TEACHERS’ EVOLVING MODELS OF THE UNDERLYING CONCEPTS OF RATIONAL NUMBER

 

Karen Koellner Clark                           Roberta Y.  Schorr

Georgia State University                       Rutgers University

     kkoellner@gsu.edu                                   schorr@rci.rutgers.edu

Abstract:  This multi-tiered teaching experiment describes three teachers’ evolving models of the underlying concepts of rational number.   The teachers participated in a 14 week workshop study where they grappled with the skills and concepts embedded within model eliciting problems.   They created curriculum maps to illustrate their understanding of the interrelatedness of the skills and concepts found in the problems.   Further, they implemented these problems in their middle school classrooms and documented student thinking to better understand the problems from their students'  perspectives.   The results illustrate the different ways the teachers revised and refined their ways of thinking about their own content knowledge, pedagogical content knowledge and knowledge of their students’ mathematical thinking.

 

Objectives

This study focused on middle school teachers’ cyclic models or conceptual schemes that evolve as they grapple with the concepts underlying particular model eliciting problems (see Lesh & Doerr, 1998) and how this impacts their content knowledge, pedagogical content knowledge and knowledge of student’s mathematical thinking.   Specifically we will focus on three teachers’ mapping of curriculum, identification of concepts and skills, and their attention to student thinking in their classrooms. 

Research Design and theoretical Framework

The study reported here is part (one tier) of a larger research project that utilized a research design referred to as a multi-tiered teaching experiment (Kelly & Lesh, 1999).   This research design was chosen as it produces auditable trails of documentation that focus on multiple levels of development in this case at the student and teacher level.   Within the multi-tiered teaching experiment, some of the most important key events focus on sequences of model eliciting problems in which participants are repeatedly challenged to reveal, test, and refine, or revise important aspects of their ways of thinking.   Model eliciting problems are designed to produce constructions, explanations, or descriptions that are conceptual tools and are in themselves the most important goals of the problem solving episode (Lesh & Doerr, 1998).   That is, to a large extent, the process is the product.   Consequently, the product explicitly reveals significant information about the reasoning processes that produced it.   In this way these problems, the central feature of the research design, promote learning; yet, at the same time, a byproduct of learning is that auditable trails of documentation emerge that reveal important aspects about the nature of the construct being developed.   Thus, in the study reported in this proposal, we specifically focused on the teachers’ evolving conceptions regarding the underlying concepts of particular model eliciting problems reported in the mapping of curriculum, as well as how they revise and refine their ways of thinking about their own content knowledge, pedagogical content knowledge and knowledge of their students’ mathematical thinking.

This study examined three middle school mathematics teachers with varying levels of experience that were teaching in suburban schools.   The three teachers involved in the study were enrolled in a masters degree program in Atlanta, Georgia with one of the researchers serving as the instructor.   The teachers volunteered to take part in this study with the understanding that they would consider meaningful forms of mathematics instruction, specifically model eliciting problems, to supplement their mathematics curriculum.   They agreed to identify underlying concepts and skills of each model eliciting problem to begin mapping their curriculum for one year.   Moreover, they agreed to focus their attention on their students’ mathematical thinking to better understand the nature of mathematics and their students’ perceptions of the skills and concepts in particular problems. 

The foundation and underlying premise of this investigation was that teachers need appropriate experiences and materials from which to build new models of instruction, learning and assessment (Schorr & Alston, 1999).  They also must be afforded with opportunities to construct deeper understanding of the mathematical concepts they are expected to teach and an increased awareness of the ways in which children learn (Carpenter & Lehrer, 1999; Schorr, Maher, & Davis, 1997; Janvier, 1996; Cobb, Wood, Yackel, & McNeal, 1993.)

Teachers met weekly with researchers in workshop environments where they were presented with one model eliciting problem every other week for 14 weeks.   They collaboratively grappled with solving the problems as well as identifying and visually mapping key concepts, skills and important mathematical ideas that were embedded within the rich problem.   They used national, state and school standards to further document the types of concepts represented in each problem.   They went on to categorize smaller problems sets and sets of symbolically represented problems that were aligned with the model eliciting problems that could serve as follow up problems or homework sets.   Again these smaller problems were combined within the mapping as well.   After sharing their own ideas and representations, they agreed to use these problems in their own classrooms. 

During classroom implementation with researchers present, teachers were encouraged to recognize and analyze student interpretations as they were continually revised and refined.  This in turn aided their understanding of the skills and concepts embedded within the model eliciting problem.  Independently they would reflect, revise and refine their own thinking about the mapping of the concepts and skills found within the model eliciting problems.  They would bring their thoughts and ideas back to share with their colleagues in subsequent workshop sessions.  Studying each other’s mappings as a group afforded the opportunity to both consider the development of their ideas, to discuss students’ thinking in regards to particular model eliciting problems as well as discuss the pedagogical implications of using model eliciting problems in their classrooms.

In this study, the researchers’ goal was to simultaneously stimulate and document changes in teacher knowledge and knowledge of student thinking.  This was accomplished through the use of model eliciting problems in which the teachers were repeatedly challenged to reveal, refine, revise, and extend important aspects of their ways of thinking.  In turn this provided the researchers with an opportunity to stimulate changes in the teachers’ classroom practices that were twofold (Carpenter & Lehrer, 1999).  The first is by challenging them to construct a deeper understanding of their own curriculum.  Secondly by helping them become more familiar with student-generated ways of thinking. 

Data Collection and Analysis

The data for the study include: (a) the teachers’ curriculum mapping about what concepts and skills they identify as important, (b) transcripts from two semi-structured interview sessions and informal questions in regards to what they perceive to be the main ideas or key concepts in model eliciting problems and students’ mathematical thinking, (c) teachers’ work from model eliciting problems that were collected during working group sessions, (d) transcripts from teacher workshop sessions, (5) teachers’ reflections on their own and other teachers work, (e) student work from model eliciting problems that were analyzed during teacher workshop sessions, and (f) researcher field notes taken while working with teachers in classrooms and workshop sessions.

Each workshop session was video-taped and audio-taped.  The audio-tapes were fully transcribed and the videotapes were used to record nonverbal communication important to the analysis, such as a teacher using his or her hands to explain his or her thinking to the rest of the group. 

Codes emerged progressively throughout data collection.  It appeared that the codes fit together to form a coordinated analysis.  First, it appeared that the teachers were able to glean a deeper understanding of the underlying concepts and interrelationships of rational number by (a) solving the student model eliciting problems collaboratively, (b) constructing their own concept maps to use as their own curriculum guides, and (c) by trying to better understand their students thinking.  Second, the teachers went through modeling cycles when identifying the underlying concepts of a particular model eliciting problem.  Their modeling cycles went from more naïve conceptual understandings to more complex over time. 

Results

Results indicated quite conclusively that for teachers the model eliciting task of constructing concept maps that illustrated underlying concepts and skills of particular model eliciting problems for students were a valuable form of information about the growth and acquisition of deeper understandings of the middle school curriculum they currently teach.  Teachers constructed multiple modeling cycles in regards to their perceptions of the interrelationships of concepts and skills.  These modeling cycles appeared to increase in stability and sophistication throughout the workshop sessions and throughout the 14 week investigation.  The teachers’ solutions illustrated that through the course of the investigation their understanding of the interrelatedness of middle school mathematics concepts became more sophisticated as these ideas were continually identified, tested and refined.

The way in which the teachers conceived of the interrelationships of rational number was determined by the ways in which their workshop group came to understand the different model eliciting problems, their ability to discriminate between interrelated skills, and their ability to document student thinking and student strategies.  After the workshop session where small groups of three teachers solved the student model eliciting problems and documented underlying skills on their concept maps they went ahead and implemented these problems in their own classroom with the support of a researcher.  It was at this time that the teachers were able to listen to student thinking and better grasp a concept from student perspectives as well as identify student misconceptions.  To this end, the teachers were empowered to aid students in grasping a concept that they indicated would have been overlooked beforehand.  Moreover, they gained a more sophisticated, deeper understanding of the underlying skills and concepts of rational number documented on their concept map as well as different strategies used by their students.  

Project Implications of Teaching and Research

When helping teachers glean deeper understanding of mathematical concepts it appears that this inquiry process, using model eliciting problems, is effective for several reasons.  First, teachers develop an understanding of the interrelatedness of mathematics curriculum when solving a model eliciting problem like constructing a concept map where the teachers focus is to not only attend to student thinking but to construct a tool that can be used subsequently for multiple purposes.  Moreover, having teachers attend to students' mathematical thinking as well as the underlying concepts and skills of the student model eliciting problems may be a critical means to helping them build a deeper understanding of rational number and algebraic and functional reasoning to inform their teaching.

As teachers begin to use model eliciting problems or the ideas of models and modeling, they typically begin to analyze how their students think mathematically.  Thus, it is important that they have instilled a deeper understanding of mathematics themselves, but further they need to know how to use this information to inform their own teaching.

Model eliciting problems can be designed to be thought revealing activities and because of this they provide powerful tools to help teachers examine, test, refine and revise their own ways of thinking.  They allow and recognize the need for teachers to develop their own ways of adapting and using successful teaching strategies as well as provide a means to deepen their own content knowledge, pedagogical content knowledge, and knowledge of student thinking.

References

Carpenter, T.  & Lehrer, R.  (1999).  Teaching and learning mathematics with understanding.  In E. Fennema & T.  A.  Romberg (Eds.), Mathematics classrooms that promote understanding.  (pp.  19-32).  Mahwah, NJ: Lawrence Erlbaum.

 

Cobb, P., Wood, T., Yackel, E., & McNeal, B.  (1993).  Mathematics as procedural instructions and as meaningful activity: The reality of teaching for understanding.  In R.B. Davis and C.A.  Maher (Eds.), Schools, mathematics and the world of reality (pp.  119-134).  Needham Heights, MA: Allyn and Bacon.

 

Janvier, (1996).  Constructivism and its consequences for training teachers.  In L.P.  Steffe, P.  Nesher, P. Cobb, G.A.  Goldin, & B.  Greer (Eds.), Theories for mathematical learning (pp.  449-463).  Hillsdale, NJ: Lawrence Erlbaum Associates.

 

Kelly, A.  & Lesh, R.(Eds.) (1999).  Handbook of research design in mathematics and science.  Mahwah, NJ: Lawrence Erlbaum.

 

Lesh, R.  & Doerr, H.  (1998).  Symbolizing, communicating, and mathematizing: Key components of models and modeling.  In P. Cobb & E.  Yackel (Eds.), Symbolizing, communicating, and mathematizing.  Hillsdale, NJ: Lawrence Erlbaum.

 

Schorr, R.  Y., Maher, C.  A., & Davis, R.  B.  (1997).  A framework for assessing teacher development.  In E.  Pehkonen (Ed.), Proceedings of the 21st International Conference for the Psychology of Mathematics Education (pp.136-143).  Lahti, Finland.

 

Schorr, R.Y.  & Alston, A.  S.  (1999) Teachers’ evolving ways of thinking about their students’ work.  In F.  Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting for the Psychology of Mathematics Education-North American Chapter (pp 778-784).  Cuernavaca, Morelos, Mexico.  Columbus, Ohio: ERIC.