Promoting Student Learning and Productive Persistence in Developmental Mathematics: Research Frameworks Informing the Carnegie Pathways.

This paper focuses on two research based frameworks that inform the design of instruction and promote student success in accelerated, developmental mathematics pathways. These are Learning Opportunities productive struggle on challenging and relevant tasks, deliberate practice, and explicit connections, and Productive Persistence promoting students' academic and social mindsets, and good strategies. These frameworks are the foundations of the highly successful Carnegie Pathways (Statway and Quantway) two distinct pathways that take students who place into developmental mathematics through college level mathematics in one year. In this paper, we describe these research based frameworks and discuss examples of high impact practices derived from them. In 2009, the Carnegie Foundation for the Advancement of Teaching engaged a network of practitioners, researchers, designers/developers, and institutional leaders to design and implement two pathways that aim to accelerate community college students' progress through their entire developmental mathematics sequence and a college level course for credit in a single year Statway(r) and Quantway(r) Statway integrates developmental mathematics and college level statistics. Quantway covers developmental mathematics and college level quantitative reasoning. The Pathways have been remarkably successful, helping thousands of students achieve success in college level mathematics in a single year and tripling the success rate for college credit completion for students who place into developmental mathematics in half the time (Sowers Yamada, 2015) Central to the effectiveness of the Pathways is instruction that incorporates two key catalysts for powerful student learning: (1) the Learning Opportunities productive struggle, deliberate practice, and explicit connections; and (2) Productive Persistence promoting students' tenacity and good strategies. In this paper, we discuss these research based frameworks and how they inform instruction designed to promote student learning, engagement, and persistence in developmental mathematics. We provide examples of high impact practices derived from these frameworks used by the Pathways network. The Pathways Learning Opportunities The National Research Council in How People Learn (2005) determined that there are three basic principles of learning: 1) New understandings are constructed on a foundation of existing or prior understandings; 2) The brain forms cognitive schema or networks that are important to emphasize in the learning process; and 3) The ability to self monitor or possess skills of metacognition enhance learning. The Pathways instructional system addresses the essence of these fundamental principles with "learning opportunities" derived from key research findings in the learning sciences, psychology, and cognitive science that inform the design of Pathways curriculum and instruction productive struggle on challenging and relevant tasks, explicit connections to concepts, and deliberate practice. Productive Struggle Derived in part from research on mathematical sense making and the development of robust conceptual understandings in mathematics, productive struggle refers to opportunities for students to grapple with important mathematical ideas. NADE Digest Winter 2016 Volume 9 Issue 1 31 We use the word struggle to mean that students expend effort to make sense of mathematics, to figure something out that is not immediately apparent. We do not use struggle to mean needless frustration or extreme levels of challenge created by nonsensical or overly difficult problems..The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed. (Hiebert Grouws, 2007,  The ultimate goal of productive struggle is to encourage students to make meaning of mathematical content for themselves. In Pathways instruction, productive struggle most often occurs in collaborative learning settings in which students explore rich mathematical tasks as they develop strategies to investigate the problem situation or question. Students who are productively struggling are engaged and inquiring, repeatedly making guesses and judgments about how to use mathematics to approach the given situation. Promoting productive struggle involves posing tasks that require substantive mathematical thinking and giving students both the time and encouragement within the classroom culture to engage with the problem. Explicit Connections By explicit attention to connections, we mean that connections among mathematical facts, procedures, and ideas should be addressed explicitly. This could include discussing the mathematical meaning underlying procedures, asking questions about how different solution strategies are similar to and different from each other, considering the ways in which mathematical problems build on each other or are special (or general) cases of each other, attending to the relationships among mathematical ideas, and reminding students about the main point of the lesson and how this point fits within the current sequence of lessons and ideas.  A review of findings from across multiple studies some teacher centered, others student centered suggest that teaching for conceptual understanding leads to improvement not only in conceptual understanding but also in procedural skill. The reverse has not been found to be true (Hiebert Grouws, 2007) Thus, when we suggest that the focus of Pathways instruction is on concepts, we are not suggesting that knowledge of procedures is unimportant, but rather that instruction focused on concepts is the better way to achieve both learning outcomes. Deliberate Practice The literature suggests that repeating a behavior over and over is not an effective method of reaching maximal levels of performance. Pashler (2008) writes that "most current mathematics texts mass practice problems relating to a given topic into one problem set presented immediately following textual presentation of that topic. Our data suggest that at least for promoting retention this may be a grievous error" (p. 189) Research further demonstrates that performance is best increased as a result of deliberate, spaced efforts aimed at improvement. As opposed to massed repetition, deliberate practice consists of tasks that are invented to overcome gaps in understanding, apply what is learned, and deepen understanding and facility with key concepts. These activities are highly structured and designed to improve performance and strengthen understanding. Deliberate practice requires effort and individuals are motivated to practice because practice improves performance (Ericsson et al. 2008) For these reasons, the Pathways are not characterized by long series of similar problems, but rather by carefully chosen questions that guide students to a deeper understanding of concepts. NADE Digest Winter 2016 Volume 9 Issue 1 32 Instruction supporting the Learning Opportunities: Problem Cycle Routine The key to realizing the potential of the learning opportunities for students is effective instruction. Modeled after the typical structure of mathematics lessons in Japan (Shimizu, 1999) in which student engagement in rich problems and facilitated discussion of student solutions are key drivers, we have developed an instructional routine, the Problem Cycle, with four phases that faculty can adopt and use strategically to implement lessons in a way that supports the learning opportunities (productive struggle and explicit connections, specifically) Table 1 specifies the purpose and key features of each phase.