FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb:...
Transcript of FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY ... Maths Literacy... · page 3 of 30 ieb:...
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 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:
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 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.
Pa
ge 5
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
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.
Pa
ge 6
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
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.
Pa
ge 7
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
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 .
Pa
ge 8
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
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.
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.
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).
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.
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
.
Pa
ge 1
3 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
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.
Pa
ge 1
4 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
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.
Pa
ge 1
5 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
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
).
Pa
ge 1
6 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
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.
Pa
ge 1
7 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
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
Pa
ge 1
8 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
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
.
Pa
ge 1
9 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
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.
Page 20 of 30
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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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK
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.
Page 22 of 30
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.
Page 23 of 30
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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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK
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.
Page 25 of 30
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.
Page 26 of 30
IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK
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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IEB: ASSESSMENT QUALITY PARTNER TO THE QCTO FOUNDATIONAL LEARNING COMPETENCE MATHEMATICAL LITERACY: CURRICULUM FRAMEWORK
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.