A study of geometric thinking, focusing on English Language Learners (ELLs), examines effects on teachers, teaching, and students following middle grade teachers’ participation in a 40-hour, year-long professional development (PD) program.  Three research questions are addressed:

1. What effects does teachers’ participation in the PD have on their geometric content knowledge, and its application in attending to students’ mathematical thinking and communication?

2. What effects does teachers’ participation in the PD have on teachers’ instructional practices known to benefit ELLs?

3.What impact on ELLs’ geometric problem-solving is evident when teachers participate in the PD?

In the past three decades, the number of U.S. children from households where the native language is not English grew from 9% to 19% (Firestone et al., 2006).  Students labeled "Limited English Proficient" comprise 9.6% of the student population (4,500,000 students) (Abedi, 2004). Many are taught mathematics in English.  Testing demands have led many schools to separate language work from mathematics work (Firestone et al., 2006), often resulting in limited active engagement by ELLs in mathematics instruction (Brenner, 1998).  How teachers approach integrating language and content influences ELLs’ success with mathematical work (Brenner, 1998; Secada & De La Cruz, 1996).

To enhance mathematics learning opportunities for ELLs, research stresses:

1. Integrating content and academic language development (e.g., August et al., 2005; Calderon, 2007; Garrison et al., 2006; Snow, 2007). 
2. Using multimodal mathematical communication to reinforce learning of mathematical representation, language, and norms of communication (e.g., Gerlic & Jausovic, 1999; Khisty & Chval, 2002; Moschkovich, 2002).
3. Maintaining high cognitive demand for all students, especially ELLs (e.g., Bransford, Brown, & Cocking, 2000; Henningsen & Stein, 1997).

Design

The study’s randomized block design involves teacher groups in 26 sites (approximately 200 teachers) assigned to participate in PD in 2009-10 (Cohort A) or 2010-11 (Cohort B). All teachers responded to an assessment of geometric content knowledge for teaching, analysis of a transcript of classroom discussion, and an instructional practice questionnaire. Teachers provided these data three times: baseline, follow-up (Cohort A immediately post-treatment, Cohort B immediately pre-treatment), and final (Cohort A one year post-treatment, Cohort B immediately post-treatment). This design supports two comparisons to assess efficacy: treatment (Cohort A) versus control (Cohort B) for 2009-10, and post-treatment versus pre-treatment (Cohort B). It also supports comparisons to assess the delay or decay of impacts, comparing outcomes a year following versus immediately after treatment (Cohort A).

A randomly selected subset of teachers participated in site visits (12 Cohort B teachers pre-treatment; 11 Cohort A teachers post-treatment). The site visits included observation of 3 consecutive days’ geometry lessons, and 2 video-recorded student problem-solving sessions involving pairs of ELLs from the teachers’ classrooms. Each pair worked on one of two problems: measuring area of irregular figures, or creating parallelograms within constraints. The design enables treatment-control comparisons for these two measures.

Analysis and Findings

All data collection was completed in Summer, 2011. Analyses for the three presentations that will report findings are currently underway as described.

Presentation 1 reports effects on teacher knowledge from two measures. The first is a 25-item multiple choice assessment requiring knowledge of middle grades geometry content (properties, transformations, and spatial measurement), as applied in teaching contexts. Factor analysis indicated a single scale.

The second measure asks teachers to analyze a classroom discussion transcript enhanced with animations of gestures and use of physical models. Rating of responses focuses on interpreting student thinking, and attending to use of language and multimodal communication. Two researchers, blind to treatment condition and time, trained to establish consistent interpretations and inter-rater reliability, rate each response.

ANOVAs compare the quantitative scores for these two instruments. Qualitative examples drawn from teachers’ analysis of the transcript will illustrate results.

Presentation 2 reports results related to teaching; two measures provide evidence. First, the teacher questionnaire included three scales measuring the frequency of teachers’ practices to: develop students’ academic language, employ multimodal communication, and teach for high cognitive demand expectations. Factor analysis supported appropriateness of the first two scales; the third came from the well-established Surveys of the Enacted Curriculum (Porter, 2002).

Second, during classroom observations one researcher focused on documenting examples of the instructional practices of interest. First-phase analysis summarizes the use of each practice across three lessons attending to:  purposeful use of the practice, quality of implementation, and engagement of ELLs. Second-phase analysis involves qualitative comparison of summaries for teachers who had and had not yet experienced the PD. Qualitative comparisons will be validated using a Turing Test to determine if experts outside our team, blind to condition, differentiate pre-PD and post-PD observation summaries.

Presentation 3 reports evidence of impacts on ELL students’ problem-solving strategies from two measures. Students taught by teachers post-PD are hypothesized to have more opportunities to engage in geometric thinking and representing/discussing their mathematical ideas, and will consequently do so when solving challenging geometry problems. First, students’ instructional opportunities to engage in geometric thinking will be described. During classroom observations one researcher focused on instructional opportunities for students to engage in four habits of geometric thinking (reasoning with relationships, investigating invariants, generalizing geometric ideas, balancing exploration and reflection). Analysis of these data involves the same two-phase qualitative procedure, with a Turing Test for validation, described for Presentation 2.

The student problem-solving measure provides evidence regarding the nature and prevalence of ELLs’ geometric thinking and mathematical communication. Video data will be analyzed by researchers blind to condition. For each video, one researcher will summarize the nature of students’ use of geometric thinking, their problem-solving strategies, and communication of mathematical ideas; a second researcher will review the summary to ensure agreement. These summaries will be compared qualitatively for students of teachers in the two PD Cohorts. 

Session Organization

After an introduction, methods and findings for each question will be presented, with one of three discussants sharing reactions after each presentation. The last half hour will involve small group discussions, facilitated by presenters and discussants.  Participants will choose a group for one of the three questions. Key points from discussions will be shared to conclude.

References

Abedi, J. (2004). The No Child Left Behind Act and English language learners: Assessment and accountability issues.  Educational Researcher, 33(1), 4-14.

August, D., Carlo, M., Dressler, C, & Snow, C. (2005). The critical role of vocabulary development for English language learners. Learning Disabilities Research & Practice, 20(1), 50-57.

Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

Brenner, M.E. (1998). Development of mathematical communication in problem solving groups by language minority students.  Bilingual Research Journal 22( 2, 3, & 4), 103-128

Calderon, M. (2007). Teaching reading to English language learners, Grades 6-12: A framework for improving achievement in the content areas.  Thousand Oaks, CA: Corwin Press

Firestone, W.A., Martinez, M.C., & Polovsky, T. (2006). Teaching mathematics and science to English Language Learners: The experience of four New Jersey elementary schools.  New Jersey Math Science Partnership. Retrieved from http://hub.mspnet.org/index.cfm/13070

Garrison, L. Amaral, O. & Ponce, G. (2006). UnLATCHing mathematics instruction for English learners.  NCSM Journal of Mathematics Education Leadership, 9(1), 14-24.

Gerlic, I. & Jausovic, N. (1999) Multimedia: Differences in cognitive processes observed with EEG.  Educational Technology Research and Development 47(3).  5-14

Henningsen, M. & Stein, M.K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524-549.

Khisty, L.L., & Chval, K. (2002). Pedagogic discourse and equity in mathematics: When teachers’ talk matters.  Mathematics Education Research Journal, 14(3), 154-168.

Moschkovich, J. (2002). A Situated and Sociocultural Perspective on Bilingual Mathematics Learners. Mathematical Thinking & Learning, 4, 189-212.

Porter, A. C. (2002, October). Measuring the content of instruction: Uses in research and practice. Educational Researcher 31(7), 3-14.

Secada, W. G., & De La Cruz, Y. (1996). Teaching mathematics for understanding to bilingual students. In J. L. Flores (Ed.), Children of la frontera: Binational efforts to serve Mexican migrant and immigrant students. (ERIC Document Reproduction Service No. ED393646)

Snow, C. (September 6, 2007). Learning all-purpose academic words webinar. Retrieved from http://www.schoolsmovingup.net/cs/wested/view/rs/7662