CK : Content Knowledge PK : Pedagogical Knowledge PCK : Pedagogical Content Knowledge LK : knowledge of Learners and their Characteristics .

()CKPKPCK LKShulman (1986)VVVShulman (1987)VVVVVVVTamir (1991)VVVVVVReynolds (1992)VVVV(l996)VVVV(2000)VVV(2002)VVVVV(2003)VVVVV(2006)VVVVVV

Fennema & Franke1992, p.162

MK : Knowledge of Mathematics PK : Pedagogical Knowledge MTK : Knowledge of Mathematics Teaching LK : Knowledge of Learners Cognitions in Mathematics .MTKMTK

MKPKMTKLMKFennema Franke(1992)VVVVThompson (1992)VVCochran, DeRuiterKing (1993)VVVV 2001VV2000VVV2003VVVVVV2004VVVVVVV2006VVVVVVV

MKTMathematical Knowledge for Teaching

From Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students., by Hill et al., 2008, Journal for Research in Mathematics Education, 39(4), 377.

AAMTStandards for Excellence in Teaching Mathematics in Australian SchoolsSETM10NCATEStandards for Elementary Mathematics SpecialistsSEMS14INTASCModel Standards for Beginning Teachers of MathematicsMSBTM10NBPTS715Middle Childhood through Early Adolescence / Mathematics StandardsMCEAMS12()NCETMTDATDA Professional Standards for Mathematics TeachersTDAPSMT33()

2012203217-239

A.A-1 2A-2 4A-3 3A-4 3B.B-1 2B-2 2B-3 2B-4 2C.C-1 2C-2 2C-3 2

1. 2. 12342012203217-239()

A-1 A-1-1 3~4444A-1-2 3444A-2 A-2-1 1~32~444A-2-2 2~32~43~44A-2-3 2~33~444A-2-4 2344A-3 A-3-1 2344A-3-2 3444A-3-3 2344A-4 A-4-1 233~44A-4-2 2~33~444A-4-3 2~33~444

()1. 2. 12342012203217-239

B-1 B-1-1 1344B-1-2 1444B-2 B-2-1 1344B-2-2 1344B-3 B-3-1 12~344B-3-2 44B-4 B-4-1 12~444B-4-2 114

()1. 2. 12342012203217-239

C-1C-1-1 2~3444C-1-2 144C-2C-2-1 2~33~44C-2-2 2~34C-3C-3-1 2~344C-3-2 4

(professional dialogue)(curriculum development)(peer supervision)(peer coaching)(action research)

Types of Cooperative Development (Glatthorn, 1987)

TypeFocusDistinguishing features1. Professional dialogueReflection about practiceGuided discussion, focusing on teaching as thinking2. Curriculum developmentProduction of materialsCollaborative development of curriculum, using naturalistic processes3. Peer supervisionAnalysis of teachingObservation of instruction, followed by analysis and feedback4. Peer coachingMastery of skillsDevelopment of specific skills, usually based on models of teaching and supported with staff development5. Action researchSolving of problemsDevelopment and implementation of feasible solution to teacher-identified problems

Isaacs (1999) --Listening --Suspending --Respecting --Voicing

Glatthorn1987

Scharmer (1998) --/politeness (Talking nice)/breakdown (Talking tough) /inquiry (Reflective dialogue)/flow (Generative dialogue)

1.2.3.

based on a 5-point scale 4.94.64.6

Scharmer study (1998)/ Politeness/Breakdown/Inquiry /Flow

vs.1.2.3.

based on a 5-point scale 4.64.5

109/21209/28310/05410/12510/19610/26711/02811/09911/16

A / B

Stage Date109/21AB209/28ABB310/05AB410/12AB

Stage Date510/19AB610/26ABB711/02AB811/09ABB911/16AB

Level 0 Kimberly H.A., Karen C. F.,& Miriam G. S.(2004) .Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education. 35(2), 81-116.

A. QuestingB. Explaining mathematical thinkingC. Source of mathematical ideasD. Responsibility for learning

Level 1 Kimberly H.A., Karen C. F.,& Miriam G. S.(2004) .Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education. 35(2), 81-116.()

A. QuestingB. Explaining mathematical thinkingC. Source of mathematical ideasD. Responsibility for learning()

Level 2 Kimberly H.A., Karen C. F.,& Miriam G. S.(2004) .Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education. 35(2), 81-116.()

A. QuestingB. Explaining mathematical thinkingC. Source of mathematical ideasD. Responsibility for learning

Level 3 Kimberly H.A., Karen C. F.,& Miriam G. S.(2004) .Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education. 35(2), 81-116.()

A. QuestingB. Explaining mathematical thinkingC. Source of mathematical ideasD. Responsibility for learning()

2010-312-333

2010-312-333

Level 0Level 1Level 2Level 3Level 4

Level 0 Level 1 ()

Level 2 ()

Level 3 ()

Level 4 ()

1. 2. 3.

1.2.3.

1.2. 3.4.

problem-based learning [PBL] 3.4.5.6.

1.2.

2009

/20075223-47

765

2009(N=85)

N35N16N34MinMaxMSMinMaxMSMinMaxMS354.46.561254.44.892254.35.734354.34.591354.50.632254.47.706254.31.676354.13.719254.35.812254.17.618354.00.730254.29.871253.91.702253.94.929253.88.880253.77.770253.81.981253.41.821153.631.003253.50.894253.53.929253.80.833253.441.031253.50.788253.94.725354.00.632253.94.776354.34.591354.25.577354.32.684254.37.731354.31.602354.29.719354.26.701354.31.602354.21.687354.34.725354.44.629254.56.705254.14.879354.13.806254.29.760254.00.939153.941.181153.941.043253.97.822254.00.816254.09.933153.74.980153.751.183153.681.007153.741.010253.56.964153.531.022

--

01N3425N386N13MinMaxMSMinMaxMSMinMaxMS254.41.783254.42.642354.38.650354.35.646254.45.686454.54.519354.35.597254.16.886454.54.519354.21.687254.11.863454.38.506254.09.712253.79.905353.77.725253.76.781253.55.828253.541.050153.591.019253.55.921253.54.877253.62.985253.63.786253.54.776254.00.778253.84.638354.15.801354.35.646354.26.644454.38.506354.38.652254.24.751354.46.660354.26.618354.21.741354.31.630354.41.609354.50.688254.38.961254.03.904354.26.760354.46.660154.031.029153.891.034254.001.000253.97.969253.95.769354.38.768153.651.098153.681.042354.00.707153.381.101153.76.971353.85.689

2009.06217-239

/20075223-47

-/M.

M.M-0M-0-1M-1M-1-1M-1-2 M-2M-2-1 M-2-2 M-2-3 M-2-4 M-3M-3-1 M-3-2 M-3-3 M-4M-4-1 M-4-2 M-4-3

-/A./T.

A.A-0A-0-1A-1A-1-1 A-1-2 A-2A-2-1A-2-2 A-3A-3-1 A-3-2 A-4A-4-1 A-4-2 T.T-0T-0-1T-1T-1-1 T-1-2 T-2T-2-1T-2-2 T-3T-3-1 T-3-2

-/H.

H.H-0H-0-1H-1 H-1-1H-1-2H-2H-2-1H-2-2H-2-3H-3H-3-1H-3-2H-3-3H-4H-4-1H-4-2

2010(N=84) M.A.T.H.

N=39N=16N=29MinMaxMSMinMaxMSMinMaxMSM.354.28.605354.19.834354.59.568454.69.468454.63.500454.83.384454.64.486354.63.619354.52.574354.64.537254.31.873354.41.628354.59.549354.44.629354.55.632454.74.442254.50.894454.79.412454.62.493354.63.619354.59.568454.64.486454.56.512354.52.574354.59.595354.63.619354.38.622354.79.469454.75.447354.55.572454.64.486454.63.500354.48.688454.64.486254.44.814354.62.561454.56.502154.381.025354.59.568

M.A.T.H.

N=39N=16N=29MinMaxMSMinMaxMSMinMaxMSA.254.23.667154.001.095354.24.577354.41.637153.941.181354.17.711354.51.601354.50.632354.55.572354.15.670154.001.033253.93.842354.31.694354.38.619354.34.670354.69.521454.63.500454.76.435253.85.587153.871.088254.03.731254.15.630354.19.655354.45.632354.62.590354.37.719354.69.604T.353.95.686153.941.181353.93.593354.33.577154.001.317354.10.673354.41.677354.44.629254.24.786354.64.537254.06.772254.48.738354.59.595354.37.619354.48.574454.51.506354.31.704354.28.702354.54.600454.50.516354.28.751

M.A.T.H.

N=39N=16N=29MinMaxMSMinMaxMSMinMaxMSH.354.03.628253.94.854353.93.651354.51.556254.37.885354.52.634454.69.468454.75.447354.61.567454.62.493154.371.025354.34.670454.49.506154.371.025354.38.677354.74.498454.56.512354.62.677354.46.643154.251.065354.55.632354.13.656154.061.124254.28.702354.41.595154.191.109354.38.677454.62.493154.631.025354.62.622354.21.615154.311.078354.34.670

M.A.T.H.

01N=3425N=316N=19MinMaxMSMinMaxMSMinMaxMSM.354.26.666354.48.626354.37.684454.65.485454.71.461454.89.315354.53.563454.61.495354.68.582354.41.657254.52.677354.63.597354.50.615354.55.568354.63.597354.65.544254.74.631454.79.419354.41.609454.74.445454.74.452454.59.500454.55.506354.63.597354.47.563354.52.626354.63.684354.71.524454.74.445354.63.597354.44.613354.65.551454.74.452454.59.500254.55.675354.68.582454.62.493154.42.807354.58.607

M.A.T.H.

01N=3425N=316N=19MinMaxMSMinMaxMSMinMaxMSA.254.12.686154.19.792354.32.749254.24.781154.23.845354.26.806354.44.613354.55.624454.63.496253.97.717153.97.983354.32.582354.32.727354.32.653354.37.597354.65.544454.74.445454.74.452253.91.753153.77.805354.16.602254.09.712354.26.575454.58.507354.47.748454.71.461354.63.597T.153.76.890254.00.632354.16.688154.12.844154.29.824354.16.688254.18.797354.52.570354.42.692254.35.734254.52.724454.63.496354.44.613354.55.568354.58.607354.32.638354.42.620354.47.612354.47.662354.42.672354.42.607

M.A.T.H.

01N=3425N=316N=19MinMaxMSMinMaxMSMinMaxMSH.353.94.694253.97.706354.05.621354.47.615254.52.677354.47.697454.68.475454.65.486354.72.575354.53.563154.39.844354.53.612354.41.557154.35.839354.58.607354.62.604354.74.514354.63.597354.38.652154.42.886354.63.597354.15.702154.13.885354.26.733354.44.660154.29.824354.32.749354.65.544154.58.807354.63.597354.32.684154.16.820354.37.684

M.A.T.H.

01N=5423N=223N=8MinMaxMSMinMaxMSMinMaxMSM.354.41.659354.23.685454.50.535454.81.392454.50.512454.75.463354.69.507354.36.581454.63.518254.54.665354.36.658454.63.518354.69.507354.18.664454.63.518254.74.589454.68.477454.63.518354.61.564454.59.503454.63.518454.65.482354.45.596454.50.535354.59.567354.32.716454.63.518454.72.452354.64.658454.75.463354.61.596454.55.510454.50.535254.69.577354.36.581454.63.518154.59.687354.36.581454.63.518

M.A.T.H.

01N=5423N=223N=8MinMaxMSMinMaxMSMinMaxMSA.154.28.763254.00.756454.13.354154.20.786254.27.935454.38.518354.57.570354.41.666454.50.535154.04.889354.05.653354.13.641354.35.677354.27.703454.38.518454.74.442354.59.590454.75.463153.83.818354.00.617454.25.463254.22.634354.27.703454.50.535354.61.627354.45.671454.88.354T.153.89.816354.05.722354.00.535154.15.899354.27.550354.25.707254.39.712354.41.666354.00.756254.44.744354.41.590454.88.354354.46.605354.59.590454.63.518354.39.627354.41.590354.38.744354.39.685354.50.598454.62.518

M.A.T.H.

01N=5423N=223N=8MinMaxMSMinMaxMSMinMaxMSH.253.94.685354.14.640353.75.707254.41.714454.64.492454.63.518354.70.503454.64.492454.62.518154.44.744354.59.590454.38.518154.43.742354.45.596454.38.518354.72.492354.59.666354.50.756154.50.771354.36.727454.38.518154.17.841354.09.684454.38.518154.31.820354.45.596454.38.518154.63.734454.59.503454.63.518154.24.799354.32.646454.38.518

***1.Isaacs Listening Suspending Respecting Voicing2.Scharmer differ in terms of (1) the presence or absence of self-reflection and (2) the primacy of wholeness or parts Self-reflectiveNon-reflectivePartsWhole / / / / This teaching learning group developed their dialogue behavior based on Isaacs four basic dialoguing behaviors: listening, suspending, respecting, and voicing (Isaacs, 1999), and Scharmers four developmental stages of professional dialogues:Politeness,Breakdown, Inquiry, Flow (Scharmer, 1998). (Isaacs1999ListeningSuspendingRespectingVoicingScharmer1998/politenessTalking nice/breakdownTalking tough/inquiryReflective dialogue/flowGenerative dialogue )

*Theoretical Side--Reading Discussion Videos

Practical Side/ Actual Teaching Problems--Classroom Teaching Textbook

ReflectingSharing

Professional GrowthImproving Teaching**