The document discusses key issues in mathematics curriculum development based on research findings. It highlights important transitions students must make between grades 6-10, such as moving from whole numbers to real numbers. Research shows students often memorize procedures without understanding, and recommends teaching proportional reasoning from an early age. The South African CAPS curriculum does not adequately address rational number concepts and proportional reasoning. Future curriculum development requires input from mathematicians and educators based on research findings.

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.