Ron Tzur

An Integrated Study of Children's Construction of Improper Fractions and the Teacher's Role in Promoting That Learning

In this constructivist teaching experiment with 2 fourth graders I studied the coemergence of teaching and childrens construction of a specific conception that supports the generation of improper fractions. The childrens posing and solving tasks in a computer microworld promoted a modification in their fraction schemes. They advanced from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole (the iterative fraction scheme). In this article I report an intertwined analysis of the childrens construction of this multiplicative relationship and an examination of the teachers adaptation of learning situations (tasks) and teacher-learner interactions to fit within the constraints of the childrens mathematical activity.

Explicating a Mechanism for Conceptual Learning: Elaborating the Construct of Reflective Abstraction

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflec- tion on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970). The mechanism is an elaboration of Piagets (2001) reflective abstrac- tion and is potentially useful for addressing some of the more intractable problems in teaching mathematics. Implications of the mechanism for lesson design are discussed and exemplified.

Becoming a Mathematics Teacher-Educator: Conceptualizing the Terrain through Self-Reflective Analysis.

My purpose in this article is to contribute tothe conceptualization of the complex terrainthat often is indiscriminately termedmathematics teacher educator development.Because this terrain is largely unresearched, Iinterweave experience fragments of my owndevelopment as a mathematics teacher educator,and reflective analysis of those fragments, asa tool to abstract notions of generalimplication. In particular, I postulate aframework consisting of four stages ofdevelopment that are distinguished by thedomain of activities ones reflections mayfocus on and the nature of those reflections.Drawing on this framework, I presentimplications for mathematics teacher educatordevelopment and for further research.

Interaction and Children's Mathematics

Abstract In this article we propose the beginnings of a constructivist model of mathematical learning that supersedes Piagets and Vygotskys views on learning. First, we analyze aspects of Piagets and Vygotskys grand theories of learning and development. Then, we formulate our superseding model, which is based on the interrelations between two types of interaction in constructivism—the basic sequence of action and perturbation, and the interaction of constructs in the course of re-presentation or other previously constructed items. When teaching children, we base our interactions with them on the schemes we infer by observing the childrens interactions in a medium. This emphasis makes contact with both Piagets and Vygotskys ideas of spontaneous development. In our model, learning is understood as being spontaneous rather than provoked. To maintain our emphasis on spontaneity, we separate the unintentionality of the learner from the intentionality of the teacher. To ground our model, we describe how ...

An Integrated Research on Children's Construction of Meaningful, Symbolic, Partitioning-related Conceptions and the Teacher's Role in Fostering That Learning

Abstract A teaching experiment was conducted with two fourth graders to study the co-emergence of teaching and childrens construction of fraction knowledge. The childrens learning, i.e., modifications in their fraction schemes, was fostered through working on tasks in a computer microworld. The children advanced from thinking about a unit fraction as one of several equal parts in a whole (the equipartitioning scheme) to operating with a unit fraction as a symbolized, iterable part the magnitude of which is based on the numerosity of the partitioned whole (the partitive fraction scheme). The paper interweaves an analysis of childrens construction of partitioning-related symbolic conceptions of fractions with an analysis of the teaching—planning and using tasks—that fosters such an advancement by introducing fraction words and numerals in the context of the childrens partitioning activities.

Moving students through steps of mathematical knowing: An account of the practice of an elementary mathematics teacher in transition☆

Abstract We present an account of a sixth-grade teachers practice as she responds to the challenges of current reform initiatives. We analyzed classroom observations and interviews to understand how the teacher, Ivy, teaches and thinks about teaching mathematics to her students. For Ivy, mathematical meaning is available in particular experiences. She creates these experiences for her students by leading them through a predetermined sequence of steps of mathematical knowing. This account contributed to our postulation of a perspective on mathematics learning that we refer to as perception-based, in which the goal of instruction is to create opportunities for students to perceive, first hand, mathematics that exists as part of an external reality. We examine implications of this account of Ivys practice for mathematics teacher development.

Is teaching parallel algorithmic thinking to high school students possible?: one teacher's experience

All students at our high school are required to take at least one course in Computer Science prior to their junior year. They are also required to complete a year-long senior project associated with a specific in-house laboratory, one of which is the Computer Systems Lab. To prepare students for this experience the lab offers elective courses at the post-AP Computer Science level. Since the early 1990s one of these electives has focused on parallel computing. The course enrolls approximately 40 students each year for two semesters of instruction. The lead programming language is C and topics include a wide array of industry-standard and experimental tools. Since the 2007-2008 school year we have included a unit on parallel algorithmic thinking (PAT) using the Explicit Multi-Threading (XMT) system. We describe our experiences using this system after self-studying the approach from a publicly available tutorial. Overall, this article provides significant evidence regarding the unique teachability of the XMT PAT approach, and advocates using it broadly in Computer Science education.

Developing New Understandings of PDS Work: Better Problems, Better Questions

Abstract Through sharing examples, the authors demonstrate how the analysis of long-term Professional Development School (PDS) problems and their evolution can serve as one indicator of growth in the PDS. Three persistent problem areas are identified: (a) building trust and relationships between university and school personnel, (b) reconceptualizing existing coursework to fit in the PDS context, and (3) making inquiry a central feature of the PDS. The historical evolution of these problem areas is traced through three phases of PDS development over a six-year period, including PDS Planning, PDS Pilot Year, and PDS Institutionalization. The authors conclude that, through careful analysis, PDS problems can be celebrated and utilized as one measurement of growth in PDS work rather than bemoaned and utilized to characterize PDS work as unstable and fragile. Finally, the authors call for other PDS practitioners across the nation to share their PDS problems publicly, beginning a national dialogue about the ways in which PDS problems lead to new and better PDS work.

Probabilistic latent class models for predicting student performance

Predicting student performance is an important task for many core problems in intelligent tutoring systems. This paper proposes a set of novel probabilistic latent class models for the task. The most effective probabilistic model utilizes all available information about the educational content and users/students to jointly identify hidden classes of students and educational content that share similar characteristics, and to learn a specialized and fine-grained regression model for each latent educational content and student class. Experiments carried out on large-scale real-world datasets demonstrate the advantages of the proposed probabilistic latent class models.

An Intelligent Tutor-Assisted Mathematics Intervention Program for Students With Learning Difficulties

The Common Core Mathematics Standards have raised expectations for schools and students in the United States. These standards demand much deeper content knowledge from teachers of mathematics and their students. Given the increasingly diverse student population in today’s classrooms and shortage of qualified special education teachers, computer-assisted instruction may provide supplementary support, in conjunction with the core mathematics instruction, for meeting the needs of students with different learning profiles. The purpose of this study was to explore the potential effects of the Please Go Bring Me-Conceptual Model-Based Problem Solving (PGBM-COMPS) intelligent tutor program on enhancing the multiplicative problem-solving skills of students with learning disabilities or difficulties in mathematics.