douglas h. clements university at buffalo, suny...douglas h. clements university at buffalo, suny...
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Douglas H. ClementsUniversity at Buffalo, SUNY
Learning Trajectories in Mathematics Education
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Math. Ed. Needs
•Clear descriptions of
• content
•how students learn that content
•how to teach that content
•All account for learning
Math. Ed. Needs
• Approaches to this triad have been diverse
• Discuss each, especially focusing on learning trajectories’connective role
• Avoid misconstrual
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Learning Trajectories:3 Parts
1.Goal
2.Developmental Progression
3. Instructional Activitiesdescriptions of children’s thinking and learning in a specific mathematical domain, and a related, conjectured route through a set of instructional tasks designed to engender those mental processes hypothesized to move children through a developmental progression of levels of thinking … (special issue, MTL)
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
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Learning Trajectories
• Some only emphasize developmental progression
• “Learning progressions” ambiguous here*
• Power and uniqueness of LT construct stems from the inextricable interconnection of the 3 components
*standards may benefit from this
Developmental Progressions
• Many ways: RME one of most longest standing, comprehensive, and innovative projects
• “Developmental Research”—integration of design and research
Developmental Progressions
• LTs historical development of math used as a heuristic
• Inspired too by students' informal solution strategies
• Original design, mental activities as students work through instructional activities (Koeno)
Developmental Progressions
• …identified within theoretically- and empirically-grounded models of children’s thinking, learning, and development
• Hierarchic Interactionalism
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
So, What’s New?
• Curriculum stems from Latin “racecourse”
• If so, what’s really new?
• All do share a family resemblance
• LTs build on past but uniquely contribute…
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Early Educational Psychology
• Sequences based on the accumulation of connections (Thorndike, 1922)
• Curricular as connections between simple situations (addends) and responses (sum), later to multidigit addition…even mathematical reasoning.
• Conceptual learning not the focus
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Bloom and Gagné
• Types of learning
• Some (S-R, Thorndike’s “bonds”) prerequisite to others (discrimination, concept, & rule learning, & problem solving).
• Assembled in “learning hierarchies”—sequences of pairs of skills (prerequisite to higher)
• Thus, “learning routes” determined by logical and empirical task analysis
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Cognitive Analyses
• Others, more empirical research and psychological models, created “cognitive” or “rational” analyses (e.g., Resnick & Ford, 1981)
• Computer metaphors
• These led to many scope and sequences
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Cognitive Science
• In parallel, Piagetian, similar research programs identified stages
• Unfortunately, those applying them to education often oversimplied and misconstrued, emphasizing laissez–fair or outdated “discovery” approaches
Learning Trajectories
• Owe much to these efforts
• Include hierarchies, but not as limited as to “logically” determined prerequisites (not always empirically supported)
• Describe levels of thinking, not just ability to correctly respond
• Cannot be summarized by stating definition, concept or rule (cf. Gagné)
Learning Trajectories
• A single problem may be solved differently by students at different (separable) levels of thinking
• Describe how students think about a topic and why—including the cognitive actions-on-objects that constitute that thinking.
• Interactionalist, rather than transmission, view of pedagogy
Developmental Progressions
• Levels, not stages
• Domain (and topic) specific
• Shorter time and cognitive “distance”
• Nongenetic levels
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Developmental Progressions
• Levels, not stages
• Domain (and topic) specific
• Shorter time and cognitive “distance”
• Nongenetic levels
• Actions-on-objects more specific
Developmental Progressions
• Series of levels of thinking reflecting cognitive science view of knowledge as interconnected webs of concepts and skills
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Hierarchic Interactionalism and Learning Trajectories
Types of knowledge develop simultaneously
Shading = probability of instantiation
Arrows = interaction
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
• Greater specificity, consonant with developing actions-on-objects with each level of LT
• Explication and iterative revision allows researcher to test the theory by testing the curriculum
Instruction
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© D. H. Clements. Do not use or duplicate without permission.
The Curriculum Research Framework
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Research Basis
•Many claim a research basis,
•but claims often vacuous, citing theories or empirical results vaguely.
•Curriculum matters—we need a science of curriculum…
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Practice Policy Theory
Effect
Effective in achieving learning
goals?
Credible relative to alternatives?
Effect size?
Curriculum goals important?
Why effective?
Credible relative to alternatives?
Conditions
When and where?
Under what conditions?
Generalize?
Support requirements for various contexts?
Why do conditions in(de)crease effects?
How & why strategies produce
previously unattained results?
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Learning Trajectories
• Learning trajectories core of Curriculum Research Framework
• Foundation on which to build, formatively evaluate, implement, and summatively evaluate a curriculum or intervention
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Curriculum Research Framework*
•A Priori Foundation• General: Broad philosophies, theories, and
empirical results
• Subject Matter• substantive contribution, build from past, generative
• Pedagogical (e.g., computer activities)
•Learning Trajectories
• Goal, developmental progression, instructional tasks
*Clements, D. H. (2007). Curriculum research: Toward a framework for ‘research-based curricula’. Journal for Research in Mathematics Education, 38, 35–70.
A Trajectory for
Composing Geometric
Shapes
Pre-Composer
•Manipulates shapes as individuals, but unable to combine them to compose larger shape
Picture Maker
•Chooses shapes using gestalt configuration or one component such as side length; “pick and discard” strategy
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Shape Composer
•Combines to make new shapes, with anticipation. Chooses shapes using angles as well as side lengths (Intentionality: “I know what fits.”)
With Typical Instruction
With Good Instruction, Spread
With Good Instruction, Focused
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Activities
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Piece Puzzler Series
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Curriculum Research Framework
•Formative Evaluation
•Small Group•Learning trajectory’s elements evaluated --like Koeno’s
•Single Classroom•Meaning teachers and students give in progressively expanding
social contexts
• Intended and unintended outcomes; emergent in complex system
•Multiple Classrooms•Diverse group of teachers
•Support required
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Checking Learning Trajectories
• Qualitative (TEs) used to test all aspects of LTs
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
•Summative
•Small Scale
•4-10 classrooms
•Large Scale
•Scale up, studying moderators and mediators for explanatory power (contextual, implementation variables)
•Fidelity of implementation and sustainability on a large scale (of effects and implementation)
•Diffusion theory; overlapping spheres of influence models
Curriculum Research Framework
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Results: Child Assessment
• p = .000+
• T Scores:
• 50 Mean
• 10 SD 0
4.25
8.50
12.75
17.00
Control Comparison Building Blocks/TRIAD
1.11 .46
Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Building Blocks / TRIAD
• http://www.thebostonchannel.com/news/15776035/detail.html
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
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PrePost
Control
TRIAD
Rasch scores
p < .0001
ES = .72
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FallFall Year 2
Spring Year 2
ES = 1.13
Control
TRIAD
Rasch scores
p < .0001
Trajectories and Technology for Teachers
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
•Good way to answer 3 types of research questions (NRC, 2002): Descriptive, Causal, Process
•Each cycle must “work” to proceed; reveals weaknesses if not
Validity is higher: Construct validity tests more frequent, more trustworthy
Benefits of the CRF w/ LT
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
•Results-centered, minimizing seductive theory-confirming strategies that insidiously replace theory-testing strategies
•Relationships among theory, research, design, and practice more salient and accessible to reflection
•Requires knowledge re: curriculum, professional development, support, etc. —> scale up
Benefits of the CRF w/ LT
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Research Challenges
• Too little known about
• developmental progressions for topics
• connections to instruction (including “skipping”)
• instantiation in CRF’s phases, PD
Impact of LTs
• 3-treatment study—main difference between comparison and BB—LTs
• Topics on which Building Blocks children made the largest relative gains had well-developed LTs
• Formative assessment with LTs as significant mediator
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
• Developmental Progressions provide benchmarks for formative assessments
• LT’s can form a foundation for future curriculum development and…
• can be tested and refined to serve mathematics education.
Advantages of LT
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© D. H. Clements. Do not use or duplicate without permission.(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
The Common Core
© Douglas H. Clements Do not copy, use, or disseminate without permission.(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
• Learning Trajectories at the Core of the Common Core
• At least the developmental progression
What Might Be Missed
http://commoncoretools.wordpress.com/
© Douglas H. Clements Do not copy, use, or disseminate without permission.(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
• 2nd grade: “They develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers.”
• The Critical Areas of the CCSSO and the CFP are the same
What Might Be Missed
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
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(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
Large EffortsUsing LTs
• NRC report
• NMAP (somewhat)
• Common Core
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
References• Sarama, J., & Clements, D. H. (2009). Early childhood
mathematics education research: Learning trajectories for young children. NY: Routledge.
• Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. NY: Routledge.
• Clements, D. H., & Sarama, J. (2011). Early childhood mathematics intervention. Science, 333, 968-970.
• Clements, D. H., Sarama, J., Spitler, M. E., Lange, A. A., & Wolfe, C. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42(2), 127-166.
(c) Douglas H. Clements. Do not copy, use, or disseminate without permission.
• Clements, D. H., & Sarama, J. (2007). Building Blocks Curriculum, Grade PreK. SRA/McGraw-Hill.
• Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443-494.
• Clements, D. H., Sarama, J., & Liu, X. (2008). Development of a measure of early mathematics achievement using the Rasch model: The Research-based Early Maths Assessment. Educational Psychology, 28(4), 457-482.
• Clements, D. H., Sarama, J., & Wolfe, C. B. (2011). TEAM—Tools for early assessment in mathematics. McGraw-Hill.
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