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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


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)


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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


• 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)


• …identified within theoretically- and empirically-grounded models of children’s thinking, learning, and development

• Hierarchic Interactionalism

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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…


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


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


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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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


• 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)


• 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


• Levels, not stages

• Domain (and topic) specific

• Shorter time and cognitive “distance”

• Nongenetic levels

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• Levels, not stages

• Domain (and topic) specific

• Shorter time and cognitive “distance”

• Nongenetic levels

• Actions-on-objects more specific


• Series of levels of thinking reflecting cognitive science view of knowledge as interconnected webs of concepts and skills


Hierarchic Interactionalism and Learning Trajectories

Types of knowledge develop simultaneously

Shading = probability of instantiation

Arrows = interaction


• 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


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…


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?



• 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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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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Activities


Piece Puzzler Series


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


Checking Learning Trajectories

• Qualitative (TEs) used to test all aspects of LTs

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•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


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

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Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal.


Building Blocks / TRIAD

• http://www.thebostonchannel.com/news/15776035/detail.html


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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



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•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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•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


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


• 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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Large EffortsUsing LTs

• NRC report

• NMAP (somewhat)

• Common Core


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.


• 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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