Crafting Research Tools to Establish a Learning Progression on Measurement Knowledge

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Crafting Research Tools to Establish a Learning
Progression on Measurement Knowledge
Jbarrett_at_ilstu.edu November 14, 2008

• Presented by Jeffrey Barrett representing
  Illinois State University
• Presented at the DRK12 PI Conference,
  Washington, D.C.
• Childrens Measurement (Barrett, Clements
  Sarama, 2007)

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Overview of Session

• Definition of terms, critical issues for
  students, teachers, researchers, policy and
  design folks
• Potential promise of Learning Trajectory work
• Examples of Learning Trajectories in our work
• Our research method using Teaching Experiments
  to assess, intervene, promote growth
• Can we address surprises and obstacles? Changes
  in our method, and current approach allow us to
  make best use of these surprises
• Critical Issues How do these tools work? (LTs)
• Theoretical History of Trajectories and
  Progressions

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What is a Learning Progression?(or, What is the
value and use of one?)

• Learning Progressions begin around a
  developmental account of growth along a concept
  or domain set of concepts (a significant learning
  goal).
• It describes an increasingly sophisticated and
  correspondent set of ideas and strategies
  children exhibit as they grow towards complete
  knowledge of the Learning Goal.
• It usually includes assessment tasks intended to
  identify or locate the level of a given child
  along the developmental progression.
• Development is a long broad process, different
  from localized learning, which is the way
  children may either assimilate or accommodate
  tasks within their schemes for a given domain
  (see Case, 1996).

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Relating Learning Progressions and Learning
Trajectories

• Learning Trajectories include instructional
  sequences specific to each subsequent level of
  understanding, in addition to the assessment
  tasks specific to each level, all aimed at the
  learning goal (Gravemeijer, 2004 Clements
  Sarama, 2004).
• Learning Progressions are, descriptions of
  successively more sopisticated ways of reasoning
  within a content domain based on research
  syntheses and conceptual analyses that can be
  useful for improving assessments (Smith, Wiser,
  Anderson and Krajcik, 2006, p. 1).
• Other thoughts, uses of these terms?

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Potential promise of such work

• Benefits to curriculum development see
  Curriculum Research Framework (CRF) from Clements
  (2007) in JRME 10 phases, and phase 4 is
  Learning Models (I.e. LTs).
• Support improvements or design of assessments
  (formal and informal).
• Supports improvements in classroom instruction,
  professional development.
• May reduce the clutter of our mathematics
  curriculum (mile wide) to focus on critical ideas
  (Curriculum Focal Points, 2006)
• Critical Research tenet of a comprehensive theory
  of learning and development (see tenets 10, 11
  and 12 of Clements Sarama, (2007), on
  Hierarchic interactionalism (p. 464-466).

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Summary so farLearning Trajectory Research

• Purpose to support Curriculum Development, and
  also Assessment Design, and Professional
  Development of Teachers
• Definition Learning Trajectories consist of
  three components
• Content goal specified within a discipline
• Accounts of students developmental progressions
  of the related concepts
• Specification of tasks pertaining directly to
  transitions from one level of the progression to
  the next.

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An Example of a Learning Trajectory on Measuring
Linear Space

• Learning Goal Using Units of Length to specify
  Linear Quantity
• Latter portion of a progression by the Level
  Titles
• Direct comparison of length
• Indirect comparison of length (using another
  object)
• Serial Order (up to 6 objects in a set to
  sequence by length)
• End to End Measurer
• Unit Repeater (and Unit Relater)
• Length Measurer
• Conceptual (internalized) ruler
• Complex (bent) path measurer, coordinates
  perimeter measures in 2D contexts

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Isolate End to End Level (handout)

• Age Kindergarten, Grade 1 six years
• Actions may use number to describe length of
  objects, but requires a linked chain of unit
  objects to be visible.
• Actions on Mental Images May be associating
  motion along a path also, but is now challenged
  to imagine a chain of units and relate that to
  the length of the object.
• Instructional Task set Relate the ruler to a
  string of unit objects relate a single line
  segment to a string of unit objects and to a
  ruler? Take away one unit at a time and talk
  about the length in sequence of decrease? Build
  back up from one, an ordered set of line segments
  by adding unit objects successively to a string
  of the unit objects.

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Tasks to check and move along End to End Level
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Layered account of Development of Length Levels
from the Trajectory
6 yrs
8 yrs
Length measurer
Unit relater and repeater
End to End measurer
Serial Orderer
Comparer, indirect
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Theoretical Account of Learning Hierarchical
Interactionism

• 12 components (Clements Sarama, 2007)
• Progressions, domain specific progressions,
  hierarchic (linear too),
• cycles of concretizing, bands of development
  relatively cohesive,
• initial bootstrap capacities (nature),
• various courses,
• increasingly sophisticated,
• nurture matters,
• Consistency of growth within the LT.
• This is largely consistent with Neo-Piagetian
  positions as stated by Robbie Case (1996), and
  integrates work of other theorists, particularly
  van Hiele and Vygotsky.

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Another example from Building Blocks Learning
Trajectory (Clements Sarama, 2003)

• The Building Blocks project for early childhood
  mathematics curriculum development was based on a
  central research guideline of Learning Trajectory
  research.
• We illustrate one case with a trajectory-focused
  website for professional development.
• (see an example) follow links from level names,
  to case-video of children exhibiting that
  strategy in response to assessment items, to
  instructional activity models for moving children
  from the level to the next.

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BBLT web example

• http//www.gse.buffalo.edu/org/triad/tbb/index.asp
  ?localparent
• This is an example of professional development
  based specifically on a set of Learning
  Trajectories constructed by Clements and Sarama
  (2001 2003) for use with PreK, K, 1 Grade level
  teachers and their students.
• This illustrates the integration of instructional
  goals, developmental progression of observable
  strategies related to the goals, instructional
  interventions and lesson activities to support
  growth along the progression.

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Another example of a learning trajectory (in
practice)

• Consider the work of three different Grade 2
  students from our cohort, March and April, 2008.
• Place the students along the trajectory
• What are issues for using the trajectory to
  suggest appropriate interventions?
• What are issues for using the trajectory to
  understand how the students are learning?

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Initial assessment (Student 1 Grade 2)Broken
Ruler Task (4 inches)
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Layered account of Development of Length Levels
from the Trajectory
6 yrs
8 yrs
Length measurer
Unit relater and repeater
End to End measurer
Serial Orderer
Comparer, indirect
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Student 1 Several days later (revisit task)

• Now the zero point along a ruler is available

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Making a record identify units (student 1)

• Here the gaps (spaces) are enumerated

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Student 2 Can you use units to find length?
(broken ruler task)

• Units are intended, but not established

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The unit identified

• Student 2 enumerating spaces as units of length

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Student 2 Make a record by drawing what happened

• Uses number labels for spaces

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Student 2 a few weeks later check on transfer
of the unit

• May need direct access to connected yellow tiles

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We asked Student 2 to set the different tools
together to coordinate compare.

• Note the tension, based on expected conservation
  of a linear object

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Student 3 Identifying CompositeUnits

• Relate the wire length unit with edge length units

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Our methodology

• Teaching Experiment (design-based research)
• Individual sessions
• Groups by level (emergent approach)
• Classroom sessions
• Clinical Interviews (structured task based)
• Micro-genetic analysis (confirmatory)
• Formal written interviews (item response theory)

Challenge How do individual teaching experiments
relate to microgenetic experiments with groups?
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Research Questions of current LT project

• How do students develop coherent knowledge and
  integrated strategies for measurement across preK
  to Grade 5?
• How are students' developing cognitive abilities
  for perceptual and numerical comparison, for
  coordinating and discriminating, for deductive
  logic, and for ordering and nesting sequences
  related to the development of knowledge and
  strategies for measurement?
• How are students' developing abilities for
  spatial thinking, algebraic reasoning, or
  proportional reasoning related to their
  measurement knowledge and strategies?
• How might students' developing representational
  fluency for measurement relate to their
  mathematical and scientific understanding of
  measurement?
• How is the development of students' developing
  knowledge of measurement (especially their use of
  rulers and other tools, with accuracy and
  precision) related to the development of their
  scientific modeling and reasoning across varying
  contexts?

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Our enacted methodology with issues to discuss

• Questions to discuss relating to methodology of
  data collection, interpretation, design cycles
• Questions related to details of sampling and
  accounting for longitudinal growth and change.
• Emerging issues of tool usage and design cycles.

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Fundamental questions for us now (Fall, 2008)

• How do we know we have enough evidence to modify
  our Learning Trajectory, either to expand or
  reduce? Unit relater and repeater seems too
  broad, as most Grade 2 students fit this
  category.
• How often should we move back and forth between
  classroom experiments and individual Teaching
  Experiment sequences?
• How best to invite teachers into the analysis of
  student thinking? How best to engage them in
  formative assessment from the trajectory?

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Further questions

• How often do we need to interview or observe a
  student to describe learning through a four-year
  longitudinal account?
• To what extent may we challenge the student in
  comparison to the extent that we support their
  ideas in scaffolds?
• What is an effective pedagogy to use within our
  cycle of TE sessions?
• Van Hieles (1986) phases of instruction suggests
  a loop supporting figural, observable operations
  phasing into internal images in preparation for
  growth to a further level.

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Surprises of Tool Adaptation and Interpretation

• Limitations of incremental imagery between end to
  end level images and unit relater and repeater
  images.
• We expected coordination among related
  representations to support and engage integration
  and growth but
• See next slide

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Theoretical History of Learning Trajectories

• Piaget logical stage development (comprehensive
  over domains and content)
• Domain specialization in reaction
• Neo-Piagetian work Case, Halford, Simon, Siegler
• Curriculum Design from Student thinking Fennema,
  Carpenter, Post Lehrer (CGI) Gravemeijer (
  RME)
• Focused Learning Trajectory work from
  experimental and constructivist frameworks
  Steffe, Confrey, Thompson, Cobb, Clements Sarama

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Possible Directions

• Horizontal integrations of related learning
  topics at a given developmental stage, by
  coordinating a set of related LTs (e.g.
  coordination of Number schemes and Spatial
  schemes)
• Vertical integrations to show long term coherence
  of curriculum and to promote improvements in
  design of articulation across years of schooling
  (Quantitative Reasoning).

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Concluding notes

• Examples of working to develop our trajectory
  emphasize the process of moving between a
  specified goal, progression, and task
  specification.
• This kind of interaction between assessment tasks
  and instructional tasks that promote growth from
  a level up to the next are demanding design
  loops.
• They demand individual clinical work and
  classroom trials of lesson sequences.
• We also rely on formal, broad assessments to help
  generalize beyond case studies with focus
  students.

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Summary

• Learning Trajectories support integrative
  development of curriculum, assessment and
  professional development for teachers
• Learning Trajectories support the Curriculum
  Research Framework to establish a research-based
  approach to Curriculum Design
• These tools and frameworks for Mathematics
  Education Research may benefit related STEM
  disciplines.
• Thank you!

jbarrett_at_ilstu.edu