List of Contents

October 2012

Editorial

Teaching-Research Guide to Problem Solving: Jump Starting Reform. ICME 12, Seoul, Korea
Bronislaw Czarnocha, William Baker,Olen Dias, Vrunda Prabhu

Problem Solving in Remedial Arithmetic: PPP presentation ICME 12, Seoul, Korea
William Baker, Olen Dias, Vrunda Prabhu, Bronislaw Czarnocha

Making Sense of Learning in Remedial Mathematics: Approach Integrating Cognition, Affect and Self-Regulatory Learning Practices, ICME 12, Seoul, Korea
Vrunda Prabhu, Peter Barbatis, James Watson

Teaching-Research for the 21st Century – Discussion Group #4, PME 36, Taipei, Taiwan
William Baker, Bronislaw Czarnocha, Olen Dias, Vrunda Prabhu

Universal and Existential Quantifiers revisited, PME 36, Taipei, Taiwan
Ruili Ye, Bronislaw Czarnocha

Development of Conceptual Structures in Remedial Mathematics. Fifth Concept Mapping Conference, Malta
Vrunda Prabhu, Peter Barbatis, James Watson

Learning Trajectories from the Arithmetic/Algebra Divide. NA-PME, Kalamazoo, Mich.
Bronislaw Czarnocha, William Baker, Olen Dias, Vrunda Prabhu

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Editorial

The Summer of 2012 was very rich for the editors of MTRJ and their TR Teams from Hostos CC and Bronx CC of CUNY. Together, this issue of MTRJ presents interests, understanding and achievements of the Bronx Mathematics Teaching-Research Team during the Summer 2012 and informs about the future directions (below).

Two issues come to the fore:

• making sense or understanding in mathematics, and
• Learning Trajectories as the underlying framework of Common Core Standards in Mathematics.

The two issues have a lot in common; teaching along learning trajectories facilitates making connections between the concepts involved in the particular learning trajectory. The clarity of those connections is understanding of mathematics, or of what Barody (2003) calls “adaptive expertise” that is “well connected knowledge”. And what is equally important is that both teacher – the designer of instruction and the student – the “discoverer of mathematics” develop those connections, often almost simultaneously. One of the questions of assessment is how to measure understanding that is how to measure the degree of connections between the concepts, how to measure the adaptivity of the expertise. The questions of measurability of understanding comes to the fore because of the repeated instances of elimination of “understanding” as one of the Student Learning Outcomes in remedial mathematics courses under a pretext of non-measurability of understanding. The consequences of this elimination can be devastating. There are several posts discussing the issue by one of the editors (BC) on the CUNY Mathblog at https://cunymathblog.commons.gc.cuny.edu