National Council of Teachers of Mathematics 2012 Research Presession

Please note: The NCTM conference program is subject to change.

1146-

Tuesday, April 24, 2012: 8:30 AM
Franklin Hall 6 (Philadelphia Marriott Downtown)
Joanne J. LaFramenta , University of Florida, Gainesville, FL
Thomasenia Lott Adams , University of Florida, Gainesville, FL
Criticism of the mathematics education program in the United States has often reflected that it is “a mile wide and an inch deep” (Schmidt, McKnight, & Raizen, 1998).  Recommendations for improvement have maintained that students should be offered the opportunity to learn mathematics in depth and teachers should teach in that manner.  A comprehensive review of the literature of mathematics education revealed that this practice, teaching mathematics in depth (TMinD) was not described or defined, although it was taken to be a beneficial pedagogical program.  However, many studies had been conducted in mathematics education to identify effective instructional practices that promoted the vision of mathematics understanding and achievement as outlined in the Principles and Standards (NCTM, 2000).  This researcher asked, “What does it mean to teach mathematics in depth?”

From a close reading of the literature since publication of the Standards, a tentative framework for TMinD emerged, drawn from the results of studies that investigated implementation of these reforms.  Several elements seemed to be essential for effective mathematics instruction: classroom discourse, accessing student thinking, flexible grouping practices, and instructional tasks.  Certainly, these instructional activities have always been part of the mathematics lesson, but reform mathematics instruction describes their use in new ways.  Moreover, the theoretical perspective of Standards-based instruction acknowledges the sharing of authority over mathematical truth within a community of learners as concepts are introduced.  If students are to learn mathematics with understanding, they must have opportunities to integrate new knowledge with existing knowledge.

A southeastern state legislature adopted mathematics standards that were derived from NCTM’s Curriculum Focal Points (CFP).  The structure of these standards presented a very different experience for the teachers of elementary school mathematics, organized as they are about three Big Ideas for each grade level.  The legislative guidelines for publishers of instructional materials mandated that mathematics be taught in depth.  This study was interested in the experience of the teachers as they implemented this instructional practice.  Therefore, the theoretical perspective of this study is constructivist.  The teachers who are practicing the pedagogy of TMinD are the holders of the information of interest.  With their colleagues, students, and administrators, they would be constructing knowledge as they implemented TMinD.  The study was designed to access this knowledge from the primary informants.  

Research questions were created to provide a foundation for the essential task of developing a framework for TMinD.  These three questions grounded the design of the study: (1) how do elementary mathematics teachers perceive TMinD; (2) how do they actualize TMinD; (3) how do peripheral mathematics educators (curriculum resource teachers, instructional mathematics coaches, and mathematics teacher educators) describe and develop TMinD?  The researcher was a volunteer in the classrooms of a fifth grade team for four months as they implemented a new set of mathematics standards and a new curriculum that mandated TMinD.  Student achievement would be assessed in a new state-wide standardized test at the end of the school year. 

Qualitative research methods associated with Charmaz’s (2006) interpretation of grounded theory were used to collect and analyze data.  Each teacher was interviewed three times and responded to six journal prompts.  The transcripts and journal entries were coded during the collection process, becoming the preliminary data analysis.  This primary data was supported by the researcher’s field notes of her classroom visits and the analysis of interviews with the six peripheral participants.  Situational analysis (Clarke, 2005, 2009) facilitated combining codes from these diverse data sources to develop the findings as seen through the lens of the researcher’s subjective position as former teacher, mature woman, and researcher.

  Two important findings emerged from the data collected from the fifth grade teachers.  The first is that the conceptualization of TMinD is strongly influenced by the teacher’s orientation toward a learning perspective.  The teachers use those practices which they believe will contribute most to an increase in student understanding of a particular topic.  The second finding to emerge is that the actualization of TMinD is contingent upon balancing the dual forces of the pacing guide (an external scheduling mechanism) and the teachers’ desire to teach for mastery.  Teachers create their instructional plans according to their assessment of existing student comprehension and understanding.  They are seeking mastery of the topic, but the school administration’s pacing guide pressures the tempo of their plans.

These findings were synthesized with the review of the literature and the analysis of secondary data from the peripheral participants to develop a framework for TMinD.  Although elements of effective mathematics practices had been identified by the research results of the last two decades, it became apparent from this study that TMinD does not take place without consideration of elements that emerged from the data.  One of these elements is detail: teachers who teach in depth teach a limited number of significant ideas with great detail.  A second is mastery: TMinD is teaching for mastery during this school year. Prior expectation of instruction in many schools in the United States was teaching for exposure.  A third element is time: to accomplish the goal of mastery requires time for presentation of the topics in detail and time for the individual students to grapple with the new information and integrate it into their knowledge base.  Implications of these findings for the field address teacher preparation and professional development, especially as more states adopt the Common Core State Standards (CCSS) which builds instruction around important key ideas and teaching these ideas in depth. TMinD requires the use of many instructional practices heretofore part of successful mathematics instruction, but these elements will need to be incorporated on a greater scale with altered focus, particularly as states implement CCSS. 

This proposal is intended to be an interactive paper session that will relate to other studies of CCSS and implementation of standards built upon the suggestions of CFP.  Discussion topics for roundtable break-outs may include implications of these results for practice, preservice teacher training, and further research. 

List of References

Charmaz, K. (2006). Constructing grounded theory: A practical guide through qualitative analysis. Los Angeles: SAGE.

Clarke, A. E. (2005). Situational Analysis: Grounded theory after the postmodern turn. Thousand Oaks, CA: SAGE.

Clarke, A. E. (2009). From grounded theory to situational analysis: What's new? Why? How? In J. Morse, P. N. Stern, J. Corbin, B. Bowers, K. Charmaz & A. E. Clarke (Eds.), Developing Grounded Theory: The Second Generation (first ed., pp. 194-235). Walnut Creek, CA: Left Coast Press.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education: Executive summary: U.S. National Research Center for the Third International Mathematics and Science Study, Michigan State University.

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