Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach

Teaching and Teacher Education xxx (xxxx) xxx

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Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach Erica Litke * University of Delaware, USA

h i g h l i g h t s
Algebra lessons focused on teaching procedures but aspects of this instruction supported students’ learning opportunities.
Types of learning opportunities include giving meaning to procedures and supporting procedural flexibility.
Teachers also provided students opportunities to make connections within and across algebraic representations, topics, and ideas.
Small adjustments that deepen practices in these areas represent a possible incremental approach to improving instruction.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 January 2019 Received in revised form 2 January 2020 Accepted 23 January 2020 Available online xxx

Algebra is considered key to success in secondary mathematics, yet instruction remains mostly teachercentered and procedurally oriented, with limited opportunities for students to develop algebraic understanding. This study draws on a large sample of video-recorded ninth grade U.S. algebra lessons to explore the nature of learning opportunities that may help deepen students’ algebraic understanding. I highlight aspects of opportunities to learn algebraic procedures and describe instruction that can enrich these opportunities. I posit that this holds promise as an incremental improvement approachdrelatively small adjustments in teachers’ current practices that can serve as a bridge to more ambitious teaching. © 2020 Elsevier Ltd. All rights reserved.

Keywords: Mathematics education Algebra Teaching practices Instructional improvement

1. Introduction Improving student outcomes in algebra has been a global focus in recent decades (e.g., Agudelo-Valderrama, Clarke, & Bishop, 2007; Kanbir, Clements, & Ellerton, 2018; Kaur, 2014; Leung,
& Clarke, Holton, & Park, 2014; MacGregor, 2004; Novotna
, 2014; Prendergast & Treacy, 2018; Stacey, Chick, & Hospesova Kendal, 2004). In the U.S., research has shown that school algebra is considered a pivotal gatekeeper to higher-level mathematics and a predictor of later academic success (Adelman, 2006; Stein, Kaufman, Sherman, & Hillen, 2011). Proficiency in algebra cannot be achieved without high-quality instructiondinstruction that creates learning opportunities to help students develop conceptual understanding of algebraic ideas and fluency with algebraic procedures. Investigations into algebra classrooms worldwide have

* University of Delaware, School of Education, 105D Willard Hall Education Building, Newark, DE, 19716, USA E-mail address: [email protected].

shown variability in emphasis and pedagogy, both between and within countries (Agudelo-Valderrama et al., 2007; Leung et al.,
, 2014; Smith, 2011). In order to 2014; Novotn
a & Hospesova improve students’ learning opportunities, it is useful to understand specific aspects of teaching that have the potential to support students in learning algebraic ideas. In the U.S., efforts at instructional improvement have largely focused on ambitious mathematics teachingdinstruction that provides students opportunities to reason about mathematics, explain their thinking, and engage with mathematics in contextualized ways through authentic problems (Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010). Through evolving standards and curricula, reformers have provided supports that aim to help teachers position students as the primary sense makers, promote discourse, and support the development of conceptual understanding (National Council for Teachers of Mathematics [NCTM], 2000; National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010; Polikoff, 2015; Saxe, Gearhart, & Nasir, 2001; Stein, Engle, Smith, & Hughes, 2008). Yet

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Please cite this article as: Litke, E., Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach, Teaching and Teacher Education, https://doi.org/10.1016/j.tate.2020.103030

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results from investigations into classroom practice suggest that these reforms have been slow to take root and that procedurallyoriented, teacher-centered instruction largely persists in mathematics classrooms in the U.S. (Hiebert et al., 2005; Kane & Staiger, 2012; Litke, 2020). Given these current practices in the U.S. and in other countries (e.g., Kaur, 2014; Leung et al., 2014; Prendergast & Treacy, 2018), it may be time to reconsider the emphasis on wholesale reform of instructional practice in mathematics, a daunting task for any teacher, and consider alternative approaches. Rather than focus solely on where practice should be, instructional reform might instead also consider meeting teachers where they are and encouraging incremental improvements (Janssen, Westbroek, Doyle, & Van Driel, 2013). Identifying the kinds of learning opportunities that might benefit from smaller instructional adjustments, particularly those that build upon and extend teachers’ existing practices, may hold promise for not only improving students’ learning opportunities in the current environment, but also serve as a bridge to more ambitious changes sought by the field (Janssen et al., 2013; Star, 2016). Furthermore, these improvement efforts could be grounded in the specific content taught. Situating improvement teaching algebra in the context of instructional practices that support students’ learning of algebra specifically is applicable across national contexts and may hold greater potential for instructional changes to be taken up and instantiated by teachers. A focus on incremental improvement, however, raises questions about what specific aspects of instruction might be improved, requiring a fine-grained description of current practices and an analysis of the kinds of small adjustments that could improve these practices. While some have argued for a modular approach to improvement by focusing on a set of core teaching practices (e.g., Janssen, Grossman, & Westbroek, 2015), I am interested in aspects of instruction that are specifically tied to the content taught and that might be leveraged in an incremental improvement approach. To address this, I conducted a video-based observational study of algebra classrooms in the U.S. context. I aimed to identify and describe the types of learning opportunities created by teachers through their instruction in order to consider small adjustments to instruction upon which teachers might build. By learning opportunities, I mean the ways in which students encounter algebra content in the classroom. I use instruction to include the particular in-themoment aspects of teaching that engage teachers and students in interactions around algebra content. These interactions can occur in both teacher- or student-centered classrooms, and thus are potential key foci for incremental improvement efforts across contexts. Specifically, I ask: What are key instructional practices across lessons in this sample that could enrich the conceptual aspects of students’ learning opportunities in algebra? The results of this study highlight practices already happening in classrooms that might be deepened, with small adjustments, to yield richer learning opportunities for students. 2. Background 2.1. Contemporary visions of algebra teaching Researchers and policy makers have pushed for studentcentered instruction in mathematics that engages students in open-ended investigation, productive struggle, and mathematical discussion (Boaler, 2002; Chazan, 2000; Hiebert & Grouws, 2007; Kieran, 2007; Stein et al., 2008). In addition, contemporary visions for understanding in algebra suggest that students’ learning opportunities must expand beyond experiencing the subject as “generalized arithmetic”dan extrapolation of rules and

computation with numbersdto include studying patterns and structure, functions and relationships, and mathematical modeling of real-life contexts (Kieran, 2007; NCTM, 2008, 2014; Smith, 2014; Stephens, Ellis, Blanton, & Brizuela, 2017). These changes require a pedagogical shift away from more traditional, teacher-centered approaches (Star, 2016). The hoped-for result of these changes has been algebra classrooms in which students engage in productive mathematical struggle and experience high cognitive demand tasks (e.g., Chazan, 2008; Hiebert & Grouws, 2007; Stacey & Chick, 2004), while simultaneously developing procedural fluency (Kieran, 2007, 2013; Smith, 2014). Yet large-scale observational studies provide evidence that this reform vision has been slow to translate into the expected changes in instructional practice. A study of 4th and 5th grade mathematics lessons across five U.S. metropolitan districts found that while lessons included some reform-oriented practices, teacher-centered instructional formats were the norm (Hill, Litke, & Lynch, 2018). In a study of approximately 3000 U.S. classrooms spanning 4the9th grades, researchers from the Measures of Effective Teaching (MET) Project found little evidence of mathematical sense-making or teachers using student ideas in mathematics lessons (Kane & Staiger, 2012). Another study of ninth grade algebra lessons using the MET data found that student-centered, inquiry-oriented instruction was rare (Litke, 2020). The TIMSS Mathematics Video Study, a comparative study of eighth-grade mathematics lessons, found that in eighth-grade classrooms in the U.S., instruction was largely teacher-led, procedurally focused, and characterized by fragmentation and little cognitive challenge (Hiebert et al., 2005). Teachers seldom provided opportunities for students to develop conceptual understanding by connecting procedures with concepts or by connecting multiple representations of mathematical ideas (Hiebert et al., 2005; Jacobs et al., 2006). These findings are not unique to the U.S. context; while instruction in some countries focuses more on conceptual understanding by asking students to draw connections among mathematical ideas (e.g., Hiebert et al., 2005; Smith, 2011), other research suggests that the instruction typical of U.S. classrooms is familiar in other nations as well (e.g., Agudelo-Valderrama et al., 2007; Leung et al., 2014). 2.2. Moving to envisioned instruction by building on current practice in algebra In light of longstanding and persistent teacher-centered instructional approaches that create limited learning opportunities for students (Hiebert, 2013; Stigler & Hiebert, 2018), some researchers have begun to argue for a more incremental improvement approach, one that develops bridges between teachers’ extant practices and desired innovations, working within the system in which teaching operates (Davis, Palincsar, Smith, Arias, & Kademian, 2017; Hiebert et al., 2005; Janssen et al., 2013; Star, 2016). For example, based on results of the TIMSS video study, Hiebert et al. (2005) advocated for focusing U.S. improvement efforts not on wholesale changes from teacher-centered classrooms to student-centered ones, but rather on extending current practices to yield more ambitious and richer learning opportunities for students. Improvement efforts might thus focus on more fine-grained instructional practices that can serve as manageable, yet meaningful bridges toward larger, more ambitious changes (Davis et al., 2017; Janssen et al., 2013). Furthermore, doing so within a content domain might better resonate with teachers, engaging them in an improvement process that links to (and values) existing practice (Kieran, Krainer, & Shaughnessy, 2012). In this study, I develop descriptions of different kinds of learning opportunities created by contemporary teaching practices in algebra. In this conceptualization, instructional practices create

Please cite this article as: Litke, E., Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach, Teaching and Teacher Education, https://doi.org/10.1016/j.tate.2020.103030

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opportunities for students to learn important mathematical ideas (Hiebert & Grouws, 2007; Stigler & Hiebert, 2018). Although these opportunities do not guarantee that learning will occur, they are a key enabling condition in the classroom ecology. Thinking about algebra specifically, this approach requires an understanding of which practices might support student learning opportunities in algebra and necessitates an examination of instruction grounded in what we know about both the learning of algebra content and teaching practices that support opportunities to learn that content. There are certainly other valuable ways to think about examining instruction in mathematics that are not specifically rooted in the content taught. For example, some researchers have focused on supporting teachers in learning high-leverage mathematical practices, improving general classroom practices, improving practices related to curriculum and its enactment, or increasing attention to mathematical processes, all equally viable lenses through which to examine instruction and instructional improvement (e.g., Grossman, Hammerness, & McDonald, 2009; Hill et al., 2008; €nig, & Schlesinger & Jentsch, 2016; Schlesinger, Jentsch, Kaiser, Ko €meke, 2018). I am interested, however, in better understandBlo ing existing instructional practices within the algebra domain in light of the types of learning opportunities we hope that teaching will support. Considering content-focused practices that support student learning opportunities holds promise for decisions about which practices may serve as levers for change and ultimately as bridges to more ambitious teaching within the algebra context. One such leverage point in algebra may be how procedures are taught. The degree to which algebra instruction should focus on concepts or procedures (or both) has been a longstanding debate, yet it is likely that the two types of knowledge are intertwined and develop in tandem (see, for example, Hiebert & Grouws, 2007; Kieran, 2013; Rittle-Johnson, Siegler, & Alibali, 2001). In algebra, procedures are prevalent, even with the current emphasis on teaching for understanding (Kilpatrick & Izs
ak, 2008). The goal is not to abandon procedures but rather to teach procedures in ways that connect them with their conceptual underpinnings. For example, if students understand how and why a procedure works, they are better able to adapt procedures flexibly to new situations (Hiebert, 2003). Comparing and contrasting multiple procedures for solving the same problem promotes both procedural fluency and conceptual understanding (Rittle-Johnson & Star, 2009) and affords students the opportunity to see and understand algebraic structure (see also, Star & Rittle-Johnson, 2009). Providing students opportunities to connect procedures, representations (e.g., graphs, tables, and equations), and algebraic ideas supports students’ learning of the embedded mathematical relationships and promotes a deeper understanding of expressions, equations, and the related procedures (Chazan & Yerulshlamy, 2003; Moschkovich, Schoenfeld, & Arcavi, 1993; Star et al., 2015). Another approach is to consider how to support students in making sense of new algebraic ideas. Algebra is more abstract than the mathematics students have previously encountered, something that can cause difficulty for students (Booth, 1988; Hiebert & Grouws, 2007; Rakes, Valentine, McGatha, & Ronau, 2010). Students can develop meaning for these abstractions when they connect symbolic representations for algebraic ideas and procedures to their concrete or numerical analogs (Kieran, 2007). Booth et al. (2017) recommend connecting familiar and concrete representations of mathematical ideas to their abstract and symbolic counterparts in order to promote both learning and transfer. Concreteness fadingdexplicitly moving from concrete to more abstract representationsdhas also been shown to help students develop meaning for more abstract ideas (Fyfe, McNeil, Son, & Goldstone, 2014). Finally, making coherent connections between

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and across mathematical topics can help students see how the procedures and concepts they are learning now fit into a larger mathematical context. This can help infuse meaning into the current topic and help students see its importance and motivate the topic under study (Booth, 1988; Hiebert & Grouws, 2007), particularly when the content is relatively abstract (Thompson, 2013). The literature discussed above provides insight into potentially promising instruction we might see in an investigation of teaching in algebra classrooms. These practices provide guidelines for envisioning how instruction heavily focused on procedures could be expanded to help students see connections between procedures and associated concepts. There is some evidence that such instruction exists in isolated schools or classrooms (e.g., Boaler & Staples, 2008; Chazan, 2000) or when teachers are involved in researcher-led initiatives (e.g., Rittle-Johnson & Star, 2009). My goal in this article is to examine, through a larger-scale, exploratory study of algebra teaching, which practices are occurring in algebra classrooms more broadly, highlighting those that might, in turn, be adjusted in an incremental process to enrich the learning opportunities for students. 3. Methods 3.1. Data The study relies on secondary analyses of data from the MET project, a U.S.-based, multi-year study that partnered with approximately 3000 teachers in grades four through nine across six metropolitan districts (Kane & Staiger, 2012). Teachers were recruited by project staff and volunteered to participate. Participating teachers were similar to non-participating teachers on demographics and years of experience and contributed up to four video-recorded lessons per year over the 2009e2011 school years. MET project personnel facilitated the video-recording of lessons using a stationary panoramic digital video camera operated remotely by teachers or school staff. Teachers uploaded their videos to a secure website (Bill & Melinda Gates Foundation, 2010a) to which I was granted access after the conclusion of the study. Teachers chose which lessons to record, although they were encouraged to choose at least two lessons that represent core topics in their subject area. I therefore do not claim that instruction is representative of all teachers in partner districts or of sample teachers’ typical practice. Ho and Kane (2013) analyzed a subset of MET video and found that teacher-chosen videos were rated higher on average than non-teacher chosen videos on observational measures of instructional quality. Thus, project video likely depicts higher quality instruction than randomly selected lessons would yield. However, the videos remain appropriate for my purposes because they illustrate aspects of instruction in U.S. algebra classrooms could be enriched with relatively small adjustments. 3.2. Sample For this study, I began with the available ninth grade algebra videos from 81 teachers across 49 schools from the first year of MET study across five of the partner districts.1 MET also administered a Content Knowledge for Teaching Algebra assessment (CKT) that

1 The MET Project includes six partner districts; however, one district did not include ninth grade math teachers as project participants. While there were more than 81 ninth grade algebra teachers in the project, MET gave participating teachers the option of allowing non-MET researchers access to their video for secondary analyses at the conclusion of the study. Approximately one third of teachers did not consent to make their videos available. For information regarding recruitment of teachers and the larger MET study, see Kane and Staiger (2012).

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focused on the specialized content knowledge that enables teachers to effectively teach algebra content to students (Bill & Melinda Gates Foundation, 2010b). I excluded teachers missing CKT scores because I used this measure for the purposes of sampling (see below). In Table 1, I show that this sample is largely similar in demographic composition to the full ninth grade sample with some differences in the percentage of teachers who were Black, white, or Latinx. Among teachers for whom CKT scores were available, the mean, median, and range of CKT scores were similar between those teachers with and without viewable video. I constructed a subsample for analysis by ranking the 81 teachers by their CKT scores and selecting a random stratified sample of 24 teachersdeight each from the highest, middle, and bottom quintiles. Past research has found that instructional quality varies sharply according to teachers’ mathematical knowledge (Charalambous & Hill, 2012), and by sampling in this manner I hoped to maximize the variability in algebra instruction included in the analysis, increasing the chances of finding illustrations of current practices that could be enriched with small changes. These 24 teachers contributed a total of 75 lessons for further study. 3.3. Data analysis The goal of this exploratory analysis was to develop descriptions of features of instruction in an algebra context that had potential to support student learning of algebraic ideas and be leveraged for incremental improvement efforts. Video data provide a number of analytic advantages for this work. As opposed to live observation, recorded lessons permit researchers to pause and re-watch classroom moments they may have missed. They also allow for iterative analytic processes in which themes from video analysis are refined and applied to new lessons (Hamre, Pianta, Mashburn, & Downer, 2007; Jacobs, Kawanaka, & Stigler, 1999; Learning Mathematics for Teaching Project, 2011). To analyze video, I worked with a research team comprised of myself and three additional math educators, all with algebra teaching experience (one of whom is also a mathematics education researcher). Together, we engaged in an iterative cycle of thematic analysis using open coding and informed by the literature (Corbin & Strauss, 2008), in which we watched video, identified and discussed emerging instructional features, wrote analytic memos, and watched and re-watched lesson video. To start this process, I

randomly selected a teacher from the subsample, randomly selected a video from that teacher, and randomly assigned that video to a member of the research team. Each researcher watched their assigned video in its entirety, with no knowledge of the teacher’s CKT score. Researchers wrote a brief lesson summary for each lesson, identifying the topic of the lesson, recording a narrative description of the lesson, and identifying mathematical strengths and weaknesses of the lesson. The goal was not consistency across raters, but rather to develop a narrative description of each lesson to inform further analysis and discussion. After watching their assigned video, researchers nominated segments of the video, ranging between 2 and 10 min in length, for the research group to watch together and study in further detail in order to ultimately create thematic categories. In early rounds of analysis, researchers noted the ubiquity of instruction around algebraic procedures in the sample. This led to an emphasis on segments of instruction that included explicit connections between procedures and concepts (Hiebert & Grouws, 2007). We also reviewed the literature on procedural knowledge to help us better understand what teachers were doing when they taught procedures and relied on the video to categorize how these practices were enacted. As the process progressed, researchers nominated segments that reflected other emergent findings. Once a segment was nominated for group discussion, each member of the research team independently watched the segment and recorded their observations. We met weekly to discuss a set of segments and after each meeting, I coded meeting notes for common themes and to generate an evolving list of practices that I brought back to the research team to discuss and refine further. We re-watched segments as necessary to clarify descriptions and applied emergent categories to the next round of researchernominated video segments to further hone descriptions, noting the ways that the instructional segment did or did not represent a particular category. For example, in developing the description of how teachers created an opportunity for students to give meaning to algebraic procedures, we discussed segments in which this opportunity was fully developed, segments in which it was only partially instantiated, and segments in which it was absent. The purpose here was not inter-rater agreement, but rather to develop detailed descriptions of categories of learning opportunities in lesson segments. In doing this, we prioritized instructional practices that the literature reviewed above suggested supported

Table 1 Baseline characteristics of sample ninth grade math teachers, year 1.

Teacher demographics % female % white % Black % Latinx % other race Mean years of experience (Range)a Mean years of experience in district (Range)a % master's degree or morea Teacher mathematical knowledge for teaching (CKT) Mean (sd) Median Min score Max score Range of scores a

All teachers (n ¼ 233)

Teachers with CKT Scores (n ¼ 141)

Teachers with CKT and video (n ¼ 81)

56.7 (8% missing) 52.8 27.9 5.2 6.0 7.5 (0e35) n ¼ 75 6.1 (0e35) n ¼ 185 27.9 n ¼ 139

57.5 (7% missing) 55.3 27.7 5.7 4.3 6.5 (0e33) n ¼ 46 5.9 (0e33) n ¼ 113 26.3 n ¼ 83

60.5 (5% missing) 65.4 22.2 6.2 1.2 6 (0e24) n ¼ 28 5.7 (0e24) n ¼ 63 32.1 n ¼ 56

N/A N/A N/A N/A N/A

61.7 (14.3) 62.9 22.9 97.1 74.2

62.7 (14.22) 62.9 22.9 97.1 74.2

Reported for teachers for whom this information was available.

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students learning opportunities in algebra, but relied on the segment videos to better understand aspects of these practices and the ways in which these practices were enacted. We repeated this iterative process weekly with a new set of nominated segments, applying emerging codes to video segments, clarifying themes, nominating new segments, and refining codes as necessary by re-watching relevant video and reviewing literature. Throughout this process, we viewed segments to both deepen our understanding of emergent learning opportunity types and to identify new types salient in the data. For example, while the literature suggested the importance of supporting students in making sense of algebraic abstractions, the coding of the lesson segments illustrated the specific ways in which teachers made connections between more and less familiar mathematical ideas to support students in understanding algebraic abstractions, generating and clarifying one of the categories presented in the findings below. As we began to feel we were approaching saturation, I analyzed the researcher-written lesson summaries for all 75 lessons and coded these for the types of learning opportunities refined in research group discussions and using open coding, taking care to note both convergence and divergence with the results from group discussions. Video from divergent cases were brought to the research team to further refine our descriptions. We continued to watch video until saturation, when we were confident that additional video from the sample was not adding further information to our descriptions (Corbin & Strauss, 2008; Guest, Bunce, & Johnson, 2006)dalthough we ensured that we had watched and discussed at least one lesson from each of the 24 teachers. 4. Findings The goal of the analysis was to understand and describe instruction with respect to the types of learning opportunities salient in lessons in this sample. While this process did not generate an exhaustive list of learning opportunities, the open coding structure allowed for a detailed picture of specific opportunities oriented toward algebra. Below, I describe five types of learning opportunities, illustrating each with vignettes from sample lessons. These types of learning opportunities represent the broader categories developed from the coding process described above and were selected to represent enacted instructional practices that have the potential to support students’ opportunity to learn in the algebra context specifically. I present these results not to make claims about their prevalence in the data, nor to present either an idealized or a typical description of instruction, but rather to describe types of learning opportunities that were salient.2 4.1. Giving meaning to algebraic procedures The teaching of procedures was ubiquitous in sample lessonsdfor example, teachers instructed students how to execute a particular algorithm, how to complete an algebraic manipulation, or how to employ a formula. In this context, some teachers created opportunities for students to make connections to underlying mathematical concepts and develop mathematical meaning around the procedures. One way in which teachers created opportunities for students to connect procedures to underlying concepts was by attending to the meaning of individual steps in a procedure. For example, in teaching students the procedure for solving systems of equations by substitution, a teacher explained why it was possible

2 For an analysis of how often and at what level of quality these practices occur across lessons in the data, see Litke (2020).

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to replace the y in one equation with the right-hand expression from the second equation, emphasizing the idea and property of equivalence. A second way that teachers did this was by attending to the meaning of the solution that resulted from a procedure. For example, after teaching students to solve a system of equations using the elimination method, a teacher underscored that the resulting x and y values represented the coordinates of the point of intersection of the graphs represented by the two linear equations, and thus the two values that make both equations true. Finally, teachers helped students develop the meaning behind procedures by pointing to the mathematical properties underlying a given procedure. For example, when teaching students how to multiply two binomials, a teacher underscored that the multiplication used in the mnemonic FOIL, was really a variant of the distributive property. When teachers did not provide students learning opportunities to create meaning for procedures, instruction focused on learning procedures as recipes to be followed. For example, in one such lesson, a teacher introduced the procedure for graphing linear inequalities, telling the students to first graph the inequality as if it were a linear equation. Next, students were instructed that the inequality symbol determined whether they should use a dashed or solid line. Finally, the teacher instructed students to select a test point and plug that point into the inequality for x and y. The teacher told students that if the inequality remained true, they should shade the side of the line that included the test point and if it were false they should shade the other side. While this description of the procedure is clear and correct, simply listing the steps does not connect the procedure to the underlying mathematical ideas and does not provide students the opportunity to make conceptual sense of linear inequalities or connect them to linear equations. In contrast, to illustrate what learning opportunities look like when they support making meaning of procedures, I present a brief vignette from a lesson on linear inequalities taught by a different teacher. Here, the teacher introduced the procedure for graphing inequalities with an example: x þ y < 2. The teacher first asked the students to graph the line x þ y ¼ 2: Next, the teacher directed students to choose a coordinate pair and plug it into the original inequality to determine whether their particular point satisfied the inequality. In sharing the points they had chosen, students noticed that points located below and to the left of the graphed line satisfied the inequality and continued to suggest points in that area, introducing the idea of the shaded region. One student asked if the reason for shading the graph was to show that many points make the inequality true. The teacher responded by highlighting this idea and elaborated: We’re talking about not a line, but a whole regiondan area where the points x þ y are less than two. As a matter of fact, all of the points below this line and to the left … all of these points [draws multiple points in the region] satisfy the inequality x þ y < 2 … So when we graph an inequality, we end up with a whole region or area and what we do is shade that area. We’re not just talking about points on a line. We’re talking about a whole region, or area, where all the points fit the inequality. And we shade that whole area. In this exchange, the teacher explained that the shaded region was the solution to the inequality, emphasizing that it was comprised of all points that made the inequality true. Next, the teacher turned to the boundary line of the shaded region at x þ y ¼ 2, explaining that if the inequality had been x þ y  2, it would be inclusive of the line x þ y ¼ 2 because it would contain the equality. The teacher used this idea to explain why x þ y < 2 should be graphed with a dashed (as opposed to solid) line, as it does not

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contain the equality. The conceptual aspect of this learning opportunity was created as the teacher attended both to the meaning of individual steps of the procedure and the solution generated by the procedure.

4.2. Moving toward procedural flexibility In addition to developing conceptual understanding of algebra through examining the meaning of procedures, algebraic understanding also can be developed by focusing on the flexibility with which procedures can be used (Star, 2005). In this sample, teachers created learning opportunities that had potential to support the development of students’ flexibility both within and across procedures. Focusing within a single algebraic procedure, teachers noted multiple pathways through the procedure. For example, in a lesson on simplifying rational expressions with radicals, students were shown they could either rationalize the denominator at the outset of the problem or simplify other aspects of the expression first and then attend to the radicals. The teacher demonstrated that both pathways produced the same solution. Teachers also attended to key conditions for proceduresdnoting what must be true for a particular procedural decision to be made. For example, in reviewing the procedure for solving a system of equations using elimination, a teacher emphasized the role of the sign of the coefficients in the two equations, explaining that when a variable appears in both equations with the same coefficients but with opposite signs (e.g., þ3x and  3x), it is possible to add the equations together to eliminate the variable. However, when both the coefficients and their signs are the same (e.g., þ 3x and þ 3x), students needed to make a choice to either subtract the two equations or multiply one equation by negative one and then add the equations together. With respect to flexibility across procedures, students in some lessons examined when one particular procedure might apply over another or explicitly compared multiple procedures. For example, in one lesson on graphing linear equations, a teacher highlighted how the form of the given equation might lead students to choose one method for graphing over another. For an equation written in standard form (Ax þ By ¼ C), finding the x- and y-intercepts might be the most efficient method for graphing. However, if the equation were written in slope intercept form (y ¼ mx þ b), graphing from the x- and y-intercepts would be less efficient, requiring additional algebraic manipulation. Teachers also supported flexibility by comparing multiple procedures for their affordances and/or limitations. For example, after teaching students the procedure for solving a system of equations by substitution, one teacher reminded students that they had recently solved similar systems by graphing, noting that while graphing might be easier or more efficient for some problems, in other cases the algebraic method may be a better fit. When attention to flexibility was absent from students’ learning opportunities, teachers presented procedures as absolute rules with little room for choice or consideration of applicability. For example, in presenting a lesson on graphing linear equations, one teacher told students that they “have to first make sure your equation is in standard form before you graph it.” While the stated goal of this particular lesson was graphing from equations in standard form, the idea that there were other procedures for graphing a line was not only omitted, but students were repeatedly told to manipulate equations that could more easily be graphed from other forms (e.g., using point slope form). This type of rigidity was characterized by a lack of discussion or exploration of alternative methods or solution pathways. To illustrate further the learning opportunities that did support

procedural flexibility, I present a vignette from a lesson on solving systems of equations using elimination. Here, the teacher presented a system of equations (4xe4y ¼ 8 and  8x þ y ¼ 19) with the goal of teaching the procedure for multiplying one equation by a constant to allow for the elimination of one of the variables. The teacher demonstrated this procedure to students, recording the work on the board (see Fig. 1). In this example, the procedure was presented in a relatively straightforward manner: multiply both sides of the second equation by four; add the two equations in order to eliminate y; solve for x; substitute the value for x into the first equation in order to solve for y. Once the teacher finished solving for x and y, a student suggested that there was another way to solve the problem. The teacher responded: Teacher: You can, which way did you do it? Student: I multiplied the top line by two. Teacher: You multiplied the top one by two so you could get positive eight and negative eight. It doesn’t matter which way. You can choose. Okay, you can choose which way you want to do it … Now, you could have, if you wanted to, you could have used the substitution method. You could have moved this [negative] 8x to the other side and y would have been equal to positive 8x plus 19. Then you would substitute it in. So you have options. Here, the teacher explicitly reinforced two possible pathways through the same procedure (in this case, multiplying the bottom equation by four as originally done or multiplying the top equation by two as the student suggested). The teacher also emphasized how students might have solved the problem using a different method entirelydsubstitutiondreinforcing that with some algebraic manipulation this was a viable alternative that yielded the same solution. The teacher went on to remind students that they could also have considered using the graphing method, graphing each equation and finding their point of intersection. The teacher cautioned students that the graphing method might not be the best method in all situations, highlighting examples where the solution did not contain integer values. In this segment of instruction, the teacher created a richer learning opportunity for students by focusing not only on teaching a particular procedure, but also on cueing students to the characteristics of the problem that motivated the selection of that method over others, while still emphasizing multiple approaches to the same problem.

4.3. Making connections across representations A third type of learning opportunity resulted from connecting multiple representations of algebraic ideas. Algebraic relationships can be expressed using multiple representational forms, with graphs, tables, contexts, and equations being prominent. Teachers in this sample created opportunities for students to connect these representations, both within and across problems. They also made connections across representations within families of functions, for example, examining how changes in an equation resulted in changes in the graphical representation. It is important to note that there is some overlap between this type of opportunity and those discussed previously because connecting across representations sometimes occurred in the context of teaching algebraic procedures (such as how to solve a system of linear equations). Indeed, these types of connections hold the potential for creating learning opportunities for students to integrate concepts with procedures. However, this opportunity also was created when procedures were not taught (e.g., in a discussion of what it means for a relationship

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Fig. 1. System of equations example solved by multiplying the bottom equation by four, using the elimination method to solve for x, and substituting to solve for y.

to be linear) and could contribute enough to students’ understanding of algebraic ideas (Chazan & Yerulshlamy, 2003; Moschkovich et al., 1993) that I present it as a distinct learning opportunity. In some cases, the opportunity to make connections across representations was not offered to students. This occurred when teachers presented only a single representation or displayed multiple representations but did nothing to support students in making connections across them. For example, one teacher solved a system of linear equations in two variables algebraically using the substitution method. The teacher walked students through the algebraic manipulation necessary to solve the system and the class determined the correct values for x and y. However, the teacher did not relate the process or the solution to the graph of the system in any way; treating it instead as an algebraic problem to be solved algorithmically. This example is particularly interesting because the algebraic solution represents the coordinates of the point of intersection of the two lines comprising the system. To illustrate the ways in which teachers create opportunities for students to connect representations, I present two vignettes. The first is from a lesson on graphing linear equations. In this lesson, the teacher reinforced the relationships among the equation, table, and graph of a single linear function. The teacher recorded the three representations side-by-side. Pointing to the table, the teacher explained to students that the numbers in the table generated from the equation provided the coordinates for the graph of the line and were also solutions to the algebraic equation. The teacher then indicated the locations on the graph of the coordinate pairs listed in the table, pointing between the two representations and emphasizing, “every point on this line has a corresponding x- and y-coordinate.” Next, the teacher pointed between the graph and the equation and said, “Any point on this line, the corresponding x- and y-coordinate will fit into that equation and make it work.” Finally, the teacher gestured between the table of values, the equation, and the graph and said, “This is the solution set to that linear equation. A linear equation graphs out as a … straight line.” This type of explicit connection may provide students the opportunity to give mathematical meaning to linear equations and to navigate a potential point of confusion around form and structure of linear equations. A second vignette illustrates the way teachers could create opportunities for students to see how changes in a symbolic representation of functions are reflected in graphical representations. In a lesson on graphing linear equations, the students used hand-held graphing calculators while the teacher’s calculator screen was projected onto a large monitor at the front of the room. The teacher first asked students to graph y ¼ x using their calculators. After discussing what that graph looked like, the teacher asked, “If I want the graph to get steeper, what am I going to do to the x? If I want a

steeper slope to what I have, what should I put in?” Students offered a number of suggestions and the class settled on y ¼ 3x. The teacher then asked students to graph this on their graphing calculators. Displaying the graphs of y ¼ x and y ¼ 3x simultaneously, the teacher reinforced that the coefficient of the x determines the steepness of the line. Next, the teacher connected the y-intercept of the graph to the equation: Teacher: Let’s say this time that I want my next line to have the same slope but this time I want it to go through this fourth number here. [Points at 4 on the y-axis]. What am I going to put in there to make it go through this fourth number right here, this fourth dot? What am I going to put in here? To make it the same slope but to go through positive 4? Student: 4x? Teacher: 4x is just going to make it steeper. I can’t mess with the x. I’m trying to make it go through here. Same slope, just going through this point. Student: x negative 4. Teacher: Put x negative 4? Try x negative four. Put xe4 and see what happens. Hit graph. You had the right idea but it went through 4. If you put xe4 and it goes through 4 what do you think I have to put to make it go throughd Student: x þ 4 In this vignette and throughout this lesson, the teacher and students graphed each student suggestion, examining the relationship between each equation and the corresponding graph to develop students’ intuition about what the numbers in linear equations represented and how they transformed the graph. In each example, the teacher helped students connect representational forms explicitly, drawing students’ attention to the relevant changes (e.g., pointing to the graph and stating “it goes through 4”). 4.4. Building connections across topics A fourth type of learning opportunity that could enrich students’ conceptual understanding occurred when teachers connected the mathematics in a specific lesson to other topics in the algebra curriculum or to the broader domain of mathematics. One way teachers did this was to connect the day’s lesson explicitly to earlier content, linking the mathematics under study to topics the class had previously studied. In one such lesson, the teacher introduced the procedure for solving systems of linear equations using elimination and then paused after eliminating one variable, remarking,

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“Now that we have eliminated one variable, this equation looks like the single-variable algebra equations we already know how to solve.” With this short remark, the teacher encouraged students to notice that the process for solving problems with two variables relies on manipulating the equations such that they resemble problems they had already learned how to solve. Simplifying complex problems to those that are easier to solve using previously learned strategies is an important insight enabled by algebraic thinking. Teachers also encouraged students to connect current content to future topics of study, often by previewing how the day’s lesson connected to later topics in the curriculum. For example, after working on plotting points from an equation onto the coordinate plane in order to generate the graph of a line, one teacher commented to the class how this exercise set them up for future units of study, particularly the connection between graphing points and graphing lines saying: “After this topic, we’re going to go into slope. So, what you’re learning right now is the beginning of slope … You [will] have an equation, you give values, you graph it … That’s the next topic we’re going to do. This is the introduction to slope.” In doing this, the teacher connected the day’s lesson to future topics in the course, developing a thread that tied together multiple topics. This type of learning opportunity was absent when teachers presented material as discrete, compartmentalized, and disconnected from the broader domain of algebra. For example, in a lesson on trigonometric ratios, the teacher began with a warm-up in which students were asked to identify information from right triangles drawn and labeled on the board (e.g., “How long is the side opposite angle R? Which side is adjacent to angle A? What is the length of the side opposite angle R divided by the length of the hypotenuse?). After recording students’ answers to the final question, the teacher immediately transitioned to presenting three trigonometric ratios (sine, cosine, and tangent) on an overhead projector transparency. The teacher made no connection to how what the students had done earlier in the lesson reflected a trigonometric ratio or to the role of the right triangle in generating these ratios. Students used the trigonometric ratios to find the missing side lengths of triangles, but were given no opportunity to see how this part of the lesson fit into earlier parts of the lesson or into the larger mathematical storyline. To illustrate further how teachers created learning opportunities for students to connect mathematical topics, I present a vignette from a lesson in which a teacher introduced systems of linear inequalities. The lesson began with a series of warm-up problems that reviewed previously learned material. The teacher asked students to graph two linear equations: y ¼ 2x þ 4 and y ¼  3x þ 2. The teacher paused to remind students that in previous lessons, they had used their knowledge of graphing lines to solve systems of linear equations and that in this lesson they would use similar ideas to solve systems of linear inequalities. The teacher reviewed the solutions to the warm-up problems and reiterated this connection, stating: You need to know how to graph a linear equation … [The curriculum] gave you like three or four different formulas you have to use [for graphing]. Point slope formula and then they wanted you to graph it. They introduced slope intercept form y ¼ mxþ b and they wanted you to graph it. The next step that they gave you was a system. A system is when you had two math problems put together on the same graph. They told you if you have two problems on one graph you’ve got to find out where they cross, you’ve got to find out one solution. After reviewing the arc of the linear equations unit, the teacher built upon the warm-up, using the two graphed linear equations to

demonstrate that together, they create a system of equations and emphasizing that their point of intersection represented the solution to that system. Next, the teacher explicitly connected the warm-up problem to the day’s objective: What we’re going to talk about today is not only having two problems on one graph, but we’re also going to change the equal sign to an inequality. So our lesson for today is actually putting two problems on a graph [points to graphed solution of system of linear equations] where both problems … are not equal [points to equals sign in the system] but are either greater than or equal to or less than or equal to. In this vignette, students were encouraged to activate prior knowledge (graphing linear equations), connect that knowledge to a new topic (solving systems of linear inequalities), and link the sections of the lesson together. As a result, the students were offered opportunities to develop clear connections between solving a system of linear equations and graphing and solving systems of inequalities. 4.5. Making connections explicit between more and less familiar mathematical ideas A final type of learning opportunity that was salient among lessons in this sample occurred when teachers connected concepts, definitions, and symbolic formulas to more familiar examples, representations, or ideas. One of the main reasons students struggle with algebra is that it is more abstract than the mathematics they have previously encountered (Booth, 1988; Hiebert & Grouws, 2007; Rakes et al., 2010). Instruction that encourages students to connect algebraic abstractions to numerical examples, concrete representations, or more familiar ideas may provide opportunities for students to build their understanding. One way that teachers created these opportunities was by using numerical examples to aid in the development of algebraic rules and properties. For example, in one lesson on the properties of exponents, one teacher worked with students to develop the rule for simplifying expressions with negative exponents. The teacher first wrote on the board a series of consecutive numerical exponentiation problems (e.g., 34, 33, 32, and 31) and asked students to simplify each expression. Students were next asked to use their answers to extend the pattern and solve 30, 31, and 32, and together, the class used the pattern to develop the generalized rule that xn ¼ x1n . While this approach does not explain why this rule holds, it does allow students to ground this rule in the familiar number system. When this type of learning opportunity was absent, teachers presented an algebraic rule, formula, or process with no connections to familiar ideas or to concrete or numeric underpinnings. For example, in a lesson on simplifying expressions with exponents, a teacher told students that there were a series of rules to follow depending on the format of the expression. The teacher wrote each m rule on the board (e.g., xm ,xn ¼ xmþn , xxn ¼ xmn , etc.), and then 2 5
asked students to simplify an expression for each rule (e.g., xxyy3 : Students completed these problems, but were given no opportunity to differentiate across the rules or make meaning of why they worked. To illustrate instruction in which such opportunities were offered to students, I present a short example from a different lesson in which students simplified rational expressions using the properties of exponents. During the lesson, the teacher asked stu5 4

dents to simplify the expression xx3 yy2 , but students struggled to do so

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correctly. To assist students in understanding the process for simplifying such expressions, the teacher wrote 55 and 22 on the board and asked students to simplify these fractions, which they did with apparent ease. The teacher returned to the algebraic example, writing it in expanded form as x,x,x,x,x,y,y,y,y and asked x,x,x,y,y students how they could use what they knew about simplifying

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and 22 to help them simplify the algebraic expression. The teacher explained to students that just as any number divided by itself equaled one, an unknown divided by itself also equaled one. This connection could help students understand the “cancelling out” of the appropriate x’s and y’s in the process of simplification. Teachers also created opportunities for students to understand algebraic concepts and processes by using analogies. To illustrate how this created learning opportunities for students to make sense of abstract algebraic ideas, I present a vignette from a lesson on solving systems of equations using substitution. Here, the teacher used an analogy of exchanging a dollar for an equivalent amount of change to help students make sense of the idea of algebraic substitution: Teacher: If I have a dollar bill. Let’s pretend. And Student A says, “Hey, can I get some change?” and I give him four quarters, is it okay?

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had the same value. The teacher emphasized how, just like with the four quarters and ten dimes, substituting one for the other did not change the value. Later in the lesson, the analogy resurfaced when a student expressed confusion about where the components of the rewritten equation came from. The teacher highlighted the process of substitution in the example, pointing simultaneously at both x þ 2y ¼ 10 and x þ 2ð2xe3Þ ¼ 10 and traveling along each equation: Teacher: Lookdx [points to first equation with left hand], x [points to second equation with right hand], plus [points to first equation], plus [points to second equation], 2 [points to first equation], 2 [points to second equation]. And instead of giving you the dollar [points to y in x þ 2y ¼ 10], I gave you the coins [points to ð2xe3Þ in x þ 2ð2xe3Þ ¼ 10]. Many students: Ohhhhhh. Teacher: I gave you the change. Instead of giving you the dollar here, I gave you the four quarters. In returning to the analogy, the teacher emphasized that the substitution was permissible because the expression 2x  3 had the “same value” as y, highlighting equivalence while grounding this concept in a familiar idea for the students.

Students: Yeah. Teacher: [Draws a dollar bill and then draws four circles next to it to represent quarters] Yes, right. It’s the same thing. Now, if you say, “Can you change my dollar?” And I give you ten dimes, it’s okay right? It’s the same. [Draws a dollar bill equal to ten smaller circles representing dimes] Students: Yeah. Teacher: You see how those are exactly the same? Now, could I say this? [Erases dollar bill in both examples and replaces them each with a y]. Could I say this? Students: Yeah. Teacher: It’s exactly the same. Now, could I say this? If I give you four quarters it’s the same thing as giving youd Student: Ten dimes. Teacher: Ten dimes. So, you can conclude because this [points to four quarters] is equal to y and this [points to ten dimes] is equal to y. Student: They are equal to each other. Teacher: Exactly, these two are equal to each other. After presenting this analogy, the teacher wrote a system of two linear equations on the board: y ¼ 2xe3 and x þ 2y ¼ 10. The teacher returned to the idea of dollars and change to demonstrate using the algebraic process of substitution to solve a linear system of this type: So here’s my dollar [points to the y in [y ¼ 2xe3] and here’s my change [circles 2xe3], it’s 2xe3. So now, if I replace everywhere I have a y [with 2xe3], I get rid of the y variable. So, your first step is to replace anywhere you see a y with the change. So, you get x þ 2ð2xe3Þ ¼ 10. The teacher used the idea of exchanging a dollar for its equivalence in coins to explain algebraic substitution, saying that it was possible to substitute the y in the second equation with the 2xe3 from the first equation because, like with the dollar and coins, they

5. Discussion and implications Contemporary descriptions of mathematics instruction in many countries generally note the absence of reform-oriented practices and a persistence of teacher-centered instruction (e.g., Hiebert et al., 2005; Kane & Staiger, 2012; Prendergast & Treacy, 2018). This is true despite efforts aimed at developing teachers’ practice to include more ambitious mathematics instruction. Some have argued that connecting improvement efforts more closely to teachers’ extant practice may allow for a bridge to the types of student-centered, reform-oriented practices the field wishes to see (Janssen et al., 2013; Star, 2016). If contemporary algebra instruction is largely teacher-centered and focused on the teaching of procedures, as it appears to be in the U.S. context (e.g., Litke, 2020), then understanding how teachers might better support students’ learning opportunities in that context could serve as such a bridge. In this study, I identified learning opportunities that could help students integrate conceptual understanding with learning procedures and that could, with small adjustments, yield even richer opportunities. By small adjustments, I mean modifications to teaching that are largely consistent with what teachers are already doing. I am not suggesting the field abandon the larger push for more ambitious practices. Rather, the approach I am describing represents an acknowledgement that, in the context of classrooms not already featuring mathematical discussion, productive struggle, or high cognitive demand tasks, asking teachers to make smaller adjustments might be a way to bridge the gap between the instruction we do see and that which we wish to see. Simply because an adjustment may be small does not imply that change will be easy. But by focusing on smaller changes that deepen the learning opportunities provided by existing practices, teachers and administrators might find instructional improvement to be more practical (Janssen et al., 2013) and therefore more likely to be adopted. To more fully understand this approach, consider the relationship between procedures and concepts in the context of algebra. Procedures are ubiquitous in algebra and procedural knowledge is an important component of algebraic understanding (Kieran, 2013; Star et al., 2015). Creating learning opportunities that focus only on conceptual understanding might be counterproductive, having the unintended effect of turning attention away from high quality

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teaching of procedures. An alternative strategy might be to begin where teachers aredteaching proceduresdand focus on helping them expand their teaching to include conceptual ideas, using this to move them toward more ambitious practices. For this approach to work, teachers will need guidance on how to do this work (Cohen, 2018). Practices that support learning opportunities like those identified in this study represent possible examples of such guidance. For example, enhancing the teaching of procedures by embedding meaning is consistent with contemporary thinking on the interweaving of concepts with procedures in algebra (Kieran, 2013; Star, 2005). Teaching flexibility and efficiency in the use and application of procedures may also be important to build procedural and conceptual knowledge (Star, 2005; Star & Rittle-Johnson, 2009). Small adjustments might focus on aspects of these practices, for example supporting teachers in emphasizing connections to underlying concepts, noting key decision points within a procedure, or making connections and comparisons across multiple procedures. To illustrate how an incremental improvement approach grounded in the types of learning opportunities described here might bridge to more ambitious practices, consider a hypothetical teacher teaching a lesson on how to solve a system of equations using the elimination method. This hypothetical teacher directs the presentation of the content and presents the procedure for solving a system of equations using elimination in a step-by-step fashion with minimal student participation and little attention to the concepts underlying the procedure. For example, the teacher might instruct students to first place one equation above the other equation. Next, the teacher might explain that it is not possible to solve for two variables simultaneously, and in the elimination method, students will add the two equations together to eliminate one of the variables. The teacher might caution students that to do this, the coefficients of the variable they plan to eliminate must be the same in both equations, but with opposite signs. The teacher might then demonstrate the next stepdadding the two equations together, eliminating one of the variables (x) and solving for the other (y). Finally, the teacher might instruct students to plug in the answer they determined for y into one of the original equations to then solve for x. After reminding the students to express their answer as an ordered pair, the teacher might encourage the students to check their work by plugging in the values they found for x and y into the original equations. In an incremental improvement approach building on the learning opportunities created by the instruction described above, a coach, administrator, or professional development facilitator might encourage the teacher to include an explanation of why adding the two equations is possible or why the coefficients might need to be the same but with different signs. The teacher might also be coached to emphasize the purpose of the procedure (e.g., finding the point of intersection of the two lines), making connections to the graphical representation of the system (e.g., showing the graphical solution and connecting the equations to their graphs). These adjustments are relatively consistent with a more teachercentered approach, but would move toward enriching the learning opportunities in small ways, allowing students to see connections to underlying concepts. From here, the teacher might be supported to go more deeply in supporting procedural flexibility by exploring with students both adding and subtracting the two equations, comparing these methods for their affordances and limitations. Alternatively, the teacher might connect the procedure for solving the system using elimination to other methods for solving the same system (e.g., substitution or graphing). Comparing multiple solution methods in this way might then provide the instructional conditions necessary to seed a mathematical

discussion or facilitate instruction that is higher in student cognitive demand. This scenario focuses on deepening the opportunities for sensemaking of algebraic procedures, supporting procedural flexibility, and connecting across representations, demonstrating an approach that could accumulate incremental improvements that ultimately bridge to more ambitious teaching. There are reasons to believe that such an approach would be effective. Decomposing teaching into intermediate grain-size practices such as these and supporting teachers to improve in their enactment, may help particular aspects of practice to become more automatic for teachers. This, in turn, allows teachers to focus on more non-routine or in-depth practices (Janssen et al., 2015). In addition, research by Janssen et al. (2013) found that initial changes connected to existing practices were generative for some teachers, leading to exploration of deeper changes in practice. Before concluding, I note that while the particular learning opportunities described in this study are aligned with theories around algebra teaching and learning, they are not exhaustive. They represent a subset of the learning opportunities that contribute to students’ conceptual understanding and that could be enriched in this way. Additional research with classrooms that provide more student-centered learning opportunities might expand the types of opportunities available for enrichment. Such research could also expand our understanding of the kinds of small adjustments in teaching that could accomplish this goal. In addition, future studies should consider the relationship between particular learning opportunities and student learning outcomes. Work with teachers might then focus on those instructional practices that best support the creation of these opportunities. Other research in practicebased teacher education settings might consider whether and how descriptions of these learning opportunities could be used with novice teachers to identify and support the development of these practices. Mathematics education researchers emphasize the need for ambitious learning opportunities that allow students to engage in high cognitive demand tasks as well as engage in mathematical discourse and analysis (e.g., Jackson, Garrison, Wilson, Gibbons, & Shahan, 2013). The results presented here and elsewhere did not uncover evidence of these learning opportunities and the instructional practices that create them (e.g., Litke, 2020). On the one hand, this may be a limitation of the datadfor example, the MET project did not provide consistent information regarding the tasks on which students worked, inhibiting a systematic analysis of task cognitive demand. However, it may also be the case that these aspects of instruction were not salient in this sample. If this is the case, it underscores the need to focus on learning opportunities that already occur and that could be deepened in the service of working toward more ambitious changes. I note that the study sample is non-random and not nationally representative, and as such, I cannot say the extent to which these findings generalize beyond classrooms in the sample. However, one contribution of this study is to name and describe existing learning opportunities that could be deepened in small ways to yield richer student learning. Reconceptualizing improvement as the accumulation of incremental changes (Star, 2016) requires an understanding of which aspects of instruction might be the focus of such efforts. I hope that the results of this analysis can provide insight into learning opportunities and associated teaching practices in algebra in ways that will be of use to both researchers engaged in work with more representative samples, and to practitioners engaged in the work of instructional improvement. In addition, by focusing on the types of practices teachers are already engaging in, researchers can work with teachers as co-producers of improvement efforts (e.g., Kieran et al., 2012).

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Instructional improvement at scale is complex and difficultdthe nature of teaching is culturally engrained and has proven resistant to change (Cohen, 2011; Gallimore, 1996; Hiebert, 2013; Stigler & Hiebert, 2018). Indeed, there is evidence across disciplines that attempts at large scale reform result in instructional changes that are simply incorporated into teachers’ existing schema, with little substantive changes in learning opportunities for students (e.g., Cohen, 1990; Cuban, 2013). The results of this study provide evidence of specific, discipline-oriented practices that teachers may already be engaging in that can be deepened to improve student learning opportunities in algebra and serve as a potential bridge to more ambitious teaching. A clearer understanding of how these opportunities are created in the context of contemporary teaching provides insights into possible leverage points for instructional improvement. Declaration of competing interest None. CRediT authorship contribution statement Erica Litke: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing. Acknowledgements This article stems from research conducted for my doctoral dissertation at the Harvard Graduate School of Education. This research did not receive any specific grant funding. I am grateful to Heather C. Hill, Jon Star, and David K. Cohen for their support and guidance in the early stages of this research. I wish to thank Johanna Barmore, Samantha Booth, and Delaina Martin for research assistance and ICPSR at the University of Michigan for facilitating data access. Thank you to Jim Hiebert for invaluable guidance on earlier drafts of this manuscript and to the anonymous reviewers for supportive and helpful feedback. References Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school though college. Washington, DC: United States Department of Education. Agudelo-Valderrama, C., Clarke, B., & Bishop, A. J. (2007). Explanations of attitudes to change: Colombian mathematics teachers’ conceptions of the crucial determinants of their teaching practices of beginning algebra. Journal of Mathematics Teacher Education, 10(2), 69e93. Bill & Melinda Gates Foundation. (2010a). Classroom observations and the MET project. Author. Retrieved from www.metproject.org. Bill & Melinda Gates Foundation. (2010b). Content knowledge for teaching and the MET project. Author. Retrieved from www.metproject.org. Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education, 33(4), 239e258. Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. Teachers College Record, 110(3), 608e645. Booth, J. L., McGinn, K. M., Barbieri, C., Begolli, K. N., Chang, B., Miller-Cotto, D., … Davenport, J. L. (2017). Evidence for cognitive science principles that impact learning in mathematics. In D. C. Geary, D. B. Berch, R. Ochsendorf, & K. M. Koepke (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts (Vol. 3, pp. 297e325). Cambridge, MA: Elsevier/Academic Press. Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford, & A. P. Shulte (Eds.), The ideas of algebra, Ke12. 1988 yearbook of the national Council of teachers of mathematics (pp. 299e306). Reston, VA: National Council of Teachers of Mathematics. Charalambous, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship. Journal of Curriculum Studies, 44(4), 443e466. Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom. New York, NY: Teachers College Press. Chazan, D. (2008). The shifting landscape of school algebra in the United States. In

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Please cite this article as: Litke, E., Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach, Teaching and Teacher Education, https://doi.org/10.1016/j.tate.2020.103030