David Slavit

AN ALTERNATE ROUTE TO THE REIFICATION OF FUNCTION

This paper presents an alternate perspective for utilizing the action/process/object framework when discussing student development of conceptions of function. After a review of related theories, a property-oriented view of function is described which is based on visual aspects of functional growth. The theory is supported with data on student learning. The property-oriented view of function incorporates and extends previously described frameworks used in analyzing functional understandings, including the covariance and correspondence views (Confrey & Smith, 1991; Thompson, 1994). The property-oriented view differs from the covariance view in that less emphasis is placed on the manner in which the variables are changing and more emphasis is placed on the properties that result from these changes. The property-oriented view differs from a correspondence view in that functional properties such as invertibility and domain give rise to a different kind of thinking about functions than do properties such as symmetry, linearity, continuity, etc. Implications for further research and curriculum development are also provided.

Leading Deep Conversations in Collaborative Inquiry Groups

Abstract Collaborative inquiry groups, such as professional learning communities and lesson study groups, are proliferating in schools across the United States. In whatever form, the potential for impacting student learning through this collaborative work is expanded or limited by the nature of teachers’ conversations. Polite, congenial conversations remain superficially focused on sharing stories of practice, whereas collegial dialogue probes more deeply into teaching and learning. Examples of talk taken from collaborative teacher inquiry groups are used to illustrate these important differences. Specific recommendations are provided, including the role that teacher leaders can play in adopting and modeling specific strategies that support the use of more substantive professional conversation.

The Role of Operation Sense in Transitions from Arithmetic to Algebraic Thought.

As attention to the development of algebraic understandings at early grade levels increases, theory and empirical support for these efforts are needed. This paper outlines a theoretical perspective for studying student understandings of mathematical operations, with a particular focus on addition. The notion of operation sense is defined using a perspective that incorporates the construction of mental objects. In the context of addition, it is argued that operation sense can be used to describe student development of additive concepts as well as transitions into algebraic ways of thinking. The report of a case study on the development of a young boy is then provided. The investigation attempts to instantiate the framework in regard to student development of an understanding of addition. Evidence was found that his attainment of aspects of operation sense supported transitions into algebraic ways of thinking, including a finite group setting and use of addition on unknown and arbitrary quantities. Limitations of the framework are discussed.

Collaborative inquiry among science and mathematics teachers in the USA: professional learning experiences through cross‐grade, cross‐discipline dialogue

This article presents five case studies of grade six to grade eight science and mathematics teachers engaged in supported collaborative inquiry. The research focus is on teachers’ growth trajectories related to their participation in professional learning communities supported through a professional development project. There were noticeably different trajectories in the teachers’ progress through the inquiry cycle and a variation in the degree of collaboration in different professional learning communities. Some of the successes and challenges are attributed to the cross‐grade, cross‐disciplinary nature of the collaboration; others to teachers’ existing relationships with each other prior to the beginning of this professional development project. The case study data indicate that dialogic inquiry grounded in classroom‐based data is a key element in teachers’ professional growth.

Requiring Collaboration or Distributing Leadership

Distributed leadership allows teachers to share their expertise and create a collective responsibility for improving student learning — and teaches district leaders the value of inviting teachers into leadership.

Teacher Groups’ Conceptions and Uses of Student-Learning Data

This article measures and discusses six teacher groups’ (a) time spent working with data and (b) time devoted to particular kinds of inquiry activities, and explores various contextual factors that influence these results. The authors make use of a framework useful in describing and analyzing the stance taken by teachers when they engage with student-learning data. Their findings suggest that most teacher groups spend the vast majority of their time collecting and analyzing data, with little time devoted to exploring potential data sources and reflecting on implications of their data analysis. Furthermore, “time on task” is less important than stance in determining the nature of the inquiry activity. Implications of these results are explored.

Using Web-Based Materials in Large-Scale Precalculus Instruction.

This paper outlines an attempt at integrating web-based activities into a precalculus course at a large university. A discussion of the development of the web-based activities is initially provided. Distinctions are made between transpositional uses of web-based technology and uses that take advantage of the full potential of the technological medium.A report of a study that investigated the effects of the use of the activities in four classrooms follows. The report focuses on the use of the activities by two instructors, only one of whom received mentoring in that regard. The unmentored instructor entered the course with an instructional philosophy more compatible to the goals of the activities than the mentored instructor, and it was found that this led to more appropriate uses of the activities during instruction, despite a lack of mentoring. The difficulties of the unmentored instructor were not inherent in the technology, but in the pedagogic incongruences between the goals of the instructor and the web-based activities. Evidence was also found that these instructional differences impacted how the two groups of students approached the tasks, as well as differences in their understandings of the underlying mathematical ideas of covariance and rate of change. Implications and recommendations are then given.

Small Groups Leading to Big Groups.

ABSTRACT This paper describes the responses of two different classes to small group-based instruction. A remedial algebra class, despite achievement gains, expressed displeasure and frustration in the small group format. An Elementary Education Mathematics Content course showed great enthusiasm and success to the instructional approach. Explanations are presented for the differences in the learning behaviors, including individual differences and changes in the overall instructional approach.

The Development of Teacher Knowledge in Support of Student Mathematical Inquiry

Abstract Applying the Mathematical Knowledge for Teaching framework, we discuss the components of teacher knowledge that might be useful in supporting mathematical inquiry, and examine ways in which we strive to develop this knowledge within a middle grades mathematics program for undergraduate students who are prospective teachers. Using sample activities from multiple courses in the program, we offer general principles of instruction for supporting mathematical inquiry at all grade levels. We contend that an awareness and application of the multiple facets of mathematical knowledge for teaching can be critical to supporting mathematical inquiry across the K-16 spectrum.

Jumping on the STEM Bandwagon: How Middle Grades Students and Teachers Can Benefit from STEM Experiences.

Abstract Given the current emphasis on science, technology, engineering, and math (STEM) education and its key attributes, middle school is an optimal time to implement STEM-based curricula. However, the interdisciplinary and open-ended nature of STEM projects often makes implementation difficult. In this article, we describe a professional development project aimed at preparing middle grades teachers to implement STEM Design Challenges with their students. We discuss the resources that supported project teachers in navigating identified implementation challenges and provide an example of a sixth-grade team’s efforts to engage all learners in STEM experiences. Recommendations and examples in this article can support other middle school teams working to enact STEM education.