**Schoenfeld, A. H. (2020). Addressing Horizontal and Vertical gaps in Educational Systems. European Review. DOI: https://doi.org/10.1017/S1062798720000940**

Students are often ill-prepared for the leap in expectations within disciplines when they make the transition from secondary to university-level instruction. Moreover, another equally large challenge exacerbates these vertical disciplinary gaps. At all levels, instruction takes place largely in disciplinary silos – in language and literature, history, mathematics, science, and so on. Learning goals in these silos are often phrased in very different language, e.g., “becoming a reader” (or writer) in language arts, “inquiry” in science, and “problem solving” in mathematics. Such horizontal gaps result in instruction being far less coherent from the student’s perspective than it might be. The Teaching for Robust Understanding framework, known as TRU, may provide a means of addressing both kinds of gap. TRU focuses on the nature of learning environments that support the development of students as powerful learners. Through the lens of the TRU framework one can see commonalities across disciplines and across grade levels, and shape instructional practices accordingly. Within any particular discipline, treating students as active sense makers and arranging learning environments to provide opportunities for active sense making may help to bridge the gap between secondary and higher education.

**Schoenfeld, A. H. (2020). Mathematical practices, in theory and practice. ZDM, 52(6), 1163-1176. Doi: https://doi.org/10.1007/s11858-020-01162-w**

Descriptions of mathematical thinking have an extended lineage. Sometimes accurate and sometimes not, sometimes misinterpreted and sometimes not, characterizations of mathematical thought processes have inspired and at times misled people interested in designing or framing mathematics instruction. Challenges the field faces in conceptualizing mathematics instruction include: What can be warranted as legitimate mathematical practices? Which aspects of mathematical practices are relevant and appropriate for K-16 instruction? What kinds of support are necessary? What is viable at scale? This paper provides a description of relevant history from the Western literature, bringing readers up to the present. It then addresses two key issues related to contemporary curricula: the framing of the mathematical enterprise as being fundamentally inquiry oriented and the need for curricula and instruction to reflect such mathematical values; and the characteristics of mathematical classrooms that support students’ development as powerful mathematical thinkers. An emphasis is on problem solving, a major component of “thinking mathematically.” The paper concludes with a description of practices that are currently under-emphasized in instruction and that would profit from greater attention.

The Teaching for Robust Understanding (TRU) Framework provides *principles* for powerful learning environments. It does not prescribe particular ways of teaching; rather, it provides a growing set of tools for problematizing and reflecting on teaching, with an eye toward enhancing teaching practices and classroom environments. Because there is latitude in implementing TRU (there is no prescribed “right way” to teach, and there are many ways to enhance current teaching practices along the TRU dimensions), teacher educators working with TRU have the latitude to adapt TRU to their local contexts and to build tools for them. In this chapter, six sets of early adapters/developers briefly describe their work with TRU and reflect on their learning as a result of that work. Each group addresses the issues they faced in making TRU their own, and the adaptations they made to fit the needs of their particular teacher learning communities.

This chapter describes the synthesis of two powerful approaches to professional development, based on the Teaching for Robust Understanding (TRU) framework and Lesson Study. The synthesis is known as TRU-Lesson Study. In TRU-based professional development, groups of teachers negotiate their visions of teaching and learning collaboratively by reflecting on artifacts of practice using the TRU framework. In math-focused Lesson Study (LS), teachers work together to design, teach, and reflect on a lesson that focuses on key mathematical issues and students’ engagement with them. TRU-Lesson Study, like Lesson Study, profits from teachers’ concerted attention to lesson design and reflection on the hypotheses reflected in the design. Like TRU professional development, it supports teachers to work together explicitly on key dimensions of classroom practice. This paper describes TRU-Lesson Study and provides descriptions of how it plays out in practice.

This article undertakes a reframing of the concept of teacher knowledge. It argues that in order to help teachers create more powerful learning environments, a much more general framing is required—one that incorporates a teacher’s perceptions, inclinations and orientations as well as their understandings and related proficiencies. A main point of departure is the Teaching for Robust Understanding (TRU) framework, which focuses on essential dimensions of classroom practice. Questions of teacher knowledge are reframed as: “How can we reconceptualize teacher knowledge (perhaps better, teacher proficiency) so that it encompasses the broad range of perceptions, orientations, understandings and proficiencies that support teachers in crafting learning environments from which students emerge as knowledgeable, flexible, and resourceful thinkers and problem solvers? How can it be organized so that it can be worked on productively? This paper explores these issues. It employs the TRU framework as the initial mechanism for reframing while drawing on Schoenfeld’s (How we think. Routledge, New York, 2010) work on teachers’ decision making and Schoenfeld and Kilpatrick’s (International handbook of mathematics teacher education, volume 2: tools and processes in mathematics teacher education. Sense Publishers, Rotterdam, pp 321–354, 2008) work on teacher proficiency to suggest what should be included in an expanded framing of teacher knowledge.

As STEMeducation matures, the field will profit from tools that support teacher growth and that support rich instruction. A central design issue concerns domain specificity. Can generic classroom observation tools suffice, or will the field need tools tailored to STEM content and processes? If the latter, how much will specifics matter? This article begins by proposing desiderata for frameworks and rubrics used for observations of classroom practice. It then addresses questions of domain specificity by focusing on the similarities, differences, and affordances of three observational frameworks widely used in mathematics classrooms: Framework for Teaching, Mathematical Quality of Instruction, and Teaching for Robust Understanding. It describes the ways that each framework assesses selected instances of mathematics instruction, documenting the ways in which the three frameworks agree and differ. Specifically, these widely used frameworks disagree on what counts as high quality instruction: questions of whether a framework valorizes orderly classrooms or the messiness that often accompanies inquiry, and which aspects of disciplinary thinking are credited, are consequential. This observation has significant implications for tool choice, given that these and other observation tools are widely used for professional development and for teacher evaluations.

*Teaching for Robust Understanding of Essential Mathematics* is based on a presentation at an international symposium entitled “Essential Mathematics for the Next Generation: What and How Students Should Learn” held in Tokyo in 2016. The paper addresses these issues: (1) What mathematics should we teach? and, (2) What are the most important dimensions of rich instructional environments? It describes the Teaching for Robust Understanding (TRU) Framework, and ways in which TRU-related tools can be used in teacher preparation and professional development.

**Schoenfeld, A.H. (2015). Thoughts on scale. ZDM, the international journal of mathematics education, 47, 161-169. DOI: 10.1007/s11858-014-0662-3. **

This essay reflects on the challenges of thinking about scale – of making sense of phenomena such as continuous professional development (CPD) at the system level, while holding on to detail at the finer grain size(s) of implementation. The stimuli for my reflections are three diverse studies of attempts at scale – an attempt to use ideas related to professional development in two different countries, the story of how research did or did not frame a nationwide attempt at undergirding CPD, and a fine-grained study of the quality of a dozen mentors’ implementation of CPD. The challenge is to “see the forest for the trees,” to be able to situate such diverse studies within a larger framework. The bulk of this article is devoted to offering such a framework, the Teaching for Robust Understanding (TRU) framework, which characterizes five fundamentally important dimensions of powerful learning environments. At the most fine-grained level, TRU applies to classrooms, establishing goals for instruction. But, more generally, it applies to all learning environments, and thus characterizes important aspects of CPD. The paper addresses issues related to the kinds of systemic coherence necessary to make progress on professional development at scale.

This article, and my career as an educational researcher, are grounded in two fundamental assumptions: (1) that research and practice can and should live in productive synergy, with each enhancing the other; and (2) that research focused on teaching and learning in a particular discipline can, if carefully framed, yield insights that have implications across a broad spectrum of disciplines. This article begins by describing in brief two bodies of work that exemplify these two fundamental assumptions. I then elaborate on a third example, the development of a new set of tools for understanding and supporting powerful mathematics classroom instruction (and by extension, powerful instruction across a wide range of disciplines) – the TRU framework. In doing so, this paper situates the corpus of work on TRU in a much larger R&D framework.

This paper describes the genesis of the TRU framework. It explores the dialectic between theorizing teachers’ decision-making and producing a workable, theoretically grounded scheme for classroom observations. One would think that a comprehensive theory of decision-making would provide the bases for a classroom observation scheme. It turns out, however, that, although the theoretical and practical enterprise are in many ways overlapping, the theoretical underpinnings for the observation scheme are sufficiently different (narrower in some ways and broader in others) and the constraints of almost real-time implementation so strong that the resulting analytic scheme is in many ways radically different from the theoretical framing that gave rise to it. This essay characterizes and reflects on the evolution of the observational scheme. It provides details of some of the failed attempts along the way, in order to document the complexities of constructing such schemes.