1. No category

Learning Trajectories:
A research lens for enhancing
formative assessment.
Caroline B. Ebby, CPRE, University of Pennsylvania [email protected]
Marge Petit, Marge Petit Consulting, MPC [email protected]
S
Take Aways
S  Effective math instruction and planning for instruction requires
careful attention to evidence of student thinking.
S  Using learning trajectories/progressions to understand evidence of
student thinking provides actionable information based on
research on how students learn specific mathematics concepts
S  The “essence” of formative assessment is the relentless attention
to evidence of student understanding and intentional and
systematic use of the evidence for planning and instruction.
(Popham, 2012)
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Formative Assessment
Feedback (1)
Assessment
Learner
Goal
Teacher
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Ebby and Petit April 2015
Learning Trajectories
S  “Learning progressions are descriptions of the successively
more sophisticated ways of thinking about a topic that can
follow one another as children learn about and investigate a
topic” (NRC, 2007, p. 214)
S  Developmental progressions of strategies, concepts and
levels of student thinking in particular mathematical
domains
S  Links research on learning and instructional practice
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Ebby and Petit April 2015
Formative Assessment
Feedback (1)
Assessment
Learner
Goal
Learning Trajectory
Teacher
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Ebby and Petit April 2015
Calculus
P
r
a
c
ti
c
e
s
High
School
Functions and
Algebra
Equations and
Expressions
Operations and
Algebraic Thinking
M
a
t
h
e
m
a
ti
c
a
l
The Number
System
Fractions
6-8
Proportionality
Multiplicative Reasoning
Additive Reasoning
6
3-5
K-2
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Looking at Student Work
S Review the samples of
student work
S What do you notice? Make
some observations
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Ebby and Petit April 2015
An Example:
The OGAP
Multiplicative
Reasoning
Progression
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Ebby and Petit April 2015
The OGAP Sort
Multiplicative
Strategies
Fractional!
Transitional
Strategies
Transitional!
Additive
Strategies
Early!Fractional!
Nonmultiplicative
Non0!Fractional!
!
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Instructional Implications
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Students move back and forth…
Middle School
Application
… as they interact with
new contexts, different
magnitude numbers,
number of factors,
and more.
Non-multiplicative
Concept development over time
Petit, Laird, Marsden, & Ebby, in
press 2015
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CCSSM
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Learning Trajectory-Oriented
Formative Assessment
Feedback (1)
Assessment
Learner
Goal
Learning Trajectory
Teacher
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Ebby and Petit April 2015
Citations
S  Hulbert, E. Petit, & Laird, R. (2013). Ongoing assessment project: Professional
development materials. Moretown, VT: Ongoing Assessment Project.
S  Popham, W. J. (2012). Forward. In E. Wylie, A. Gullickson, K. Cummings, L.
Noakes, K. Norman, & S. Veeder (Eds.), Improving formative assessment to
empower student learning (pp. ix-xii). Thousand Oaks, CA: Corwin Press.
S  Petit, Laird, Marsden, & Ebby (in press 2015). A Focus on Fractions: Bringing
Research to the Classroom. Routledge, NY, NY.
S  Petit, Hulbert, & Laird (2013). OGAP Multiplicative Reasoning Framework.
Moretown, Vermont.
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