Keywords

1 Introduction

A growing body of research around the world, represented in part by the work shared in Topic Study Group 38, is seeking to uncover and conceptualize the multiple ways teachers interact with resources to design and enact mathematics instruction and to identify the capacities involved in doing this work. In this chapter, I explore these issues theoretically and empirically. First, I discuss some conceptual and theoretical issues related to curriculum resources , including the meaning of curriculum as an adjective, how resources can be understood as a genre of tools used by teachers, and what is involved in reading and using them. I then discuss what I mean by teacher-resource interactions and the capacity involved in this work. Finally, I offer an illustrative example of one elementary teacher’s interactions with a mathematics curriculum resource to explore how research on teachers’ interactions with particular curriculum resources can inform our understanding of this capacity.

2 What Are Curriculum Resources?

In this TSG, mathematics teaching and learning resources refers to a genre of materials and tools, including, but not limited to textbooks, designed to guide, support, and enhance mathematics teaching and learning in schools. For many years, the textbook, accompanied in some countries by a teacher’s guide, served as the primary instructional resource found in mathematics classrooms. In 2017, the list of types of resources available is long and diverse, including, but not limited to print, digital, and online materials and tools used either periodically or over an extended period of time. Some resources are designed to be used to support and guide instruction; others are resources taken up by teachers and deployed as instructional tools.

2.1 Distinguishing Different Types of Resrouces

There is an equally long list of terms used to refer to teaching and learning resources. I begin by clarifying some terms that may assist in discussions of different types of resources.

I use the term instructional resource to refer to tools provided to, appropriated by, or generated by teachers to guide or support instruction. Instructional resources represent a broad category of artifacts, as shown in the largest region in Fig. 4.1. These resources include curriculum resources and others that, alone, are not curricular in nature. I discuss the meaning of curriculum below.

Fig. 4.1
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Relationships among different types of instructional resources

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Five or ten years ago, I might have used the term materials instead of resources. This term may be a holdover from print materials; it may also be unique to the United States. In using the term resources, I am following the lead of many of my European and South African colleagues who use the term resources to include the wide range of possible types of tools, including print or digital instructional materials, simulations, videos, interactive tools, and the like. I also find Jill Adler’s (2000) ideas about resource use as “re-sourcing” constructive. Describing teachers’ work as re-sourcing hints at the active nature of the work. By using resource as “both noun and verb, as both object and action,” Adler focuses attention on teachers working with resources in practice (p. 207).

The term resource also draws attention to a critical aspect of teachers’ work—interpreting and transforming available artifacts for one’s own instructional purposes. Gueudet and Trouche (2009) use the term documentational genesis to describe this process. Informed by Rabardel’s (1995) articulation of how artifacts, when used for specific purposes, become instruments, endowed with human purpose, and the participatory perspective on teachers’ use of curriculum resources (Remillard 2005), Gueudet and Trouche argue that teachers appropriate and transform available resources as they design instructional plans. The documents , which are the products of this work, are then used by teachers. In the next round of usage, the documents are transformed further, informed by the experience of the first round. In this way, the documentational genesis process is a dialectic and dynamic one.

I use the term curriculum resources to refer to print or digital artifacts designed to support a program of instruction and student learning over time. The term curriculum refers to the course or pathway on which learners are guided. Resources that attend to sequencing or mapping students’ learning over a period of time, such as a lesson sequence , a set of lessons, a year of instruction, or more, I argue, are curricular in nature. From this perspective, as shown in Fig. 4.1, all curriculum resources are examples of instructional resources , but not vice versa. This aspect of sequencing, appears to be an important component of curriculum resource design, as it proposes an intended learning progression for particular mathematical domains. Choppin (2011) has identified these learning sequences as a critical element of many curriculum programs that is not always made visible to the teacher. Moreover, Sleep (2009) identifies identifying learning sequences as an important feature of content-specific curriculum knowledge . I argue that sequencing or curriculum mapping (Remillard 2016) is an under-appreciated aspect of curriculum design and curriculum knowledge .

The curriculum resource domain in Fig. 4.1 includes three examples of such resources. Student texts or textbooks and teacher’s guides are curriculum resources designed for teachers’ use. The term textbook is most frequently used in the field to refer to the resource designed for the students’ consumption (Fan et al. 2013). Textbooks include exercises, problems, and other tasks for students, along with worked examples, and definitions. Many curriculum resources also include guidance prepared specifically for the teacher, often referred to as the teacher’s guide. In the U.S. and a number of other countries, guides are written to communicate to the teacher and support them in shaping instruction (Remillard et al. 2016). In cases where there are no written teacher’s guides, teachers draw on their own experience and the student text to make inferences about the intent of instruction plans. Documents , drawing on Gueudet and Trouche (2009), refer to the products of teachers’ curriculum design work, which approximate what Remillard and Heck (2014) refer to as the teacher intended curriculum.

In the analysis presented in this chapter, I focus on the relationship between these three types of curriculum resources . I look closely at one teacher’s guide, to understand what is involved in interpreting it, and how one teacher appropriates the resources it contains. First, I conceptualize Teacher’s guides as a genre of communication and the craft involved in using them.

2.2 Teacher’s Guides as a Genre of Communicaton

Curriculum resources are cultural artifacts (Vygotsky 1978). They are the products of cultural activity and reflect norms, values, and practices specific to the local cultural context (Pepin and Haggarty 2001; Pepin et al. 2013). In this sense, they hold cultural knowledge. Curriculum resources designed for teachers are intended as a resource for teachers, but making sense of them requires a process of interpretation.

I think of teacher’s guides as a genre of communication within a larger class of written and visual communication. They are designed to offer information, instructions, and suggestions that will aid in the construction of curriculum in the classroom. In essence, they are meant to guide action and decision making. As a genre, they communicate with readers with action in mind. In the case of mathematics, these actions are contingent on the development of mathematical understanding in others.

The notion of genre is elaborated by Ongstad (2006) in his semiotic analysis of communication in mathematics and mathematics education. “Genre precisely presupposes much of what can be expected in the kind of communication in question” (p. 262). Its familiarity conjures a “zone of expectation” and aids in how one makes sense of any form of communication, textual, or discursive. For teachers, the familiarity they might have with curriculum resources as a genre influences how they read, interpret, and use them.

As I discuss below, not all reading and interpreting of teacher’s guides, despite their familiarity, is straightforward. Furthermore, teacher’s guides are likely to contain unfamiliar elements as well. Ongstad uses the term “rheme” to identify the unfamiliar or new ideas in a familiar form, those that fit outside of the zone of expectation. Reading teacher’s guides involves an interaction between the familiar (the theme) and the new (the rheme) in which the theme contextualizes and aids in the interpretation of the rheme (p. 263).

To further examine the genre of teacher’s guides, I draw on Brown’s (2009) sheet music metaphor for curriculum resources . They are both “static representations” of intended activity and the “means of transmitting and producing” it, but not the activity itself. In other words, they are not the music or the curriculum, but offer guidance to the agent responsible for enactment. As such, they serve as “an interface between the knowledge, goals, and values of the author and the user” (p. 21).

Curriculum resources and sheet music are also similar in that “they are intended to convey rich ideas and dynamic practices,” but they do so “through succinct shorthand that relies heavily on interpretation” (Brown 2009, p. 21). They often rely upon “culturally shared notational rules, norms, and conventions” and “reflect common or existing practices.” At the same time, curriculum materials are often designed to “influence common practice by introducing innovative approaches and ideas.” Most critically, and often overlooked, curriculum materials “require craft in their use; they are inert objects that come alive only through interpretation and use by a practitioner” (p. 22).

2.3 The Craft of Resource Use: Pedagogical Design Capacity

Seeing curriculum resources as a genre highlights the complexity of reading and using curriculum resources. As Brown (2009) points out, curriculum resources tend to represent complex and multifaceted ideas in succinct shorthand. Teachers must read and interpret a variety of components of curriculum resources and determine their meanings and implications relevant to their teaching context. Engaging in this type of analysis involves substantial ability on the part of teachers.

Brown (2009) introduced the term pedagogical design capacity (PDC) to refer to a teacher’s ability to perceive and mobilize curriculum resources in order to “craft instructional episodes” (p. 29). He points out that, together, perceiving and mobilizing include knowing and doing, making determinations and then acting on them. The concept of PDC is important because it signals that the task of using resources is not straightforward and involves skills that teachers need to develop. PDC is also mediated by the particular resource a teacher is using. Some resources are more demanding for teachers to read and mobilize because of their complexity or unfamiliarity; others may include additional, “educative,” supports that increase the transparency or explicitness of guidance included (Davis and Krajcik 2005; Stein and Kim 2009). Unfortunately, because the work of using resources is not well understood, insufficient attention has been paid to how to delineate these capacities and help teachers develop them. Moreover, at least in the U.S., the myth that the best teachers do not rely on resources, but develop lessons on their own, works against nurturing these capacities (Remillard and Taton 2015).

Over the last several years, researchers in the ICUBiTFootnote 1 project have studied elementary teachers’ interactions with curriculum resources in the U.S., with the goal of elaborating PDC. In order to understand what is involved in perceiving resources, we have analyzed mathematics teacher’s guides to uncover and elaborate the succinct content of these guides that teachers read, interpret, and reason about in order to use. We then observed and interviewed teachers using these resources to understand the process of mobilizing them to design and enact instruction.

In the analysis that follows, I examine one teacher’s interactions and use of a single lesson from the teacher’s guide, drawn from the ICUBiT data set. My aim is to illustrate how analyzing curriculum resources, both teacher’s guides and related documents associated with use, can shed light on aspects of PDC.

3 A Case of Teacher-Resource Interactions

Elsewhere, I have argued that in the process of using them, teachers interact with their curriculum resources (Remillard 2005). This perspective challenges the assumption shared by some educational researchers and decision makers that professionally designed curriculum resources decrease the demands on the teacher using them; teachers simply pick them up and use them. As discussed earlier, evidence from research suggests that using curriculum resources is a dynamic process involving reading, interpretation, appropriation, and design (Brown 2009; Gueudet and Trouche 2009; Remillard 2005). I think of it as a participatory process, through which teachers actively partner with resource designers, the resources they designed, and, potentially, other teachers. In our research on teacher-resource interactions, the ICUBiT team adopted a participatory perspective . After a brief description of our methods, I provide an analysis of the multifaceted and layered nature of one curriculum resource, focusing on a single lesson. I then describe and analyze one teacher’s interactions with and use of this lesson.

3.1 Methods

The team analyzed five elementary mathematics curriculum programs used in the U.S., focusing on the mathematical and pedagogical demands placed on teachers and how authors communicate with teachers about different aspects of the intended curriculum. Using the lesson as a unit of analysis, we identified the different components of the resource and the ways mathematical ideas were represented. We also coded all written and visual aspects of the teacher’s guide for what the authors communicate to the teacher about. Looking across teacher’s guides for the five different programs, we compared the different amounts and types of information provided to teachers. We also identified a set of common forms of communication used across the guides. In the analysis presented in this chapter, I use a single curriculum program, called Mathematics in Focus (MiF) (Kheong et al. 2010). MiF was a modified version of one of the mathematics programs used in Singapore, developed by Marshall Cavenish, for sale in the U.S.

Data collection of teachers using the curriculum resources relied on a teaching set methodology (Cobb et al. 2009; Simon and Tzur 1999), which involves collecting video records of multiple lessons along with associated artifacts and then using specific events or practices observed in the data as a basis for teacher interviews. The ICUBiT study collected teaching sets for 25 teachers from four states in the U.S. The teacher discussed in this chapter, Maya Fiero,Footnote 2 was a 4th grade teacher in an elementary school in the eastern United States. She was in her tenth year of teaching and was using MiF for the second year.

Two teaching sets were collected for each teacher, one in the fall and one in the spring. The teaching set included 3 video recorded lesson observations, a completed curriculum reading log (CRL) for the lessons taught during the week of observation, and a follow-up interview. Prior to the fall teaching set, each teacher completed an introductory interview, during which the teachers provided information about professional background and curriculum use. The CRLs consisted of a copy of the relevant lesson in the teacher’s guide on which teachers used coloured highlighters to indicate which parts of the guide they read and how they planned to use them: General reading (yellow); plan to use (green); other portions that were helpful when planning, include those to be adapted (orange). During the follow-up interview, the interviewers asked teachers to respond to questions about the observed lessons and the CRL.

3.2 Findings from Curriculum Analysis

Brown (2009) described curriculum resources as conveying “rich ideas and dynamic practices through succinct shorthand” (Brown 2009, p. 21). Further, as Brown points out, the representations of these forms are static, although they intend to communicate dynamic practices. Our analysis of all five curriculum guides was aimed at uncovering how they communicate with teachers and what is involved in perceiving their content and messages.

Our analysis surfaced three layers of communication that teachers routinely encounter and must interpret when designing instruction: mathematical-instructional objects, pedagogical guidance and insights, curricular sequences . I use the term layered to refer to the way concepts and ideas are packaged in instructional activities, which are sequenced to develop over time.

Figure 4.2 shows the first page of a Grade 4 lesson in a teacher’s guide taken from Mathematics in Focus (Lesson 3.2, p. 86). This lesson introduces an approach to multiplying two-digit numbers by two-digit multiples of 10. The column on the left-hand side of the page provides an overview of the lesson, detailing the objectives and the various resources the teacher might draw on from elsewhere. At the bottom of the left-hand margin there is a description of a short activity, entitled “5-minute Warm up.” An image from the students’ book is captured on the upper right-hand side, showing three ways to represent and solve, first, 4 × 10 and then 3 × 20: (a) a story context, (b) a place value chart, and (c) an equation-based approach to multiplying by multiples of 10. Beneath the student page are suggestions for the teacher to use when teaching the lesson.

Fig. 4.2
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Excerpt from MATH IN FOCUS: The Singapore Approach (Kheong et al. 2009, Grade 4, Lesson 3.2, p. 86). Copyright © 2009 by Houghton Mifflin Harcourt Publishing Company. All rights reserved. Reprinted by permission of the publisher

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3.2.1 Mathematical-Instructional Objects

The first layer, mathematical-instructional objects, are mathematical ideas packaged for the purpose of instruction. These objects include tasks, activities, strategies, models, and representations, are designed for direct engagement with students and are intended to facilitate their learning of mathematics. Lessons in curriculum guides typically consist of a variety of tasks and activities designed for students to do, which are seen as the primary vehicle through which the learning takes place and by which teachers assess students’ understanding (Doyle 1983; Stein et al. 1996). The story problems in Fig. 4.2 (upper right-hand side) provide one example of tasks. Others include the “5-Minute Warm Up” at the beginning of the lesson, during which students work with partners to practice multiplying “1-digit numbers by tens and hundreds mentally” and the set of “Guided Practice” problems on the following page (not pictured), which include 14 × 10, 9 × 40, and 47 × 80.

Mathematical-instructional objects are the primary vehicles through which students encounter and engage with mathematical ideas, but as Sleep (2009) emphasizes, the mathematical point is not always clearly articulated. In other words, there is a distinction between instructional activities and the key mathematical points underlying them. Later, I discuss the mathematical complexity embedded in the mathematical-instructional objects in Fig. 4.2.

3.2.2 Pedagogical Guidance and Insights

The second layer of communication is pedagogical guidance and insights. To differing extents, teacher’s guides provide information teachers might use to plan lessons and follow during the lesson. This layer is intended for the teacher, not the student. All lessons include statement of the mathematical learning goal or objective. Teacher’s guides also include instructional suggestions on what to do or say during the lesson. For example, guidance that accompanies the page in Fig. 4.2 includes the following recommendation: “Help students recall the strategy for multiplying a number by tens by working through the examples in the Student Book” (p. 86). I think of this type of guidance as directing teachers’ actions or moves (Remillard 2013).

Other guidance provides information about the rationale behind particular design decisions, how students might or should respond to various tasks, or the underlying mathematical ideas (Davis and Krajcik 2005; Stein and Kim 2009). The guidance that accompanies the 5 Minute Warm Up that begins the lesson, includes the following rationale: “This activity helps recapitulate the previous lesson and provides a warm-up for this lesson” (p. 86). Several of the teacher’s guides we examined include examples of common student errors or anticipate difficulties they might have along with ways teachers might address these. The guidance with Fig. 4.2 includes: “For students who cannot visualize multiplying by tens, use a place value chart to show the connection” (p. 86).

3.2.3 Curricular Sequences

The third layer of communication, curricular sequences, is structural in nature and refers to how the mathematical content is organized for learning over a variety of different timeframes. Curriculum materials offer a sequenced learning pathway for the development of identified mathematical goals (Choppin 2011). I use the term learning pathway to refer to the planned development of mathematical ideas and related skills over time, within a lesson, within a year, or over multiple years (Sleep 2009). The sequences built into these pathways are critical to the designed curriculum.

One marker of the sequence built into Lesson 3.2 includes the warm-up described above. This short activity is designed to activate students’ fluency with multiplying by 10 or 100, which will be drawn on in this lesson. The sequencing is also evident in the way the lesson introduces the equation-based structure with 4 × 10 and 3 × 20 before using the same structure to multiply larger numbers, including multiples of 100. Over the next several lessons, the curriculum uses the structure for multiplying by multiples of ten to introduce the steps of multiplying any two 2-digit numbers using the standard, multiplication algorithm. Common pathways built into all of the curriculum programs we analyzed included movement from use of visual or concrete models to abstract representations of procedures and relationships and gradual increase in the size or complexity of number students are expected to deal with. Other learning pathways we identified included connections across topics, such as the fractions and decimals or different operations.

Understanding curriculum resources as containing multiple layers of communication has roots in didactic transposition theory (Chevallard 1988), the notion that in order to be packaged for the purpose of teaching, ideas, such as mathematical concepts, are structured into pedagogical forms. It is representations of these forms that are encountered by teachers and from which the mathematical and pedagogical ideas must be perceived. Perceiving curriculum resources involves reading on each of these three levels, uncovering the embedded mathematical and pedagogical meanings , and assessing their relevance to the goals at hand.

3.2.4 Unfamiliar Mathematical and Pedagogical Approaches

Because it was based on curriculum resources used in Singapore, MiF tended to present complex mathematical concepts using approaches not typical of instructional approaches used in the U.S. Lesson 3.2, excerpted in Fig. 4.2, provides an example of one such strategy presented in MiF and is typical of the way the authors build on mathematical structures throughout the program. The symbolic approach shown to multiply any number by a multiple of 10 is not commonly taught in the U.S. It uses an equivalent relationship to guide students through steps of rewriting the original expression in equivalent forms in order to solve: 4 × 10 can be written as 4 × 1 ten; and then 4 and 1 can be multiplied, resulting in 4 tens, or 40, also shown in the place value chart. Although it is likely that students will know that 4 × 10 is 40, the authors appear to be building a structure that will be applicable to all cases involving multiplying by multiples of 10, such as the second problem, 3 × 20, or 24 × 300, a problem that appears on the following page in the lesson.

This structure relies on several foundational mathematical ideas that underlie the structure of numbers and multiplication and will be critical to algebraic thinking. One idea is that numbers can be decomposed into workable components that can facilitate operating on them (e.g., 20 = 2 × 10); further, any number with a zero in the ones place can be considered as a certain number of tens (e.g., 20 = 2 tens); a third, related idea is that “tens” refers to units (e.g., 2 tens means 2 units of ten), which then can be counted as a group (e.g., 3 × 2 tens means 3 groups of 2 tens). By rewriting 10 as 1 ten, the curriculum implicitly emphasizes ten as a unit; rewriting 3 × 20 as 3 × 2 tens emphasizes that the problem involves 3 sets of 2 tens, also illustrated by the model on the left, showing tens as circles and each package of 20 crayons as 2 tens. This model further represents the equivalent relationship between 3 × (2 tens) and 6 tens, as well as illustrates the associative property of multiplication: 3 × (2 tens) = (3 × 2) tens, i.e., 3 × (2 × 10) = (3 × 2) × 10. The final step in the approach, using the understanding of sets of ten (or 100s) to multiplying a number by 10 (or 100) by adding zeros, assumes an understanding of iterating composite units, a concept foundational to multiplicative reasoning (Ulrich 2015).

3.2.5 Minimal Transparency

Our analysis of how the authors of MiF communicated with teachers revealed a tendency to provide directive guidance, offering suggested actions or moves teachers might make. At the same time, the authors provided minimal explanation or descriptions of the rationale behind the instructional approach of the intended sequencing of the tasks. For this reason, we describe this approach to communicating with teachers as lacking transparency about the mathematical or pedagogical designs. Few of the mathematical ideas detailed above are discussed explicitly in the teacher’s guide, although the role of the associative property of multiplication is mentioned later in the lesson.

3.3 The Work of Perceiving Curriculum Resources

My aim in analyzing the layered content of a curriculum resource and its approach to communicating with teachers was to illustrate the complexity of curriculum design work from both the curriculum designer’s perspectives and that of the teacher using it. Perceiving the affordances of designed curriculum resources, a key component of PDC, involves identifying the mathematical purpose or point underlying mathematical-instructional objects, considering the rationale behind recommended pedagogical actions, and mapping the learning pathways underlying curricular sequences. When teachers perceive these affordances, even when they are not explicitly stated in the resource, they are better situated to use them to design instruction. The following section provides an illustration of one teacher’s perception and mobilization of some of the resources in this lesson.

3.4 One Teacher’s Interaction with the Teacher’s Guide

When we observed and interviewed Ms. Fiero during the 2011–12 school year, her school had just adopted MiF. She was one of the teachers who had piloted the program the previous year. She told us, “Math is my subject. I like math. I like to teach it.”

3.4.1 Ms. Fiero’s View of the Resource

Ms. Fiero explained that, after using MiF during the first year, she “really liked it,” although many of her colleagues did not. When asked what she liked, she provided two reasons. First, she liked that the approach to teaching math was somewhat different than what she was familiar with. She especially liked that the guide almost always offered more than one way to approach a problem. “I want to give them other strategies,” she explained. “Some kids can’t solve one [problem/task] one way, but they can solve it another way.” Her second reason was that the guide provided a lot of details on how to introduce the strategies to students. These details helped her because many of the strategies introduced in MiF were unfamiliar to her. For this reason, she said that she read every detail in the teacher’s guide and tried to use most of it. This approach is supported by her detailed highlights and annotations in the CRL. She found that it was helping her develop “a new mindset. . . I’m a lot older; my mind doesn’t think this way either,” she said, “but I find it interesting and I get excited about it.”

3.4.2 Ms. Fiero’s Reading of Lesson 3.2

As is mentioned above, Ms. Fiero appeared to read the teacher’s guide thoroughly. An excerpt from her CRL is shown in Fig. 4.3. We asked teachers to mark in yellow all parts of the guide that they read when planning their lesson. We asked them to mark in green parts that they planned to use during the lesson and we asked them to highlight the parts in orange that influenced their planning or that they found helpful. This might include parts that they planned to modify. Ms. Fiero not only marked the parts she planned to use, but added additional comments.

Fig. 4.3
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Maya Fiero’s Curriculum Reading Log for Lesson 3.2, p. 86

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It is worth nothing that the 5-minute warm-up is marked in orange. When asked about it during the interview, she said that they had spent some time multiplying by tens and hundreds the previous day. “We just built on the multiplying by tens and just you know, we have hundreds with two zeros instead of one zero. And they seemed to get that pretty well. . . I didn’t want to go too in-depth because like we spent a little bit of time on this and I walked them through. . . I felt that was sufficient.” As I describe in the brief synopsis of Ms. Fiero’s lesson below, she replaced this 5-minute warm-up with a multiplication practice sheet, which included several 2- by 1-digit multiplication exercises.

3.4.3 Synopsis of Lesson 3.2

The lesson began, as is typical for Ms. Fiero, with the 2- by 1-digit multiplication exercises, presented in a vertical format. As they practiced multiplying by the number in the ones place and then the tens place Ms. Fiero emphasized the use of “basic facts,” such as 3 times 6 = 18, in 62 × 3. The students spent about 12 min on the exercises and then Ms. Fiero led the class in reviewing them on the board. She then guided students through a review of the previous day’s multiplication work. Twenty-five minutes into the lesson, Ms. Fiero began the introduction shown in Fig. 4.2. They began with a short discussion of title, objective, and key vocabulary listed on the first page of the student workbook, all items marked in the CRL. She then moved the class to considering how to multiply numbers by 10. The guidance in the teacher’s guide states: “Help students recall the strategy for multiplying a number by tens by working through the examples in the Student Book.” Rather than using the examples in the student book, Ms. Fiero, reminded the students that they had multiplied numbers by 10 the previous day. She wrote 81 × 10 on the white board to illustrate. For the next several minutes, the students struggled to provide an answer. One students said, “We haven’t learned that yet.”

Ms. Fiero responded: “We don’t know how to do 2-digit multiplication. No we don’t, yet. But do we know how to multiply things by 10? She called on several students who offered incorrect answers. Then, she called on a student who said: “You multiply the first number by the one, the 10, and then you just add the zero at the end.”

Ms. Fiero then moved the class onto the two examples on the page 86 (Fig. 4.2). She copied the following from the student book, guiding them through each step:

$$ \begin{aligned} 4 \times 10 & = 4 \times 1\;{\text{ten}} \\ & = 4\;{\text{tens}} \\ & = 40 \\ \end{aligned} $$

Students had difficulty with several aspects of this approach. They appeared to be confused by representing the two equivalent expressions: 4 × 10 = 4 × 1 ten or 3 × 20 = 3 × 2 tens. Rather than represent 4 × 10 in an equivalent form, they wanted to provide an “answer” to the right of the equal sign. In response, Ms. Fiero focused on the meaning of the equal sign in the number sentence. She asked questions like, “What does the equals sign mean?” and emphasized that it indicated that “both sides are equal or the same value.” She also told the students, “When you get into algebra and things when you’re bigger, you’re gonna have tons of stuff on this side equal to tons of stuff on this side.”

Another difficulty students had during this part of the lesson involved recognizing that 4 × 10 was indeed equivalent to 4 tens. A related difficulty students had, especially when the numbers in the problems increased in size, was converting a number of tens, such as 6 tens (or 18 tens, on the following page) to 60 (or 180). Ms. Fiero’s approach was to encourage them to recall “basic fact” that when multiplying by 10, they could simply add a zero.

The teacher’s guide included the following suggestion: “For students who cannot visualize multiplying by tens, use a place-value chart to show the connection.” Even though the place value charts were on the student page, Ms. Fiero did not refer students to these models once. Instead, she went through several examples, emphasizing the process of rewriting the values in equivalent forms and reiterating the meaning of the equal sign. As they practiced similar exercises on the following page, she announced, “Remember the equals sign is very, very important. Gets a bad wrap, it just hangs out there. But it tells you what you need to equal.”

3.4.4 Analysis of Lesson 3.2

In the analysis, I focus on how Ms. Fiero perceived and leveraged elements in the teacher’s guide and what she appeared to have missed. It is evident that she grasped the structural approach introduced in this lesson, seeing it as an opportunity to emphasize the meaning of the equivalence and as a precursor to algebra. When asked about this approach in the follow-up interview, she emphasized:

I think that’s something they should have further on down and I know they’ve learned about the equal sign, but they don’t seem to understand the importance. And then you know when you go to algebra and you have a huge thing over on this side you know, I think that’s going to be important.

Ms. Fiero expressed some surprise that her students had difficulty with this approach. She attributed their struggles to the use of the unfamiliar notation used by MiF. “The way they reword and set up problems differently really does throw them off.” She did not appear to appreciate that students may have been struggling with the multiplicative meaning of 4 tens or the relationship between 4 × 10 and 4 tens. During the interview, Ms. Fiero was puzzled by this particular confusion. She indicated that students did work on place value in other grades and for several lessons at the beginning of fourth grade. She wondered if they were “still caught up and confused on how Math in Focus presents itself.”

Ms. Fiero appeared to have missed an important element of the curricular sequencing built into the designed lesson. The 5-minute warm-up activity was intended to provide students with review and practice multiplying numbers by ten. This was one of the steps that students struggled with during the lesson. It is possible that, had she assigned students the warm-up in the guide, Ms. Fiero may have been alerted to their difficulty and made modifications to build the foundation for this task.

She also opted to not use or refer to the visual models (place value charts) on the student page to emphasize the relationship between 4 × 10 and 4 tens, even though she had marked it as something she would use in the CRL. When asked about this during the interview, she explained that she thought the representation used on the page was somewhat confusing.

4 Discussion

The analysis of lesson 3.2 in the MiF teacher’s guide, alongside the analysis of how Maya Fiero read, interpreted, and used it to design and enact a lesson, illustrates and provides insight into several important aspects of teacher-resource interactions , the complexity of reading curriculum resources as a genre, and the craft involved in using them to design instruction. Using Ms. Fiero’s interaction with the MiF excerpt, I discuss these aspects and consider the meaning of pedagogical design capacity (PDC).

The analysis of the lesson excerpt from the MiF teachers’ guide illustrates several aspects of the curriculum resource genre of communication. First, it offers different types of information, which teachers are to read and interpret. The example illustrates how a complex set of mathematical concepts, tasks, and representations are intertwined with pedagogical suggestions and explanation. Further, these concepts and tasks are intentionally sequenced. Second, these forms are represented succinctly and often with limited transparency about their intent. Third, in this genre of communication, complex mathematical and pedagogical ideas are often communicated through straightforward directives, such as “Help students recall the strategy for multiplying a number by tens by working through the examples in the student book.” These directives do not necessarily offer insight into their intent or the rationale behind them.

Because of the succinctness and general lack of transparency of this genre, teachers are expected to do significant interpretive work (Remillard and Kim 2017). The analysis of MiF suggests that these characteristics were especially problematic, placing greater demands on the teacher. The brief explication of Maya Fiero’s interpretations of the resource revealed that some aspects of her PDC were more robust than others. Her grasp of the foundational mathematical ideas underlying the unfamiliar approach to multiplying by factors of ten appeared to be sufficiently strong. She saw the mathematical importance of restating the expressions in equivalent forms, the value of understanding the meaning of the equal sign, and the ways this approach was a precursor to algebraic manipulation. At the same time, she did not seem to appreciate the meaning of multiplication as iterating composite units; nor did she fully understand the developmental progression needed for children to work flexibly within multiplicative structures (Ulrich 2015). As a result, Ms. Fiero did not recognize or find value in other components of the lesson aimed at helping students understand the meaning of multiplying composite units of ten.

The areas where Ms. Fiero’s understandings appeared to be more fragile are characteristic of what Ball et al. (2008) identify as specialized content knowledge and knowledge of content and students, because they involve forms of content knowledge deeply connected to how it is learned. These possible gaps in Ms. Fiero’s understanding of the development of multiplicative reasoning may help to explain why she underestimated the challenge it would present to students and did not interpret the warm-up activity as a useful precursor to the focal tasks of the lesson.

It is important to emphasize that PDC is not a static capacity residing in individual teachers. It is mediated by characteristics of the resource. As I have argued, curriculum resources are layered and complex to use. Nevertheless, some curriculum authors provide more transparency and explication than others to guide teachers’ interpretations and assist them in anticipating student difficulties (Davis and Krajcik 2005; Stein and Kim 2009). My analysis of the excerpt of the MiF teacher’s guide raises questions about the extent to which the curriculum resource was designed to support teachers to activate their PDC. As Ms. Fiero pointed out in her interview, the teacher’s guide provides extensive guidance on what the teacher might say or demonstrate during a lesson. At the same time, it communicates with teachers primarily through directing their pedagogical actions. It provides little in the way of transparency or other educative supports that might provide insights into the underlying mathematical ideas or design rationale.

5 Concluding Thoughts

The analysis of Maya Fiero’s interpretation and use of the MiF excerpt underlines the complex characteristics of curriculum resources and illustrates that PDC is an interaction between affordances of the resource and the interpretive capacities of the teacher. These findings have implications for research on resource use and design. Studying or assessing PDC involves examining the teacher’s interpretive interactions with the resource and accounting for characteristics that both the teacher and the resource bring to and leverage in the interaction. Here, I return to Ongstad’s (2006) insights about the genre of curriculum resources to elaborate this process and its challenges. He argues that teachers’ interpretive work is often mediated by familiar aspects of the genre, such as the common forms and structures used to communicate. These familiar forms conjure a “zone of expectation” that aid the interpretive process. Ongstad also posited that these familiar forms also support teachers’ sense making when curriculum resources present novel components or approaches. In this light, recall that Ms. Fiero was in her second year of using MiF, and she described it as using a number of unfamiliar approaches and representations. The novel approaches, together with the lack of transparent supports in MiF may have constrained, rather than supported, her interpretive work. Her case raises an important question about resource design: Can curriculum resources be designed to support teachers’ interpretive work with resources that use unfamiliar approaches and representations?