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The choirs and orchestras of secondary mathematics

This resource explores the concepts of trigonometry, straight line graphs, and calculus in secondary mathematics. It provides explanations, conventions, and limitations of these topics, as well as suggestions for whole-school approaches to teaching them. Prior knowledge, helpful and unhelpful factors are also discussed.

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The choirs and orchestras of secondary mathematics

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  1. The choirs and orchestras of secondary mathematics Anne Watson East London Maths Forum June 2008

  2. Trigonometry A 68 mm 2.6 cm θ C B

  3. Trigonometry Triangle properties (not names): right angle; orientation; angle-sum Labelling (conventions): vertex letters are not algebra; lengths are not accurate; θ is ‘theta’ and often used for angles; what is the curved line? Concepts: ratio; inverse function; transforming equations etc. …

  4. What prior knowledge is helpful? What is unhelpful? • Images • Language • Methods • Contributory concepts • Assumptions • Limitations ………………. etc.

  5. Whole school approaches to creating the orchestra

  6. Straight line graphs

  7. Internet images of straight line graphs • Nearly all positive • Negatives go through zero or are about fitting lines to data points • Key Maths says ‘top left to bottom right’

  8. Straight line graphs • Graphs: represent a quantitative relationship; relation between line and coordinate pairs (conventions and meaning) • Reading a graph: axes; continuity; (conventions and meaning) • Dimensions of variation & range of possible changes • etc.

  9. What prior knowledge is helpful? What is unhelpful? • Images • Language • Methods • Assumptions • Limitations ………………. etc.

  10. Whole school approaches to creating the orchestra

  11. Calculus y = (x2 + 1) e -x

  12. Calculus

  13. Calculus • Use of letters: variables; unknowns; parameters; constants • Functions: expressions; relations between functions • Graphs: … • Similarity: … • etc. …

  14. What prior knowledge is helpful? What is unhelpful? • Images • Language • Methods • Assumptions • Limitations ………………. etc.

  15. Whole school approaches to creating the orchestra

  16. Vertical integration of key ideas in the curriculum • Continuity • How to read mathematics • Different meanings of letters • Ratio • Similarity • Functions and graphs • etc.

  17. Algorithm … Algorithm – you can use it Algorithm – why teach anything more? Algorithm – don’t abuse it Algorithm – why teach anything more? I will test you To see if you know it Then I’ll give you Your next target

  18. Thankyou for thinking anne.watson@education.ox.ac.uk www.education.ox.ac.uk/people/academics Anne Watson: Raising Achievement in Secondary Mathematics, Open University Press 8th Annual Institute of Mathematics PedagogyJuly 28th to 31stCuddesdon near Oxfords.elliott@shu.ac.ukJohn Mason, Malcolm Swan, Anne Watson

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