Teachers’ Learning of Mathematics in the Presence of Technology: Participatory Cognitive Apprenticeship

Mara Alagic SITE 2002 1

Teachers’ Learning of Mathematics in the

Presence of Technology:Participatory Cognitive

Apprenticeship

Mara AlagicWichita State University


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


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


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)


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


What to differentiate

?

TO IMPROVE

TO IMPROVE



CONTENT KNOWLEDGE

STUDENTS LEARNING

CONTENT

PROCESS

PRODUCT

CI 752RIDEAS FOR

INTERVENTIONS

CI 752RGOAL


ENVIRONMENT


THROUGH TECHNOLOGY


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


CONTENT

PROCESSPRODUCT

ENVIRONMENT


What does technology bring

in?

TO IMPROVE

TO IMPROVE


ENVIRONMENT



CONTENT KNOWLEDGE

STUDENTS LEARNING


THROUGH TECHNOLOGY

CI 752RIDEAS FOR

INTERVENTIONS

CI 752RGOAL

TECHNOLOGY CAPABILITIES

TOOLS

INFORMATION VEHICLE

CONTEXT

SOCIAL MEDIUM

INTELLECTUAL PARTNER

...


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)


TOOLS

INFORMATION VEHICLE

CONTEXT

SOCIAL MEDIUM


...

TECHNOLOGY BASED LEARNING ENVIRONMENT


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)


“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


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


The cognitive apprenticeship framework

(Berryman, 1989, 1993) ContentPedagogical methods ScaffoldingContext


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


PCASTLE framework Content:

MATH content and “tricks of the trade” cognitive management strategies learning strategies

Pedagogical methods Scaffolding Context


PCASTLE framework Content Pedagogical methods - employed to

encourage and support this learning environment: both students= group and independent work

Scaffolding Context


PCASTLE framework Content Pedagogical methods Scaffolding: Sequencing of learning

activities by classroom “experts” so that learners tasks are increasingly complex

Context


PCASTLE framework Content Pedagogical methods Scaffolding Context: The sociology of learning

environment providing some motivational characteristics resembling real-world situations


“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


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


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?


Activities: Three Levels•Learner’s Level

•Challenging goal in the context•Network Level

•Broadening interactions•Local Level

•“Lectures” and group-work


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.


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 ]


• 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 + …


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?


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


Question 2

How are you planning to differentiate

instruction in your classroom as aresult of experiences in this class?


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)


… 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


Learning Cycle(Bybee’s 5Es)

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


Learning Cycle PCASTLEPCASTLE

• ENGAGE • EXPLORE • EXPLAIN - TERM INTRODUCTION:

PCASTLE PCASTLE - PParticipatory CCognitive AApprenticeship SSituated in the TTechnology-based LLearning EEnvironment

• EXPAND• EVALUATE


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


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?


IF TIME, ...

http://education.twsu.edu/faculty/mara_alagic.html


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


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 :)


The Whole Picture

TO IMPROVE

TO IMPROVE

DIALECTIC BETWEEN PERCEPTION AND

CONCEPTUALIZATION

ENVIRONMENT



COMPONENTS

MENTAL MAPS

TRANSFER

POSSIBLE STAGES OF KNOWLEDGE

DEVELOPMENT


CONTENT KNOWLEDGE

CORRESPONDENCE BETWEEN

TWO REPRESENTATIONS

SOME REPRESENTATIONS

BETTER THAN OTHERS

STUDENTS LEARNING

CONTENT

PROCESSPRODUCT

TOOLS

INFORMATION VEHICLE

CONTEXT

SOCIAL MEDIUM


...

REPRESENTATION

ENVIRONMENT

MENTAL MAPS

TRANSFER

REPRESENTATION

CI 752RIDEAS FOR INTERVENTIONS

CI 752RGOAL


ENVIRONMENT


THROUGH TECHNOLOGY: MAPPING (FOR) LEARNING

ENVIRONMENT


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

Documents

Teachers’ Learning of Mathematics in the Presence of Technology: Participatory Cognitive Apprenticeship