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Title: Getting Inside the CIA Troika: The Role of Cognitive Theory and Research in the Design of Mathematic

Getting Inside the C-I-A TroikaThe Role of
Cognitive Theory and Researchin the Design of
Mathematics and ScienceTeaching Learning
  • Jim Pellegrino
  • University of Illinois at Chicago

  • How theories of learning knowing impact
    Curriculum, Instruction Assessment
  • Knowledge about How People Learn
  • Implications for the C-I-A triad
  • Issues of Bridging Research and Practice
  • Some Final Thoughts

Questions, Discussion, Issues of Application
The Curriculum-Instruction-Assessment Triad
Whats Implicit in the C-I-A Triad
Theory of Learning Knowing
20th Century Theories of Learning Some
  • Dominant theory -- behaviorism associationism
  • Generic model of learning and knowledge
  • Knowledge conceptualized as discrete elements
  • Consequences
  • Curriculum is viewed as a set of separate pieces
  • Instruction is largely a process of knowledge
    telling -- transmission model -- learning is
  • Assessment relies on simple statistical and
    conceptual schemes -- problems are independent,
    item difficulty is the primary concern, and
    scores are computed on unidimensional IRT scales
    designed to maximally differentiate among persons

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Anyone, Anyone????
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The Importance of a Contemporary View of How
People Learn
  • Constitutes the theory knowledge base that sits
    at the core of the C-I-A triangle
  • It impacts how we should conceptualize each of
    the three vertices -- Curriculum, Instruction,
  • It is critical to how we should think about their
    correspondence and interaction

Advances in Sciences ofThinking Learning
  • The most critical implications for C-I-A are
    derived from study of the nature of expertise and
    the development of competence in specific
    curriculum domains.
  • The nature of expertise
  • The influence of pre-existing knowledge
  • Learning with understanding
  • Metacognitive knowledge skill
  • Multiple paths to competence
  • Situated knowledge and expertise

Domain-Based Models as Core
HPL Domain-Based Models of Learning
An Ongoing Concern
  • Status of knowledge about HPL in various
    instructional domains across K-16
  • Scope and richness of the knowledge base
  • Theory and empirical data on multiple aspects of
    HPL in a given domain or subdomain
  • Some domains have rich data and theory whereas
    others remain impoverished
  • In the absence of specific data and theory we can
    still rely on general principles about HPL

The Nature of Expertise
  • Shift in theorizing From an emphasis on
    expertise as general thinking skills to
    expertise as knowledge-rich thinking and problem
  • Illustrate what the results of successful
    learning expertise look like

What Do You Notice Understand?
What Did You Notice Understand?What Would
You Report About What You Saw?
A Novice
An Expert in Bio-Imaging
Expertise in Physics
  • Novices
  • These are inclined plane problems
  • Experts
  • This can be viewed as a conservation of
    energy problem (Chi, Feltovich and Glaser, 1981)

Summary of What We Know About the Nature of
  • Experts have well-organized knowledge
  • their knowledge is organized to support
    understanding (qualitative before quantitative)
    and it is conditionalized for use.
  • Experts have fluent access to their knowledge and
    recognize patterns chunks.
  • Such knowledge expertise is acquired over time
    and depends on multiple, contextualized
  • Implications -- Wisdom cant be taught directly
    and instruction must be directed towards the
    gradual acquisition of understanding expertise.
    Assessments must be designed to capture the
    multiple and discriminating features of expertise
    and its emergence.

The Impact of Pre-existing Knowledge
  • Shift in theorizing From an emphasis on students
    as being blank slates or having little relevant
    knowledge in subject-matter areas to an
    understanding of individuals as natural
    scientists observers with complex views about
    the world -- the constructive nature of knowing
  • Illustrate with a constructivist parable drawn
    from a childrens book

Lionnis Fish is Fish
The Fishs Image of Birds
The Fishs Image of Cows
The Fishs Image of People
Some Analogs to Fish is Fish
  • Young children who believe the earth is flat.
  • Physics students who assume force of the hand
    when a ball is thrown into the air
  • Electrical engineering students who use water
    pipe analogies for electricity
  • Student beliefs that history is about the good
    guys vs the bad guys
  • Students at multiple ages beliefs about seasons
    -- distance from sun not tilt

Summary of What We Know About Students
Pre-existing Knowledge
  • Students come to the classroom with
    pre-conceptions about how the world works. If
    their initial understanding is not engaged they
    may fail to grasp the new concepts and
    information that are taught, or they may learn
    them for purposes of a test but revert to their
    preconceptions outside the classroom.
  • Implications - Teachers must draw out and work
    with the preexisting understandings that their
    students bring with them. Assessment practices
    must be designed to assist in this process of
    tapping student mental models.

Learning With Understanding
  • Problem of transfer -- Judds 1908 dart throwing
    experiment with children
  • Learning facts versus understanding their context
    and significance
  • Geography -- memorizing names and places vs
    understanding how natural features define borders
    importance of water -- transfer across
  • History -- facts and dates versus understanding
    conditions of life
  • Biology -- properties of arteries and veins
    versus how those properties relate to the
    functions they serve

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Summary of What We Know About Knowledge
  • To develop competence in an area of inquiry,
    students must (a) have a deep foundation of
    factual knowledge, (b) understand facts and ideas
    in the context of a conceptual framework, and (c)
    organize knowledge in ways that facilitate
    retrieval and application.
  • Implication - Teachers must teach some subject
    matter in depth, providing many examples in which
    the same concept is at work and providing a firm
    foundation of factual knowledge. Assessment must
    be designed to capture the multiple components of
    organized knowledge.

MetacognitionTaking Control of Ones Learning
  • Experts and good learners know what they know
    and what they dont know -- this often takes the
    form of an internal conversation
  • Studies of self explanation and learning outcomes
  • Modeling of strategic control -- studies of
    reciprocal teaching strategies for reading
  • Metacognition is context and subject specific
  • Science -- hypotheses and predictions
  • History -- intentions of authors

Summary of What We Know About the Importance of
  • Expert learners monitor their own state of
    understanding as they solve a problem, read a
    passage, or perform a task. Students can be
    helped to learn to take control of their own
    learning by defining learning goals and
    monitoring their progress in achieving them.
    These Metacognitive skills can be taught but they
    must be connected to specific content.
  • Implication - The teaching of metacognitive
    skills should be integrated into the curriculum
    in a variety of subject areas. Assessment of such
    skills and strategies is an important concern.

Multiple Paths to Competence
  • Not all students learn in the same way or follow
    the same paths to competence.
  • Students problem solving strategies become more
    effective over time and with practice
  • the growth process is not a simple, uniform
    progression, nor is there movement directly from
    erroneous to optimal solution strategies.
  • Implication - Instruction and assessment should
    be sensitive to the specific strategies students
    use and where they fall on a continuum of
    efficiency and appropriateness for a given domain.

Situated Knowledge Expertise (1)
  • Knowledge frequently develops in a highly
    contextualized and inflexible form, and often
    does not transfer very effectively.
  • Transfer depends on the development of an
    explicit understanding of when to apply what has
    been learned.
  • Implication - The design of instruction should
    include conditions to foster the development of
    transfer. Assessment of knowledge and skill must
    take into account whether tasks or situations are
    tests of near, far or zero transfer and why.

Situated Knowledge Expertise (2)
  • There are important relationships among learners
    and the contexts in which they learn which define
    part of knowing and expertise.
  • Expert performers, through interactions with
    peers, build communities of practice and
    understanding which are distributed and build on
    the learning of others.
  • Implication - Students abilities to participate
    in the socially organized practices of thinking
    and inquiry in a domain need to be incorporated
    explicitly into the design of instruction and the
    assessment of learning.

Some Major Implications of HPL for the C-I-A
  • Curriculum --Transcending false dichotomies about
    the goals of instruction
  • e.g., facts vs thinking skills
  • Focus on the organization of concepts
  • Instruction -- Helps bring order to chaos
  • debates about constructivism instructional
  • Assessment -- The process of Knowing What
    Students Know
  • Why assessment is critically important
  • What to assess and how to do so

Knowing, Learning, Teaching
  • Constructivism is a theory of the nature of
    knowing, including how we understand and learn
    it is not a theory of pedagogy.
  • There are many misconceptions regarding this
    point, e.g., ideas that we should never lecture
    or engage in direct teaching.
  • Critical to understand the complex relationships
    between theories of knowing and learning and
    theories of instructional design and teaching.

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ESEA/NCLB Key Requirements
  • Annual assessments of all students in Math and
    Reading for Grades 3-8, and once in grades 9-12,
    beginning no later than 2005/2006 academic year
  • Math and Reading annual assessments must be
    aligned with state academic content and
    achievement standards
  • Annual assessment of students in science no less
    than once in each of grades 3-5, 6-9 and 10-12,
    beginning no later than 2007/2008 academic year
  • Adequate Yearly Progress
  • 100 of students must meet or exceed a
    proficient level of academic achievement by the
    2013-2014 academic year
  • Establish intermediate goals for ongoing
    improvement over the 12 year period
  • Reporting in multiple categories for multiple
    demographic groups

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Opportunities To Rethink and Refashion
Educational Assessment
  • Advances in the cognitive sciences illuminate
    important aspects of learning and understanding
    that should be major targets of instruction and
  • Advances in measurement and technology expand the
    capability to collect and interpret more complex
    forms of data and evidence
  • Merger of the three holds promise for promoting
    an enhanced science and practice of educational
    assessment that can assist learning and teaching

Assessment as a Process of Reasoning From
EvidenceThe Assessment Triangle
  • cognition
  • model of how students represent knowledge
    develop competence in the domain
  • observations
  • tasks or situations that allow one to observe
    students performance
  • interpretation
  • method for making sense of the data

Must be coordinated!
Two Important Themes
  • What is variable purpose and context of
    assessment use
  • situations should not be treated exactly the same
    and one size does not fit all assessment needs
  • Formative - to aid student learning
  • Summative - to determine overall achievement
  • Program evaluation - to judge program impact
  • What is constant the principles underlying any
    assessment activity
  • how we conceive each of the 3 elements of the
    assessment triangle and their interplay are
    critical -- none of the 3 can be taken for granted

Scientific Foundationsof Educational Assessment
  • Advances in the Sciences of Thinking and Learning
    -- the cognition vertex
  • informs us about what observations are important
    and sensible to make
  • Advances in Measurement and Statistical Modeling
    -- the interpretation vertex
  • Informs us about how to make sense of the
    observations we have made

Translating Science into Engineering Principles
of Assessment Design
  • Assessment design should be based upon a model of
    student learning and a clear sense of the
    inferences about student competence that are
    desired for the particular context of use.
  • Design is recursive process -- starting with the
    Student Model
  • The student model suggests the most important
    aspects of student achievement that one would
    want to make inferences about and provides clues
    about the types of tasks that will elicit
    evidence to support those inferences.

Why Cognitive Models of Content Knowledge are
  • Tell us what are the important aspects of
    knowledge that we should be assessing.
  • Give deeper meaning and specificity to standards
  • Give us strong clues as to how such knowledge can
    be assessed
  • Suggest what can and should be assessed at points
    proximal or distal to instruction
  • Can lead to assessments that yield more
    instructionally useful information -- within and
    across levels and contexts
  • Can guide the development of systems of
  • Comprehensive, Coherent Continuous

Aspects of Student Models
  • Domain specific and empirically based
  • Identifies cognitive performances that
    differentiate expert and novice learners
  • Lays out one or more typical progressions toward
    competence including milestones or landmark
    performances along the way.
  • Can be at various levels of detail grain size
    depends on assessment purpose
  • Valuable information source

Some Selected Examples of Applicable Student
  • Development of number sense (Case Griffin)
  • Subtraction bugs (Brown Burton)
  • Cognitively Guided Instruction (Carpenter et al.)
  • ACT-R theory and domain models for algebra
    geometry (Anderson, Koedinger, Corbett et al.)
  • Facets in physics (Minstrell Hunt)
  • Middle School Math through Applications (Greeno
    et al.)
  • Science inquiry -- SEPUP/BEAR (Wilson)
  • Australias Develop. Prog. Maps (Masters)

What Will Be Needed toLeave No Child Behind
  • Better balance and coordination between
    large-scale classroom assessment practices
  • Instructional assessment materials that
    incorporate knowledge about domain specific
    learning trajectories
  • Not using what we currently know
  • Teacher education -- preservice inservice
  • Specifics of quality practices and how they work
  • General pedagogical shift regarding assessment
  • Technology supports
  • Make the assessment process doable and manageable
  • Technology Assessment Project URLs
  • Educating policymakers -- limits and uses of tests

Recommendations Regarding Assessment Policy
  • Policy makers are urged to recognize the
    limitations of current assessments, and to
    support the development of new systems of
    multiple assessments that would improve their
    ability to make decisions about education
    programs and the allocation of resources.
  • Important decisions should not be based on single
    test score
  • Systems should measure growth in achievement over
  • Emphasis should be shifted from assessment of
    learning to an increased importance of assessment
    for learning.
  • Assessments at classroom and large-scale levels
    should grow out of shared knowledge base about
    learning and knowing

Coming soon in U.S. are More Tests --- But will
they help children learn?
Relating Teaching, Learning Assessment
Formative Assessment
Classroom Teaching Learning
High-Stakes Summative Tests
Level of Impact
Information Feedback
Why Focus on Classroom Formative Assessment?
  • As instruction is occurring, teachers need
    information to evaluate whether their teaching
    strategies are working.
  • They also need information about the current
    understanding of individual students and groups
    of students so they can identify the most
    appropriate next steps for instruction.
  • Students need feedback to monitor their own
    learning success and to know how to improve.
  • Black Wiliam (1998) reviewed impact of
    formative assessment practices on learning
    outcomes -- effect sizes ranging from .5 - 1.0

Whats Needed for Formative Assessment to Work
  • Sadlers 3 interconnected elements
  • A clear view of the learning goals
  • Information about the present state of the
  • Actions to close the gap
  • The major challenge is always that of Knowing
    What Students Know
  • Need conceptually rich systems that link
    curriculum, instruction and assessment

Opportunities Affordedby Technology
  • Computer and telecommunications technologies
    provide powerful new tools necessary to meet many
    of the design and implementation challenges
    implied by the merger of current cognitive models
    and measurement methods
  • going beyond conventional practices for item
  • implementing a range of task designs and item
  • tapping a broader repertoire of cognitive skills
    and knowledge
  • recording and scoring complex aspects of behavior

Enhanced Linkages Among the Three Elements of
the Triangle
Examples of Enhanced Assessment Triangle Linkages
  • Tasks with principled connections to theory
  • theory-based item generation - Mathematics Test
    Creation Assistant
  • Tasks tapping complex knowledge reasoning
  • concept organization - Concept Mapping Software
  • complex problem solving scenarios - Architecture
    and Dental Exams, AmericaQuest environment
  • Scoring and interpretation of multiple aspects of
    student performance on a wide range of tasks
  • Text analysis and scoring - LSA
  • Analysis of complex solution strategies - IMMEX

Strengthening Connections Among
Domain-Based Models of Learning Understanding
Examples of EnhancedC-I-A Linkages
  • Implementing effective formative assessment
  • intelligent tutors and other software programs
    like DIAGNOSER, IMMEX, and Summary Street can
    help teachers assess students learning and
    interpret the ways in which students solve
  • Assessment embedded in technology-enhanced
    learning environments
  • Situations where learning occurs through a
    process of inquiry, and collaborative,
    problem-based activity focused on aspects of the
    curriculum often difficult to teach
  • Integration of cognitive theory with
    instructional design and embedded assessment
  • examples Biologica, SMART, WISE

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Selected URLs
  • http// -- The Diagnoser
    software tools for science and other content
    areas -- see also http//
  • http// -- The IMMEX tools and
  • http// -- Latent Semantic
    Analysis of text The Summary Street Project
  • http// -- Web Based Inquiry in
    Science Education website
  • http// -- Biologica
    project at Concord Consortium
  • http// -- Concept
    mapping software tools and other information

Systems of Assessments
  • Comprehensive - utilize a range of measurement
    approaches that together produce information that
    can be combined and reported at different levels
  • Coherent - the conceptual base for the student
    models underlying design of the multiple
    assessments should be compatible. There also
    needs to be alignment between curriculum,
    instruction and assessment so that everything is
    working toward a common set of learning goals.
  • Continuous - Sequential observations, over time
    must be conceptually linked so that growth can be
    measured. Assessment is viewed as a cumulative
    process rather than a drop-in-from-the-sky event

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Foundation for NumberCentral Conceptual
  • Ability to verbally count number words
  • Ability to count with one-to-one correspondence
  • Ability to recognize quantity as set size
  • Ability to mentally simulate sensorimotor
  • Structures are grafted onto the system of
    written numerals


Low SES kids often lag behind in conceptual
development of number -- consequences for math
achievement increase with age
Income-to-needs and child cognitive ability Deep
poverty and math ability (PIAT-Math), NLSY-CS
data set. Source Brooks-Gunn, B.J., G.J.
Duncan, and P.R. Britto (1999) Are socioeconomic
gradients for children similar to those for
adults? P. 109 in Developmental Health and the
Wealth of Nations, D.P. Keating and C. Hertzman,
eds. New York Guilford Press.
Number WorldsK-2 Integration of Curriculum,
Instruction Assessment
  • Grounded in a theory of how childrens conceptual
    understanding of number progresses -- central
    conceptual structures
  • Includes 78 games activities that allow each of
    the critical understandings to be firmly put into
  • Integral part is the Number Knowledge test to
    assess the individual childs current

Evaluation of Number Worlds Program Impact
  • 3 year longitudinal study grades K -2
  • Treatment control groups from low-income, high
    risk urban communities
  • Normative comparison group -- middle income,
    magnet school


Mental Models Misconceptions in High School
  • Jim Minstrells work on mapping FACETS of
    knowledge and understanding in multiple areas of
  • DIAGNOSER program for use in instruction.
  • A facet is a convenient unit of thought, an
    understanding or reasoning, a piece of content
    knowledge or strategy seemingly used by the
    student in making sense of a particular
    situation. (Minstrell, 1992, pg. 2)

Elements of Minstrells Facets-Based
Instructional System
Elicitation Questions
  • Questions designed to elicit as many of the
    facets within a cluster as possible
  • Open-ended format
  • Embedded in an everyday context
  • Testable situation

Example Elicitation Question
Glass case with air removed
Scale reading 10.0 lbs
Scale reading _____ lbs
A Facet Cluster for Separating Effects of a Fluid
Medium from Gravitational Effects
  • 310 - pushes from above and below by a
    surrounding fluid medium lend a slight support
    (Pretest 3)
  • 311 - a mathematical formulaic approach (e.g.,
    rho x g x h1 - rho x g x h2 net buoyant
  • 314 - surrounding fluids dont exert any forces
    or pushes on objects
  • 315 - surrounding fluids exert equal pushes all
    around an object (Pre 35)
  • 316 - whichever surface has greater amount of
    fluid above or below the object has the greater
    push by fluid on the surface
  • 317 - fluid mediums exert an upward push only
    (Pretest 13)
  • 318 - surrounding fluid mediums exert a net
    downward push (Pre 29)
  • 319 - weight of an object is directly
    proportional to medium pressure on it (Pretest

Connections to Learning Goals
Developing ideas Applying ideas
Diagnostic Questions
  • Questions grouped in sets that span the facets in
    the relevant cluster
  • Each question is diagnostic of several facets
    within the relevant cluster
  • Assigned in the middle of a unit, after several
  • Diagnose facets that may still need targeted
    activities for the whole class or for individuals
  • Created to supplement (not replace) teacher

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BEAR Assessment System
IEY Items scoring guide
Measurement model
4 progress variables
Progress Variables from the IEY Curriculum
  • Designing and Conducting Investigations--designing
    a scientific experiment, performing laboratory
    procedures to collect data, recording and
    organizing data, and analyzing and interpreting
    the results of an experiment.
  • Evidence and Tradeoffs--identifying objective
    scientific evidence, as well as evaluating the
    advantages and disadvantages of different
    possible solutions to a problem on the basis of
    the available evidence.
  • Understanding Concepts--understanding scientific
    concepts (such as properties and interactions of
    materials, energy, or thresholds) in order to
    apply the relevant scientific concepts to the
    solution of problems.
  • Communicating Scientific Information--effectively,
    and free of technical errors, organizing and
    presenting results of an experiment or explaining
    the process of gathering evidence and weighing
    tradeoffs in selecting a solution to a problem.

IEY Item
  • You are a public health official who works in the
    Water Department. Your supervisor has asked you
    to respond to the public's concern about water
    chlorination at the next City Council meeting.
    Prepare a written response explaining the issues
    raised in the newspaper articles. Be sure to
    discuss the advantages and disadvantages of
    chlorinating drinking water in your response, and
    then explain your recommendation about whether
    the water should be chlorinated.

  • As an edjucated employee of the Grizzelyville
    water company, I am well aware of the controversy
    surrounding the topic of the chlorination of our
    drinking water. I have read the two articals
    regarding the pros and cons of chlorinated
    water. I have made an informed decision based on
    the evidence presented the articals entitled The
    Peru Story and 700 Extra People May bet Cancer
    in the US. It is my recommendation that our
    towns water be chlorin treated. The risks of
    infecting our citizens with a bacterial diseease
    such as cholera would be inevitable if we drink
    nontreated water. Our town should learn from the
    country of Peru. The artical The Peru Story
    reads thousands of inocent people die of cholera
    epidemic. In just months 3,500 people were
    killed and more infected with the diease. On the
    other hand if we do in fact chlorine treat our
    drinking water a risk is posed. An increase in
    bladder and rectal cancer is directly related to
    drinking chlorinated water. Specifically 700
    more people in the US may get cancer. However,
    the cholera risk far outweighs the cancer risk
    for 2 very important reasons. Many more people
    will be effected by cholera where as the chance
    of one of our citizens getting cancer due to the
    water would be very minimal. Also cholera is a
    spreading diease where as cancer is not. If our
    town was infected with cholera we could pass it
    on to millions of others. And so, after careful
    consideration it is my opion that the citizens of
    Grizzelyville drink chlorine treated water.

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The Way Things Typically Work
The Way Things Need To Work
Applying an HPL KWSK Frame of Reference
toEducational Projects
  • Using the principles to evaluate the conceptual
    and operational basis of existing curricula,
    materials, and instructional designs
  • Using the principles to design new curricula,
    materials and instructional designs and to
    evaluate their effectiveness

Two General Concerns/Questions
  • Given the wide range of instructional materials,
    designs and/or programs currently available or
    under development for a given area of education,
    we need to ask
  • In what ways are they consistent or
    inconsistent with contemporary principles of
    learning, assessment Instructional design?
  • How might they be modified and improved to be
    more consistent?

  • What is the nature of the conceptual coherence
    among curriculum-instruction-assessment for the
    target domain and population?
  • What assumptions, if any, have been made about
    the development of competence and expertise in
    the domain and how is this reflected in the
    sequencing of content and/or in the selection of
    instructional strategies and learning activities?
  • How does the instructional program take into
    account peoples prior knowledge, beliefs, and
    preconceptions. Does it provide appropriate
    mechanisms for responding to such knowledge

  • Is the development of metacognitive skills a
    component of the program? How might it be more
    effectively integrated into the design?
  • What role(s) does assessment play in the design
    of the materials and instructional strategies and
    is it guided by a substantive theory of knowing
    and understanding in the subject area domain?
    What levels of knowing and understanding are the
    apparent targets of assessment?

Working in Pasteurs Quadrant
Limitations of Research
  • Rarely does one study produce an unequivocal and
    durable result multiple methods, applied over
    time and tied to evidentiary standards, are
    essential to establishing a base of scientific
    knowledge.Shavelson Towne, 2002

Complexity of the Study Domain
  • The character of education not only affects the
    research enterprise, but also necessitates
    careful consideration of how the understanding or
    use of results can be impeded or facilitated by
    conditions at different levels of the system.
    Organizational, structural, and leadership
    qualities all influence how the complex education
    system works in practice.Shavelson Towne,

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Projections Regarding the Future
  • Learning environments will change profoundly -
    its happening as we speak
  • Individuals will exert significant control over
    their learning and learning environments
  • What students can come to know and understand
    will increase dramatically
  • Linguistic and cultural variation among learners
    can be accommodated as a normal part of the
    teaching-learning setting
  • Expectations regarding the outcomes from
    education will increase several fold

Summary Points About Learning, Teaching,
Assessment, Technology
  • There is a developing science of learning
  • It has major implications for all aspects of
    schooling -- curriculum, instruction, assessment,
    plus preservice and inservice education, and
    faculty professional development
  • It provides a basis for knowing when, how and why
    to use various instructional strategies it can
    guide intelligent design and use of new
    curricular materials as well as information