This page provides a collection of downloadable research papers related to the TRU Framework, in reverse chronological order of publication. Click on the title to download the paper.
Schoenfeld, A. (2023). A theory of teaching. In Anna Praetorius and Charalambos Charalambous (Eds.), Theorizing Teaching: Bringing Together Expert Perspectives to Move the Field Forward., pp. 159-189. New York: Springer Nature. Springer Open Access https://link.springer.com/book/10.1007/978-3-031-25613-4?sap-outbound-id=93FC9FFDDC8AE3A92A79BF46FFBAB3E51C5B109A. eBook ISBN 978-3-031-25613-4; Print ISBN 978-3-031-25612-7.
Whether one can claim to have a theory of teaching depends on what one takes to constitute teaching and what one means by theory. This chapter characterizes both. Given those characterizations, I claim that we already have a theory of teaching, which specifies that teachers’ in-the-moment classroom decisions can be modeled by attending to three major factors: the resources at the teachers’ disposal (both their knowledge and material resources), their orientations (beliefs, preferences, values, etc.), and their goals (which exist at multiple levels and change dynamically according to evolving events). Beyond that, the Teaching for Robust Understanding (TRU) framework indicates that the following five dimensions of learning environments are consequential and comprehensive – the degree to which the environment: (1) offers affordances for rich engagement with content; (2) operates within the students’ zone of proximal development; (3) supports all students in engaging with core content; (4) provides opportunities for students to contribute to classroom discourse and develop a sense of agency and disciplinary identity; and, (5) reveals and responds to student thinking. Combining these two theoretical frames yields a theoretical specification of what has been called “ambitious teaching.” There is much more to be concerned with, however. In general, the field’s understanding of relevant knowledge and resources for ambitious teaching is weak, a problem exacerbated by the widespread adoption of virtual instruction due to the presence of Covid-19. Moreover, little is understood regarding teachers’ developmental trajectories. Such knowledge will be necessary to establish effective long-term professional development efforts.
Schoenfeld, A. H. (2023). Learning in Math(s). In: Tierney, R.J., Rizvi, F., Erkican, K. (Eds.), International Encyclopedia of Education, 4th Edition, vol. 6. Elsevier. https://dx.doi.org/10.1016/B978-0-12-818630-5.14086-2. ISBN: 9780128186305.
A comprehensive characterization of learning in mathematics calls for the careful examination of (a) the nature of learning, (b) what is intended to be learned, and (c) the nature of learning environments and the impact they have on learning. Researchers’ understandings of these aspects of learning have evolved over the 20th and early 21st centuries, with significant growth in recent decades as mathematics education coalesced and matured as a discipline. In brief, this paper surveys and exemplifies: characterizing learning in general, and specifically in mathematics; learning goals and learning outcomes; Mathematical thinking and learning; antecedents; Current understandings regarding mathematical content, processes and practices; metacognitive behavior; orientations and identity; productive learning environments; and, implications and prospects.
Schoenfeld, A. H. (2023). Rethinking mathematics education. In Greer, B., Skovmose, O., & Kollosche, D. Breaking images: Iconoclastic perspectives on Mathematics and its education. Open Book Publishers (https://www.openbookpublishers.com/section/125/1)
I am now concluding my 50th year as a professional mathematics educator. That benchmark provides an opportunity to reflect on the emergence of ideas and understandings over the past five decades, and the persistence of challenges that the field continues to face.
To quote from the opening page of A Tale of Two Cities, “It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of light, it was the season of darkness, it was the spring of hope, it was the winter of despair.” On the one hand, our intellectual advances have been extraordinary. We understand thinking, teaching, and learning in ways that transcend previous understandings. In this chapter we take a chronological tour through such discoveries – the nature of problem solving, of teaching, of powerful learning environments. On the other hand, social and institutional progress have been hard to come by. Schools and classrooms reflect the structural and racial ills of American society; mathematics instruction, while potentially meaningful and useful in people’s lives, has little to do with the kinds of sense making it could support. If anything, school mathematics’ distance from meaningful issues in people’s lives serves to reify current structures rather that to problematize and challenge them. The chapter concludes with a proposal to address this state of affairs.
Schoenfeld, A.H. (2023). On Problems, Problem-Solving, and Thinking Mathematically. In: R. Leikin (Ed.), Mathematical challenges for all: Research in mathematics education. Springer, Cham. https://doi.org/10.1007/978-3-031-18868-829.
This paper revisits some of the key ideas that have shaped my problem-solving courses through the years. I then discuss the specifics of how my students and I worked through some problems this year, and why. Ultimately, problems are the raw materials for mathematical construction; if they are rich in potential, then like fine wood, metal or gems, many different things can be made of them. The question is what might be made – and how, and why.
Schoenfeld, A. (2022). Why are Learning and Teaching Mathematics so Difficult? In M. Danesi, (ed). Handbook of Cognitive Mathematics. New York: Springer Nature. https://doi.org/10.1007/978-3-030-44982-7_10-1.
Decades ago Hans Freudenthal referred to the school mathematics experienced by most students as the “fossilized remains” of reasoning processes. Indeed, the facts and procedures of school mathematics may seem as frightening to some as the fossilized remains of a tyrannosaurus rex, although they are empty; like dinosaur skeletons, they bear only partial resemblance to the real thing. The challenge is to see the substance behind the structure and to understand how the mathematics fits together. That is a matter of mathematical thinking, reasoning, and problem solving – the how and the why beneath the fossilized surface. Opportunities for such understandings are accessible through mathematical sense making, but they are rare in schools. This review indicates that there is more to learning and understanding mathematical content and practices than it would appear. Moreover, understanding mathematics is only one component of effective or “ambitious” teaching – better framed as the creation of mathematically rich and equitable learning environments. The challenge is to create robust learning environments that support every student in developing not only the knowledge and practices that underlie effective mathematical thinking, but that help them develop the sense of agency to engage in sense making. This implicates issues of race and equity, which are a challenge not only in classrooms but in society at large; structural and social inequities permeate the schools. Major obstacles to addressing the challenges of powerful mathematics within schools include a general absence of curricular support for rich and meaningful mathematics, instructional practices that do not invite students into mathematics, assessments that fail to focus on thinking, professional development that focuses on what the teacher does rather than the students’ learning opportunities and experiences, and a vastly inequitable cultural context both outside and inside schools. This chapter points to existence proofs that at least some these challenges can be addressed, while documenting the substantial challenges to making progress at scale.
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.
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.
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.
Schoenfeld, A. H., Baldinger, E., Disston, J., Donovan, S., Dosalmas, A., Driskill, M., Fink, H., Foster, D., Haumersen, R., Lewis, C., Louie, N., Mertens, A., Murray, E., Narasimhan, L., Ortega, C., Reed, M., Ruiz, S., Sayavedra, A., Sola, T., Tran, K., Weltman, A., Wilson, D., & Zarkh, A. (2019). Learning with and from TRU: Teacher Educators and the Teaching for Robust Understanding Framework. In K. Beswick (Ed.), International Handbook of Mathematics Teacher Education, Volume 4, The Mathematics Teacher Educator as a Developing Professional (pp. 271-304). Rotterdam, the Netherlands: Sense publishers.
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.
Schoenfeld, A., Dosalmas, A., Fink, H., Sayavedra, A., Weltman, A., Zarkh, A, Tran, K., & Zuniga-Ruiz, S. (2019). Teaching for Robust Understanding with Lesson Study. In Huang, R., Takahashi, A., & Ponte, J.P. (Eds.), Theory and Practices of Lesson Study in Mathematics: An international perspective (pp. 136-162). New York: Springer. ISBN 978-3-030-04031-4
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.
Schoenfeld, A. H. (2019). Reframing Teacher Knowledge: A Research and Development Agenda. In J. Star, H. Hill, & F. Depaepe (Eds.). Expertise to develop students’ expertise in mathematics: Bridging teachers’ professional knowledge and instructional quality. ZDM: Mathematics education, Volume 52, Issue 1, 2020. DOI: 10.1007/s11858-019-01057-5
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.
This paper provides an overview of the teaching for robust understanding (TRU) Framework, its origins, and its evolving use. The core assertion underlying the TRU Framework is that there are five dimensions of activities along which a classroom must do well, if students are to emerge from that classroom being knowledgeable and resourceful disciplinary thinkers and problem solvers. The main focus of TRU is not on what the teacher does, but on the opportunities the environment affords students for deep engagement with mathematical content. This paper’s use of the TRU framework to highlight salient aspects of three classroom videos affords a compare-and-contrast with other analytic frameworks, highlighting the importance of both the focus on student experience and the mathematics-specific character of the analysis. This is also the first paper on the framework that introduces a family of TRU-related tools for purposes of professional development.
Schoenfeld, A. H., Floden, R. B., and the algebra teaching study and mathematics assessment projects. (2018). On Classroom Observations. Journal of STEM Education Research https://doi.org/10.1007/s41979-018-0001-7
As STEM education 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.
This article characterizes my use of video as a tool for research, design and development. I argue that videos, while a potentially overwhelming source of data, provide the kind of large bandwidth that enables one to capture phenomena that one might otherwise miss; and that although the act of taping is in itself an act of selection, there is typically enough shown in a video that it rewards multiple watching and supports the kinds of arguments over data that are essential for theory testing and replication. In pragmatic terms, video presents phenomena in ways that have an immediacy that is tremendously valuable. I discuss ways in which videos help students and teachers focus on phenomena that might otherwise be very hard to grapple with. This article begins with a brief review of my uses of video, almost 40 years ago, for research and development in problem solving. It then moves to the discussion of very fine-grained research on learning and decision making. The bulk of the article is devoted to a discussion of the teaching for robust understanding (TRU) framework, which was derived in large measure from the extensive review of classroom videotapes, and which serves as the basis for an extensive program of pre-service and in-service professional development. The professional development relies heavily on the use of videos to convey the key ideas in TRU, and to help teachers plan and review instruction.
Schoenfeld, A. H. (2017). Teaching for Robust understanding of essential mathematics. In T. McDougal, (Ed.), Essential Mathematics for the Next Generation: What and How Students Should Learn (pp. 104-129). Tokyo, Japan: Tokyo Gagukei University
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.
This article begins with a series of meta-level comments on the enterprise of observing and theorizing classroom teaching, using as a springboard a frequently discussed model from the literature. It then provides a series of commentaries on the empirical articles in this issue (volume 48, issue 1) of ZDM, with a main focus on methodological issues. It concludes with a brief return to the meta-level, with a comment on the field’s use of the term “model”.
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.
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