Mathematics Education Research: Can the Field Deliver?
Research Advisory Committee, Standards Impact Research Group
The terrain for
mathematics education research has shifted considerably since last year when
NCTM Research Advisory Committee (RAC) reported on its activities in JRME
(RAC, 2001). New federal legislation (i.e., the Elementary and Secondary
Education Act [ESEA], 2001) includes the phrase "scientifically based
research" repeatedly (see House of Representatives Bill HR 1614 IH). The
National Research Council (NRC) has
released a report, Scientific Research in Education (NRC, 2002b), which
provides guiding principles for scientific inquiry and discusses design for conducting scientific
research, in response to increasing controversy about what counts as
"scientific" in educational research. The RAND Corporation has
supported the work of a Mathematics Study Panel to propose a strategic
research and development program in mathematics education.
The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems
Wim Van Dooren, Lieven Verschaffel, Patrick Onghena
The study reported here investigated the arithmetical and algebraic problem-solving strategies and skills of preservice primary school and secondary school teachers in Flanders, Belgium, both at the beginning and at the end of their teacher training. The study then compared these aspects of the preservice teachers' own problem-solving behavior with the way in which they evaluated students' arithmetical and algebraic solutions to problems. Future secondary school teachers clearly preferred the use of algebra, both in their own solutions and in their evaluations of students' work, even when an arithmetical solution seemed more evident. Some future primary school teachers tended to apply exclusively arithmetical methods, leading to numerous failures on difficult word problems, whereas others were quite adaptive in their strategy choices. Taken as a whole, the evaluations of the preservice primary school teachers were more closely adapted to the nature of the task than those of their secondary school counterparts.
Applying Covariational Reasoning While Modeling Dynamic Events: A Framework and a Study
Marilyn Carlson, Sally Jacobs, Edward Coe, Sean Larsen, Eric Hsu
The article develops the notion of covariational reasoning and proposes a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. It also reports on an investigation of high-performing 2nd-semester calculus students' ability to reason about covarying quantities in dynamic situations. The study revealed that these students were able to construct images of a function’s dependent variable changing in tandem with the imagined change of the independent variable, and in some situations, were able to construct images of rate of change for contiguous intervals of a function's domain. However, students appeared to have difficulty forming images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations. These findings suggest that curriculum and instruction should place increased emphasis on moving students from a coordinated image of two variables changing in tandem to a coordinated image of the instantaneous rate of change with continuous changes in the independent variable for dynamic function situations.
Secondary School Mathematics Teachers' Conceptions of Proof
Eric J. Knuth
Recent reform efforts call on secondary school mathematics teachers to provide all students with rich opportunities and experiences with proof throughout the secondary school mathematics curriculum-opportunities and experiences that reflect the nature and role of proof in the discipline of mathematics. Teachers' success in responding to this call, however, depends largely on their own conceptions of proof. This study examined 16 in-service secondary school mathematics teachers' conceptions of proof. Data were gathered from a series of interviews and teachers' written responses to researcher-designed tasks focusing on proof. The results of this study suggest that teachers recognize the variety of roles that proof plays in mathematics; noticeably absent, however, was a view of proof as a tool for learning mathematics. The results also suggest that many of the teachers hold limited views of the nature of proof in mathematics and demonstrated inadequate understandings of what constitutes proof.