Learning Trajectories
in Mathematics
A Foundation for Standards,
Curriculum, Assessment, and
Instruction
Consortium for Policy Research
in Education (CPRE)
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Prepared by
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Phil Daro
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Frederic A. Mosher
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CPRE, Sr. Research Consultant
Tom Corcoran
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CCSS, member of lead writing team
CPRE, Co-director
January 2011
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Learning Trajectories
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typical, predictable sequences of thinking
that emerge as students develop
understanding of an idea
modal descriptions of the development of
student thinking over shorter ranges of
specific math topics
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Learning Trajectories
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learning progressions which characterize
paths children seem to follow as they
learn mathematics.
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Piaget’s Genetic Epistemology
Vygotsky’s Zone of Proximal Development
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Development of Learning
Trajectories vs. CCSS
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Learning Trajectories begin by defining a starting
point based on children’s entering understanding
and skills and then working forward
CCSS were begin at the level of college and
career ready standards backwards down through
the grades. This mapping is based on a logical
rendering of the set of desired outcomes needed
to define pathways or benchmarks to the
standard.
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Learning Trajectories
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Are too complex and too conditional to
serve as standards. Still learning
trajectories point to the way to optimal
learning sequences and warn against the
hazards that could lead to sequence
errors.
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Shape Composing Trajectory
Based on Doug Clements’ & Julie Sarama’s in Engaging Young Children In
Mathematics (2004).
Pre-Composer
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Free exploration
with shapes
Manipulation of
shapes as
individuals
No combining of
shapes to compose
larger shapes
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Picture Composer
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Matches shapes
Puts several shapes
together to make one
part of a picture
Uses “pick and discard”
strategy, rather than
intentional action
Notices some aspects
of sides but not angles.
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Picture Maker
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Moves from using
“pick and discard”
strategy to placing
shapes intentionally.
Good alignment of
sides and improving
alignment of angles
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Shape Composer
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Combines shapes to make new shapes
with anticipation.
Chooses shapes using angles as well as
side length.
Intentionality based.
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Substitution Composer
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Creates different ways to fill a frame
emphasizing substitution relationships.
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Learning Trajectory for
Composing Geometric Shapes
1. Pre-composer: Free exploration with shapes
2. Picture maker: Makes one part of a picture
(arms on pattern block person but not legs)
3. Shape composer:
More advanced. Chooses
shapes with certain angles and length of sides. “I
know that will fit!”
4. Substitution composer:
yet more advanced.
Can take hexagon outline and fill it in different
ways to make a hexagon with pattern blocks.
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Trajectories can be used
to develop instructional tasks that:
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support student movement of
understanding from one level to another
in specific ways
elicit and assess student understandings
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The blank puzzle illustrates the type of
structure that will challenge and help a child
move their skills along the trajectory
Picture Maker Example
Some Trajectories
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Present a continuum of tasks that are well
connected and build on each other in specific
ways over time
Present tasks that connect across topical areas
of school math
Offer detailed guidance to teachers in
understanding the capacities and misconceptions
of their students at different points in their
learning of a particular topic.
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Aim of Trajectories
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Are chronologically predictive
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In the sense of what students do (or are able to with
appropriate instruction) move successfully from one
level to the next
Yield positive results
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for example deepened conceptual understanding and
transferability of knowledge and skills as determined
by assessment
Have learning goals that are mathematically valuable
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align with broad agreement on what math students ought to
learn (as reflected in the CCSS)
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Trajectories Might Serve CCSS
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by defining more clearly the agreed upon
goals for which specific learning
trajectories must still be developed
because they describe pivotal concepts of
school math
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Getting the sequence right is not
guaranteed
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It involves testing hypothesized
dependency of one idea on another, with
particular attention to areas where
cognitive dependencies are potentially
different from logical dependencies as a
mathematician sees them
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Learning Trajectory Researchers
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Are answering questions about when
instruction should follow a logical
sequence of deduction from precise
definitions and when instruction that
builds on a more complex mixture of
cognitive factors and prior knowledge is
more effective
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Value of Learning Trajectories
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Offer a basis for identifying interim goals that students
should meet
Provide understandable points of reference for designing
assessments that point to where students are, rather
than merely their final score.
Adaptive instruction thinking your sole goal is to gather
actionable information to inform instruction and student
learning, not to grade or evaluate achievement
Could help teachers manage a wide variety of individual
learning paths by identifying a more limited range of
specific types of reasoning for a given type of problem.
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Number Core Trajectory
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Seeing how many objects there are
(cardinality)
Knowing the number word list (one, two,
…)
1-1 correspondences when counting
Written number symbols
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Multiplicative Reasoning and
Rational Number Reasoning
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Equi-partitioning
Multiplication and division
Fraction as number
Ratio and Rate
Similarity and Scaling
Linear and Area measurement
Decimals and Percents
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Multiplication Strategies
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Count all
Additive calculation
Count by
Patterned based
Learned products
Hybrids of these strategies
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Spatial Thinking
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In, on, under, up and down
Beside and between
In front of, behind
Left, right
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Measurement
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Compare sizes
Connect number to length
Measurement relating to length
Measuring and understanding units
Length-unit iteration
Correct alignment with ruler
Concept of the zero point
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