# Patricio HerbstUniversity of Michigan | U-M · School of Education

Patricio Herbst

Ph. D.

## About

172

Publications

27,909

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2,730

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Introduction

Pat Herbst is a Professor of Education and Mathematics at the University of Michigan. He does research on mathematics teaching and teaching knowledge, with a particular interest in the teaching of geometry at the secondary and tertiary levels. A secondary focus of his work is the development and use of rich media technologies in the development of materials for teacher education. Pat is happy to share pdf copies of his writing with those who write requests directly to him (that is, not through the Research Gate app). Please write to pgherbst@umich.edu and indicate which piece you are interested in.

**Skills and Expertise**

Additional affiliations

September 1999 - present

Education

September 1993 - August 1998

## Publications

Publications (172)

In this paper, we examine the relationships between teachers’ subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching algebra 1. We administered the same instrument to two different samples of teachers−high school practicing teachers and community college faculty−who...

The concept of teacher noticing has been a powerful methodological tool for understanding teachers’ decision-making and professional judgment. In this line of inquiry, researchers usually try to identify elements of classroom practice salient to teachers. Data about teacher noticing and decision making can be collected at scale through the use of s...

This study analyses College of Education and Human Development an online scenario-based instrument in which secondary mathematics teachers from across the United States were presented with episodes of mathematics instruction and then asked to make a decision at a critical juncture. The theory of practical rationality suggests that the decisions of...

In this chapter, we contribute some insights from project SimTeach (Herbst P, Chieu VM, SIMTEACH: What can practical knowledge modeled in a teaching simulator contribute to support mathematics teacher learning? National Science Foundation Grant, EHR, DRL-1420102, 2014) concerning the role that technologically mediated teaching simulations can play...

This study proposes task of teaching as an organizer of dimensionality in teachers’ subject matter knowledge for teaching (SMK) and investigates it in the context of measuring SMK for teaching high school geometry (SMK-G). We hypothesize that teachers use different SMK-G in different aspects of their teaching work and that such differences can be s...

We investigated how US secondary mathematics teachers expect students to present geometric proofs at the board. We analyzed video records of geometry classrooms and found students to be engaged in a practice that we call proof transcription—i.e., mark-for-mark reproductions of previously completed proofs that were not reasoned reconstructions of ar...

We investigate teachers' decision making in contexts where they could choose to provide students more authentic experiences with proving. Specifically, we investigate their preferences to depart from norms about what proof problems to assign to students. Scenario-based instruments consisting of two sets of items reflecting two hypothesized norms in...

We show how we used a national distribution of responses from 416 practicing teachers to items of a test of mathematical knowledge for teaching geometry to estimate changes in knowledge by a group of 11 practicing teachers who participated in a 2-year practice-based professional development programme. To draw a reliable interpretation of the change...

OPEN ACCESS LINK: http://www.learntechlib.org/p/216903/
ABSTRACT: We describe how we further developed StoryCircles to support teacher learning online during COVID-19. In StoryCircles, a facilitator gathers teachers to collectively represent a lesson through iterative phases of scripting, visualizing, and arguing about alternatives. We share new i...

How should we expect growing understandings of the nature of mathematical practice to inform classroom mathematical practice? We address this question from a perspective that takes seriously the notion that mathematics education, as a societal enterprise, is accountable to multiple sets of stakeholders, with the discipline of mathematics being only...

Novel tasks are valuable, yet discussions of novel tasks are often challenging to manage. In this paper, we describe a U.S. high school geometry lesson taught by two teachers in six different classes, in which students were assigned a novel task that the teacher framed as an instance of two familiar instructional situations — constructing a diagram...

This chapter addresses the role of technological tools in mathematics teacher
learning within a perspective that conceives of this learning as practice-based and
work-specific. The notion of practice-based and work-specific mathematics teacher
education is envisioned as a just-in-time endeavor emphasizing important continuities
between prospective...

I discuss design considerations behind a problem that was used in a classroom research project, the “ranking triangles task” (Herbst, 2003), in the context of illustrating what a modeling perspective may look like in creating opportunity to learn in high school geometry. Quite often in scholarship that reports on design experiments in mathematics i...

This paper reports on an ongoing project aimed at developing an inter-institutional system of professional support for the improvement of the Geometry for Teachers (GeT) courses that mathematics departments teach to preservice secondary teachers. In alignment with the literature on improvement science (see Bryk et al., 2015; Lewis, 2015), it is ess...

Agency has been identified as a personal asset for teachers that may enable them to competently manage their classrooms (Stein, Kintz, & Miness, 2016). We explored how tools from systemic functional linguistics (Halliday & Matthiessen, 1994) can be used to track the agency instructors express. We interviewed graduate student teaching assistants (GT...

Inside the discipline, mathematical work consists of the interplay between stating and refining conjectures and attempting to prove those conjectures. However, the mathematical practices of conjecturing and proving are traditionally separated in high school geometry classrooms, despite some research showing that students can successfully navigate t...

As educational researchers and policy makers advocate for algebra classrooms to become places where students share ideas and compare approaches to solving problems, the question remains open as to whether and how much these ideas have penetrated actual practice. For example, how do algebra teachers respond to a correct, but unexpected way of solvin...

Using mixed-effects regression, we analyzed teachers’ responses to a multimedia survey of instructional practices in posing proof problems in geometry. Teachers described and rated for appropriateness three different ways of involving students in deciding what to prove, including one in which the teacher chooses the givens and the conclusion to pro...

This paper reports on a project aimed at developing a system of professional support for the improvement of the Geometry for Teachers course that mathematics departments teach to preservice secondary teachers. We share data from interviews with 20 instructors to report on how they perceive their position of geometry instructors and the work they do...

This chapter explores a way of describing the teacher education curricular materials being developed by mathematics teacher educators through their interaction with the LessonSketch online platform (www.lessonsketch.org). We briefly describe the platform and the larger project and then using the experiences created by two fellows illustrate the kin...

This chapter concludes the collection of chapters, each of which expands on the papers presented during the Topic Study Group on the teaching and learning of secondary school geometry at ICME-13. In articulating a vision for where the field of secondary school geometry could go in the near future, the chapter revisits issues of methodologies for da...

This chapter introduces the book by providing an orientation to the field of research and practice in the teaching and learning of secondary school geometry. The introduction then outlines the chapters in the book, each of which expands on the papers presented during the Topic Study Group on the teaching and learning of secondary school geometry at...

This book presents current perspectives on theoretical and empirical issues related to the teaching and learning of geometry at secondary schools. It contains chapters contributing to three main areas. A first set of chapters examines mathematical, epistemological, and curricular perspectives. A second set of chapters presents studies on geometry i...

We investigated how secondary mathematics teachers check student geometry proofs. From video records of geometry teachers checking proofs, we conjectured that teachers have different expectations for details that follow from written statements than for details that are conveyed by diagrams. To test our conjectures, we randomly assigned 44 secondary...

Based on data collected from the administration of a multimedia questionnaire to a nationally-distributed sample of secondary mathematics teachers, we investigated such teachers’ stances on the norm of high school geometry instruction that the teacher is responsible for providing the propositions (‘given’ and ‘prove’ statements) that students are a...

For more than 30 years, education researchers have been trying to characterize and differentiate between the practice and thinking of expert and novice teachers (Borko, Roberts, & Shavelson, 2008). Berliner (1986) identified four challenges in studying expertise: (1) The shortcomings of (then) common methods of studying teaching expertise (e.g., th...

There is growing interest in the field of education for leveraging emerging digital technologies to support teachers' learning in online or blended settings. This paper builds on Clarke and Hollingsworth's (2002) Interconnected Model of Professional Growth by investigating an alternative instantiation of professional experimentation. In particular,...

A program of research on the practical rationality of mathematics teaching is introduced in Spanish, starting from problematizing teachers’ decision making in instruction. References to specific articles covering 15 years of research on the subject are provided. The notions of instructional exchange, instructional situation, situational norm, and p...

This paper describes how the notion of instructional situation can serve as a cornerstone for a subject specific theory of mathematics teaching. The high school geometry course in the U.S. (and some of its instructional situations — constructing a figure, exploring a figure, and doing proofs) is used to identify elements of a subject-specific langu...

The notion of reference point has been proposed by Wise, Padmanabham, and Duffy to allude to artifacts shared in an online learning experience and that participants might refer to as they interact online. Representations of instructional practice in the forms of video records, animations, and so on, are examples of such reference points that have b...

The notion of practice-based teacher education has recently been offered to mitigate the long-standing tension between theory and practice in teacher education. In practice-based teacher education, preservice teachers study a core set of teaching practices by examining decompositions of practice and learn these practices through varied and multiple...

In this essay, the author frames the contents of the book by connecting dialogues and storytelling to the notion of representations of practice, which he examines from a semiotic perspective. The author argues that a constructionist notion of representation supports taking the dialogues and storyboards presented in this book as representations of p...

This book shows how the practice of script writing can be used both as a pedagogical approach and as a research tool in mathematics education. It provides an opportunity for script-writers to articulate their mathematical arguments and/or their pedagogical approaches. It further provides researchers with a corpus of narratives that can be analyzed...

For decades, mathematics educators have been interested in engaging K-12 students in the practice of creating and using mathematical models. What might this look like in the context of geometry? Inspired by claims that students come to secondary school with knowledge of three-dimensional space that can be leveraged to engage them in modeling, we wo...

We investigated secondary mathematics teachers' attitudes toward alternative ways of managing instruction on geometric proofs. Participants assigned to different experimental conditions viewed storyboard episodes of instruction. Some episodes showed instruction we hypothesized teachers would recognize as routine. Other episodes showed instruction t...

This chapter conceptualizes and illustrates StoryCircles, a form of professional education that builds on the knowledge of practitioners and engages them in collective, iterative scripting, visualization of, and argumentation about mathematics lessons using multimedia. The drive to invent and study new forms of professional education for mathematic...

Graduate student teaching assistants (GTAs) are responsible for the instruction of undergraduate students in critical introductory courses, but are not yet in the position of professors. Given their unique status, we ask if there are differences in how graduate students and professors express their agency when speaking about their responsibilities...

This paper is part of a three-year inquiry that supports and investigates the work of groups of mathematics teacher educators using technological tools to design and implement multimedia practice-based teacher education curriculum materials. This paper describes the kinds of activities, interactions, and tools used by mathematics teacher educators...

We discuss affordances and liabilities of using a storyboard to depict a written case of a teacher’s dilemma that involves race, opportunity to learn, and student community. We rely on reflections by the teacher educator who authored the written case and later depicted it as a storyboard to use it with his preservice teachers (PSTs). The analysis i...

The primary goal of this paper is to investigate whether a computer-based simulation can detect the difference between novice and expert teachers’ decision-making in mathematics instruction, which is complex in nature. The design of the simulation is grounded in a sociological perspective on practical rationality of mathematics teaching. The simula...

In this paper, we report on the potential of StoryCircles, a process of collaboratively representing a lesson using a storyboarding tool, for engaging teachers in substantive discussions about teaching mathematics. The analysis focuses on the interactions amongst a group of inservice teachers during which they wrestled with several dilemmas of prac...

This study describes an investigation exploring relationships between pre-service teachers' (PSTs') mathematical knowledge for teaching geometry (MKT-G) and their educational experiences. Our data from 108 pre-service teachers from 6 universities suggest that PSTs' experience in the classroom had the most significant effect on their MKT-G scores. T...

A challenge inherent in improving mathematics instruction is whether new practices will be adopted and sustained by practitioners. In order to better understand the circumstances under which mathematics teachers are willing to depart from customary instruction, we report on findings from a study in which K-12 mathematics teachers were asked to eval...

Secondary geometry teachers have been shown to recognize norms of instructional situations such as installing a new theorem (Herbst, Nachlieli, & Chazan, 2011) or doing proofs (Herbst, Aaron, Dimmel, & Erickson, 2013). In the latter case, the diagrammatic register norm describes the customary way in which a diagram and notation are used to present...

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching) is an exciting new series of texts for teacher education which aims to advance the learning and teaching of mathematics by integrating mathematics content with the broader research and theoretical base of mathematics education. The Learning and Teaching of Geometry in Secondary Sch...

How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk, I consider the proposal that geometry instruction could be improved by infusing it...

We examine responses from a national sample of high school mathematics teachers to a questionnaire, which had been developed to study teachers’ recognition of a system of hypothesized norms that stipulate that geometry proof problems are to be posed using a diagrammatic register. We report on the psychometric properties of the questionnaire, as wel...

This study presents an analysis of whether geometry teachers create opportunities for student discussion when engaging students in proving as revealed through the use of a multimedia survey instrument. We presented 42 secondary mathematics teachers with 8 multimedia narratives set in the situation of doing proofs in high school geometry and asked t...

Records of what takes place in mathematics classrooms are indispensable in mathematics teacher education and in research on mathematics teaching. We describe here the LessonSketch environment, an online platform for creating and analyzing representations of mathematics classrooms. We describe the authoring and analytic tools of LessonSketch with a...

This chapter focuses on how digital technologies provide affordances for Practice-Based Mathematics
Teacher Education (PBMTE) and maps needed research on technologically mediated teacher education.
The chapter elaborates on Grossman’s pedagogies of practice, describing how they can be enacted in
digital environments. Using LessonSketch as a prototy...

In this study, we examine the use of a nondescript cartoon graphics for representing emotion in professional scenarios. The results from fifteen inservice teachers who took 33 forced-choice items identifying emotions of 20 photos of human actors and 13 nondescript cartoon graphics of students represented in classrooms will be discussed. Findings sh...

We investigate the potential of a mode of professional development called StoryCircles for engaging pre-service teachers (PSTs) in iterative approximations of the work of teaching (Grossman et al., 2009) in settings of reduced complexity. StoryCircles are a facilitated process of collaboratively representing a lesson using a multimedia storyboardin...

This paper reports on a study of two norms of an instructional situation in high school geometry - Geometric Calculation in Algebra (GCA) - that consisted of confronting practitioners with representations of practice as a way to stimulate them to relay their knowledge of practice. In the situation of GCA, students are provided with a diagram in whi...

[Link to download (available until June 5, 2016): http://www.sciencedirect.com/science/article/pii/S0742051X16300488]
We investigate the quality of user interaction in forums that used animated classroom stories to support mathematics teachers' conversations. An analysis of pairs of posts connected by a parent-child relationship helped understand c...

In this paper, we argue that to prepare pre-service teachers for doing complex work of teaching like leading classroom mathematics discussions requires an implementation of different pedagogies of teacher education in deliberate ways. In supporting our argument, we use two frameworks: one curricular and one pedagogical. The curricular framework is...

Background: This paper reports on a study examining teachers’ perceptions of students they observed in an animated episode and who were engaged in the work of making conjectures in a geometry classroom. We examined eight conversations among subsets of 29 experienced geometry teachers with respect to how they described students and the mathematical...

We discuss findings from a study of 226 K-12 mathematics teachers who rated the appropriateness of instructional actions in a scenario-based assessment. The teaching actions shown in the scenarios were designed to represent mathematics teachers fulfilling their obligation to the discipline of mathematics. Our analysis of the participants' follow-up...

This paper reports on a study of the instructional situation in high school Geometry that Hsu (2010) called Geometric Calculation in Algebra (GCA). In particular, we conducted a virtual breaching experiment in order to examine the extent to which high school teachers recognized breaches of two norms that we conjectured to describe geometry teachers...

This paper reports on a study of the instructional situation in high school Geometry that Hsu (2010) called Geometric Calculation in Algebra (GCA). In particular, we conducted a virtual breaching experiment in order to examine the extent to which high school teachers recognized breaches of two norms that we conjectured to describe geometry teachers...

This paper describes instruments designed to use multimedia to study at scale the instructional decisions that mathematics teachers make as well as teachers’ recognition of elements of the context of their work that might influence those decision. This methodological contribution shows how evidence of constructs like instructional norm and professi...

We describe how multimedia scenarios delivered online can be used in instruments for the study of professional knowledge. Based on our work in the study of the knowledge and rationality involved in mathematics teaching, we describe how the study of professional knowledge writ large can benefit from the capacity to represent know-how using multimedi...

Erratum to: Educ Stud Math (2015)DOI 10.1007/s10649-015-9599-xThe original version of this article contained an error in the presentation of Theorem 2 (Ceva’s Theorem). The corrected presentation is given below.Theorem 2 (Ceva’s Theorem)Consider triangle ABC with points P on\( \overleftrightarrow{AB} \), Q on\( \overleftrightarrow{BC} \), and R on\...

Although mathematical practice has traditionally valued two distinct kinds of mathematical work—referred to by Gowers (2000) as theory building and problem solving—activity in classrooms appears to be organized largely around the latter, rather than the former. This study takes up the question of whether there is a customary role for theory buildin...

[Draft version: http://deepblue.lib.umich.edu/bitstream/handle/2027.42/110700/SITE2015-ChieuBoileauHerbst.pdf?sequence=1&isAllowed=y] LessonSketch is an online, multimedia-based learning environment that supports practice-based teacher education. A work in progress since the development of its initial components in 2008 and its launch in 2011, Less...

Geometry diagrams use the visual features of specific drawn objects to convey meaning about generic mathematical entities. We examine the semiotic structure of these visual features in two parts. One, we conduct a semiotic inquiry to conceptualize geometry diagrams as mathematical texts that comprise choices from different semiotic systems. Two, we...

[Draft version: http://deepblue.lib.umich.edu/bitstream/handle/2027.42/108487/JTE-FORUMANALYSIS_DB.pdf?sequence=1&isAllowed=y] It has been common to use video records of instruction in teacher professional development, but participants have rarely been encouraged to evaluate teachers and students’ actions in those records, allegedly because evaluat...

This chapter addresses the mathematics education doctorate and, in particular, the dissertation or thesis. The core of this chapter is an interview with Jeremy Kilpatrick in an effort to document Jeremy’s own advising style. The interview is situated within an attempt to bring out more general issues that are in play in how a mathematics education...

[Link to the full version: http://www.citejournal.org/vol14/iss4/mathematics/article1.cfm] This paper builds on Grossman's notion of approximations of practice as scaled-down opportunities for preservice teachers to learn to teach by doing. The authors propose the use of media rich, collaborative web-authoring tools for preservice teachers to creat...

We investigate online discussions among teachers. It is important to support novice teachers in examining and reflecting on different aspects of teaching practice. It is also important to enable them to consider and propose alternative teaching actions. We have used animated classroom stories in online forums to enhance teachers’ reflection and con...

LessonSketch (www.lessonsketch.org) is a unique web-based platform that focuses on the use of interactive and rich media to support scenario-based learning and assessment. Rich media include video records, audio tracks, animations, storyboards, drawings and scanned pictures in the forms of still images, text, and combinations of some of them. The s...

This paper documents efforts to develop an instrument to measure mathematical knowledge for teaching high school geometry (MKT-G). We report on the process of developing and piloting questions that purported to measure various domains of MKT-G. Scores on a piloted set of items had no statistical relationship with total years of experience teaching,...

This table identifies strategies and moves a mathematics teacher can use when designing work assignments for students (homework, seatwork) and when making decisions regarding what problems to review in public and how to review each problem.

Mathematical argumentation in classrooms has been linked with students’ increased understandings of and achievement in mathematics. A key component of mathematical argumentation is the teacher’s use of questioning strategies. In particular, effective questioning asks students to explain and justify their work and probes student understanding. Howev...

This study compared conversations among groups of teachers of high school
geometry that had been elicited by a representation of instruction (either a video or an
animation) and facilitated with an open-ended agenda. All artifacts used represented
instruction scenarios that departed from what, according to prior work, had been
hypothesized as norma...

This two-page decomposition of practice articulates some of the strategies that are useful for mathematics teachers to set up norms for classroom mathematical work. The material is often used as part of the curriculum for the teaching of secondary methods in teacher preparation.

Recommendations that teachers promote argument and discourse in their mathematics classrooms
anticipate researchers’ needs for methods for examining and analyzing such talk. One form of
discourse is oral arguments, including proofs. We ask: How can we track the development of an oral
argument by a teacher and her/his students?We illustrate amethod...

This paper investigates how critical events or reference points in animated classroom stories can support teacher learners' online discussions about their professional practice. Research has indicated positive impact of shared artifacts or reference objects such as video records of teaching practice on the quality of teachers' conversations in both...

This study explores interactions with diagrams that are involved in geometrical reasoning; more specifically, how students publicly make and justify conjectures through multimodal representations of diagrams. We describe how students interact with diagrams using both gestural and verbal modalities, and examine how such multimodal interactions with...

This paper investigates how critical events or reference points in animated classroom stories can support teacher learners’ online discussions about their professional practice. Research has indicated positive impact of shared artifacts or reference objects such as video records of teaching practice on the quality of teachers’ conversations in both...

We describe the development of measures of teachers' recognition of an instructional norm: that proof problems in high school geometry are presented in a diagrammatic register. A first instrument required participants to openly respond to depictions of classroom scenarios in which the norm was breached. A second instrument was a survey that require...

We compared effects of video and animated representations of teaching practice among teacher candidates. Sixty-one participants completed a questionnaire rating the representation in terms of its genuineness and its effectiveness enabling users to project to it their own experiences, to reflect, to conceive of alternative actions, and to notice asp...

Using a multimedia questionnaire we explore the extent to which secondary mathematics teachers recognize a hypothesized norm of doing proofs in geometry???that the teacher is in charge of providing the 'given' and the 'prove.' We also explore whether teachers who recognize the norm make a negative appraisal of its breach and find that geometry teac...

We examined the effect of reference to critical events in an animated classroom story on the quality of teachers??? comments in an online learning experience. We analyzed data using systemic functional linguistics and logistic regression. We found statistically significant evidence that participants made more reflective and evaluative comments and...

This paper examines the correlation between critical events in an animated classroom episode in geometry, which we take as an example of reference points, and the nature of teacher learners’ comments within an online learning experience.

This chapter describes how the authors have utilized digital graphics and Web 2.0 technologies to de- sign an information technology environment, LessonSketch. In LessonSketch teachers can learn about mathematical practice in instruction through the transaction of representations of practice. The authors describe the main features of LessonSketch,...