Maria L. Blanton

Teaching and Learning Proof Across the Grades : A K-16 Perspective

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proof

Using Classroom Discourse To Understand a Prospective Mathematics Teacher's Developing Practice.

This is an interpretive study of a prospective mathematics teacher’s emerging practice during the professional semester. A Vygotskian (Mind in Society, Harvard University, Cambridge, MA, 1978 (Cole et al., Trans.; original work published 1934); Thought and Language, Massachusetts Institute of Technology, Cambridge, MA, 1986 (Kozulin, Trans.; original work published 1934)) perspective was used to examine the nature of classroom discourse and its role in Mary Ann’s (pseudonym) development while student teaching. Results indicate that early classroom discourse mediated Mary Ann’s teaching toward a traditional paradigm of giving information. Moreover, her subsequent e!orts to cultivate dialogic discourse generated con#ict that positioned students as mediators of her practice. Ultimately, experiencing the power and diversity of students’ ideas contributed to shifts in Mary Ann’s early forms of practice. ( 2001 Elsevier Science Ltd. All rights reserved.

Exploring a Pedagogy for the Supervision of Prospective Mathematics Teachers.

Our investigation explored a pedagogy for supervision through a case study of one prospective middle school mathematics teacher during her student teaching semester. Classroom observations by the university supervisor, teaching episode interviews between the supervisor and student teacher, and focused journal reflections by the student teacher were coordinated to challenge the student teachers existing models of teaching. The emerging pedagogy of the teaching episodes, a central focus of this study, was characterized by (a)the use of open-ended questions that centered the student teacher in the process of sense making; (b) a shift away from the supervisors direct, authoritative evaluations of the student teachers practice; (c) a sustained focus throughout supervision derived from the student teachers classroom experiences; and(d) an effort to maintain sensitivity to the student teachers zone of proximal development. We found our approach to be coincident with the notion of instructional conversation (IC)advanced by Gallimore and Goldenberg (1992).The nature of the teaching episodes seemed to open the student teachers zone of proximal development so that her practice of teaching could be mediated with the assistance of a more knowing other.

Children's Use of Variables and Variable Notation to Represent Their Algebraic Ideas.

In this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity, children as young as six years of age can use variable notation in meaningful ways to express relationships between co-varying quantities. In this article, we argue that the early introduction of variable notation in children’s mathematical experiences can offer them opportunities to develop familiarity and fluency with this convention as groundwork for ultimately powerful means of representing general mathematical relationships.

Optimal active pointwise control of vibrating thin plates

Abstract The optimal control of a class of self-adjoint distributed parameter systems using a combined open-closed loop control mechanism is considered. In particular, the proposed method involves the application of a finite number of actuators and sensors to dampen actively the undesirable transient vibrations of a rectangular plate. Necessary and sufficient conditions for optimality of the open-loop control are obtained by a variational approach in conjunction with the convexivity of the quadratic performance index as a system of integral equations. Parameters describing the closed-loop control are determined numerically by a minimization algorithm. Numerical results for a simply-supported plate are given to illustrate the effectiveness of the proposed control.

A first grade student’s exploration of variable and variable notation / Una alumna de primer grado explora las variables y su notación

Abstract This paper presents a case study of a first grade student to illustrate the diversity of her understandings related to variables and variable notation. While prior research has documented secondary school students’ difficulties with variables and variable notation, we identify many productive understandings in this much younger student, leading us to question the prevailing argument that students might have difficulties with variables due mostly to their own limitations. We draw our data from a teaching experiment that explored functional relationships. Individual interviews were carried out with a subset of the students in the experiment prior to, as well as mid-way through and at the end of the experiment. This paper focuses on a set of three interviews with one of the first grade students. We illustrate the shifts that occurred in the student’s understandings about variables and variable notation across as well as within each of the three interviews.

Optimal active pointwise control of thin plates via state-control parametrization

A direct method to solve an optimal control problem by parametrizing both state and open-loop control variables is developed. This technique is designed to suppress the undesirable vibrations of a rectangular plate by open-closed loop control applied at discrete points in space. The closed-loop mechanism is assumed to be proportional to displacement. The measure of performance of the structure is taken as a combination of its total energy and the penalty terms describing the expenditure of the open-closed loop forces used in the control process. Using modal expansion and an appropriate transformation, the optimal control of a distributed parameter system is reduced to the optimal control of a linear time-varying lumped parameter system. Next, a computationally efficient formulation to evaluate the optimal control and trajectory of the structure, based on shifted Legendre polynomial approximations of the state variable and each open-loop control variable, is developed. This converts the linear quadratic pr...

Cycles of Generalizing Activities in the Classroom

This study considers classroom situations in which students and the teacher co-contribute to promoting generalization. It specifically focuses on the ways in which students and a teacher in one classroom engage in generalizing arithmetic. Generalized arithmetic is an important route into early algebra (Kaput in Algebra in the Early Grades. Routledge, New York, 2008); its potential as a way to deepen students’ understandings of concepts of school arithmetic makes it an important focus of early algebra research. In the analysis we identified generalizations around properties of arithmetic and the actions that promoted these types of generalizations, and then considered the relationship between these actions . Analysis revealed that generalizations became platforms for further generalization.

Implementing a Framework for Early Algebra

In this chapter , we discus s the algebra framework that guides our work and how this framework was enacted in the design of a curricular approach for systematically developing elementary-aged students’ algebraic thinking. We provide evidence that, using this approach, students in elementary grades can engage in sophisticated practices of algebraic thinking based on generalizing, representing, justifying, and reasoning with mathematical structure and relationships. Moreover, they can engage in these practices across a broad set of content areas involving generalized arithmetic; concepts associated with equivalence, expressions, equations, and inequalities; and functional thinking.

Exploring Kindergarten Students’ Early Understandings of the Equal Sign

ABSTRACT This study explores kindergarten students’ early notions of mathematical equivalence in the United States. In particular, it uses qualitative methods to examine the understandings children hold about the equal sign prior to formal instruction and how these understandings shift throughout an 8-week classroom teaching experiment designed to develop relational thinking about this symbol. Findings suggest that, even prior to formal instruction, young children hold an operational view of the equal sign that can persist throughout instruction. This early and persistent operational perspective underscores the critical need to design mathematical experiences in kindergarten, and even preschool, that will orient students towards a relational understanding of the equal sign upon its introduction in first grade.