Research on Learning Trajectories in Mathematics and … Mathematics and Science ... •...

Scaling Up Digital Design Studies | NC State University | College of Education

Research on Learning Trajectories in Mathematics and Science

Jere Confrey, Joseph D. Moore Distinguished University Professor,

Alan MaloneyResearch Scientist

NSF Methods at Midday February 13th, 2015


Questions about Learning Trajectories

•  What are they (and what are they not)?

•  Is there a difference between LTs in math and science?

•  How standardized are they for any particular topic?

•  Why develop particular ones?

•  How do you recognize indicators of robust LTs?

•  What structures, theories, and methodologies apply?

•  What use are they?

•  What advice do we have for future directions?


ExamplesMath Science

Masters & Forster (1997): Coun@ng and Ordering

Mohan (2009): carbon cycling in socio-‐ecological systems

Clements, Wilson & Sarama (2004): Composi@on of geometric figures

Catley et al (2005): Evolu@on Furtak (2012): Natural Selec@on

Clements & Sarama (2009): Early mathema@cs (pre-‐K to grade 2)—coun@ng, geometric shapes.

Lehrer and Schauble 2014: data modeling modelingdata.org

Confrey et al. (2009, 2014): Equipar@@oning Shavelson et al., 2005 Buoyancy (Floa@ng and Sinking)

BaUsta, M. (2007); BarreW et al (2012) : Length and length measurement. Songer et al (2009): Biodiversity

Lehrer et al. (2014): Sta@s@cal Reasoning (K-‐8); modelingdata.org

Smith, et al., 2006) Atomic Molecular Theory of MaWer

Cobb et al. (2003): Reasoning about Distribu@ons

Duncan, Rogat & Yarden (2009): Modern Gene@cs

Blanton; Knuth: Early Algebraic Reasoning Schwarz et al (2009): Scien@fic Modeling

Kalchman & Koedinger (2005): Func@ons Structure, Proper@es, and Transforma@on of MaWer. (CPRE Report, Rogat et al. 2011)

TurnOnCCMath, LTs for K-‐8 Mathema@cs. Confrey et al (2011)

Flow of MaWer and Energy in Ecosystems. (CPRE Report, Rogat et al. 2011)


What is a learning trajectory?

•  Focus is on learning as an active process of building and modifying ideas,•  Possible path or set of paths from prior knowledge to more

sophisticated reasoning,•  Establish target,•  Establish base,•  Describe intermediate steps;

–  communal and individual successive reorganizations of content within an engineered instructional environment.

•  States of “likelihood”, not regimented steps or psycholoigical stages;•  Include predictable landmarks and obstacles•  Populates an unsettled relationship (messy middle) to authorized

knowledge in a discipline, brought to local closure through instruction.


Learning Trajectory within a Conceptual Corridor

Confrey (2006) Design Studies Chapter

Cambridge Handbook of the Learning Sciences


What are they not?

•  Logical deconstructions of prerequisites of a domain concept,

•  Thought experiments about sequencing, absent empirical investigation of learning,

•  A linear sequence of curricular tasks based solely on a literature review,

•  A whole discipline

•  Development au naturel.


How standardized are they for any particular topic?

They are:•  Likelihoods and common behaviors.•  Investigated and engineered (science and engineering

based)•  Collectively built by students, scaffolded by teachers, and

emergent•  Task- and tool-dependent •  Dependent on significant opportunities for interaction,

discussion, and divergent thinking •  Grounded in literature explaining similarities and

differences


Are there differences between the nature of LTs in math and science?

•  Science LP Definitions:

•  “… empirically grounded and testable hypotheses about how students’ understanding of, and ability to use, core scientific concepts and explanations and related scientific practices grow and become more sophisticated over time, with appropriate instruction” (p. 8)


Are there differences between the LTs in math and science?

•  Progressions tend to be in science and trajectories in math [nuances in definition of “progressions” esp. in science]

•  Progressions tend to be bigger chunks of material

•  Some distinguish progressions (curriculum) from trajectories (developmental sequences) [Clements and Sarama; Lehrer]

•  Structure of LPs reflects the structure of approved and “resolved” knowledge in the scientific field.

•  Many science LPs have huge meta-theme as” the big idea”

•  Variation in emphasis on student- emergence about models

–  Some ascribe how students come to know predefined models

–  Others emphasize how children bring informal knowledge and experience and link to formal models progressively


Distinction in science LPs

•  Distinction Curriculum and instruction type vs. cognition and instruction type (Shavelson, 2009)

•  Curriculum and instruction type: Sequences of concepts based on logical analyses of big ideas in science to serve as foundations for curricular units.

•  Cognition and instruction type: Maps out a progression for how students come to understand scientific concepts from the naïve everyday ideas to scientific accurate.

“This [curriculum and instruction] type of learning progression is not empirically validated and is highly dependent on context, meaning that the curriculum is being used will influence how students will perform with respect to the learning progression. Songer and Gotwals (2012) among others have argued that learning progressions [ cognition and instruction type] should not represent content alone, but should be in some way integrated with scientific practices that support student learning of science ideas” Furtak 2012


An example from math•  Equipartitioning (or splitting) indicates cognitive behaviors

that have the goal of producing equal-size groups (from collections) or pieces (from continuous wholes) as “fair shares” for each of a set of individuals.

•  Equipartitioning is not breaking, fracturing, fragmenting, or segmenting in which there is the creation of unequal parts.

•  Target understanding: Generalization that a items shared among b people is the same as a/b=a÷b (CC Standard 5.NF.3)


Why develop an equipartitioning learning trajectory?

•  multiplication and division unduly delayed until 3rd grade as derived from addition and subtraction,

•  remarkable competence evident in fair sharing and the discourse surrounding it,

•  different roots postulated for counting and splitting from a 3-year teaching experiment (grade 3-5)(Confrey, 1991),

•  different cases of fair sharing researched separately lacking integration, and

•  weak performance on the target (a divided by b = a/b) as a big idea


Evenly Divisible Collec4ons

Wholes

Equipar44oning Collec4ons 1.EQP.A

Reassembly of Equipar44ons 1.EQP.B, 2.G.3

Qualita4ve Compensa4on 1.G.3

Composi4ons of splits; factor-‐pairs 2.G.2, 2.EQP.D

Con4nuity Principle of Equipar44oning 2-‐4.EQP.C

Mul4ple Wholes

Factor-‐based changes in equipar44ons (quan4ta4ve compensa4on) 2-‐3.EQP.E

Emergent Rela4ons and Proper4es in

Collec4ons and Single Wholes

Prac4ces applied to Collec4ons and

Wholes Naming the Results of Equipar44oning 1.EQP.A, 2.EQP.B,, 2.G.3,

3.G.2, 3.NF.1

Jus4fying the Results of Equipar44oning 1.EQP.A, 1.G.3, 2.G.3, 3.G.2

Equipar44oning Wholes 1.G.3, 2.G.3

Property of Equality of Equipar44oning (PEEQ) 2.G.3

Realloca4on (Collec4ons) 2-‐3.EQP.F

Distribu4ve strategies: equipar44oning over breaking or fracturing. 4-‐5.EQP.A

Generaliza4on of a objects shared among b sharers as a/b

Equipar44oning Mul4ple Wholes 4.EQP.A

Co-‐spliOng 2-‐4.EQP.A

The Equipar@@oning Learning Trajectory and the Standards


Close listening and observing


Equipartitioning Wholes


Example of New Construct: PEEQ

•  Property of Equality of Equipartitioning (PEEQ): If two congruent shapes are each split for the same number of persons, then the size of the shares from one of the shapes is equal to the size of the shares from the other shape, regardless of the shape of the shares.


Read the following problem, and predict how students will solve it.


transitivity argument: shares from both splits are still ½ of the same size brownie)…

…or shows equivalence by compensation or decomposition

Student demonstrates equivalent results of two strategies


How do you recognize indicators of robust LTs?•  Synthesis of Literature, integrating disparate cases

–  Sharing Collections (Hunting & Sharpley, 1991; Pepper, 1991);

–  Sharing Wholes (Piaget, 1971; Pothier & Sawada, 1983; Confrey et al., 2009; Empson & Turner, 2006);

–  Sharing Multiple Wholes (Charles & Nason 2000; Lamon 1966)

•  Identifying a foundation for whole trajectory: three criteria for equipartitioning-

–  equal-sized,

–  correct number,

–  exhaust the whole (Piaget)

•  Finding new constructs – –  re-assembly as the reverse of partitive vs. quotative/measurement division. “Times

as many” and “1/nth of”


How do you recognize indicators of robust LTs?

•  Utilizing collective socio-mathematical activity: -- naming and justifying (“times as many or much”)

•  Leveraging misconceptions: Composition of splits – seeing how repeated splitting played out

•  Encountering Surprises: Easier to do 7 shared among 4 than 4 shared among 7


Look-fors

•  A robust learning trajectory should include: new constructs, surprises, options, testable hypotheses, and data


What kinds of structures, evidence, and validation apply?

•  Theory

•  Methodology

•  Evidence Claims

•  Utility arguments


What theories?

•  Generally constructivist or soci-cultural or both

•  Emergent instructional theories that scaffold content (not just supporting learning but fostering it)

•  Grand and humble theories that do work


Strategies

Conven@ons

Proper@es

Sophis@cated Strategies

Generaliza@ons STRU

CTURE


What are the phases of LT Research and Development?

•  Selecting the target (big idea, difficult to learn, affected by new tools or ways of thinking)

•  Literature review and/or synthesis•  Conducting initial clinical interviews

–  Design tasks, sequences, and tools

•  Short design cycles with instructional supports–  Pre- and post assessments–  Materials development, tools, and/or assessment to elicit thinking

Establishing classroom norms –  Creating data collection methods

•  Longer design studies

•  Scale up


Design Study Methodology

Five features of Design Studies:

1.  generating theory

2.  implementing highly interventionist strategies,

3.  iterative design across three phases of the study (planning, implementation, and retrospective analysis)

4.  placing “humble” domain-specific theories in harm’s way (being willing to revise or reject depending on observations and results)

5.  requiring theories to be accountable to the activity of designàrequiring them to successfully do work if they are to survive.

Cobb, Confrey, diSessa, Lehrer & Schauble 2003)


Item Validation for assessing LTs

•  Design for item validation

– ~4800 children across K-7

–  33 test forms, with overlapping items for calibration


Method: Item Analysis

Each X represents 10 student cases.

Numbers indicate the item number; Colors represent location of item in the framework for understanding

KEY Strategies and representations: Red

Mathematical Practices: Red Emergent Properties: Green

Generalizations: Blue

Wright Map from IRT analysis: Grades 2-3-4-5-6


IRT: difficulty estimates—proficiency levels


To what use has the EQP LT been put?

•  LTBI-Learning Trajectory-based Instruction project (NSF)

•  MOOCs on equipartitioning (Maloney)

•  CCSS—incorporated principles of EQP

•  TurnOnCCMath: LTs for K-8 mathematics to interpret CCSS

•  Licensing to commercial firm for curriculum product development

•  Gates Foundation Learning Map work


What use are LTs?

Valued for:•  Informing standards development•  Designing professional development (Clements & Sarama,

Wilson, Mojica, Learning Trajectory-Based Instruction)•  Developing formative assessment practices (Heritage et

al., Furtak)•  Building diagnostic assessments (Confrey, Izsak, Lehrer/

Wilson)•  Curriculum development (Clements et al., )•  Large-scale assessment (Alonzo et al)


Caveat on Professional Development Use

•  “.. several of the teachers seemed to use the learning progressions simply as catalogs of misconceptions to be “squashed”, rather than drawing upon the developmental affordances offered by a learning progression.” (Futak, 2012)


Looking forward to a new genre

•  Deeply digital learning systems

•  Learning Maps and related assessment systems (supported by Gates)

•  Examples: SUDDS, Dynamic Learning Maps, Enlearn, CRESST, GlassLabs, New Classrooms

•  Facilitating navigation, depth of knowledge, and analytical activity

•  Demo map


NSF’s role?•  LTs/LPs provide a foundation for STEM ed, right grain-size

•  Need additional research in specific topic areas:–  Less-investigated topics and big ideas,

–  high school topics: LTs require integration of multiple strands (such as RLCs)

•  STEM Centers to support programmatic efforts including libraries of exemplars, means to leverage for applied uses, related tasks and assessments access–  Library of how LT/LP work can be conducted programatically

–  Topic mini-centers (Smith, J., MSU: length and area)

–  Modelingdata.org: Rich Lehrer’s IES site is a great example (PW modeling)


NSF’s role?

•  Should fit into a larger context of continuous school improvement and equity (Bryk and Gomez) and deeply digital strategies (Cyberlearning) with clear links to:–  curriculum

–  Through-course or diagnostic assessment measures

–  Professional development

–  Targeted accelerated learning for low performers


References•  Alonzo, A. C. & Steedle, J. T. (2009). Developing and assessing a force and motion learning progression. Science

Education, 93(3), 389–421.

•  Battista, M. T. (2006). Understanding the development of students’ thinking about length. Teaching children mathematics 13(3), 140-147.

•  Catley, K., Lehrer, R., & Reiser, B. (2005). Tracing a prospective learning progression for developing understanding of evolution. Paper Commissioned by the National Research Council Board of Testing and Assessment, Committee on Test Design for K-12 Science Achievement. Washington, DC: National Research Council.

•  Clements, D. H. & Sarama, J. (2014). Learning trajectories: Foundations for effective, research-based education. In Maloney, A. P., Confrey, J. & Nguyen, K. H. (Eds.) Learning over time: Learning trajectories in mathematics education. Charlotte: Information Age Publishers, pp. 1-29.

•  Clements, D. H., Wilson, D. C. & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning 6, 163-184.

•  Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.

•  Cobb, P. (1999) Individual and Collective Mathematical Development: the Case of Statistical Data Analysis. Mathematical Thinking and Learning 1(1), 5-43.

•  Cobb, P., McClain, K. & Gravemeijer, K. (2003). Learning about statistical variation. Cognition and Instruction 21(1), 1-78.

•  Confrey, J., Maloney, A. P., Nguyen, K. H. & Rupp, A. A. (2014). Equipartitioning, a foundation for rational number reasoning. In Maloney, A. P., Confrey, J. & Nguyen, K. H. (Eds.) Learning over time: Learning trajectories in mathematics education. Charlotte: Information Age Publishers, pp. 61-96.

•  Duncan, R. G., Rogat, A. D. & Yarden, A. (2009). A learning progression for deepening students’ understandings of modern genetics across the 5th–10th grades. Journal of Research on Science Teaching 46(6), 655-674.


References•  Forster, M. & Masters, G. (2004). Bridging the Conceptual Gap between Classroom Assessment and System

Accountability. In M. Wilson (ed.), Towards Coherence. •  Fosnot, Catherine and Dork, Marteen (2002) Young Mathematicians at Work: Constructing Fractions, Decimals, and

Percents. Portsmouth: Heinemann.

•  Furtak, E. M. (2012) Linking a learning progression for natural selection to teachers’ enactment of formative assessment. Journal of Research in Science Teaching 49(9), 1181-1210.

•  Gravenmeijer, K. (1994) Developing Realistic Mathematics Education, Utrecht, the Netherlands: CD-B Press.

•  Heritage, M 2008. Learning progressions: supporting instruction and formative assessment. Paper prepared for the Formative Assessment for Teachers and Students (FAST) State Collaborative on Assessment and Student Standards (SCASS) of the Council of Using Learning Progressions to Monitor Progress Across Grades. Washington DC: Council of Chief State School Officers (CCSSO).

•  Hess, Karin, Methods and strategies: Learning progressions to monitor progress across grades Science and Children 47(6), NSTA, Feb 2010.

•  Kalchman, M. & Koedinger, K. R. (2005). Teaching and learning functions. In National Research Council (Ed.), How Students Learn. Washington, D. C.: National Academies Press. 351- 393.

•  Lehrer, R., Kim, M.-J., Ayers, E. & Wilson, M. (2014). Toward establishing a learning progression to support the development of statistical reasoning. In Maloney, A. P., Confrey, J. & Nguyen, K. H. (Eds.) Learning over time: Learning trajectories in mathematics education. Charlotte: Information Age Publishers, pp. 31-59.

•  Maloney, A. P., Confrey, J. & Nguyen, K. H. (Eds.) Learning over time: Learning trajectories in mathematics education. Charlotte: Information Age Publishers, pp. 61-96.

•  Masters, G., & Forster, M. (1997). Developmental assessment. Victoria, AU: The Australian Council for Educational Research Ltd.

•  Mohan, L., Chen, J. & Anderson, C. W. (2009). Developing a multi-year learning progression for carbon cycling in socio-ecological systems. Journal of Research on Science Teaching 46(6), 675-698.


References•  NRC (2007). Duschl, R, Schweingruber, H. & Shouse, A. (Eds) Taking Science to school: Learning and Teaching Science in

Grades K-8. Washington D.C: National Academies Press.

•  Rogat, A., Anderson, C. W., Foster, J., Goldberg, F., Hicks, J., Kanter, D., Krajcik, J., Lehrer, R., Reiser, B., & Wiser, M. (2011). Developing learning progressions in support of the new science education standards: A RAPID workshop series. Consortium for Policy Research in Education, Teachers College.

•  Sarama, J. & Clements, D. H. (2009a). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.

•  Schwarz. C. V., Reiser, B. J., Davis, E. A., Kenyon, L., Achér, A., Fortus, D., Shwartz, Y., Hug, B. & Krajcik, J. (2009). Developing a learning progression for scientific modeling: Making scientific modeling accessible and meaningful for learners. Journal of Research on Science Teaching 46(6), 632-654.

•  Shavelson, R. J. & Kurplus, A. 2012. Reflections on learning progressions. In Alonzo, A. C. & Gotwals, A. W. (Eds.) Learning Progressions in Science. Rotterdam: Sense Publishers.

•  Smith, C. L., Wiser, M., Anderson, C. W. & Krajcik, J. (2006). Implications of research on children’s learning for standards and assessment: A proposed learning progression for matter and atomic-molecular theory. Measurement 14(1&2). 1-98.

•  From Shavelson, R., Stanford Educational Assessment Laboratory (SEAL) and Curriculum Research & Development Group (CRDG). (2005). Embedding assessments in the FAST curriculum: The romance between curriculum and assessment. Final Report.


•  Questions?

•  Thank you.

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