teachers possess?
In order to (1) assess which resources, especially mathematical knowledge, teachers draw upon in the work of evaluating and noticing students’ work and thinking (addressing my first research question) and to (2) understand how engaging with student work encourages teachers to think about mathematical topics (addressing my second research question), it was important that I have insight into the mathematical knowledge of my participants. Thus, it was important that I (i) focus the teachers’ noticing on a small selection of targeted mathematical concepts and (ii) assess their understanding of these particular concepts. The area of mathematics I targeted is
trigonometry. More specifically, I targeted the concepts of radian, unit circle, and the sine and cosine functions, central ideas within the study of trigonometry.
Thus, as a consequence of necessary data collection toward addressing my main two research questions, I was provided an opportunity to address the following preliminary research question, What content knowledge of radian, unit circle, and the
sine and cosine functions do secondary mathematics preservice, in-service, and student teachers possess?
Summary of Research Questions
0. What content knowledge of radian, unit circle, and the sine and cosine functions do secondary mathematics preservice, in-service, and student teachers posess? 1. How do secondary mathematics preservice. in-service, and student teachers attend
to, interpret, and respond to hypothetical students’ work and thinking in trigonometry?
a. What do teachers do when asked to notice? To what extent is their focus on students’ thinking?
b. What resources do teachers draw upon to interpret students’ thinking based on the students’ written work?
c. What resources do teachers draw upon to respond to students based on the students’ written work?
d. What are the goals of teachers’ proposed instructional responses, and how are they informed by their interpretations of students’ thinking? 2. In what ways does the hypothetical students’ work on trigonometry problems
challenge and/or further the preservice, in-service, and student teachers’ mathematical thinking?
a. What mathematics do teachers engage in as a result of examining the hypothetical students’ work that they hadn’t engaged in by simply working on the problem?
b. Which possible characteristics of the problems, student work, or interview prompts, might be related to the teachers’ mathematical engagement?
CHAPTER II
BACKGROUND LITERATURE
Theoretical Framework: A Situative Perspective
Guiding my work, including research questions, data collection, and analysis, is the belief that all learning and knowing is situated. That is, I hold a situative perspective through which I assume learning, knowledge, and beliefs to be inseparable from context and situation in which they are enacted. Thus, in the case of teachers, knowledge is important only to the extent that it is appropriately situated within the context of the work of teaching (Borko, Peressini, Romagnano, Knuth, Yorker, Wooley, Hoevrmill, &
Masarik, 2000). Whereas from a cognitive perspective, knowledge is acquired in one setting and transferred to another, from a situative perspective, the consideration of transfer is more complicated (Perisinni, Borko, Romagnano, Knuth, & Willis, 2004). This distinction has implications for my research.
From a situative perspective, researchers would not ask whether or how
knowledge transfers across situations, but rather, “ask questions about the consistency of patterns of participation across situations, conditions under which successful
participation in an activity in one type of situation facilitates successful participation in other types of situations, and the process of recontextualizing resources and discourses in new situations" (Perissini, Borko, Romagnano, Knuth, &Willis, 2004, p.70). Thus, rather than attempting to correlate certain proxies for teacher knowledge (such as mathematics test scores) with quality of teaching, within the situative tradition one might investigate which resources teachers draw upon in the actual work of teaching. In the case of my own work this means looking at how the resources acquired by preservice
and in-service teachers through various experiences such as college mathematics courses and working with students, are recontextualized in analyzing and responding to student work and thinking. For example, if I gather information about a teacher’s
knowledge of a specific procedure, I would be interested in how the teacher uses this knowledge in deciding whether or not the student is correct, how serious the student’s errors are, and how to engage the student to improve his or her understanding.
Teachers often complain that their preparation programs were too removed from the work of teaching (Putnam & Borko, 2000). From, a situative perspective, learning is situated and thus it seems essential to situate teachers’ preparation in the authentic tasks of teaching like analyzing student work and thinking. Putnam and Borko (2000) argue that while some may suggest that a situative perspective implies that all learning experiences for teachers should take place in actual classrooms, “for some purposes, in fact, situating learning experiences for teachers outside of the classroom may be important- indeed essential- for powerful learning” and that the situative perspective simply “focuses researchers' attention on how various settings for teachers' learning give rise to different kinds of knowing” (p.6). For example, in addressing the second research question of this study, I investigate how analyzing hypothetical students’ work outside of an actual classroom may give rise to different kinds of mathematical
knowledge and thinking.
Professional Mathematics Teacher Noticing Defining Noticing
The idea of professional noticing is not a new idea, nor is it unique to mathematics teaching. In 1994, Goodwin termed professional vision to explain
perceptual frameworks developed within particular professions. Similarly, in a general sense, “professional noticing is an ability to recognize and act on key indicators
significant to one’s profession” (p. 380, Schack, Fisher, Thomas, Eisenhardt, Tassell, & Yoder, 2013). Mason (2002) discussed intentional noticing, or noticing of a particular profession, and noted that with respect to teaching in particular, “Every act of teaching depends on noticing: noticing what children are doing, how they respond, evaluating what is being said or done against expectations and criteria, and considering what might be or done next” (p. 7).
By applying Goodwin’s (1994) idea of professional vision to mathematics education, Sherin and van Es have carried out significant work, bringing attention to mathematics teacher noticing (Sherin & van Es, 2005, 2009; van Es & Sherin, 2002, 2006, 2008). In general mathematics educators define teacher noticing to include two main processes: (i) attending to particular events in an instructional setting and (ii)
making sense of events in an instructional setting (Sherin, Jacobs, & Philipp, 2011).
When attending to particular events teachers choose, either consciously or
subconsciously, to focus on certain aspects of the classroom (and choose for how long to focus) and also which aspects not to pay attention to. These choices have significant impact on the trajectory of any given class. In contrast to attending, making sense of events involves interpretation, “relating observed events to abstract categories and characterizing what they see in terms of familiar instructional episodes” (Sherin, Jacobs, & Philipp, 2011, p.5). More recently, some researchers (Jacobs, Lamb, & Philipp, 2010; Jacobs, Lamb, Philipp, & Schappelle, 2011; Kazemi et al., 2011) have argued that making sense of classroom events involves not only interpretation but also, deciding
how to respond, a skill that “reflects intended responding, not the actual execution of the
response” (Jacobs, Lamb, Philipp, & Schappelle, 2011, p.99). The argument for inclusion in the noticing framework is that responding is conceptually linked to both attending and interpreting, and thus, it is valuable to investigate these skills in conjunction with each other.
There are additional variations in how researchers conceptualize and utilize the framework of noticing resulting from differences in: (i) focus, or object, of noticing and (ii) specific content. In this particular study, I focused on the content of trigonometry,
namely radian, unit circle, and the sine and cosine functions. With respect to the focus of noticing, I sought to investigate teachers’ interactions with students’ work and thinking. Building on previous more general research on noticing (Sherin & van Es, 2005, 2009; van Es & Sherin, 2002, 2006, 2008), Jacobs, Lamb, and Philipp (2010) introduced the construct of professional noticing of children’s mathematical thinking as “a set of three interrelated skills: attending to children’s strategies, interpreting children’s understanding, and deciding how to respond on the basis of children’s understandings” (p.172). By attending to children’s strategies, the authors referred to teachers’ attention to the mathematical details in the students’ strategies. I defined attending (as well as interpreting) slightly differently. In the following paragraphs I will explain and describe these choices further.
In my own study, some participants made interpretive claims about students not on the basis of the students’ strategies, but on the students’ result or particular
diagrams the students’ drew. Further, in my study, I presented the teachers’ with
strategies, and thus in order to pay attention to students' strategies or what the students
did, the teachers often needed to pay attention to students’ thinking. Thus, instead of
considering specific attention to strategy, I considered attention more generally than Jacobs, Lamb and Philipp (2010). In particular, I considered the participants’ attention to both the mathematical details of the written work as well as their attention to thinking, or any aspects of a student’s strategies or understandings that was not explicitly written on the paper. I was particularly interested in the extent to which the participants’ attention or focus was on students’ thinking.
Jacobs, Lamb, and Philipp (2010) included “interpreting children’s understanding” as the second skill of noticing, but they did not provide a definition of “understanding.” I found it more helpful to include any interpretive claims (i.e. claims that were not
immediately obvious based on the students’ work) within the category of interpreting. In particular, I used interpreting to encapsulate both interpreting students’ thinking about the problem as well as interpreting the students’ more general ways of knowing.
As I discussed, while I find their work helpful and informative to my study, I do not adopt the exact construct used by Jacobs, Lamb, and Philipp’s (2010). I consider
professional noticing of students’ written mathematical work and thinking to involve: (1) attending to students’ written work and thinking, (2) interpreting students’ thinking and ways of knowing, and (3) deciding to respond. Note that my modified construct involves
the same skills as Jacobs, Lamb and Philipp’s (2010) however the three skills are slightly restructured to minimize ambiguity and maximize its utility in making sense of my own data. This construct guided my first research question, data collection and analysis; in particular, I looked specifically at these three skills in an attempt to (i)
describe these skills as exhibited by my participants, (ii) draw connections between these skills, and (iii) extrapolate which resources teachers draw upon in carrying out these skills. Jacobs and colleagues investigated teachers’ professional noticing of
children’s mathematical thinking using data from a cross-sectional study, which I
discuss in the following section dedicated to research related to professional noticing of students’ mathematical thinking.
Research Related to Professional Noticing of Students’ Mathematical Thinking
In this section I provide background as to what research has been done with respect to teachers’ noticing of students’ thinking, as well as how this research
influences my own study. In reviewing the literature I found four main categories of such projects: (i) studies examining the relationship between noticing students’ thinking and quality of teaching, (ii) studies aimed at developing or improving noticing of students’ thinking, (iii) studies aimed at describing and evaluating teachers’ noticing students’ thinking, and (iv) studies aimed at examining the relationship between teacher
knowledge and noticing. My own study falls in the latter two categories, and thus, these are discussed in more detail. In the following, I briefly address the first two categories and then elaborate further on the third. The fourth category of research is summarized in a later section within the section on the relationship between knowledge and noticing.
Studies examining the relationship between noticing students’ thinking and quality of teaching. Several research projects have found that attending to students’
mathematical thinking has a positive effect in the classroom (Chamberlin, 2005). In particular, Cognitively Guided Instruction (CGI) research illustrated that in classrooms in which teachers’ attended to students’ thinking, (i) the teachers were transformed into
learners (ii) students could solve a wider variety of problems and use a wider range of strategies to solve those problems, (iii) there was a shift in emphasis from skills to concepts and problem solving, and (iv) students had higher levels of conceptual
understandings (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Franke & Kazemi, 2001). Additional research has also shown that attention to children’s thinking can have a positive affect on student learning (Kersting, Givvin, Sotelo, & Stigler, 2010).
Sherin, Jacobs, & Philipp (2011) highlighted three additional teaching and
research practices advocated for by the mathematics education community that indicate the importance of professional noticing: (i) adaptive and responsive teaching, (ii)
learning from teaching, and (iii) decomposing practice. In addition, research and policy emphasizing formative assessment in mathematics teaching also highlights the
relevance of noticing. While noticing is recognized as an important teaching skill, it proves to be difficult for teachers (Bray, 2011; Santaga, 2005; Schleppenbach, Flevares, Sims, & Perry, 2007; Stevenson & Stigler, 1992; Stigler & Hiebert, 1999). Thus, many mathematics educators have attempted to improve preservice and
inservice teachers’ noticing through professional development, methods courses, and research interventions. Several of these attempts are discussed in the following section.
Studies aimed at developing or improving noticing of students’ thinking. In
reviewing the literature related to noticing, I found that most studies were designed to improve teachers’ noticing. Further, most of these studies aimed to do so by increasing participants’ engagement with student thinking. Although my study does not aim to develop teachers’ noticing, but rather describe their current noticing to inform future
development, I find it necessary to include some background on this form of research due to it’s prevalence, in order to further situate my own work within the overall work done with respect to professional noticing. In addition, while it is not the focus of my research, this literature suggests that the participants in my own study may become better at noticing through their prompted noticing in the interview settings.
Current research on developing teachers’ noticing of student thinking advocates most strongly for interaction with classroom artifacts. More specifically, it is well
established that professional development and teacher preparation programs should involve investigating students’ work (Ball, 1997; Chamberlin, 2005; Crespo, 2000; Driscoll & Moyer, 2001). Ball argued that having teachers discuss artifacts of teaching and learning, including student work, held “promise for equipping teachers with the intellectual resources likely to be helpful in navigating the uncertainties of interpreting student thinking” (Ball, 1997, p.808). In the following, I discuss three particular studies that provide evidence of the benefit of using students’ work. I take care to acknowledge the precise ways in which each project utilized artifacts as well as other factors that may have contributed to the development of teachers’ noticing.
In the Linked Learning in Mathematics Project (LLMP), middle school teachers gained a deeper understanding of students’ algebraic thinking after examining students’ work (Driscoll & Moyer, 2001). Their improvement in making sense of students’
algebraic thinking cannot simply be attributed to the students’ work; two other significant aspects of the program included: (i) discussion of the students’ work in small groups, and (ii) guidelines for analyzing algebraic thinking which were given to the teachers, helping them to focus on particular algebraic habits of mind (Driscoll & Moyer, 2001).
Within the research and development project, Cognitively Guided Instruction (CGI), Franke and Kazemi (2001) also observed benefits using focused teacher group discussions centered around students’ work. In particular, they placed twelve K-5 teachers in four work groups, and each month the teachers would pose a mathematics problem to their own students, and then bring their students’ work to a work group discussion meeting facilitated by the researchers. Crespo (2000) also used authentic student work in her attempt to improve preservice elementary teachers’ ability to make sense of students thinking via a mathematical letter exchange program. In this program, the preservice teachers exchanged letters with fourth grade students who attempted to make sense of a particular mathematics problem. Crespo (2000) noted changes in focus from correctness (evaluation) to meaning and changes in responses from quick and conclusive to thoughtful and tentative.
Other programs designed to develop teachers’ noticing utilized video as opposed to students’ written work. For example, McDuffie and colleagues attempted to support prospective K-8 teachers’ noticing by engaging them in video analysis of excerpts of mathematics lessons (Roth McDuffie, Foote, Bolson, Turner, Aguirre, Bartell, Drake, & Land, 2014). The mathematics teacher educators structured the video analyses by providing background reading and focusing the teachers’ discussion on four aspects of a mathematics lesson: teaching, learning, task, as well as power and participation. They observed increased depth of noticing and a move from attending primarily to what they saw, or teacher moves, especially teachers’ interactions with students, to interpreting the importance and effects of the moves or teacher-student interactions (Roth McDuffie et al., 2013). In another recent example, Schack and colleagues reported successfully
developing the professional noticing abilities of preservice elementary teachers by having them engage with video excerpts of interviews with children doing mathematics (Schack et al., 2013). In addition to observing video, the preservice elementary teachers also engaged in group discussions and were provided instruction regarding professional noticing.
There were significantly more studies aimed at improving teachers’ noticing than there were studies aimed at understanding how teachers’ make sense of students’ work and thinking, and how they decide how to respond to students. I argue that programs designed to improve teachers’ noticing might be better informed by having a better understanding, or picture, of how teachers typically attend to, interpret, and respond to students’ work and thinking and then building on their pre-existing noticing. Studies that attempt to contribute to such a picture are discussed in the following section (Shack et al., 2013).
Studies aimed at describing teachers’ noticing of students’ thinking. Since
their construct of professional noticing of children’s mathematical thinking as well as their research methodologies and findings were influential on my own study, I begin this section by discussing, in detail, the work of Jacobs and colleagues (Jacobs, Lamb, & Philipp, 2010; Jacobs & Philipp, 2010; Jacobs, Lamb, Philipp, & Schappelle, 2011).
Jacobs and colleagues investigated teachers’ noticing of children’s thinking using data from a cross-sectional study, “Studying Teachers’ Evolving Perspectives” (STEP) of 131 preservice and in-service K-3 teachers (Jacobs, Lamb, & Philipp, 2010; Jacobs & Philipp, 2010; Jacobs, Lamb, Philipp, & Schappelle, 2011). Jacobs, Lamb & Philipp (2010) used two assessments to capture participants’ noticing. In one they were asked
to watch a video clip of a classroom, and in the other they were given a set of written student work. After observing either the video or student work, the participants were asked to write responses to various prompts, namely: (i) Please describe in detail what
you think each child did in response to the problem; (ii) Please explain what you learned about these children’s understandings; and (iii) Pretend that you are the teacher of these children. What problem or problems might you pose next? (Jacobs, Lamb, &
Philipp, 2010). These three prompts were designed to correspond with each of the three skills of noticing the authors had outlined – attending, interpreting, and responding – respectively. Jacobs and colleagues also report using a video of a student named Rex solving subtraction and addition word problems (Jacobs & Philipp, 2010; Jacobs, Lamb, Philipp, & Schappelle, 2011). Similarly, participants were asked to respond in writing to