Senior phase comments 1

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Caroline Long's comment on Maths Senior phase CAPS

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Senior phase comments 1

  1. 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. 2. Curriculum <ul><li>Re-packaging the NCS curriculum? </li></ul><ul><ul><li>Will this strategy solve the problem of learning and teaching mathematics? </li></ul></ul><ul><li>Curriculum – a document of central importance </li></ul><ul><ul><li>It influences the mathematical experiences of children have through the guidance and support for teachers. </li></ul></ul><ul><li>It should include the best that mathematics education research has to offer. </li></ul><ul><ul><li>For example: Rational Number Project has done extensive analysis applicable at all levels. </li></ul></ul><ul><li>Responsibility for the curriculum </li></ul><ul><ul><li>Mathematicians together with mathematics education specialist and specialist teachers for sound knowledge base – the substantive aspect of the curriculum. </li></ul></ul><ul><ul><li>Teachers are responsible for the technical-professional implementation of the curriculum. </li></ul></ul><ul><ul><li>Cycles of review and piloting are essential </li></ul></ul>
  3. 3. Levels of curriculum (Thijs & Van den Akker, 2009)
  4. 4. Key transitions from Grade 6 to 10 (Usiskin, 2005) <ul><li>These are amongst others, from; </li></ul><ul><ul><li>whole number to real number; </li></ul></ul><ul><ul><li>number to variable; </li></ul></ul><ul><ul><li>from patterns to functions; </li></ul></ul><ul><ul><li>inductive arguments to deductive arguments; </li></ul></ul><ul><ul><li>from informal description to formal definition of mathematics ideas; </li></ul></ul><ul><ul><li>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. </li></ul></ul><ul><li>Because learners have not made these transitions they are </li></ul><ul><ul><li>&quot;forced to memorize their way through algebra and geometry and functions&quot;; </li></ul></ul><ul><ul><li>&quot;expected to think formally but they do not know what this [thinking] means&quot;; </li></ul></ul><ul><ul><li>&quot;assumed to understand the properties of real numbers, but they are still thinking in terms of whole numbers&quot; (p. 4). </li></ul></ul>
  5. 5. What does research tell us? <ul><li>Kieren tradition – analysis of mathematical concepts, investigates acquisition by learners and conducts teaching design experiments with implications for instruction </li></ul><ul><ul><li>Kieren (1976). On the mathematical, cognitive and instructional foundations of rational numbers. </li></ul></ul><ul><li>Children learn from their total experience and they bring their observations and learning to the classroom. </li></ul><ul><li>Learning in the early grades affects the understanding of later concepts </li></ul><ul><ul><li>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. </li></ul></ul><ul><li>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). </li></ul>
  6. 6. Rational number project (1979 – 2010) <ul><li>Fraction and rational number </li></ul><ul><ul><li>sometimes used interchangeably but .. </li></ul></ul><ul><ul><li>. . . NOT the same thing </li></ul></ul><ul><li>Rational number </li></ul><ul><ul><li>formal mathematical concept, with definitions, operations and theorems </li></ul></ul><ul><ul><li>understanding of rational number is a long term process </li></ul></ul><ul><li>Fraction </li></ul><ul><ul><li>a concept, for example half, </li></ul></ul><ul><ul><li>a symbol ¾ which have at least 5 different meanings. </li></ul></ul>
  7. 7. Five meanings (at least) of a fraction symbol (Lamon, 2007)
  8. 8. Proportional reasoning <ul><li>Capstone of primary school and cornerstone of high school (Lesh et al, 1988) </li></ul><ul><li>Children have intuitive understanding of proportional reasoning – this has to be developed starting from FP </li></ul><ul><li>Levels of cognitive development and levels of complexity are to be found in research </li></ul><ul><ul><li>Qualitative reasoning precedes quantitative reasoning </li></ul></ul><ul><li>Lack of fluency with proportional reasoning seen as one on the reasons for failure at tertiary level </li></ul>
  9. 9. Percentage <ul><li>Problems with percentage related to ONLY teaching part-whole understanding of fraction </li></ul><ul><li>Covers the different notions underpinning rational number, and has additional complexity </li></ul><ul><li>See Parker & Leinhardt, (1995). Percent: a privileged proportion </li></ul>
  10. 10. Senior Phase CAPS document - progression
  11. 11. Senior phase document – time allocation
  12. 12. Further comments <ul><li>Rational number, ratio and rate </li></ul><ul><ul><li>Grades 7, 8 – ratio and rate ( one week, no progression indicated ) </li></ul></ul><ul><ul><li>Grades 9 – ratio and rate ( one week ), add direct and indirect proportion (one week) </li></ul></ul><ul><li>Rational number concept built up through experiences with common fractions, decimal fractions, ratios, rate, throughout the year. </li></ul><ul><li>Develop an understanding the rational number can have different representations. </li></ul><ul><li>No mention of proportional reasoning (see Lampen document) </li></ul>
  13. 13. Conclusion <ul><li>Development of a curriculum takes time </li></ul><ul><li>Key research must be considered as in the alternative curriculum (previous slide) </li></ul><ul><li>Mathematicians and mathematics education specialists are responsible for substantive aspects . </li></ul><ul><li>Teachers for implementation of technical- professional aspects. </li></ul><ul><li>Next steps towards planning the mathematical future of our children require radical redirection. </li></ul>

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