Learning Trajectories Learning Trajectories in Mathematicsin Mathematics

A Foundation for Standards, A Foundation for Standards, Curriculum, Assessment, and Curriculum, Assessment, and

InstructionInstruction

22

Consortium for Policy Research Consortium for Policy Research

in Education (CPRE)in Education (CPRE) Prepared by Prepared by

Phil DaroPhil Daro CCSS, member of lead writing teamCCSS, member of lead writing team

Frederic A. MosherFrederic A. Mosher CPRE, Sr. Research Consultant CPRE, Sr. Research Consultant

Tom CorcoranTom Corcoran CPRE, Co-director CPRE, Co-director

January 2011January 2011

33

Learning TrajectoriesLearning Trajectories

typical, predictable sequences of typical, predictable sequences of thinking that emerge as students thinking that emerge as students develop understanding of an ideadevelop understanding of an idea

modal descriptions of the modal descriptions of the development of student thinking over development of student thinking over shorter ranges of specific math shorter ranges of specific math topicstopics

44

Learning TrajectoriesLearning Trajectories

learning progressions which learning progressions which characterize paths children seem to characterize paths children seem to follow as they learn mathematics.follow as they learn mathematics. Piaget’s Genetic EpistemologyPiaget’s Genetic Epistemology Vygotsky’s Zone of Proximal Vygotsky’s Zone of Proximal

DevelopmentDevelopment

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Development of Learning Development of Learning Trajectories vs. CCSSTrajectories vs. CCSS

Learning Trajectories begin by defining a Learning Trajectories begin by defining a starting point based on children’s entering starting point based on children’s entering understanding and skills and then working understanding and skills and then working forwardforward

CCSS were begin at the level of college CCSS were begin at the level of college and career ready standards backwards and career ready standards backwards down through the grades. This mapping is down through the grades. This mapping is based on a logical rendering of the set of based on a logical rendering of the set of desired outcomes needed to define desired outcomes needed to define pathways or benchmarks to the standard.pathways or benchmarks to the standard.

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Learning TrajectoriesLearning Trajectories

Are too complex and too conditional Are too complex and too conditional to serve as standards. Still learning to serve as standards. Still learning trajectories point to the way to trajectories point to the way to optimal learning sequences and warn optimal learning sequences and warn against the hazards that could lead against the hazards that could lead to sequence errors.to sequence errors.

Shape Composing Trajectory

Based on Doug Clements’ & Julie Sarama’s in Engaging Young Children In Mathematics (2004).

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Pre-ComposerPre-Composer

Free exploration Free exploration with shapeswith shapes

Manipulation of Manipulation of shapes as shapes as individualsindividuals

No combining of No combining of shapes to shapes to compose larger compose larger shapesshapes

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Matches shapes Matches shapes Puts several shapes Puts several shapes

together to make together to make one part of a picture one part of a picture

Uses “pick and Uses “pick and discard” strategy, discard” strategy, rather than rather than intentional actionintentional action

Notices some Notices some aspects of sides but aspects of sides but not angles.not angles.

Picture ComposerPicture Composer

1010

Moves from using Moves from using “pick and discard” “pick and discard” strategy to placing strategy to placing shapes shapes intentionally. intentionally.

Good alignment of Good alignment of sides and sides and improving improving alignment of alignment of anglesangles

Picture MakerPicture Maker

1111

Shape ComposerShape Composer

Combines shapes to make new Combines shapes to make new shapes with anticipation. shapes with anticipation.

Chooses shapes using angles as well Chooses shapes using angles as well as side length. as side length.

Intentionality based.Intentionality based.

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Substitution ComposerSubstitution Composer

Creates different ways to fill a frame Creates different ways to fill a frame emphasizing substitution emphasizing substitution relationships.relationships.

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Learning Trajectory for Learning Trajectory for Composing Geometric Composing Geometric

ShapesShapes1.1. Pre-composer:Pre-composer: Free exploration with Free exploration with

shapesshapes

2.2. Picture maker:Picture maker: Makes one part of a picture Makes one part of a picture (arms on pattern block person but not legs)(arms on pattern block person but not legs)

3. Shape composer: 3. Shape composer: More advanced. More advanced. Chooses shapes with certain angles and length Chooses shapes with certain angles and length of sides. “I know that will fit!”of sides. “I know that will fit!”

4. Substitution composer: 4. Substitution composer: yet more yet more advanced. Can take hexagon outline and fill it advanced. Can take hexagon outline and fill it in different ways to make a hexagon with in different ways to make a hexagon with pattern blocks.pattern blocks.

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Trajectories can be usedTrajectories can be used

to develop instructional tasks that:to develop instructional tasks that: support student movement of support student movement of

understanding from one level to understanding from one level to another in specific waysanother in specific ways

elicit and assess student elicit and assess student understandingsunderstandings

The blank puzzle illustrates the type of structure that will challenge and help a child move their skills along the trajectory

Picture Maker Example

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Some TrajectoriesSome Trajectories

Present a continuum of tasks that are well Present a continuum of tasks that are well connected and build on each other in connected and build on each other in specific ways over timespecific ways over time

Present tasks that connect across topical Present tasks that connect across topical areas of school mathareas of school math

Offer detailed guidance to teachers in Offer detailed guidance to teachers in understanding the capacities and understanding the capacities and misconceptions of their students at different misconceptions of their students at different points in their learning of a particular topic.points in their learning of a particular topic.

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Aim of TrajectoriesAim of Trajectories

Are chronologically predictiveAre chronologically predictive In the sense of what students do (or are able to In the sense of what students do (or are able to

with appropriate instruction) move successfully with appropriate instruction) move successfully from one level to the nextfrom one level to the next

Yield positive resultsYield positive results for example deepened conceptual understanding for example deepened conceptual understanding

and transferability of knowledge and skills as and transferability of knowledge and skills as determined by assessmentdetermined by assessment

Have learning goals that are mathematically Have learning goals that are mathematically valuablevaluable

align with broad agreement on what math students align with broad agreement on what math students ought to learn (as reflected in the CCSS)ought to learn (as reflected in the CCSS)

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Trajectories Might Serve Trajectories Might Serve CCSSCCSS

by defining more clearly the agreed by defining more clearly the agreed upon goals for which specific learning upon goals for which specific learning trajectories must still be developed trajectories must still be developed because they describe pivotal because they describe pivotal concepts of school mathconcepts of school math

2020

Getting the sequence right is Getting the sequence right is not guaranteednot guaranteed

It involves testing hypothesized It involves testing hypothesized dependency of one idea on another, dependency of one idea on another, with particular attention to areas with particular attention to areas where cognitive dependencies are where cognitive dependencies are potentially different from logical potentially different from logical dependencies as a mathematician dependencies as a mathematician sees themsees them

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Learning Trajectory Learning Trajectory ResearchersResearchers

Are answering questions about when Are answering questions about when instruction should follow a logical instruction should follow a logical sequence of deduction from precise sequence of deduction from precise definitions and when instruction that definitions and when instruction that builds on a more complex mixture of builds on a more complex mixture of cognitive factors and prior cognitive factors and prior knowledge is more effectiveknowledge is more effective

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Value of Learning Value of Learning TrajectoriesTrajectories

Offer a basis for identifying interim goals that students Offer a basis for identifying interim goals that students should meetshould meet

Provide understandable points of reference for Provide understandable points of reference for designing assessments that point to where students designing assessments that point to where students are, rather than merely their final score. are, rather than merely their final score.

Adaptive instruction thinking your sole goal is to Adaptive instruction thinking your sole goal is to gather actionable information to inform instruction gather actionable information to inform instruction and student learning, not to grade or evaluate and student learning, not to grade or evaluate achievementachievement

Could help teachers manage a wide variety of Could help teachers manage a wide variety of individual learning paths by identifying a more limited individual learning paths by identifying a more limited range of specific types of reasoning for a given type of range of specific types of reasoning for a given type of problem.problem.

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Number Core TrajectoryNumber Core Trajectory

Seeing how many objects there are Seeing how many objects there are (cardinality)(cardinality)

Knowing the number word list (one, Knowing the number word list (one, two, …)two, …)

1-1 correspondences when counting1-1 correspondences when counting Written number symbolsWritten number symbols

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Multiplicative Reasoning and Multiplicative Reasoning and Rational Number ReasoningRational Number Reasoning

Equi-partitioning Equi-partitioning Multiplication and divisionMultiplication and division Fraction as numberFraction as number Ratio and RateRatio and Rate Similarity and ScalingSimilarity and Scaling Linear and Area measurement Linear and Area measurement Decimals and PercentsDecimals and Percents

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Multiplication StrategiesMultiplication Strategies

Count allCount all Additive calculationAdditive calculation Count byCount by Patterned basedPatterned based Learned productsLearned products Hybrids of these strategiesHybrids of these strategies

2626

Spatial ThinkingSpatial Thinking

In, on, under, up and downIn, on, under, up and down Beside and betweenBeside and between In front of, behindIn front of, behind Left, rightLeft, right

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MeasurementMeasurement

Compare sizesCompare sizes Connect number to lengthConnect number to length Measurement relating to lengthMeasurement relating to length Measuring and understanding unitsMeasuring and understanding units Length-unit iterationLength-unit iteration Correct alignment with rulerCorrect alignment with ruler Concept of the zero pointConcept of the zero point