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Multiple Representations in a Communication-Enhanced Environment
Overview
We report on a cluster randomized trial implementing SimCalc MathWorlds¨ in Algebra 2 classrooms across six districts in Massachusetts as part of a five study program of longitudinal research across four years. We discovered that non-honors students can learn more complex mathematical ideas in Algebra 2 particularly related to reasoning across multiple representations. This aligns with the NCTM's priority area of instructional interventions. Educational Significance & Theoretical Perspective
We have addressed the mathematics of change and variation a core school mathematics strand (NCTM, 2000) that is representationally demanding, that is studied at many levels by all students, from Pre-Algebra through Calculus and that can serve to energize and contextualize the core ideas of algebra in ways that lay a conceptual base for calculus (Kaput & Roschelle, 1998).
Ethnographical studies of high school students reveal a world of alienation with strongly negative responses to standard practices. Students exhibit consistently positive responses to alternative modes of instruction and content (Boaler, 2002), particularly those that build upon intrinsic instead of external motivation. Our perspective is grounded in this work focusing on increasing motivation through active participation. We do this by designing mathematically-meaningful tasks combining software and wireless technologies.
Research Design & Question
SimCalc combines two innovative technological ingredients: (1) dynamically-linked representations and (2) wireless networks that enhance student participation and mathematical communication in the classroom. Our materials fuse these two important ingredients through new curriculum materials. These activities focus on linear and quadratic functions, linearly varying speed, covariation, and slope-as-rate. The activities bring important mathematical discussions to the surface because of the participatory nature of the activities, e.g., your group number defines your speed, or slope of your linear function.
We replace existing curriculum with activities that allow students to create functions through various representations in SimCalc on the TI-83 Plus or TI-84 Plus graphing calculator and work with dynamic representations of these functions through the animations of characters whose motion is driven by the defined functions. These functions are collected by a teacher into the SimCalc software running in parallel on a computer using the TI-Navigatorª Wireless network for class discussion and generalization.
The research questions for the overall study focus on: do we see learning gains following an implementation of a 6 to 8-week set of materials in Algebra 2?
Data Collection & Analysis
We collected pre- and post-test data from 268 control students across 16 classrooms and 298 treatment students across 15 classrooms from 6 school districts that represent a wide variety of achievement and SES levels in Massachusetts in our Algebra 2 Main Study (2009-2010). Sixteen teachers participated in the study. We report here on learning gains in non-honors classes where we see most gains. Student learning was measured using a content test that the research team developed using standardized test items and validated by the external evaluation team at the Donahue Institute.
In analyzing student learning across classrooms (cluster) assigned to the treatment (SimCalc) vs. Control ("business-as-usual" Algebra), we looked at total scores for each test, and on content test sub-categories including M1 (simple, procedural, one-step problems), M2 (conceptual and procedurally more complex, multi-step problems), and multiple representations.
Summary of Findings
<> Treatment students (n = 172, Mdn =
2) have higher learning gains compared to control students (n = 234, Mdn = 0),
U = 25,595, p = .000, r = .234.
In particular, we see some of the
highest and most statistically significant gains in the categories of multiple
representations and M2 in the treatment group and not in the control group. For change in multiple representations,
Treatment (n = 172, Mdn = 1) and Control (n = 234, Mdn = 0), U = 23,021.0, p =
.012, r = .125.
For change in M2 items, Treatment
(n = 172, Mdn = 1) and Control (n = 234, Mdn = 0), U = 24,561.5, p < .000, r
= .190.
One
M2 and multiple-representation item on the algebra 2 content test pertains to
algebraic expressions across representations. This content item involves interpreting the behavior of a
quadratic function graph represented by an algebraic expression and includes a
narrative component for students to interpret.
The SimCalc group gains
considerably more on this item as calculated by an Odds Ratio. The estimated
risk of students with a SimCalc intervention getting this wrong on a post-test
was 54% (ORx100%) of that without (control), with a confidence interval of [.31,
.93], p = .013. In
addition, the treatment group has 5 other items with significant odds ratios
where the control group had none.
We believe that these are
significant results as they not only demonstrate the effectiveness of combining
dynamic representations and wireless communication systems in classrooms in
important mathematical topics but that regular non-honors students can learn
significantly more on complex forms of tasks. This is especially important in
light of the Common Core Initiative. One of the Standards for Mathematical
Practices includes "formulating a model by creating and selecting geometric,
graphical, tabular, algebraic, or statistical representations that describe
relationships between the variables" (NGA & CCSSO, 2010).
We will focus our
session on presenting specific areas of mathematical learning and associated
classroom videos to demonstrate enhanced participation that led to these
results for audience analysis and discussion.
References
National Council of Teachers of Mathematics.
(2000). Principles and standards for
school mathematics. Reston, VA: Author.
