Teachers

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Teachers Learning of Mathematics in the Presence
of TechnologyParticipatory Cognitive
Apprenticeship

• Mara Alagic
• Wichita State University

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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

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What to differentiate?
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Classroom Environment

• Classrooms - promote understanding
• Reflective teachers - applying reflective
  decision-making processes
• A standards-based curriculum the teachers
  understanding of it
• Students - variety of experiences, capabilities,
  learning profiles, social contexts, and
  confidences in their ability to learn

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What does technology bring in?
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Technology Supporting Learners Meaning Makingof
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)

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Impact of Technology on Student Learning . . .

• . . .
• depends on how the technology is
• used and therefore implies the need
• for different instructional processes
• and 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 - learners original ideas
  staged discrepant or confirming experiences
• Emphasize learning in context

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The cognitive apprenticeship framework(Berryman,
1989, 1993)

• Content
• Pedagogical methods
• Scaffolding
• Context

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PCASTLE model
social
Situated
based on
constructivism
learning
focuses on
learner
emphasis
activities
exemplified in
apprenticeships
Anchored
Conversation
instruction
Action
mimic
Participatory
cognitive
learn through
apprenticeships
cognitive
engages
community
have context
apprenticeship
of learners
classroom
learning
derived from
supported by
coaching
community of
computer
scaffolding
practitioners
tools
USING
alternative
involves
assesment
modeling
modeling
Central
cognitive
Legitimate
inquiry
skills
peripheral
participation
demonstrates
reflects
participation
process
thinking
process
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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

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PCASTLE framework

• Content
• Pedagogical methods
• Scaffolding Sequencing of learning activities by
  classroom experts so that learners tasks are
  increasingly complex
• Context

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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

• Learners 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 confidence
• and abilities at using appropriate
• technology as you teach mathematics
• in your own classroom? On a scale of
• 1 to 10 with 10 as high.

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• 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 you
• wanted in IC technology to enhance
• teaching of mathematics, what other
• obstacles would you need to
• overcome, if any?

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Question 2

• How are you planning to differentiate
• instruction in your classroom as a
• result 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(Bybees 5Es)

• ENGAGE
• EXPLORE
• EXPLAIN - TERM INTRODUCTION
• EXPAND
• EVALUATE

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Learning Cycle PCASTLE

• ENGAGE
• EXPLORE
• EXPLAIN - TERM INTRODUCTION PCASTLE -
  Participatory Cognitive Apprenticeship Situated
  in the Technology-based Learning Environment
• EXPAND
• EVALUATE

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EXPAND/EVALUATENext 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

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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?

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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

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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 )

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The Whole Picture
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Situated Learning