The design of a system to support exploratory learning of algebraic generalisation

This paper charts the design and application of a system to support 11-14 year old students' learning of algebraic generalisation, presenting students with the means to develop their understanding of the meaning of generality, see its power for mathematics and develop algebraic ways of thinking. We focus squarely on design, while taking account of both technical and pedagogical issues and challenges, and provide an account of how we have designed and built a system with a very close fit to our knowledge of students' difficulties with the subject matter. We report the challenges involved in building a system that is both intelligent and exploratory, a learning environment in which both student and teacher are supported without explicit tutoring.

[1]  E. Jablonka.  International group for the psychology of mathematics education , 2008 .

[2]  Manolis Mavrikis,et al.  Using Qualitative Data Analysis Software to analyse students’ computer‐mediated interactions: the case of MiGen and Transana , 2011 .

[3]  Seymour Papert,et al.  Teaching Children to be Mathematicians vs. Teaching About Mathematics. Artificial Intelligence Memo Number 249. , 1971 .

[4]  Celia Hoyles,et al.  Bob‐A suitable case for treatment? , 1993 .

[5]  Annemarie Hauf,et al.  Computers in education , 1983 .

[6]  Antonija Mitrovic,et al.  Revisiting Ill-Definedness and the Consequences for ITSs , 2009, AIED.

[7]  Alexandra Poulovassilis,et al.  Design of Teacher Assistance Tools in an Exploratory Learning Environment for Algebraic Generalization , 2012, IEEE Transactions on Learning Technologies.

[8]  Celia Hoyles.  Microworlds/Schoolworlds: The Transformation of an Innovation , 1993 .

[9]  Alexandra Poulovassilis,et al.  The Design of Teacher Assistance Tools in an Exploratory Learning Environment for Mathematics Generalisation , 2010, EC-TEL.

[10]  L. Healy,et al.  Charting the microworld territory over time: design and construction in mathematics education , 2010 .

[11]  Celia Hoyles,et al.  Mathematics Education and Technology-Rethinking the Terrain , 2010 .

[12]  Patrick W. Thompson.  Mathematical microworlds and intelligent computer-assisted instruction , 1986 .

[13]  Manolis Mavrikis,et al.  Sequence Detection for Adaptive Feedback Generation in an Exploratory Environment for Mathematical Generalisation , 2010, AIMSA.

[14]  Etienne Wenger,et al.  Communities of Practice: Learning, Meaning, and Identity , 1998 .

[15]  Manolis Mavrikis,et al.  Layered Development and Evaluation for Intelligent Support in Exploratory Environments: The Case of Microworlds , 2010, Intelligent Tutoring Systems.

[16]  Celia Hoyles,et al.  Windows on Mathematical Meanings: Learning Cultures and Computers , 1996 .

[17]  Alexandra Poulovassilis,et al.  The Conceptual and Architectural Design of a System Supporting Exploratory Learning of Mathematics Generalisation , 2009, EC-TEL.

[18]  Manolis Mavrikis,et al.  Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation , 2009 .

[19]  R. Mayer.  Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. , 2004, The American psychologist.

[20]  George D. Magoulas,et al.  Adaptive Modelling of Users' Strategies in Exploratory Learning Using Case-Based Reasoning , 2010, KES.

[21]  Alexandra Poulovassilis,et al.  Layered Learner Modelling in ill-defined domains : conceptual model and architecture in MiGen , 2010 .

[22]  Michèle Artigue,et al.  Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work , 2002, Int. J. Comput. Math. Learn..

[23]  E. Goldenberg,et al.  Habits of mind: An organizing principle for mathematics curricula , 1996 .

[24]  Manolis Mavrikis,et al.  Not all wizards are from Oz: Iterative design of intelligent learning environments by communication capacity tapering , 2010, Comput. Educ..

[25]  Manolis Mavrikis,et al.  A LEARNING ENVIRONMENT TO SUPPORT MATHEMATICAL GENERALISATION IN THE CLASSROOM , 2009 .

[26]  Greg P. Kearsley,et al.  Artificial intelligence and instruction: Applications and methods , 1987 .

[27]  Manolis Mavrikis,et al.  Students’ justification strategies on the equivalence of quasi-algebraic expressions , 2011 .

[28]  Joanna McGrenere,et al.  Affordances: Clarifying and Evolving a Concep , 2000, Graphics Interface.

[29]  MavrikisManolis,et al.  The design of a system to support exploratory learning of algebraic generalisation , 2012 .

[30]  Luc Trouche,et al.  Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students’ Command Process through Instrumental Orchestrations , 2004, Int. J. Comput. Math. Learn..

[31]  Celia Hoyles,et al.  Messing Up: Reflections on Introducing Cabri Géomètre , 1994 .

[32]  Richard E. Clark,et al.  Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching , 2006 .

[33]  Ton de Jong,et al.  Scientific Discovery Learning with Computer Simulations of Conceptual Domains , 1998 .

[34]  Roy Fielding,et al.  Architectural Styles and the Design of Network-based Software Architectures"; Doctoral dissertation , 2000 .

[35]  Manolis Mavrikis,et al.  DEVELOPING A MICROWORLD TO SUPPORT MATHEMATICAL GENERALISATION , 2009 .

[36]  Manolis Mavrikis,et al.  Sowing the seeds of algebraic generalization: designing epistemic affordances for an intelligent microworld , 2013, J. Comput. Assist. Learn..

[37]  H. Neil,et al.  A suitable case for treatment , 1995, The Lancet.

[38]  M. Goos.  A sociocultural analysis of learning to teach , 2005 .

[39]  Manolis Mavrikis,et al.  Revisiting Pedagogic Strategies for Supporting Students’ Learning in Mathematical Microworlds , 2008 .

[40]  Joanna McGrenere,et al.  Affordances: Clarifying and Evolving a Concep , 2000, Graphics Interface.

[41]  Jeremy Roschelle,et al.  SIMCALC : Accelerating Students’ Engagement With the Mathematics of Change , 2000 .

[42]  Richard Noss.  Special Issue on knowledge transformation, design and technology : Three papers from the Technology-Enhanced Learning Research Programme , 2013, J. Comput. Assist. Learn..

[43]  George D. Magoulas,et al.  A Case-Based Reasoning Approach to Provide Adaptive Feedback in Microworlds , 2010, Intelligent Tutoring Systems.

[44]  George D. Magoulas,et al.  Challenges for intelligent support in exploratory learning: the case of ShapeBuilder , 2008 .