Advances in domain-centered models of student learning (learning trajectories), technology, and formative assessment have stimulated development of diagnostic assessment systems in mathematics and science.  These hold great promise as means to provide guidance for teachers and improve student learning, as illustrated at a recent national conference (November, 2010) involving experts from mathematics and science education, measurement and assessment, technology design, and teacher education to explore development and design of "Interactive Diagnostic Assessment Systems" or IDAS (Author1 et al 2011).  The systems are technologically enabled environments for documenting student learning in relation to empirically-derived cognitive models, designed to leverage information in real time for instructional decision-making guidance. Reliance on a learning progression provides teachers with a scientific basis on which to respond formatively to student thinking and foster interaction and discourse (Heritage 2010).

The equipartitioning learning trajectory modules ("ePackets") of our IDAS prototype, LPPSync, have now been completed.  They have been piloted with a group of 12 urban 2^nd through 4^th graders in a two-week teaching experiment, with other ongoing in-class use.  Project staff will report on findings.  We have reported elsewhere on the construct validity of this trajectory (Author1 et al., in review).  Delivered on touch-screen networked devices via secure browser connections, LPPSync incorporates sequestered diagnostic assessments, collaborative practice environments that support peer-to-peer communication and work-sharing, and various types of instantaneous individual and summary feedback reports to students and teachers. 

This work session will explore the relationship of LPPSync design to system content and behavior and to accompanying curriculum.  Participants will consider how this IDAS can be integrated into instruction and can facilitate teacher and researcher understanding of the nuanced emergent properties and generalizations that students make when working with virtual manipulatives.  At hands-on stations, project staff will introduce participants to LPPSync ePackets, the assessments and practice situations that focus on different subconstructs of the equipartitioning learning trajectory. 

Session structure:

Introduction (15 minutes):  LPPSync Project Leaders will introduce participants to recent progress on the design and implementation of LPPSync Equipartitioning Diagnostic e-Packets.

Parallel hands-on sessions (60 minutes): Four stations, each led by project staff member. Session participants will work with one or more diagnostic packets:  Reallocation of equipartitioned collections, Composition, the Property of Equality of Equipartitioning, and Covariation. 

Discussion (15 minutes):  Project leaders, staff, and participants

Note:  iPads will be provided. However, participants are encouraged to bring their own iPads to the work session.

Focus questions:

1.     How can the use of an IAS be embedded within a routine of class practice?

2.     How does the introduction of an equipartitioning curriculum change our understanding of the learning trajectory and its related diagnostic assessments?

3.     What evidence is produced on student learning by the use of the IDAS, and how do teachers report using these data?

LPPSync interactive stations.

Project staff will guide participants in use of individual LPPSync ePackets, address design of specific ePackets in relation to the learning trajectory and the system configuration, and report on specific empirical results.  Individual ePackets listed below in order from lower to higher proficiency levels of Equipartitioning learning trajectory.

1. Reallocation (packet 3) refers to strategies and justifications used in generating a new set of fair shares from an existing equipartitioned collection upon the arrival or departure of one or more of the persons sharing.  This proficiency level is organized around three parameterized tasks—reallocation upon departure of a person, reallocation upon arrival of a new person, and remediating uneven shares, and involve integral numbers of items in the original and final shares. There are three reallocation tasks, which are reallocation departure, reallocation arrival and remediating uneven shares tasks. These cases anticipate a nascent form of the distributive property (equipartitioning distributed over breaking).  This station will focus on students' strategies on these three types of reallocation tasks, examining how the task attributes affects student performance.

2. Understanding Composition of equipartitioning (packet 4) is a key stepping stone to learning both division and multiplication and area measurement (Pothier & Sawada 1981; Author1 2008; Author3 & Author1 2010). This station will explore practice with the equipartitioning concept of composition of splits (an m-split composed with an n-split to yield an m x n split) and how the diagnostic assessment can help students and teachers be alert to the misconception that an m-split composed with an n-split yields m + n fair shares.

3. PEEQ (the Property of Equality for Equipartitioning) is an emergent property within equipartitioning.  This property involves situations in which identical wholes are each split for the same number of parts, but in ways that generate fair shares that are not geometrically congruent (for example, two identical rectangular brownies, each split for 2 persons, but one split vertically, and the other split diagonally).  Students come to recognize and justify the equality of such non-congruent shares, with proficiency demonstrated by the type of justification students employ.  Lower level justification is by visual inspection only, while more sophisticated levels invoke the reasoning that an n-split performed on identical wholes must yield equal parts, regardless of orientation of the split.  This foreshadows the logic required in the transitive property of equality. We conjecture that proficiency in sharing wholes using multiple methods and in naming the shares is a key stepping stone to understanding PEEQ.

4. Covariation is an upper level of the equipartitioning learning trajectory that employs "distribution" problems (Streefland 1991), that ask students to determine how a number of people and pizzas can be fairly distributed across multiple tables so that all persons receive the same fair share.  This early construct of covariation illustrated by LPPSync ePacket 6 reconceptualizes Streefland's diagrams, supporting students to discover and use the ratio unit, or the "littlest recipe" (the smallest whole number ratio within a set of proportions; Author1 & Author, 1994), to make distinctions with building-up strategies (Lo & Watanabe, 1997), and to extend the notion of ratio unit to the unit ratio (quantity per one)--a precursor to understanding fractions (Author1 & Author, 2005).