In AIM-TRU, teachers collaboratively analyze high quality OER lessons called Formative Assessment Lessons to deepen content and pedagogical knowledge and support shifts in practice aligned with the Teaching for Robust Understanding (TRU) framework. The central feature of these lessons is that they are built to foster rich classroom conversations around deep mathematical ideas, making them an ideal vehicle for video cases.
Our project has the dual focus of creating video cases as partner institutions support teacher participants to implement and videotape Formative Assessment Lessons in their classrooms, and then studying the use of the video cases and related impact during professional development sessions that bring teachers together with a focus on the particular Formative Assessment Lesson. Specifically, the PD learning cycle we’ve developed includes a deep dive into the mathematical content central to the lesson through the lens of Big Mathematical Ideas (Charles, 2005), followed by an examination of video excerpts showing students engaged in rich mathematical activity and sets of reflective questions based on the TRU framework (Schoenfeld, 2014). The video cases form the backbone of our PD cycle. We have created a small set of video cases that are stored in a custom microsite hosted at the University of Michigan’s Teaching and Learning Exploratory (TLE), with plans to make the library of video cases generated by this project freely accessible to others who want to use it for professional development.
Our research thus far includes study of use and implementation in four regions of the country. Our preliminary data indicates that analyzing the video cases afforded rich learning experiences for teachers and caused them to further implement the instructional materials in their classroom, and advocate for the use of Formative Assessment Lessons with their peers. This project will provide opportunities for broader participation as different groups of teachers analyze the cases, providing crucial data on how teachers develop their knowledge through our PD model and what, if any, impact on classroom practice can be attributed to their participation in the professional learning community. The Professional Development Cycle page of this website provides an overview of the learning cycle and select slide decks developed for use with PLCs using our professional learning cycle.
Vision for high-quality instruction
Much has been written about the need to change the way math is taught in the United States, moving away from a procedural-based approach, towards one focused on problem-solving and student thinking. Buchbinder, Chazan, and Capozzoli (2014) summarize the goals for reform nicely:
"Students are expected not simply to rely on a taught method as justification for their solutions (Lampert, 1990a) but rather to develop their own solution methods and provide justifications for those methods…..where teachers are expected to implement tasks that promote reasoning and problem solving, with mathematical discourse focusing on ‘analyzing and comparing student approaches and arguments’” (NCTM, 2014, p. 10).
Rationales for this vision for change in mathematics education include arguments based on connections to mathematics as a discipline and opportunities for justification that are present when multiple solutions are side by side (Schoenfeld, 1992), as well as cognitively based arguments for building instruction on a basis of informal student solution methods to foster sense making (e.g., Nathan & Koedinger, 2000) and examination of students’ ways of thinking (e.g., Carpenter, Fennema, & Franke, 1996). Central to this vision of instruction through problem solving is the notion that both correct and incorrect student solutions are worthy of inspection and discussion (Silver, Ghousseini, Gosen, Charalambous, & Font Strawhun, 2005).”
Through problem-solving and sense making, the ultimate goal is to help students understand big ideas that are central to the learning of mathematics, and that link numerous mathematical understandings into a coherent whole (Charles, 2005).
Challenges to realizing the vision
Teaching in ways that all students come to understand big ideas through problem solving and sense-making is extremely challenging, as this type of instruction requires a repertoire of complex knowledge and skills. Deep content knowledge is central to reform teaching:
“...we expect the teacher to understand why a particular topic is particularly central to a discipline whereas another may be somewhat peripheral” (p. 9). Shulman also talks of vertical knowledge as “familiarity with the topics and issues that have been and will be taught in the same subject area during the preceding and later years in school, and the materials that embody them” (Shulman, 1986, p. 10).
Content knowledge on its own, however, is not enough. Schoenfeld (2014, 2018), synthesized decades of research to specify five dimensions of instruction (see below) that are necessary and sufficient for teaching that supports students’ growth towards becoming “powerful” mathematical thinkers. In addition to content, these dimensions include cognitive demand; equitable access to content; agency, ownership, and identity; and formative assessment. Collectively, these five dimensions are referred to as the Teaching for Robust Understanding (TRU) framework.