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Senior phase comments 1


Caroline Long's comment on Maths Senior phase CAPS

Caroline Long's comment on Maths Senior phase CAPS

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  • 1. Curriculum and Assessment Policy Statement (CAPS) Senior Phase document University of the Witwatersrand, Author Feedback session, 7 th October, 2010 Caroline Long, Centre for Evaluation and Assessment, (CEA)
  • 2. Curriculum
    • Re-packaging the NCS curriculum?
      • Will this strategy solve the problem of learning and teaching mathematics?
    • Curriculum – a document of central importance
      • It influences the mathematical experiences of children have through the guidance and support for teachers.
    • It should include the best that mathematics education research has to offer.
      • For example: Rational Number Project has done extensive analysis applicable at all levels.
    • Responsibility for the curriculum
      • Mathematicians together with mathematics education specialist and specialist teachers for sound knowledge base – the substantive aspect of the curriculum.
      • Teachers are responsible for the technical-professional implementation of the curriculum.
      • Cycles of review and piloting are essential
  • 3. Levels of curriculum (Thijs & Van den Akker, 2009)
  • 4. Key transitions from Grade 6 to 10 (Usiskin, 2005)
    • These are amongst others, from;
      • whole number to real number;
      • number to variable;
      • from patterns to functions;
      • inductive arguments to deductive arguments;
      • from informal description to formal definition of mathematics ideas;
      • from a view of mathematics as a set of memorized facts to seeing mathematics as interrelated ideas accessible through a variety of means. p. 4.
    • Because learners have not made these transitions they are
      • "forced to memorize their way through algebra and geometry and functions";
      • "expected to think formally but they do not know what this [thinking] means";
      • "assumed to understand the properties of real numbers, but they are still thinking in terms of whole numbers" (p. 4).
  • 5. What does research tell us?
    • Kieren tradition – analysis of mathematical concepts, investigates acquisition by learners and conducts teaching design experiments with implications for instruction
      • Kieren (1976). On the mathematical, cognitive and instructional foundations of rational numbers.
    • Children learn from their total experience and they bring their observations and learning to the classroom.
    • Learning in the early grades affects the understanding of later concepts
      • for example the early teaching of fractions as only part of a whole ONLY, interferes with later understanding of a concept such as percentage increase.
    • Learners can be taught a procedure, but they do not necessarily remember it in the way it was taught and neither can they apply the procedure correctly when confronted with a parallel problem (Hart, 1981; 1984).
  • 6. Rational number project (1979 – 2010)
    • Fraction and rational number
      • sometimes used interchangeably but ..
      • . . . NOT the same thing
    • Rational number
      • formal mathematical concept, with definitions, operations and theorems
      • understanding of rational number is a long term process
    • Fraction
      • a concept, for example half,
      • a symbol ¾ which have at least 5 different meanings.
  • 7. Five meanings (at least) of a fraction symbol (Lamon, 2007)
  • 8. Proportional reasoning
    • Capstone of primary school and cornerstone of high school (Lesh et al, 1988)
    • Children have intuitive understanding of proportional reasoning – this has to be developed starting from FP
    • Levels of cognitive development and levels of complexity are to be found in research
      • Qualitative reasoning precedes quantitative reasoning
    • Lack of fluency with proportional reasoning seen as one on the reasons for failure at tertiary level
  • 9. Percentage
    • Problems with percentage related to ONLY teaching part-whole understanding of fraction
    • Covers the different notions underpinning rational number, and has additional complexity
    • See Parker & Leinhardt, (1995). Percent: a privileged proportion
  • 10. Senior Phase CAPS document - progression
  • 11. Senior phase document – time allocation
  • 12. Further comments
    • Rational number, ratio and rate
      • Grades 7, 8 – ratio and rate ( one week, no progression indicated )
      • Grades 9 – ratio and rate ( one week ), add direct and indirect proportion (one week)
    • Rational number concept built up through experiences with common fractions, decimal fractions, ratios, rate, throughout the year.
    • Develop an understanding the rational number can have different representations.
    • No mention of proportional reasoning (see Lampen document)
  • 13. Conclusion
    • Development of a curriculum takes time
    • Key research must be considered as in the alternative curriculum (previous slide)
    • Mathematicians and mathematics education specialists are responsible for substantive aspects .
    • Teachers for implementation of technical- professional aspects.
    • Next steps towards planning the mathematical future of our children require radical redirection.