Becoming a Reflective Mathematics Teacher

A Guide for Observations and Self-Assessment

Alice F. Artzt, Eleanor Armour-Thomas, Frances R. Curcio, Theresa J. Gurl, Mara Markinson, Mara P. Markinson

• Publisher: Routledge
• Year: 2015
• eBook ISBN: 9781317691983
• Edition: 3
• Topic: Bildung
• Subtopic: Bildung Allgemein

Ideal for preservice mathematics teachers who are taking methods courses or are student teaching, this research-based, activity-oriented guide offers a highly effective framework for teacher reflection and self-assessment. Highlighting inquiry-based, learner-centered teaching and grounded in a cognitive perspective, Becoming a Reflective Teacher of Mathematics, Third Edition features:

• Detailed observation instruments for observing other teachers
• Reflective activities that provide a structure for beginning teachers to think about their teaching
• Guidelines and instruments for supervisors to use when observing, conferencing with, and assessing beginning or student teachers

The Third Edition of Becoming a Reflective Teacher of Mathematics is aligned with the latest standards for teaching mathematics including the Common Core State Standards-Mathematics, and the latest assessments for mathematics teacher certification which place a high priority on reflective practice. Thoroughly revised and updated throughout, the Third Edition continues to provide preservice and in-service mathematics teachers with practical ideas for developing and honing reflective and self-analytical skills needed to advance and improve instruction.

Part I
Philosophical Basis for the Model

1
Toward an Understanding of Student-Centered Teaching

You have decided to join the teaching profession at a time when there is a great need for highly qualified, well-prepared mathematics teachers. Expectations of teachers for the 21st century are very different from what they were as little as a decade ago. The purpose of this chapter is to give you a vision of what high-quality teaching looks like and the issues you must consider to teach that way. A description of instructional practice in mathematics, complexities of teaching, challenges facilitating student-centered, assessment-based teaching, reflective practice, and professional development set the stage for the remaining chapters in this book.

Instructional Practice in Mathematics

What does it mean to be a competent mathematics teacher? What do you envision when you think of a good mathematics lesson? Picture two lessons involving probability. In the first lesson, the teacher presents a clear lecture providing all the definitions and rules. The students take copious notes, and after watching the teacher model how to do a few problems, they do some practice problems at their seats, which are then reviewed as a whole-class discussion. Afterward, the teacher brings the class together and answers questions about how to do the problems. In the second lesson, the teacher engages the students in an unfair game of dice. The students must make conjectures regarding the fairness of the game. They work in pairs to figure out why the game is unfair and then have a class discussion during which they inadvertently construct the concepts of probability underlying the game. The teacher facilitates discussion that helps the students organize and formalize their ideas. The students complete exit slips in which they must explain why the game was unfair. The next day, in anticipation of further developing probabilistic concepts, the teacher groups students heterogeneously, based on their responses on the exit slips, making sure that each group includes a student who understood the concept well, alongside students who might have revealed a weaker grasp of the concepts.

As you may notice, student involvement is maximized in the second lesson. In contrast to the first lesson where the students are passive recipients of information, students in the second lesson are actively engaged in the problem-solving process with one another. In such a lesson, the teacher oversees, guides, and facilitates the students’ construction of new knowledge. Endorsement for this latter type of teaching comes from the literature on how students learn mathematics. That is, research suggests that learning is an active problem-solving process in which social interaction plays a critical role (Cobb, 1986; Vygotsky, 1978). Learning is facilitated when learners are encouraged to link new information to their prior knowledge and thereby generate new understandings (Fennema, Carpenter, & Peterson, 1989; Greeno, 1989; Lampert, 1986; Noddings, 1990; von Glasersfeld, 1987). Moreover, the learning of individual students is assessed at the end of the lesson and used to inform appropriate instruction for the next day’s lesson. The advantage of such data-driven instruction is beginning to be supported by both research (Carlson, Borman, & Robinson, 2011) and teachers’ perceptions of improvements made by their students when using such approaches (Means, Padilla, & Gallagher, 2010). The use of data to inform instruction is becoming a requirement for effective teaching, and it is sometimes included in teacher evaluation rubrics (e.g., Danielson, 2008). Standards documents, such as those produced by the Interstate Teacher Assessment and Support Consortium (CSSO, 2013) and some teacher certification examinations, such as the edTPA (Stanford Center for Assessment, Learning and Equity [SCALE], 2014), which is a performance assessment, also address the use of data for instructional design.

In line with this research, the Professional Standards for Teaching Mathematics (National Council of Teachers of Mathematics [NCTM], 1991) and the Principles and Standards for School Mathematics (NCTM, 2000) suggest that teachers must create opportunities that stimulate, guide, and encourage students to make connections among mathematics concepts, construct mathematical ideas, solve problems through reasoning, and take responsibility for their own learning. The research supportive of student-centered teaching is also consistent with standards for beginning teachers that were developed by InTASC (CSSO, 2013), is included in teaching evaluation rubrics (Danielson, 2008), and is part of some teacher certification examinations such as the edTPA.

Teaching Is More Than Meets the Eye

Although it is true that a competent teacher must engage in classroom behaviors that are likely to promote student learning of mathematics, this is not the full story of professional competence. There is growing recognition that teaching involves more than what teachers actually do in the classroom and extends to the driving forces behind the teacher’s actions: their cognitions. For example, if you were to interview the two teachers described in the previous section about their lessons, what differences would you expect to find? What issues do you think they would discuss? For example, which teacher would be more apt to focus on covering the content? Which teacher would be more apt to focus on student understanding?

For almost two decades, researchers have been building frameworks and models that seek to understand the mind and related actions of a teacher (e.g., Artzt & Armour-Thomas, 1998; Fennema & Franke, 1992; Schoenfeld, 1998, 2013; Simon, 1997). It is now quite evident that teachers’ knowledge, beliefs, and goals directly influence their instructional practice, and ultimately student achievement (Baumert et al., 2010; Cross, 2009; Hill, Rowan, & Ball, 2005, Love & Kruger, 2005). We examine teacher decision making across three stages of teaching: preactive (i.e., planning prior to enacting the lesson), interactive (i.e., monitoring and regulating during the lesson), and postactive (i.e., evaluating and revising after the lesson is over). These components form a network of overarching cognition that directs and controls the instructional behaviors of teachers in the classroom. To examine both areas of teaching (i.e., instructional practice and cognitions), we developed a model of two interrelated frameworks grounded in a student-centered perspective of teaching. The first framework, which we call the Instructional Practice Framework (IPF), consists of indicators of instructional practice and is described in detail in Chapter 2. The other framework, the Teacher Cognition Framework (TCF), consists of indicators of teacher cognitions and is elaborated in Chapter 3.

The Challenge of Facilitating Student-Centered Teaching

Although there is broad agreement in the mathematics education and research communities concerning the cognitive view of learning and its implications for teaching, research has not yet fully revealed how to help teachers develop student-centered practices in the classroom, and the cognitions that guide and shape such practices is in its infancy.

One difficulty is that not enough is yet known about how to change the deeply entrenched beliefs about mathematics learning and teaching that are based largely on our own experiences as students. For example, one longstanding yet erroneous belief about teaching is that the role of the teacher is to transmit mathematical content, demonstrate procedures for solving problems, and explain the process of solving sample problems. Although there may be some instances when such an approach may be appropriate, there is much more to teaching for student understanding.

Despite the research literature to the contrary, a commonly held belief about learning is that students learn the content by listening well to the teacher and remembering what they are told (Cooney, 1999). In this rather narrow view, students show that learning has occurred by applying the demonstrated procedures and working problems similar to the ones introduced earlier by the teacher.

Another difficulty involves the unpredictability of change, even among teachers given the same learning experiences intended to transform their teaching. Predicting change can be inaccurate because each prospective teacher comes to a learning situation with an existing cognitive structure that includes certain personal beliefs, knowledge, and goals. Although a group of teachers might be exposed to the same learning experiences, what one teacher notices and acts on depends on how the experiences filter through her or his unique existing cognitive structure. Consequently, different interpretations of the same experiences could lead to differential patterns of change. In short, understanding new teachers’ trajectories is still an inexact science.

Despite these difficulties, cognitive models of teacher change maintain that teachers move through pathways that culminate in student-centered practice (e.g., Cooney, 1993; Cooney & Shealy, 1997; Fennema et al., 1996; Franke, Fennema, & Carpenter, 1997). In reviewing these studies, Goldsmith and Schifter (1997) identified three stages of teaching. The initial stage is characterized by traditional instruction, where the teacher is driven by the belief that students learn best by receiving clear information transmitted by a knowledgeable teacher. In subsequent stages of instruction, the teacher is more focused on helping students build on what they understand and less focused on helping them in the sole acquisition of facts. The instruction is founded on the teacher’s belief that students should take greater responsibility in their own learning. In what Goldsmith and Schifter call the final stage, instruction is in line with high-quality teaching. We prefer to refer to this stage as an advanced stage of teaching to indicate that there is always potential for future growth. At this advanced stage, the teacher arranges experiences for students in which they actively explore mathematical topics, learning both the hows and whys of mathematical concepts and processes. The teacher is motivated by the belief that, given appropriate settings, students are capable of constructing deep and connected mathematical understanding.

We address the question of teacher change in our mathematics education program by requiring prospective teachers to use a reflection and self-assessment procedure. We believe that if teachers are to become truly student centered in their teaching, they must view themselves as agents in their own learning and development. They must be willing and able to take responsibility for their actions in the classroom by giving careful consideration to what they intend to do not only before and during the lesson but after the lesson as well. To recognize that one must be committed to thinking deeply about various aspects of one’s teaching is to climb the first step on the ladder of change. More than 70 years ago, Dewey (1933) defined this type of thinking as reflective thinking: “Active, persistent, and careful consideration of any belief or supposed form of knowledge in the light of the grounds that support it…” (p. 9; original in italics).

Reflection, though necessary, is not sufficient for transformative teaching. Teachers must also be willing and able to acknowledge problems that may be revealed as a result of the reflective process. Moreover, they must explore the reasons for the acknowledged problem, consider more plausible alternatives, and eventually change their thinking and subsequent action in the classroom. We argue that over time, the habitual use of reflective and self-assessment processes about learning experiences leads to transformation in teaching.

Definition of Reflection and Self-Assessment

The two strands that compose our approach are reflection and self-assessment. The first, reflection, is defined as thinking about one’s own teaching. It involves the thoughts teachers have before, during, and after the actual enactment of a lesson. Although there are many definitions of reflection (e.g., Dewey, 1933; Schon, 1983; Van Manen, 1977; von Glasersfeld, 1991), the construct resembles several of the thinking processes identified by Pólya (1945) and Schoenfeld (1987) in their conceptions of mathematical problem solving: understanding, analyzing, planning, monitoring and regulating, and looking back. In Pólya’s model of problem solving, expert problem solvers think about the problem to be solved by using thinking processes indicative of understanding, analyzing, and planning. We contend that teachers must engage in similar types of thinking before the actual enactment of a lesson. For example, they must think about their goals for the students and thereby activate their knowledge and beliefs about the students, pedagogy, and the content itself. Teachers also must think about the difficulties students are likely to encounter in their efforts to attain the goals; consequently, teachers must think about the strategies they will need to consider in conducting the anticipated lesson. One possible approach is beginning the planning process with the desired end result and goals in mind, and then determining the appropriate instruction and assessments necessary to reach those goals. This approach is referred to as “backwards design” (Wiggins & McTighe, 2005).

Similar to mathematical problem solving, the monitoring and regulating phase of instruction entails the reflective process of deciding whether the methods being used are indeed leading to the accomplishment of the goals. For effective instruction, teachers must probe students’ understanding using informal and formal formative assessment strategies that allow them to gauge the level and quality of each student’s understanding and then adjust instruction based on the different needs of the students. Effective differentiated instruction is completely dependent on the teacher’s ability to engage in this critical reflective process that occurs during the implementation of a lesson.

Continuing with the problem-solving analogy, we view Pólya’s “looking back” as another type of reflective thinking that teachers engage in after conducting a lesson. In other words, just as an expert problem solver reexamines the steps taken to solve a problem, a teacher, on completing a lesson, must rethink lesson goals and reconsider what the teacher and the students said, did, and felt during the lesson. Using informal and formal formative and summative assessment strategies can help teachers engage in accurate reflective decision making. This reflective phase is likely to uncover difficulties or problems that, if the teacher does not address, may impede progress toward self-improvement in teaching. How do teachers increase the likelihood that problematic aspects of their teaching are indeed revealed during this reflective process? And, as importantly, what conjectures or judgments do they make about such difficulties?

The second strand in our procedure is self-assessment. Self-assessment describes the kinds of evaluative questions teachers ask themselves as they reflect on their teaching after completing lessons.

Taking personal responsibility and control of one’s learning is a hallmark of academic excellence. A critical factor in this type of learning that researchers define as self-regulated (e.g., Harris, 1979; Paris & Newman, 1990; Zimmerman, 1990) is self-assessment. When students are in the habit of asking themselves such questions as “What do I need to do?” “How am I doing?” and “How well did I do?” in relation to academic goals, their learning is more proficient than that of students who do not habitually ask themselves these questions. Indeed, most contemporary models of human cognition consider self-assessment an important dimension of good thinking (Brown, 1978; Flavell, 1981; Schoenfeld, 1987; Sternberg, 1986). Likewise, we believe that self-assessment plays a pivotal role in enabling teachers ...

Table of contents

• Cover Page
• Half Title page
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface to the Third Edition
• A Note to Professors
• Part I Philosophical Basis for the Model
• Part II How to Use the Model
• Part III Evidence—The Model in Action
• Appendix A Research Results of Exploratory Study
• Appendix B Observation Guidelines
• Appendix C Observation Charts Made by Preservice Teachers
• Appendix D Guides and Forms for Supervised Observations
• Appendix E Portfolio Assignment and Rubrics
• References
• Author Index
• Subject Index