| This study was conducted to investigate and to understand secondary mathematics teachers' efforts to make targeted mathematical ideas (TMIds) of school algebra accessible and comprehensible to students. Influenced by TIMSS researchers' use of close-up and wide-angle lenses (Hiebert et al., 2003) to examine and discuss data (e.g., video, lesson artifacts) about lessons taught by teachers from many countries, this study made use of close-up and several wide-angle lenses to examine and to understand data from the efforts of seven middle school and high school teachers. Data include approximately 60 hours of video recordings of the teachers teaching lessons (i.e., 3 to 5 lessons per teacher), observation-related interviews, and problem-solving interviews, as well as lesson and interview artifacts.;Initial examination of data was done using the lens of a conceptual framework whose development was based upon research supporting two perspectives of teaching: Teaching is an application of knowledge for teaching mathematics and teaching is communication of mathematical ideas. These perspectives are reflected in the four constructs that make up this initial framework: decompressing, trimming, bridging, and framing. The first three constructs originated in the Knowledge of Algebra for Teaching (KAT) and are described by KAT researchers as ways in which teachers use their KAT. Framing emerged from research literature in fields for which communication is an object of study. Loosely speaking, framing is the ways in which teachers make TMIds accessible and comprehensible to students.;Data analysis was conducted in four rounds, each round examining the data using a different lens. In Round 1, the data are examined using the initial conceptual framework as a lens (i.e., close-up lens) to identify instances of framing, and resulted in the identification of more than fifty instances of framing. After the instances of framing were identified, the framings were sorted and re-examined for patterns that cut across the framings. Round 2 resulted in the identification of 11 characteristics of framing. The 50+ instances of framing were also sorted into framings that reflected the use of decompressing, trimming, bridging, and nonmathematical elements and/or approaches. The fourth set of framings was examined more closely to determine how the nonmathematical elements were employed. This round of analysis (... |
