Learning Trajectories in Mathematics

1
Learning Trajectories in Mathematics

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

2
Consortium for Policy Research in Education
(CPRE)

• Prepared by
• Phil Daro
• CCSS, member of lead writing team
• Frederic A. Mosher
• CPRE, Sr. Research Consultant
• Tom Corcoran
• CPRE, Co-director
• January 2011

3
Learning Trajectories

• 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

4
Learning Trajectories

• learning progressions which characterize paths
  children seem to follow as they learn
  mathematics.
• Piagets Genetic Epistemology
• Vygotskys Zone of Proximal Development

5
Development of Learning Trajectories vs. CCSS

• Learning Trajectories begin by defining a
  starting point based on childrens 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.

6
Learning Trajectories

• 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.

7
Shape Composing Trajectory
Based on Doug Clements Julie Saramas in
Engaging Young Children In Mathematics (2004).
8
Pre-Composer

• Free exploration with shapes
• Manipulation of shapes as individuals
• No combining of shapes to compose larger shapes

9
Picture Composer

• 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.

10
Picture Maker

• Moves from using pick and discard strategy to
  placing shapes intentionally.
• Good alignment of sides and improving alignment
  of angles

11
Shape Composer

• Combines shapes to make new shapes with
  anticipation.
• Chooses shapes using angles as well as side
  length.
• Intentionality based.

12
Substitution Composer

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

13
Learning Trajectory for Composing Geometric Shapes

• Pre-composer Free exploration with shapes
• 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.

14
Trajectories can be used

• to develop instructional tasks that
• support student movement of understanding from
  one level to another in specific ways
• elicit and assess student understandings

15
The blank puzzle illustrates the type of
structure that will challenge and help a child
move their skills along the trajectory
16
Picture Maker Example
17
Some Trajectories

• 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.

18
Aim of Trajectories

• Are chronologically predictive
• 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
• for example deepened conceptual understanding and
  transferability of knowledge and skills as
  determined by assessment
• Have learning goals that are mathematically
  valuable
• align with broad agreement on what math students
  ought to learn (as reflected in the CCSS)

19
Trajectories Might Serve CCSS

• 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

20
Getting the sequence right is not guaranteed

• 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

21
Learning Trajectory Researchers

• 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

22
Value of Learning Trajectories

• 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.

23
Number Core Trajectory

• Seeing how many objects there are (cardinality)
• Knowing the number word list (one, two, )
• 1-1 correspondences when counting
• Written number symbols

24
Multiplicative Reasoning and Rational Number
Reasoning

• Equi-partitioning
• Multiplication and division
• Fraction as number
• Ratio and Rate
• Similarity and Scaling
• Linear and Area measurement
• Decimals and Percents

25
Multiplication Strategies

• Count all
• Additive calculation
• Count by
• Patterned based
• Learned products
• Hybrids of these strategies

26
Spatial Thinking

• In, on, under, up and down
• Beside and between
• In front of, behind
• Left, right

27
Measurement

• 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