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Developing knowledge for teaching from experience: Mathematics teaching and professional development in the United States of America Erik Jacobson

In this chapter, I draw on data from a larger project (Izsák et al. 2010; Orrill and Brown 2012) that investigated mathematics teacher learning in the context of a professional development (PD) course, focused on content knowledge for teaching. After reviewing literature on teachers’ professional knowledge and learning, I present contrasting cases of two teachers with similar backgrounds and teaching experience who attended the course. These teachers both had high knowledge scores on the PD course post-test, but observational data collected after the course revealed unexpected limitations when they taught lessons that they had designed and that applied what they had learned in the PD course. The resulting opportunities for student learning were quite divergent. These findings suggest two questions that I explore in the next sections of the chapter. First, how could teachers with such similar PD course post-test scores – ostensibly a measure of their learning – have such different enacted knowledge? To address this question, I analyze these teachers’ responses to individual items on the knowledge test and argue that their similar overall scores in fact masked fine-grained differences in their actual PD learning outcomes that had consequences for the observed instruction. These results illustrate how conclusions about teacher learning in professional development experiences that are based on single-score assessments may mislead. The second question is this. How could two teachers with such similar teaching experience and backgrounds learn such different things from the same PD course? Both teachers in this study followed the same route to licensure;

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they both were in their second year of teaching; and they both taught middle grades mathematics in single-subject classes using curriculum materials from the same publisher. To address the second question, I present a narrative analysis of the teachers’ own stories of learning to teach and argue that the PD course was a learning experience that these teachers saw differently because they had different goals for professional improvement, goals shaped in part by their professional relationships in the different schools and districts where they worked. Thus, the same amount of prior teaching experience was qualitatively different in ways that influenced these teachers’ learning in the PD course. In the final section, I discuss implications of these results for in-service teacher education and for assessing teacher learning.

Teachers’ professional knowledge and the PD course content Until the last decade, large-scale empirical support for the claim that teachers use professional knowledge that goes beyond knowledge of the discipline they teach had been scarce, despite strong arguments (e.g. Fennema and Franke 1992) and compelling case studies (e.g. Ball 1990). Researchers have tried for almost a century to link mathematics teacher knowledge with student learning (see Hill et al. 2007, for a review). The most promising recent work to establish such links has built on Shulman’s (1986) notion of pedagogical content knowledge (PCK), defined as an amalgam of content knowledge and pedagogical knowledge that includes, for example, pedagogically useful representations of disciplinary concepts. In mathematics education, researchers have identified examples of content knowledge that is rarely used except by teachers, such as the mathematical analysis of non-standard student work. The concept carries over to teaching other school subjects; for example, English language teachers must engage in literary analysis of their students’ compositions, although by different means and to different ends than those of the professional critic or academic scholar. The term content knowledge for teaching describes both content knowledge and PCK used in all aspects of teachers’ work (Ball et al. 2008). Success in identifying and measuring PCK and the broader domain of content knowledge for teaching has enabled researchers to establish empirical relationships between teachers’ knowledge and student achievement (e.g. Baumert et al. 2010, Hill et al. 2005). In the wake of this progress, measures of content knowledge for teaching are being developed by the Educational Testing

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Service and may soon be piloted on teacher licensure tests in the United States (Phelps and Gitomer 2012). However, little is known about how teachers learn content knowledge for teaching. What it means to learn and how learning occurs both depend on the nature of the knowledge that is learned. As Petrou and Goulding (2011) observe, the Ball et al. (2008) framework of teachers’ knowledge and related measurement development work (e.g., Hill et al. 2004) is primarily concerned with teachers’ individual cognition, although it is also clearly meant as a description of applied knowledge grounded in the practice of teaching rather than a description primarily in terms of mental schema. To understand how content knowledge for teaching might be learned from experience, it is useful to extend explanations of individual minds to include teachers’ broader social and cultural contexts. One important aspect of content knowledge for teaching is knowledge of common student conceptions and misconceptions about mathematics. This kind of knowledge could be captured in propositional statements (gleaned from case studies or survey research, perhaps), yet it is more likely to be learned through teachers’ (social) interactions with students around the tasks and activity of school mathematics. This learning is also likely to be mediated by interactions with other, more experienced teachers. A novice teacher might share a student comment or error he or she finds peculiar with a more experienced colleague, who would reinterpret the experience for the novice as a case of something expected, something that teachers know about students. In this sense, learning to teach is tied up with acquiring the professional identity of teacher in a community of practice (Stein et al. 1998; see also Olsen in this volume). Another kind of content knowledge for teaching supports teachers in choosing and deploying appropriate representations and tasks. Propositional accounts of this knowledge are unfeasible because of the sheer number of tasks and representations, especially in the US, where decisions about curricular resources are made at the local level and multiple publishers vie for market share. Moreover, given the diverse communities served by US public education, the story or situation described in a task may be appropriate for some students but inaccessible for others, and knowledgeable teachers in specific schools may have rephrased the tasks for their students in unique ways. It may be useful to consider this knowledge at the school level, rather than the teacher level, as situated, and belonging to a collective practice as well as located within individuals. Teacher learning of this knowledge can then be understood by means of participation in a community that includes (via the history of

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cultural artefacts – the tasks and representations) members widespread in time and space (Ellis 2009, Lave and Wenger 1991). The PD course on which this chapter is based was focused on helping teachers develop content knowledge for teaching mathematics, and in particular focused on drawn representations of rational numbers (Izsák et al. 2010, 2012, Lee et al. 2011; Orrill and Brown 2012). One example of content knowledge for teaching with special relevance for this chapter is knowledge of the ‘referent unit’ for fractions; that is, the whole to which each fraction refers. Explicit attention to referent unit is critical for working with and representing fractions and percentages. However, the referent unit is often implicit in the performance of expert users of mathematics, and appropriate attention to referent units can be a challenge for teachers (e.g., Izsák et al. 2010, Lee et al. 2011). The PD course emphasized explicit attention to the referent unit and the diversity of student thinking on the topic.

The participants and the observed lessons On the face of it, teaching experience seems a plausible opportunity for learning content knowledge for teaching. This is especially true for alternatively certified teachers who often have little preparation before beginning to teach and who must learn to teach on-the-job. Thus these teachers offer a particularly promising population for understanding how teaching experience can support the learning of content knowledge for teaching. In the present study, both participants were alternative-route middle grades teachers. They were similar to each other in many characteristics that policy research uses to classify teachers. Diane had majored in psychology as an undergraduate and had worked for seven years in marketing; Brian had majored in economics and had been recruited by Teach for America (TFA) immediately after graduating. (Pseudonyms are used for both participants.) Neither teacher had had opportunities for supervised professional practice before beginning to work full time as a classroom teacher. Both worked in the same state, teaching mathematics in the middle grades, and both engaged in frequent collegial activity with other teachers in their school and district. Both were finishing coursework in master’s degree programmes and teaching with provisional certification. After the PD course, both had scores on a test of content knowledge for teaching that were about one standard deviation above the mean score of a convenience sample of 201 in-service middle

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grades teachers from 13 districts in four US states (see Izsák et al. 2010 for details). In one assignment, participants were asked to design a lesson that implemented ideas from the course. The analysis in this section focuses on how Brian and Diane taught the lessons they designed. The data sources for this analysis included video records of the lessons, paired with post-lesson interviews. In this section, I summarize the use of representations, the obstacles faced and resultant student learning opportunities in Brian’s and Diane’s lessons. In particular, I found that the teachers differed in their enacted knowledge of the referent unit for fractions and percentages and in their knowledge related to student thinking.

Brian’s lesson In the observed lesson, Brian introduced a representational tool for proportions that was used extensively during the PD course: the double number line (two parallel number lines with a common origin). In the PD course, the double number line representation was used over several weeks for problems involving ratios of fractions and decimals, but not percentages. Brian adapted the double number line representation for use with percentages and modelled for his students how to use the representation to solve problems involving percentages. Brian explicitly called attention to referent units at many points during the observed lesson. For example, he emphasized that a question was asking what ¼ mile was as a percentage of a 2-mile race (¼ is 12.5 per cent of 2), rather than as a percentage of a single mile (¼ is 25 per cent of 1). This emphasis helped sensitize the students to the importance of identifying the referent unit in problem situations. Brian did have some difficulty making the double number line representation accessible to all of his students. After introducing the representation and assigning problems for students to work on in pairs, Brian helped a student who had incorrectly labelled one of the number lines (Figure 14.1). Brian tried several conceptually based remediation strategies, such as relating intervals on the number line to dollars and quarters, fractional amounts with which the student would have had direct experience. The remediation, however, did not address the 1/n pattern apparent in the student’s work, evidence that the student was viewing denominators as whole numbers – a common error. There was no evidence in the data that Brian made any effort to understand how the student made sense of the situation or to use the student’s thinking as a starting point.

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Figure 14.1 One student’s incorrect labelling of fractions on a number line in Brian’s class. Note the sequence 0, 1 , 1 , 1 , 1 , 1 , 1 , … on the upper line. 2 3 4 5 6 7

Diane’s lesson In the observed lesson, Diane presented an invented representation of equivalent fractions that she said was inspired by her participation in the PD course (see Figure 14.2). This representation was not used in the PD course, but features of Diane’s interaction with the model (such as partitioning drawn regions into equal parts) were common strategies for interacting with the area representation for fractions and fraction arithmetic that the PD course had highlighted. Diane’s invented representation was mathematically valid but pedagogically problematic. A misleading correspondence between the diagram and label made it difficult to identify the referent unit. The second column in Figure 14.2, shows the student referent unit error frequently observed with this representation: interpreting the denominator as the number of pieces in just the lower circle rather than the number of pieces in both circles. Diane further obscured the referent unit by asking the students to partition the regions in the

Figure 14.2 A reproduction of Diane’s representation of equivalent fractions

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representation without linking explicitly the resulting pieces of the region to the original referent unit. This omission meant that the students frequently lost track of critical mathematical information while using the representation. Diane was attuned to student errors and on several occasions she made a clear effort to understand what the students were thinking by asking questions about their work. On balance, however, Diane’s instruction using her representation limited the students’ opportunities to learn the content. In two observed episodes of remediation, Diane instructed the students to verify their work by counting, whereas multiplying would have been more useful for understanding the mathematical relationship between equivalent fractions. Crucially, her choice of language (‘halves’ – see Figure 14.3) with one student obscured the fact that the pieces under discussion were halves of ninths and thus eighteenths of the referent unit rather than halves of the referent unit. A student would have had to sort out these implicit meanings in Diane’s speech – a difficult task – in order to take advantage of this remediation. There were large differences between these teachers’ knowledge around selecting and using drawn representations across the observed lessons. The teachers’ use of representations also revealed differences in their enacted knowledge of the referent units of percentages and fractions. Brian’s instruction highlighted the referent unit, whereas Diane’s obscured it. Conversely (and in spite of students frequently sharing ideas in both classes), Brian did not engage with non-standard student thinking during his lesson, whereas Diane often did. Diane claimed that the advantage of her representation of equivalent fractions was that students ‘are seeing the transition from one fraction to the next without actually having to think about it’. In the section on teaching experience, I argue

Figure 14.3 Diane remediating the student’s error: ‘How many halves do we have here? [Counting] one, two, three …’

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that this statement characterized Diane’s general goals for instruction, and I describe how Diane’s goals for instruction and professional improvement differed from Brian’s.

Understanding differences in learning by considering knowledge at a finer grain size In the last section, I presented analysis that uncovered differences in the knowledge held and used by Brian and Diane during the observed lessons they taught after completing the PD course. These differences were surprising because both teachers did relatively well on the PD post-test of content knowledge for teaching, and because the PD course, the test and the lessons were all focused on drawn representations and dealt explicitly with referent units and non-standard student thinking. In this section, I present an analysis of participants’ written responses to individual test items and of data from item response interviews to explain the differences between these teachers’ instruction. I found that although the teachers had very similar overall test scores, data at the item level provide strong evidence of fine-grained differences in what teachers’ had learned by the end of the PD course. These differences were consequential for instruction. Diane (29 correct) and Brian (26 correct) had similar, relatively high knowledge scores on the PD post-test as compared with the sample of 201 in-service middle grades teachers. The similar scores were based on different items; Brian answered four items correctly that Diane had not, and Diane answered seven items correctly that Brian had not. The study used mixture Rasch methods to analyze the scores and identify two latent (unobserved) subgroups of teachers who differed in their item responses patterns. A qualitative analysis of teachers attending the PD – and a similar control group – revealed that the latent subgroups differed in their ability to appropriately attend to the referent unit of fractions and percentages (Izsák et al. 2010). Brian was classified in the subgroup associated with greater referent unit proficiency, and Diane was classified in the subgroup associated with less referent unit proficiency. This classification is congruent with the obstacles and limitations faced by Diane as she enacted her lesson. The classification of Brian and Diane into different latent subgroups helps explain observed differences in their instruction and suggests that similar differences in instruction (explicit and accurate identification of referent unit) might well extend to other teachers in the latent subgroups who participated in the PD course.

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Differences in Brian’s and Diane’s knowledge – and thus differences in what they had learned by the end of the PD course – can also be inferred from clinical interviews during which the participants explained their responses to individual items on the PD course post-test. Brian’s responses suggested that he had strong content knowledge for teaching proportions and number lines. For example, he reasoned proportionally to answer one item correctly that many other interviewed teachers answered incorrectly. Another item required an interpretation of a number line representation, and Brian demonstrated a sophisticated knowledge of recursive partitioning that many of the other interviewees (including Diane) did not demonstrate. Izsák et al. (2008) showed that recursive partitioning is important for supporting students’ understanding of operations with fractions that are represented on number lines. By contrast, Brian incorrectly evaluated student-invented solutions and strategies – he argued that some mathematically valid student work was invalid. This assertion suggests that perhaps Brian had not developed models of students’ mathematical thinking that were distinct from his own ways of thinking, a key milestone in the development of content knowledge for teaching (Silverman and Thompson 2008). Diane did very well on items involving student thinking, which is congruent with the observations of her instruction suggesting that she was able to follow students’ mathematical work and use it during instruction. Diane answered referent unit items on the PD course post-test correctly whenever those items included student thinking or involved percentages, but she did poorly on referent unit problems that involved area representations of fraction multiplication or division. These results may explain why Diane did not to deal appropriately with issues of referent unit in the observed lesson: the representation she used was a variation of the area representation that she had had trouble with on the PD post-test.

Understanding teaching experience by considering professional context The analysis presented in the last section explained the limitations of the post-test knowledge score: by examining responses on individual items, I demonstrated that these teachers – who had similar overall scores – actually had large differences in what they had evidently learned from the PD course. It remains to explain how teachers with similar backgrounds and teaching experience could

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have had these large differences in content knowledge for teaching after the PD course. At first, I conjectured that these differences were simply due to the fact that the participants in this study attended different sections of the PD course taught by different instructors. Ultimately, however, this explanation was unsatisfactory. Although there were some differences between the two sections (Orrill and Brown 2012 and Izsák et al. 2012 each describe a different section), the tasks and structure of the two sections were the same. In both sections, more than half of the 14-week PD course focused on teachers’ knowledge of referent units and on their facility with using and producing mathematically valid representations for fractions and proportions. In this section, I work towards a more plausible explanation of the teachers’ different learning outcomes by presenting a narrative analysis (Chase 2007) of the teachers’ own stories of learning to teach. I drew on retrospective interviews and used process coding and the constant comparative method (Charmaz 2006) to identify key features in each teacher’s experience. I found that Brian’s and Diane’s different professional networks supported different goals for professional improvement. I find it likely that these goals led Brian and Diane to attend to (and subsequently learn and enact during instruction) different aspects of the PD course. Brian’s story of learning to teach had a clear trajectory of improvement and progress: He could point to specific, deliberate changes in his teaching practice between his first and second year. The changes he described suggest how he was learning about content, students and teaching from his experience working as a teacher. For example, Brian discovered that some of his teaching strategies were ineffective and was motivated to redouble his efforts and use new strategies. Experiencing that frustration of [students] not understanding really helped me … some things they’re just not going to understand by using pencil and paper. They need to see it; they need to touch it. I knew it was going to be more of a struggle to teach … [I needed] to have more manipulatives and more strategies and more ways to teach it than just one simple way.

Referring to his the first year of teaching, he said, ‘I literally was just on the board, writing numbers … I would pull the manipulatives out if it was an emergency.’ By his second year, Brian said that he would ‘introduce a lot of things with manipulatives so [the students] were able to see it from the start’. Diane reported similar challenges in her first year. She said, ‘[I] did a lot of backtracking in my first year. There was introducing an idea, going through it and then realizing [the students] really didn’t get it and having to go back.’

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Unlike Brian, Diane seemed to have experienced little payoff from her efforts at improvement. She commented: The first year when I did [decimal notation], there was these misconceptions. So the second year I changed what I said, and then there were new misconceptions. So it was kind of – it created new problems from trying to reteach it.

Unlike Brian, who aimed to develop an arsenal of multifaceted and adaptive teaching strategies, Diane described an orientation toward professional improvement that valorized simplicity and efficiency. Describing how her teaching practice changed between her first and second year, Diane said, ‘I don’t think it’s been that drastic of a difference. I just think it’s me being more effective with my time management.’ Later in the interview, she explained, ‘I’ve learned a lot of things that have made [teaching] easier and more effective.’ Diane was the only fifth-grade mathematics teacher at her school, but she regularly worked with other teachers in the district. About once a month, she met with the fourth-grade and sixth-grade teachers in her school to pass on the mathematics information from the school board. ‘I do communicate with them and tell them what’s new, what’s expected, what we have to put in our math folders … and stuff like that.’ She was also part of a professional learning community organized by the district that was focused on improving teachers’ assessment practices. For example, she had learned about rubrics in this group and found their use appealing because ‘it is just really easy to go, “yes, yes, no,” and then tally the points’. Brian’s professional learning was supported by his work with colleagues. During his first year, a scheduling conflict prevented Brian from attending weekly meetings with the other sixth-grade mathematics teachers at his school. He described the isolation he felt during that year, saying, ‘I had nobody else to meet with; it was really just me.’ At the end of his first year, the schedule changed and Brian was able to meet weekly with the rest of the sixth-grade mathematics teachers at his school. They discussed their weekly lessons and gave each other feedback. If one teacher’s students had not learned a particular concept, the other teachers offered strategies they had found successful in the past. Brian said that these common planning times were ‘a lot of the reason that I had so much change from first year to second year’. Across the cases, it is evident that Brian described a clear trajectory of professional growth in terms of knowledge and practice, but Diane characterized her identity as a teacher in more static terms. Brian was motivated by his students’ lack of understanding and had a clear goal of increasing his students’

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appreciation and understanding of mathematics. By contrast, Diane wanted to make mathematics easy and simple for her students, much as she was motivated to make her teaching more streamlined and efficient. This difference might be characterized as a difference in productive disposition for teaching: mathematics teachers’ orientation toward – and their related beliefs and attitudes about – the subject of mathematics, teaching and learning it, and their own professional growth (Kilpatrick et al. 2001). It is perhaps no coincidence that Brian met weekly with other sixth-grade teachers to discuss lesson plans and student learning; his goals to improve student understanding were likely shaped and certainly supported by this professional community. Diane may have had inadequate resources to identify what needed to change and what promising strategies to try next. Colleagues with experience teaching the same content might have helped her troubleshoot her instruction of decimals, and might have helped her notice what was problematic about her representation of equivalent fractions used in the observed lesson. Diane valued her invented representation because it was easy – students could use it ‘without having to think’. Brian’s goal of student understanding led him to value the double number line representation as a tool that might help some students understand when other tools failed. The PD course activities likely presented divergent opportunities for Brian’s and Diane’s professional learning because of the differences in their productive disposition for teaching and in their professional contexts.

Implications for teacher education The public discourse in the United States around education, and mathematics education in particular, has focused more and more in recent years on the role of teachers and the importance of teachers’ knowledge; teachers are increasingly seen as key levers for improving educational quality in recent legislation and related policy initiatives in the United States. For example, the No Child Left Behind Act of 2001 stipulated ‘highly qualified’ teachers in Title-I schools, and the American Recovery and Reinvestment Act of 2009 encouraged improved teacher effectiveness through the Race to the Top funding competition. Professional development courses are widely seen as an important means to support teacher learning. Hill (2011) used data from a national sample of middle grades teachers and tests like the one used in this study to examine teacher learning related to professional development, and she reported small gains in teacher knowledge.

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The cases presented in this chapter demonstrate how single-score tests used to evaluate what teachers learn during professional development (even those measures that target a narrow content domain such as multiplicative reasoning in middle grades school mathematics) mask the complexity of what teachers have actually learned and how teachers might use what they have learned in practice. Other psychometric possibilities for assessing teacher learning exist, including the mixture Rasch analysis described above and diagnostic classification, which reports teacher ‘mastery’ with respect to discrete knowledge categories (e.g. Rost 1990; Rupp et al. 2010). These options may be more useful for teacher education than single-score assessments. The problem that this study uncovers is not simply that of how to accurately assess teachers’ learning in professional development courses. In addition to finding that similar test scores did not reflect similar learning, I found that similar professional development experiences for teachers with similar backgrounds and teaching experience afforded different opportunities for learning. Brian and Diane are largely indistinguishable on the variables that large-scale surveys frequently use to characterize teachers. Yet, there is evidence that sociocultural factors played a significant role in these teachers’ learning outcomes after two years of teaching, including those associated with the PD course. This chapter provides evidence that learning from experience is contingent on teachers’ goals for professional improvement. The cases presented in this chapter demonstrate the importance of teachers’ professional context for teacher education policy. Just as in Hodgen’s (2011) account of the constraints experienced by a master teacher in an interview setting – and surprisingly with content for which she had been involved in curriculum design and preparing teachers – this study suggests that knowledge for teaching is ‘stretched over’ (Lave 1988) practitioners and their social and cultural resources. That is, aspects of knowledge for teaching may not be located within individuals but between them. If this is the case, then attempts to understand how this knowledge is learned that focus only on inferred changes taking place within a teacher’s mind may be insufficient. The results presented in this chapter call into question the wisdom of relying on single-score measures of individual teachers’ knowledge to assess teachers’ learning when teaching itself is a social practice. The results of this study also challenge teacher educators and policy makers to identify the conditions – in addition to promising content and delivery – that influence whether teachers learn in professional development courses. It may be more helpful for teacher educators to understand how learning in professional development interacts

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with teachers’ prior professional experience and with teachers’ own goals for professional learning than it is to try to determine whether a particular professional development course is effective in and of itself. A theoretical model for learning content knowledge for teaching that incorporates both individual and sociocultural perspectives on cognition is needed to pursue these promising avenues for future research and practice.

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Acknowledgement The work reported here was supported by a grant from the National Science Foundation under grant number DRL-0633975. The results are my views and do not necessarily represent the views of the funding agency. I wish to thank the members of the ‘Does it It Work’ research team for their effort in collecting and analyzing some of the data reported here and for commenting on previous drafts; all errors that remain are my own.

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