Primary Mathematics: Extending Knowledge in Practice
Publication Year: 2008
Still the biggest concern for many on initial teacher training courses is the acquisition of subject knowledge and the ability to translate that into effective teaching. This book addresses this – building on the core subject knowledge covered in the Achieving QTS series and relating it to classroom practice. It supports trainees in extending and deepening their knowledge of Maths and demonstrating how to apply it to planning and implementing lessons. Practical and up-to-date teaching examples are used to clearly contextualize subject knowledge. A clear focus on classroom practice helps trainees to build confidence and develop their own teaching strategies.
- Front Matter
- Back Matter
- Subject Index
First published in 2008 by Learning Matters Ltd
Reprinted in 2011 (twice)
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission in writing from Learning Matters.
© 2008 Alice Hansen
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ISBN: 978 1 84445 0541
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The Author[Page vii]
Alice Hansen is a Principal Lecturer in Primary Mathematics Education at the University of Cumbria. She has taught extensively at primary level in England and abroad. Alice has particular interests in how children construct geometrical definitions, the design of mathematical tasks and the use of ICT to enhance mathematics teaching and learning.[Page viii]
Objectives Index[Page 115]The Early Years Foundation StageEarly Learning Goals: Problem Solving, Reasoning and Numeracy
Objective Page Recognise numerals 1 to 9 8 Use developing mathematical ideas and methods to solve practical problems 8 Count reliably up to ten everyday objects 19 Find one more or one less than a number from 1 to 10 33 Use language such as ‘circle’ or ‘bigger’ to describe the shape and size of solids and flat shapes 68 Use everyday words to describe position 68 Sort familiar objects to identify their similarities and differences 100National CurriculumKey Stage 1 Objective Page Ma2.1a: communicate in spoken, pictorial and written form, at first using informal language and recording, then mathematical language and symbols 6, 8 Ma2.1e: use the correct language, symbols and vocabulary associated with number and data 21 Ma2.2b: create and describe number patterns, explore and record patterns 6 related to addition and subtraction, and then patterns of multiples 2, 5 and 10 explaining the patterns and using them to make predictions; recognise sequences … recognise the relationship between halving and 52 doubling [Page 116] Ma2.2c: read and write numbers to 20 at first and then to 100 or beyond … recognise that the position of the digit gives its value and know what each digit represents, including zero as a place holder … 21 Ma2.3a: understand subtraction as both ‘take away’ and ‘difference’ and use the related vocabulary; recognise that subtraction is the inverse of addition; give the subtraction corresponding to an addition and vice versa; use the symbol ‘=’ to represent equality; solve simple missing number problems 49 Ma2.3c: develop rapid recall of number facts: know addition and subtraction facts to 10 and use these to derive facts with totals of 20 … 36 Ma2.5a: solve a relevant problem by using simple lists, tables and charts to sort, classify and organise information 102 Ma2.5b: discuss what they have done and explain their results 102 Ma3.2a: describe properties of shapes that they can see or visualise using the related vocabulary 70 Ma3.2b: observe, handle and describe common 2D and 3D shapes; name and describe the mathematical features of common 2D and 3D shapes … 70 Ma3.4a: estimate the size of objects … compare and measure objects using uniform non-standard units … 85 Ma3.4c: estimate, measure and weigh objects; choose and use simple measuring instruments, reading and interpreting numbers, and scales to the nearest labelled division 87Key Stage 2 Objective Page Ma2.1a: make connections in mathematics and appreciate the need to use numerical skills and knowledge when solving problems in other parts of the mathematics curriculum 45 Ma2.1b: break down a more complex problem or calculation into simpler steps before attempting a solution; identify the information needed to carry out the tasks 12, 15 Ma2.1c: select and use appropriate mathematical equipment, includingICT 15 Ma2.1g: use notation diagrams and symbols correctly within a given problem 11 Ma2.1e: make mental estimates of the answers to calculations; check results 40 [Page 117] Ma2.2b: recognise and describe number patterns, including two- and three-digit multiples of 2, 5 or 10, recognising their patterns and using these to make predictions; make general statements … 37, 59 Ma2.2c: … multiply and divide decimals by 10 or 100 45 Ma2.2d: understand unit fractions then fractions that are several parts of one whole, locate them on a number line and use them to find fractions of shapes and quantities 26 Ma2.2g: recognise approximate proportions of a whole and use simple fractions and percentages to describe them, explaining their methods and reasoning 30 Ma2.3g: halve and double any two-digit number 52 Ma2.3h: multiply and divide, at first in the range 1 to 100, then for particular cases of larger numbers by using factors, distribution or other methods 43 Ma2.3i: use written methods to add and subtract positive integers less than 1,000 … 55 Ma2.3k: use a calculator for calculations involving several digits, including decimals; use a calculator to solve number problems; know how to enter and interpret money calculations and fractions; know how to select the correct key sequence for calculations with more than one operation 62 Ma2.4a: choose, use and combine any of the four number operations to solve word problems involving numbers in ‘real life’, money or measures of length, mass, capacity or time, then perimeter and area 57 Ma3.1d: use checking procedures to confirm that their results of geometrical problems are reasonable 79 Ma3.1e: recognise simple spatial patterns and relationships and make predictions about them 75 Ma3.2b: visualise and describe 2D and 3D shapes and the way they behave, making more precise use of geometrical language, especially that of triangles … 79 Ma3.3b: transform objects in practical situations; transform images using ICT; visualise and predict the position of a shape following a rotation, reflection or translation 76 Ma3.4a: recognise the need for standard units of length, mass and capacity, choose which ones are suitable for a task, and use them to make sensible estimates in everyday situations; convert one metric unit to another … 91, 94 [Page 118] Ma3.4b: read scales with increasing accuracy; record measurement using decimal notation 87 Ma3.4c: recognise angles as greater or less than a right angle or half-turn, estimate their size and order them … 72 Ma3.4e: find perimeters of simple shapes; find areas or rectangles using the formula, understanding its connection to counting squares and how it extends its approach; calculate the perimeter and area of shapes composed of rectangles 96 Ma4.2a: solve problems involving data 106, 108 Ma4.2d: know that mode is a measure of average and that range is a measure of spread, and to use both ideas to describe data sets 111Primary Framework for MathematicsUsing and applying mathematics Objective Page Year 1: Describe simple patterns and relations involving numbers or shapes; decide whether examples satisfy given conditions 6 Year 1: Describe ways of solving puzzles and problems, explaining choices and decisions orally or using pictures 6 Year 4: Represent a puzzle or problem using number sentences, statements or diagrams; use these to solve the problem; present and interpret the solution in the context of the problem 11 Year 5: Solve one-step and two-step problems involving whole numbers and decimals and all four operations, choosing and using appropriate calculation strategies, including calculator use 13 Year 6: Tabulate systematically the information in a problem or puzzle; identify and record the steps or calculations needed to solve it, using symbols where appropriate; interpret solutions in the original context and check their accuracy 15Counting and understanding number Objective Page Year 2: Read and write two-digit and three-digit numbers in figures and words … 21 Year 5: Express a smaller whole number as a fraction of a larger one; relate fractions to their decimal representations 26 Year 6: Express one quantity as a percentage of another; find equivalent percentages, decimals and fractions 30[Page 119]Knowing and using number facts Objective Page Year 1: Derive and recall all pairs of numbers with a total of 10 … 36 Year 4: Derive and recall multiplication facts up to 10 ×10, the corresponding division facts and multiples of numbers to 10 up to the tenth multiple 37 Year 4: Use knowledge and understanding of rounding, number operations and inverses to estimate and check calculations 40 Year 5: Identify pairs of factors of two-digit whole numbers and find common multiples 43 Year 6: Use knowledge of place value … to derive related multiplication and division facts involving decimals 45Calculating Objective Page Year 1: Understand subtraction as ‘take away’ and find a ‘difference’ by counting up … 49 Year 1: Solve practical problems that involve combining groups of 2, 5 or 10 52 Year 2: Use practical informal methods and related vocabulary to support multiplication and division 52 Year 3: Find unit fractions of numbers and quantities 52 Year 4: Refine and use efficient written methods to add and subtract two-digit and three-digit whole numbers and £pD.p 55 Year 4: Multiply and divide numbers to 1,000 by 10 and then 100, understanding the effect; relate to scaling up or down 57 Year 4: Find fractions of numbers, quantities or shapes 52 Year 5: Find fractions using division 52 Year 6: Relate fractions to multiplication and division 52, 59 Year 6: Use a calculator to solve problems involving multi-step calculations 62[Page 120]Understanding shape Objective Page Year 2: Visualise common 2D shapes and 3D solids; identify shapes from pictures of them in different positions and orientations; sort, make and describe shapes, referring to their properties 70 Year 4: Know that angles are measured in degrees and that one whole turn is 360; compare and order angles less than 180° 73 Year 5: Complete patterns with up to two lines of symmetry; draw the position of a shape after a reflection or translation 76 Year 6: Visualise and draw on grids of different types where a shape will be after reflection, after translation, or after rotation through 90° or 180° about its centre or one of its vertices 76 Year 6/7: Extend knowledge of properties of triangles and quadrilaterals and use these to visualise and solve problems, explaining reasoning with diagrams 79Measuring Objective Page Year 1: Estimate, measure, weigh and compare objects, choosing and using suitable uniform non-standard or standard units and measuring instruments 85 Year 2: Read the numbered divisions on a scale, and interpret the divisions between them 87 Year 4: Choose and use standard metric units and their abbreviations when estimating, measuring and recording length, weight and capacity; know the meaning of ‘kilo’, ‘centi’ and ‘milli’… 91, 94 Year 5: Interpret a reading that lies between two unnumbered divisions on a scale 87 Year 6: Calculate the perimeter and area of rectilinear shapes; estimate the area of an irregular shape by counting squares 96Handling data
Objective Page Year 1: Answer a question by recording information in lists and tables; present outcomes using practical resources, pictures, block graphs or pictograms 103 [Page 121] Year 1: Use diagrams to sort objects into groups according to a given criterion; suggest a different criterion for grouping the same objects 103 Years 1–4: Answer a question by identifying what data to collect; organise, present, analyse and interpret the data in tables, diagrams, tally charts, pictograms and bar charts, using ICT where appropriate 108 Year 5: Answer a set of questions … 106 Year 6: Solve problems by collecting, selecting, processing and interpreting data, using ICT where appropriate; draw conclusions and identify further questions to ask 106 Year 6: Describe and interpret results and solutions to problems using the mode, range, median and mean 111
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