Preservice Teachers' Knowledge for Teaching Algebra

For Equity: A Preliminary Report

Successful teachers must have deep knowledge of mathematics and of the diverse characteristics and needs of students. The proposed Symposium will present preliminary results of a 5-year NSF-funded design experiment (Lamberg & Middleton, 2009) that endeavors to design, develop, and test prototype activities to enhance preservice teachers' knowledge for teaching algebra for equity. As shown in Figure 1, each activity includes: (1) A Mathematics Problem and Teaching for Equity Challenge, (2) Collaboration, (3) Second Life (SL) Simulated Teaching, and (4) Assessment.   

Figure 1. Components of prototype activities for teaching algebra for equity.

Methods

Each of the following papers drew from the participants, procedures, and data sources used in the first design stage of the project.  

Participants: There were 7 sophomores, 10 juniors and 1 senior preservice teachers (PTs) in a required Problem Solving Course at a southwestern Research I university; 14 White females, 1 African-American female, and 3 White males. 

Procedures: The Problem Solving Course consisted of (1) problem solving heuristics (Polya, 2004), (2) strategies for teaching diverse students, and (3) SL teaching experiences. Instruction for diversity was guided by Hypothetical Learning Trajectories (HLTs) developed by the research team which contained four instructional schemes: Situated Learning, Cultural Relevance, Innovation/Change, and Critical Pedagogy (Authors, 2011). PTs chose one of these schemes to design and teach a problem solving lesson in SL.

Data Sources: Participants completed the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) (Enochs, Smith, & Huinker, 2000), and the Knowledge for Algebra Teaching for Equity (KATE) test developed by the authors. The KATE test contained 20 items adapted from the Cultural Awareness and Beliefs Inventory (CABI) (Roberts-Walker, 2007), and three open-ended problems to assess knowledge about teaching algebra. Participants completed 1) pre-interviews on their education experience and perceptions on equity; 2) post interviews on the SL teaching experience, and 3) post interviews on their experiences in the problem solving course.

Presentation 1 (15 minutes):

Hypothetical Learning Trajectories for Student Learning

 and Equity in Algebra Teaching

Hypothetical learning trajectories (HLT) have attracted the attention of educators to improve mathematics instruction, assessment, curricula, and standards (Daro, Mosher, & Corcoran, 2011). The HLT used in this project follows the model of Simon and Tzur (2004), whose assumptions about their use is that generation of an HLT is based on understanding of the current knowledge of the students. A focus of this paper is to use the data collected from PTs to further understand and revise the proposed HLT schemes.

PTs who chose the Culturally Relevant scheme “strongly agreed” with CABI items about their awareness of culturally responsive teaching behaviors, integrating cooperative learning opportunities, strengths of diverse students, and comfort in working with students having different values and beliefs from theirs.

Most of the MTEBI items aligned with the Innovation/Change Scheme of the equity HLT.  The average gain on the MTEBI was 3.9 points with scores ranging from -7 to +21 (out of a possible 44 points) suggesting that participants made progress on the equity HLT. Participants showed higher gains on the personal subscale than they did on the general subscale.

Presentation 2 (15 minutes):

Effects of Problem Solving Equity Challenges on Knowledge

 for Teaching Algebra and on Perceptions about Equity

NCTM's (2000) Equity Principle requires that teachers develop effective methods for supporting the learning of mathematics for all students. The goal of success for all can be achieved by providing opportunities for PTs' to engage in experiences with diversity issues through a simulated classroom environment in which math content, student questions, and other variables are controlled. The research question was how PTs' cultural awareness and teaching beliefs were changed through the prototype activities and SL experiences.

            The results indicated that there was improvement on 17 of 20 CABI items from pretest to posttest, and significant (p<.05) improvement in four items that dealt with comfort in teaching students of color or who live in poverty. 12 of the 18 PTs improved on algebraic problem solving and posing ability. As shown in Figure 2, the improvements on diversity and on algebra knowledge for teaching were not always aligned. Although some of these results may have been due to high pretest scores on the CABI for some PTs, resulting in possible “ceiling” effects, further research is needed to assure that improvements in both of these important outcomes coincide.

Figure 2. Individual gain scores on the CABI for each participant, ranked from highest to lowest gain.

Presentation 3 (15 minutes)

Knowledge for Teaching Algebra for Equity from the Perspective of

Two Prospective Teachers

According to Sleeter (2001), most PTs lack experience in multicultural classes, failing to meet the expectations of working with diverse students. Recent research suggests that Second Life (SL) can offer a valuable learning environment for undergraduate students (e.g., Greenberg, Nepkie, & Pence, 2009; O'Connor & Sakshuag, 2009).

            In this case study, we investigated the effects of instruction on teaching algebra for equity from the perspective of two PTs. The research question was: How was the PTs' knowledge of teaching algebra for equity and their perceptions on SL teaching enhanced? Participants were selected for contrasting gains on CABI and problem solving; Participant A demonstrated high gains on CABI and low gains on problem solving, while participant B showed the reverse.

            A thematic analysis of the pre- and post interviews produced themes in relation to the participants' gains on algebra and CABI. The intervention helped enhance both participants' algebra knowledge on mathematical understanding, problem solving strategies, clarity in the solution, and completeness of the problem. A plausible explanation for A's small improvement on problem solving is that he has been a low achiever in mathematics since high school.  Participant A showed distinctive improvement on understanding diverse learners, whereas B's cultural awareness and beliefs were less impacted. Interviews with B revealed she held high-level cultural awareness and beliefs prior to the intervention. It is possible that the intervention is less likely to impact participants who have high-level cultural awareness and beliefs. Finally, SL teaching provided low-risk teaching scenarios for the participants. Except for some constraints on the authenticity of the classroom, both participants indicated that the SL teaching experience was helpful for their development in preparing for teaching. 

Presentation 4 (15 minutes):

Lessons Learned from Simulated Teaching in Second Life

In developing the SL classroom simulations and observing initial efforts of the PTs in developing lessons for diverse students, some challenges and affordances were revealed (Dalgarno & Lee, 2010).  For example, participants addressed student-avatars' mathematical questions but often ignored or glossed over questions about the cultural context of the problem, indicating that they might have over-estimated their openness and cultural awareness on the KATE beliefs pretest items. Debriefing discussions indicated that participants' opinions of SL changed markedly after they taught their lessons in SL.  Some commented that the SL experience might offer them a good transition to teaching in a real classroom. One PT who had a previous unfavorable classroom experience commented that he would have liked to have taught the SL lesson prior to entering an actual classroom. 

            Overall PTs saw great value in the SL lesson experiences. They suggested that the SL training be modified or shortened, recalling their frustration with losing instructional time due to technical and network issues. PTs commented that training should focus on what they needed to know to deliver lessons in SL. They also expressed interest in seeing more of the middle school avatars' work during lessons while understanding that some capabilities may go beyond the limits of the program.

Audience Participation (30 minutes)

References

Authors (2011, accepted). Journal of Negro Education, Special Issue.

Dalgarno, B., & Lee, M. J. (2010). What are the learning affordances of 3-D virtual environments? British Journal of Educational Technology, 41(1), 10-32. doi:10.1111/j.1467-8535.2009.01038.x

Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics. (Research Report t # RR-68). Philadelphia, PA: Consortium for Policy Research in Education

Enochs, L. G., Smith, P. L., & Huinker, D. (2000). Establishing factorial validity of the mathematics teaching efficacy beliefs instrument. School Science and Mathematics, 100(4), 194-202.

Greenberg, J., Nepkie, J., & Pence, H. E. (2009). The SUNY Oneonta Second Life Music

        Project. Journal of Educational Technology Systems, 37(3), 251-258.

Lamberg, T. & Middleton, J. A. (2009).  Design research perspectives on transitioning from individual microgenetic interviews to a whole-class teaching experiment.  Educational Researcher 38(4), 233-245.

National Council of Teachers of Mathematics. (2000). Principles and standards for school     mathematics. Reston, VA: Author.

O'Connor, E. A., & Sakshaug, L. E. (2009). Preparing for Second Life: Two teacher

        educators reflect on their initial foray into virtual teaching and learning. Journal 

        of Educational Technology Systems, 37(3), 259-271.

Polya, G. (2004). How to solve it: A new aspect of mathematical method. Princeton University        Press: Princeton , NJ.

Roberts-Walter, P. F. (2007). Determining the validity and reliability of the cultural   

        awareness and beliefs inventory. Unpublished PhD, Texas A&M University,

        College Station, TX.

Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking & Learning, 6(2), 91-104.

Sleeter, C. E. (2001). Preparing teachers for culturally diverse schools. Research and the overwhelming presence of whiteness. Journal of Teacher Education, 52(2), 94-106.