is an independent K-8 education consultant. He edited Learning Over Time: Learning Trajectories in Mathematics Education and led numerous professional development workshops and MOOCs on CCSS-M from a learning trajectories perspective. He co-authored the mathematics educational software applications Function Probe and Graphs ’n Glyphs. He has previously been a research scientist, and holds a doctorate in biological sciences from Stanford University.

Confrey J. & Maloney A. (2006) From constructivism to modeling. In: Stewart S. M., Olearski J. E. & and Thompson D. (eds.) Proceedings of the Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (METSMaC). Middle East Teachers of Science, Mathematics and Computing, Abu Dhabi: 3–28. Fulltext at https://cepa.info/3880

This paper traces the development of constructivism as a theory of epistemology and learning, and identies ten key principles of this “grand theory.” It identies the need to further develop bridging theories that more closely link to empirical evidence. Within these bridging theories, it identies primary themes: grounding in action, activity and tools, alternative perspectives, student reasoning patterns and developmental sequences, student-invented representations, socioconstructivist norms, etc., that are useful in linking theory and practice. Finally, it discusses how these ideas have been evolving into a view of modelling as an orientation to mathematics and science instruction, and identies this approach as a successor to constructivist theories.

Panorkou N. & Maloney A. (2015) Authors’ Response: Planting Seeds of Mathematical Abstraction. Constructivist Foundations 10(3): 352–354. Fulltext at https://cepa.info/2150

Upshot: We consider that elementary students’ situated activities with geometric transformations and animation contain the seeds of complex, and eventually, mathematically generalizable and abstract reasoning. Further studies can explore such technologically-based activities’ potential as building blocks for flexible, creative, and formalized knowledge.

Panorkou N. & Maloney A. (2015) Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment. Constructivist Foundations 10(3): 338–347. Fulltext at https://cepa.info/2146

Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies.