Curricular Treatments of Length Measurement in the United States: Do They Address Known Learning Challenges?

Extensive research has shown that elementary students struggle to learn the basic principles of length measurement. However, where patterns of errors have been documented, the origins of students’ difficulties have not been identified. This study investigated the hypothesis that written elementary mathematics curricula contribute to the problem of learning length measurement. We analyzed all instances of length measurement in three mathematics curricula (grades K–3) and found a shared focus on procedures. Attention to conceptual principles was limited overall and particularly for central ideas; conceptual principles were often presented after students were asked to use procedures that depended on them; and students often did not have direct access to conceptual principles. We also report five groupings of procedures that appeared sequentially in all three curricula, the conceptual principles that underlie those procedures, and the conventional knowledge that receives substantial attention by grade 3

THE “INSERTION ” ERROR IN SOLVING LINEAR EQUATIONS

This proposed research investigates a particular phenomenon that occurred during a study of students ’ flexibility in solving linear equations (Star, 2004). 160 6th graders participated in five hours (over five days) of algebra problem solving. In the first hour, the students were given a brief lesson on four different steps that could be used to solve algebraic equations (adding to both sides, multiplying on both sides, distributing, and combining like terms). Students then spent three hours solving a series of unfamiliar linear equations with minimal facilitation. 23 students (randomly selected from all participants) were interviewed while working individually with a tutor/interviewer. On the last day of the project, students completed a post-test. Analyses of students ’ work made apparent an interesting type of error, named “insertion”, in 12 (7.5%) students ’ of which three were interviewed. The insertion error was evident when 2x + 10 = 4x + 20 became 4x − 2x + 10 = 4x − 4x + 20. Similarly, 2(x + 5) = 4(x + 5) became 2 – 2(x + 5) = 4 – 2(x + 5). Interestingly, this error has not previously been reported nor classified in the literature on linear equation solving (e.g., Matz, 1980; Payne & Squibb, 1990). Out of the many proposed classifications of students ’ rule-based errors in computational or algebraic problems ( Matz, 1980; Payne & Squibb, 1990; Sleeman, 1984), Ben-Zeev’s (1998

Do We Do Dewey? Using a Dispositional Framework to Examine Reflection Within Internship Professional Development Plans

Our revised secondary teacher education professional development plan (PDP) project required preservice teachers to identify their teaching beliefs, use these beliefs to analyze practice, and create an action plan centered on a research question from this analysis. We predicted these plans would show evidence of Dewey\u27s (1964) reflective dispositions (open-mindedness, whole-heartedness, responsibility). We had two research questions to examine this assertion: 1) To what degree did our pre-service teachers\u27 PDPs show evidence of Dewey\u27s dispositions toward reflective practice? 2) What were the trends for each of these dispositions? Overall, the PDPs showed evidence of all reflective dispositions; however, the dispositions were unevenly represented. From this study, we recommend a more longitudinal and personalized approach to the PDP to make it more integral to the development of preservice teachers