SimCalc Connected Classrooms: New Forms of Learnin

Slide1 : SimCalc Connected Classrooms: New Forms of Learning, New Forms of Teaching Stephen J. Hegedus Sara Dalton shegedus@umassd.edu sdalton@umassd.edu SimCalc Research Projects, Department of Mathematics University of Massachusetts, Dartmouth merg.umassd.edu www.simcalc.umassd.edu Developed under NSF-based Grants: REC-0087771, Understanding Classroom Interactions Among Diverse, Connected Classroom Technologies; REC-0337710, Representation, Participation and Teaching in Connected Classrooms; REC-0228515, Scaling Up SimCalc: Professional Development for Leveraging Technology to Teach More Complex Mathematics, Phase I; REC-0437861, Scaling Up Middle School Mathematics Innovations, Phase II

Democratizing Access : Democratizing Access Mathematical alienation Motivation repressed via opaque classroom objectives Curriculum restrictions Classroom participation is an expectation rather than a phenomenological artifact of productive learning

SimCalc MathWorlds : SimCalc MathWorlds Dynamic interactive representations that are linked, e.g. edit a position function and automatically see velocity graphs update Graphically and algebraically editable functions Import motion data and re-animate (CBR & CBL2) Simulations are at the heart of SimCalc - executable representations (Moreno, 2001)

SimCalc Connected MathWorlds -The Product : SimCalc Connected MathWorlds - The Product The historic evolutions of two software into one integrated product SMW for the TI 83+/84+ - Version 5.0 SMW for the Desktop PC (cross-platform for non-connected work) - Version 3.0 Mental Model for users: Microsoft Office - Can be used in integrated ways or independently - documents can be written to be used in other applications

SimCalc “Connected” MathWorlds : SimCalc “Connected” MathWorlds New generation of SimCalc that increases participation, motivation and learning Exploits wireless networks to allow the aggregation of student work in mathematically meaningful ways Teachers have powerful classroom management tools to focus attention and pedagogical agenda Student work becomes contextualized into a class of contributions for comparison and generalization Mathematical thinking goes from a local to a social activity

Three fundamental powers of connectivity : Three fundamental powers of connectivity To harvest students work to examine variation and common misconceptions (error analysis) To aggregate students work in a mathematically meaningful way – use natural variation to examine parametric variation (i.e. each students varies a parameter) To focus on connections across representations, i.e. students work with representation A (e.g. a velocity graph) and the teacher displays/works with representation B (e.g. a position graph) - cf. Kaput 1991

Slide8 : Parallel Software From Student Device To Teacher Display Executable Representations

Dynamic Mathematics : Dynamic Mathematics Dynamic representations are a new access route to new visions of mathematical ideas and problem solving Connectivity is a foundation to allow public collaboration, mutual expression in dynamic media, physical expression through time and space via gesture, discourse and action, and social cognition.

Slide10 : Software Connectivity Hardware Communication Infrastructure Representational Infrastructure Dynamic Environment

Representational Infrastructure (inherently or explicitly mathematical)  + Communication Infrastructure (explicitly not mathematical, i.e. a generic hardware/protocol, yet inherently mathematical) : Representational Infrastructure (inherently or explicitly mathematical) + Communication Infrastructure (explicitly not mathematical, i.e. a generic hardware/protocol, yet inherently mathematical)

Demonstration : Demonstration SimCalc MathWorlds: Using it to lay the foundation for Calculus and broadening access for more students Demonstrate simulations in computer software then calculator software Y=MX+B Dealing with Rate graphs - Averages, Mean Value Theorem, Fundamental Theorem of Calculus

CMW Supports Three New Classes of Functions : CMW Supports Three New Classes of Functions Class 1: Piecewise editable functions graphically and algebraically 1. Piecewise Linear Functions 2. Piecewise Quadratic Functions

Class 2: Parametrically Defined Functions : Class 2: Parametrically Defined Functions 1. Linear: Y = MX + B 2. Quadratic: Y + AX2 + BX + C 3. Quadratic (product of roots): Y= A(X– alpha)(X – beta)

Class 2: Parametrically Defined Functions : Class 2: Parametrically Defined Functions 4. Exponential: Ae(BX+C) 5. Periodic: Y=Asin(BX+C)+D

Class 3: Sampled Data: CBR & CBL2 : Class 3: Sampled Data: CBR & CBL2 • Ability to support a wide variety of probes • Ability to disconnect on collection and use a variety of smoothing methods • Ability to count in seconds/minutes/hours to offer “faster” animation

Classroom Management: a fundamental design principle in a representationally-rich environment : Classroom Management: a fundamental design principle in a representationally-rich environment Aggregation/Receiving – allows two forms of agency in the classroom/distributed agency Post-Connectivity: Data management vs Representational management - role of filters to assess students’ progressive understanding (i.e. representational timestamps) and systematically generate public reasoning and generalization Note: This is not always about allowing students to have ownership of the public display space (cf. Stroup) - we tightly control this Design challenges and solution strategies - roster as a central ordering principle

Exploiting Connectivity : Exploiting Connectivity Facilitate work-flow, Aggregate student constructions to: i. vary essential parameters on a per-student basis, ii. elevate student attention from single objects to parameterized families of objects, Provide opportunity for generalization and expose common thought-patterns (e.g. errors) Students make personally meaningful mathematical objects to be publicly shared and discussed Students project their personal identity into the objects and constructed motions Students math and social experience are deeply intertwined Teacher are in a central role to orchestrate whole class of events

Slide19 : Students experience and contributions are embedded in a social workspace Mathematical structure and understanding can be emergent, e.g. What do you expect to see before I show you the ... Representational infrastructure includes data management systems to manage the flow of information and examination of mathematical sub-structures; such power serves a variety of pedagogical needs, and sustains pedagogical flexibility Some Top-Level Thoughts

Forms of Interaction : Forms of Interaction We are focusing on interactions cycles particularly classifying what types of questions teachers ask, how often, and how students respond, contrasting with non-CC contexts across teachers. Interaction between oral and public workspace. How the teacher interacts with the inscriptions within the software (hide/show for pedagogical and curricular purposes) and as annotations, i.e how the teacher uses a white board marker on top of the displayed window. Both alter focus of attention. New forms of questions emerging from CC phenomenology: e.g. What do you expect to see when I show the motions/graphs? Where are you in the public display? How does your group’s motions/graphs compare with another group?

Teacher Collaboration and Examining Practice : Teacher Collaboration and Examining Practice Communities of Practice: Teachers meet to discuss impact on students learning and their practice – what videos of each others classes and answering structured questions Epistemic distancing – being more than reflective of ones teaching but about its structure, its evolution, making knowledge about teaching from a distance Building school-capacity: Enable teachers to learn while teaching vs. external professional development, develop flexible modes of instruction, develop metacognitive awareness of learning-opportunities through flexible curriculum-centered technology (adaptive expertise)

Impact on Learning : Impact on Learning Results pre-post from interventions hons class: increase from 0.56 to 0.76 (alpha<0.0001) - similar performance to regular high school students Highlight past and future interventions Demonstrate students engagement at the start and end of an intervention – 2 video clips from Julie’s class (one with Luke and Sackrace, then the arrows clip) Correlations between attitude surveys and test scores

Conclusions : Conclusions Research-to-date shows positive impact on mathematical knowledge (necessary and advanced) AND participation and motivation to do mathematics (attitude & behavioral data) Over 6 years of design & experimentation has produced a software environment that redefines the educational landscape of the mathematics classroom in the 21st century Dynamic Representations and in-class communication infrastructure + mathematically meaningful activities = powerful opportunities for MORE students.

Dynamic Mathematics:SimCalc MathWorldsand Classroom Connectivity : Dynamic Mathematics: SimCalc MathWorlds and Classroom Connectivity Dr Stephen J. Hegedus shegedus@umassd.edu Professor of Mathematics, Department of Mathematics University of Massachusetts, Dartmouth merg.umassd.edu www.simcalc.umassd.edu Developed under NSF-based Grants: REC-0087771, Understanding Classroom Interactions Among Diverse, Connected Classroom Technologies; REC-0337710, Representation, Participation and Teaching in Connected Classrooms; REC-0228515, Scaling Up SimCalc: Professional Development for Leveraging Technology to Teach More Complex Mathematics, Phase I; REC-0437861, Scaling Up Middle School Mathematics Innovations, Phase II