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Using Mathematical Learning Goals to Analyze Teacher Noticing
Abstract
Teacher noticing of student mathematical thinking is increasingly seen as an important construct, but challenges remain in operationalizing and assessing teachers’ analyses of their classrooms. In this chapter, we present a methodology for analyzing teachers’ professional noticing of student mathematical thinking based on its alignment to mathematical learning goals. This process entails first deconstructing a mathematical learning goal into its conceptually important pieces (known as subgoals). Then, researchers can look for references to these subgoals in teachers’ attending, interpreting, and deciding (the three skills of noticing). When teachers reference conceptual subgoals of a learning goal in their noticing, it indicates their attention to students’ reasoning about the important mathematical ideas of a lesson. This method of data analysis can be used across a variety of contexts and allows for greater precision in understanding teacher noticing by focusing on its mathematical content and attention to relevant student thinking. In this theoretical chapter, we describe this research methodology (and the process of deconstructing learning goals and using subgoals), justify its appropriateness as a measure of teacher noticing, and provide examples from our own and others’ work to illuminate its use.
... Morris et al. (2010) suggest that the best claims should align with the mathematical learning goal, beginning by dissecting learning goals into their component mathematical ideas, called subgoals, and comparing student responses to those subgoals. High-quality evaluation of student thinking will address as many of the conceptually important mathematical ideas (subgoals) as possible, allowing teachers to make judgments about the full range of ideas in the learning goal and to pay attention to gaps in mathematical knowledge (Morris 2006;Morris et al. 2010;Spitzer and Phelps-Gregory 2017). ...
... For our first category of codes, addressing research question one, we coded PTs' claims about student thinking using a coding scheme that categorized PTs' claims based on their level of specificity and alignment to the learning goal (see Table 1 for descriptions and examples of these codes). In alignment with our conceptual framework and following the work of other researchers (Meikle 2014;Morris et al. 2010;Spitzer and Phelps 2017), we consider PTs' claims about subgoals of the learning goal as the highest quality claim, because these claims are both specific about the kinds of student thinking inferred and relevant to the learning goal (as a reminder, see Fig. 1 for the goal and one possible subgoal decomposition). The other three types of claims are either poorly aligned with the learning goal (focusing on irrelevant mathematics or pure procedural knowledge) or too broad to be useful (generic claims about math). ...
This study examined elementary and secondary prospective teachers’ (PTs’) abilities to analyze a classroom lesson in order to make claims about student thinking around specific mathematical learning goals based on relevant and revealing evidence. Previous research suggests PTs have some skills in analyzing evidence but apply them inconsistently. Our goal was to describe in more detail the strengths and weaknesses in PTs’ ability to analyze evidence of student thinking. Results indicate that PTs can make some appropriate claims about student learning in a lesson transcript, but more often make overly broad and general claims. PTs were able to support their claims with specific student work but often used poorly aligned evidence. PTs also often explicitly recognized the shortcomings of evidence from the lesson transcript, but then relied on that evidence to make claims about student thinking. Finally, PTs’ background, such as number of teacher education courses completed, does not appear to strongly influence their ability to make claims and support them with evidence, though secondary PTs were more likely to recognize the limitations of evidence than elementary PTs. These results have implications for teacher educators, pointing to the importance of designing interventions to help PTs look beyond the most visible and salient features of a lesson when analyzing student thinking.
This paper reports an investigation into what 13 pre-service teachers (PTs) enrolled in a secondary mathematics methods unit in one university noticed when providing written feedback on peers’ lesson plans. Drawing on the first two elements of the curricular noticing framework (Dietiker et al., 2018), we identify aspects of the lesson plans the PTs comment upon in their feedback (attending) and how they make sense of those aspects (interpreting). Results highlight three main themes of PT noticing from the lesson plans: The pedagogical approach, the nature of tasks, and the learning intentions. We discuss how the context of the methods unit and the design of the activity impacted the PTs’ feedback.
In this paper, we investigate the links between prospective elementary teachers’ (PTs’) ability to notice student mathematical thinking, their mathematical knowledge for teaching (MKT) and their ability to decompose learning goals into component conceptual parts (Morris et al., 2009). Previously, researchers have proposed theoretical connections between MKT and teachers’ professional noticing, but empirical support for these relationships has been limited. Results of this mixed-methods study indicate that PTs who scored higher on an MKT assessment outperformed their peers in terms of attending to student thinking but had similar performance in terms of interpreting that thinking. However, PTs who were able to conceptually unpack a learning goal into subconstructs performed higher-quality interpretations of student thinking. We hypothesize that the skill of decomposing learning goals may allow PTs to apply their mathematics knowledge to successfully interpret student work. These results have implications for understanding how noticing and MKT may develop in relation with each other.
The purpose of this study was to analyze the changes in pre-service teachers’ noticing through an elementary mathematics methods course along with a practicum. For this purpose, the comments pre-service teachers made on an entire video-based mathematics lesson were collected three times over the semester. Their comments were analyzed in terms of topic, actor, stance, evidence, and alternative strategy. The results showed that the pre-service teachers’ noticing abilities were slightly changed after learning mathematical content and pedagogy related to teaching elementary mathematics. Substantial changes in their noticing occurred after a four-week practicum and subsequent discussions on their own lesson planning, implementation, and reflections. This study has implications for designing a mathematics methods course to develop teacher expertise and refining a teacher noticing analytic framework.
In supporting their students’ learning in the classroom, noticing is an important professional skill for teachers, encompassing perceiving and interpreting relevant incidents, as well as ad hoc decision-making. While noticing is an established research topic in other domains, it remains largely neglected in relation to the geography teacher. The paper compares the noticing skills of geography teachers at three different stages of their professional development. Qualitative content analysis of teachers’ think-aloud responses to video vignettes of geography teaching revealed that the three groups differed in terms of number of incidents perceived, interpretive stance, and decision-making.
The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.
Mathematics Teacher Noticing is the first book to examine research on the particular type of noticing done by teachers---how teachers pay attention to and make sense of what happens in the complexity of instructional situations. In the midst of all that is happening in a classroom, where do mathematics teachers look, what do they see, and what sense do they make of it? This groundbreaking collection begins with an overview of the construct of noticing and the various historical, theoretical, and methodological perspectives on teacher noticing. It then focuses on studies of mathematics teacher noticing in the context of teaching and learning and concludes by suggesting links to other constructs integral to teaching. By collecting the work of leaders in the field in one volume, the editors present the current state of research and provide ideas for how future work could further the field.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
In this article, we describe features of learning goals that enable indexing knowledge for teacher education. Learning goals are the key enabler for building a knowledge base for teacher education; they define what counts as essential knowledge for prospective teachers. We argue that 2 characteristics of learning goals support knowledge-building efforts. First, learning goals should be targeted; they must be sufficiently well specified to suggest interventions for supporting learners in achieving them and to indicate the types of evidence needed to determine if the goals have been achieved. Second learning goals should be shared; they must be mutually understood and committed to by all participants in the knowledge-building process. By drawing on our experience in our local efforts to develop knowledge for preparing prospective elementary mathematics teachers, we illustrate how targeted and shared learning goals support efforts to build a knowledge base for teacher education. We then propose the use of learning progressions for supportingknowledge-building endeavors across sites.
The authors propose a framework for teacher preparation programs that aims to help prospective teachers learn how to teach from studying teaching. The framework is motivated by their interest in defining a set of competencies that provide a deliberate, systematic path to becoming an effective teacher over time. The framework is composed of four skills, rooted in the daily activity of teach- ing, that when deployed deliberately and systematically, constitute a process of creating and test- ing hypotheses about cause-effect relationships between teaching and learning during classroom lessons. In spite of the challenges of acquiring these skills, the authors argue that the framework outlines a more realistic and more promising set of beginning teacher competencies than those of traditional programs designed to produce graduates with expert teaching strategies.
This study explores the use of video clips from teachers' own classrooms as a resource for investigating student mathematical thinking. Three dimensions for characterizing video clips of student mathematical thinking are introduced: the extent to which a clip provides windows into student thinking, the depth of thinking shown, and the clarity of the thinking. Twenty-six video clips were rated as being low, medium, or high on each dimension. Corresponding teacher discussions of each video were then examined to identify the ways in which clip dimensions served as catalysts for more and less productive teacher conversations of student mathematical thinking. Findings include first, that, under certain circumstances, both low-and high-depth clips lead to productive discussions. Second, high-depth clips in which student thinking is sustained only briefly do not typically lead to productive discussions. Third, in cases where windows and depth are both high, clips that are either low or high in clarity resulted in productive conversations of student thinking on the part of teachers.
Professional noticing of students’ mathematical thinking in problem solving
involves the identification of noteworthy mathematical ideas of students’ mathematical
thinking and its interpretation to make decisions in the teaching of mathematics. The goal
of this study is to begin to characterize pre-service primary school teachers’ noticing of
students’ mathematical thinking when students solve tasks that involve proportional and
non-proportional reasoning. From the analysis of how pre-service primary school
teachers notice students’ mathematical thinking, we have identified an initial framework
with four levels of development. This framework indicates a possible trajectory in the
development of primary teachers’ professional noticing.
This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.
For orchestrating whole-class discussions, note these suggestions to fine tune problem-solving techniques into cognitively challenging tasks.
It can be difficult for teachers to make in-the-moment decisions about which solution strategies to cognitively challenging tasks should be included in the whole-class discussion (Stein, Engle, Smith, & Hughes, 2008). Teachers can purposefully select and sequence the solution strategies to help create a whole-class discussion that promotes the mathematical learning goal. An intervention was implemented in a middle school methods course that aimed to understand preservice Teachers' (PSTs') competencies in formulating rationales for their selecting and sequencing choices.
Learning algebra is critical for students in the USA today, yet many students in the USA struggle in algebra classes. Researchers claim that one reason for these difficulties is that algebra classes often focus on symbol manipulation and procedures above, and many times at the expense of, a more conceptual understanding of the content. Teaching algebra in a more conceptual way, however, can be quite challenging for teachers. This study explores the development of teacher noticing as a way to help teachers learn to broaden their views of algebra, pay attention to a wide range of student algebraic thinking, and reason about students’ ideas in substantive ways. This study takes place in the context of a video club in which seven preservice teachers watched and discussed video excerpts from algebra classes over an 8-week period. A framework for noticing student algebraic thinking was created and used to structure the discussions in the video club. In addition, a new, online video-tagging tool was used to document the development of the teachers’ ability to notice student algebraic thinking. Results suggest that participating in the video club helped teachers more consistently attend to substantive student algebraic thinking and to reason about this thinking they noticed in deeper ways.
This study examined changes in the beliefs and instruction of 21 primary grade teachers over a 4-year period in which the teachers participated in a CGI (Cognitively Guided Instruction) teacher development program that focused on helping the teachers understand the development of children's mathematical thinking by interacting with a specific research-based model. Over the 4 years, there were fundamental changes in the beliefs and instruction of 18 teachers such that the teachers' role evolved from demonstrating procedures to helping children build on their mathematical thinking by engaging them in a variety of problem-solving situations and encouraging them to talk about their mathematical thinking. Changes in the instruction of individual teachers were directly related to changes in their students' achievement. For every teacher, class achievement in concepts and problem solving was higher at the end of the study than at the beginning. In spite of the shift in emphasis from skills to concepts and problem solving, there was no overall change in computational performance. The findings suggest that developing an understanding of children's mathematical thinking can be a productive basis for helping teachers to make the fundamental changes called for in current reform recommendations.
Researchers have advocated building a knowledge base for teacher education based on practitioner knowledge. In this article, the authors, two teacher educators, present knowledge gained from their systematic study of two lessons to help prospective teachers learn to study teaching. They offer a detailed look at the lessons' successes and shortcomings as well as potential revisions. They also reflect on their use of a model for systematically studying lessons, sharing details about their learning during this process. This article attempts to provide a visible, tangible product for other teacher educators interested in helping prospective teachers examine evidence of student learning.
The goal of this study is to develop the professional noticing abilities of prospective elementary school teachers in the context of the Stages of Early Arithmetic Learning. In their mathematics methods course, ninety-four prospective elementary school teachers from three institutions participated in a researcher-developed five-session module that progressively nests the three interrelated components of professional noticing—attending, interpreting, and deciding. The module embeds video excerpts of diagnostic interviews of children doing mathematics (representations of practice) to prepare the prospective teachers for similar work. The module culminates with prospective teachers implementing similar diagnostic interviews (approximations of practice) to gain experience in the three component skills of professional noticing. A pre- and post-assessment was administered to measure prospective teachers’ change in the three components. A Wilcoxon signed ranks test was conducted and found the prospective elementary school teachers demonstrated significant growth in all three components. Selected prospective elementary school teacher responses on the pre- and post-assessment are provided to illustrate sample growth in the prospective teachers’ abilities to professionally notice. These results, the first in an ongoing study, indicate the potential that prospective teachers can develop professional noticing skills through this module. Continued data collection and analysis from the ongoing study by these authors and future, longer-term emphasis on professional noticing for prospective teachers should be studied.
Research about lesson study and other forms of collaborative, practice-based professional development has tended to focus on identifying structures that support teacher learning. This study builds on this work by examining preservice elementary teachers’ efforts to conduct lesson studies in mathematics. Analysis showed that the structures of the lesson study project supported some groups in developing mathematical and equity lenses for looking at teaching; however, the use of these lenses simultaneously led to productive and problematic learning. This finding suggests that future research focus on challenges present in collaborative, practice-based work even when productive structures are in place.
The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an essential starting point for studying and improving one's teaching. Thirty K-8 preservice teachers completed 4 written tasks. Each task specified a learning goal and then asked the preservice teachers to complete a teaching activity with this goal in mind. For example, preservice teachers were asked to evaluate whether a student's responses to a series of mathematics problems showed understanding of decimal number addition. The results indicate that preservice teachers can identify mathematical subconcepts of learning goals in supportive contexts but do not spon- taneously apply a strategy of unpacking learning goals to plan for, or evaluate, teaching and learning. Implications for preservice education are discussed.
In this study, we examined prospective middle school mathematics teachers’ reflective thinking skills to understand how they
learned from their own teaching practice when engaging in a modified lesson study experience. Our goal was to identify variations
among prospective teachers’ descriptions of students’ thinking and frequency of their interpretations about how teaching affected
their students’ learning. Thirty-three participants responded to open-ended questionnaires or interviews that elicited reflections
on their own teaching practice. Prospective teachers used two forms of nuance when describing their students’ thinking: (1)
identifying students’ specific mathematical understandings rather than general claims and (2) differentiating between individual
students’ thinking rather than characterizing students as a collective group. Participants who described their students’ thinking
with nuance were more likely to interpret their teaching by posing multiple hypotheses with regard to how their instruction
affected their students’ learning. Implications for supporting continued growth in reflective thinking skills are discussed
in relation to these results.
Better data: Using classroom evidence to improve teaching incrementally
- C M Phelps
- F S Shore
- S M Spitzer
Noticing matters. A lot. Now what?
- A H Schoenfeld
Deciding how to respond on the basis of children’s understanding
- V R Jacobs
- L L Lamb
- R A Philipp
- B P Schappelle
- VR Jacobs
Prospective teachers’ use of lesson experiments: The effects of an online discussion board
- S Spitzer
- C Phelps
Reflections on the study of teacher noticing
- B Sherin
- J R Star
From teacher noticing to a framework for analyzing and improving classroom lessons
- R Santagata
Annual perspectives in mathematics education 2014: Using research to improve instruction
- C M Phelps
- F S Shore
- S M Spitzer
- CM Phelps