Coordinating Social and Individual Aspects of Generalizing Activity: A Multi-tiered 'Focusing Phenomena' Study

Principal Investigator: 
Co-Investigator: 
Project Overview
Background & Purpose: 

The goals of this project are to account for: (a) the ways in which features of classroom environments influence what students attend to mathematically; (b) the particular mathematical focus that emerges in a classroom; and (c) how attention-focusing interactions are related to the ways in which students generalize or transfer their learning experiences. In order to pursue these theoretical goals, a series of empirical studies were conducted in San Diego, California and in Madison, Wisconsin involving middle school and high school students learning about the topics of linear functions and quadratic functions. It is anticipated that this research will result in benefits for teachers and their students, by demonstrating how the durable concepts that students take away from instruction are influenced by many subtle and often unintentional aspects of teaching practices involving the nature of how certain mathematical properties come to be the object of focus when multiple sources of information compete for students’ attention.

Setting: 

A series of empirical studies were conducted in San Diego, California and in Madison, Wisconsin involving middle school and high school students learning about the topics of linear functions and quadratic functions.

Research Design: 

The research design for this project is comparative, and is designed to generate evidence that is descriptive (observational, design research) and associative/correlational (grounded theory to establish conceptual conceptions). The project used teacher-designed and researcher-designed curricular interventions. For each study, there were 2 to 4 comparative instructional conditions. This project collected original data using observation videography and face-to-face semi-structured interviews. 10 clinical interview protocols were developed for the project. The project primarily used methods adapted from Grounded Theory (ala Strauss and Corbin) to conduct qualitative analysis of videotaped classroom and interview data. We also created a conceptual and methodological framework for the research project, called the Focusing Interactions Framework, which guided the analysis of videotaped classroom data.

Findings: 

There is growing interest in the field of mathematics education in the processes of 'noticing' and 'attention-focusing.' This interest arises primarily from the research on teachers and teacher professional development. As teacher educators began relying on the use of classroom videotapes, research by van Es and others indicates that teachers don't notice what their trainers expect when watching video. Fewer studies have been conducted in the field of mathematics education on students' 'noticing' behavior. In fact, we unaware of any work other than our own that demonstrates a connection between what students come to notice mathematically and features of the instructional environment which support that particular focus.

The findings from one of our empirical studies contribute to the field's understanding of the role of “noticing” in learning and generalizing mathematics. Specifically, students from two 7th grade classes came to 'see' and reason about transfer situations involving linear functions in distinctly different ways. For example, 88% of students in Session 1, when presented with a table of linear data in a water pumping situation, coordinated the two quantities, used beginning levels of ratio reasoning, and correctly determined the pumping rate (what we recognize as the slope of the function). In contrast, only 33% of students in Session 2 reasoned similarly, instead focusing primarily on a single quantity or on additive relationships. These different ways of 'seeing' were linked with the objects of focus in that emerged in the classroom. In Session 1, a major shift of focus appears to have occurred, from a less sophisticated focus on two quantities in an uncoordinated manner to a more sophisticated focus on a composed unit of quantities. This trend reverses itself in Session 2, from a more sophisticated focus on a functional relationship between two quantities during Lesson 4 to a less sophisticated focus on growth as an additive increase in just one quantity for the rest of the unit. More importantly, an analysis of the focusing interactions in both classrooms reveals how instructional features such as annotating representations, renaming, and the nature of language use support these differential shifts in focus. This research provides a useful contrast between the types of foci that are related to productive student generalizations and those that unwittingly afford less powerful student generalizations.

These findings can benefit teachers and their students, by demonstrating how the durable concepts that students take away from instruction are influenced by many subtle and often unintentional aspects of teaching practices. Recommendations for reform in mathematics education have emphasized the use of particular pedagogical actions (e.g., questioning versus telling), organizational strategies (e.g., the use of collaborative learning groups), inquiry-oriented curricular materials, and artifacts. However, both teachers for Sessions 1 and 2 utilized reform-oriented curriculum, collaborative groups, computer software and hands-on materials. Instead, many micro-features of instruction came into play in these two sessions, which worked to support productive mathematical ideas in one case but which worked to undermine the implementation of reform practices in another case. Just as there is a range of alternative conceptions and misconceptions that students construct for any given mathematical topic, there is a range of possible mathematical foci to which teachers can direct students' attention. Being sensitive to these multiple foci can help teachers identify potential traps and suggest generative alternatives.

Publications & Presentations: 

Charles, R., & Lobato, J. (2009, April). Essential understandings book series: Professional development tools for engaging teachers with mathematics. Paper presented at the annual conference of the National Council of Supervisors of Mathematics. Washington D.C.

Ellis, A.B. (2007). Unexpected connections across function families: Students’ generalizations about quadratic data. In the Proceedings of the 29th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

Ellis, A. (2007, April). Attention focusing and student noticing: Implications for conception development. In A. Ellis (Chair), Missing Links in the Implementation of Mathematics Education Reforms: “Attention Focusing” and “Noticing”. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Ellis, A. (2009, April). Calculator-Based Exploration of Quadratic Relationships by Students. Poster presented at the annual meeting of the American Educational Research Association, San Diego.

Ellis, A., & Grinstead P. (2009) Hidden Lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Manuscript under review by the Journal of Mathematical Behavior.

Ellis, A., & Ely, R.(2009). Why Mathematics Knowledge for Teaching Matters: Examining Three Student Strategies. Manuscript under review by the Mathematics Teacher.

Hohensee, C. (2007). Student' emerging understanding of concepts related to similarity and slope. Unpublished master's thesis, San Diego State University, San Diego, California.

Lobato, J. (2006). Alternative perspectives on the transfer of learning: History, issues, and challenges for future research. Journal of the Learning Sciences, 15 (4), 431-450.

Lobato, J. (2007, June). How is what we “notice” mathematically in classrooms socially organized? Invited presentation as part of The Maseeh Mathematics and Statistics Colloquium Series, Portland State University.

Lobato, J. (2007, April). Beyond the typical recommendations for mathematics reform: Attention-focusing matters! In A. B. Ellis (chair), Missing links in the implementation of mathematics education reforms: "Attention focusing" and "noticing." Symposium conducted at the annual meeting of the American Educational Research Association, Chicago, IL.

Lobato, J. (2007, February). The transfer of learning: A simple shift in assumptions with multiple implications for teaching undergraduate mathematics. Paper presented at the Tenth Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Lobato, J. (2007, November). Attention-focusing and the transfer of learning. Invited address to the inauguaral conference of the International Mind, Brain & Education Society (IMBES), Dallas, Texas.

Lobato, J., (2008). On learning processes and the National Mathematics Advisory Panel Report, Educational Researcher, 37(9), 595-601.

Lobato, J. (2008, June). The importance of “noticing” in mathematics classrooms. Research symposium hosted by Seoul National University, South Korea.

Lobato, J. (2008, June). The social organization of noticing: An adaptation of Goodwin to mathematics classrooms. Research symposium hosted by Capital Normal University, Beijing, China.

Lobato, J. (2008, June). The role of executive attention in mathematics learning. Research symposium hosted by East Capital Normal University, Shanghai, China.

Lobato, J. (2008, May). The Role of “Noticing” in the Transition to Algebra. Invited presentation to the conference, Critical Issues in Education: Teaching and Learning Algebra, Mathematical Sciences Research Institute, Berkeley, California.

Lobato, J. (2006, November). Relationships between attention-focusing and the “transfer” of learning across two instructional approaches to rates of change. In S. Alatorre, J. L. Cortina, M. Sáiz , and A. Méndez, Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education, Universidad Pedagógica Nacional.

Lobato, J., & Ellis, A. B. (in press). Essential understandings: Ratios, proportions, and proportional reasoning. In R. M. Zbiek (Series Ed.), Essential understandings. Reston, VA: National Council of Teachers of Mathematics (NCTM).

Lobato, J., & Hohensee, C. (2008, March). The psychological and social organization of “noticing” mathematical regularities. In A. Izsak (chair), Learning as interactively constituted phenomena: New approaches in mathematics education. Symposium conducted at the annual meeting of the American Educational Research Association, New York.

Lobato, J., & Rhodehamel. B. (2009, April). Focusing Interactions” as an Alternative Transfer Process. In R. Engle (chair), Transfer in Context: Empirical Investigations of Novel Explanatory Mechanisms for Mediating the Transfer of Learning. Symposium conducted at the annual meeting of the American Educational Research Association, San Diego.

Tillema, E., Lobato, J., Ellis, A., Hohensee, C. (2009, April). Eighth graders reasoning about quadratic functions. Symposium conducted at the Research Presession of the annual meeting of the National Council of Teachers of Mathematics, Washington D.C.

In preparation:
Ellis. A. The coordination of prior and current influences: Making sense of the mechanisms for generalization. In preparation for the American Educational Research Journal.

Ellis A., & Grinstead, P. Pattern-seeking activities: Are all strategies created equal? In preparation for Mathematics Teaching in the Middle School.

Lobato, J. Coordinating psychological and social aspects of generalizing activity. In preparation for Cognition and Instruction.

Lobato, J., & Rhodehamel, B. Professional vision. In preparation for the Journal for Research in Mathematics Education.

Lobato, J., & Rhodehamel, B. Attention-focusing as a Transfer Mechanism. In preparation for the Journal of the Learning Sciences.