A process was developed to create Web-based video models of effective instructional practices for use in teacher education settings. Three video models, created at three university sites, demonstrated exemplary implementation of specific, evidence-based strategies in reading, math, and science. Video models of strategies were field tested with preservice and practicing teachers working with diverse student populations. The authors provide an explanation of the video development process and present field-test data that demonstrate the influence of video modeling on teacher learning.

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Ms. Carlisi teaches middle school mathematics for students who were retained at least 1 year and who were identified as at risk of failure in mathematics. About half of her class of 14 students has identified learning disabilities and/or emotional and behavioral difficulties. Ms. Carlisi worries constantly that she is not going to be able to help them “catch up” by the time they take the mandated state high-stakes standardized assessment. Ms. Carlisi knows her students need to build proficiency in fractions. Her students continue to demonstrate difficulties despite previous instruction. She decides to use Mathematics Dynamic Assessment (MDA) to help her students. MDA is an informal mathematics assessment process that integrates four research-supported assessment practices for struggling learners: Assessment of students' interests and experiences. Concrete-representational-abstract assessment within authentic contexts. Error pattern analyses. Flexible interviews. The MDA process provides teachers with important information about what students do and do not understand about foundational mathematics concepts, students' levels of understanding and abilities to express their understandings, and where students are in the learning sequence (frustration, instructional, mastery). The data collected through the MDA process provide teachers with an in-depth evaluation of their students' mathematical understandings and thinking that allows teachers to plan their instruction to address students' specific mathematical learning needs. Ms. Carlisi structures an MDA in the area of fractions with an emphasis on comparing fractions, an area in which her students demonstrate difficulty.

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This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States.

The national council of teachers of Mathematics advocates a balanced approach of teaching both procedural and conceptual knowledge (NCTM 2000). In practice, however, students with special needs often receive a great deal of algorithmic instruction because mastering algorithms is what we “see” them struggle with the most. Even with a heavy dose of algorithmic instruction, many of these students still have difficulty performing algorithms efficiently. Furthermore, without developing conceptual understanding while learning algorithms, these students will never understand foundational mathematical concepts.