Curriculum and Assessment Policy

Statement (CAPS) Intermediate Phase

document

Caroline Long, University of Pretoria

Curriculum

Re-packaging the NCS curriculum?• Will this 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

• Selected groups at each phase comprising both teachers and subject specialists.

• Cycles of review and piloting are essential

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 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 problem (Hart, 1981; 1984).

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 can mean many different

things.

Five interpretations of a fraction symbol

Adapted from Lamon, (2007, p. 654) Alternatives to part-whole fraction ( Interpretations of 3/4 Meaning Classroom activities Part-whole comparisons “3 parts out of 4 equal parts”

¾ means three parts out of four equal parts of the unit, with equivalent fractions found by thinking of the parts in terms of larger or smaller chunks

2

5.1

16

12

4

3

(whole pies) (quarter pies) (pairs of pies)

Producing equivalent fractions and comparing fractions

Measure “3 (4

1-units)”

4

3 means a distance of 3 (

4

1-units) from 0

on the number line or 3 (4

1-units) of a

given area

Reading metres and gauges

Operator “4

3of something”

4

3is a rule that tells us how to operate on a

unit. Multiply by 3 and divide by 4, or divide by 4 and multiply the result by 3.

Area models for multiplication and division.

Quotient “3 divided by 4”

4

3is the amount each person receives when

4 people share a 3-unit of something

Partitioning

Ratio “3 of A are compared to 4 of B” 3:4 is a relationship in which 3 A’s are compared, in a multiplicative rather than an additive sense, to 4 B’s.

Proportional reasoning

Capstone of primary school and cornerstone of high school

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

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

Intermediate Phase CAPS document

1.2 Fractions• Different conceptions of fraction not made explicit• Operator concept in G5 and G6 (G4 notes)• Problem solving is stated – requires guidance as to problem

types 1.3 Ratio and rate

• No progressive development across grades (also in NCS) • Ratio and rate examples• Proportional reasoning (simple proportion to multiple

proportion) 1.5 Properties of rational number

• Lists the properties (Properites of zero and 1, in Grade 6)• Rational number concept is developed through experiences

with many different contexts• Density of rational number field and equivalence – critical

concepts (RED indicates lack in CAPS)

Further comments

Fractions, rational number, ratio and rate• Grades 4, 5 – only in third term (one week

for ratio and rate)• Grades 6 – fractions in second, ratio and

rate in third term (one week) Proportional reasoning needs to be

developed through work with fractions, ratios, rate, throughout the year.

Concept of percent is built up through use from early grades.

Alternative curriculum – Grade 4

An alternative curriculum designed and ready for piloting … and engagement with teachers, is structured as follows.

Term 1• 3 notions of rational number made explicit (see page 6)• Rate introduced

Term 2• 4 sections developing fraction concept• Ratio and rate

Term 3• Fraction development continued• Proportional reasoning introduced

Term 4• Development of fraction concept• Proportional sharing

Conclusion

Development of a curriculum takes time

Key research must be considered as in the alternative curriculum (previous slide)

Gautrain problems detected? Go back to the drawing board. There is too much at stake.

Next steps towards planning the mathematical future of our children require radical redirection.