Primary Mathematics: Extending Knowledge in Practice


Alice Hansen

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    The Author

    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.

  • Objectives Index

    The Early Years Foundation Stage
    Early Learning Goals: Problem Solving, Reasoning and Numeracy
    Recognise numerals 1 to 98
    Use developing mathematical ideas and methods to solve practical problems8
    Count reliably up to ten everyday objects19
    Find one more or one less than a number from 1 to 1033
    Use language such as ‘circle’ or ‘bigger’ to describe the shape and size of solids and flat shapes 68
    Use everyday words to describe position68
    Sort familiar objects to identify their similarities and differences100
    National Curriculum
    Key Stage 1
    Ma2.1a: communicate in spoken, pictorial and written form, at first using informal language and recording, then mathematical language and symbols6, 8
    Ma2.1e: use the correct language, symbols and vocabulary associated with number and data21
    Ma2.2b: create and describe number patterns, explore and record patterns6
    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 and52
    Ma2.2c: read and write numbers to 20 at first and then to 100 or beyondrecognise 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 problems49
    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 information102
    Ma2.5b: discuss what they have done and explain their results102
    Ma3.2a: describe properties of shapes that they can see or visualise using the related vocabulary70
    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 division87
    Key Stage 2
    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 curriculum45
    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 tasks12, 15
    Ma2.1c: select and use appropriate mathematical equipment, includingICT15
    Ma2.1g: use notation diagrams and symbols correctly within a given problem11
    Ma2.1e: make mental estimates of the answers to calculations; check results40
    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 10045
    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 quantities26
    Ma2.2g: recognise approximate proportions of a whole and use simple fractions and percentages to describe them, explaining their methods and reasoning30
    Ma2.3g: halve and double any two-digit number52
    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 methods43
    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 operation62
    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 area57
    Ma3.1d: use checking procedures to confirm that their results of geometrical problems are reasonable79
    Ma3.1e: recognise simple spatial patterns and relationships and make predictions about them75
    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 translation76
    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
    Ma3.4b: read scales with increasing accuracy; record measurement using decimal notation87
    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 rectangles96
    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 sets111
    Primary Framework for Mathematics
    Using and applying mathematics
    Year 1: Describe simple patterns and relations involving numbers or shapes; decide whether examples satisfy given conditions6
    Year 1: Describe ways of solving puzzles and problems, explaining choices and decisions orally or using pictures6
    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 problem11
    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 use13
    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 accuracy15
    Counting and understanding number
    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 representations26
    Year 6: Express one quantity as a percentage of another; find equivalent percentages, decimals and fractions30
    Knowing and using number facts
    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 multiple37
    Year 4: Use knowledge and understanding of rounding, number operations and inverses to estimate and check calculations40
    Year 5: Identify pairs of factors of two-digit whole numbers and find common multiples43
    Year 6: Use knowledge of place value … to derive related multiplication and division facts involving decimals45
    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 1052
    Year 2: Use practical informal methods and related vocabulary to support multiplication and division52
    Year 3: Find unit fractions of numbers and quantities52
    Year 4: Refine and use efficient written methods to add and subtract two-digit and three-digit whole numbers and £pD.p55
    Year 4: Multiply and divide numbers to 1,000 by 10 and then 100, understanding the effect; relate to scaling up or down57
    Year 4: Find fractions of numbers, quantities or shapes52
    Year 5: Find fractions using division52
    Year 6: Relate fractions to multiplication and division52, 59
    Year 6: Use a calculator to solve problems involving multi-step calculations62
    Understanding shape
    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 properties70
    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 translation76
    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 vertices76
    Year 6/7: Extend knowledge of properties of triangles and quadrilaterals and use these to visualise and solve problems, explaining reasoning with diagrams79
    Year 1: Estimate, measure, weigh and compare objects, choosing and using suitable uniform non-standard or standard units and measuring instruments85
    Year 2: Read the numbered divisions on a scale, and interpret the divisions between them87
    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 scale87
    Year 6: Calculate the perimeter and area of rectilinear shapes; estimate the area of an irregular shape by counting squares96
    Handling data
    Year 1: Answer a question by recording information in lists and tables; present outcomes using practical resources, pictures, block graphs or pictograms103
    Year 1: Use diagrams to sort objects into groups according to a given criterion; suggest a different criterion for grouping the same objects103
    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 appropriate108
    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 ask106
    Year 6: Describe and interpret results and solutions to problems using the mode, range, median and mean111


    Ainley, J, Pratt, D and Hansen, A (2006) Connecting engagement and focus in pedagogical task design. British Educational Research Journal, 32(1), 23–38.
    Amin, Z and Eng, KH (2003) Basics in medical education. London: World Scientific Publishing.
    Anghilieri, J and Baron, S (1999) Playing with the materials of study: poleidoblocs. Education, 3–13, 27(2), 57–64.
    Askew, M, Brown, M, Rhodes, V, Wiliam, D and Johnson, D (1997a) Effective teachers of numeracy: report of a study carried out for the Teacher Training Agency. London: King's College, University of London.
    Askew, M, Brown, M, Rhodes, V, Wiliam, D and Johnson, D (1997b) Effective teachers of numeracy in primary schools: teachers’ beliefs, practices and pupils’ learning. Paper presented at the British Educational Research Association Annual Conference, 11–15 September, University of York.
    ATM (2003) Young children learning mathematics. Derby: ATM.
    Aubrey, C (1997) Mathematics teaching in the Early Years: an investigation of teachers’ subject knowledge. London: Falmer Press.
    Ball, DL and Bass, H (2000) Interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. In Boaler, J (ed.), Multiple perspectives on the teaching and learning of mathematics. Westport, CT: Ablex, pp83–104.
    Ball, DL, Hill, HH and Bass, H (2005) Knowing mathematics for teaching: who knows mathematics well enough to teach third grade, and how can we decide?American Mathematical Educator, Fall, 14–46.
    Baroody, AJ and Ginsburg, HP (1983) The effects of instruction on children's understanding of the ‘equals’ sign. The Elementary School Journal, 84(2), 198–212.
    Barrett, JE and Clemson, DH (1998) Analysing children's length strategies with two-dimensional tasks: what counts for length? In Berenson, SB, Dawkins, KR, Blanton, M, Coulombe, WN, Kolb, J, Norwood, K and Stiff, K (eds), Proceedings of the 12th Annual North American Chapter of the International Group for the Psychology of Mathematics Education. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environment Education, Vol. 1, pp321–7.
    BBC News (2000) Mars probe canyon crash theory. (accessed 6 January 2008).
    BBC (2008) Average conditions: Cape Town, South Africa, (accessed 3 January 2008).
    Bills, L, Ainley, J and Wilson, K (2006) Modes of algebraic communication – moving from spreadsheets to standard notation. For the Learning of Mathematics, 26(1), 41–7.
    Carruthers, E and Worthington, M (2005) Making sense of mathematical graphics: the development of understanding abstract symbolism. European Early Childhood Education Research Journal, 13(1), 57–79.
    Cass, M, Cates, D, Jackson, C and SmithM. (2003) Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. Learning Disabilities Research and Practice, 18(2), 112–20.
    Chakrabarti, DK (2004) Indus civilization sites in India: new discoveries. Mumbai: Marg Publications.
    Clarke, B, Clarke, DM and Home, M (2006) A longitudinal study of children's mental computation strategies. In Novotná, J, Moraová, H, Krátká, M and Stehlíková, N (eds), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague: PME, Vol. 2, pp329–36.
    Clements, DH and Battista, MT (1992) Geometry and spatial reasoning. In Grouws, DA (ed.), Handbook of research in mathematics teaching and learning. New York: Macmillan, pp420–64.
    Clements, DH, Battista, MT and Sarama, J (1998) Development of geometric and measurement ideas. In Lehrer, R and Chazan, D (eds), Designing learning environments for developing understanding of geometry and space. Hillsdale, NJ: Lawrence Erlbaum Associates, pp201–47.
    Cole, C and Newson, G (1996) Primary children's views on using calculators in school. Mathematics Education Review, 7, January.
    Davies, N, Connor, D and Spencer, N (2003) An international project for the development of data handling skills of teachers and pupils. Journal of Applied Mathematics and Decision Sciences, 7 (2), 75–83.
    DfEE (1999) The National Curriculum: handbook for primary teachers in England. London: DfEE/QCA.
    DfES (2001) The framework for teaching mathematics and the approach to calculation: Unit 5. London: DfES.
    DfES (2004) Every Child Matters: change for children. London: DfES. Available from: (accessed 12 November 2007).
    DfES (2006) Primary Framework for Teaching Mathematics. London: DfES.
    DfES (2007a) The Early Years Foundation Stage. Effective practice: observation, assessment and planning. London: DfES. Available from: (accessed 14 January 2008).
    DfES (2007b) The Early Years Foundation Stage. Effective practice: outdoor learning. London: DfES Publications.
    DfES (2007c) The Early Years Foundation Stage: setting the standards for learning, development and care for children from birth to five. London: DfES.
    Doerr, H (2006) Examining the tasks of teaching when using students’ mathematical thinking. Educational Studies in Mathematics, 62(1), 3–24.
    Doerr, H and English, LD (2006) Middle grade teachers’ learning through students’ engagement with modeling tasks. Journal of Mathematics Teacher Education, 9(1), 5–32.
    Donaldson, J and Scheffler, A (2003) The snail and the whale. London: Macmillan Children's Books.
    Dougherty, B and Slovin, H (2004) Generalised diagrams as a tool for young children's problem solving. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. Bergen, Norway: PME, Vol. 2, pp295–302.
    Edwards, D (1991) Discourse and the development of understanding in the classroom. In Boyd-Barrett, O and Scanlon, E (eds), Computers and learning. Wokingham: Addison-Wesley.
    English, L and Watters, JJ (2005) Mathematical modelling with 9-year-olds. In Chick, HL and Vincent, JL (eds), Proceedings of the 29th Conference of the International group for the Psychology of Mathematics Education. Melbourne: PME. Vol. 2, pp297–304.
    Eyles, A (1999) Paying children to do their maths!Primary Mathematics, 2 (3).
    Ferrara, F, Pratt, D and Robutti, O (2006) The role and uses of technology of algebra and calculus. In Gutierrez, A and Boero, P (eds), Handbook of research on the psychology of mathematics education. Rotterdam: Sense Publishers, pp305–46.
    Fischbein, E (1993) The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–62.
    Fischbein, E (1994) The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity. In Biehler, R et al. (eds), Didactics of mathematics as a scientific discipline. Dordrecht: Reidel, pp231–45.
    Fisher, C (2004) Multiplying and dividing decimals by powers of ten. (accessed 6 January 2008).
    Foster, R (1997) It's that goat page!Education3–13, 25(1), 44–8.
    French, D (2004) Teaching and learning geometry: issues and methods in mathematical education. London: Continuum.
    Frobisher, L, Monaghan, J, Orton, A and Orton, J (1999) Learning to teach number – aHandbook for students and teachers in the primary school. London: Stanley Thornes.
    Fujita, T and Jones, K (2006) Primary trainee teachers’ understanding of basic geometrical figures in Scotland. In Novotná, H, Moraová, H, Krátká, M and Stehlíková, N (eds), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague: PME, Vol. 3, pp129–36.
    Furner, JM and Marinas, CA (2007) Geometry sketching software for elementary children: easy as 1, 2, 3. Eurasia Journal of Mathematics, Science and Technology Education, 3(1), 83–91.
    Fusion, K (1986) Teaching children to subtract by counting up. Journal for Research in Mathematics Education, 17(3), 172–89.
    Fusion, K (1992) Research on learning and teaching addition and subtraction of whole numbers. In Leinhardt, G, Putman, R and Hattrup, R (eds), Analysis of arithmetic for mathematics teaching. Hillsdale, NJ: Lawrence Erlbaum Associates, pp52–187.
    Fusion, K and Willis, GB (1988) Subtracting by counting up: more evidence. Journal for Research in Mathematics Education, 19(5), 402–20.
    Galili, I (2001) Weight versus gravitational force: historical and educational perspectives. International Journal of Science Education, 23(10), 1073–93.
    Garrick, R (2002) Pattern-making and pattern play in the nursery: special organisation. Paper presented at the Annual Conference of the British Educational Research Association, University of Exeter, England, 12–14 September.
    Gelman, R (1978) Counting in the preschooler: what does and does not develop. In RSSiegler (ed.), Children's thinking: what develops?Hillsdale, NJ: Erlbaum.
    Gelman, R and Gallistel, CR (1978) The child's understanding of number. Cambridge, MA: Harvard University Press.
    Gray, EM and Tall, DO (1994) Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–41.
    Greeno, JG (1980) Some examples of cognitive task analysis with instructional implications. In Snow, RE, Frederico, P and Montague, WE (eds), Aptitude, learning and instruction, Vol. 2: Cognitive process analysis of learning and problem-solving. Hillsdale, NJ: Lawrence Erlbaum Associates, pp1–21.
    Groves, S and Stacey, K (1996) Redefining early number concepts through calculator use. In Mulligan, J and Mitchelmore, M (eds), Children's number learning. Adelaide: Australian Association of Mathematics Teachers, pp205–26.
    Hansen, A (ed.) (2005) Children's errors in mathematics. Exeter: Learning Matters.
    Hansen, A (2007) Using models and images to support children's mathematical thinking. In Drews, D and Hansen, A (eds), Using resources to support mathematical thinking. Exeter: Learning Matters.
    Hasegawa, J (1997) Concept formation of triangles and quadrilaterals in second grade. Educational Studies in Mathematics, 32, 157–79.
    Heirdsfield, A and Cooper, TJ (2004) Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers. Journal of Mathematical Behavior, 23(4), 443–63.
    Hembree, R and Dessart, DJ (1992) Research on calculators in mathematics education. In Fey, JT (ed.), Calculators in mathematics education: 1992 yearbook of the National Council of Teachers of Mathematics. Reston, VA: NCTM, pp22–31.
    Hershkowitz, R (1990) Psychological aspects of learning geometry. In Nesher, P and Kilpatrick, J (eds), Mathematics and cognition. Cambridge: Cambridge University Press, pp70–95.
    Hershkowitz, R, Ben-Chaim, D, Hoyles, C, Lappan, G, Mitchelmore, M and Vinner, S (1990) Psychological aspects of learning geometry. In Nesher, P and Kilpatrick, J (eds), Mathematics and cognition: a research synthesis by the International Group for the Psychology of Mathematics Education. Cambridge: Cambridge University Press, pp70–95.
    Hughes, M (1986) Children and number: difficulties in learning mathematics. Oxford: Blackwell.
    Jones, I and Pratt, D (2005) Three utilities for the equal sign. In Chick, HL and Vincent, JL (eds), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Melbourne: PME, Vol. 3, pp185–92.
    Jones, I and Pratt, D (2006) Connecting the equals sign. International Journal of Computers for Mathematical Learning, 11(3), 301–25.
    Jones, K (2003) Research bibliography: four-function calculators. MicroMath, 19(1), 33–4.
    Kafai, YB, Franke, ML, Ching, CC and Shih, JC (1998) Game design as an interactive learning environment for fostering students’ and teachers’ mathematical enquiry. International Journal of Computers for Mathematical Learning, 3(2), 149–84.
    Koshy, V and Murray, J (2002) Unlocking numeracy. London: David Fulton.
    Lehrer, R, Jenkins, M and Osana, H (1998) Longitudinal study of children's reasoning about space and geometry. In Lehrer, R and Chazan, D (eds), Designing learning environments for developing understanding of geometry and space. Hillsdale, NJ: Lawrence Erlbaum Associates, pp137–67.
    Lesh, R. and English, LD (2005) Trends in the evolution of models and modelling perspectives on mathematical learning and problem solving. In Chick, HL and Vincent, JL (eds), Proceedings 29th Conference of the International Group for the Psychology of Mathematics Education. Melbourne: PME, Vol. 1(1), pp192–6.
    Lesh, R. and Zawojewski, JS (2007). Problem solving and modeling. In FLester (Ed.) Second Handbook of research on mathematics teaching and learning. Greenwich, CT: Information Age Publishing.
    Margolinas, C, Coulange, L and Bessot, A (2005) What can the teacher learn in the classroom?Educational Studies in Mathematics, 59 (1–3), 205–34.
    Mokros, J and Russell, SJ (1995) Children's concepts of average and representativeness. Journal for Research in Mathematics Education, 26(1), pp20–39.
    Montague-Smith, A (1997) Mathematics in nursery education. London: David Fulton.
    Morris, H (2001) Issues raised by testing trainee primary teachers’ mathematical knowledge. Mathematics Education Research Journal, 3, 37–47.
    Mullet, E and Gervais, H (1990) Distinction between the concepts of weight and mass in high school students. International Journal of Science Education, 12(2), 217–26.
    Munn, P (1997) Children's beliefs about counting. In Thompson, I (ed.), Teaching and learning early number. Buckingham: Open University Press.
    Murray, J (2003) Mental mathematics. In Koshy, V, Ernest, P and Casey, R (eds), Mathematics for primary teachers. London: Routledge.
    National Numeracy Strategy (1999) Primary Framework for literacy and mathematics. London: NNS.
    Nickson, M (2004) Teaching and learning mathematics: a guide to recent research and its applications.
    2nd edition
    . London: Continuum.
    Noss, R (1987) Children's learning of geometrical concepts through Logo. Journal for Research in Mathematics Education, 18(5), 343–62.
    Ofsted (2001) The National Numeracy Strategy: the second year. An evaluation by HMI. Available from: (accessed 14 August 2007).
    Onslow, B (2002) Esso family math (Grades 2–5),
    3rd edition
    . Cited in Sauer, R (2002) Estimation. Ontario: GTK Press.
    Ouseley, H and Lane, J (2006) Early Years Foundation Stage: response to the consultation on a single quality framework for services to children from birth to five. Every Child Matters: change for children. London: Blink. Available from (accessed 12 November 2007).
    Platz, D (2004) Challenging young children through simple sorting and classifying: a developmental approach. Education, Fall. Available from: (accessed 13 January 2008).
    PNS (2006) Renewing the Primary Framework for Mathematics Guidance Paper: oral and mental work in mathematics. Available from: (accessed 8 January 2008).
    PNS (2007a) Guidance Paper – Calculation. Available at: (accessed 10 January 2008).
    PNS (2007b) The use of calculators in the teaching and learning of mathematics. Available at: (accessed 10 January 2008.
    Pound, L (1999) Supporting mathematical development in the early years. Buckingham: Open University Press.
    QCA (1999) Mental calculation strategies. London: QCA.
    QCA (2008) Key Stage 2 assessment and reporting arrangements 2008. Available from (accessed 10 January 2008).
    Raiker, A (2007) Assessment for learning. In Jacques, K and Hyland, R (eds), Professional studies: primary and early years.
    3rd edition
    . Exeter: Learning Matters, pp46–59.
    Reys, RE, Suydam, MN and Lindquist, MM (1995) Helping children learn mathematics. Needham Heights: Allyn & Bacon.
    Revs, RE, Lindquist, MM, Lambdin, DV, Smith, NL and Suydam, MN (2006) Helping children learn mathematics.
    8th edition
    . Boston: John Wiley & Sons.
    Robinson, A (2007) The story of measurement. London: Thames & Hudson.
    Rodrigues, S (1994) Data handling in the primary classroom: children's perception of the purpose of graphs. Research in Science Education, 24(1), 280–6.
    Rowland, T, Martyn, S, Barber, P and Heal, C (2001) Investigating the mathematics subject matter knowledge of pre-service elementary school teachers. In van den Heuvel-Panhuizen, M. (ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education. Utrecht: Freudenthal Institute, Utrecht University, Vol. 4, pp121–8.
    Rudduck, J and Flutter, J (2004) How to improve your school: giving pupils a voice. London: Continuum Press.
    Ruthven, K (1998) The use of mental, written and calculator strategies of numerical computation by upper primary pupils within a ‘calculator-aware’ number curriculum. British Educational Research Journal, 24(1), 21–42.
    Saads, S and Davis, G (1997) Spacial abilities, van Hiele levels and language used in three dimensional geometry. Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education. Lahti, Finland: PME, Vol. 4. pp104–11.
    Scheuneman, JD, Camara, WJ, Cascallar, AS, Wendler, C and Lawrence, I (2002) Calculator access, use, and type in relation to performance on the SAT I: Reasoning Test in mathematics. Applied Measurement in Education, 15(1), 95–112.
    Schlessinger Media (2004) Zeros. Multiplying and dividing by 10, 100 and 1000. Available at: (accessed 6 January 2008).
    Schliemann, A, Brizuela, B, Carraher, D, Earnest, D, Goodrow, A, Lara-Roth, S and Peled, I (2003) Algebra in elementary school. In Pateman, NA, Dougherty, BJ and Zilliox, J (eds), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education held jointly with the 25th Conference of PME-NA. Hawai'i: PME, Vol. 4, pp127–34.
    Schmandt-Besserat, D (1999) The history of counting. New York: Harper Collins.
    Shulman, L (1986) Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
    Shulman, L (1987) Knowledge and teaching: Foundations of the new reform. Harvard Education Review, 57, 1–22.
    Sigler, L (2003) Fibonacci's Liber Abaci: Leonardo Pisano's book of calculation. New York: Springer-Verlag.
    Smidt, S (2005) Observing, assessing and planning for children in the Early Years. London: Routledge.
    Streefland, L (1991) Fractions in realistic mathematics education. Dordrecht: Kluwer Academic.
    Streefland, L (1993) Fractions: a realistic approach. In Carpenter, TP, Fennema, E and Romberg, TA (eds), Rational numbers: an integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates.
    TDA (2007a) Professional standards for teachers: qualified teacher status. London: TDA.
    TDA (2007b) Editorial. tdaNews, May.
    TDA (2008) Subject knowledge booster courses. Available from: (accessed 11 January 2008).
    Thompson, I (1999) Teaching and learning early number. Buckingham: Open University Press.
    Thompson, I (2003), Enhancing primary mathematics teaching. London: Open University Press.
    Tobin, K (1987) The role of wait time in higher cognitive level learning. Review of Educational Research (ERIC Clearinghouse on Reading and Communication Skills), 57(1), 69–95.
    Vandersteen, G (2002) Children's own methods of recording number. Mathematics in School, November, pp2–8.
    Wallace, P (2005) Blending instructional design principles with computer game design: the development of Descartes’ Cove. Proceedings of the Association for the Advancement of Computing in Education, Educational Multimedia and Hypermedia, Montreal, Canada. Chesapeake, VA: AACE, pp402–7.
    Watson, J (2007) The role of cognitive conflict in developing students’ understanding of average. Educational Studies in Mathematics, 65(1), 21–47.
    Watson, J and Moritz, JB (2000) The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2 (1 & 2), 11–50.
    Wilson, K, Ainley, J and Bills, L (2005) Spreadsheets, pedagogic strategies and the evolution of meaning for variable. In Chick, HL and Vincent, JL (eds), Proceedings of the 29th Annual Conference of the International Group for the Psychology of Mathematics Education. Melbourne: PME, Vol. 4, pp321–8.
    Wood, E (2007) Reconceptualising child-centred education: contemporary directions in policy, theory and practice in early childhood. FORUM: For Promoting3–19Comprehensive Education, 49(1), 119–34.
    Worthington, M and Carruthers, E (2003) Research uncovers children's creative mathematical thinking. Primary Mathematics (Mathematics Association), 7(3), 21–5.
    Worthington, M and Carruthers, E (2005) The art of children's mathematics: the power of visual representation. Paper presented at Roehampton University's ‘Art in Early Childhood: Creativity, Collaboration, Communication’ Conference, 7–9 July.
    Wu, H (2002) On the teaching of fractions. Berkley, CA: University of California. Available at: (accessed 1 November 2006).
    Yackel, E and Cobb, P (1996) Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–77.

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