SimCalc: Democratizing Access to Advanced Mathematics

Historically, what people can learn is co-determined by the representational infrastructure for knowledge building. When Latin was the required medium of knowledge building, few could engage in scholarly activities; without the change to the vernacular, nearly universal access to higher education would not be possible. The highly compact, abstract, and opaque symbolism of mathematics presents similar barriers to the necessary democratization of access to important mathematics.  Over the course of a program of research lasting more than 20 years and involving contributors from institutions throughout the United States and worldwide (Hegedus & Roschelle, 2013), the representationally innovative design of SimCalc Mathworlds® has provided affordances for novel and effective approaches to teaching important algebraic and calculus-related ideas. When integrated with appropriate curricular workbooks, teacher professional development, and other instructional factors, dynamic representation has enabled diverse populations to learn more advanced mathematics. Research has included both design research as well as large-scale experiments involving hundreds of teachers and thousands of students; overall, the approach also has an unusually strong base of empirical support. We focus on lasting, essential design contributions of this body of work with a special emphasis on the dialectic relationship between affordances of technology and curricular progressions.

[1]  A L Hammond,et al.  Computer-assisted instruction: two major demonstrations. , 1972, Science.

[2]  Nikol Rummel,et al.  Promoting Learning in Mathematics: Script Support for Collaborative Problem Solving with the Cognitive Tutor Algebra , 2007 .

[3]  R. Lehrer,et al.  Technology and mathematics education , 2008 .

[4]  H. Rittel,et al.  Dilemmas in a general theory of planning , 1973 .

[5]  Andrea A. diSessa.  An Overview of Boxer. , 1991 .

[6]  B. White.  ThinkerTools: Causal Models, Conceptual Change, and Science Education , 1993 .

[7]  Alan H. Schoenfeld,et al.  Mathematical thinking and problem solving , 2016 .

[8]  William R. Penuel,et al.  Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures , 2008 .

[9]  James J. Kaput,et al.  An Introduction to the Profound Potential of Connected Algebra Activities: Issues of Representation, Engagement and Pedagogy. , 2004 .

[10]  Ellen Yi-Luen Do,et al.  Posey: instrumenting a poseable hub and strut construction toy , 2008, Tangible and Embedded Interaction.

[11]  Mark Guzdial.  A media computation course for non-majors , 2003 .

[12]  Jeremy Roschelle,et al.  Getting to Scale with Innovations that Deeply Restructure How Students Come to Know Mathematics , 2008 .

[13]  Chris Rasmussen,et al.  PME Special Issue: Bodily Activity and Imagination in Mathematics Learning , 2004 .

[14]  Michael Eisenberg,et al.  Computer-assisted pop-up design for children: computationally enriched paper engineering , 2006 .

[15]  Deborah G. Tatar,et al.  The Design Tensions Framework , 2007, Hum. Comput. Interact..

[16]  Ricardo Nemirovsky,et al.  Body Motion and Graphing , 1998 .

[17]  Luis Moreno-Armella,et al.  Intersecting representation and communication infrastructures , 2009 .

[18]  M. Hegarty,et al.  Types of visual–spatial representations and mathematical problem solving. , 1999 .

[19]  Phil Vahey,et al.  Leveraging Handhelds to Increase Student Learning: Engaging Middle School Students with the Mathematics of Change , 2004, ICLS.

[20]  Ian Westbury.  A Nation at Risk , 1984 .

[21]  RoschelleJeremy,et al.  Handhelds Go to School , 2003 .

[22]  Margaret Dickey-Kurdziolek,et al.  “They Need to Be Solid in Standard Skills First”: How Standards Can Become the Upper Bound , 2013 .

[23]  Paul Drijvers,et al.  Digital Technology in Mathematics Education: Why It Works (Or Doesn't) , 2013 .

[24]  Jennifer M. Shephard,et al.  Spatial versus object visualizers: A new characterization of visual cognitive style , 2005, Memory & cognition.

[25]  Pierre Dillenbourg,et al.  Technology for Classroom Orchestration , 2010 .

[26]  Stephen Hegedus,et al.  Learning and Participation in High School Classrooms , 2013 .

[27]  Patrick Jermann,et al.  Classroom orchestration: The third circle of usability , 2011, CSCL.

[28]  Harold Abelson,et al.  Turtle geometry : the computer as a medium for exploring mathematics , 1983 .

[29]  Paul Cobb,et al.  The Evolution of Mathematical Practices: A Case Study , 1999 .

[30]  Harold Abelson,et al.  Boxer: a reconstructible computational medium , 1986, CACM.

[31]  Janet Bowers,et al.  Linking Algebraic Concepts and Contexts: Every Picture Tells a Story , 2000 .

[32]  Daniel L. Schwartz,et al.  A time for telling , 1998 .

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

[34]  Celia Hoyles,et al.  Developing new notations for a learnable mathematics in the computational era , 2002 .

[35]  Ricardo Nemirovsky.  On ways of symbolizing: The case of Laura and the velocity sign , 1994 .

[36]  H. Doerr,et al.  You have printed the following article : Creating Meaning for and with the Graphing Calculator , 2008 .

[37]  Celia Hoyles,et al.  Rethinking the mathematics curriculum , 1999 .

[38]  Jeremy Roschelle,et al.  The role of scaling up research in designing for and evaluating robustness , 2008 .

[39]  Daniel L. Schwartz,et al.  Rethinking transfer: A simple proposal with multiple implica-tions , 1999 .

[40]  Robert B. Kozma,et al.  Learning Chemistry Through the Use of a Representation-Based Knowledge Building Environment , 2002 .

[41]  Phil Vahey,et al.  FIVE KEY CONSIDERATIONS FOR NETWORKING IN A HANDHELD-BASED MATHEMATICS CLASSROOM , 2005 .

[42]  Frank Fischer,et al.  Scripting Computer-Supported Collaborative Learning : cognitive, computational, and educational perspectives , 2007 .

[43]  Uri Wilensky,et al.  Networked Gridlock: Students Enacting Complex Dynamic Phenomena with the HubNet Architecture , 2000 .

[44]  Jeremy Roschelle,et al.  The Mathematics of Change and Variation from a Millennial Perspective: New Content, New Context , 2013 .

[45]  Alexander Repenning,et al.  Agentsheets: A Medium for Creating Domain-Oriented Languages , 1995, Computer.

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

[47]  Caitlin Kelleher,et al.  Using storytelling to motivate programming , 2007, CACM.