grid algebra - mixed attainment maths...küchemann (1981, p104) identified six categories of letter...
TRANSCRIPT
Mixed Attainment Mathematics Conference Saturday 17th June 2017
Tom Francome – [email protected] - @TFrancome
Grid Algebra:
Developing fluency with
formal notation
• Teach on PGDipEd at University of Birmingham
• Previously head of maths at KNGS
• Mixed attainment groups
• Best Practice in Grouping
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
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.
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.
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)
Approaches to algebra
• Functional
• Relating pictures to rules
• Structural
• Manipulating symbols (expanding, factorising, collecting terms, solving equations …)
– (Kieran 2006; Kirshner 2001)
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)
Theoretical frameworks
• Arbitrary and Necessary
– (Hewitt, 1999; 2001a; b)
• Subordination
– (Hewitt, 1994; 1996; 2015)
• Embodied Cognition
– (Varela et al. 1993)
Why we undertook the research project
Why we undertook the research project
Why we undertook the research project
Why we undertook the research project
Why we undertook the research project
Why we undertook the research project
https://www.ncetm.org.uk/blogs/8505
Reflections
• An opportunity to reflect with colleagues on your teaching of algebra
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