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). 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.
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. Roberts-Walter,
P. F. (2007). Determining the validity and reliability of the cultural awareness and
beliefs inventory. Unpublished PhD, 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.
This symposium describes the first phase of a project to design, develop, and test technology-enriched teacher preparation strategies that address equity in algebra learning. Results indicate that Second Life simulations can be rich settings for prospective teachers to develop mathematics teaching skills and apply their ideas about diversity.
Session Type: Research Symposium