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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK AQP for Foundational Learning Competence FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK CONTENTS A. INTRODUCTION TO THE FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY FRAMEWORK Sets out the definition, overall purpose, elements, levels of complexity and scope of contexts of Foundational Mathematical Literacy B. THE FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY FRAMEWORK Sets out the organising principles, curriculum elements and outcomes, including descriptions of scope and contexts C. FACILITATOR GUIDELINES, ASSESSMENT GUIDELINES AND ACCREDITATION SPECIFICATIONS Gives general guidelines on methodology and assessment

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Page 1: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

AQP for Foundational Learning Competence

FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

CONTENTS

A. INTRODUCTION TO THE

FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY FRAMEWORK

Sets out the definition, overall purpose, elements, levels of complexity and scope of contexts of Foundational Mathematical Literacy

B. THE FOUNDATIONAL

LEARNING COMPETENCE MATHEMATICAL LITERACY FRAMEWORK

Sets out the organising principles, curriculum elements and outcomes, including descriptions of scope and contexts

C. FACILITATOR GUIDELINES,

ASSESSMENT GUIDELINES AND ACCREDITATION SPECIFICATIONS

Gives general guidelines on methodology and assessment

Page 2: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

Page 2 of 30

IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

SECTION A INTRODUCTION TO THE FOUNDATIONAL LEARNING

COMPETENCE MATHEMATICAL LITERACY FRAMEWORK 1. DEFINITION OF FOUNDATIONAL MATHEMATICAL LITERACY

Foundational Mathematical Literacy (FML) is the minimum, generic mathematical literacy that will provide learners with the necessary foundation to: x access learning at National Qualifications Framework (NQF) Levels 2, 3 and 4 for

occupations and trades; and x engage meaningfully in real-life situations.

Important note: Where particular occupations or trades require mathematics beyond FML, or applications in contexts that are specific to occupations or trades, these requirements are addressed within the core of qualifications related to those occupations and trades.

2. OVERALL PURPOSE OF FML

People who have met all the requirements of FML are able to: Make sense of and solve problems in real contexts by responding to information about mathematical ideas that is presented in a variety of ways.

This overall purpose of FML is clarified below:

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Page 3 of 30

IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

3. ELEMENTS OF FML AND THEIR PURPOSES

Element Purpose

1. Number Use numbers in a variety of forms to describe and make sense of situations, and to solve problems in a range of familiar and unfamiliar contexts.

2. Finance Manage personal finances using financial documents and related formulae

3. Data and chance Collect, display and interpret data in various ways and solve related problems.

4. Measurement Make measurements using appropriate measuring tools and techniques and solve problems in various measurement contexts.

5. Space and shape Describe and represent objects and the environment in terms of spatial properties and relationships.

6. Patterns and relationships

Describe, show, interpret and solve problems involving mathematical patterns, relationships and functions.

Solving problems means that the learner can:

x Define the goals to reach.

x Analyse and make sense of the problem situation.

x Plan how to solve the problem.

x Execute the plan.

x Interpret and evaluate the results, justify the method and solution.

Real contexts means:

x In everyday life.

x At work.

x In further learning.

Make sense of and solve problems in real contexts by responding to information about mathematical ideas that is presented in a variety of ways.

Responding means:

x Identifying or locating.

x Acting upon.

x Ordering, sorting and comparing.

x Counting.

x Estimating.

x Computing.

x Measuring.

x Modelling.

x Interpreting.

x Communicating about.

Variety of ways means in:

x Objects and pictures.

x Numbers and symbols.

x Formulae.

x Diagrams and maps.

x Graphs.

x Tables and spreadsheets.

x Texts.

Mathematical ideas means:

x Number and quantity.

x Space and shape.

x Patterns and relationships.

x Data and chance.

x Measurements.

Page 4: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

Page 4 of 30

IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

4. LEVELS OF FML

In keeping with the generally accepted and broad definition of mathematical literacy as the application of basic mathematics in different real-life contexts, the following statements about levels can be made about FML: x The level of mathematics involved is roughly equivalent to NQF Level 1. x The level of complexity of the contexts ranges from NQF Level 1 to NQF 4. Learners

need to straddle these levels as demanded by particular situations defined in the FML Framework.

Although the framework gives examples of the sort of contexts in which learners are required to apply mathematics, it makes no distinctions between the levels of those contexts. Therefore, the level of FML is not pegged at any particular level but is regarded as foundational to learning for occupations and trades at NQF Levels 1 – 4. From an administrative point of view, the FML Curriculum is part of the Foundational Learning Competence Part Qualification at NQF Level 2.

5. SCOPE OF CONTEXTS

The FML Framework identifies a range of contexts as a guide to the sort of contexts that could be used in learning and assessment activities. This does not mean that the list is complete, and it is possible that the range of contexts will be expanded in later drafts. The framework does not try to cover the full range of contexts that people will meet in life, work and learning. The framework is based on the assumption that particular occupational contexts will be addressed within those particular occupations. However, FML programmes should expose learners to a sufficiently wide range of contexts in order to ensure a solid foundation. The programmes should also help learners to develop their skills of transferring what they know from familiar to unfamiliar contexts. Therefore, when learners find themselves in unfamiliar contexts, whether in the world of work or within occupational learning areas, they should be able to respond to further learning within those new contexts with some coaching.

6. LEARNING ASSUMED TO BE IN PLACE

It is assumed that learners starting FML learning programmes are already competent in communications and in mathematical literacy at Adult Basic Education and Training (ABET) Level 3.

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B: A

SSES

SMEN

T Q

UA

LITY

PA

RTN

ER T

O T

HE

QC

TO

FOU

ND

ATI

ON

AL

LEA

RN

ING

CO

MPE

TEN

CE

MA

THEM

ATI

CA

L LI

TER

AC

Y: C

UR

RIC

ULU

M F

RA

MEW

OR

K

SEC

TIO

N B

T

HE

FM

L F

RA

ME

WO

RK

CL

AR

IFIC

AT

ION

OF

EL

EM

EN

TS:

PU

RPO

SE, S

KIL

LS,

RE

QU

IRE

D S

TA

ND

AR

DS

OF

PER

FOR

MA

NC

E,

EX

AM

PLE

S O

F C

ON

TE

XT

S, G

UID

ELIN

ES,

SC

OPE

AN

D C

ON

TE

XT

S

Ele

men

t 1: N

umbe

r

Purp

ose:

Use

num

bers

in a

var

iety

of f

orm

s to

desc

ribe

and

mak

e se

nse o

f situ

atio

ns, a

nd to

solv

e pro

blem

s in

a ran

ge o

f fam

iliar

and

unf

amili

ar c

onte

xts.

Skill

s:

1.1

U

se n

umbe

rs to

mak

es se

nse o

f and

des

crib

e sit

uatio

ns

1.2

Read

, int

erpr

et a

nd u

se d

iffer

ent n

umbe

ring

conv

entio

ns in

diff

eren

t con

text

s and

iden

tify

the

way

s in

whi

ch d

iffer

ent c

onve

ntio

ns w

ork

1.3

Inte

rpre

t and

use

num

bers

writ

ten

in e

xpon

entia

l for

m in

clud

ing

squa

res a

nd c

ubes

of n

atur

al n

umbe

rs a

nd th

e sq

uare

and

cub

e ro

ots o

f per

fect

squa

res a

nd c

ubes

1.

4

Do

calc

ulat

ions

in v

ario

us si

tuat

ions

usin

g a

varie

ty o

f tec

hniq

ues

1.5

So

lve p

robl

ems i

nvol

ving

ratio

and

pro

porti

on

1.6

So

lve p

robl

ems i

nvol

ving

frac

tions

, dec

imal

s and

per

cent

ages

R

equi

red

Stan

dard

s of P

erfo

rman

ce:

x Pr

oble

m-s

olvi

ng st

rate

gies

are b

ased

on

a co

rrect

inte

rpre

tatio

n of

the

cont

ext.

x Ca

lcul

atio

ns ar

e per

form

ed a

ccur

atel

y an

d ac

cord

ing

to th

e con

vent

ions

gov

erni

ng th

e ord

er o

f ope

ratio

ns.

x M

etho

ds a

re p

rese

nted

in a

cle

ar, l

ogic

al a

nd st

ruct

ured

man

ner,

usin

g m

athe

mat

ical

sym

bols

and

nota

tion

cons

isten

t with

mat

hem

atic

al co

nven

tions

. x

Met

hods

use

d ar

e ef

ficie

nt, l

ogic

al, i

nter

nally

con

siste

nt a

nd ju

stifie

d by

the

cont

ext.

x So

lutio

ns a

re e

valu

ated

and

val

idat

ed in

term

s of t

he c

onte

xt, a

nd n

umbe

rs ro

unde

d ap

prop

riate

ly to

the p

robl

em si

tuat

ion.

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SMEN

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UA

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PA

RTN

ER T

O T

HE

QC

TO

FOU

ND

ATI

ON

AL

LEA

RN

ING

CO

MPE

TEN

CE

MA

THEM

ATI

CA

L LI

TER

AC

Y: C

UR

RIC

ULU

M F

RA

MEW

OR

K

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

1.1

Use

num

bers

to m

akes

sens

e of

and

desc

ribe

situa

tions

x

Und

ersta

nd n

umbe

r con

cept

and

its a

ssoc

iate

d la

ngua

ge.

x D

o ba

sic m

enta

l cal

cula

tions

. x

Use

num

bers

to co

unt,

orde

r and

estim

ate.

x

Use

nam

es a

nd sy

mbo

ls fo

r num

bers

. x

Und

ersta

nd su

bset

s of w

hole

num

bers

(odd

, eve

n, sq

uare

, squ

are r

oots,

fact

ors a

nd m

ultip

les)

. x

Use

pos

itive

and

neg

ativ

e nu

mbe

rs a

s dire

ctio

nal i

ndic

ator

s: o

–1

0 °C

to in

dica

te 1

0 °C

bel

ow fr

eezi

ng.

o

–R30

0 to

indi

cate

ove

rdra

ft.

o

–40

m to

indi

cate

40

m b

elow

sea l

evel

. x

Add

and

subt

ract

pos

itive

and

neg

ativ

e nu

mbe

rs w

ithin

real

con

text

s e.g

., te

mpe

ratu

re, d

epos

its a

nd o

verd

rafts

in a

ban

k.

x U

se fr

actio

ns to

esti

mat

e siz

e or p

ortio

n (s

uch

as 'w

e ha

ve u

sed ¼

of o

ur al

loca

ted

time')

.

1.2

Read

, int

erpr

et a

nd u

se d

iffer

ent

num

berin

g co

nven

tions

in

diffe

rent

cont

exts

and

iden

tify

the

way

s in

whi

ch d

iffer

ent

conv

entio

ns w

ork

x U

se th

e dec

imal

com

ma a

nd p

oint

and

the

thou

sand

s sep

arat

or c

orre

ctly

: o

2,

567,

890.

00.

o

2 56

7 89

0,00

. o

13

.1 in

cric

ket r

efer

s to

13 o

vers

and

1 b

all o

f the

nex

t ove

r. Th

is is

not t

he sa

me

as 1

3,1

in th

e de

cim

al sy

stem

. x

Use

num

eral

s and

wor

ds in

terc

hang

eabl

y to

indi

cate

num

ber:

1 50

2 00

0 =

one

mill

ion

five

hund

red

and

two

thou

sand

. x

Read

and

inte

rpre

t num

bers

com

mun

icat

ed in

scie

ntifi

c no

tatio

n fo

r ver

y sm

all a

nd v

ery

big

num

bers

e.g

., 4,

56 ×

107 =

45

600

000.

x

Com

pare

and

use

diff

eren

t tim

e no

tatio

ns:

o

a.m

./p.m

. o

24

-hou

r clo

ck.

o

Ana

logu

e and

dig

ital.

1.3

Inte

rpre

t and

use

num

bers

w

ritte

n in

exp

onen

tial f

orm

in

clud

ing

squa

res a

nd c

ubes

of

natu

ral n

umbe

rs a

nd th

e sq

uare

an

d cu

be ro

ots o

f per

fect

squa

res

and

cube

s

x 32 =

9.

x 23 =

8.

x 9

= 3

.

x 3

8 =

2.

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SMEN

T Q

UA

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PA

RTN

ER T

O T

HE

QC

TO

FOU

ND

ATI

ON

AL

LEA

RN

ING

CO

MPE

TEN

CE

MA

THEM

ATI

CA

L LI

TER

AC

Y: C

UR

RIC

ULU

M F

RA

MEW

OR

K

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

1.4

Do

calc

ulat

ions

in v

ario

us

situa

tions

usin

g a

varie

ty o

f te

chni

ques

x Es

timat

e so

lutio

ns to

cal

cula

tions

and

car

ry o

ut m

enta

l cal

cula

tions

by

brea

king

up

and

reco

mbi

ning

num

bers

: R23

4,60

+ R

419,

70 =

± R

650.

x

App

ly d

istrib

utiv

e [4(

6 +

3) =

4.6

+ 4

.3] a

nd as

soci

ativ

e pr

oper

ties [

(2.3

). 4

= 2.

(3.4

)] to

sim

plify

cal

cula

tions

whe

re p

ossib

le a

nd/o

r use

ful.

(It is

not

nec

essa

ry to

kno

w th

e na

mes

of t

he p

rope

rties

.) x

Wor

k fle

xibl

y w

ith n

umbe

rs to

bre

ak u

p an

d re

com

bine

them

to si

mpl

ify ca

lcul

atio

ns.

x D

o ca

lcul

atio

ns co

rrect

ly u

sing

tech

niqu

es ap

prop

riate

to th

e pro

blem

. The

se co

uld

incl

ude p

aper

, men

tal a

nd/o

r cal

cula

tor m

etho

ds,

spre

adsh

eets

and

stand

ard

or n

on-s

tand

ard

algo

rithm

s. Sp

read

shee

ts co

uld

be a

n op

tion

but n

ot an

aut

omat

ic to

ol.

x A

pply

the c

orre

ct o

rder

of o

pera

tions

in ca

lcul

atio

ns a

s per

conv

entio

ns i.

e., B

OD

MA

S an

d Le

ft-to

-Rig

ht ru

le fo

r ope

ratio

ns o

f the

sam

e ord

er

e.g.,

12 ÷

4 ×

2 =

3 ×

2 =

6; 4

– 3

+ 5

= 1

+ 5

= 6

. x

Use

the

follo

win

g fu

nctio

ns o

n a b

asic

calc

ulat

or (a

four

-func

tion

calc

ulat

or):

o

addi

tion,

subt

ract

ion,

mul

tiplic

atio

n an

d di

visio

n;

o

perc

enta

ge;

o

squa

re ro

ot;

o

mem

ory;

and

o

'cl

ear'

and

'clea

r all'

key

s. x

Estim

ate

reas

onab

le v

alue

s for

unk

now

ns in

pro

blem

s bas

ed o

n ex

perie

nce

of th

e con

text

in o

rder

to so

lve t

he p

robl

ems a

nd/o

r pre

dict

a

solu

tion.

x

Roun

d nu

mbe

rs u

p, d

own

or o

ff (i.

e., t

runc

ated

) app

ropr

iate

ly a

ccor

ding

to th

e con

text

or i

nstru

ctio

n. L

earn

ers m

ust s

how

that

they

are

aw

are

of th

e im

plic

atio

ns o

f rou

ndin

g to

o ea

rly o

r too

late

in a

cal

cula

tion.

x

Use

the l

aws o

f exp

onen

ts co

rrect

ly. L

earn

ers a

re n

ot e

xpec

ted

to m

anip

ulat

e exp

ress

ions

invo

lvin

g th

ese l

aws,

but s

houl

d be

abl

e to

appl

y th

ese

law

s whe

n th

ey a

ppea

r in

num

eric

al e

xam

ples

: o

an ×

am =

an + m

. o

an ÷

am =

an –

m.

o

a0 = 1

. o

a –

n = 1

/ an .

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SMEN

T Q

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PA

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O T

HE

QC

TO

FOU

ND

ATI

ON

AL

LEA

RN

ING

CO

MPE

TEN

CE

MA

THEM

ATI

CA

L LI

TER

AC

Y: C

UR

RIC

ULU

M F

RA

MEW

OR

K

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

1.5

Solv

e pro

blem

s inv

olvi

ng ra

tio

and

prop

ortio

n x

Shar

e ac

cord

ing

to a

give

n ra

tio.

o

e.g.,

Mar

y w

orke

d fo

r 2 h

ours

, Sus

an w

orke

d fo

r 3 h

ours

. The

y ea

rn R

400

betw

een

them

. Sha

re th

e m

oney

acc

ordi

ng to

the

hour

s the

y w

orke

d.

x M

ix a

ccor

ding

to a

giv

en ra

tio.

o

e.g.,

A c

hem

ical

is m

ade

up o

f A a

nd B

in th

e rat

io o

f 2:5

. How

muc

h of

A a

nd B

are

need

ed to

mak

e 3

litre

s of t

he c

hem

ical

? x

Incr

ease

or d

ecre

ase

amou

nts a

ccor

ding

to a

give

n ra

tio.

o

e.g.,

Incr

ease

R45

0 by

one

fifth

. x

Det

erm

ine a

ratio

bet

wee

n qu

antit

ies.

o

e.g.,

ther

e ar

e 60

men

and

75

wom

en in

a ro

om. W

hat i

s the

ratio

of m

en to

wom

en?

x Co

nver

t bet

wee

n un

its w

ithou

t los

ing

the

ratio

: R to

$, m

m to

km

. x

Dire

ct p

ropo

rtion

. o

e.g

., ca

r ren

tal i

s cha

rged

per

day

. The

mor

e day

s you

hire

the

car,

the

mor

e it

costs

, in

dire

ct p

ropo

rtion

to th

e dai

ly ra

te.

x In

vers

e pro

porti

on.

o

e.g.,

the

mor

e peo

ple

we

have

on

a jo

b, th

e le

ss ti

me

it ta

kes.

1.6

So

lve p

robl

ems i

nvol

ving

fra

ctio

ns, d

ecim

als a

nd

perc

enta

ges

x Ex

pres

s one

qua

ntity

as a

frac

tion,

per

cent

age a

nd/o

r dec

imal

of a

noth

er q

uant

ity.

x Ex

pres

s par

ts of

a w

hole

as f

ract

ions

, per

cent

ages

and

dec

imal

s of a

who

le.

x Fi

nd a

giv

en si

mpl

e fra

ctio

n, p

erce

ntag

e or d

ecim

al o

f a q

uant

ity o

r sha

pe.

x Ch

ange

a qu

antit

y or

shap

e by

incr

easin

g or

dec

reas

ing

it by

frac

tions

, per

cent

ages

or d

ecim

als e

.g.,

an it

em co

sts R

400

excl

udin

g V

AT,

how

m

uch

does

it co

st in

clud

ing

VA

T?

x Fi

nd o

ut th

e orig

inal

am

ount

or s

ize o

f a q

uant

ity o

r sha

pe w

hen

give

n th

e in

crea

sed

or d

ecre

ased

am

ount

or s

hape

and

the

fract

ion,

pe

rcen

tage

or d

ecim

al in

crea

se o

r dec

reas

e e.g

., an

item

costs

R32

5 in

clud

ing

14%

VA

T. H

ow m

uch

was

the

item

bef

ore

VA

T?

x D

escr

ibe

situa

tions

or c

hang

es b

y ex

pres

sing

and

calc

ulat

ing

one q

uant

ity a

s a fr

actio

n of

ano

ther

, and

reco

gnise

situ

atio

ns fo

r usin

g th

ese

desc

riptio

ns.

x D

escr

ibe

situa

tions

invo

lvin

g pa

rts o

f a w

hole

and

reco

gnise

situ

atio

ns fo

r usin

g th

ese d

escr

iptio

ns.

x U

se fr

actio

ns, d

ecim

als a

nd p

erce

ntag

es a

s mea

sure

s of p

arts

of a

who

le:

o

¾ o

f the

mix

ture

is w

ater

. o

75

% o

f the

mix

ture

is w

ater

. o

0,

75 o

f the

mix

ture

is w

ater

. x

Expr

ess f

ract

ions

in eq

uiva

lent

form

s and

conv

ert b

etw

een

fract

ions

, dec

imal

s and

per

cent

ages

with

and

with

out a

calc

ulat

or:

o

43=

86 =

100

75 =

75%

= 0

,75.

Page 9: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

Pa

ge 9

of 3

0 IE

B: A

SSES

SMEN

T Q

UA

LITY

PA

RTN

ER T

O T

HE

QC

TO

FOU

ND

ATI

ON

AL

LEA

RN

ING

CO

MPE

TEN

CE

MA

THEM

ATI

CA

L LI

TER

AC

Y: C

UR

RIC

ULU

M F

RA

MEW

OR

K

Ele

men

t 2: F

inan

ce

Purp

ose:

Man

age p

erso

nal f

inan

ces u

sing

finan

cial

doc

umen

ts an

d re

late

d fo

rmul

ae.

Skill

s:

2.1

Re

ad a

nd in

terp

ret f

inan

cial

info

rmat

ion

pres

ente

d in

a ra

nge o

f doc

umen

ts in

per

sona

l and

fam

iliar

con

text

s 2.

2

Iden

tify,

cla

ssify

and

reco

rd so

urce

s of i

ncom

e an

d ex

pend

iture

2.

3

Plan

and

mon

itor p

erso

nal f

inan

ces

2.4

Ev

alua

te o

ptio

ns w

hen

purc

hasin

g pr

oduc

ts an

d se

rvic

es

2.5

D

eter

min

e the

impa

ct o

f int

eres

t, de

prec

iatio

n, in

flatio

n, d

efla

tion

and

taxa

tion

on p

erso

nal f

inan

ces

R

equi

red

Stan

dard

s of P

erfo

rman

ce:

x In

terp

reta

tions

of p

erso

nal f

inan

cial

doc

umen

ts ar

e con

siste

nt w

ith re

cord

ed fa

cts.

x Fi

nanc

ial i

nfor

mat

ion

is re

cord

ed an

d or

gani

sed

clea

rly, a

ccur

atel

y an

d ac

cord

ing

to g

ener

al fi

nanc

e-re

cord

ing

tech

niqu

es a

nd p

rinci

ples

. x

Pers

onal

bud

gets

refle

ct th

e fin

anci

al si

tuat

ion

in su

ffici

ent d

etai

l for

pla

nnin

g an

d m

onito

ring

pers

onal

fina

nces

. x

Costs

of p

rodu

cts a

nd se

rvic

es a

re e

valu

ated

usin

g a

varie

ty o

f iss

ues.

Thes

e inc

lude

affo

rdab

ility

, per

sona

l nee

ds a

nd ac

cura

cy o

f adv

ertis

ing

clai

ms.

x Pe

rson

al fi

nanc

es a

re m

onito

red

in te

rms o

f var

ious

influ

ence

s, in

clud

ing

inco

me,

exp

endi

ture

, inv

estm

ents,

loan

s, ta

xatio

n an

d in

flatio

n.

Page 10: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

Pa

ge 1

0 of

30

IEB

: ASS

ESSM

ENT

QU

ALI

TY P

AR

TNER

TO

TH

E Q

CTO

FO

UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

2.1

Read

and

inte

rpre

t fin

anci

al

info

rmat

ion

pres

ente

d in

a ra

nge

of d

ocum

ents

in p

erso

nal a

nd

fam

iliar

cont

exts

x D

ocum

ents

incl

ude:

o

Pa

y sli

ps (i

nclu

ding

ded

uctio

ns a

nd in

crea

ses)

. o

Ch

eque

s.

o

Rece

ipts.

o

Ba

nk st

atem

ents.

o

A

ccou

nts o

r inv

oice

s.

x Fi

nanc

ial i

nfor

mat

ion

incl

udes

: o

Ba

lanc

es.

o

Cred

it (in

com

e) a

nd d

ebit

(exp

ense

) ite

ms.

o

Bene

ficia

ries o

r rec

ipie

nts.

o

Paym

ents

or c

redi

ts ow

ing.

o

D

ate o

r tim

e pe

riods

of d

ocum

ents.

2.2

Iden

tify,

cla

ssify

and

reco

rd

sour

ces o

f inc

ome a

nd

expe

nditu

re

x So

urce

s of i

ncom

e cou

ld in

clud

e:

o

Sala

ries,

wag

es a

nd c

omm

issio

ns.

o

Gift

s and

poc

ket m

oney

. o

Bu

rsar

ies a

nd lo

ans.

o

Savi

ngs.

o

Inte

rest.

o

In

herit

ance

s.

x Ex

pend

iture

coul

d in

clud

e:

o

Livi

ng e

xpen

ses (

e.g.

, foo

d, re

ntal

and

clo

thin

g).

o

Acc

ount

s (e.

g., m

unic

ipal

serv

ices

). o

Co

mm

unic

atio

n (e

.g.,

tele

phon

e an

d em

ail).

o

Fe

es (e

.g.,

colle

ge a

nd b

ank

fees

). o

In

sura

nce (

e.g.

, med

ical

aid

, per

sona

l lia

bilit

y an

d ve

hicl

e ins

uran

ce).

o

Pers

onal

taxe

s. o

Lo

an re

paym

ents

(e.g

., ba

nk lo

ans,

burs

arie

s and

hire

pur

chas

e, in

clud

ing

inte

rest)

. o

En

terta

inm

ent.

2.3

Plan

and

mon

itor p

erso

nal

finan

ces

x In

com

e and

exp

endi

ture

stat

emen

ts.

x Pe

rson

al b

udge

ts (g

ener

al a

nd fo

r per

sona

l pro

ject

s, tri

ps, e

vent

s, sig

nific

ant p

urch

ases

and

oth

ers)

. x

Costi

ng o

f ite

ms (

such

as r

oofin

g til

es fo

r a p

artic

ular

area

, app

lianc

es o

r pai

ntin

g co

sts).

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Pa

ge 1

1 of

30

IEB

: ASS

ESSM

ENT

QU

ALI

TY P

AR

TNER

TO

TH

E Q

CTO

FO

UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

2.4

Eval

uate

opt

ions

whe

n pu

rcha

sing

prod

ucts

and

serv

ices

x Co

mpa

re cr

edit

card

s with

diff

eren

t int

eres

t rat

es fo

r diff

eren

t per

iods

of t

ime.

x

Estim

ate

the

long

-term

cos

ts of

mak

ing

low

er m

onth

ly cr

edit

card

pay

men

ts.

x Ev

alua

te a

dver

tisin

g cl

aim

s for

pro

duct

s and

serv

ices

and

mak

e de

cisio

ns o

n be

st pu

rcha

se o

ptio

ns e.

g., c

ompa

re c

ell p

hone

pac

kage

s. x

Eval

uate

diff

eren

t sav

ing

and

purc

hasin

g op

tions

. The

se w

ill in

clud

e sa

ving

acc

ount

s, lo

ans,

hire

pur

chas

e, b

uyin

g on

cred

it an

d m

ortg

age

bond

s.

2.5

D

eter

min

e the

impa

ct o

f int

eres

t, de

prec

iatio

n, in

flatio

n, d

efla

tion

and

taxa

tion

on p

erso

nal

finan

ces

x Co

mpa

re si

mpl

e an

d co

mpo

und

inte

rest.

x

Expl

ore

the

effe

ct o

f dep

reci

atio

n.

x In

vesti

gate

the

varia

bles

invo

lved

in lo

ans a

nd in

vestm

ents

e.g.

, int

eres

t rat

es, d

urat

ion.

x

Iden

tify

the

impa

ct o

f inf

latio

n an

d de

flatio

n on

per

sona

l fin

ance

s by

com

parin

g th

e buy

ing

pow

er o

f inc

ome

from

yea

r to

year

. x

Iden

tify

tax

liabi

lity

asso

ciat

ed w

ith d

iffer

ent l

evel

s of i

ncom

e by

usin

g ta

x ta

bles

. Fo

r ass

essm

ent p

urpo

ses,

lear

ners

shou

ld b

e gi

ven

the r

elev

ant f

orm

ulae

. Dur

ing

lear

ning

act

iviti

es, t

he u

se o

f for

mul

ae sh

ould

not

be a

t the

ex

pens

e of d

evel

opin

g th

eir u

nder

stand

ing

thro

ugh

inve

stiga

ting

the

impa

cts b

y us

ing

pen

and

pape

r and

/or s

prea

dshe

ets a

nd o

ther

tool

s.

Ele

men

t 3: D

ata

and

Cha

nce

Purp

ose:

Col

lect

, dis

play

and

inte

rpre

t dat

a in

var

ious

way

s and

solv

e re

late

d pr

oble

ms.

Skill

s:

3.1

Col

lect

dat

a 3.

2 C

lass

ify, o

rgan

ise

and

sum

mar

ise

data

3.

3 D

ispl

ay d

ata

3.4

Ana

lyse

and

inte

rpre

t dat

a to

dra

w c

oncl

usio

ns a

nd m

ake

pred

ictio

ns

3.5

Det

erm

ine

and

inte

rpre

t cha

nce

Req

uire

d St

anda

rds o

f Per

form

ance

: x

Dat

a col

lect

ion

tech

niqu

es ar

e app

ropr

iate

to th

e co

ntex

t, ty

pe o

f dat

a an

d pu

rpos

e. x

Dat

a is c

lass

ified

, org

anise

d an

d su

mm

arise

d ap

prop

riate

ly so

that

it p

rom

otes

mea

ning

ful a

naly

sis.

x D

ata i

s disp

laye

d us

ing

tech

niqu

es a

ppro

pria

te to

the

type

of i

nfor

mat

ion

and

cont

ext.

x D

ata d

ispla

ys ar

e con

siste

nt w

ith co

llect

ed d

ata,

pro

mot

e eas

e of i

nter

pret

atio

n an

d m

inim

ise b

ias.

x In

terp

reta

tions

and

pre

dict

ions

are

ver

ified

by

the d

ata a

nd o

bser

ved

trend

s. Th

ey ta

ke in

to ac

coun

t pos

sible

sour

ces o

f erro

r and

dat

a m

anip

ulat

ion.

x

Sim

ple p

roba

bilit

ies a

re d

eter

min

ed a

ccur

atel

y an

d sta

tem

ents

of c

hanc

e are

corre

ctly

inte

rpre

ted

in co

ntex

t.

Page 12: FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb: assessment quality partner to the qcto foundational learning competence mathematical

Pa

ge 1

2 of

30

IEB

: ASS

ESSM

ENT

QU

ALI

TY P

AR

TNER

TO

TH

E Q

CTO

FO

UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

Exa

mpl

es o

f con

text

s:

x H

uman

righ

ts, so

cial

, eco

nom

ic, c

ultu

ral,

envi

ronm

enta

l and

pol

itica

l mat

ters

are e

xam

ples

. The

y ca

n be

use

d to

: o

ca

rry o

ut q

ualit

y co

ntro

l by

usin

g a

repr

esen

tativ

e sa

mpl

e on

whi

ch to

bas

e dec

ision

s. o

w

rite a

repo

rt ab

out a

qua

ntita

tive

issue

and

incl

ude

tabl

es a

nd c

harts

. Mak

e a p

rese

ntat

ion

on th

e m

ater

ial t

o pe

ers o

r sup

erio

rs.

o

read

and

liste

n to

repo

rts cr

itica

lly a

nd a

sk in

telli

gent

que

stion

s.

o

prod

uce a

sche

dule

or t

ree d

iagr

am fo

r a p

roje

ct.

o

criti

cally

inte

rpre

t sta

tistic

al c

harts

and

gra

phs p

rese

nted

in th

e m

edia

. o

us

e sp

read

shee

ts to

inve

stiga

te d

iffer

ent s

ales

opt

ions

and

pre

pare

gra

phs t

hat i

llustr

ate t

hese

opt

ions

. o

lo

ok fo

r pat

tern

s in

data

to id

entif

y tre

nds i

n co

sts, s

ales

and

dem

and.

o

re

ad st

atist

ical

tabl

es in

repo

rts o

f loc

al sp

ortin

g ev

ents

and

unde

rsta

nd h

ow th

ey w

ere

gene

rate

d.

o

appl

y kn

owle

dge o

f pro

babi

lity

to ri

sks i

n he

alth

and

fina

ncia

l (su

ch a

s gam

blin

g) is

sues

affe

ctin

g th

e lo

cal c

omm

unity

. o

pl

ay g

ames

of c

hanc

e, u

nder

stand

ing

scor

ing

and

statis

tics.

o

ju

dge t

he li

kelih

ood

of a

n ev

ent o

r sto

ry, l

ike

that

of a

n U

nide

ntifi

ed F

lyin

g O

bjec

t (U

FO),

bein

g tru

e.

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

3.1

Colle

ct d

ata

x Id

entif

y so

urce

s of d

ata

(e.g

., pe

ers,

fam

ily, n

ewsp

aper

s, bo

oks,

mag

azin

es o

r int

erne

t).

x U

se ap

prop

riate

dat

a co

llect

ion

tech

niqu

es (o

bser

ving

, int

ervi

ewin

g, a

dmin

ister

ing

writ

ten

ques

tionn

aire

s, co

nduc

t foc

us g

roup

disc

ussio

ns,

cond

uct s

urve

ys, c

arry

out

exp

erim

ents)

and

instr

umen

ts or

instr

umen

ts (e

.g.,

chec

klis

t; da

ta fo

rms;

mea

surin

g in

stru

men

ts; i

nter

view

gu

ide;

che

cklis

t; qu

estio

nnai

re; t

ape

reco

rder

; que

stio

nnai

re).

x

Use

acce

pted

sam

plin

g te

chni

ques

. x

Und

ersta

nd so

urce

s of b

ias i

n da

ta c

olle

ctio

n.

x U

nder

stand

eth

ical

con

sider

atio

ns.

3.2

Clas

sify,

org

anise

and

su

mm

arise

dat

a

x Cl

assif

y co

llect

ed in

form

atio

n as

: o

Ca

tego

rical

(e.g

., m

ale/

fem

ale;

dog

/cat

; typ

e of c

ar).

o

Q

uant

itativ

e (nu

mer

ical

): –

Disc

rete

(e.g

., nu

mbe

r of s

iblin

gs o

r num

ber o

f pai

rs o

f sho

es)

– Co

ntin

uous

(e.g

., he

ight

, wei

ght a

nd ra

infa

ll).

o

Qua

litat

ive (

not n

umer

ical

). x

Org

anise

dat

a us

ing

tally

tabl

es, f

requ

ency

tabl

es, t

wo-

way

tabl

es o

r ste

m-a

nd-le

af d

iagr

ams.

x A

naly

se u

ngro

uped

dat

a us

ing

mod

e, m

edia

n an

d m

ean.

Lea

rner

s sho

uld

be a

ble

to in

terp

ret t

hese

mea

sure

s (e.

g., t

he m

edia

n is

the

mid

-poi

nt

of a

dist

ribut

ion.

Hal

f the

dat

a is

belo

w th

e m

edia

n an

d ha

lf is

abov

e it)

. x

Org

anise

dat

a int

o ap

prop

riate

cla

ss in

terv

als w

here

nec

essa

ry.

x Ca

lcul

ate r

elat

ive

frequ

ency

and

per

cent

age

frequ

ency

.

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FO

UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

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CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

3.3

Disp

lay

data

x

Pie c

harts

. x

Bar g

raph

s (sin

gle b

ar a

nd m

ultip

le b

ar g

raph

s).

x Li

ne a

nd b

roke

n-lin

e gr

aphs

.

3.4

Ana

lyse

and

inte

rpre

t dat

a to

draw

conc

lusio

ns a

nd m

ake

pred

ictio

ns

x Re

ad a

nd c

ritic

ally

inte

rpre

t dat

a in

tally

tabl

es, f

requ

ency

tabl

es, t

wo-

way

tabl

es, s

tem

-and

-leaf

dia

gram

s, ba

r gra

phs,

doub

le b

ar g

raph

s, pi

e ch

arts

and

line

grap

hs.

x Ch

oose

the

mos

t sui

tabl

e m

easu

re o

f cen

tral t

ende

ncy

(mea

n, m

edia

n or

mod

e).

x M

ake

use

of m

easu

res o

f spr

ead

(rang

e, qu

artil

es, i

nter

-qua

rtile

rang

e).

x Sh

ow a

n aw

aren

ess o

f the

sour

ces o

f erro

r and

dat

a m

anip

ulat

ion

(gro

upin

g, sc

ale

and

choi

ce o

f sum

mar

y sta

tistic

s).

x Ex

plai

n th

e m

isuse

of s

cale

s in

diag

ram

s as a

sour

ce o

f erro

r and

bia

s. x

Expl

ain

the

misu

se o

f gro

upin

g in

tabl

es a

nd d

iagr

ams a

s a so

urce

of e

rror a

nd d

ata

man

ipul

atio

n.

x Sh

ow a

n aw

aren

ess o

f the

diff

eren

t fea

ture

s and

appl

icat

ions

of g

raph

s and

tabl

es:

o

Pie c

harts

reve

al th

e pr

opor

tions

bet

wee

n th

e diff

eren

t cha

ract

erist

ics o

f the

info

rmat

ion

but d

o no

t rev

eal t

he p

opul

atio

n or

sam

ple

size.

o

Ba

r gra

phs r

evea

l the

pop

ulat

ion

or sa

mpl

e siz

e but

do

not a

lway

s sho

w th

e rel

atio

nshi

p be

twee

n th

e cat

egor

ies c

lear

ly.

o

The c

hoic

e of s

cale

on

the

axes

, and

/or t

he p

oint

at w

hich

the a

xes c

ross

, im

pact

on

the

impr

essio

n cr

eate

d by

the

grap

h.

o

Tabl

es w

ill o

ften

cont

ain

mor

e in

form

atio

n th

an g

raph

s, bu

t tre

nds o

r pat

tern

s are

less

eas

y to

obs

erve

. x

Ans

wer

the q

uesti

ons f

or w

hich

the

info

rmat

ion

was

col

lect

ed.

3.5

Inte

rpre

t and

det

erm

ine

chan

ce (l

imite

d to

sim

ple

prob

abili

ties i

n sy

mm

etric

al

situa

tions

)

x D

eter

min

e the

pos

sible

out

com

es (e

.g.,

usin

g tre

e dia

gram

s) a

nd ca

lcul

ate

the p

roba

bilit

y of

eac

h po

ssib

le o

utco

me.

x

Det

erm

ine a

ctua

l out

com

es a

nd th

eir r

elat

ive

frequ

enci

es.

x U

nder

stand

exp

ress

ions

of p

roba

bilit

y.

x Pe

rform

sim

ple e

xper

imen

ts an

d co

unt t

he fr

eque

ncie

s of t

he a

ctua

l out

com

es.

x U

se th

e fre

quen

cies

of t

he a

ctua

l out

com

es to

calc

ulat

e the

rela

tive

frequ

ency

of e

ach.

x

Calc

ulat

e pro

babi

lity

base

d on

hist

oric

al e

vent

s.

x Ca

lcul

ate t

he p

roba

bilit

y of

non

-occ

urre

nce,

inde

pend

ent a

nd d

epen

dent

eve

nts,

mut

ually

exc

lusiv

e eve

nts a

nd m

ultip

le in

depe

nden

t eve

nts.

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EWO

RK

Ele

men

t 4: M

easu

rem

ent

Purp

ose:

Mak

e m

easu

rem

ents

usi

ng a

ppro

pria

te m

easu

ring

tool

s and

tech

niqu

es a

nd so

lve

prob

lem

s in

vario

us m

easu

rem

ent c

onte

xts.

Skill

s:

4.1

Es

timat

e an

d m

easu

re q

uant

ities

usin

g m

easu

ring

instr

umen

ts in

var

ious

cont

exts,

pay

ing

atte

ntio

n to

sign

ifica

nt fi

gure

s and

mar

gins

of e

rror

4.2

Ca

lcul

ate q

uant

ities

in m

easu

rem

ent c

onte

xts p

ayin

g at

tent

ion

to si

gnifi

cant

figu

res a

nd m

argi

ns o

f erro

r 4.

3

Solv

e m

easu

rem

ent p

robl

ems i

n va

rious

pra

ctic

al a

nd n

on-p

ract

ical

con

text

s R

equi

red

Stan

dard

s of P

erfo

rman

ce:

x M

easu

ring

instr

umen

ts m

eet t

he ac

cura

cy re

quire

men

ts of

the

cont

ext.

x Re

adin

gs ar

e acc

urat

ely

reco

rded

with

in a

ppro

pria

te m

argi

ns o

f erro

r and

usin

g ap

prop

riate

uni

ts.

x Ca

lcul

atio

ns ar

e per

form

ed a

ccur

atel

y, k

eepi

ng u

nits

cons

isten

t. x

Conv

ersio

ns b

etw

een

units

are

accu

rate

and

appr

opria

te to

the

cont

ext.

x So

lutio

ns to

pro

blem

s are

val

idat

ed a

ccor

ding

to th

e con

text

, inc

ludi

ng c

onte

xtua

lly a

ppro

pria

te ro

undi

ng a

nd u

se o

f uni

ts.

Exa

mpl

es o

f con

text

s:

x W

eigh

and

mea

sure

item

s and

kee

p re

cord

s. x

Mea

sure

and

reco

rd m

edic

ine d

oses

(use

ratio

and

pro

porti

ons)

. x

Read

and

inte

rpre

t rec

ipes

invo

lvin

g m

easu

rem

ents.

x

An

item

, qua

ntity

or l

iqui

d is

prov

ided

and

mea

sure

d (a

box

, for

exa

mpl

e, m

ust b

e w

eigh

ed).

x Si

tuat

ions

whe

re a

certa

in q

uant

ity m

ust b

e m

easu

red

(300

g of

a q

uant

ity, f

or e

xam

ple,

mus

t be

wei

ghed

out

). x

Mea

sure

the

time

it ta

kes a

per

son

to ru

n or

wal

k ar

ound

a fi

eld;

mea

sure

the d

istan

ce; c

alcu

late

the

spee

d.

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UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

4.1

Es

timat

e an

d m

easu

re q

uant

ities

us

ing

mea

surin

g in

strum

ents

in

vario

us co

ntex

ts, p

ayin

g at

tent

ion

to si

gnifi

cant

figu

res

and

mar

gins

of e

rror

x Q

uant

ities

incl

ude:

o

Le

ngth

, wid

th, h

eigh

t, de

pth,

dist

ance

. o

Ti

me

inte

rval

s. o

Te

mpe

ratu

re.

o

Wei

ght (

mas

s in

scie

ntifi

c la

ngua

ge).

o

Ang

les.

x M

easu

ring

instr

umen

ts in

clud

e ru

lers

, tap

e m

easu

res,

scal

es, c

lock

s, th

erm

omet

ers,

capa

city

-mea

surin

g in

strum

ents,

pro

tract

ors,

wat

ches

and

sto

pwat

ches

. x

Iden

tify

and

take

acc

ount

of p

ossib

le so

urce

s of m

easu

rem

ent e

rrors

. 4.

2

Calc

ulat

e qua

ntiti

es in

m

easu

rem

ent c

onte

xts p

ayin

g at

tent

ion

to si

gnifi

cant

figu

res

and

mar

gins

of e

rror

x Pe

rimet

ers o

f pol

ygon

s and

circ

les.

x A

reas

of t

riang

les,

rect

angl

es a

nd ci

rcle

s. x

Are

as o

f pol

ygon

s by

divi

ding

them

into

tria

ngle

s and

rect

angl

es.

x V

olum

es o

f tria

ngul

ar a

nd re

ctan

gula

r pris

ms a

nd c

ylin

ders

. x

Surfa

ce ar

ea o

f tria

ngul

ar a

nd re

ctan

gula

r pris

ms a

nd c

ylin

ders

. x

Capa

city

of r

ecta

ngul

ar p

rism

s.

x D

ensit

y of

shap

es.

4.3

So

lve

mea

sure

men

t pro

blem

s in

vario

us p

ract

ical

and

non

-pr

actic

al co

ntex

ts

x Co

nver

t bet

wee

n co

mm

on u

nits

of m

easu

re (w

ithou

t the

ben

efit

of a

conv

ersio

n ta

ble)

: o

Be

twee

n m

illim

etre

s, ce

ntim

etre

s, m

etre

s and

kilo

met

res.

o

Betw

een

mill

ilitre

s (cu

bic

cent

imet

res)

and

litre

s. o

Be

twee

n m

illig

ram

s, gr

ams,

kilo

gram

s and

tonn

es.

o

Betw

een

seco

nds,

min

utes

, hou

rs a

nd d

ays.

x Co

nver

t bet

wee

n SI

uni

ts an

d Im

peria

l uni

ts (w

ith th

e ben

efit

of a

conv

ersio

n ta

ble)

. x

Conv

ersio

n ra

tes (

such

as S

I uni

ts to

Impe

rial u

nits)

. x

Cons

umpt

ion

rate

s (su

ch a

s kilo

met

res p

er li

tre).

x

Dist

ance

, tim

e an

d sp

eed

(suc

h as

kilo

met

res p

er h

our).

x

Cost

rate

s (su

ch a

s Ran

ds p

er k

ilogr

am o

r Ran

ds p

er m

etre

).

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AR

NIN

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OM

PETE

NC

E M

ATH

EMA

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LUM

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AM

EWO

RK

Ele

men

t 5: S

hape

and

spac

e Pu

rpos

e: D

escr

ibe

and

repr

esen

t obj

ects

and

the

envi

ronm

ent i

n te

rms o

f spa

tial p

rope

rties

and

rela

tions

hips

. Sk

ills:

5.

1

Iden

tify

and

wor

k w

ith g

eom

etric

figu

res a

nd so

lids,

incl

udin

g cu

ltura

l for

ms a

nd p

rodu

cts

5.2

A

naly

se th

e pr

oper

ties o

f geo

met

ric fi

gure

s and

solid

s 5.

3

Dra

w g

eom

etric

shap

es a

nd c

onst

ruct

mod

els o

f sol

ids

5.4

U

se th

e Th

eore

m o

f Pyt

hago

ras t

o so

lve

prob

lem

s inv

olvi

ng m

issi

ng le

ngth

s in

know

n ge

omet

ric fi

gure

s and

solid

s 5.

5

Dra

w d

iffer

ent v

iew

s of s

impl

e, re

gula

r obj

ects

in re

al-li

fe si

tuat

ions

5.

6

Rea

d, in

terp

ret a

nd u

se p

lans

and

road

map

s to

show

and

mak

e se

nse

of re

al lo

catio

ns, d

ista

nces

and

rela

tive

posi

tions

R

equi

red

Stan

dard

s of P

erfo

rman

ce

x Tw

o- a

nd th

ree-

dim

ensio

nal s

hape

s are

acc

urat

ely

desc

ribed

and

com

pare

d in

term

s of t

heir

mai

n pr

oper

ties.

x

The l

angu

age o

f sha

pe a

nd sp

ace i

s use

d in

cont

ext.

x D

raw

ings

and

repr

esen

tatio

ns o

f sha

pes a

nd so

lids a

re co

nsist

ent w

ith th

e ac

tual

shap

es a

nd so

lids.

x Th

e rel

ativ

e siz

es o

f dra

wn

shap

es a

re co

nsist

ent w

ith th

e sc

ale

and

sizes

of t

he sh

apes

and

solid

s sho

wn.

x

Dra

win

gs o

f obj

ects

from

diff

eren

t vie

wpo

ints

are c

onsis

tent

with

the

shap

e of t

he o

bjec

t fro

m th

ose

view

s. x

Map

s are

use

d ef

fect

ivel

y to

giv

e, an

d m

ake

sens

e of r

eal l

ocat

ions

, dist

ance

s and

rela

tive p

ositi

ons.

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OM

PETE

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EMA

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ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

5.1

Id

entif

y an

d w

ork

with

ge

omet

ric fi

gure

s and

solid

s, in

clud

ing

cultu

ral f

orm

s and

pr

oduc

ts

x Id

entif

y ba

sic g

eom

etric

shap

es.

x Id

entif

y an

d w

ork

with

bas

ic tr

ansf

orm

atio

ns (t

rans

latio

ns, r

efle

ctio

ns a

nd ro

tatio

ns).

x

Iden

tify

and

wor

k w

ith re

gula

r and

irre

gula

r sha

pes s

uch

as:

o

Poly

hedr

ons,

sphe

res,

cylin

ders

. o

Sh

apes

of,

and

deco

ratio

ns o

n, c

ultu

ral p

rodu

cts s

uch

as d

rum

s, po

ts, m

ats,

build

ings

, nec

klac

es, a

rchi

tect

ure

and

tess

ella

tions

.

5.2

A

naly

se th

e pr

oper

ties o

f ge

omet

ric fi

gure

s and

solid

s

x Th

e pro

perti

es o

f tria

ngle

s, qu

adril

ater

als,

regu

lar a

nd ir

regu

lar p

olyg

ons,

circ

les a

nd p

olyh

edro

ns ar

e ana

lyse

d ac

cord

ing

to th

e fo

llow

ing,

as

appl

icab

le:

o

Leng

th, b

read

th, h

eigh

t, w

idth

. o

Pe

rimet

er.

o

Dia

gona

l.

o

Are

a.

o

Ang

le.

o

Cent

re, r

adiu

s, di

amet

er, c

ircum

fere

nce.

o

V

olum

e.

o

Perp

endi

cula

r.

o

Para

llel.

o

Li

nes o

f sym

met

ry.

o

Rota

tiona

l sym

met

ry.

o

Wei

ght.

5.3

D

raw

geo

met

ric sh

apes

and

co

nstru

ct m

odel

s of s

olid

s x

Dra

w a

pla

n e.

g., o

f you

r ow

n liv

ing

spac

e an

d m

odify

it to

suit

your

nee

ds; d

raw

a fl

oor p

lan

of y

our l

earn

ing

envi

ronm

ent.

Show

the a

ctiv

ity

area

s.

5.4

U

se th

e The

orem

of P

ytha

gora

s to

solv

e pro

blem

s inv

olvi

ng

miss

ing

leng

ths i

n kn

own

geom

etric

figu

res a

nd so

lids

x A

ladd

er le

anin

g ag

ains

t a w

all w

ith th

e le

gs so

me

dista

nce a

way

from

the

base

of t

he w

all o

ffers

an

exam

ple o

f a co

ntex

t for

wor

king

with

th

e The

orem

of P

ytha

gora

s.

5.5

D

raw

diff

eren

t vie

ws o

f sim

ple,

re

gula

r obj

ects

in re

al-li

fe

situa

tions

x Le

ft, to

p an

d fro

nt v

iew

s of c

omm

on, r

egul

ar sh

apes

such

as d

esks

, she

lves

, tab

les,

com

pute

r scr

eens

, tel

evisi

ons a

nd h

uts.

5.6

Re

ad, i

nter

pret

and

use

pla

ns

and

road

map

s to

show

and

m

ake

sens

e of r

eal l

ocat

ions

, di

stanc

es a

nd re

lativ

e pos

ition

s

x M

ake

use

of m

ap c

oord

inat

es a

nd g

rid re

fere

nces

to lo

cate

pla

ces a

nd p

ositi

ons

x U

se b

earin

gs a

nd c

ompa

ss p

oint

s to

find

and

show

dire

ctio

n

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ALI

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TH

E Q

CTO

FO

UN

DA

TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

Ele

men

t 6: P

atte

rns a

nd r

elat

ions

hips

Pu

rpos

e: D

escr

ibe,

show

, int

erpr

et a

nd so

lve

prob

lem

s inv

olvi

ng m

athe

mat

ical

pat

tern

s, re

latio

nshi

ps a

nd fu

nctio

ns.

Skill

s:

6.1

Inve

stiga

te, c

ompl

ete,

exte

nd a

nd g

ener

ate p

atte

rns

6.2

Wor

k w

ith a

nd in

terp

ret a

rang

e of

repr

esen

tatio

ns o

f rel

atio

nshi

ps in

clud

ing

wor

ds, e

quat

ions

, tab

les o

f val

ues a

nd g

raph

s 6.

3 Re

pres

ent r

elat

ions

hips

to so

lve

prob

lem

s and

com

mun

icat

e or i

llustr

ate r

esul

ts R

equi

red

Stan

dard

s of P

erfo

rman

ce:

x Pa

ttern

s are

iden

tifie

d an

d de

scrib

ed in

term

s of t

he re

latio

nshi

ps b

etw

een

the e

lem

ents

of th

e pat

tern

. x

Gen

eral

isatio

ns re

gard

ing

patte

rns a

re id

entif

ied

and

thei

r val

idity

is v

erifi

ed a

nd e

xpla

ined

corre

ctly

. x

Patte

rns a

re c

ompl

eted

and

ext

ende

d in

man

ners

cons

isten

t with

the p

atte

rns.

x Re

latio

nshi

ps b

etw

een

inde

pend

ent a

nd d

epen

dent

var

iabl

es a

re re

pres

ente

d cl

early

and

effe

ctiv

ely

thro

ugh

tabl

es a

nd g

raph

s to

faci

litat

e ana

lysis

and

pro

blem

solv

ing.

x

Solu

tions

to p

robl

ems i

nvol

ving

pat

tern

s and

rela

tions

hips

are

valid

ated

in c

onte

xt.

Exam

ples

of f

unct

iona

l rel

atio

nshi

ps (i

ndep

ende

nt a

nd d

epen

dent

var

iabl

es):

x Th

e tim

e it

take

s for

a jo

urne

y de

pend

s on

the d

istan

ce a

nd th

e sp

eed

trave

lled.

x

The

med

icin

e dos

age

that

chi

ldre

n re

ceiv

e is

dete

rmin

ed b

y th

eir w

eigh

t or a

ge.

x Th

e cos

t of e

lect

ricity

is d

eter

min

ed b

y co

nsum

ptio

n.

x Th

e fe

e fo

r a b

ank

trans

actio

n is

dete

rmin

ed b

y th

e va

lue o

f the

tran

sact

ion.

x

The

volu

me

of p

aint

nee

ded

to c

ompl

ete

a job

is d

eter

min

ed b

y th

e are

a of t

he w

all t

o be

pai

nted

and

the

spre

adin

g ra

te o

f the

pai

nt.

x Th

e tim

e w

hich

a lig

ht b

ulb

burn

s det

erm

ines

the a

mou

nt o

f coa

l tha

t mus

t be

burn

t to

gene

rate

the e

lect

ricity

.

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E Q

CTO

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TIO

NA

L LE

AR

NIN

G C

OM

PETE

NC

E M

ATH

EMA

TIC

AL

LITE

RA

CY

: CU

RR

ICU

LUM

FR

AM

EWO

RK

SKIL

LS

GU

IDEL

INES

, SC

OPE

AN

D C

ON

TEX

TS

6.1

In

vesti

gate

, com

plet

e, ex

tend

an

d ge

nera

te p

atte

rns

x U

se a

rang

e of t

echn

ique

s to

dete

rmin

e m

issin

g an

d/or

add

ition

al te

rms i

n a p

atte

rn, i

nclu

ding

the

rela

tions

hip

betw

een

cons

ecut

ive

term

s. o

N

umer

ical

pat

tern

s inc

lude

: –

Cons

tant

diff

eren

ce p

atte

rns (

dire

ct p

ropo

rtion

and

line

ar re

latio

nshi

ps),

such

as t

he co

st of

a nu

mbe

r of i

tem

s. –

Cons

tant

ratio

pat

tern

s (in

vers

e pr

opor

tion

and

expo

nent

ial r

elat

ions

hips

), su

ch a

s gro

wth

pat

tern

s. o

G

eom

etric

pat

tern

s inc

lude

the

tilin

g pa

ttern

s use

d in

hom

es. T

hese

are

studi

ed to

det

erm

ine

how

man

y of

the

diffe

rent

tile

s are

nee

ded

to co

mpl

ete a

pat

tern

.

6.2

W

ork

with

and

inte

rpre

t a ra

nge

of re

pres

enta

tions

of

rela

tions

hips

incl

udin

g w

ords

, eq

uatio

ns, t

able

s of v

alue

s and

gr

aphs

x Re

cogn

ise, c

reat

e an

d de

scrib

e va

rious

pat

tern

s and

func

tiona

l rel

atio

nshi

ps.

x Re

cogn

ise re

latio

nshi

ps b

etw

een

inpu

t and

out

put v

aria

bles

in te

rms o

f the

ir nu

mer

ical

, gra

phic

al, v

erba

l and

sym

bolic

repr

esen

tatio

ns, a

nd

conv

ert b

etw

een

the

vario

us re

pres

enta

tions

. x

Iden

tify

gene

ralis

atio

ns o

f pat

tern

s fro

m g

iven

exp

ress

ions

i.e.

, sel

ect t

he co

rrect

gen

eral

isatio

n fro

m a

num

ber o

f giv

en g

ener

alisa

tions

o

N

umer

ic a

nd g

eom

etric

pat

tern

s. o

Ra

te o

f cha

nge

(the c

onne

ctio

n be

twee

n th

e slo

pe o

f a li

ne a

nd c

onsta

nt ra

te o

f cha

nge)

. x

Gen

erat

e pa

ttern

s usin

g di

ffere

nt g

ener

alise

d m

athe

mat

ical

form

s: o

G

raph

s.

o

Form

ulae

. x

Use

exp

ress

ions

and

oth

er ru

les f

or e

xpre

ssin

g pa

ttern

s. x

Conv

ert f

rom

one

bas

ic fo

rm o

f rep

rese

ntat

ion

to a

noth

er to

reve

al fe

atur

es o

f pat

tern

s and

rela

tions

hips

. x

Com

plet

e a ta

ble o

f val

ues b

y re

adin

g va

lues

from

a gr

aph

e.g.

, a li

near

or q

uadr

atic

func

tion

disp

laye

d on

a g

raph

. x

Plot

a gr

aph

from

the

valu

es in

a ta

ble o

f val

ues e

.g.,

a lin

ear o

r qua

drat

ic fu

nctio

n.

x Co

mpl

ete a

tabl

e of v

alue

s for

a g

iven

form

ulae

and

/or a

des

crip

tion

of a

rela

tions

hip.

6.3

Re

pres

ent r

elat

ions

hips

to so

lve

prob

lem

s and

com

mun

icat

e or

illus

trate

resu

lts

x Ch

oose

and

dev

elop

a re

pres

enta

tion

from

tabl

es a

nd g

raph

s. x

Dev

elop

and

use

sim

ple l

inea

r alg

ebra

ic e

quat

ions

to so

lve p

ract

ical

pro

blem

s. Th

ese

shou

ld b

e lim

ited

to si

tuat

ions

whe

re li

near

equa

tions

ap

pear

in co

ntex

t; th

ey sh

ould

not

be

used

in th

e abs

tract

. x

Sele

ct a

nd u

se si

mpl

e alg

ebra

ic fo

rmul

ae to

solv

e pro

blem

s. Th

ey sh

ould

incl

ude c

hang

ing

the

subj

ects

of fo

rmul

ae a

nd su

bstit

utin

g gi

ven

valu

es fo

r the

unk

now

ns.

x So

lve s

imul

tane

ous e

quat

ions

in re

al li

fe p

robl

em c

onte

xts:

o

e.g

., w

ork

out w

heth

er it

is m

ore b

enef

icia

l to

get p

aid

a ba

sic sa

lary

plu

s com

miss

ion

or a

fixe

d sa

lary

. o

e.g

., co

mpa

re th

e co

st of

hiri

ng c

ars f

rom

diff

eren

t ren

tal c

ompa

nies

with

diff

eren

t dai

ly c

harg

es a

nd ra

tes p

er k

ilom

etre

. x

Der

ive i

nfor

mat

ion

from

tabl

es a

nd g

raph

s and

ans

wer

rele

vant

que

stion

s on

the d

ata.

x

Use

appr

opria

te a

xes t

o pr

esen

t inf

orm

atio

n on

gra

phs.

x In

terp

ret t

he tr

ends

and

feat

ures

of g

raph

s sho

win

g m

athe

mat

ical

and

real

-life

situ

atio

ns.

x D

evel

op a

nd so

lve c

onte

xtua

lised

pro

blem

s usin

g va

rious

repr

esen

tatio

ns su

ch a

s gra

phs a

nd ta

bles

to u

nder

stand

the

purp

oses

and

use

fuln

ess

of e

ach.

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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

SECTION C FACILITATOR GUIDELINES, ASSESSMENT GUIDELINES AND

ACCREDITATION SPECIFICATIONS 1. FACILITATOR GUIDELINES

There are five key principles that should guide developers and facilitators of FML learning programmes: x FML programmes are learner-centred and discovery-based; x FML programmes promote communication and participation; x FML programmes are contextually relevant; x FML programmes focus on problem solving; and x FML programmes use tools such as calculators, spreadsheets and standard algorithms

to maximise efficiency, but without sacrificing understanding of the underlying mathematical concepts and processes.

1.1 FML programmes are learner-centred and discovery-based

Traditional approaches to teaching mathematics and mathematical literacy have tended to be authoritative and prescriptive. The teacher acted as the mathematical authority and tried to transfer knowledge into passive receptacles by prescribing methods or recipes. Such learning approaches required learners to interpret or make sense of the teacher's way of understanding a particular concept or of solving a particular problem type, usually within abstract or foreign contexts. This approach encouraged a narrow form of interpretive learning. Learners were expected to understand others' interpretations of knowledge. The negative effects of such an approach are as follows: x Learners look to the authority for answers and do not attempt to solve problems

themselves. Learners become receptive, passive and reliant. x The priorities of trying to understand what certain methods actually achieve and

why they produce certain results are not emphasised. x Learners attempt to learn by rote. x Learners are generally unable to tackle new problems without direction from the

authority. x Learners often build up a fear and dislike for mathematics. This translates into

poor performance.

By contrast, progressive ways of developing mathematical literacy rely on learner-centred, discovery-based approaches where realistic problem solving, within accessible contexts, becomes central.

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The learner-centred, discovery-based approach acknowledges that learners have prior knowledge and life experiences and that they are not just empty vessels waiting to be filled. The basic principle of this philosophy is that learners construct new knowledge by applying known situations and concepts or by creating new ones. Learning happens as learners engage in a series of carefully designed learning activities. Problems are solved in a social environment which involves sharing, discussion, comparison and debate. Learners make discoveries for themselves, share experiences with others, engage in helpful debates about methods and solutions, invent new methods, articulate their thoughts and borrow ideas from their peers to solve problems. Using this approach, the facilitator provides learning activities and mediates the learning process.

Learners should be encouraged to look for and develop their own techniques for solving problems. They should also be encouraged to share their understanding of situations. Communication plays a vital role both in making sense of situations and in sharing with others what has been learnt. Learners should be allowed to develop an understanding of the principles and ideas in each situation before they are introduced to formal terms, symbols and concepts. Formal concepts should only be introduced when learners are able to execute tasks whose nature, meaning and purpose they already know. This introduction, although brief compared to the extensive amount of literature available, has highlighted the learner-centred approach to designing and offering FML programmes. The approach is presented because it supports the broader aims of the FML Framework vis-à-vis achieving the purposes of FML. Such an approach: x promotes self-reliance and self-esteem; x promotes the confidence to tackle new problems; x encourages learners to develop problem-solving strategies; x reduces anxiety and pressure; x leads to a real understanding of concepts; x reduces systematic (or silly) errors; x promotes logical thinking; and x leads to a more lasting understanding and ability to apply what is learnt.

1.2 FML programmes promote communication and participation

Developers and facilitators of FML learning programmes should recognise the importance of mathematical communication. Mathematical communication is an over-arching process. It includes understanding, expressing and conveying ideas mathematically in order to reflect on and clarify one's thinking, to make convincing arguments, and to reach decisions. Communication: x is essential for understanding; x provides the foundation for learning; x is the bridge to finding and exchanging ideas, to identifying problems, and to

seeking and finding solutions to these problems; x is essential to working collaboratively; and x is the link that makes other skills effective.

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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

Mathematical communication is supported through well-designed and carefully monitored group processes. Learners cooperate to learn in group work. One of the most powerful resources to assist the learning process is group work because learning is essentially a social process. From a very early age, we learn within a social environment, and this resource should be exploited fully in the learning situation. Facilitators should create a learning environment and learning culture that promotes the verbalisation of ideas, of strategies and of what learners understand. A most powerful reinforcement of their understanding occurs as learners find words to express what is inside their heads. The search for words to articulate thoughts helps to clarify problems and contributes toward greater understanding. Discussing different ideas and approaches often leads learners to healthy educational debates that transcend the mere mathematics of the problem because it incorporates social, cultural and environmental issues. Discussion of this nature also allows for the transfer of ideas between learners. These could be used as they are or be reworked and reformulated by others. The verbalisation process also provides the facilitator with a good assessment tool about how well learners are progressing with a particular topic. It allows the facilitator to make decisions about pace: whether to move on, dwell on a topic, or go back. The verbalisation process assists in: x formulating ideas; x clarifying problems; x reinforcing understanding; x transferring ideas; and x assessing progress.

Varied forms of group work should be used. Sometimes learners should work in small groups (2 or 3), sometimes in large groups (5 or 6) and sometimes in one large group. At other times, they could even work individually. Facilitators should allow learners to use different forms of group work. Sometimes they could work through a problem together from start to finish. At other times, they could work on their own and consult each other from time to time. On other occasions, they could simply compare work at the end. Whatever form is used, facilitators should encourage as much verbalisation as possible.

1.3 FML programmes are contextually relevant

The key to promoting a real and lasting understanding of mathematical concepts is to provide rich and comprehensive learning activities that require problem solving in real-life contexts. Mathematically literate people make sense of situations, solve problems, and offer and validate solutions within specific contexts. In real-world applications, people need to be able to work within a range of different contexts and transfer the skills they developed in familiar contexts to unfamiliar contexts. However, studies show that these skills are not easily transferred. We cannot assume that learners will automatically transfer skills learnt in familiar contexts to unfamiliar contexts.

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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

Thus, one goal in a mathematical literacy programme should be to expose learners to a wide range of contexts in order to develop a solid foundation in the mathematics involved and to improve transfer skills. However, when learners find themselves in unfamiliar contexts, whether in the world of work or in occupational learning programmes, they will generally need some form of coaching to apply mathematical learning to those particular contexts. For this reason, although FML programmes should expose learners to a variety of contexts, such programmes will not necessarily enable learners to apply their skills in all contexts. However, it is reasonable to expect, when learners are confronted by unfamiliar contexts – either in occupational learning programmes or at work – that they will receive the assistance they need to apply their knowledge to the new contexts. Note: The FML Framework does not aim to cover the full range of contexts that people will meet in life, work and learning. Furthermore, the framework is based on the assumption that particular occupational contexts will be addressed within those particular occupations. Given the nature of mathematical literacy and the principles of adult learning, it is recommended that learning programmes are based on activities that are: x realistic and relevant to the context of learners; and x vary in complexity.

Relevant, realistic and context-led learning activities Learning activities should involve contexts that are relevant to the learners. There should be clear connections to other disciplines, to real life, to the workplace and to the various areas of mathematics. Learners need to make social, historical or personal links and connect with other learning areas. Mathematics connects to virtually everything, and good learning activities make these connections explicit. When the learning activities relate to the workplace or real-life situations, then mathematical literacy becomes more meaningful and learners will be able to make sense of, and remember, the concepts involved. Learners need to be able to make links between what they already know and new learning. The emphasis in mathematical literacy is on working within a variety of contexts. Thus, learning programmes should enable learners to work within real-life contexts, while at the same time ensuring the development of those mathematical concepts that are fundamental to mathematical literacy. Furthermore, learners learn more effectively when they encounter problems in contexts that invite full use of their personal strengths and do not rely simply on a narrow range of skills.

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Varying complexity Different kinds of problems and different contexts demand more complex problem-solving skills and learners should be equipped to deal with these more complex levels.

Some of the factors that determine the complexity of the problems include: x how familiar the contexts are; x the instruments and language used; x related concepts; x the number of steps involved; x how organised the data is; x the extent to which the information is complete and accurate; x the degree of certainty or uncertainty; x the number of concurrent factors; and x how routine the problem solving process is.

Problem solving becomes progressively more complex when we consider the mathematical concepts themselves. For example, when we think about the concept of measurement, we can see that it gets more complicated as we move: x from measuring length (such as distance and height), either directly or by

observing similarity; x to measuring area (of regular and irregular shapes), by using squares, cutting

and rearranging surface area and using scales; x to measuring volume and capacity; x to working out dimensions; x to changing the shapes of containers while keeping volumes the same (such as

by changing cylinders to prisms); and finally x to measuring pressure in relation to volume (such as in car tyres).

When we consider problem types we may require learners to find missing dimensions if only some are given. We might also ask learners to calculate how many smaller containers fit into a bigger one. In the latter example, we need to consider the levels of accuracy we require in the measurement, what percentage waste is acceptable, whether a better fit would be possible if the geometric shape were changed, and so on. We might also consider other measurements such as 'business confidence', inflation or growth in property values. In all instances, we would create an index to measure against. In real life, people have to cope with problem-solving situations of varying levels of complexity. Sometimes the problems can be solved in one or two easy steps. On other occasions people require several steps. Sometimes all the relevant information is readily at hand. At other times problem-solvers have to gather information, identify missing information and evaluate its relevance and usefulness. This is a reality about real-life problem solving, and thus, learning activities in FML programmes should include problems of varying levels of complexity.

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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK

1.4 FML programmes focus on problem solving

Problem-solving activities combine the interest in, and usefulness of, mathematics in applications that build on learners' own experiences.

There are two main considerations in problem solving: x Learners learn through problem solving. Studies show that learners are able to

develop a real and lasting understanding of mathematics when they learn through an approach that has problem solving as its central feature. Learners are better able to understand the various concepts when they encounter them in problems they can identify with. They are also better able to make the necessary links between related concepts. Thus, problem solving ought to be a major part of any learning methodology for mathematical literacy.

x Learners need to learn about problem solving. Problem solving is important as a methodology. It is also a critical skill in its own right. While problem solving is not the only skill to develop in mathematical literacy, it is a critical skill. Furthermore, problem solving becomes a central component of mathematical literacy, which is defined as the application of mathematics to real-life contexts.

Problem solving, therefore, needs to be a central feature of Foundational Mathematical Literacy because of its value as a methodology to aid learning and because of its importance as a skill. Learners should be confronted by a large variety of problem types and they should be allowed to solve the problems using their own ideas and methods.

The steps in the problem solving process have been frequently described (cf. Polya, 1945/1980) as: x defining the goals; x analysing the situation and constructing a mental picture of it; x devising a strategy and planning the steps to take; x executing the plan, including its control (and, if necessary, modifying the

strategy); and x evaluating the result.

The table below illustrates how these steps have been expanded:

Defining the goals

x Set goals. x Decide which goals must be reached and specify the reasons for the

decision. x Recognise which goals/wishes are contradictory and which are

compatible. x Assign priorities to goals/wishes.

Analysing the situation

x Select, obtain and evaluate information. Decide: o what information is required, what is already available, what is still

missing, and what is not needed; o where and how you can obtain the information; and o how you should interpret the information.

x Identify the people (and with what knowledge and skills) who should be involved in solving the problem.

x Select the tools to be used. x Recognise conditions (such as time restrictions) that need to be taken into

account.

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Planning the solution

x Decide on the steps to be taken. x Decide on the sequence of steps. x Coordinate work and deadlines. x Make a comparative analysis of alternative plans and decide which plan is

most suitable for achieving the goals. x Adapt the plan to changed conditions if necessary. x Decide on a plan

Executing the plan x Carry out the individual steps.

Evaluating the results

x Assess whether, and to what extent, the target has been reached. x Recognise the mistakes made. x Identify the reasons for the mistakes. x Assess the consequences of the mistakes

(Reeff, J-P., Zabal, A. & Blech, C. 2006. The Assessment of Problem-Solving Competencies A draft version of a general framework. Deutsches Institut für Erwachsenenbildung).

1.5 FML programmes use tools such as calculators, spreadsheets and

standard algorithms to maximise efficiency, but without sacrificing understanding of the underlying mathematical concepts and processes

The experience of mathematics, for many learners, is nothing more than learning and applying standard algorithms that work efficiently. Standard algorithms, calculators and spreadsheets are very useful for accurate computations. However, they can be counter-productive from an educational perspective. Educational objectives ought to prioritise thinking skills and understanding, and the blind use of standard algorithms can lead to the opposite. Although the use of standard algorithms, calculators and spreadsheets has an important place in mathematical literacy, facilitators should take care to ensure that learning activities promote thinking and understanding. Therefore, facilitators should control the use of these tools. It is important for learners to develop computational strategies that they understand. This enables them to make sense of what they are doing. Devising their own computational strategies promotes mathematical thinking. Thus, there will be times when facilitators should avoid the use of tools such as standard algorithms, calculators and spreadsheets to challenge the learners to devise (and share with others) their computational strategies. This should form a significant part of mathematical literacy sessions. The logic and concepts that underpin the algorithms then become clear. There may be times when facilitators choose to reveal or teach standard algorithms, but facilitators should decide when it would be useful and appropriate. Often, learners' own strategies are just as efficient as standard algorithms. In other words, it is not an objective of mathematical literacy to teach standard algorithms, but they are useful tools for quick computations and so can be incorporated when needed. The important thing is to reveal these tools and strategies to learners only after they have gained an understanding of the key concepts and processes and can appreciate the need for methods that are more efficient.

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On the other hand, there will be times when the learning activity will be compromised by the absence of quick, efficient tools such as standard algorithms, calculators or spreadsheets. For example, when learners are engaged in complex problem-solving exercises, we want to promote the higher-order skills of thinking, reasoning and decision making. We do not want learners to become distracted by the intricacies of difficult calculations at those times. We should not pose problems merely to give practice in computation. Rather, we should set problems that will enable learners to develop logical problem-solving strategies and to provide meaningful contexts through which to learn new concepts. Efficient tools become important then because they allow learners to focus on higher-order problem-solving skills.

2. ASSESSMENT GUIDELINES

INTERNAL CONTINUOUS ASSESSMENT Facilitators of FML programmes should engage in continuous assessment of learner progress to ensure that gaps are discovered early and suitable learning activities are used to address such gaps. Assessments should be designed to address each of the six elements comprehensively, as well as to ensure meaningful integration across the elements. The following tables provide facilitators with a useful reference point for what learners should be achieving in relation to each element.

Number Use numbers in a variety of forms to describe and make sense of situations, and to solve problems in a range of familiar and unfamiliar contexts.

What learners must be able to do: Criteria against which performance must be measured:

x Use numbers to describe and make sense of real-life situations

x Read, interpret and use different numbering conventions in different contexts and identify the ways in which different conventions work

x Interpret and use numbers written in exponential form including squares and cubes of natural numbers and the square and cube roots of perfect squares and cubes

x Do calculations in various situations using a variety of techniques

x Solve problems involving ratio and proportion x Solve problems involving fractions, decimals and

percentages

x Problem-solving strategies are based on a correct interpretation of the context.

x Calculations are performed accurately and according to the conventions governing the order of operations.

x Methods are presented in a clear, logical and structured manner, using mathematical symbols and notation consistent with mathematical conventions.

x Methods used are efficient, logical, internally consistent and justified by the context.

x The degree of accuracy of solutions is justified by the context.

x Solutions are evaluated and validated in terms of the context, and numbers are rounded appropriately to the problem situation.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

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Finance Manage personal finances using financial documents and related formulae. What learners must be able to do: Criteria against which performance must be measured:

x Read and interpret financial information presented in a range of documents in personal and familiar contexts

x Identify, classify and record sources of income and expenditure

x Plan and monitor personal finances x Evaluate options when purchasing products and

services x Determine the impact of interest, depreciation,

inflation, deflation and taxation on personal finances

x Interpretations of personal financial documents are consistent with recorded facts.

x Financial information is recorded and organised clearly, accurately and according to general finance-recording techniques and principles.

x Personal budgets reflect the financial situation in sufficient detail for planning and monitoring personal finances.

x Costs of products and services are evaluated using a variety of issues. These include affordability, personal needs and accuracy of advertising claims.

x Personal finances are monitored in terms of various influences, including income, expenditure, investments, loans, taxation and inflation.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

Data & Chance Collect, display and interpret data in various ways and solve related problems.

What learners must be able to do: Criteria against which performance must be measured:

x Collect data from various sources and in various ways

x Classify, organise and summarise data x Display data using tables, graphs and charts x Analyse and interpret data to draw conclusions

and make predictions x Determine and interpret simple chance in

everyday contexts

x Data collection techniques are appropriate to the context, type of data and purpose.

x Data is classified, organised and summarised appropriately so that it promotes meaningful analysis.

x Data is displayed using techniques appropriate to the type of information and context.

x Data displays are consistent with collected data, promote ease of interpretation and minimise bias.

x Interpretations and predictions are verified by the data and observed trends. They take into account possible sources of error and data manipulation.

x Simple probabilities are determined accurately and statements of chance are correctly interpreted in context.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

Measurement Make measurements using appropriate measuring tools and techniques and solve problems in various measurement contexts.

What learners must be able to do: Criteria against which performance must be measured:

x Estimate and measure quantities using measuring instruments in various contexts, paying attention to significant figures and margins of error

x Calculate quantities in measurement contexts paying attention to significant figures and margins of error

x Solve measurement problems in various practical and non-practical contexts

x Measuring instruments used meet the accuracy requirements of the context.

x Readings are accurately recorded within appropriate margins of error and using appropriate units.

x Calculations are performed accurately, keeping units consistent.

x Conversions between units are accurate and appropriate to the context.

x Solutions to problems are validated according to the context, including contextually appropriate rounding and use of units.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

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Shape &

Space Describe and represent objects and the environment in terms of spatial properties and relationships.

What learners must be able to do: Criteria against which performance must be measured:

x Identify and work with geometric figures and solids, including cultural forms and products

x Analyse the properties of geometric figures and solids

x Draw geometric shapes and construct models of solids

x Use the Theorem of Pythagoras to solve problems involving missing lengths in known geometric figures and solids

x Draw different views of simple, regular objects in real-life situations

x Read, interpret and use plans and road maps to show and make sense of real locations, distances and relative positions

x Two- and three-dimensional shapes are accurately described and compared in terms of their main properties.

x The language of shape and space is used in context.

x Drawings and representations of shapes and solids are consistent with the actual shapes and solids.

x The relative sizes of drawn shapes are consistent with the scale and sizes of the shapes and solids shown.

x Drawings of objects from different viewpoints are consistent with the shape of the object from those views.

x Maps are used effectively to give and make sense of real locations, distances and relative positions.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

Patterns & Relationships

Describe, show, interpret and solve problems involving mathematical patterns, relationships and functions.

What learners must be able to do: Criteria against which performance must be measured:

x Investigate, complete, extend and generate simple number and geometric patterns

x Work with and interpret a range of representations of relationships including words, equations, tables of values and graphs

x Represent relationships to solve problems and communicate or illustrate results

x Patterns are identified and described in terms of the relationships between the elements of the pattern.

x Patterns are expressed in general terms where possible.

x Patterns are completed and/or extended in keeping with the general patterns.

x Relationships between independent and dependent variables are represented clearly and effectively through tables and graphs to facilitate analysis and problem solving.

x Solutions to problems involving patterns and relationships are validated in context.

NOTE: Refer to the FML Framework for the full description of scope and contexts to be covered.

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EXTERNAL SUMMATIVE ASSESSMENT General Information The Foundational Learning Competence Assessment is a national assessment offered in the two learning areas of Foundational Communication and Foundational Mathematical Literacy. Its purpose is to provide a quick and efficient means to benchmark the broad competence level of an individual in the two Foundational Learning areas, in support of successful occupational training. This national assessment has the following features: x It is a machine scored, item-based, multiple choice format assessment. x It is available at regular intervals, with quick delivery of results. x It is administered by an external Assessment Quality Partner appointed by the QCTO. x Successful candidates are awarded a statement of results by the QCTO. Success in the Foundational Learning assessment in both learning areas is compulsory for final award of any occupational qualifications at NQF Levels 3 and 4. Curriculum designers for occupations at NQF level 2 will have the option to decide whether FML is a requirement for award or not. Candidates can enter the assessments before or during their occupational training. If successful in a learning area, they do not need to undertake a Foundational Learning programme in that area. If unsuccessful, they undertake the relevant learning programme and then re-take the relevant Foundational Learning Assessment. The Foundational Learning Competence Assessment for Foundational Mathematical Literacy The purpose of this assessment is to determine whether a learner has sufficient competence and skills in mathematical literacy to engage successfully with formal occupational training up to NQF Level 4. The assessment is based on the Foundational Mathematical Literacy Curriculum according to a construct developed by the assigned Assessment Quality Partner in such a way as to ensure the sample of questions fairly represents all six elements and contains a spread of questions at varying levels of complexity. Providers should advise learners who are below ABET Level 3 in English language competence that they are unlikely to be able to deal with the literacy demands of the test. Learners in programmes should be prepared for the assessment through: x Familiarisation with the format and instructions of the paper. x Practise in reading and understanding multiple choice questions, in terms of what these are

assessing and how to 'think through' the options given. x Practise in using time efficiently in order to complete the paper.