Grid Algebra - Mixed Attainment Maths...Küchemann (1981, p104) identified six categories of letter...
Transcript of Grid Algebra - Mixed Attainment Maths...Küchemann (1981, p104) identified six categories of letter...
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Mixed Attainment Mathematics Conference Saturday 17th June 2017
Tom Francome – [email protected] - @TFrancome
Grid Algebra:
Developing fluency with
formal notation
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• Teach on PGDipEd at University of Birmingham
• Previously head of maths at KNGS
• Mixed attainment groups
• Best Practice in Grouping
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https://www.atm.org.uk/Interactive-Journal---MT218i/mt218i-grid-algebra-in-practice/70199
Mixed Attainment Mathematics Conference Sheffield Hallam University – Saturday 17th June 2017
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Some known algebra issues…
• Learners’ incomplete understanding of “=” – Operational understanding / Relational understanding /
Substitutive understanding (Jones & Pratt, 2012)
• Process/Object – 2𝑥 + 5 – Process - product dilemma (Kieran, 1989) – Process - object dichotomy (Sfard & Linchevski, 1994) – Proceptual thinking (Gray & Tall, 1994)
• Introduction of letters – Küchemann, D. (1981) Algebra. In K. M. Hart (Ed),
Children's understanding of mathematics: 11-16, London: John Murray. p.102-19.
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Introduction of 𝑥
Küchemann (1981, p104) identified six categories of letter usage (in hierarchical order):
1. Letter evaluated: the letter is assigned a numerical value from the outset;
2. Letter not used: letter is ignored, or at best acknowledged existence but without given meaning;
3. Letter as object: shorthand for an object or treated as an object in its own right;
4. Letter as specific unknown: regarded as a specific but unknown number, and can be operated on directly;
5. Letter as generalised number: seen as being able to take several values rather than just one;
6. Letter as variable: representing a range of unspecified values, and a systematic relationship is seen to exist between two sets of values.
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30 years on…
• many secondary students leave school with only a limited understanding of algebra
• 1976 2008 results have got worse
• 12% do A level, at 14 only 3% think about variables
– (Hodgen et al. 2009; National Mathematics Advisory Panel 2008)
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Approaches to algebra
• Functional
• Relating pictures to rules
• Structural
• Manipulating symbols (expanding, factorising, collecting terms, solving equations …)
– (Kieran 2006; Kirshner 2001)
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Notation but also Conceptual Understanding
• 189 Y5 students had significantly better conceptual understanding of algebra after 3 hours of Grid Algebra lessons in a RCT comparing to a functional approach.
– (Jones, I., Bisson, M., Gilmore, C. Inglis, M., 2017)
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Theoretical frameworks
• Arbitrary and Necessary
– (Hewitt, 1999; 2001a; b)
• Subordination
– (Hewitt, 1994; 1996; 2015)
• Embodied Cognition
– (Varela et al. 1993)
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Why we undertook the research project
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Why we undertook the research project
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Why we undertook the research project
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Why we undertook the research project
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Why we undertook the research project
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Why we undertook the research project
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https://www.ncetm.org.uk/blogs/8505
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Reflections
• An opportunity to reflect with colleagues on your teaching of algebra
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References
• Hart, K. M. (1980). Secondary School Children's Understanding of Mathematics. A Report of the Mathematics Component of the Concepts in Secondary Mathematics and Science Programme.
• Gray, E. M. and Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education 25(2), pp. 115-141.
• Hewitt, D (2016) Designing Educational Software: The Case of Grid Algebra, Digital Experiences in Mathematics Education, 2(2), ISSN: 2199-3246. DOI: 10.1007/s40751-016-0018-4.
• Hewitt, D (2015) The economic use of time and effort in the teaching and learning of mathematics. In Oesterle, S and Allan, D (ed) 2014 Annual Meeting of the Canadian Mathematics Education Study Group, University of Alberta, Edmonton, Canada, pp.3-23, ISBN: 978-0-86491-381-4.
• Hewitt, D (2014) A Symbolic Dance: The Interplay Between Movement, Notation, and Mathematics on a Journey Toward Solving Equations, Mathematical Thinking and Learning, 16(1), pp.1-31, ISSN: 1098-6065. DOI: 10.1080/10986065.2014.857803.
• Hewitt, DPL (2013) Introduction of letters and solving linear equations using Grid Algebra, Mathematics Teaching, pp.6-10. • Hewitt, DPL (2013) Learning algebraic notation and order of operations using Grid Algebra software, Mathematics Teaching, 232, pp.21-24. • Hewitt, D (2012) Young students learning formal algebraic notation and solving linear equations: Are commonly experienced difficulties
avoidable?, Educational Studies in Mathematics, 81(2), pp.139-159, ISSN: 0013-1954. DOI: 10.1007/s10649-012-9394-x. • Hewitt, DPL and Hayton, P (2007) Grid Algebra, Association of Teachers of Mathematics. • Hewitt, DPL (2010) The role of subordination and fading in learning formal algebraic notation and solving equations: the case of Year 5 students. In Pinto,
MMF and Kawasaki, TF (ed) 34th Conference of the International Group for the Psychology of Mathematics Education, Belo Horizonte, Brazil, pp.81-88.Hewitt, DPL (2001) Arbitrary and Necessary: Part 3 Educating Awareness, For the Learning of Mathematics: an international journal of mathematics education, 21(2), pp.37-49, ISSN: 0228-0671.
• Hewitt, DPL (2001) Arbitrary and Necessary: Part 2 Assisting Memory, For the Learning of Mathematics: an international journal of mathematics education, 21(1), pp.44-51, ISSN: 0228-0671.
• Hewitt, DPL (1999) Arbitrary and Necessary: Part 1 a Way of Viewing the Mathematics Curriculum, For the Learning of Mathematics: an international journal of mathematics education, 19(3), pp.2-9, ISSN: 0228-0671.
• Hewitt, D. (1996). Mathematical fluency: the nature of practice and the role of subordination. For the Learning of Mathematics 16(2), pp. 28-35. • Hewitt, D. (1994). The principle of economy in the teaching and learning of mathematics. Unpublished PhD dissertation. The Open University, Milton
Keynes. • Hodgen, J., Kuchemann, D., Brown, M. and Coe, R. (2010). Children's understandings of algebra 30 years on: what has changed? In V. Durand-Guerrier, S.
Soury-Lavergne and F. Arzarello (Eds), Proceedings of the CERME 6, (pp. 539-548). Lyon, France: Institut National De Recherche Pédagogique. • Jones, I., Bisson, M., Gilmore, C. Inglis, M., (2017), Measuring conceptual understanding in Randomised Controlled Trials: can comparative judgement help?
(In press) • Jones, I. and Pratt, D. (2012). A substituting meaning for the equals sign in arithmetic notating tasks. Journal for Research in Mathematics Education 43(1),
pp. 2-33. • Kieran, C. (1989). The early learning of algebra: a structural perspective. In S. Wagner and C. Kieran (Eds), Research Issues in the Learning and Teaching of
Algebra, Reston, Virginia, USA: National Council of Teachers of Mathematics, Lawrence Erlbaum Associates, pp. p33-56. • Küchemann, D. (1981) Algebra. In K. M. Hart (Ed), Children's understanding of mathematics: 11-16, London: John Murray. p.102-19. • Sáenz-Ludlow, A. and Walgamuth, C. (1998). Third graders' interpretations or equality and the equal symbol. Educational Studies in Mathematics 35, pp.
153-187. • Sfard, A. and Linchevski, L. (1994). The gains and pitfalls of reification - the case of algebra. Educational Studies in Mathematics 26, pp. 191-228.
Tom Francome – [email protected] - @TFrancome