragraph 13 of this document. Where gaps in trainees' subject knowledge are identified, providers of ITT must make arrangements to ensure that trainees gain that knowledge during the course and that, by the end of the course, they are competent in using the mathematics in their teaching. It is likely that many aspects of Section e can be taught quite quickly. ITT providers will decide how best to teach the content of Section C, but, where appropriate, much might be covered through the use of supported self-study, or through guided reading prior to the course. While some of the content may require direct teaching, some could also be taught alongside aspects of Sections A and B of the curriculum.
The ITT National Curriculum for primary mathematics does not attempt to cover everything that needs to be taught to trainee teachers. It is expected that providers of ITT will include in their courses, other aspects of mathematics, not specified in this curriculum.
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This Annex specifies a curriculum. It does not constitute a course model or a scheme of work. There is no intention to impose on providers of ITT the way in which the curriculum should be delivered and assessed, nor to specify the materials or activities which should be used to support the training. Although the curriculum is set out in separate sections, there is no expectation that providers will teach these discretely. There is considerable flexibility for providers of ITT to integrate aspects of the three sections of the curriculum in order to provide coherent, intellectually stimulating and professionally challenging courses of primary ITT.
As part of all courses, providers of ITT are required to prepare trainees to teach the National Curriculum for pupils and understand statutory requirements. The In National Curriculum for primary mathematics does not, therefore, repeat the content of the pupils' curriculum but rather sets out the core of what it is that trainees need to be taught, know and be able to do if they are to teach the pupils' National Curriculum effectively. While the ITT National Curriculum for primary mathematics includes the key areas contained in the pupils' National Curriculum for mathematics, it is not set out under the same headings as the pupils' National Curriculum, nor does it give equal emphasis to each area. In setting out a core ITT curriculum for primary mathematics, greater detail has been given on those areas of mathematics teaching where research and inspection evidence indicates that trainees need to focus most if they are to teach effectively. The curriculum gives particular attention to the teaching of number.
Providers must ensure that only those trainees who have shown that they have the knowledge, understanding and skills to teach effectively are judged to have successfully completed a course leading to Qualified Teacher Status. Detailed requirements are set out in the Standards for the Award of Qualified Teacher Status (Annex A to this Circular).
A. Knowledge and Understanding Required by Trainees to Secure Pupils' Progress in Mathematics
1. Early mathematics (nursery and reception)
As part of all courses, trainees must be taught the Importance of pupils in nursery and reception classes acquiring the basic mathematical concepts necessary for later progression in mathematics, including the knowledge, understanding and skills needed to:
a. recognise and use numbers, including:
i. counting;
ii. appropriate recording of numerals and counting;
iii. understanding the value of small numbers, and combining them;
iv. becoming familiar with larger numbers;
b. compare, as a basis for recognising relationships, eg. in measures and shapes;
c. order and sequence, as a basis for understanding number, spatial relationships and measures;
d. identify the properties of, and sort, sets of objects, numbers and shapes as a basis for classification;
e. establish invariant properties as a basis for work in number, measures and shape;
f. use mathematical language to describe shape, position, size and quantity, eg. circle, in front of, bigger than, more than, and become familiar with the language associated with carrying out simple number operations;
g. use their developing understanding to carry out simple number operations and to solve practical problems.
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2. Progression in pupils' mathematics
a. In order to ensure that pupils develop a progressively more powerful, abstract and precise understanding of mathematics, all courses must ensure that trainees are taught that pupils' progress depends upon teaching which enables them to go beyond their concrete experiences in order to establish general concepts and develop the use of flexible and efficient mental and written procedures.
b. In order to understand the high expectations that teachers should have of their pupils, to aid planning and to ensure that trainees know how pupils are progressing in mathematics, all courses must ensure that trainees are taught the essential stages of development and progression in pupils' mathematics.
As part of all courses, trainees must be taught the Importance of ensuring that pupils progress from:
i. using informal mathematical vocabulary, to using precise and correct mathematical vocabulary, notation and symbols;
ii. counting, ordering and sorting small whole numbers, to using and approximating numbers within the extended number system and using the number operations to calculate accurately and efficiently;
iii. guessing unknown numbers as a basis for trial and improvement and forming simple statements with unknowns, to solving simple equations using inverse operations, manipulating algebraic symbols, and constructing general expressions;
iv. sorting and classifying shapes and identifying properties, to transforming shapes, recognising invariant properties and using precise geometric language;
v. using simple drawings and diagrams to represent mathematical ideas, to using conventional diagrams, graphs and notation;
vi. measuring with non-standard units, to understanding the systems of measurement in common use and using them to measure and calculate angle, length, mass, area, volume, capacity, speed and time, give approximate answers and estimate;
vii. collecting discrete data by counting and recording these with simple diagrams and graphs, to handling both discrete and continuous data, classifying, representing and interpreting the data, employing more sophisticated graphical forms and summary statistics.
3. Key aspects of mathematics underpinning progression
In order to understand how to develop pupils' mathematics, all courses must ensure that trainees know and understand the following key aspects of mathematics. They must be taught how and why the different elements work, how they are connected and how they underpin pupils' progress in developing understanding of, and skills in, mathematics.
a. Structures and operations, including:
i. the structure of number eg. order and size;
ii. the conceptual links between different aspects of number eg. place value, zero, fractions, powers of ten, and how the relationship between these provides a conceptual framework for decimals;
iii. the nature of the four operations and the relationships between them eg. subtraction as the inverse operation of addition; multiplication as repeated addition; if ab = c (and a ≠ 0) then c ÷ a = b;
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iv. the precedence of number operations eg. in the absence of brackets, multiplication and division take priority over addition and subtraction;
v. the manipulation of numbers and part numbers eg. multiplying and dividing by 3, 30 and then 300 etc.; using the sequence 1/2, 1/4, 1/8, 1/16 to find 7/16;
vi. the basic rules of arithmetic
- a + b = b + a; a x b = b x a (the commutative laws) eg. that 4 x 3 = 3 x 4 [but 4 ÷ 3 ≠ 3 ÷ 4];
- (a + b) + c = a + (b + c); (a x b) x c = a x (b x c) (the associative laws) eg. that (2 + 4) + 1 = 2 + (4 + 1) [but (2 - 4) - 1 ≠ 2 - (4 - 1)];
- a x (b + c) = (a x b) + (a x c); a x (b - c) = (a x b) - (a x c) (the distributive laws) eg. that 2 x (5 + 3) = (2 x 5) + (2 x 3) [but 2 ÷ (5 + 3) ≠ (2 ÷ 5) + (2 ÷ 3)];
vii. the effect of operations eg. knowing when and why multiplying by a number results in a larger value and when it does not, (1/2 of 1/2 is 1/4); repeated transformations in geometry;
viii. how and why algorithms work, including standard and non-standard written methods for the four rules of arithmetic, including subtraction by decomposition and subtraction by equal addition, long multiplication, long and short division;
b. Equivalence, including:
i. numbers represented in equivalent forms eg. fractions/decimals/percentages (1/2 of 6 = 0.5 of 6 = 50% of 6); 362 = 300 + 60 + 2; 78 = 7 x 10 + 8; 36 = 2² x 3²; and how and why to move between different representations eg. converting fractions to decimals and explaining the reasons for the equivalence;
ii. that forms of notation can be equivalent eg. divide "÷", "/"; multiply "x". ".", 5(4), ab; 2³ = 2 x 2 x 2; and that the same form of notation can represent different concepts eg. that 1/4 can mean: 1/4 of a whole; 1/4 of a number; 1 ÷ 4; 1/4 on a number line; ratio; a scale factor etc;
iii. that algebraic expressions can be equivalent eg. that 2n + 2 is equivalent to 2(n + 1);
iv. that certain mathematical terms relate to equivalent operations eg. the use of a scale factor, enlargement or reduction, involves multiplication;
v. that a mathematical term can define a class of items that meet specified properties eg. "pentagon" represents all 5-sided polygons;
vi. that, in some cases, ordinary words used in a mathematical sense can mean different things eg. and does not mean add in the sentence "find the difference between 6 and 4" but it does in "total 3 and 4"; and that use of the same language may require different operations eg. if £20 is shared equally between 4 children, how much will each child get (division); and if 4 children shared equally a sum of money and each gets £6, how much did they share? (multiplication);
vii. that combinations of operations may or may not lead to equivalent outcomes eg. the final position of a shape when reflected then rotated may be different if rotated then reflected;
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viii. conservation of properties under operations eg. angles in a triangle remain unchanged after scaling; some transformations lead to congruence;
c. Classification, including:
i. properties of numbers eg. odd, prime, squares, factors, triangular numbers;
ii. properties of shapes eg. categorising shapes according to properties, such as regular polygons with equal sides and equal angles; the relationships between the interior/exterior angles and sides of 2-D shapes; the faces, edges and vertices of 3-D shapes;
iii. summary statistics of sets of data eg. range, median, mode, mean;
iv. comparison and ordering eg. ordering containers by capacity, finding numbers greater than a hundred or less than -2, using relationships, such as "in proportion to", "similar to", "congruent to";
d. Diagrammatic, graphical and algebraic representation, including:
i. construction of 2-D and 3-D shapes;
ii. representation of algebraic relationships using co-ordinate systems eg. transformation of shapes, scale factor; mappings such as x → + x + 2; functions; converting Fahrenheit to Celsius using C = 5/9 (F-32);
iii. representation of discrete and continuous data eg. using tally charts, frequency diagrams, line graphs, pie charts, scattergrams, tables and charts;
iv. interpretation and prediction eg. recognising trends, effects of sample choice, extrapolation from data to predict;
v. representing a numerical problem as an equation, which can then be solved;
e. Methods and applications, including:
i. use of correct mathematical terminology eg. product, sum, mean, zero, perpendicular;
ii. the difference between mathematical conventions and inherent mathematical properties eg. the order of mathematical operations is a convention, but the fact that multiplying two negative numbers gives a positive answer (-3x - 4 = + 12), is inherent in the mathematical structure;
iii. selecting and using the most appropriate and efficient mathematical methods to solve problems;
iv. testing, conjecturing and justifying eg. stating and showing that the sum of any two odd numbers is even; testing whether, and understanding why, the sum of 3 consecutive numbers is always divisible by 3; explaining why multiplication does not always make numbers bigger;
v. the application of number to shape and space, data handling and measures eg. the use of percentage to compare properties in samples of different sizes; the relationship between measurement and the concept of ratio;
vi. the use of mathematics across the curriculum eg. collecting, presenting and interpreting data in science, history and geography; measuring in science, and measuring and using properties of shape in design, technology and art;
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vii. reasoning and proof eg. reasoning that the exterior angles of a polygon sum to 360° and proving it through the use of geometrical properties.
4. As part of all courses, trainees must be taught the importance of engaging pupils' interest in, and developing their enthusiasm for, mathematics.
B. Effective Teaching and Assessment Methods
5. As part of all 3-8 courses and 3-11 courses, trainees must be taught:
a. how to teach the early stages of mathematics to pupils in nursery and reception classes through:
- high quality, interactive oral work;
- demonstration; and
- structured practical activities
eg. using construction activities, physical activities, stories, rhymes, songs and everyday experiences
in order to develop pupils' understanding of number, measures and shape and space, their awareness of simple number operations, and their use of mathematical language.
6. As part of all courses, trainees must be taught:
a. how to teach accurate and rapid mental calculation, through ensuring that pupils:
i. identify and use the properties of number and the relationships between them: size (including estimation and approximation). order and equivalence;
ii. understand the operations of addition, subtraction, multiplication and division;
iii. have instant recall of number facts, including multiplication tables;
iv. use known number facts to derive others;
v. build effective strategies for dealing with mental calculations and use and adapt these for more complex cases;
vi. use a variety of mental strategies including:
- rearranging numbers eg. putting the larger number first for addition - recognising that 7 + 23 is the same as 23 + 7;
- using repeated operations eg. finding 1/9 by finding 1/3 of 1/3;
- halving and doubling eg. 14 x 3 = 7 x 6 = 42; 15% of 60 is 10% of 60 (=6), plus 50% of 6 (= 3) to give 9;
- recognising and using near doubles or halves eg. 16 + 17 = (16 x 2) + 1 or (17 x 2) - 1; realising that 24 - 13 is the same as 24 - 12 - 1 = 11;
- using patterns of similar calculations eg. since 25 x 4 = 100 then 26 x 4 is 4 more, working out the 6x table by doubling the 3x table;
- partitioning eg. 27 + 34 = 20 + 30 + 7 + 4 = 50 + 11 = 61;
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- bridging eg. through the nearest 10, or a multiple of 10, working out mentally that 158 + 17= 175 and explaining that 158 + 2 + 15 = 160 + 15 = 175;
b. how to teach efficient standard and non-standard written and part-written methods of computation for calculations too complex to be undertaken by mental methods, through ensuring that pupils:
i. understand and know how to use at least one standard method of calculation for each operation;
ii. practise and refine written methods derived from mental methods;
iii. contrast the efficiency of standard and non-standard methods of calculation and assess the extent to which they apply to particular problems;
iv. are presented with calculations in a format which allows them to select and use the most efficient method for any calculation eg. presenting calculations in horizontal format which encourages a choice of method, rather than presenting calculations set out vertically, which implies that a standard written method must be used;
v. select and use the most efficient standard or non-standard written method for the calculation in hand, recognising where a mental method is more efficient eg. that 2001-1999 is best done mentally;
vi. determine a correct sequence of operations to use for calculations;
vii. use mental and written methods to approximate expected answers to computations and check for reasonableness and accuracy;
c. how to teach the solving of numerical problems involving more than one operation, through ensuring that pupils:
i. read, interpret and simplify problems;
ii. determine which operations are needed to solve problems;
iii. select the most appropriate mental, partial written, written or calculator strategy;
iv. carry out accurately the mathematical operations required to solve the problem;
v. check their answers for reasonableness and accuracy;
vi. present solutions logically, whether orally or in writing, as appropriate;
d. how to teach the appropriate and efficient use of calculators, especially when working with large numbers and realistic data, ensuring that they are not used to replace mental calculation;
e. how to teach the foundations of algebra, through ensuring that pupils:
i. are taught how to make general statements using words, pictures and symbols that they can record and interpret;
ii. move from using words and pictures to represent values, to using letters to represent unknowns and variables;
iii. make succinct general statements arising from examples and observations, and begin to recognise the power of algebra to do so;
iv. use the equals sign accurately in different mathematical statements or sentences, so that they understand where and why the sign must be used, and that the quantities or expressions on either side must be equivalent;
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f. how to teach shape and space and measures, through ensuring that pupils:
i. handle and construct shapes in order to begin to classify them, identify their properties in order to form generalised concepts of 2-D and 3-D shapes, and use the associated vocabulary accurately;
ii. understand that angular measure describes the amount of turn, before identifying, constructing and measuring angles;
iii. appreciate the effect of enlargement and reduction of 2-D and 3-D shapes, recognising which properties are conserved and which change, including the relationship between length, area and volume;
g. how to teach data handling, through ensuring that pupils are given the kinds of examples and tasks which lead them to appreciate the advantages and limitations, according to context, of different forms for representing discrete and continuous data;
h. the ways in which information technology can be used to support mathematics teaching, including its potential for use by pupils in:
i. the rapid exploration and manipulation of data eg. through using a spreadsheet to find a mean and draw a bar chart;
ii. working with dynamic images;
iii. learning from immediate feedback eg. entering an instruction and changing it in the light of an observed response, for example, when building a shape on the screen using Logo;
iv. developing logical thinking eg. writing short programs or using a spreadsheet;
v. practice and reinforcement eg. using software designed to "teach" a particular skill and receiving rapid assessment feedback;
i. how to plan and pace individual mathematics lessons and sequences of lessons in the short, medium and longer term which:
i. ensure that the introduction of any new topic incorporates the essential features of the mathematical concepts which pupils must ultimately acquire;
ii. include sufficient time dedicated to the systematic and regular teaching of number, including mental work eg. a dedicated numeracy hour;
iii. secure deeper understanding of the connections within and between different areas of mathematics, including through purposeful enquiry within mathematics;
iv. allow rigorous application of mathematical knowledge and understanding to new and real contexts and problems;
j. how to select and use materials, including:
i. how to evaluate, choose and use mathematical resources effectively to support and enhance teaching, including textbooks, mathematics schemes, teachers' resource books, materials, apparatus, calculators, software, educational broadcasts, visits and real life materials and situations;
ii. how to identify when it is appropriate to use apparatus to support progression in mathematics and when it is not;
k. how to lead oral work while teaching whole classes or groups which:
i. has pace and variety, and flows well from one section of the mathematics lesson to the next;
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ii. includes sufficient teacher exposition, direct instruction and effective questioning to secure the involvement of the whole class and to enable pupils to contribute actively to discussion, and allows time for pupils to think through answers before a response is demanded;
iii. includes teachers' questions which are adjusted and targeted to ensure that pupils of all abilities are engaged;
iv. encourages and enables pupils to show their understanding of mathematical operations, and elicits any misconceptions they may have;
v. requires pupils to provide clear explanations to the teacher and to other pupils, as well as giving answers, and encourages them to ask questions;
vi. provides clear feedback, indicating how pupils' work can be improved and remedying misconceptions;
l. how to structure their teaching and use interactive methods with whole classes, groups and individuals, including:
i. introducing the lesson to command attention, to set out what mathematics is to be taught and, where appropriate, to review and draw upon previous work;
ii. using skilfully framed open and closed, oral and written questions which elicit answers from which pupils' mathematical understanding can be judged and giving clear feedback to take pupils' learning forward;
iii. using oral and mental work, in particular to develop and extend pupils' use of mathematical vocabulary and accurate and rapid recall of number facts;
iv. giving clear instructions eg. how to measure angles using a protractor; how to present a mathematical argument;
v. providing clear explanations when introducing an area of mathematics work and when resolving pupils' misconceptions and errors;
vi. demonstrating and illustrating mathematics using appropriate resources and visual displays eg. effective use of an OHP or board; showing the structure of place value using appropriate apparatus; using spreadsheets for data handling;
vii. providing opportunities for follow-up, guided practice and consolidation in mathematics, including how to:
- use diverse activities on a mathematical topic in order to consolidate and extend understanding;
- provide pupils with opportunities to solve problems through applying mathematical knowledge, understanding and skills to new situations;
- intervene constructively, eg. to monitor progress or inject pace and challenge, and not just when pupils request help;
- provide corrective instruction for pupils who have not grasped the material being taught;
viii. summarising and reviewing during and towards the end of lessons the mathematics that has been taught and what pupils have learnt, and using this to engage pupils in the presentation of their work, to identify and rectify misunderstandings, and to give pupils insight into the next stage of their learning.
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7. Common errors and misconceptions in mathematics
As part of all courses, trainees must be taught:
a. to recognise common pupil errors and misconceptions in mathematics, and to understand how these arise, how they can be prevented, and how to remedy them, including, among others:
i. counting on 3 from 7 to get 9 as result of starting with the 7;
ii. reading 206 as 26 as a result of misunderstanding about the number system and place value;
iii. misunderstanding the order of the subtraction operation eg. 3 - 7 = 4;
iv. an expectation that the outcome of division always gives a smaller value eg. 4 ÷ 1/2 = 2;
v. lining up columns of' numbers for operations against a left or right hand margin, irrespective of the position of the decimal point;
vi. thinking that numbers are larger if there are more decimal digits eg. 3.16 is larger than 3.2;
vii. stating that two identical angles are unequal because the length of the arms are different in each, as a result of thinking that an angle is the distance between the ends of the lines;
viii. misreading the scale on a ruler, starting at 1, rather than 0, as a result of not understanding that the measure starts from 0;
ix. not using the scale when interpreting a graph, treating the graph as a picture rather than a scaled representation;
x. thinking that, when throwing a die, a 6 is harder to get than other numbers, through not understanding the nature of equally likely and independent events;
b. to avoid teaching mathematics in ways which contribute to or exacerbate pupils' misconceptions by, for example:
i. recognising that if pupils are taught to add a zero when multiplying by 10, they may also assume the rule works with decimals and numbers less than 1 eg. avoiding errors such as 2.3 x 10 = 2.30 or 2.3 x 10 = 20.3;
ii. making it clear that it is the relative position of the digits which is altered, and not the decimal point which moves, when multiplying and dividing by powers of 10;
iii. ensuring that algebraic symbols are used to represent values and not as shorthand for words eg. 2a + 3b is not shorthand for 2 apples plus 3 bananas;
iv. recognising that if examples of geometric shapes are always presented in the same orientation this may limit pupils' concept of these shapes;
v. recognising the need to use precise mathematical vocabulary or notation eg. avoiding the use of "take away" as the general word for subtraction, or avoiding misuse of the "=" sign to carry on part of the same calculation (eg. 3 x 10-2 written as 3 x 10 = 30 - 2 = 28, rather than 3 x 10 - 2 = 30 - 2 = 28).
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8. In order to understand how to evaluate and assess their teaching and pupils' learning in mathematics, all courses must ensure trainees are taught:
a. how to use formative, diagnostic and summative methods of assessing pupils' progress in mathematics, including how to:
i. make effective use of assessment information on pupils' attainment and progress in their teaching and in planning future lessons and sequences of lessons;
ii. set up assessment activities so that specific mathematical assessment can be undertaken for all pupils, including the very able, those who are not yet fluent in English and those with special educational needs, through preparing oral and written questions and setting up activities and tests which check for:
- misconceptions and errors in mental arithmetic and in written methods of calculation;
- understanding of mathematical concepts and the connections between different mathematical ideas;
iii. make summative assessments of individual pupils' progress and achievement in mathematics and present the outcomes for reporting purposes through the use of National Curriculum tests, baseline assessment where relevant, teacher assessment and other forms of individual pupil assessment, including the appropriate use of standardised tests;
b. how to recognise the standards of attainment in mathematics they should expect of their pupils, including:
i. the expected demands in relation to each relevant level description for KS1 and KS2 in mathematics and how to judge levels of attainment against these;
ii. how to identify under-achieving and very able pupils in mathematics;
iii. how national, local, comparative and school data about achievement in mathematics can be used to identify under-achievement and to set clear expectations and targets;
c. how inspection and research evidence, and international comparisons on the teaching of mathematics, can inform their teaching.
9. Opportunities to practise
As part of all courses, trainees must be given opportunities to practise, in taught sessions and in the classroom, those methods and skills described above.
C. Trainees' Knowledge and Understanding of Mathematics
10. All trainees enter a course of primary initial teacher training with a minimum qualification of GCSE Grade C (or its equivalent) in mathematics. However, the equivalence of such qualifications does not necessarily reflect a common range and depth of study. The mathematics qualifications held by trainees may not be sufficient to ensure they feel confident about, and are competent in, the mathematics they have studied and which they are required to teach.
11. Providers should audit trainees' knowledge, understanding and skills in mathematics against both the mathematics content specified in the KS1 and KS2 programmes of study and that set out in paragraph 13 below. Where gaps in trainees' subject knowledge and understanding are identified, providers must make arrangements, for example, through supported self-study, to ensure that trainees gain that knowledge and understanding during the course and that, by the end of the course, trainees are competent in using the mathematics specified.
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12. In relation to the mathematics set out in paragraph 13, as part of all courses trainees must be given opportunities to:
a. understand, and use correctly, mathematical terms which, in addition to those in the National Curriculum Mathematics Order, are necessary to enable trainees to be precise in their explanations to pupils, to discuss primary mathematics at a professional level, and to read inspection and classroom-focused research evidence with understanding;
b. identify how the different areas of mathematics relate to each other in order to provide coherence and progression in mathematics, and to make conceptual links across the subject and consider the implications of this for their teaching;
c. solve problems that require the use and application of mathematics;
d. use technology such as calculators and computers when appropriate, recognise when they might be inappropriate and become aware of their strengths and limitations;
e. enjoy mathematics so that they can teach it with enthusiasm.
13. Subject knowledge and understanding
This section has been divided into two columns. The left-hand column specifies the mathematical knowledge and understanding which all trainees are required to demonstrate by the end of their course, in order to underpin effective teaching. Trainees should be able to make conceptual links between the aspects of mathematics listed in the left-hand column.
The right-hand column has been included to indicate the relevance of the required subject knowledge to the KS1 and KS2 programmes of study, which trainees are required to teach.
As part of all courses, trainees must demonstrate that they know and understand | To underpin the teaching of Key Stage 1 and Key Stage 2 programmes of study, including: |
a. number and algebra
i. the real number system:
the arithmetic of integers, fractions and decimals;
forming equalities and inequalities and recognising when equality is preserved;
the distinction between a rational number and an irrational number; making sense of simple recurring decimals. | for example: the order and size of numbers; place value; the relationship between different representations, eg. fractions, decimals, percentages, and determining which representation is most appropriate; extending the number system to negative numbers, fractions and decimals; effects of multiplying positive numbers less than one; methods of computation, including the interpretation of remainders: 1/3 = 0.3'; 1/9 = 0.1'; 1/3 + 1/9 = 0.4'. |
ii. Indices:
representing numbers in index form including positive and negative integer exponents;
standard form. | for example: finding and recognising squares and cubes of numbers; ways of representing very large and very small numbers; developing understanding of lace value based on powers of ten (integers and decimal fractions). |
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As part of all courses, trainees must demonstrate that they know and understand | To underpin the teaching of Key Stage 1 and Key Stage 2 programmes of study, including: |
iii. number operations and algebra:
using the associative, commutative and distributive laws;
use of cancellation to simplify calculations;
using the multiplicative structure of ratio and percentage to solve problems;
finding factors and multiples of numbers and of simple algebraic expressions;
constructing general statements;
manipulating simple algebraic expressions and using formulae;
knowing when numerical expressions and algebraic expressions are equivalent;
number sequences, their nth terms and their sums. | for example: using efficient written methods of computation eg. 8 x 5 x 2 = 8 x (5 + 2) = 8 x 10 = 80; using algebraic structure to develop flexible, efficient methods of mental calculation eg. 17 x 9 = (10 + 7) x 9 = 10 x 9 + 7 x 9; sharing 12 in the ration 1:2; moving from boxes and words to letters and symbols, eg. from 13 + [] = 17 to 13 + n = 17; exploring number sequences, recognising and explaining patterns; expressing a general term of a number sequence as an algebraic expression. |
iv. equations, functions and graphs:
forming equations and solving linear and simultaneous linear equations, finding exact solutions;
interpreting functions and finding inverses of simple functions;
representing functions graphically and algebraically;
understanding the significance of gradients and intercepts;
interpreting graphs, and using them to solve equations. |
for example: representing general statements about numbers in algebraic form eg.
- recognising that a number is a multiple of 5 and that 5 therefore is a factor of that number, and representing it as 5n; - recognising that an even number can be represented by 2n and an odd number by 2n+1; finding numbers that satisfy different conditions eg. - a number plus three equals eighteen. What is the number? - the sum of two numbers is 27 and their difference is four, what are the numbers? - the sum of a number and its square is 2 less than 14, what is the number? linear relationships and simple mappings; recognising the relationship between co-ordinates of points on a straight line; drawing graphs to find approximate solutions of equations; using graphical representation of data to make predictions; understanding the different kinds of graphical representation in mathematics, across the curriculum, and in real situations. |
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As part of all courses, trainees must demonstrate that they know and understand | To underpin the teaching of Key Stage 1 and Key Stage 2 programmes of study, including: |
b. mathematical reasoning and proof
the correct use of =, ≡, →, ∴ ;
the difference between mathematical reasoning and explanation, as well as the proper use of evidence;
following rigorous mathematical argument;
familiarity with methods of proof, including simple deductive proof, proof by exhaustion and disproof by counter-example. | for example:
demonstrating and checking a particular case; the dangers of drawing conclusions after an event has occurred a few times; recognising the difference between something that happens occasionally and something that will always happen; using experimental evidence to determine likelihood and to predict; proving, for example, that numbers divisible by 6 are also divisible by 3 (deduction); proving, for example, that there are only 11 unique nets of cubes (exhaustion); disproving, for example, that any quadrilateral with sides of equal length is a square (counter-example). |
c. measures
understanding that the basis of measures is exact and that practical measurement is approximate;
standard measures and compound measures, including rates of change;
the relationship between measures, including length, area, volume and capacity;
understanding the importance of choice of unit and use of proportion. | for example: the relationships between imperial and metric measures in daily use; compound measures; work in shape and space including calculation and measuring length, area, volume, capacity; measuring time; measuring angles in degrees, half turn, quarter turn (right angles); measuring on scale diagrams; using given measurements to produce accurate diagrams. |
d. shape and space
Cartesian co-ordinates in 2-D;
2-D transformations;
angles, congruence and similarity in triangles and other shapes;
geometrical constructions;
identifying and measuring properties and characteristics of 2-D shapes;
using Pythagoras' theorem;
recognising the relationships between and using the formulae for the area of 2-D shapes; including rectangle and triangle, trapezium, and parallelogram; | for example: using co-ordinates to study locations; relationships between co-ordinates of related points on a line or in a shape; making and moving shapes; recognising the names and characteristics of transformations eg. translation, rotation, reflection, enlargement; visualising transformations; visualising and drawing nets of solids; understanding the properties of position, direction and movement; identifying conservation following certain transformations; understanding and using the properties of shapes, including symmetry; calculating perimeters and ares of simple shapes; measuring perimeters and areas of more complicated shapes; |
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As part of all courses, trainees must demonstrate that they know and understand | To underpin the teaching of Key Stage 1 and Key Stage 2 programmes of study, including: |
d. shape and space (cont'd)
the calculation of the area of circles and sectors, the length of circumferences and arcs;
recognise, understand and use formulae for the surface area and volume of prisms;
identifying 3-D solids and shapes and recognising their properties and characteristics. | for example: introducing the relationship between circumference and diameter and finding the area of a circle; identifying the numbers of faces, edges and vertices of solid shapes and the relationship between these; sorting solid objects according to specified characteristics; comparing the volumes of different objects; finding volumes of cuboids; finding, by practical methods, volumes of more complicated shapes; recognising reflective symmetries of 3-D shapes. |
e. probability and statistics
using discrete and continuous data and understanding the difference between them;
tabulating and representing data diagrammatically and graphically;
interpreting data and predicting from data;
finding and using the mean and other central measures;
finding and using measures of spread to compare distributions;
using systematic methods for identifying, counting and organising events and outcomes;
understanding the difference between probability and observed relative frequencies;
recognise independent and mutually exclusive events. | for example: counting (discrete data), measuring (continuous data); tabulating results from a survey; selecting tables and graphs to display different types of data and justifying choice; using a spreadsheet or data handling package; interpreting data in other subjects; understanding and using measures of average, the mode, the median and the mean in relevant contexts, and the range as a measure of spread; using tree diagrams to sort and list outcomes; listing the total scores possible when using two dice or two spinners; recognising where probabilities must be estimated and where they can be based on assumed equally likely outcomes; recognising that if a head results from the toss of a coin this does not influence the outcome of the next toss; recognising that when selecting a card from a pack, drawing a spade and drawing a heart are mutually exclusive events. |
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Annex D
Requirements for all Courses of Initial Teacher Training
Introduction
This Annex sets out requirements for all courses of initial teacher training (ITT). These requirements come into force for all courses from 1 September 1998 (10).
The document is divided into four sections:
A. Trainee Entry and Selection Requirements
This section sets out the entry requirements for all courses of initial teacher training and details the selection criteria to be applied by all providers. These are minimum requirements and providers will wish to develop additional criteria.
B. Course Length and Coverage
This section lists the types of course which may be offered and the minimum requirements for each type of course. The minimum requirements for all courses of primary ITT are set out at paragraphs 2.3.1, 2.3.2 and 2.3.3. Where, in addition to the specified minimum, providers of primary In choose to offer one or mere non--core, non-specialist subjects, the standards trainees must meet before being awarded Qualified Teacher Status are set out in the Standards for the Award of Qualified Teacher Status (Annex A) at A.2.g. and in the introduction to Section B. Providers may also wish to offer more limited coverage of other subjects, than that required for non-core, non-specialist subjects, eg. a few hours of taster training in a foundation subject, safety training in PE and/or design & technology. The nature and extent of any such training can be recorded on the newly qualified teacher's TTA Career Entry Profile. The specified types of course provide a basis for further continuing professional development. It is expected that teachers will continue to develop and broaden the range of their expertise throughout their careers.
C. Partnership Requirements
This section applies to training which takes place in partnership between schools (11) and higher education institutions or other providers, and sets out requirements relating to the involvement of schools, including the amount of time which trainees must spend in schools.
D. Quality Assurance Requirements
This section applies to all courses of initial teacher training. It sets out the arrangements which providers must put in place to ensure that training is of high quality, is regularly reviewed and that the award of Qualified Teacher Status is securely based.
Similar requirements apply to employment-based routes into teaching, such as the planned Graduate Teacher Programme.
10 The requirements at B.i.2.1.3, 2.1.4 and 2.1.5 apply 10 all those awarded QTS from May 1998.
11 Throughout this document, requirements relating to partnerships between HEIs and "schools" apply equally to partnerships between HEIs and Further Education colleges, VI form colleges and departments within schools and colleges.
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A. Trainee Entry and Selection Requirements
I. General
1.1 In the case of all courses of initial teacher training (ITT), providers are required to ensure that:
1.1.1 all entrants are able to communicate clearly and grammatically in spoken and written standard English;
1.1.2 all entrants have attained the standard required to achieve at least a grade C in the GCSE examination in mathematics and English; (12)
1.1.3 all those born on or after 1 September 1979 who enter primary and KS2/3 courses of ITT after 1 September 1998, have attained the standard required to achieve at least a grade C in a GCSE examination in a science subject (including combined science). The science qualifications which were approved by the Secretary of State under Section 400 of the Education Act 1996 for the year in which the qualification was awarded are acceptable to meet this requirement (12);
1.1.4 all entrants meet the Secretary of State's requirements for physical and mental fitness to teach as detailed in the relevant Circular (currently DFE 13/93);
1.1.5 entrants have not previously been excluded from teaching or working with children;
1.1.6 systems are in place to seek information on entrants' criminal backgrounds which might prevent employment as a teacher or with children or young persons (currently set out in DfEE Circular 11/95). Guidance can be found in DFE Circular 9/93;
1.1.7 selection procedures include representatives from those centrally involved in the training process, including school staff;
1.1.8 all trainees possess the personal, intellectual and presentational qualities suitable for teaching; providers should seek evidence of relevant experience with children;
1.1.9 as part of selection procedures, all candidates admitted to a course have been seen at an individual or group interview (13);
1.1.10 in order to ensure a high rate of course completion and award of Qualified Teacher Status, selection procedures and data, including entry qualifications, completion rates, newly qualified teachers' destinations and employers' satisfaction with newly qualified teachers, are monitored and action is taken to ensure that high calibre entrants are recruited to courses of ITT.
II. Postgraduate courses
1.2 In the case of postgraduate courses of ITT, in addition to the requirements set out at 1.1, providers should satisfy themselves that:
1.2.1 entrants hold a degree of a United Kingdom university or a higher education institution with degree awarding powers, or a degree of the CNAA, or a qualification recognised to be equivalent to a UK or CNAA degree;
1.2.2 the content of entrants' previous education provides the necessary foundation for work as a teacher in the phase(s) and subject(s) they are to teach.
III. Undergraduate courses
1.3 In addition to the requirements set out at 1.1, providers should satisfy themselves that:
1.3.1 entrants fulfil the academic requirement for admission to first degree studies; and
12 For prospective trainees without standard qualifications, providers should set their own equivalence tests. The TTA will audit samples of tests to ensure that standards are appropriate.
13 There is no need to interview all those who apply for courses and who are eligible.
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1.3.2 in the case of two-year courses, entrants have satisfactorily completed the equivalent of at least one year of full-time higher education studies. The content of entrants' higher education studies must provide the necessary foundation for work as a teacher in the phase(s) and subject(s) they are to teach.
B. COURSE LENGTH AND COVERAGE
I. Types of course
2.1 All providers must:
2.1.1 ensure, where applicable, that courses comply with requirements set out in any relevant ITT National Curriculum which is in force;
2.1.2 ensure that course content, structure and delivery, and the assessment of trainees, are designed to develop trainees' knowledge, skills and understanding to ensure that the standards for the award of Qualified Teacher Status are met; (14)
2.1.3 ensure that courses involve the assessment of all trainees against all the standards specified for the award of Qualified Teacher Status;
2.1.4 ensure that trainees meet all the standards specified for the award of Qualified Teacher Status before successfully completing a course of ITT;
2.1.5 ensure that all those trainees who successfully complete a course of ITT leading to Qualified Teacher Status receive a TTA Career Entry Profile.
2.2 All primary ITT courses must prepare trainees to teach at least one specialist subject and ensure that trainees are assessed against the relevant standards in relation to subject knowledge set out in the Standards for the Award of Qualified Teacher Status (Annex A), Section A.2. Specialist courses may also include advanced study of subject pedagogy and the foundations of preparation for subject co-ordination. The particular areas of strength which trainees acquire through specialist subject study can be recorded on the TTA Career Entry Profile.
2.3 Courses must cover one of the age ranges below:
2.3.1 3-8 - these courses must include specialist training for early years (nursery and reception) (15), the core subjects across KS1 and KS2 as specified in the ITT National Curriculum, and at least one specialist subject (16) across KS1 and KS2; in addition they must equip trainees to teach across the full primary curriculum in this age range;
2.3.2 3 or 5-11 - as a minimum, these courses must cover the core subjects across KS1 and KS2 as specified in the primary ITT National Curriculum, must equip trainees to teach across the entire primary age range with an emphasis on 3- or 5-8 or 7-11, and must include at least one specialist subject across KS1 and KS2. 3-11 courses must include specialist training for early years (nursery and reception); (16 & 17)
14 Qualified Teacher Status is awarded on successful completion of a course of ITT with a TTA accredited provider. This award is either concurrently with or after the award of a first degree of a UK university or a higher education institution with degree awarding powers, or a degree of the CNAA, or a qualification recognised to be equivalent to a UK or CNAA degree.
15 Additional specialist standards relating to early years (nursery and reception) for trainees on 3-8 courses and 3-11 courses are included in the Standards for the Award of Qualified Teacher Status (Annex A).
16 A specialist subject could be one of the core subjects or an additional subject.
17 Where providers choose to offer one or more non-core, non-specialist subjects in addition to the specified minimum, trainees being assessed for Qualified Teacher Status should be able to demonstrate a secure knowledge of the subject to a standard equivalent to at least level 7 of the pupils' National Curriculum, and meet all the other required standards, but, if necessary, with the support of a teacher experienced in the subject concerned. For RE, the required standard is broadly equivalent to the end of Key Stage statements for Key Stage 4 in SCAA's Model Syllabuses for RE. The newly qualified teacher's TTA Career Entry Profile can indicate priorities for induction in each of these subjects. Providers may also wish to offer more limited coverage of other subjects than that required for non-core, non-specialist subjects, eg. a few hours of taster training in a foundation subject, safety training in PE and/or design and technology. The nature and extent of any such training can be recorded on the trainee's TTA Career Entry Profile.
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2.3.3 7-11 - as a minimum, these courses must cover the core subjects across KS1 and KS2 as specified in the primary ITT National Curriculum, and must include at least one specialist subject across KS1 and KS2; (16 & 17)
2.3.4 7-14 - as a minimum, these courses must cover the core subjects as specified in the primary ITT National Curriculum, and a specialist subject at KS2 and KS3; (16)
2.3.5 11-16 or 18 - these courses must cover at least one specialist subject;
2.3.6 14-19 - these courses must cover at least one specialist subject, the 14-19 qualifications framework, including the relevant KS4 and post-16 examination syllabuses and vocational courses, and the relevant key skills required by 14-19 qualifications.
II. Length of postgraduate courses
2.4 The minimum amount of time which will be spent on courses of ITT is:
2.4.1 38 weeks for all full-time primary postgraduate courses;
2.4.2 36 weeks for all other full-time postgraduate courses.
C. Partnership Requirements
3.1 In the case of all courses of ITT, higher education institutions and other non-school trainers must work in partnership with schools (11) ensuring that:
3.1.1 schools are fully and actively involved in the planning and delivery of ITT, as well as in the selection and final assessment of trainees. The full partnership should regularly review and evaluate the training provided;
3.1.2 the division and deployment of available resources has been agreed in a way which reflects the training responsibilities undertaken by each partner;
3.1.3 effective selection criteria for partnership schools have been developed which are clear and available to all partners and trainees,and which take account of indicators such as OFSTED reports, test and examination results, exclusion rates, commitment to and previous successful experience of involvement in ITT;
3.1.4 where partnership schools fall short of the selection criteria set, providers must demonstrate that extra support will be provided to ensure that the training provided is of a high standard;
3.1.5 where schools no longer meet selection criteria, and extra support to ensure the quality of the training process cannot be guaranteed, procedures are in place for the de-selection of schools;
3.1.6 effective structures and procedures are in place to ensure efficient and effective communication across partnerships.
Time spent in schools
3.2 The amount of time spent by trainees in schools during their training, excluding school holidays, must be at least:
3.2.1 32 weeks for all four-year undergraduate courses;
3.2.2 24 weeks for all three-year undergraduate courses;
3.2.3 24 weeks for all full-time two-year secondary and KS2/3 undergraduate courses;
3.2.4 24 weeks for all full-time secondary and KS2/3 postgraduate courses;
3.2.5 18 weeks for all full-time primary postgraduate and two-year primary undergraduate courses;
3.2.6 18 weeks for all part-time postgraduate courses.
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D. Quality Assurance Requirements
4.1 For all courses of ITT, providers must ensure that:
4.1.1 the quality of provision across all aspects of the course is of a consistently high standard and complies with all the requirements set out in this Annex;
4.1.2 the training process is kept under regular review to ensure that the division of training responsibilities continues to reflect the strengths of those involved, that the standards and quality of the training process are identified and that, where necessary, action is taken to secure improvements;
4.1.3 trainees are given opportunities to observe good teachers at work and to work alongside them, to participate in teaching with expert practitioners in their chosen phase(s) and subject specialism(s), and to undertake substantial and sustained periods of class teaching in more than one school, observing, teaching and assessing pupils of differing abilities across the full age range for which they are being trained;
4.1.4 the roles and responsibilities of all those involved in ITT are set out clearly and are available to all participants, including trainees;
4.1.5 all those involved in training understand their roles and responsibilities and have the knowledge, understanding and skills needed to discharge these competently;
4.1.6 only those schools (11) and teachers who can offer appropriate training and support for trainees are used to provide ITT;
4.1.7 there are sufficient books, information technology resources and other specialist teaching resources, relevant to the age ranges and subjects offered, to enable all trainees to develop their knowledge, understanding and skills to at least the standard required for the award of Qualified Teacher Status;
4.1.8 the competence of trainees is rigorously, accurately and regularly assessed in order to evaluate their progress towards achieving the standards required for Qualified Teacher Status and to enable training to be focused on trainees' achievement of those standards;
4.1.9 internal and independent external moderation procedures are in place to ensure consistent, reliable and accurate assessment against the standards for Qualified Teacher Status;
4.1.10 quality issues raised through internal and external moderation are investigated, and that the outcomes of these investigations are used to establish appropriate short term, medium term and long term priorities for improving courses;
4.1.11 plans for course improvement are acted upon and monitored, evaluated and reviewed against criteria for success, and that targets are demonstrably met;
4.1.12 information about the effectiveness of newly qualified teachers in their first year of teaching is collected and used to improve training courses.