teachers’ learning of mathematics in the presence of technology: participatory cognitive...
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Teachers’ Learning of Mathematics in the Presence of Technology: Participatory Cognitive Apprenticeship. Mara Alagic Wichita State University. current school reform for standards-based teaching that supports integration of technology (NCTM, 2000) development of the IC technologies - PowerPoint PPT PresentationTRANSCRIPT
Mara Alagic SITE 2002 1
Teachers’ Learning of Mathematics in the
Presence of Technology:Participatory Cognitive
Apprenticeship
Mara AlagicWichita State University
Mara Alagic SITE 2002 2
Paradigm shift: Changes in the teaching/learning of the (school) mathematics introduced by • current school reform for standards-
based teaching that supports integration of technology (NCTM, 2000)
• development of the IC technologies• information from cognitive sciences
about the nature of human learning
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Designing Mathematics Learning Environments
• address the needs of an increasingly complex student body
• recognize need for standards-based teaching/learning
• “balance” availability and the development of the IC technologies
• understand the nature and development of long term memory
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Teachers’ beliefs• Teachers’ beliefs about teaching
and learning of mathematics & integration of technology shape ...
• Using ongoing assessment of student readiness, interests, and learning profile teachers (can) differentiate content, process, products, and/or environment (Tomlinson, 1999)
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Goal
Interventions
TO IMPROVE
TO IMPROVE
TECHNOLOGY BASED LEARNING
ENVIRONMENT
DIFFERENTIATING INSTRUCTION
TEACHERS'PEDAGOGICAL
CONTENT KNOWLEDGE
STUDENTS LEARNING
DIFFERENTIATING MATHEMATICS INSTRUCTION
THROUGH TECHNOLOGY
CI 752RIDEAS FOR
INTERVENTIONS
CI 752RGOAL
Mara Alagic SITE 2002 6
What to differentiate
?
TO IMPROVE
TO IMPROVE
DIFFERENTIATING INSTRUCTION
TEACHERS'PEDAGOGICAL
CONTENT KNOWLEDGE
STUDENTS LEARNING
CONTENT
PROCESS
PRODUCT
CI 752RIDEAS FOR
INTERVENTIONS
CI 752RGOAL
TECHNOLOGY BASED LEARNING
ENVIRONMENT
DIFFERENTIATING MATHEMATICS INSTRUCTION
THROUGH TECHNOLOGY
Mara Alagic SITE 2002 7
Classroom Environment• Classrooms - promote understanding• Reflective teachers - applying reflective
decision-making processes • A standards-based curriculum & the teacher’s
understanding of it• Students - variety of experiences, capabilities,
learning profiles, social contexts, and confidences in their ability to learn
DIFFERENTIATING INSTRUCTION
CONTENT
PROCESSPRODUCT
ENVIRONMENT
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What does technology bring
in?
TO IMPROVE
TO IMPROVE
TECHNOLOGY BASED LEARNING
ENVIRONMENT
DIFFERENTIATING INSTRUCTION
TEACHERS'PEDAGOGICAL
CONTENT KNOWLEDGE
STUDENTS LEARNING
DIFFERENTIATING MATHEMATICS INSTRUCTION
THROUGH TECHNOLOGY
CI 752RIDEAS FOR
INTERVENTIONS
CI 752RGOAL
TECHNOLOGY CAPABILITIES
TOOLS
INFORMATION VEHICLE
CONTEXT
SOCIAL MEDIUM
INTELLECTUAL PARTNER
...
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Technology Supporting Learners’ Meaning Making
of MATHS• Tool - knowledge construction• Information vehicles - exploring knowledge • Context - learning by doing, representing and
simulating real-world problems and contexts• Social medium - learning through conversation• An intellectual partner - learning-by-
reflecting• . . . (Wenglinsky, 1998)
TECHNOLOGY CAPABILITIES
TOOLS
INFORMATION VEHICLE
CONTEXT
SOCIAL MEDIUM
INTELLECTUAL PARTNER
...
TECHNOLOGY BASED LEARNING ENVIRONMENT
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Impact of Technology on Student Learning . . .
. . . depends on how the technology isused and therefore implies the needfor different instructional processesand better preparation of teachers
(Jonassen, Peck, & Wilson, 1999)
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“Mathematics teachers need to do mathematics in environments
supported by diverse technologies”
• Link to the core lessons & and to the skills the student is working to acquire
• Model best practices by using technology • Empower teachers & students to learn by
doing, communicate and exchange ideas locally/globally through the use of technologies
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Successful Apprenticeship Programs (Berryman,1989)
• Focus on application of the knowledge & skills• Weave together specific knowledge with the
development of general basic skills and problem-solving strategies
• Stimulate questions - learner=s original ideas + staged discrepant or confirming experiences
• Emphasize learning in context
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The cognitive apprenticeship framework
(Berryman, 1989, 1993) ContentPedagogical methods ScaffoldingContext
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PCASTLE model exemplified in
engages
supported by
have context
mimic
USING
reflects demonstrates
learn through
derived from
involves
focuses on
based on
emphasis
Anchored instruction
cognitive apprenticeships
Situated learning
learner activities
Participatory cognitive
apprenticeshipcommunity of learners
coaching
modeling
scaffolding
classroom learning
apprenticeships
alternative assesment
computer tools
inquirymodeling cognitive
skills
thinking process
process
Conversation & Action
community of practitioners
Legitimate peripheral
participation
Centralparticipation
social constructivism
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PCASTLE framework Content:
MATH content and “tricks of the trade” cognitive management strategies learning strategies
Pedagogical methods Scaffolding Context
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PCASTLE framework Content Pedagogical methods - employed to
encourage and support this learning environment: both students= group and independent work
Scaffolding Context
Mara Alagic SITE 2002 17
PCASTLE framework Content Pedagogical methods Scaffolding: Sequencing of learning
activities by classroom “experts” so that learners tasks are increasingly complex
Context
Mara Alagic SITE 2002 18
PCASTLE framework Content Pedagogical methods Scaffolding Context: The sociology of learning
environment providing some motivational characteristics resembling real-world situations
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“Technology in the Mathematics Classroom K-
12”• All four grade bands were represented • 19 teachers• Motivation• Previous experiences • The computer lab - state-of-the-art • Technical support
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The content of the course • “Current changes” in the teaching of the
school mathematics (NCTM 2000, NRC 2000)» standards-based» learning for understanding» differentiating instruction» integrating technology
• Variety of computer programs for learning and doing mathematics
• Web resources for developing of mathematics teaching and learning
• Portfolio - performance assessment
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Integrating Technology: Differentiating Instruction
• conceptual understanding of integration of IC technologies
• differentiating instruction through the use of technology
• On which motives these teachers base their decision concerning to what extent, if at all, they are willing to differentiate instruction through the use of PCASTLE model in their own classroom?
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Activities: Three Levels•Learner’s Level
•Challenging goal in the context•Network Level
•Broadening interactions•Local Level
•“Lectures” and group-work
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Confidence Question
How would you rate your confidenceand abilities at using appropriatetechnology as you teach
mathematicsin your own classroom? On a scale of1 to 10 with 10 as high.
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Confidence Level Change
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
students
conf
iden
ce le
vel (
1-10
)
Series1 Series2
M e a n 1 = 3 . 7 M e a n 2 = 5 . 4R a n g e 1 = [ 1 , 8 ] R a n g e 2 = [ 2 , 1 0 ]
Mara Alagic SITE 2002 25
• Participants– primary - 2– elementary - 5– middle - 6– high - 5– college - 1
• ICT use– 11 NO - classroom– 8 YES - classroom– 8no + 2yes - Lab
available– 16 - personal use
• Projects (required)– 2 LOGO– 4 CAS – 2 Maple – 5 on dynamic geometry– 6 on spread sheets
• Everyone was exposed to all of the above + Concept Maps + variety of software on CDs and exploration through the Internet + …
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Question 1: Obstacles
If in the fall you had everything youwanted in IC technology to enhanceteaching of mathematics, what otherobstacles would you need toovercome, if any?
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time knowledge tech support pck othertime tech support pck
knowledge not comfortable with ICTtime knowledge tech support
tech support pcktime knowledge curriculum materialstime tech support pck adapting lesson planstime tech support pck
knowledge have to learn with studentstime
knowledge tech supporttime knowledge pck too much work
tech support pck textbookstime knowledge tech support
tech support students know moretime tech support students know moretime pcktime knowledge making choices time pck textbookstime tech support overwhelmed-start slowly
7+5+1 6+1+1 4+3+4 2+4+2 19+13+8
total # 13 8 11 8 40
1st 37% 32% 21% 11% 19
2nd 38% 13% 27% 50% 13
3rd 8% 13% 36% 25% 8
Q1: OBSTACLES
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Question 2
How are you planning to differentiate
instruction in your classroom as aresult of experiences in this class?
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Guide on the Side or... • ICT= source for DI (2 teachers)• already “differentiating” - ICT just an
additional opportunity (4 teachers)• training “How to ...?” before they can
even think of DI (4 teachers)• need for appropriate curriculum
materials & training (5 teachers)
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… or Sage on the Stage?
• Maize: centers with 4-6 computers - shared some ideas for DI (3 teachers)
• 2 other teachers that have already been using GCs very actively expressed an enthusiasm in preparing their materials for DI; reporting some positive results
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Learning Cycle(Bybee’s 5Es)
• ENGAGE • EXPLORE • EXPLAIN - TERM INTRODUCTION• EXPAND• EVALUATE
Mara Alagic SITE 2002 32
Learning Cycle PCASTLEPCASTLE
• ENGAGE • EXPLORE • EXPLAIN - TERM INTRODUCTION:
PCASTLE PCASTLE - PParticipatory CCognitive AApprenticeship SSituated in the TTechnology-based LLearning EEnvironment
• EXPAND• EVALUATE
Mara Alagic SITE 2002 33
EXPAND/EVALUATE:Next steps
• This summer - new group of teachers• Both quantitative and qualitative data is
being gathered• Online discussion group established• Research questions for classrooms
action research to determine• changes that these teachers are taking into
their classrooms• effects on student learning
Mara Alagic SITE 2002 34
Future: Searching for Reasonable Research Questions
and Appropriate Methods • What types of teacher knowledge are needed in
order to provide successful use of technological tools in their mathematics classroom?
• Are there particular stages in the development of such knowledge?
• What influences this development? • What kind of impact is this experience going to
have on students learning in their classroom?
Mara Alagic SITE 2002 35
IF TIME, ...
http://education.twsu.edu/faculty/mara_alagic.html
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What Did Teachers Say: Support and Impediments
• Paradigm shift - What students can DO? • Changing learning environment • Teachers’ involvement • Inadequate technological resources• Absence of appropriate curriculum
materials• Limited time/saving time
Mara Alagic SITE 2002 37
Class-initiated Discussions• skills for utilizing IC technologies• conceptual understanding of mathematical
representations• differentiating instruction - obstacles have been
noted• explicit plans for implementation small steps• a teacher as “all knowing” - pupils “more
comfortable around technology than we are”• high school teachers - graphing calculators • the issue of “memorizing multiplication table”
was not missed, as usual :)
Mara Alagic SITE 2002 38
The Whole Picture
TO IMPROVE
TO IMPROVE
DIALECTIC BETWEEN PERCEPTION AND
CONCEPTUALIZATION
ENVIRONMENT
DIFFERENTIATING INSTRUCTION
TECHNOLOGY CAPABILITIES
COMPONENTS
MENTAL MAPS
TRANSFER
POSSIBLE STAGES OF KNOWLEDGE
DEVELOPMENT
TEACHERS'PEDAGOGICAL
CONTENT KNOWLEDGE
CORRESPONDENCE BETWEEN
TWO REPRESENTATIONS
SOME REPRESENTATIONS
BETTER THAN OTHERS
STUDENTS LEARNING
CONTENT
PROCESSPRODUCT
TOOLS
INFORMATION VEHICLE
CONTEXT
SOCIAL MEDIUM
INTELLECTUAL PARTNER
...
REPRESENTATION
ENVIRONMENT
MENTAL MAPS
TRANSFER
REPRESENTATION
CI 752RIDEAS FOR INTERVENTIONS
CI 752RGOAL
TECHNOLOGY BASED LEARNING
ENVIRONMENT
DIFFERENTIATING MATHEMATICS INSTRUCTION
THROUGH TECHNOLOGY: MAPPING (FOR) LEARNING
ENVIRONMENT
Mara Alagic SITE 2002 39
focuses on
based on
requires
exemplified in
engages
supported by
have context
mimic
engages
requires
are outcome of
describes
call on
calls on
has goal
involves
using
using
usingusing
reflects
demonstrates
learn through
emphasiizes
derived from
involves
engages
requires
Situated learning
Anchored instruction
learner activities
social constructivism
relinquishing authority
situated cognition
cognitive apprenticeships
community of learners
coaching
modeling
scaffolding
classroom learning
apprenticeships
Situated plans
Situated actions
responses in context
Informal learning
everyday cognition
alternative assesment
computer tools
inquiry
modeling cognitive
skills
thinking process
process
Conversation
community of practitioners
Legitimate peripheral
participation
SituatedLearning