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November 2006, Volume 37, Issue 5

FEATURES

Conventional Wisdom
Steve Williams
After my initiation into research as a graduate student, I began to hear what many may think is the “conventional wisdom” regarding JRME. I hear it more often now as editor while fielding inquiries about potential submissions: “I know JRME usually doesn't publish quantitative research but ..."

Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems With Graphing Software
Michal Yerushalmy
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences between the work of these less successful students and the traditional  problem-solving patterns of less successful students. These less successful students used the graphing software to obtain a broader view, to confirm conjectures, and to complete difficult operations. However, they delayed using symbolic formalism, and  most of their solution attempts focused on numeric and graphic   representations. Their process of reaching a solution was found to be relatively long, and the graphing software tool was often not used at all because it did not support symbolic formulation  and manipulations.

Pedagogical Content Tools: Integrating Student Reasoning and Mathematics in Instruction
Chris Rasmussen, Karen Marrongelle
Teaching in a manner consistent with reform recommendations is a challenging and often overwhelming task. Part of this challenge involves using students’ thinking and understanding as a basis for the development of mathematical ideas (cf. NCTM, 2000). The purpose of this article is to address this challenge by developing the notion of pedagogical  content tool. A pedagogical content tool is a device such as a graph, diagram, equation, or verbal statement that a teacher intentionally uses to connect to student thinking while moving the mathematical agenda forward. We tender two examples of pedagogical content tools: Transformational record and generative alternative. These two pedagogical content tools are put forth as instructional counterparts to the Realistic Mathematics Education (RME) design heuristics of emergent models and guided reinvention,  respectively. We illustrate the pedagogical content tools of transformational record and generative alternative by drawing on examples from two classroom teaching experiments in undergraduate differential equations.

Teaching as Assisting Individual Constructive Paths Within an Interdependent Class Learning Zone: Japanese First Graders Learning to Add Using 10
Aki Murata, Karen Fuson
The framework of Tharp and Gallimore (1988) was adapted to form a ZPD (Zone of Proximal Development) Model of Mathematical Proficiency that identifies two interacting kinds of learning activities: instructional conversations that assist understanding  and practice that develops fluency. A Class Learning Path was conceptualized as a classroom path that includes a small number of different learning paths  followed by students, and it permits a teacher to provide assistance to students at their own levels. A case study illustrates this model by describing how one teacher in a Japanese Grade 1 classroom assisted student learning of addition with teen totals by  valuing students’ informal knowledge and individual approaches, bridging the distance between their existing knowledge and the new culturally valued method, and giving carefully structured practice. The teacher decreased assistance over time but increased  it for transitions to new problem types and for students who needed it. Students interacted,  influenced/supported one another, and moved forward along their own learning paths within the Class Learning Path.