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DOCUMENT RESUME ED 394 793 SE 058 057 TITLE The Common Curriculum Framework for K-12 Mathematics: Western Canadian Protocol for Collaboration in Basic Education. INSTITUTION Alberta Dept. of Education, Edmonton. REPORT NO ISBN-0-7732-1780-0 PUB DATE Jun 95 NOTE 593p. PUB TYPE Guides Non-Classroom Use (055) EDRS PRICE MF03/PC24 Plus Postage. DESCRIPTORS *Educational Objectives; Elementary Secondary Education; Foreign Countries; *Mathematics Curriculum; Mathematics Instruction; *Standards IDENTIFIERS *Canada ABSTRACT This document provides a common base for the K-9 (grades 10-12 are still under development) mathematics curriculum expectations mandated by the western Canadian provinces and territories of Manitoba, Saskatchewan, Alberta, British Columbia, Yukon Territory, and the Northwest Territories. The framework identifies beliefs about mathematics, general and specific student outcomes, and illustrative examples agreed upon by these six jurisdictions. The following aspects are discussed: (1) mathematical processes based on the standards of the National Council of Teachers of Mathematics: communication, connections, estimation and mental mathematics, problem solving, reasoning, technology, and visualization; (2) nature of mathematics: change, constancy, dimension, number, pattern, quantity, relationships, shape, and uncertainty; (3) strands: number, patterns and relations, shape and space, and statistics and probability; and (4) student expectations. The majority of the document contains general outcomes and specific outcomes with illustrative examples for each grade from kindergarten through grade 9. (MKR) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ***********************************************************************
Transcript
  • 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.

    ***********************************************************************

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    16

    stud

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    172

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  • Posi

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    n ap

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    and

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    Wes

    tern

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    adia

    n Pr

    otoc

    ol, J

    une

    1995

    is3

    19In

    trod

    uctio

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  • III.

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    icat

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    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


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