-
DOCUMENT RESUME
ED 394 793 SE 058 057
TITLE The Common Curriculum Framework for K-12 Mathematics:Western Canadian Protocol for Collaboration in BasicEducation.
INSTITUTION Alberta Dept. of Education, Edmonton.REPORT NO ISBN-0-7732-1780-0PUB DATE Jun 95NOTE 593p.PUB TYPE Guides Non-Classroom Use (055)
EDRS PRICE MF03/PC24 Plus Postage.DESCRIPTORS *Educational Objectives; Elementary Secondary
Education; Foreign Countries; *MathematicsCurriculum; Mathematics Instruction; *Standards
IDENTIFIERS *Canada
ABSTRACTThis document provides a common base for the K-9
(grades 10-12 are still under development) mathematics curriculumexpectations mandated by the western Canadian provinces andterritories of Manitoba, Saskatchewan, Alberta, British Columbia,Yukon Territory, and the Northwest Territories. The frameworkidentifies beliefs about mathematics, general and specific studentoutcomes, and illustrative examples agreed upon by these sixjurisdictions. The following aspects are discussed: (1) mathematicalprocesses based on the standards of the National Council of Teachersof Mathematics: communication, connections, estimation and mentalmathematics, problem solving, reasoning, technology, andvisualization; (2) nature of mathematics: change, constancy,dimension, number, pattern, quantity, relationships, shape, anduncertainty; (3) strands: number, patterns and relations, shape andspace, and statistics and probability; and (4) student expectations.The majority of the document contains general outcomes and specificoutcomes with illustrative examples for each grade from kindergartenthrough grade 9. (MKR)
***********************************************************************
Reproductions supplied by EDRS are the best that can be madefrom the original document.
***********************************************************************
-
The
Com
mon
Cur
ricu
Gm
Fram
ewor
k
K 1
2 M
AT
HE
MA
TIC
S(
1012
is u
nder
dev
elop
men
t )
Wes
tern
Can
adia
nPr
otoc
ol f
or C
olla
bora
tion
inB
asic
Edu
catio
n
PE
RM
ISS
ION
TO
RE
PR
OD
UC
E A
ND
DIS
SE
MIN
AT
E T
HIS
MA
TE
RIA
LH
AS
BE
E G
RA
N E
D B
Y
(k-
JUN
E 1
995
10 T
HE
ED
UC
AT
ION
AL
RE
SO
UR
CE
SIN
F O
RM
AT
ION
CE
N T
ER
(E
RIC
)
C't
BE
ST
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PY
AV
AIL
AB
LE
U.S
DE
PA
RT
ME
NT
Of E
DU
CA
TIO
NO
rfic
e 01
Edu
cato
nsi R
esea
rch
and
Imor
ov.m
.ro
E D
UC
A T
ION
AL
RE
SO
UR
CE
S IN
FO
RM
AT
ION
CE
NT
ER
(E
RIC
)
Ns
docu
men
t has
Dos
s m
orod
uclo
dI ,
,ed
tror
n th
e op
rson
or
orge
nsla
tron
Ong
onah
ng
CI M
ims,
cha
nges
hav
e be
an m
ad tO
orn
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prod
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n Q
ualit
y
Pom
IS o
l 0re
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oom
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mal
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oIN
I110
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f do
not n
eceS
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ly r
IprI
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nt &
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E R
I pos
dron
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nt.B
ER
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EN
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NT
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.
The
Com
mon
Cur
ricu
lum
Fra
mew
ork
K 1
2 M
AT
HE
MA
TIC
S(
1012
is u
nder
dev
elop
men
t )
Wes
tern
Can
adia
n Pr
otoc
ol f
or C
olla
bora
tion
in B
asic
Edu
catio
n
ePri
nted
on
Rec
ykd
Pape
r
-
Cat
alok
ruin
g in
Pub
licat
ion
Dat
a
The
com
mon
cur
ricu
lum
fra
mew
ork
for
K-1
2 m
athe
mat
ics
: wes
tern
Can
adia
npr
otoc
ol f
or c
olla
bora
tion
in b
asic
edu
catio
n.
ISB
N 0
-773
2-17
80-0
(10-
12 is
und
er d
evel
opm
ent)
1. M
athe
mat
icsS
tudy
and
teac
hing
. 2. M
athe
mat
icsY
ukon
Ter
rito
ryC
urri
cula
. 3. M
athe
mat
icsN
orth
wes
t Ter
rito
ries
Cur
ricu
la.
4. M
athe
mat
icsB
ritis
h C
olum
biaC
urri
cula
. 5. M
athe
mat
icsA
lber
taC
urri
cula
. 6. M
athe
mat
icsS
aska
tche
wan
Cur
ricu
la. 7
. Mat
hem
atic
sM
anito
baC
urri
cula
. I. T
itle.
IL
Titl
e: W
este
rn C
anad
ian
prot
ocol
for
colla
bora
tion
in b
asic
edu
catio
n.
QA
14.C
2.A
734
1995
510.
7
Cop
yrig
ht 0
199
5, th
e C
row
n in
Rig
ht o
f th
e G
over
nmen
ts o
f M
anito
ba, S
aska
tche
wan
, Bri
tish
Col
umbi
a, Y
ukon
Ter
rito
ry, N
orth
wea
t Ter
rito
ries
and
Alb
erta
as
repr
esen
ted
by th
e M
inis
ter
of E
duca
tion
and
Tra
inin
g, M
anito
ba;
the
Min
iste
r of
Edu
catio
n, T
rain
ing
and
Em
ploy
men
t, Sa
skat
chew
an; t
he M
inis
ter
of E
duca
tion,
Bri
tish
Col
umbi
a;th
e M
inis
ter
of E
duca
tion,
Yuk
on T
erri
tory
; the
Min
iste
r of
Edu
catio
n, C
ultu
re a
nd E
mpl
oym
ent,
Nor
thw
est
Ter
rito
ries
; and
, the
Min
iste
r of
Edu
catio
n, A
lber
ta.
Perm
issi
on is
giv
en b
y th
e co
pyri
ght o
wne
rs to
rep
rodu
ce th
is d
ocum
ent f
or e
duca
tiona
l pur
pose
s an
d on
a n
onpr
ofit
basi
s, w
ith th
e ex
cept
ion
of m
ater
ials
on
page
s 6,
7, 8
and
9 e
xcer
pted
, with
per
mis
sion
, fro
mC
urnc
ulum
and
Eva
luat
ion
Sta
ndar
ds,
Nat
iona
l Cou
ncil
of T
each
ers
of M
athe
mat
ics,
198
9, w
hich
sha
ll no
t be
repr
inte
d af
ter
June
30,
199
7, a
t whi
ch ti
me
the
NC
TM
doc
umen
t is
sche
dule
d fo
r up
datin
g.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
ii7
-
AC
KN
OW
LE
DG
EM
EN
TS
The
Com
mon
Cur
ricu
lum
Fra
mew
ork
for
K-1
2 M
athe
mat
ics
was
dev
elop
ed th
roug
hth
eco
oper
ativ
e ef
fort
s of
the
four
wes
tern
pro
vinc
es a
nd tw
o te
rrito
ries
. T1.
3se
juri
sdic
tions
wou
ld li
ke to
ack
now
ledg
e th
e fo
llow
ing
cont
ribu
tors
.
Bri
tish
Col
umbi
aK
eith
Cho
ngSi
mon
Fra
ser
Uni
vers
ityG
race
Fra
ser
Tea
cher
(R
etir
ed)
Al F
risk
Exa
min
atio
ns B
ranc
h, B
C M
inis
try
of E
duca
tion
Nor
m G
ilber
stad
Cur
ricu
lum
Bra
nch,
BC
Min
is ;.
-.y
of E
duca
tion
Ivan
Joh
nson
Past
Pre
side
nt, B
C A
ssoc
iatio
n of
Mat
hem
atic
sT
each
ers
Car
ryl K
oePr
esid
ent,
BC
Ass
ocia
tion
of M
athe
mat
ics
Tea
cher
sSc
ott M
acD
onal
dC
urri
culu
m B
ranc
h, B
C M
inis
try
of E
duca
tion
Nei
l Whi
tmor
eC
urri
culu
m B
ranc
h, B
C M
inis
try
of E
duca
tion
Alb
erta Dia
ne C
ongd
onA
lber
ta T
each
ers'
Ass
ocia
tion,
Iso
bel F
. Cox
Sch
ool,
Prai
rie
Lan
d R
egio
nal D
ivis
ion
No.
25
Lin
da C
oxA
lber
ta D
ista
nce
Lea
rnin
g C
entr
e, A
lber
ta E
duca
tion
Jack
Edw
ards
Cur
ricu
lum
Sta
ndar
ds B
ranc
h, A
lber
ta E
duca
tion
Flor
ence
Gla
nfie
ldSt
uden
t Eva
luat
ion
Bra
nch,
Alb
erta
Edu
catio
nK
ay M
elvi
lleSt
uden
t Eva
luat
ion
Bra
nch,
Alb
erta
Edu
catio
nM
ario
n O
berg
Alb
erta
Tea
cher
s' A
ssoc
iatio
n, P
arkl
and
Com
posi
te,
Gra
nde
Yel
low
head
Reg
iona
l Div
isio
n N
o. 3
5W
endy
Ric
hard
sA
lber
ta T
each
ers'
Ass
ocia
tion,
Ros
slyn
Jun
ior
Hig
h,E
dmon
ton
Scho
ol D
istr
ict N
o. 7
Hug
h Sa
nder
sC
urri
culu
m S
tand
ards
Bra
nch,
Alb
erta
Edu
catio
nE
vely
n Sa
wic
kiA
lber
ta T
each
ers'
Ass
ocia
tion,
Cal
gary
Cat
holic
Sch
ools
Mer
y T
horn
ton
Cur
ricu
lum
Sta
ndar
ds B
ranc
h, A
lber
ta E
duca
tion
Sask
atch
ewan
Dia
ne H
anso
n
Lyl
e M
arko
wsk
iB
arry
Mits
chke
Off
icia
l Min
ority
Lan
guag
e O
ffic
e, S
aska
tche
wan
Edu
catio
n, T
rain
ing
and
Em
ploy
men
t (SE
TE
)C
urri
culu
m a
nd I
nstr
uctio
n B
ranc
h, S
ET
EC
urri
culu
m a
nd I
nstr
uctio
n B
ranc
h, S
ET
E
Way
m N
sch
Con
nie
Ros
owsk
y
Max
ine
Stin
kaB
urt T
hies
sen
Man
itoba
John
Bar
sby
Dia
nne
Bur
nett
Jose
ph C
ombi
adak
is
Dav
e L
ong
Pat M
acD
onal
dH
arri
Mah
araj
Mic
hèle
Man
aigr
e-Si
ms
Gin
ette
Roy
Dia
nne
Solte
ss
Kat
hy T
etlo
ckW
ayne
Wat
tK
aren
Yam
ada
Nor
thw
est T
erri
tori
esFl
o C
ampb
ell
Geo
rge
Hin
ton
Aile
en N
ajdu
ch
Yuk
on T
erri
tory
Ian
Oos
tindi
e
Oth
er W
rite
rsM
arie
Hau
kPe
ggy
Hill
Bet
hune
Sch
ool,
Buf
falo
Pla
ins
Scho
ol D
ivis
ion
No.
21
Kam
sack
Com
preh
ensi
ve I
nstit
ute,
Kam
sack
Sch
ool
Div
isio
n N
o. 3
5C
urri
culu
m a
nd I
nstr
uctio
n B
ranc
h, S
ET
EE
van
Har
dy, S
aska
toon
Pub
lic S
choo
l Div
isio
n N
o. 1
3
St. J
ohn'
s R
aven
scou
rt S
choo
lW
inni
peg
Scho
ol D
ivis
ion
No.
1B
urea
u de
l'éd
ucat
ion
fran
gais
e, M
anito
ba E
duca
tion
and
Tra
inin
g (M
ET
)Sa
mue
l Bur
land
Sch
ool,
St. V
ital S
choo
l Div
isio
n N
o. 6
Prog
ram
Dev
elop
men
t, M
ET
Ass
essm
ent a
nd E
valu
atio
n, M
ET
Eco
le W
estd
ale
Scho
ol, A
ssin
iboi
ne S
outh
Sch
ool
Div
isio
n N
o..3
Eco
le T
aché
, Div
isio
n sc
olai
re f
ranc
o-m
anito
bain
e N
o. 4
9St
rath
mill
an S
choo
l, St
. Jam
es-A
ssin
iboi
a Sc
hool
Div
isio
n N
o. 2
Mat
hem
atic
s/Sc
ienc
e/T
echn
olog
y, M
ET
Prog
ram
Dev
elop
men
t, M
ET
Fort
Gar
ry S
choo
l Div
isio
n N
o. 5
Eco
le S
t. Jo
seph
Sch
ool,
Yel
low
knif
e C
atho
lic S
choo
lsD
iam
ond
Jenn
ess
Hig
h Sc
hool
, Sou
th S
lave
Div
isio
nal
Boa
rdC
urri
culu
m S
ervi
ces,
Dep
artm
ent o
f E
duca
tion,
Cul
ture
and
Em
ploy
men
t
Dep
artm
ent o
f E
duca
tion,
Gov
ernm
ent o
f th
e Y
ukon
Con
trib
utor
toIl
lust
rativ
e E
xam
ples
Illu
stra
tive
Exa
mpl
es
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
iiiA
ckno
wle
dgem
ents
-
TA
BL
E O
F C
ON
TE
NT
S
AC
KN
OW
LE
DG
EM
EN
TS
iii
I.B
AC
KG
RO
UN
D1
II.
INT
RO
DU
CT
ION
2
Purp
ose
of th
e D
ocum
ent
2
Bel
iefs
abo
ut S
tude
nts
and
Mat
hem
atic
s L
earn
ing
2
Goa
ls f
or S
tude
nts
2
III.
CO
NC
EPT
UA
L F
RA
ME
WO
RK
FO
R K
-12
MA
TH
EM
AT
ICS
4
Mat
hem
atic
al P
roce
sses
5
Nat
ure
of M
athe
mat
ics
12
Stra
nds
16
Stud
ent E
xpec
tatio
ns17
Sum
mar
y17
IV.
INST
RU
CT
ION
AL
FO
CU
S18
Sugg
este
d T
ime
Allo
tmen
ts18
V.
STU
DE
NT
OU
TC
OM
ES
19
Gen
eral
Out
com
es20
Gen
eral
Out
com
es a
nd S
peci
fic
Out
com
es28
Num
ber
28
Patte
rns
and
Rel
atio
ns38
Shap
e an
d Sp
ace
44
Stat
istic
s an
d Pr
obab
ility
54
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
1 1
Tab
le o
f C
onte
nts
10
-
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UT
U,
-
I. B
AC
KG
RO
UN
D
Part
ners
for
Col
labo
ratio
n in
Bas
icE
duca
tion:
Man
itoba
Sask
atch
ewan
Alb
erta
Bri
tish
Col
umbi
aY
ukon
Ter
rito
ryN
orth
wes
t Ter
rito
ries
The
Wes
tern
Can
adia
n Pr
otoc
ol f
or C
olla
bora
tion
in B
asic
Edu
catio
n K
inde
rgar
ten
to G
rade
12
was
sig
ned
inD
ecem
ber
1993
by
the
min
iste
rs o
f ed
ucat
ion
from
Man
itoba
, Sas
katc
hew
an, A
lber
ta, B
ritis
h C
olum
bia,
Yuk
on T
erri
tory
and
the
Nor
thw
est T
erri
tori
es. T
hepr
otoc
ol s
tate
s th
at th
e fo
ur w
este
rn p
rovi
nces
and
the
two
terr
itori
es a
gree
to c
olla
bora
te in
bas
ic e
duca
tion
beca
use
of th
e im
port
ance
they
pla
ce o
n:
com
mon
edu
catio
nal g
oals
high
sta
ndar
ds in
edu
catio
nre
mov
ing
obst
acle
s fo
r st
uden
t acc
ess
to e
duca
tiona
lop
port
uniti
es, w
hich
incl
udes
impr
ovin
g th
e ea
se o
ftr
ansf
er f
rom
juri
sdic
tion
to ju
risd
ictio
nop
timum
use
of
educ
atio
nal r
esou
rces
.
The
Com
mon
Cur
ricu
lum
Fra
mew
ork
for
K-1
2M
athe
mat
ics
(The
Com
mon
Fra
mew
ork)
is th
e fi
rst i
n a
seri
es o
fjoi
nt d
evel
opm
ent p
roje
cts
in b
asic
edu
catio
n. I
tha
s be
en d
evel
oped
by
the
six
min
istr
ies
of e
duca
tion
inco
llabo
ratio
n w
ith te
ache
rs, a
dmin
istr
ator
s, p
aren
ts,
busi
ness
rep
rese
ntat
ives
, pos
t-se
cond
ary
inst
ruct
ors
and
othe
rs.
The
Com
mon
Fra
mew
ork
iden
tifie
s be
liefs
abo
utm
athe
mat
ics,
gen
eral
and
spe
cifi
c st
uden
t out
com
es a
ndill
ustr
ativ
e ex
ampl
es a
gree
d up
on b
y th
e si
x ju
risd
ictio
ns.
Eac
h of
the
prov
ince
s an
d te
rrito
ries
will
det
erm
ine
whe
nan
d ho
w T
he C
omm
on F
ram
ewor
k is
to b
e im
plem
ente
dw
ithin
its
own
juri
sdic
tion.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
1B
ackg
roun
d
1514
-
II. I
NT
RO
DU
CT
ION
The
Com
mon
Fra
mew
ork
com
mun
icat
es h
igh
expe
ctat
ions
for
stu
dent
s.
Stud
ents
mus
t con
stru
ctth
eir
own
mea
ning
of
mat
hem
atic
s.
PUR
POSE
OF
TH
E D
OC
UM
EN
T
The
Com
mon
Fra
mew
ork
addr
esse
s th
e m
ajor
goa
ls o
f th
epr
otoc
ol. T
his
docu
men
t pro
vide
s a
com
mon
bas
e fo
r th
ecu
rric
ulum
exp
ecta
tions
man
date
d by
eac
h pr
ovin
ce a
ndte
rrito
ry. T
his
com
mon
bas
e w
ill r
esul
t in
cons
iste
ntst
uden
t out
com
es in
mat
hem
atic
s ac
ross
juri
sdic
tions
will
enab
le e
asie
r tr
ansf
er f
or s
tude
nts
mov
ing
from
one
juri
sdic
tion
to a
noth
er. I
ts in
tent
is to
com
mun
icat
ecl
earl
y hi
gh e
xpec
tatio
ns f
or s
tude
nts
in m
athe
mat
ics
educ
atio
n to
all
educ
atio
nal p
artn
ers
acro
ss th
eju
risd
ictio
ns a
nd f
acili
tate
the
deve
lopm
ent o
f co
mm
onle
arni
ng r
esou
rces
.
BE
LIE
FS A
BO
UT
ST
UD
EN
TS
AN
D M
AT
HE
MA
TIC
SL
EA
RN
ING
Stud
ents
are
cur
ious
, act
ive
lear
ners
who
hav
e in
divi
dual
inte
rest
s, a
bilit
ies
and
need
s. T
hey
com
e to
cla
ssro
oms
with
dif
fere
nt k
now
ledg
e, li
fe e
xper
ienc
es a
ndba
ckgr
ound
s th
at g
ener
ate
a ra
nge
of a
ttitu
des
abou
tm
athe
mat
ics
and
life.
Stud
ents
lear
n by
atta
chin
g m
eani
ng to
wha
t the
y do
; and
they
mus
t be
able
to c
onst
ruct
thei
r ow
n m
eani
ng o
fm
athe
mat
ics.
Thi
s m
eani
ng is
bes
t dev
elop
ed w
hen
lear
ners
enc
ount
er m
athe
mat
ical
exp
erie
nces
that
pro
ceed
from
the
sim
ple
to th
e co
mpl
ex a
nd f
rom
the
conc
rete
to th
eab
stra
ct. T
he u
se o
f m
anip
ulat
ives
can
add
ress
the
dive
rsity
of
lear
ning
sty
les
and
deve
lopm
enta
l sta
ges
ofst
uden
ts a
nd c
an e
nhan
ce th
e fo
rmat
ion
of s
ound
,tr
ansf
erab
le, m
athe
mat
ical
con
cept
s. A
t all
leve
ls,
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
16
stud
ents
ben
efit
from
wor
king
with
app
ropr
iate
mat
eria
ls,
tool
s an
d co
ntex
ts w
hen
cons
truc
ting
pers
onal
mea
ning
abou
t new
mat
hem
atic
al id
eas.
The
lear
ning
env
iron
men
tsh
ould
val
ue a
nd r
espe
ct e
ach
stud
ent's
way
of
thin
king
, so
that
the
lear
ner
feel
s co
mfo
rtab
le in
taki
ng in
telle
ctua
lri
sks,
ask
ing
ques
tions
and
pos
ing
conj
ectu
res.
Mat
hem
atic
s is
a c
omm
on h
uman
act
ivity
, inc
reas
ing
inim
port
ance
in a
rap
idly
adv
anci
ng, t
echn
olog
ical
soc
iety
.A
gre
ater
pro
fici
ency
in u
sing
mat
hem
atic
s in
crea
ses
the
oppo
rtun
ities
ava
ilabl
e to
indi
vidu
als.
Stu
dent
s ne
ed to
beco
me
mat
hem
atic
ally
lite
rate
in o
rder
to e
xplo
repr
oble
m-s
olvi
ng s
ituat
ions
, acc
omm
odat
e ch
angi
ngco
nditi
ons,
and
act
ivel
y cr
eate
new
kno
wle
dge
in s
triv
ing
for
self
-ful
fillm
ent.
GO
AL
S FO
R S
TU
DE
NT
S
The
mai
n go
als
of m
athe
mat
ics
educ
atio
n ar
e to
pre
pare
stud
ents
to:
use
mat
hem
atic
s co
nfid
ently
to s
olve
pro
blem
sco
mm
unic
ate
and
reas
on m
athe
mat
ical
lyap
prec
iate
and
val
ue m
athe
mat
ics
com
mit
them
selv
es to
life
long
lear
ning
beco
me
mat
hem
atic
ally
lite
rate
adu
lts, u
sing
mat
hem
atic
s to
con
trib
ute
to s
ocie
ty.
At t
he c
ompl
etio
n of
a p
rogr
am, s
tude
nts
shou
ld h
ave
deve
lope
d a
posi
tive
attit
ude
tow
ard
mat
hem
atic
s an
d ha
vea
base
of
know
ledg
e an
d sk
ills
rela
ted
to N
umbe
r, P
atte
rns
and
Rel
atio
ns, S
hape
and
Spa
ce, a
nd S
tatis
tics
and
Prob
abili
ty.
Mat
hem
atic
s ed
ucat
ion
mus
t pre
pare
stu
dent
s to
use
mat
hem
atic
s to
sol
vepr
oble
ms.
172
Intr
oduc
tion
-
Posi
tive
attit
udes
tow
ard
It is
impo
rtan
t for
stu
dent
s to
dev
elop
a p
ositi
vem
athe
mat
ics
are
attit
ude
tow
ard
mat
hem
atic
s so
that
they
can
bec
ome
impo
rtan
tco
nfid
ent i
n th
eir
abili
ty to
und
erta
ke th
e pr
oble
ms
of a
chan
ging
wor
ld, t
here
by e
xper
ienc
ing
the
pow
er a
ndus
eful
ness
of
mat
hem
atic
s. S
tude
nts
also
sho
uld
gain
an u
nder
stan
ding
and
app
reci
atio
n of
the
cont
ribu
tions
of m
athe
mat
ics,
as
a sc
ienc
e an
d as
an
art,
toci
viliz
atio
n an
d to
cul
ture
.
Stud
ents
sho
uld:
exhi
bit a
pos
itive
atti
tude
tow
ard
mat
hem
atic
sen
gage
and
per
seve
re in
mat
hem
atic
al ta
sks
and
proj
ects
cont
ribu
te to
mat
hem
atic
al d
iscu
ssio
nsta
ke r
isks
in p
erfo
rmin
g m
athe
mat
ical
task
sex
hibi
t cur
iosi
tysh
ow s
ome
enjo
ymen
t of
mat
hem
atic
al e
xper
ienc
es.
All
stud
ents
sho
uld
rece
ive
a le
vel o
f m
athe
mat
ics
educ
atio
n ap
prop
riat
e to
thei
r ne
eds
and
abili
ties.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
is3
19In
trod
uctio
n
-
III.
CO
NC
EPT
UA
L F
RA
ME
WO
RK
FO
R K
-12
MA
TH
EM
AT
ICS
Stud
ents
of
mat
hem
atic
s, r
egar
dles
s of
age
or
expe
rien
ce,
stru
ggle
to d
o m
athe
mat
ics
in s
ettin
gs th
at a
re n
ew to
them
. The
con
cept
ual f
ram
ewor
k ou
tline
d in
this
sec
tion
pres
ents
a m
ultif
acet
ed v
iew
of
mat
hem
atic
s an
d pr
esen
tsth
e di
scip
line
as s
kills
, pro
cedu
res
and
conc
epts
wov
ento
geth
er.
The
fra
mew
ork
char
t bel
ow s
how
s ho
w s
tude
nt o
utco
mes
,or
gani
zed
by g
rade
and
str
and,
are
des
igne
d to
be
infl
uenc
ed b
y M
athe
mat
ical
Pro
cess
es a
nd th
e N
atur
e of
Mat
hem
atic
s. T
hese
com
pone
nts
are
desc
ribe
d m
ore
fully
in th
is s
ectio
n.
GR
AD
E
ST
RA
ND
12
34
56
78
910
1112
Num
ber
Num
ber
Con
cept
sN
umbe
r O
pera
tions
Pat
tern
s an
d R
elat
ions
Pat
tern
sV
aria
bles
and
Equ
atio
nsR
elat
ions
and
Fun
ctio
ns
Sha
pe a
nd S
pace
Mea
sure
men
t3-
D O
bjec
ts a
nd 2
-D S
hape
sT
rans
form
atio
ns
Sta
tistic
s an
d P
roba
bilit
yD
ata
Ana
lysi
sC
hanc
e an
d U
ncer
tain
ty
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
GE
NE
RA
L O
UT
CO
ME
S, S
PE
CIF
IC O
UT
CO
ME
S A
ND
ILLU
ST
RA
TIV
E E
XA
MP
LES
toO
utlin
e K
now
ledg
e, S
kills
and
Atti
tude
s ab
out M
athe
mat
ics
.P11
% 44
PR
OO
LIM
SO
LVIN
G*
NA
TU
RE
1011
.001
1401
0011
0010
1Y,
's
olA
wis
iorf
lOrn
and- -,
s
Ilte
tiOnn
algo
kliti
o"s:
no
Con
cept
ual F
ram
ewor
k fo
r K
-12
Mat
hem
atic
s
21
-
Com
mun
icat
ion
Con
nect
ions
Est
imat
ion
and
Men
tal M
athe
mat
ics
Prob
lem
Sol
ving
Rea
soni
ngT
echn
olog
y
Vis
ualiz
atio
n
MA
TH
EM
AT
ICA
L P
RO
CE
SSE
S
The
re a
re c
ritic
al c
ompo
nent
s th
at s
tude
nts
mus
ten
coun
ter
in a
mat
hem
atic
s pr
ogra
m in
ord
er to
ach
ieve
the
goal
s of
mat
hem
atic
s ed
ucat
ion
and
to e
ncou
rage
lifel
ong
lear
ning
in m
athe
mat
ics.
Stu
dent
s ar
e ex
pect
edto
:
com
mun
icat
e m
athe
mat
ical
lyco
nnec
t mat
hem
atic
al id
eas
to o
ther
con
cept
s in
mat
hem
atic
s, to
eve
ryda
y ex
peri
ence
s an
d to
oth
erdi
scip
lines
use
estim
atio
n an
d m
enta
l mat
hem
atic
s w
here
appr
opri
ate
rela
te a
nd a
pply
new
mat
hem
atic
al k
now
ledg
e th
roug
hpr
oble
m s
olvi
ngre
ason
and
just
ify
thei
r th
inki
ngse
lect
and
use
app
ropr
iate
tech
nolo
gies
as
tool
s to
sol
vepr
oble
ms
use
visu
aliz
atio
n to
ass
ist i
n pr
oces
sing
info
rmat
ion,
mak
ing
conn
ectio
ns a
nd s
olvi
ng p
robl
ems.
The
Com
mon
Fra
mew
ork
inco
rpor
ates
thes
e se
ven
inte
rrel
ated
mat
hem
atic
al p
roce
sses
that
are
inte
nded
tope
rmea
te te
achi
ng a
nd le
arni
ng.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
5C
once
ptua
l Fra
mew
ork
for
K-1
2 M
athe
mat
ics
232"
-
Stud
ents
mus
t be
able
toco
mm
unic
ate
effe
ctiv
ely
how
an
answ
er w
asob
tain
ed
Com
mun
icat
ion
Stud
ents
nee
d to
com
mun
icat
e m
athe
mat
ical
idea
s cl
earl
yan
d ef
fect
ivel
y, o
rally
and
in w
ritin
g.
Com
mun
icat
ion
will
hel
p st
uden
ts m
ake
conn
ectio
nsam
ong
diff
eren
t rep
rese
ntat
ions
of
mat
hem
atic
al id
eas;
nam
ely,
"ph
ysic
al, p
icto
rial
, gra
phic
, sym
bolic
, ver
bal a
ndm
enta
l rep
rese
ntat
ions
." (
NC
TM
, p. 2
6)
It is
not
eno
ugh
to a
rriv
e at
an
answ
er. S
tude
nts
mus
t be
able
to c
omm
unic
ate
effe
ctiv
ely
how
the
answ
er w
asob
tain
ed. I
n ot
her
wor
ds, s
tude
nts
need
opp
ortu
nitie
s to
read
, to
expl
ore,
to in
vest
igat
e, to
wri
te, t
o lis
ten
to, t
odi
scus
s an
d to
exp
lain
idea
s in
thei
r ow
n la
ngua
ge o
fm
athe
mat
ics.
Thu
s, s
tude
nts
can
crea
te th
eir
own
links
"bet
wee
n th
eir
info
rmal
, int
uitiv
e no
tions
and
the
abst
ract
lang
uage
and
sym
bolis
m o
f m
athe
mat
ics.
" (N
CT
M, p
. 26)
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
24
NC
TM
CO
MM
UN
ICA
TIO
N S
TA
ND
AR
DS
K-4
5-8
9-12
The
stu
dy o
f m
athe
mat
ics
shoi
dd in
dude
num
erou
sop
port
uniti
es f
orco
mm
iusi
catio
n so
that
stud
ents
can
:
The
stu
dy o
f m
athe
mat
ics
shou
ld in
clud
eop
port
uniti
es to
com
mun
icat
e so
that
stud
ents
can
:
The
mat
hem
atic
scu
rric
ulum
sho
uld
incl
ude
the
cont
inue
dde
velo
pmen
t of
lang
uage
and
sym
bolis
m to
com
nutn
icat
em
athe
mat
ical
idea
s so
that
all
stud
ents
can
:
rela
te p
hysi
cal
mat
eria
ls, p
ictu
res,
and
diag
ram
s to
mat
hem
alic
alid
eas
refl
ect o
n an
dcl
arif
y th
eir
thin
king
abo
utm
athe
mat
ical
idea
s an
dsi
tuat
ions
rela
te th
eir
e. e
ryda
y la
ngua
geto
mat
hem
atic
alla
ngua
ge a
ndsy
mbo
lsre
alis
e th
atre
pres
entin
g,di
scua
sing
,re
adin
g, w
ritin
g,an
d lis
teni
ng to
mat
hem
atic
s ar
e a
vita
l par
t of
lear
ning
and
usi
ngm
athe
mat
ics.
mod
el s
ituat
ions
usin
g or
al, w
ritte
n,co
ncre
te,p
icto
rial
,gr
aphi
cal,
and
alge
brai
c m
etho
dsre
flec
t on
and
clar
ify
thei
r ow
nth
inki
ng a
bout
mat
hem
atic
al id
eas
and
situ
atio
nsde
velo
p co
mm
onun
dars
tand
ings
of
mat
hem
atic
alid
eas,
incl
udin
g th
ero
le o
f de
fini
tions
um th
e sk
ills
ofan
dire
nadi
ngte
rprv
itlin
gand
iste
toni
ng,
eval
uate
mat
hem
atic
al id
eas
disc
umm
athe
mat
ical
idem
and
mak
eco
njec
ture
s an
dco
nvin
cing
argu
men
tsap
proc
iate
the
valu
e of
mat
hem
atic
alno
tatio
n an
d its
rol
ein
the
deve
lopm
ent
of m
athe
mat
ical
idea
s.
refl
ect u
pon
and
clar
ify
thei
rth
inki
ng a
bout
mat
hem
atic
alid
eas
and
rela
tions
hips
form
ulat
em
athe
mat
ical
defi
nitio
ns a
ndex
pres
sge
nera
lizat
ions
dim
vere
d th
roug
hin
vest
igat
ions
expr
ess
mat
hem
atic
alid
eas
oral
ly a
nd in
wri
ting
read
wri
tten
pres
enta
tions
of
mat
hem
atic
s w
ithun
ders
tand
ing
ask
clar
ifyi
ng a
ndex
tend
ing
ques
tions
rel
ated
tom
athe
mat
ics
they
ham
rea
d or
baw
dab
out
appr
ecia
te th
eec
onom
y, p
ower
,an
d el
egan
ce o
fm
athe
mat
ical
nota
tion
and
itsro
le in
dm
deve
lopm
ent o
fm
atho
mat
ical
idea
s.
6
(NC
TM
, p. 2
6)(N
CT
M, p
. 78)
(NC
TM
, p. 1
40)
Con
cept
ual F
ram
ewor
k fo
r K
-12
Mat
hem
atic
s
25
-
Thr
ough
con
nect
ions
stud
ents
sho
uld
begi
n to
view
mat
hem
atic
s as
an
inte
grat
ed w
hole
.
Con
nect
ions
Stud
ents
nee
d nu
mer
ous
and
vari
ed e
xper
ienc
es in
ord
er to
appr
ecia
te th
e us
eful
ness
of
mat
hem
atic
s an
d, a
t the
sam
etim
e, to
exp
lore
con
nect
ions
with
in m
athe
mat
ics,
fro
mm
athe
mat
ics
to o
ther
dis
cipl
ines
, and
fro
m m
athe
mat
ics
toth
eir
daily
exp
erie
nces
. Whe
n m
athe
mat
ical
idea
s ar
eco
nnec
ted
to e
ach
othe
r th
roug
h co
ncre
te, p
icto
rial
and
sym
bolic
rep
rese
ntat
ions
, stu
dent
s be
gin
to v
iew
mat
hem
atic
s as
an
inte
grat
ed w
hole
.
Thi
s in
tegr
atio
n "a
llow
s st
uden
ts to
see
how
one
mat
hem
atic
al id
ea c
an h
elp
them
und
erst
and
othe
rs, a
nd it
illus
trat
es th
e su
bjec
t's u
sefu
lnes
s in
sol
ving
pro
blem
s,de
scri
bing
and
mod
elin
g re
al-w
orld
phe
nom
ena,
and
com
mun
icat
ing
com
plex
thou
ghts
and
info
rmat
ion
in a
conc
ise
and
prec
ise
man
ner.
" (N
CT
M, p
. 94)
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
26
NC
TM
CO
NN
EC
TIO
NS
STA
ND
AR
DS
K-4
5-8
9-12
The
stu
dy o
fm
athe
mat
ics
shou
ldin
clud
e op
port
uniti
esto
mak
e co
nnec
tions
so
that
stu
dent
s ca
n:
The
mat
hem
atic
scu
rric
ulum
sho
uld
incl
ude
the
inve
stig
atio
n of
mat
hem
atic
alco
nnec
tions
so
that
stud
ents
can
:
The
mat
hem
atic
scu
rric
ulum
sho
uld
incl
ude
inve
stig
atio
nof
the
conn
ectio
ns a
ndin
terp
lay
amon
gva
riou
s m
athe
mat
ical
topi
cs a
nd th
eir
appl
icat
ions
so
that
all
stud
ents
can
:
link
conc
eptu
al a
ndpr
oced
ural
know
ledg
ere
late
var
ious
repr
esen
tatio
ns o
fco
ncep
ts o
rpr
oced
ures
to o
nean
othe
rre
cogn
ize
rela
tions
hips
amon
g di
ffer
ent
topi
cs in
mat
hem
atic
sus
e m
athe
mat
ics
inot
her
curr
icul
umar
eas
use
mat
hem
atic
s in
thei
r da
ily li
ves.
see
mat
hem
atic
s as
an in
tegr
ated
who
leex
plor
e pr
oble
ms
and
desc
ribe
res
ults
usin
g gr
aphi
cal,
num
eric
al,
phys
ical
, alg
ebra
ic,
and
verb
alm
athe
mat
ical
mod
els
orre
pres
enta
tions
use
a m
athe
mat
ical
idea
to f
urth
er th
eir
unde
rsta
ndin
g of
othe
r m
athe
mat
ical
idea
sap
ply
mat
hem
atic
alth
inki
ng a
ndm
odel
ing
to s
olve
prob
lem
s th
at a
rise
in o
ther
dis
cipl
ines
,su
ch a
s ar
t, m
usic
,ps
ycho
logy
, sci
ence
,an
d bu
sine
ssva
lue
the
role
of
mat
hem
atim
in o
urcu
lture
and
soc
iety
.
reco
gniz
eeq
uiva
lent
repr
eeen
tatio
ns o
fth
e sa
me
conc
ept
rela
te p
roce
dure
sin
one
repr
esen
tatio
n to
proc
edur
es in
an
equi
vale
ntre
pres
enta
tion
use
and
valu
e th
eco
nnec
tions
am
ong
mat
hem
atic
alto
pics
use
and
valu
e th
eco
nnec
tions
betw
een
mat
hem
atic
s an
dot
her
disc
iplin
es.
7
(NC
TM
, p. 3
2)(N
CT
M, p
. 84)
(NC
TM
, p. 1
46)
Con
cept
ual F
ram
ewor
k fo
r K
-12
Mat
hem
atic
s
27
-
Men
tal m
athe
mat
ics
isth
e co
rner
ston
e fo
res
timat
ion.
Est
imat
ion
and
Men
tal M
athe
mat
ics
Stud
ents
nee
d to
kno
w w
hen
and
how
to e
stim
ate.
The
cont
ext o
f a
prob
lem
hel
ps to
det
erm
ine
whe
n it
isne
cess
ary
or d
esir
able
to h
ave
an e
xact
ans
wer
or
anes
timat
e of
that
ans
wer
. Pro
blem
con
text
s in
clud
e nu
mbe
r,pa
ttern
s an
d re
latio
ns, s
hape
and
spa
ce, a
nd s
tatis
tics
and
prob
abili
ty. T
he u
se o
f te
chno
logy
incr
ease
s th
e em
phas
ison
est
imat
ion
skill
s to
ena
ble
stud
ents
to d
eter
min
e th
ere
ason
able
ness
of
com
pute
d an
swer
s.
A v
arie
ty o
f es
timat
ion
stra
tegi
es a
ssis
ts s
tude
nts
inar
rivi
ng a
t qui
ck a
ppro
xim
atio
ns f
or e
xact
ans
wer
s.
Faci
lity
with
men
tal m
athe
mat
ics
is a
n im
port
ant o
utco
me
for
stud
ents
. A f
ocus
on
men
tal m
athe
mat
ics
forc
esst
uden
ts to
thin
k an
d im
prov
e th
eii e
ffic
ienc
y an
d ac
cura
cyin
cal
cula
ting,
incl
udin
g pe
ncil
and
pape
r ca
lcul
atio
ns.
Men
tal m
athe
mat
ics
is th
e co
rner
ston
e fo
r es
timat
ion
and
lead
s to
bet
ter
unde
rsta
ndin
g of
num
ber
conc
epts
and
num
ber
oper
atio
ns. (
Hop
e, p
p. 1
61-1
73)
Prob
lem
Sol
ving
"Pro
blem
-sol
ving
whi
ch in
clud
es th
e w
ays
in w
hich
prob
lem
s ar
e re
pres
ente
d, th
e m
eani
ngs
of th
e la
ngua
geof
mat
hem
atic
s, a
nd th
e w
ays
in w
hich
one
con
ject
ures
and
reas
onsm
ust b
e ce
ntra
l to
scho
olin
g so
that
stud
ents
can
exp
lore
, cre
ate,
acc
omm
odat
e to
cha
nged
cond
ition
s, a
nd a
ctiv
ely
crea
te n
ew k
now
ltdge
ove
r th
eco
urse
of
thei
r liv
es."
(N
CT
M, p
. 4)
Prob
lem
sol
ving
is th
e fo
cus
of m
athe
mat
ics
at a
llgr
ade
leve
ls. T
he d
evel
opm
ent o
f ea
ch s
tude
nt's
abi
lity
to s
olve
pro
blem
s is
ess
entia
l. St
uden
ts d
evel
op a
true
unde
rsta
ndin
g of
mat
hem
atic
al c
once
pts
and
proc
edur
es w
hen
they
sol
ve p
robl
ems
in m
eani
ngfu
lco
ntex
ts. P
robl
em s
olvi
ng is
to b
e em
ploy
edth
roug
hout
all
of m
athe
mat
ics
and
shou
ld b
eem
bedd
ed th
roug
hout
all
of th
e st
rand
s.
Prob
lem
sol
ving
pro
vide
s an
opp
ortu
nity
for
stu
dent
s to
be
activ
e in
con
stru
ctin
g m
athe
mat
ical
mea
ning
, to
lear
npr
oble
m-s
olvi
ng s
trat
egie
s, to
pra
ctis
e a
vari
ety
of c
once
pts
and
skill
s in
a m
eani
ngfu
l con
text
, and
to c
omm
unic
ate
mat
hem
atic
al id
eas.
Mos
t pro
blem
-sol
ving
situ
atio
ns in
the
elem
enta
ry y
ears
com
e fr
om th
e ev
eryd
ay e
xper
ienc
esof
the
stud
ent.
Stud
ents
are
abl
e to
atta
ch m
athe
mat
icai
mea
ning
to f
amili
ar a
ctiv
ities
. As
they
pro
gres
s th
roug
hsc
hool
, the
pro
blem
s be
com
e m
ore
com
plex
. The
pro
blem
sw
ill a
rise
fro
m a
n ex
plor
atio
n of
mat
hem
atic
s its
elf,
as
wel
las
fro
m th
e w
orld
aro
und
them
. Gra
dual
ly, s
tude
nts
beco
me
mor
e co
nfid
ent i
n th
eir
abili
ty to
use
and
com
mun
icat
e m
athe
mat
ics,
usi
ng c
orre
ct te
rmin
olog
y.
Prob
lem
sol
ving
is th
efo
cus
of m
athe
mat
ics
atal
l gra
de le
vels
.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
8C
once
ptua
l Fra
mew
ork
for
K-1
2 M
athe
mat
ics
;?..b
29
-
As
stud
ents
dev
elop
mat
hem
atic
ally
, the
y ar
e ab
le to
sol
vem
ore
chal
leng
ing
prob
lem
s on
an
incr
easi
ng v
arie
ty o
fto
pics
. Stu
dent
s ne
ed th
e op
port
unity
"to
sol
ve p
robl
ems
that
req
uire
them
tc w
ork
coop
erat
ivel
y (a
nd in
divi
dual
ly),
to u
se te
chno
logy
, to
addr
ess
rele
vant
and
inte
rest
ing
mat
hem
atic
al id
eas,
and
to e
xper
ienc
e th
e po
wer
and
usef
ulne
ss o
f m
athe
mat
ics.
" (N
CT
M, p
p. 7
5-76
) B
y th
etim
e st
uden
ts r
each
the
seco
ndar
y le
vel,
man
y pr
oble
m-
solv
ing
stra
tegi
es s
houl
d be
inte
rnal
ized
and
pro
blem
solv
ing
shou
ld b
e a
proc
ess
for
cons
truc
ting
and
rein
forc
ing
mat
hem
atic
al c
once
pts.
Stud
ents
sho
uld
be c
onfi
dent
and
fle
xibl
e pr
oble
mso
lver
s, u
sing
a w
ide
rang
e of
str
ateg
ies
in th
eir
wor
k,an
d ac
cept
that
som
e pr
oble
ms
have
dif
fere
ntso
lutio
ns.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
30
NC
TM
PR
OB
LE
M-S
OL
VIN
G S
TA
N6A
RD
S
K-4
5-8
9-12
The
stu
dy o
fm
athe
mat
ics
shou
ldem
phas
ize
prob
lem
solv
ing
so th
atst
uden
ts c
an:
The
mat
hem
atic
s'
curr
icul
um s
houl
din
clud
e nu
mer
ous
and
vari
ed e
zper
ienc
es w
ithpr
oble
m s
olvi
ng a
s a
met
hod
of in
quir
y, a
ndap
plic
atio
n so
that
stud
ents
can
:
The
mat
hem
atic
scu
rric
ulum
sho
uld
incl
ude
the
refi
nem
ent
and
exte
nsio
n of
met
hods
of
mat
hem
atic
al p
robl
emso
lvin
g so
that
all
stud
ents
can
:
use
prob
lem
-so
lvin
g ap
proa
ches
to in
vest
igat
e an
dun
ders
tand
mat
hem
atic
alco
nten
tfo
rmul
ate
prob
lem
sfr
om e
very
day
and
mat
hem
atic
alsi
tuat
ions
dcve
lop
end
appl
yst
rate
gies
to s
olve
aw
ide
vari
ety
ofpr
oble
ms
veri
fy a
ndin
terp
ret r
esul
tsw
ith r
espe
ct to
the
orig
inal
pro
blem
acqu
ire
conf
iden
cein
usi
ngm
athe
mat
ics
mea
ning
fully
.
use
prob
lem
-sol
ving
appr
oach
es to
inve
stig
ate
and
unde
rsta
ndm
athe
mat
ical
cont
ent
form
ulat
e pr
oble
mfr
om s
ituat
ions
with
in a
nd o
utai
dem
athe
mat
ics
deve
lop
and
appl
y a
vari
ety
of s
trat
egie
sto
sol
ve p
robl
ems,
with
em
phas
is o
nm
ultis
tep
and
nonr
outin
epr
oble
ms
veri
fy a
nd in
terp
ret
resu
lts w
ith r
espe
ctto
the
orig
inal
prob
lem
situ
atio
nge
nera
lize
solu
tions
and
stra
tegi
es to
new
pro
blem
situ
atio
nsac
quir
e co
nfid
ence
in u
sing
mat
hem
atic
sm
eani
ngfu
lly.
ure,
with
incr
easi
ngco
nfid
ence
,pr
oble
m-s
olvi
ngap
proa
ches
toin
vest
igat
e an
dun
ders
tand
mat
hem
atic
alco
nten
tap
ply
inte
grat
edm
athe
mat
ical
prob
lem
-sol
ving
stra
tegi
es to
sol
vepr
oble
ms
from
with
in a
nd o
utri
dem
athe
mat
ics
reco
gniz
e an
dfo
rmul
ate
prob
lem
sfr
om s
ituat
ions
with
in a
nd o
utai
dem
athe
mat
ics
appl
y th
e pr
oces
s of
mat
hem
atic
alm
odel
ing
tore
al-w
orld
pro
blem
situ
atio
ns.
(NC
TM
, p. 2
3)(N
CT
M, p
. 75)
(NC
TM
, p. 1
37)
9C
once
ptua
l Fra
mew
ork
for
K-1
2 M
athe
mat
ics
31
-
Rea
soni
ng h
elps
stu
dent
sto
mak
e se
nse
ofm
athe
mat
ks a
nd to
be
logi
cal i
n th
eir
thin
king
.
Rea
soni
ng
Stud
ents
nee
d to
dev
elop
con
fide
nce
in th
eir
abili
ty to
reas
on a
nd to
just
ify
thei
r th
inki
ng w
ithin
and
out
side
of
mat
hem
atic
s. T
he p
ower
of
reas
onin
g he
lps
stud
ents
tom
ake
sens
e of
mat
hem
atic
s, to
be
logi
cal i
n th
eir
thin
king
,an
d to
con
vinc
e ot
hers
.
Indu
ctiv
e re
ason
ing
help
s st
uden
ts e
xplo
re a
nd m
ake
conj
ectu
res
from
act
iviti
es th
at a
llow
gen
eral
izat
ions
fro
ma
patte
rn o
f ob
serv
atio
ns.
Ded
uctiv
e re
ason
ing
help
s st
uden
ts te
st c
onje
ctur
es a
ndbu
ild a
rgum
ents
that
ser
ve to
val
idat
e th
inki
ng.
Ded
uctiv
e re
ason
ing
build
s a
stru
ctur
ed b
ody
ofkn
owle
dge.
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
32
NC
TM
RE
ASO
NIN
G S
TA
ND
AR
DS
K-4
5-8
9-12
The
stu
dy o
fm
athe
mat
ics
shou
ldem
phas
ize
reas
onin
gso
that
stu
dent
s ca
n:
Rea
soni
ng s
hall
perm
eate
the
mat
hem
atic
scu
rric
ulum
so
that
stud
ents
can
:
The
mat
hem
atic
scu
rric
ulum
sho
uld
incl
ude
num
erou
s an
dva
ried
expe
rienc
es th
atre
info
rce
and
exte
ndlo
gica
l rea
soni
ng s
kills
so th
at a
ll st
uden
ts
draw
logi
cal
conc
lusi
ona
abou
tm
athe
mat
ics
use
mod
els,
kno
wn
fact
s, p
rope
rtie
s,an
d re
latio
nshi
psto
exp
lain
thei
rth
inki
ngju
stif
y th
eir
answ
ers
and
solu
tion
proc
esse
sus
e pa
ttern
s an
dre
latio
nshi
ps to
anal
yze
mat
hela
atic
alsi
tuat
ions
belie
ve th
atm
athe
mat
ics
mak
es s
ense
.
reco
gniz
e an
d ap
ply
dedu
ctiv
e an
din
duct
ive
reas
onin
gun
ders
tand
and
appl
y re
ason
ing
proc
esse
s, w
ithsp
ecia
l atte
ntio
n to
spat
ial r
easo
ning
and
reas
onin
g w
ithpr
opor
tions
and
grap
hsm
ake
and
eval
uate
mat
hem
atic
alcn
njec
toir
es a
ndar
guni
enta
valid
ate
thei
r ow
nth
inki
ngap
prec
iate
the
perv
asiv
e us
e an
dpo
wer
of
reas
onin
gas
a p
art o
fm
athe
mat
ics.
mak
e an
d te
st
fetO
rmnj
ectu
rula
:
coun
tere
xam
ples
follo
w lo
gica
lar
gum
ents
judg
e th
e va
lidity
of a
rgum
ents
cons
truc
t sim
ple
valid
arg
umen
ts.
10
(NC
TM
, p. 2
9)(N
CT
M, p
. 81) 33
(NC
TM
, p. 1
43)
Con
cept
ual F
ram
ewor
k fo
r K
-12
Mat
hem
atic
s
-
Tec
hnol
ogy
will
aid
stud
ents
in s
olvi
ngco
mpl
ex p
robl
ems.
Tec
hnol
ogy
Impr
ovem
ents
in te
chno
logy
, and
its
incr
ease
d av
aila
bilit
yin
sch
ools
, hav
e ch
ange
d th
e fo
cus
of m
athe
mat
ics
educ
atio
n. T
he ti
me
save
d by
usi
ng c
alcu
lato
rs o
rco
mpu
ters
to p
erfo
rm c
ompl
exca
lcul
atio
ns c
an b
e us
ed to
help
stu
dent
s be
tter
unde
rsta
nd m
athi
mat
ical
con
cept
s.St
uden
ts c
an th
en u
nder
stan
d th
e re
latio
nshi
ps a
mon
gco
ncep
ts a
nd u
se th
ese
rela
tions
hips
toso
lve
prob
lem
s.
Cal
cula
tors
and
com
pute
rs c
an b
e us
ed a
s to
ols
to:
deve
lop
conc
epts
expl
ore
and
dem
onst
rate
mat
hem
atic
al r
elat
ions
hips
and
patte
rns
orga
nize
and
dis
play
dat
aas
sist
with
sol
ving
pro
blem
s an
d th
us p
rom
ote
inde
pend
ence
enco
urag
e st
uden
ts to
be in
quis
itive
and
cre
ativ
ede
crea
se th
e tim
e sp
ent o
n te
diou
s co
mpu
tatio
nsre
info
rce
the
lear
ning
of
basi
c nu
mbe
r fa
cts
and
prop
ertie
sde
velo
p an
und
erst
andi
ng o
f co
mpu
tatio
nal a
lgor
ithm
scr
eate
geo
met
ric
disp
lays
sim
ulat
e si
tuat
ions
.
In s
ome
case
s, te
chno
logy
will
allo
w te
ache
rs to
ask
ques
tions
req
uiri
ng a
hig
h le
velo
f th
inki
ng a
nd w
ill a
llow
stud
ents
to s
olve
com
plex
, mul
tifac
eted
pro
blem
s.T
echn
olog
y ca
n fo
ster
env
iron
men
ts in
whi
ch th
e gr
owin
gcu
rioe
ity o
f st
uden
ts c
an le
ad to
rich
mat
hem
atic
aldi
scov
erie
s. I
n th
ese
envi
ronm
ents
, the
con
trol
of
expl
orin
g m
athe
mat
ical
idea
s ca
n be
turn
ed o
ver
tost
uden
ts.
Vis
ualiz
atio
n
Vis
ualiz
atio
n "i
nvol
ves
thin
king
in p
ictu
res
and
imag
esan
d th
e ab
ility
to p
erce
ive,
tran
sfor
m a
nd r
e-cr
eate
diff
eren
t asp
ects
of
the
visu
al-s
patia
l wor
ld."
(A
rmst
rong
,p.
10,
ital
ics
in o
rigi
nal)
The
use
of
imag
es in
the
stud
y of
mat
hem
atic
s pr
ovid
es s
tude
nts
with
the
oppo
rtun
ity to
unde
rsta
nd m
athe
mat
ical
con
cept
s an
d to
mak
eco
nnec
tions
am
ong
them
.
The
phy
sica
l env
iron
men
t is
full
of im
ages
. The
imag
es a
reof
3-D
obj
ects
, 2-D
sha
pes,
1-D
line
s an
d pi
ctur
es. I
nge
omet
ry, t
he s
tudy
of
a 3-
D o
bjec
t isa
ssis
ted
byvi
sual
izin
g ei
ther
the
net o
f 2-
D s
hape
s or
the
skel
eton
of
1-D
line
s re
quir
ed to
con
stru
ct th
e ob
ject
.
The
mat
hem
atic
al e
nvir
onm
ent i
s fu
ll of
imag
es. T
hese
imag
es a
re u
sed
to c
omm
unic
ate
mat
hem
atic
al c
once
pts
and
mul
tiple
sol
utio
ns to
pro
blem
s. A
t an
elem
enta
ryle
vel,
four
pile
s, e
ach
cont
aini
ng th
ree
coin
s, c
anbe
use
d to
repr
esen
t 3 +
3 +
3 +
3 =
12. R
earr
angi
ngth
e pi
les
into
fou
rro
ws
of 3
can
then
be
used
to r
epre
sent
4X
3 =
12.
Con
nect
ing
the
two
imag
es li
nks
the
proc
ess
ofm
ultip
licat
ion
with
that
of
repe
ated
add
ition
.A
t a m
ore
adva
nced
leve
l, an
alyt
ic g
eom
etry
des
crib
es f
igur
esal
gebr
aica
lly a
nd p
rovi
des
a to
ol f
or th
evi
sual
izat
ion
ofal
gebr
aic
rela
tions
. The
ana
lysi
s an
d in
terp
reta
tion
ofda
ta u
sing
a v
isua
l sum
mar
y ai
ds in
und
erst
andi
ng th
eda
ta a
nd m
akin
g pr
edic
tions
fro
m it
.
Imag
es a
re u
sefu
l in
desc
ribi
ng th
e ph
ysic
alan
d m
athe
mat
ical
envi
ronm
ent
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
11C
once
ptua
l Fra
mew
ork
for
K-1
2 M
athe
mat
ics 35
34
-
Cha
nge
Con
stan
cyD
imen
sion
Num
ber
Patte
rnQ
uant
ityR
elat
ions
hips
Shap
eU
ncer
tain
ty
Cha
nge
is a
ver
y br
oad
conc
ept.
Stud
ents
mus
tbe
com
e se
nsiti
ve to
patte
rns,
suc
h as
line
ar,
expo
nent
ial,
loga
rith
mic
and
peri
odic
.
NA
TU
RE
OF
MA
TH
EM
AT
ICS
By
enri
chin
g ou
r vi
ew o
f m
athe
mat
ics
and
the
lear
ning
envi
ronm
ent,
the
outc
omes
of
The
Com
mon
Fra
mew
ork
can
be a
ccom
plis
hed.
The
bra
in is
con
stan
tly lo
okin
g fo
r an
d m
akin
gco
nnec
tions
. "B
ecau
se th
e le
arne
r is
con
stan
tly s
earc
hing
for
conn
ectio
ns o
n m
any
leve
ls, e
duca
tors
nee
d to
orch
estr
ate
the
expe
rien
ces
from
whi
ch le
arne
rs e
xtra
ctun
ders
tand
ing.
. . .
Bra
in r
esea
rch
esta
blis
hes
and
conf
irm
s th
at m
ultip
le c
ompl
ex a
nd c
oncr
ete
expe
rien
ces
are
esse
ntia
l for
mea
ning
ful l
earn
ing
and
teac
hing
."(C
aine
, p. 5
)
The
re a
re a
dditi
onal
cri
tical
com
pone
nts
that
mus
t be
addr
esse
d in
a m
athe
mat
ics
prog
ram
bey
ond
thos
e lis
ted
asm
athe
mat
ical
pro
cess
es. T
he c
ompo
nent
s di
scus
sed
are:
Patte
rn, N
umbe
r, S
hape
, Cha
nge,
Con
stan
cy, D
imen
sion
(siz
e an
d sc
ale)
, Rel
atio
nshi
ps, Q
uant
ity a
nd U
ncer
tain
ty.
The
y ar
e us
ed to
des
crib
e m
athe
mat
ics
in a
bro
ad w
ay in
orde
r to
est
ablis
h th
e w
ide
vari
ety
of c
onne
ctio
ns th
at c
anbe
mad
e am
ong
the
vari
ous
stra
nds
used
to o
rgan
ize
the
outc
omes
cen
tral
to T
he C
omm
on F
ram
ewor
k.
Cha
nge
Cha
nge
can
be d
iscu
ssed
fro
m K
inde
rgar
ten
to G
rade
12
acro
ss m
any
aspe
cts
of m
athe
mat
ics.
The
stu
dy o
f ch
ange
is o
ften
dis
cuss
ed in
the
cont
ext o
f ca
lcul
us, b
ut is
oft
enlim
ited
to th
is c
onte
xt. H
owev
er, c
hang
e is
a m
uch
broa
der
conc
ept t
han
that
use
d in
cal
culu
s. I
n or
der
to m
ake
pred
ictio
ns, s
tude
nts
need
to d
escr
ibe
and
quan
tify
thei
r
Wes
tern
Can
adia
n Pr
otoc
ol, J
une
1995
3 6
obse
rvat
ions
, atte
mpt
to b
uild
pat
tern
s, a
nd id
entif
y th
ose
quan
titie
s th
at r
emai
n fi
xed
and
thos
e qu
antit
ies
that
chan
ge. F
or e
xam
ple,
look
at t
he p
atte
rn 4
, 6, 8
, 10,
12,
.. .
An
elem
enta
ry s
choo
l stu
dent
can
des
crib
e th
is a
s sk
ipco
untin
g by
2s,
sta
rtin
g fr
om 4
. A s
enio
r hi
gh s
choo
lst
uden
t may
des
crib
e th
is p
atte
rn a
s an
ari
thm
etic
sequ
ence
, with
fir
st te
rm 4
, and
a c
omm
on d
iffe
renc
e of
2.
Ano
ther
stu
dent
may
des
crib
e it
as a
line
ar f
unct
ion
with
adi
scre
te d
omai
n. A
ll th
ree
inte
rpre
tatio
ns a
re f
ocus
ing
onth
e ch
angi
ng s
ize
of th
e nu
mbe
rs w
ithin
the
sequ
ence
. To
be a
ble
to u
nder
stan
d ch
ange
, stu
dent
s m
ust b
ecom
ese
nsiti
ve to
pat
tern
s, s
uch
as li
near
, exp
onen
tial,
loga
rith
mic
and
per
iodi
c. (
Stee
n, p
. 184
)
Con
stan
cy
Stud
ents
are
exp
ecte
d to
com
mun
icat
e id
eas
visu
ally
, usi
ngdi
agra
ms
and
oral
and
wri
tten
wor
ds, w
hen
desc
ribi
ngco
nsta
ncy
or in
vari
ance
. Dif
fere
nt a
spec
ts o
f co
nsta
ncy
"are
des
crib
ed b
y th
e te
rms
stab
ility
, con
serv
atio
n,eq
uilib
rium
, ste
ady
stat
e, a
nd s
ymm
etry
." (
AA
AS-
Ben
chm
arks
, p. 2
70)
The
mos
t im
port
ant p
rope
rtie
s in
mat
hem
atic
s an
d sc
ienc
e re
late
to th
ose
prop
ertie
s th
at d
ono
t cha
nge
whe
n ou
tsid
e co
nditi
ons
chan
ge. E
lem
enta
rysc
hool
stu
dent
s de
al w
ith c
onst
ancy
in s
ituat
ions
whe
redi
ffer
ent m
etho
ds a
re u
sed
to s
olve
a s
ingl
e m
ultip
licat
ion
prob
lem
, suc
h as
fm
ding
the
area
of
a 3-
tile
by 4
-tile
tabl
etop
. Sec
onda
ry s
tude
nts
need
to d
eal w
ith c
onst
ancy
whe
n th
ey s
olve
the
mor
e co
mpl
icat
ed m
ultip
licat
ion
prob
lem
s th
at a
ppea
r in
det
erm
inin
g th
e nu
mbe
r of
elem
ents
pre
sent
in th
e sa
mpl
e sp
aces
of
prob
abili
typr
oble
ms.
Man
y of
thes
e si
tuat
ions
will
invo
lve
perm
utat
ions
and
com
bina
tions
.
Con
stan
cy is
des
crib
edby
the
term
s st
abili
ty,
cons
erva
tion,
equi
libri
um, s
tead
y st
ate
and
sym
met
ry.
12C
once
ptua
l Fra
mew
ork
for
K-1
2 M
athe
mat
ics
37
-
The
con
cept
of
dim
ensi