Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain restrictions, and the arcsine function

Journal of Mathematical Behavior 57 (2020) 100752

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Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain restrictions, and the arcsine function

T

Keith Webera,*, Juan Pablo Mejía-Ramosa, Timothy Fukawa-Connellyb, Nicholas Wassermanc a

Rutgers University, USA Temple University, USA c Teachers College, Columbia University, USA b

ARTICLE INFO

ABSTRACT

Keywords: Inverse function Inverse trigonometric function Real analysis Teacher education

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.

1. Introduction In many places worldwide, prospective secondary mathematics teachers are required to complete extensive coursework in undergraduate mathematics to become certified to teach secondary mathematics. This coursework usually includes advanced upperlevel coursework for mathematics majors (e.g., Conference Board of the Mathematical Sciences, 2001), with many institutions currently requiring that future mathematics teachers complete the equivalent of an undergraduate degree in mathematics (FerriniMundy & Findell, 2010). Given these requirements, it is natural to ask the following two questions: Does completing a typical course in advanced mathematics prepare one to teach secondary mathematics more effectively? How can we design advanced mathematics courses to better meet the needs of prospective secondary teachers? There are a number of ways that an advanced mathematical course can benefit a prospective mathematics teacher: There is often a considerable intersection between the mathematical objects and structures covered in advanced mathematics and secondary mathematics; for instance, the real numbers and continuous functions receive considerable attention in both real analysis and secondary mathematics. Hence, studying advanced mathematics has the potential to solidify and deepen teachers’ understandings of important objects and structures. Advanced mathematics courses provide teachers with the opportunity to experience core

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Corresponding author at: Graduate School of Education, 10 Seminary Place, New Brunswick, NJ 08901, USA. E-mail address: [email protected] (K. Weber).

https://doi.org/10.1016/j.jmathb.2019.100752 Received 31 January 2019; Received in revised form 11 November 2019; Accepted 12 November 2019 0732-3123/ © 2019 Elsevier Inc. All rights reserved.

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disciplinary practices like abstracting, generalizing, and proving—practices that these teachers will be expected to foster in their own classrooms (Common Core Content Standards Initiative, 2012; National Council of Teachers of Mathematics (NCTM), 2000). Indeed, scholars have illustrated a myriad of case studies in which advanced mathematics courses provide important benefits and insights to teachers (Even, 2011; Wasserman, 2016, 2017a; Zazkis & Mamolo, 2011). While advanced mathematics courses are potentially beneficial to all prospective mathematics teachers and genuinely beneficial to at least some teachers if they are taught in the right way, there are three arguments that suggest that advanced mathematics courses often are not helpful in preparing prospective secondary mathematics teachers. First, quantitative studies have found a weak to non-existent correlation between the number of advanced mathematics courses a teacher has completed and her students’ mathematical achievement (Darling-Hammond, 2000; Monk, 1994; for more discussion, see Wasserman, Weber, Villanueva, & MejiaRamos, 2018). Second, teachers’ self-reports indicate that they see little value in the advanced mathematics courses that they had completed. For instance, Zazkis and Leikin (2010) surveyed or interviewed 52 practicing secondary mathematics teachers about how their understanding of advanced mathematics influenced their teaching. The majority of the participants in Zazkis and Leikin’s study claimed that they rarely used their knowledge of advanced mathematics in their teaching and few could cite any specific instances of their knowledge of advanced mathematics actually informing their teaching. These results are consistent with other studies documenting that many teachers claimed that the advanced mathematics courses that they completed were not relevant toward their teaching (e.g., Goulding, Hatch, & Rodd, 2003; Rhoads, 2014; Wasserman & Ham, 2013; Wasserman et al., 2018). Third, many universities have developed “connections” or “capstone” mathematics courses for prospective teachers that aim to connect the content of advanced mathematics courses to the secondary mathematics that will be taught (Conference Board of the Mathematical Sciences, 2012; Murray & Star, 2013), which we interpret as an acknowledgement that many prospective teachers have difficulty seeing how the content from advanced mathematics courses relates to their teaching. To summarize, prospective secondary mathematics teachers are typically required to complete many courses in advanced mathematics because these courses have genuine potential to improve their teaching. However, the evidence that these advanced mathematics courses help these prospective teachers practice their craft is limited. In fact, many prospective teachers believe these courses are not helpful (although perhaps they are beneficial in ways that these teachers do not realize). These findings give urgency to the issue of how we can design advanced mathematics courses to better meet the needs of secondary mathematics teachers; the goal of the paper is to address this issue. At a broad level, we illustrate how an innovative real analysis course specifically designed for prospective secondary mathematics teachers helps them connect the content and practices from their real analysis course to their instruction of secondary mathematics. More specifically, we describe a particular module in which we connect a theorem from real analysis—if f is a continuous real-valued function,1 then f has an inverse function on an interval I if and only if f is strictly monotonic on I—to the pedagogical practice of introducing the concept of arcsine x in a secondary mathematics classroom and evaluating the solutions that students produce as they solve trigonometric equations. 2. Inverse trigonometric functions 2.1. What is an inverse function? In both the mathematics and the mathematics education literature, there are two ways to define a function and an inverse function. Most commonly in mathematics, a function is defined as consisting of a domain, a codomain, and a correspondence between the domain and the codomain such that each member of the domain is assigned exactly one element of the codomain. More formally, we can follow Bourbaki (1968) in defining a function f as a triple (F, A, B), where F is a relation from a set A to a set B (i.e. F A × B ) satisfying the following condition: For all x in A, there exists a unique y in B such that the ordered pair(x, y) is in F. The set A is called the domain of f (indeed, the condition above requires A to be the domain of the relation) and the set B is called the codomain of f. It is important to realize that a consequence of this definition (f = (F, A, B)) is that changing the codomain of a function changes the function. For instance, if f is the squaring function with domain R and codomain R and g is the squaring function with domain R and codomain [0, ∞), f and g are different functions. In particular, g is surjective while f is not. In high school mathematics and undergraduate real analysis, when one is dealing with functions defined by algebraic equations, the convention is that the codomain is assumed to be R unless otherwise stated. With this conception of function, if f = (F , A, B ) is a function, its inverse function (if it is exists) is the function f 1 = (F 1, B, A), where F 1 is the inverse relation of F (i.e. (y, x ) is a member of F 1 if and only if (x , y ) is a member of F). Or less formally, f (x ) = y if and only if f 1 (y ) = x . Inverse functions do not always exist. If f is not injective, then the inverse function for f does not exist: if x1 and x2 are distinct values of A and f (x1) = f (x2) = b , then both (b, x1) and (b, x2) would be members of F 1, so F 1 would not assign each member of B a unique value in A . Further, if f is not surjective, then its inverse function does not exist: if there is an element b in B for which there is no a in A such that f (a) = b , then for b B there is no value a A such that (b, a) F 1. However, for a given function f, an inverse function is guaranteed to exist if f is both injective and surjective. (Hence the invertibility of f depends on its codomain). This conception of function is common in undergraduate mathematics textbooks (e.g., Lipschutz, 1998, see p. 99) and in the writing of mathematics educators (e.g., Vinner & Dreyfus, 1989). There is an alternative way that functions and inverses are defined in mathematics textbooks and mathematical practice. As Forster (2003) noted, “some mathematical cultures… [say] a function is an ordered triple of domain, range, and a set of ordered 1

By a real-valued function, we mean a function whose domain and image are both subsets of R. 2

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pairs. This notation has the advantage of clarity, but it has not yet won the day” (p. 10–11). The alternative is to define a function as any set of ordered pairs that satisfy the univalence criterion: Specifically, if (x, y1) and (x, y2) are in f, then y1 and y2 are equal. In other words, f assigns each x at most one value. The inverse relation for f, denoted by f 1, is the inverse function for f if and only if f 1 is a function. In this case, f has an inverse function if and only if it is injective. In particular, f does not need to be surjective to have an inverse function. Indeed, using this definition, “surjectivity” is not a property that can be assigned abstractly to a function; one would need to specify a particular set that the function might be surjective upon. Every function is surjective onto its image and no other set. To highlight the distinction between these two conceptions of function, consider the question of whether f (x ) = e x and g (x ) = ln x are inverse functions of each other. According to the first conception of function, the answer may be that they are not: if f is interpreted to be the function with domain and codomain (as it is commonly done in Calculus), then it is not surjective onto its codomain and g is not a function from the codomain of f. According to the second conception of function, they are inverse functions: f is the set of ordered pairs (x , e x ) for all x , and its inverse relation f 1 is the set of ordered pairs (e x , x ) for all x (which also + fulfills the univalence criterion); g , the set of ordered pairs (x , lnx ) for all x , is precisely this inverse relation set of ordered pairs. This second conception of function appears in most set theory textbooks (e.g., Jech, 2000, p. 11) and some transition-to-proof textbooks (e.g., Forster, 2003). Further, at least with regards to inverses, we found that most pre-calculus and calculus textbooks and even some real analysis textbooks claim a function is invertible if and only if it is injective (e.g., Abbott, 2010;2 Stewart, 20063). This conception is also adopted in some mathematics education papers (e.g., Sajka, 2003), with other mathematics education papers claiming that f (x ) = e x and g (x ) = ln x are inverse functions (e.g., Even, 1990; Mayes, 1994). In this paper, we explore a real analysis course that prepares future secondary mathematics teachers. The textbook in our real analysis course (Abbott, 2010) and many pre-calculus and calculus textbooks (e.g., Stewart, 2006) use the second conception of function with respect to inverse functions. In particular, they define a function as being invertible if and only if it is injective. For these reasons, we adopted this convention in the instruction that we designed. 2.2. Using inverse functions to solve equations Inverse functions have the property that for all x in the domain of f, f 1 (f(x)) = x. If f is invertible, to solve an equation of the form, f(x) = a, for a variable x, one can apply the inverse function of f(x) to both sides of the equation to yield a solution:

f (x ) = a

f

1 (f

x = f

1 (a )

(x )) = f 1 (a )

For instance, if we wish to solve x3 = 8, we obtain x = 2 by taking the cube root of both sides. When the inverse function of f exists and is straightforward to identify, solving equations of the form f(x) = a is not problematic. However, if an inverse function of f does not exist (i.e. when the relation f 1 is not a function), interesting mathematical complexities arise. For instance, how should we solve an equation like x2 = 5 when f(x) =x2 is not invertible? In this paper, we focus on trigonometric equations; for instance sin(x) = .5. Here, f (x ) = sin(x ) is not injective, and consequently has no inverse function. To solve equations of this type, we find a partial inverse function by restricting the domain of sine to an interval on which f is injective. At least in the context of secondary mathematics, to find partial inverse functions, the domain of f is conventionally restricted to the largest interval on which f is injective that contains 0 and a positive number.4 For instance, one , 2 ) partial inverse for f (x ) = sin(x ) is g (x ) = arcsin(x ), which is the inverse of f | , (x ) (the restriction of f to the interval 2 2 2

(see Fig. 1 below), even though one could have defined a partial inverse for f (x ) = sin(x ) by restricting f to any interval on which f is injective (e.g.,

3 2

,

5 2

or 0,

2

).5 Note that g (x ) = arcsin(x ) is not the inverse function of f (x ) = sin(x ) but a partial inverse

, 2 ) (e.g., arcsin(sin(π)) = arcsin(0) = 0). See Fig. 1 below. function of f : indeed, g (f (x )) = x is only true on 2 Suppose g is the partial inverse function of f on a domain I. Then if we solve, f(x) = a by applying g to both sides of the equation, we obtain g(f(x)) = g(a). If g(a) exists, then g(a) will be the unique solution for f(x) = a on the domain I, but g(a) is not necessarily the 2 R is one-to-one, then we can define an inverse function f 1 on the range of f in the Specifically, Abbott (2010) writes that, “If a function f : A natural way: f 1 (y) = x where y=f(x)” (p. 140). 3 Stewart did not offer a definition of inverse function, but did claim that inverse functions can be found (when they exist) by switching the domain and range columns of a function in its tabular representation (p. 59). Stewart also wrote that “One-to-one functions are important because they are precisely the functions that admit inverse functions” (p. 61) and “If a > 0 and a ≠ 1, the exponential function f(x) = ax is either increasing or decreasing and so is one-to-one… It therefore has an inverse function f 1, which is called the logarithmic function with base a, and is denoted by loga” (p. 63, bold appeared in the original). These quotations are consistent with our second conception of functions as a set of ordered pairs. 4 We do not know why this is the convention or if others have defined the convention in the same way that we have. We only note that this seems to work in the context of secondary mathematics. 5 Again, note that there is variation in how arcsin(x) is defined. For instance, Stewart (2006) writes that, “When we try to find inverse trigonometric functions, we have a slight difficulty: Because the trigonometric function is not one-too-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become one-to-one” (p. 67). Stewart then defines the arcsine function or “inverse sine function” as we do above. However, other textbooks (e.g., Courant & John, 1999) define arcsine as a multifunction. We adopt Stewart’s convention in this paper.

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Fig. 1. Defining the arcsine function.

Fig. 2. Using symmetry and periodicity to find the solutions to sin x = .5.

unique solution in the domain of f. In secondary mathematics, we have found that there are usually multiple solutions and the student is expected to use the periodicity and/or symmetry of f to find the remaining solutions. In the concrete example where f(x) = sin(x), if we solve sin(x) = .5 by taking the arcsine of both sides, x = arcsin(.5) = is the unique solution to the equation on the interval 6

, 2

2

. We illustrate this solution in Fig. 2a below.6 Since f(x)= sin x is symmetric around the vertical line x = , we can conclude 2

3

, 2 . We illustrate these solutions in Fig. 2b below. Finally, that and 5 are the unique solutions to the equation in the interval 2 6 6 since f(x)= sin x is periodic with period 2 , we can conclude that a number a is a solution to the equation if and only if it is of the 5 form 6 + 2 k or 6 + 2 k for some integer k. We illustrate this entire solution set in Fig. 2c below. 2.3. Research on students’ and teachers’ conceptions of inverse functions While inverse functions play an important role in many secondary mathematics curricula, Paoletti, Stevens, Hobson, Moore, and LaForest (2018) noted that research on students’ and teachers’ conceptions of inverse functions is limited. Vidakovic (1996) argued that students need to coordinate three meanings of inverse function to understand the concept—specifically, (i) an inverse function reverses the function process, (ii) composing a function with an inverse function produces the identity function, and (iii) “inverse function as an action of switching for x and y and solving it for y” (p. 311) (or colloquially, switch-and-solve). In general, students and teachers do not achieve the robust understanding that Vidakovic described. Vidakovic found that many students’ understanding consists predominantly of switch-and-solve. Many scholars reported that students’ and teachers’ understanding of inverse functions is compartmentalized; that is, students and teachers may carry out algebraic rules for inverse functions in some situations (e.g., switchand-solve) and graphical rules in others (e.g., finding an inverse function by reflecting a graph across the line y = x) while seeing no connection between these two approaches (e.g., Brown & Reynolds, 2007; Engelke, Oehrtman, & Carlson, 2005; Paoletti et al., 2018; Vidakovic, 1996). One particular source of confusion is the symbol “−1” in the inverse notation f 1. As Zazkis and Kontorovich (2016) documented, this notation is a potentially powerful unifying notation in which the superscript “−1” denotes the inverse with respect to a particular operation, and set. In the case of inverse functions, f 1 typically refers to the function that is the compositional inverse of f(x) where the identity function f(x) is the identity with respect to that operation. In other contexts, such as 3-1, this typically refers to the number that is the multiplicative inverse where 1 is the multiplicative identity. While helping students appreciate this point can be valuable for students (Zazkis & Kontorovich, 2016), many prospective teachers simply conflate the meanings of the superscript “−1” in functional 1 and numerical settings, regarding f 1 (x ) = f (x ) (e.g., Engelke et al., 2005; Paoletti et al., 2018; Zazkis & Kontorovich, 2016). With regard to the status of inverse functions for functions that are not injective, we are aware of only two studies. Marmur and Zazkis (2018) presented prospective teachers with an algebraic ‘switch-and-solve’ solution to find the inverse function of a quadratic function. The prospective teachers recognized that the resulting expression, which contained the +/− symbol, was not a function (and consequently not an inverse function). However, they were reluctant to conclude that no inverse function for this function existed. Indeed, many would not assign the claim that there was an inverse function a definitive truth-value. In our own work, we asked 14 pre-service and in-service teachers whether √x was the inverse function of x2 and arcsin(x) was the inverse function of sin (x). Nine of the teachers responded yes to these questions, not mentioning the facts that x2 and sin(x) were not injective or anything 6

We include the graph for illustrative purposes. It is possible to perform this reasoning algebraically and without reference to a graph. 4

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Fig. 3. Sandoval’s framework for analyzing and reporting design research.

about domain restrictions. In this paper, we will reveal further limitations about prospective teachers’ understandings of the arcsine function. 3. Design research The data that we discuss in this paper is part of a larger design research project that we have been engaged in for the past several years. Sandoval (2014) noted that design research is not defined in terms of adherence to a prescriptive research methodology. Instead, design research is characterized mainly in terms of the epistemic commitments by the researcher. These epistemic commitments include: simultaneously developing theory and designing instruction; engaging in iterative cycles of design, enactment, analysis, and revision; and performing fine-grained analyses to link processes of enactment with outcomes of interest (Cobb, Confrey, DiSessa, Lehrer, & Schauble, 2003; Edelson, 2002; Sandoval, 2014). Design researchers have met these commitments using a broad array of research methods (Cobb et al., 2003). In this paper, we use Sandoval’s (2014) framework of conjecture mapping to report our study. A summary of Sandoval’s framework is presented in Fig. 3. Design researchers begin with high-level conjectures on how to support the kinds of learning the researchers are interested in supporting. For example, the idea that students can come to understand a concept when they are put in a situation when they need to reinvent it is an instance of a high-level conjecture. The researchers also have an initial set of learning outcomes that they aim to achieve and a method for measuring how successful they were in obtaining these outcomes (even if the desired learning outcomes and methods of measurement may change as a result of the study; cf., Simon, 1995). Examples of learning outcomes could be the ability to solve a particular class of tasks or to productively critique the argument of a peer. While high-level conjectures and learning outcomes provide the broad shape of a design research study, the bulk of the work involves the embodiment of instruction to achieve the mediating processes that support learning. By embodiment, we refer to the concrete instantiation of the high-level conjectures by explicating specific features of instructional design. The embodiment includes the tasks that the participants will complete, the way the tasks will be structured (e.g., will the students work individually or in groups? How long will they be given to complete these tasks? What scaffolding will be provided?), and the resources that students will have available to complete these tasks. (To avoid misinterpretation, Sandoval’s notion of embodiment is independent from embodied cognition). By having students complete these tasks, the researchers predict that students will engage in certain mediating processes to support their learning. These mediating processes will consist of observable behaviors such as specific types of conversations between students or the production of certain artifacts. The researchers’ assumptions about the types of mediating processes that their tasks will elicit are referred to as design conjectures. For instance, a researcher’s assumption that a particular task will lead students to create a particular solution method is an instance of a design conjecture. The researcher also predicts that the mediating processes will produce the desired learning outcomes; these predictions are referred to as theoretical conjectures. For instance, the prediction that “if students create a new solution method and engage in a classroom discussion about its virtues, then this will lead to the attainment of the learning goal that students will use that solution method to solve challenging problems” is an example of a theoretical conjecture. As mentioned above, we use Sandoval’s (2014) framework to structure this paper. In our Theoretical Perspective section, we describe our overarching model for how a real analysis course can help prospective teachers learn how to teach better. This model delineates our high-level conjectures. In the following section on Pedagogical Learning Goals, we present our learning outcomes for this study, describe how they were generated, and justify why we believe they are appropriate. In the Lesson Plan section, we summarize how we embodied the high-level conjectures in terms of specific activities and describe the mediating processes these activities were designed to elicit. Finally, after presenting a Methods section describing our research context, data collection, and analysis, we examine the extent to which our instruction was successful. In the first section of the Results, we report on the extent to which our activities elicited the desired mediating processes. In the second section of the Results section, we document the extent to which our learning goals were achieved.

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Fig. 4. Our pedagogical model of stepping up from pedagogical practice to real analysis and stepping down from real analysis to pedagogical practice.

4. Theoretical perspective Our model for how instruction of real analysis can influence the future pedagogical practices of prospective teachers was originally delineated in Wasserman, Villanueva, Fukawa-Connelly, Mejia-Ramos, and Weber (2017). We present this model in Fig. 4 as our high-level conjectures for how prospective mathematics teachers can benefit from the advanced mathematics courses that they are required to take. The left side of Fig. 4 represents building up from pedagogical practice. We begin our modules by presenting a realistic pedagogical situation from a secondary classroom. From there, we discuss the secondary mathematical concepts that are in play and problematize the mathematical challenges inherent in the situation that we provide, highlighting fundamental issues that lie beneath the surface and are handled in the advanced mathematics course students are taking. In short, we hope prospective teachers recognize that the pedagogical problems we pose are challenging because the secondary mathematics involved is complex. Then we discuss the secondary mathematics in terms of advanced mathematics. We cover the associated concepts with a formal treatment and make explicit what connections this has for high school mathematics. The right side of the diagram represents stepping down to pedagogical practice. After studying real analysis, we ask our prospective teachers to apply their newfound understanding to answer questions about secondary mathematics content and respond to pedagogical situations. We believe that our model addresses the limitations of simply teaching students advanced mathematics and hoping that they can flexibly apply what they learn in a later situation. Rather the instruction is steeped in, and motivated by, authentic pedagogical situations and supports students in making connections between advanced coursework and secondary mathematics and teaching. This model can be viewed as an instance of the more general teacher preparation model in which the mathematics that students can productively use is motivated by a pedagogical situation (c.f., Heid & Wilson, 2015). What is novel about this model is its applications to advanced mathematics, where to our knowledge, instruction based on this premise is uncommon. Further, our model emphasizes the curricular requirement that PSTs practice using the newly learned advanced mathematics in pedagogical situations.. In Wasserman et al. (2017), we describe how our theoretical model differs from the way that advanced mathematics is typically taught to prospective teachers. 5. Pedagogical learning goals: inverse functions 5.1. Choosing learning goals for our modules We developed modules with specific pedagogical learning goals that required the prospective teachers to effectively engage in the practice of teaching. We chose pedagogical learning goals for our modules based on three criteria: (i) The mathematical content that we taught is a topic commonly taught in the secondary mathematics curriculum.7 (ii) We required teachers to engage in tasks that approximated8 core pedagogical tasks that teachers actually engage in, such as giving presentations of new concepts, interpreting students’ utterances, grading students’ work, and providing students with feedback. (iii) Effectively engaging in these pedagogical practices is difficult due to the complexity of the mathematics involved. Understanding some aspect of the advanced mathematics being covered can allow the teacher to cope with the secondary mathematics and thereby engage in these pedagogical practices more effectively. In short, real analysis can help teachers do the required work.

7 We operationalized this by only including topics that were explicitly mentioned as learning goals in the Core Content State Standards in Mathematics (CCSS-M, 2010) or the Advanced Placement Calculus Exams, as these are the guiding secondary mathematics curriculum documents in the United States, where our work is situated. However, we believe the topics that we included would be important to secondary mathematics curricula in most other countries as well. 8 We say “approximate” to acknowledge that there is a difference between the tasks we pose to teachers (e.g., responding to a hypothetical students’ work presented in a comic strip independent from other teaching activities with time to reflect and formulate a response) and the tasks teachers actually engage in (e.g., responding to an actual student in real time). Our core pedagogical tasks were approximations of the High Leverage Practices identified by TeachingWorks (2015).

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We describe in more detail how we generated our pedagogical learning goals in Wasserman et al. (2017). 5.2. The learning outcomes for the inverse functions module In this paper, we focus on the Inverse Functions Module. This module had the following learning outcomes: At the broad level, we want PSTs to understand how to find inverse functions for functions that are not injective by restricting their domains and how this gets used in solving equations, which as we have documented, is difficult for prospective secondary teachers (Marmur & Zazkis, 2018; Wasserman et al., 2018). We want PSTs to be able to apply these ideas in the context of trigonometric functions. The PSTs should be able to offer a mathematically correct and pedagogically useful explanation for what the arcsine function represents and, in this case, to provide a student who presented a common erroneous solution to a trigonometric equation with feedback.9 6. The embodiment of our design principles and learning goals 6.1. A description of our lesson plan The activities and content of our real analysis modules are provided in the Appendix of this paper. In Table 1, we give a short description of each activity, we describe the mediating processes that we expected the activity to elicit, and we delineate the observable behavior that would inform us if we were successful in eliciting these mediating processes. The interested reader can request for detailed lesson plans for our modules in which we provide the motivation for the activities and the PISTs’ anticipated behavior as they engage in these activities.10 The high-level idea of our lesson plan is as follows. We ask PSTs to respond to a student who presents a common incorrect answer to a trigonometry equation (Activity 1). This classroom scenario is depicted in the form of a comic strip developed using the Depict tool (in the Lesson Sketch platform; www.lessonsketch.org). The value of using these comics is that it focuses PSTs’ attention on the generic aspects of the classroom that we think are salient (in this case, the students’ written mathematical work and justification), rather than particularities that we find irrelevant (e.g., the student’s tone of voice). See Herbst, Chazan, Chen, Chieu, and Weiss (2011) for more details about the benefits of using these comics. In Activities 2 and 3, PSTs wrestle with the relationships between continuity, monotonicity, injectivity, and invertibility. The purpose of these activities is to hint at the fact that the arcsine function is more nuanced than most students realized—in particular, arcsine is not simply the inverse function of the sine function as many PSTs believe (Wasserman et al., 2018)—and to provide students with the experience of grappling with these ideas to appreciate the main real analysis theorem of the lesson. The theorem is that if a function is continuous, it is invertible on an interval if and only if it is strictly monotonic on an interval.11 This theorem is important because it forms the basis for how non-invertible functions are treated in the secondary curriculum (since most functions studied at the secondary level are continuous). We typically find inverse functions for these functions by restricting their domain to an interval on which they are strictly monotonic. However, it is important to note that this theorem is not strictly necessary for restricting domains so that a function is invertible. One could accomplish this task using ideas from secondary mathematics about injectivity, such as the Horizontal Line Test. After PSTs apply this theorem in several contexts (Activity 5), the instructor then illustrates how this theorem can allow students to use partial inverse trigonometric functions to find one member of the solution set when solving a trigonometric equation, but not all solutions. To find the remaining solutions, one needs to consider the symmetry and periodicity of the trigonometric functions in question. Finally, in Activity 7, PSTs apply these themes in a secondary mathematics context coordinating these ideas to find all the solutions to a specific trigonometric equation. In Activity 8, the PSTs revisit the original cartoon to provide better feedback in light of the lessons they have learned. (Normally, students do this as part of their homework. In our study, we include this as a post-test item). 7. Methods 7.1. Broader research context The data reported here are part of a larger design research project to generate instructional modules that connect the learning of real analysis to the teaching of secondary mathematics. We focus here on the seventh module that informs teaching about inverse trigonometric functions, as described in section 4.1. Some of these modules, including the Inverse Function Module, were first implemented in an unpublished pilot constructivist teaching experiment (Steffe & Thompson, 2000) with three PSTs. The Inverse Function Module was refined after this implementation. All of the modules were then implemented in a special real analysis course 9 The Core Curriculum State Standards for Mathematics (2012) has goals that students should be able to produce an invertible function from a non-invertible function by restricting its domain (HSF.B.4.D), particularly with trigonometric functions (HSF.TF.B.6), and solve trigonometric equations (HSF.TF.B.7). We believe these will be goals with curricula in countries outside the United States as well. 10 The URL for the website is: http://ultra.gse.rutgers.edu. The Module discussed in this paper is Module 7. The “instructor’s copy” specifies rationales and predicted behaviors and is available upon request. 11 For clarity, “invertible” is defined in terms of the characterization used throughout the paper. It is more cumbersome, but more accurate, to replace “a function is invertible” with “a function has either an inverse function or a partial inverse function on a restricted domain”.

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Table 1 Summary of Inverse Function Module activities. Activity

Description

Anticipated behaviors and mediating processes

1. Pedagogical situation:

PSTs are shown a comic of a student presenting an incorrect solution and asked how they would respond to this student. They will then discuss the feedback they would give in a class discussion.

2. Step up to secondary mathematics.

PSTs are asked what the definition of an inverse function is and whether sin(x) and arcsin(x) are inverse functions with a whole class discussion about their answers.

3. Step up to secondary mathematics.

PSTs will be given five assertions about functions on an interval and asked if these statements are always, sometimes, or never true. The statements will concern monotonic and/or continuous functions and ask if they are injective on a domain.

4. Real analysis.

Instructor presents a proof that continuous functions are invertible on an interval if and only if they are strictly monotonic on that interval. PSTs are told the conventional domain restriction for a function is the largest interval including 0 and a positive number on which f is invertible. Participants find the conventional domain restriction of a graph of a function and revisit whether sine and arcsine are inverse functions. PSTs are shown a theorem that if f is strictly monotonic on an interval I, then the unique solution of f(x) = a (a in the range of f(x)) on I is f−1(a). PSTs are then shown an example of how to recover the missing solutions in the case of trigonometric equations PSTs are shown a graph of f(x) = sin(2x) and g(x) = .7 with the intersections labeled. PSTs are asked which solution is found by taking inverse functions, which solutions are found by the symmetry of the sine function, and which are found by the periodicity of the sine function. PSTs revisit the original cartoon.

PSTs will not see the errors in the students’ work. Rather the PSTs will respond to the presentation of the solution. If an error is located, the PSTs will justify the error in terms of the student misapplying a procedure, rather than explain it in terms of the meaning of the arcsine function. PSTs will provide a set of criteria for functions to be inverse functions (e.g., the graphs of y = f(x) and y = g(x) reflect across the line y = x), but will not be able to explain how these criteria are related. They will think that sin x and arcsin(x) are inverse functions. PSTs will answer all questions correctly, possibly with the exception of (ii) (See materials in Appendix). However, we anticipated this being a difficult process for students in which they engage in productive struggle, meaning that they will (a) change their understanding of the meaning of a definition, (b) explore or justify relationships between concepts, and/or (c) expand their example space. N/A

5. Step down to secondary mathematics.

6. Real analysis.

7. Step down to secondary mathematics.

8. Step down to pedagogy

PSTs will correctly use the ideas of the real analysis theorems to find the largest neighborhood around 0 on which f is monotonic. They will also state that arcsin(x) is the inverse of sin(x) when it is restricted to the largest neighborhood around 0 where sin(x) is monotonic. N/A

PSTs will correctly use the ideas of the previous theorem and example to answer the question correctly. The only difficulties will occur with defining a formula to find a solution from the symmetry of the sine function, which is complicated algebraically. PSTs use their understanding of the sine function to recognize and respond to the errors in the students’ approach.

for PSTs taught by the third author of this paper. Previous reports provide the methodological details of this course (McGuffey et al., 2019; Wasserman, Weber, & McGuffey, 2019), although we have not discussed the Inverse Function Module before. The data reported in this paper occurred in a real analysis course taught by the second author in a large state university in the northeast United States. At this university, prospective secondary mathematics teachers enrolled in a five-year mathematics education program in which they would earn a bachelors degree in mathematics after their fourth year, and then a masters degree in mathematics education and state certification to teach secondary mathematics after their fifth year. To earn their bachelors degree in mathematics, the prospective teachers, like all students seeking a mathematics degree, were required to complete eight junior- and senior-level mathematics courses. In particular, students seeking a mathematics degree were required to complete a course in real analysis, which is typical of university degree programs in mathematics. Students enrolled in this program would have taken some mathematics education courses that focused on content and discussed methods tangentially, but would not yet have taken a mathematics education course focusing on teaching methods. In the semester in which this course was offered, the mathematics department offered six sections of real analysis, one section of which was advertised as a special section for prospective mathematics teachers. Seventeen students enrolled in this section. In the beginning of the semester, all students were told that research was being conducted on this experimental class; all students agreed that the data that was collected in the study could be used for research purposes. Twelve of these students were enrolled in the secondary mathematics five-year certification program described above. Of the remaining five students, one student (PST4) was enrolled in a similar program for prospective physics teachers. Three students (PST1, PST2, and PST3) were mathematics majors who had a potential interest in becoming mathematics teachers. The remaining student (PST14) was a mathematics major who expressed no interest in teaching. For rhetorical convenience, we refer to all of these students as PSTs for the remainder of this paper. The class met three times a week. Roughly one out of the three weekly class meetings was devoted to implementing a real analysis module from our innovative curriculum, one of the three weekly class meetings consisted of a traditional lecture covering real analysis content that was necessary for the course but outside the content of our 12 modules (e.g., compactness and uniform

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continuity), and one of the three weekly class meetings was a workshop in which students were given practice and assistance solving problems and writing proofs. 7.2. Module implementation and data collection We implemented the Inverse Function Module in the 8th week of the semester split across two class meetings. Fifteen PSTs (PST1PST15) attended the Tuesday class meeting during which the first part of the Inverse Function Module was completed. The PSTs first spent ten minutes completing a Pre-Test. The first question on the Pre-Test asked students to write a response to the student who presented an erroneous solution to a trigonometric equation (see Appendix). The incorrect solution was as follows: sin(2x) = .7

2x=arcsin(.7) (take the arcsin of both sides) 2x = 0.7754 (evaluate arcsin(.7)) x= .3877+2 k (divide by 2 and add“+2 k”to account for additional solutions) There are two errors in the solution. The first is that the solution in the interval [π/4, 3π/4] is not accounted for (i.e., 1.183). The second is that the period of sin(2x) is π, not 2π. Procedurally, we might say that the student added the “+2πk” too late and this should have appeared in step 2. Hence, the solution should be x ≈ 0.3877 + πk and x ≈ 1.183 + πk. (Normally, we would have PSTs discuss the cartoon in Activity 1 in small groups, but in previous iterations of the course, we found it difficult to assess the efficacy of the modules since we were unable to measure individual PST’s prior knowledge from their discussions in groups). The second PreTest item asked students how they would introduce the arcsine function if they were teaching a secondary trigonometry class. A whole group discussion about their responses to Activity 1 followed. From there, we implemented Activities 2-7. Ten PSTs (PST1-PST10) attended the next Thursday class meeting, all of whom had attended the previous class meeting. We believe this relatively low attendance was due to the fact that this class meeting occurred two days before spring break. At the end of this meeting, participants were given a post-test in which they were asked to respond to a student presenting an incorrect solution to a trigonometric equation (similar, but not identical, to the Pre-Test question) and again to write a paragraph on how they would introduce the arcsine function if they were teaching a secondary trigonometry course. We asked students to complete Activity 2, Activity 3, Activity 5, and Activity 7 working in four groups. (Activity 1 was the pretest, Activities 4 and 6 were lectures on real analysis, and Activity 8 was the post-test). Three groups had a facilitator who was part of the research team. One facilitator was a mathematics education professor and the PI on the project, a second facilitator was a graduate assistant affiliated with the project, and the third was an advanced undergraduate who had completed two semesters of honors real analysis who was affiliated with the project. The role of the facilitators was to clarify questions about the tasks students were given, to prompt students to explain their reasoning or justify their solutions, and to direct students’ attention to relevant aspects of the assignment if the groups ignored them (e.g., “Have you done question c?”). The ideas from the group discussion were supposed to come from the students, but the facilitator might provide prompts to elicit ideas or seek clarification on ideas that were already present. Facilitators were not to explain any mathematical ideas or correct students as they worked on the tasks. All group work was audiotaped. 7.3. Data analysis Following Sandoval (2014), prior to implementing our Inverse Function Module, we listed specific anticipated behaviors and mediating processes that our activities would elicit. In our first round of data coding, we documented the presence or absence of these behaviors from the audio-recordings of the groups and the field notes of the facilitators. In our second round of analysis, in instances in which PSTs behaved in a manner that we did not anticipate, we built tentative hypotheses for why the participants behaved the way they did and, if necessary, why they did not engage in the mediating processes that we anticipated. For the Pre-Test and Post-Test data, we first coded PSTs’ responses for mathematical correctness. For the question where the PSTs provided the hypothetical student with feedback, we coded whether the PSTs coded for either of the set of missing solutions (i.e., the class of solutions missed by having a “+2πk” instead of “+πk” and the class of solutions that could be obtained by symmetry, “1.1831 + πk”). We then used an open coding scheme to identify common characteristics of the feedback that they provided. For the PSTs’ descriptions of how they would introduce the arcsine function to their class, we first coded for whether the PSTs’ explanations contained mathematical inaccuracies and for whether they mentioned domain restrictions in their explanations. We then used an open coding scheme to identify common characteristics in the types of explanations that the PSTs provided. To analyze whether our mediating processes were elicited, we analyzed the audio recordings of each group’s performance on the tasks. We documented whether the mediating processes that we hypothesized in Table 1 were present. We also noted the presence of any interesting and theoretically relevant behaviors that we did not anticipate, using facilitator’s field notes to inform our analysis. In the event that the mediating processes that we predicted did not transpire or other important unanticipated behaviors did transpire, we developed hypotheses for why events unfolded the way that they did.

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8. Mediating processes: PSTs’ response to our activities As noted in the previous section, we analyzed four groups of PSTs’ work on Activities 1, 2, 3, 5, and 7. A comprehensive analysis of each group would cause this paper to be unreasonably long. Thus, for the sake of brevity, we briefly describe the aspect of the activities that went as planned (i.e., elicited the mediating processes that we hypothesized would occur), sometimes providing transcripts to illustrate what these mediating processes were like. We focused our attention on cases in which our hypotheses were not confirmed. 8.1. Activity 1: the initial conversation Normally, when implementing the Inverse Function Module, we had students discuss the cartoon in Activity 1 in groups. This time, for the purposes of the Pre-test/Post-test, we had PSTs write a response to Activity 1 and then engaged in a four-minute wholeclass discussion. For the first three minutes, the class discussion went as anticipated in Table 1. PST contributions included:

• “The student is basically correct but they should have clarified that k was the set of all integers” (PST14) • “Sort of going off ideas of previous modules, the student rounded a bit too early” (PST6) • “When the student did the arcsine of both sides, it may have indicated that the student doesn’t have a good understanding of what arcsine is for because it’s more of a separate operation. I feel like the end result was the same but I would ask the student to clarify what arcsine does”. (PST8)

This is consistent with the mediating processes that we predicted in Table 1. In each of the responses, no PST challenged the correctness of the student’s answer. Indeed, PST14 and PST8 affirmed the work was basically correct. Three minutes into the discussion, one PST (14) stated: When you write arcsine down, you kind of take the assumption that you restrict that function to the domain of negative π to π, right? So to write 2πk at the arcsine step is meaningless because you haven’t yet defined the domain of arcsine. But once you’ve defined it, then you have to identify, oh yeah, it has this plus 2πk. In reality, it’s modulo 2πk. Attention to domain restrictions and the meaning of arcsine was not anticipated (see Table 1). There are three inaccuracies in PST14’s statement. First, the domain restriction of the sine function is [−π/2, π/2] (not [−π, π], an interval on which sine is not injective). Second, the domain of the arcsine is indeed defined as the range of the sine function (viz. [−1, 1]) and this will be true regardless of the domain to which we restricted sine, so long as the interval is maximal with respect to where sine is restricted. Third, we interpreted PST14’s comment about the ambiguity of arcsine as referring to the range of arcsine (or the domain of sine), so there are many possible values of arcsine, as if it were a multifunction. This view is not correct, as arcsine is defined by convention to have a range of [−π/2, π/2]. After PST14’s comment, the instructor posed the question to the class if adding +2πk instead of +πk led to the student missing solutions. When no PST answered, the instructor pivoted to showing a graph of y = sin(2x) and y = .7, illustrating the many solutions missed by the students’ work. 8.2. Activity 2: what is an inverse function? Are sine and arcsine inverse functions? For the sake of brevity, we provide only a brief description of the groups’ discussion on what an inverse function is. Three groups behaved as anticipated in Table 1; they mentioned multiple properties of inverse functions (e.g., the inverse function as “undoing”, f( f 1 (x)) = x) and occasionally some facts about inverse functions (e.g., the domain of the original function is the range of the inverse function) but they made no attempt to coordinate these observations and did not use these observations to conclude that sine and arcsine were not inverse functions. In other words, the students exhibited the compartmentalized understandings of inverse functions that are described in the literature (e.g., Engelke et al., 2005; Paoletti et al., 2018). One group mentioned in passing that √x and x2 were only inverses in a restricted sense, but did not use this insight to describe the meaning of inverse function; in fact, the group still concluded that sine and arcsine were inverse functions with no mention of domain restrictions. The remaining group of PSTs (PST14’s group) considered domain restrictions. The group decided that all functions have “inverses”, but the inverses may not be functions. (They did not explicitly use the language of “inverse relations”). Several members also remarked that the inverse would be a function if the original function was “one-to-one and onto”. PST14 said arcsine was the relation that mapped from the range of the sine function to the domain of the sine function. He concluded that the arcsine would be an inverse function on sine “if you restrict the domain in the right way”. When PST4 asked PST14 what the right way would be, he presented a graphical argument using the Horizontal Line Test to justify the restriction of the domain of sine so the sine of the left endpoint would be -1 and the sine of the right endpoint would be 1. In Table 1, we did not anticipate a group would mention domain restrictions on Activity 2, but we observe that this nonetheless still motivated the main theorem that we presented in class, which provides a set of conditions on which a function is one-to-one and therefore invertible. We believe that this group’s unanticipated behavior followed from PST14 spontaneously raising the issue of domain restrictions in the discussion from Activity 1. After this discussion was completed, the instructor presented and illustrated the definitions of one-to-one functions, onto functions, increasing and strictly increasing functions, decreasing and strictly decreasing functions, and monotonic and strictly monotonic

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functions. He also justified why functions that are not one-to-one on an interval do not have inverse functions when restricted to that interval. This presentation was a lecture with no student discussion. 8.3. Activity 3: the always-sometimes-never task The goal of the always-sometimes-never task (See Activity 3 in the Appendix) was for PSTs to think about the conceptual meaning of strictly monotonic and one-to-one functions while also pondering the relationship between continuity, strict monotonicity, injectivity, and invertibility. Our goal for this activity was not to have students produce the correct answers (indeed, the correct answer to task (iii) depends on one’s conception of whether functions with restricted domains like g(x) = ln x count as inverse functions), but to engage in a productive struggle that would lead them to discuss the meaning and relationship between these terms. The mediating processes that we anticipated in Table 1 were realized, which we illustrate with transcripts of parts of two group’s conversations below. Here is one group considering the first question. [1] PST13: So strictly increasing means that [short pause]… [2] PST9: It’s going up, dude! It’s going up. [3] PST13: so there’s never going to be a uh… [4] PST9: A stagnant point or a dropping point. [5] PST13: Well I was going to go off of the 1-to-1 aspect. Like there will never be a not 1-to-1 point. [PST6: Yeah] It will always be 1-to-1. [6] PST6: Yeah, because it’s always increasing. [7] PST9: Wait, didn’t we just give a counterexample to that? [referring to the constant function that the instructor provided, illustrating an increasing function that is not strictly increasing]. [PST13: No] [8] PST6: No, because this one says strictly increasing. If we didn’t have that, yes. [9] PST13: Strictly increasing is 1-to-1 so it will always… [S3: Always]. [10] PST9: It will always have… so you’re saying that if it’s always strictly increasing, then there will exist an inverse function… [11] PST13: It will always be 1-to-1 then. And if it’s 1-to-1, then… [12] PST9: Are you sure about that? [13] PST8: Well, we’re given that the inverse is a function, but do we know that its… [14] PST9: [Reading the directions of the exercise] Fill in the following statements with always, sometimes, never. Draw any examples or counterexamples. It doesn’t have to have an inverse function. [15] PST8: I know that if it does have an inverse, it will be a function. [16] PST9: Wait… [calling the professor] Dr. M, are we assuming that f is a function? [17] Professor: Yes. Yes. [18] PST9: Okay. Just asking. So always. [19] PST13: So if the function is strictly increasing, it will always have an inverse. We see in lines [1] through [4], the PSTs are developing a graphical interpretation of strictly increasing functions (“it’s going up”) and consequences of this graphical interpretation (“there’s never going to be a stagnant point or a dropping point”). In [7] and [8], PST6 helps clarify PST9’s confusion about whether a constant function qualifies as an increasing function or a strictly increasing function. In the exchange between [13] and [18], PST9 is clarifying his understanding of the question. In [9] through [12], the group is discussing the relationship between strict monotonicity and injectivity. We suggest that the students changed their understanding of the meaning of a definition and explored relationships between the concepts. To illustrate PSTs wrestling on the remaining question, consider another group of PSTs addressing whether a continuous function that is not strictly monotonic on an interval will have an inverse function on that interval: [1] PST12: I think four is never, just because its continuous, if its not strictly increasing or decreasing, it could be like [draws a function constantly changing direction], I guess it could be like all over the place. Well, it’s like x2 again [the group used x2 as a counterexample to the claim that continuous functions necessarily have inverses]. [2] PST7: It’s not strictly monotonic but its continuous and defined on all the points of its interval, right. [3] PST2: So I guess if we are looking for an inverse function to map to I, then yeah. Yeah. Never. [pause] [4] Facilitator: So x2 would be a counterexample, right, but that would only show you that it’s not always. [Students all agree] [5] PST2: But again, if we say it’s not strictly monotonic and … and that there’s an inverse, then we have a contradiction, right? [6] Facilitator: Not strictly monotonic and… [7] PST2: And one-to-one. [8] Facilitator: Not strictly monotonic. So wait a minute. There’s a lot of terms going on. Can you… [9] PST2: Or actually. I guess. I guess sometimes. Actually. Because we can have… [draws a graph that continuously decreases on an interval (a, b), has a jump discontinuity, and continuously increases on a second closed interval (b, c), where f(b) is strictly greater than f(a)] [10] PST12: Well what if we take the sine function? It’s continuous. It’s not strictly monotonic. [11] PST2: We can have something that goes down… and comes up here.

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[12] PST12: Is that continuous? [13] PST2: Oh, it has to be continuous. After the exchange, the PSTs all agreed that the answer was a continuous function that is not strictly monotonic on that interval will never have an inverse function on that interval, but could not justify their answer. The facilitator highlighted that PST3’s failed counterexample was useful, in that it established that discontinuous functions can have inverse functions on intervals, even if they are not strictly monotonic (that is, this counterexample can be used to expand the PSTs’ example space). So we see the PSTs were able to explain what could go wrong if a continuous function was not strictly monotonic (such as the counterexamples discussed in [1] through [3]) and recognize that both continuity and a lack of strict monotonicity needed to be in place to prevent a counterexample ([9] through [13]) they could not successfully provide a general argument. It is this general argument that the instructor provided next in his proof of the main theorem. 8.4. Activity 5: applying the theorem to secondary mathematics In Activity 4, the professor presented the main theorem that a continuous function restricted to an interval has an inverse function if and only if it is strictly monotonic on that interval. In Activity 5, the PSTs were expected to apply the lessons that they learned to recognize that the arcsine function was only the inverse of the sine function on a restricted domain. All groups were able to correctly conclude that sine would only have an inverse function on a restricted domain. (The correctness of this conclusion is dependent upon our conception of inverse function). Some groups did so by applying the main theorem covered in Activity 4, as the following transcript clearly illustrates. [The PSTs sketch a graph of the cosecant function, believing that this is the arcsine function] [1] PST5: Since f is not one-to-one, we should be able to find that domain A that they’ve been talking about. [2] PST15: Like… negative pi over 2 to… pi over 2? [3] PST5: Yes. [4] PST15: Like right here, right? [5] PST5: Yes. From here to here [from −π/2 to π/2], because that’s strictly increasing. Because it contains zero and it contains a positive number. And because it is the biggest interval. We use this excerpt to illustrate two themes. First, to find the correct domain to restrict the sine function, this group was able to apply the main theorem correctly. Second, after this dialogue, the group graphed the arcsine function as the cosecant function, which indicated that there still was some remaining disconnect with the notion of an inverse function. (Namely, the group expressed the 1 confusion reported in the literature that f 1 (x ) = f (x ) . See, e.g., Engelke et al. (2005); Zazkis and Kontorovich (2016). Because of this, the group concluded the arcsine function was the cosecant function restricted to the domain [−π/2, π/2]. This difficulty was only resolved after the instructor came to the group’s table and discussed their answer with them. This illustrates how the correct application of real analysis needs to be coordinated with an accurate understanding of secondary mathematics to yield insight into secondary mathematics, a theme we return to in the Discussion section. We can contrast this group’s performance with another group who answered the questions by relying on resources other than the main theorem: [1] PST2: You’re supposed to write the function arcsine x. Do you know domain restrictions? [2] PST12: No, that’s always just what I do. You turn it to the right and you graph it. [PST12 and PST2 both laugh]. That’s what I was always taught. [3] PST2: Does anyone remember? [4] PST12: No. Please share your wisdom with me because now I’m concerned. [5] PST7: About what? The graph of arcsine? [6] PST12: Yeah. [PST2] said my graph was wrong. [7] PST2: Oh. I said it wasn’t a function. [8] PST11: Isn’t it like the opposite though? [9] PST2: Yeah, but it has to be a function, right? Because it’s g of x? [10] PST11: But that’s so [inaudible] (referring to the sine function). That’s a function. [11] PST2: (clarifying) Well, the arcsine. [12] PST12: That’s what I’m asking. [12] PST11: That’s what I thought! It’s this way! [13] PST2: That wouldn’t be a function. [14] PST12: We have to limit it. The range. [15] PST11: Oh. This is not a function because… [pause] the vertical line test. [16] PST2: Slam! [All students laugh] [17] PST11: But we have to restrict the… range. That sounds weird. We have to restrict the range.

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[18] PST2: I’m okay with that. [inaudible cross talk] Of what function? [19] PST11: Oh, of arcsine. We restrict the domain of sine, which in turn is restricting the range of arcsine. [PST12 says this in unison]. [20] PST12: So we have to restrict the domain of sine. [PST2: To fit the] From negative pi over 2 to pi over 2. A short time later, after the students explained their work to the facilitator, the facilitator asked how their work related to the main theorem. [21] Facilitator: Can you relate that the theorem that [the professor] proved? That when a function is continuous that it has an inverse on an interval that’s monotonic. [22] PST12: Well when we’re restricting the domain of sine to negative pi over 2 to pi over 2, the domain of sine is strictly monotonic so the inverse, the arcsine, will be a function. [23] Facilitator: If it was bigger than minus pi over 2 to pi over 2, so if you extended the domain in either direction, would it still be monotonic? [All PSTS]: No. [24] PST12: Can we repeat that? Because that was a lot of words. [25] PST11: When we restrict the domain of sine to like negative pi over 2 to pi over 2 [PST12: yeah], that’s when sine of x is strictly monotonic and by [Theorem] Two, the inverse of f will be a function. [26] PST11: Okay. What we see in this excerpt is that the PSTs are able to see the need to restrict the domain of the sine function to find an inverse function, but the reasoning that they use does not invoke the main theorem. We discuss three noteworthy themes from this transcript. First, PST11 and PST12 rely on a “rule” for finding inverses to justify the common erroneous beliefs amongst students that sine and arcsine are inverse functions of each other—see lines [2], [5], and [6], particularly PST12’s comment, “that’s what I’ve always been taught.” Second, the recognition that the symmetric graph that they produced was not a function (i.e., the inverse sine relation) relied not on the main theorem or the ideas discussed in previous activities, but instead on the “vertical line test”, a heuristic taught in secondary mathematics for determining whether a graph of a relation in the Cartesian plane represents a function (line [15]). The PSTs actually initially resolve the failure of the vertical line test by restricting the range of the arcsine function (line [17]) before realizing this is tantamount to restricting the domain of sine (line [19]). Third, when the facilitator explicitly prompts the students to consider the main theorem (line [21]), they are able to relate the main theorem to their work (line [25]). This group’s response to the next task, where we provided a transformed sine function, was similar. The transformation included a horizontal dilation and both horizontal and vertical shifts. [1] PST11: I want to go from, I don’t know. I don’t know if I’m getting this. [2] PST7: I think it’s like negative pi over 8 to… [3] PST2: It needs to pass the Horizontal Line Test. [4] PST12: [Long exasperated groan] I forgot about that. [5] PST7: This is my guess. [6] PST2: Did you get pi over 8? Yeah, this is what I got too. [Other students agree] [7] Facilitator: Okay, so you, [S7], drew it a little bit to the left of the zero [S7: Yeah] and to a little bit right to the pi over 4. [S7: Mm-hmm]. And can you explain why you drew that? [8] PST7: Well, if you make the domain any bigger, h(x) will not be 1-1. So then the inverse… [9] PST11: It basically will not fit these criteria because if you extend the domain, it will not be strictly monotonic. [10] PST7: Oh! We see similar trends in this transcript. The students initially solve the problem using the Horizontal Line Test (line [3]), a heuristic used in secondary mathematics to determine when the graph of a function represents a one-to-one (and consequently invertible) function. After an appropriate interval is found, S11 justifies the domain choice in terms of the main theorem, but this was not the way the interval was generated. 8.5. Activity 7: finding specific solution values to a trigonometric equation For the sake of brevity, we discuss the group work on this activity concisely. The groups largely behaved in the manner we anticipated in Table 1 and they were able to use the ideas from class to successfully answer these questions. One point of difficulty that all groups had was in finding the solution that could be obtained by considering the periodicity of sin(2x) (i.e., Solution F—see the Appendix). The students had difficulty with the algebra of finding the x-value that would be obtained by reflecting solution E across the line x = π/4. As we did not want algebraic complications to sidetrack students, in our future iterations, we revised this module to have Activity 7 use cosine instead of sine, as reflecting across the line x = 0 makes it easy for PSTs to find the reflected solution value.

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Fig. 5. One PST’s illustration of a graphical argument to illustrate to a student how they missed a solution.

9. Learning goals: pre-test/post-test comparison 9.1. Response to a students’ incorrect solution Here we consider the performance of the ten PSTs who completed the pre-test and the post-test. On the Pre-Test, only two of the ten PSTs correctly identified at least one error in the students’ work. One PST identified the student’s work as incorrect and presented a graphical argument to identify all solutions to the trigonometric equation that the student missed. The other PST identified that the student added “+2πk” too early in their solution. The remaining eight PSTs did not correctly identify an error in the student’s work: five PSTs thought the student’s work was correct and the other three PSTs indicated the student’s work was correct until a “+2πk” was added to the solution. On the post-test, all ten PSTs indicated that the solution that the student presented was not correct. Five PSTs identified both errors in the students’ work. Four identified one error. The remaining PST indicated that the student’s work was incorrect and directed the student to “think about the restriction of values that cos(t) can give and why we add 2πk”, but this PST did not identify a specific mistake the student made. On the post-test, six of the ten PSTs provided feedback to student to help them understand why the student’s work was incorrect. Five PSTs presented a graphical argument illustrating why the student’s work missed a class of solutions, as we illustrate with PST5’s work in Fig. 5 below. We present this data to illustrate how many of the PSTs could not only correctly identify errors in a student’s solution on the post-test, but many chose to help the student understand why their work was erroneous, rather than simply say they

Fig. 6. PST3’s explanation of the arcsine function on the Pre-Test. 14

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Fig. 7. PST3’s description of how she would help secondary students understand the arcsine function on the Post-Test.

Fig. 8. PST10’s description of how she would help secondary students understand the arcsine function on the Post-Test. 15

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made a mistake in the procedure that they applied. 9.2. Explaining the arcsine function On the pre-test, only one PST mentioned the domain restriction of the sine function; she provided a mathematically correct explanation of what the arcsine function was. The other nine PSTs all described arcsine as the inverse function of sine. The response from PST3 in Fig. 6 below was typical of many responses. PST3 first reminds students that an inverse function is one that reverses the input-output relationship, then describes the arcsine function as the inverse function of sine, and finally has the students practice by computing specific values of arcsine. On the post-test, nine PSTs included in their explanation that the arcsine function was only the inverse of the sine function on the restricted domain [−π/2, π/2]. (The remaining PST only gave a vague acknowledgment that sine and arcsine were not complete inverses of each other by stating “These two relationships, sine and inverse sine, can ‘undo’ each other, to a certain extent”. We did not code the phrase “to a certain extent” as an acknowledgment of domain restrictions). Although this was not a specific learning goal of the Inverse Functions Module, we note that the PSTs’ explanations often had attributes of what are typically thought of as good instruction. Specifically, PSTs gave conceptual justifications for why domain restrictions were necessary and they often used graphical representations to do so. Consider, for instance, PST3’s and PST10’s explanations in Figs. 7 and 8 respectively. In these explanations, both PSTs invite students to recognize that the sine function could not have an inverse in general, and then to see how by restricting the domain of the sine function, an inverse function is possible. Nine of the ten PSTs provided mathematically correct explanations for why domain restrictions were necessary if the sine function was to have an inverse function. Like PST3 and PST10, five PSTs made graphical arguments. Finally, note that both PST3 and PST10 sought student involvement while providing their explanations. PST3 would have students draw the graphs of sine and arcsine, see how they were related, and explain why domain restrictions were necessary. Two other PSTs used non-graphical representations, specifically a unit circle and a table, to illustrate why the sine function could not have an inverse function. Similarly, PST10 would have the idea that the arcsine function would not be the inverse function of the sine function with unrestricted domain come from the students. The main point here is that not only were PSTs providing a mathematically correct explanation for what the arcsine function is, they were also providing explanations for why this is the case using representations appropriate for the population that they were teaching. In general, like PST3 and PST10, the PSTs would have the students participate in some way when the explanation was provided. For instance, PST3 would have the students generate the graph of the sine and arcsine functions and propose explanations for why the domain of sine needed to be restricted. We believe this suggests a general theme. The PSTs, in general, valued conceptual understanding, graphical representations, and active learning. When they developed a conceptual understanding of the topics that were discussed, they were able to apply these values to the feedback and explanations that they provided to students. 10. Discussion We conclude the paper by discussing the main themes that can be drawn from this and how these relate to our theoretical model. 10.1. Prospective teachers’ initial difficulties with the arcsine function and the solutions to trigonometric equations Although the goal of this study was not to document weaknesses in PST’s knowledge, the responses of the PST do illustrate these. When explaining how they would introduce the arcsine function to their class on the pre-test, most PSTs focused on saying the arcsine was a specific case of finding an inverse function. This would encourage their students to develop the unproductive conception that the sine and arcsine functions were unqualified inverses of one another. Further, most of the PSTs could not recognize errors in a mistaken solution to a trigonometric equation. Introducing the arcsine function and grading students’ solutions to trigonometric equations are pedagogical tasks that secondary mathematics teachers will need to engage in, so remedying this gap is important. 10.2. Achievement of the learning goals We chose two learning goals that approximated mathematical practice. First, we wanted students to provide a mathematically accurate explanation that they would give to a class of secondary students. This goal was largely successful. Nine of the 10 PSTs provided mathematically accurate explanations and the majority framed their recommendations in representation systems accessible to secondary students or in the context of student-centered activities or both. The second goal required students to provide feedback to an incorrect student solution to a trigonometric equation. This goal was partially achieved. All ten participants recognized that the solution method was wrong on the post-test and most provided feedback in terms of conceptual reasons that explained why the student’s solution was wrong. However, only five of the ten PSTs identified both errors in the student’s solution. Overall, PST’s performance on the post-test provided evidence that we partly ameliorated PST’s unproductive conceptions of the arcsine function and had them better prepared to respond to tasks that approximated their future pedagogical practice.

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10.3. Evaluating our design conjectures and theoretical conjectures In the results section, we argued that our design conjectures were largely confirmed. In Table 1, we listed behaviors we anticipated to see as PSTs worked on our activities. In the results section, we verified that PSTs did indeed engage in the predicted behaviors. An important exception was Activity 5. We predicted that PSTs would use the theorem presented in Activity 4 to choose domain restrictions for functions that were not one-to-one. In fact, some groups of students used ideas from secondary mathematics that we did not cover in our module (e.g., the Horizontal Line Test) to answer the question. Our theoretical conjecture (i.e., the relationship between the mediating processes and their learning outcomes) was that by puzzling about the nature of invertible functions, seeing the theorem presented in Activity 4, and then applying the theorem in Activity 5, participants would be able to complete the approximated pedagogical tasks on the post-test. However, our data are consistent with the claim that the theorem and proof presented in Activity 4 were expendable. In our theoretical model presented earlier in Fig. 4, we could perhaps remove the top “advanced mathematics” tier to our model and still achieve our learning goals. (We are not saying this is necessarily so, only that some of our data is consistent with this possibility). The preceding paragraph raises an important open question. Should we applaud the PST’s adaptive use of their secondary mathematics knowledge to solve a problem of secondary mathematics? Or should we be concerned that PSTs are relying on a “rule” that is often not backed with understanding? These questions are especially pertinent as the main theorem covered in Activity 4 is useful, but not necessary, for doing the work in secondary mathematics that we described. We revised our module to take a middle ground. On the one hand, we added questions for Activity 5 that ask PSTs to relate their work to the main theorem from the lesson. (We cannot count on a facilitator being there to prompt this question). We also suggest in our module that the instructor review the Horizontal Line Test, justify why it is true, and relate it to the mathematics being covered. 10.4. Connecting advanced mathematics with secondary mathematics We conclude the paper on a speculative note. Recall for Activity 5, PST5 and PST15 correctly applied the theorem in Activity 4, but nonetheless answered the question incorrectly because they believed that the inverse function of the sine function was the cosecant function. This illustrates how an understanding of advanced mathematics could not help the PSTs cope with a pedagogical situation if there were more basic gaps in their understanding (a point also made in Wasserman, 2017b). We briefly note that we observed a similar phenomenon in a prior iteration of the Inverse Functions module. In that iteration, we did not include Activities 6 and 7. We were surprised that the PSTs in this iteration of the study recognized that “taking the arcsine of both sides” only found the solution to equations of the form sin(x) = a on the interval [−π/2, π/2]. However, they did not know how to find the remaining solutions. We thought of applications of the periodicity and symmetry of the trigonometric functions as “basic secondary mathematics”; we realized that this topic needed specific attention, which we now provide in Activities 6 and 7. We use these observations to tentatively propose one account for why advanced mathematics courses sometimes might not benefit PSTs. Analysis in advanced mathematics typically is general and abstract, concerning as broad a set of objects as possible (e.g., theorems about continuous functions). Work in secondary mathematics often concerns a more narrow set of objects (e.g., how to solve a quadratic equation, how to find inverse functions of x2 and sine after their domains are restricted). The value of advanced mathematics often consists of applying abstract ideas to concrete situations and coordinating this application with one’s knowledge of secondary mathematics. This secondary mathematics material is usually not the content of an advanced mathematics class, both because this tends to concern specific objects (e.g., the periodicity or symmetries of sin x or x2) and because this material is too elementary to warrant coverage. However, if PSTs do not have a deep understanding of the secondary mathematics, they will not be able to reap the benefits of applying the more abstract ideas from advanced mathematics to secondary mathematics situations. Acknowledgments This material is based upon work supported by the National Science Foundation under collaborative grants DUE 1524739, DUE 1524681 and DUE 1524619. Any opinions, findings, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We would like to thank the reviewers for their extensive insightful and constructive criticism. Appendix. Lesson plan and activities distributed to students Activity 1: Building up. Teaching secondary mathematics Classroom Scenario:

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Source: The graphics used in this figure are © 2018 the Regents of the University of Michigan, used with permission. Images were created with the Depict tool in www.lessonsketch.org. Q1A. Evaluate the students’ work – is it correct or incorrect? Provide feedback to the students and the class if there is anything incorrect. Q1B. You are introducing the inverse sine function to the student. Discuss how you would explain this function to the student? Be specific. Activity 2: Building up. Secondary mathematics Monotonicity, domain restrictions Q2. Suppose f and g are inverse functions. What does this mean to you? Are sin⁡(x) and arcsin⁡(x) inverse functions? (Note: arcsin⁡ (x) is also denoted as sin−1(x)) Explain your answers. Activity 3: Building up. Secondary mathematics Q3. Fill in the following statements with: always, sometimes, or never. Draw any examples or counterexamples that you consider. i. If f is strictly increasing on an interval I (and defined on every point of I), it will ____________ have an inverse function. ii. If f is continuous on an interval I (and defined on every point of I), it will ___________ have an inverse function. iii. If f is continuous and strictly monotonic on an interval I (and defined on every point of I), it will ______________ have an inverse function. iv. If f is continuous and not strictly monotonic on an interval I (and defined on every point of I), it will ______________ have an inverse function. Activity 4. Advanced mathematics These proofs were presented to students: Theorem 1. Let A

. Suppose f(x) is a function from A to

. If f(x) is strictly monotonic, then f(x) has an inverse.

Proof. By definition of inverse, f(x) will have an inverse if it is one-to-one. We will prove this for the case where f(x) is strictly increasing. A similar proof follows for f(x) being strictly decreasing. 18

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Choose two distinct points x1 and x2 in A with x1 < x2. We need to show f(x1) ≠ f(x2). Since x1 < x2 and f(x) is strictly increasing, we know f(x1) < f(x2) so f(x1) ≠ f(x2), as needed. Example 1. f(x) = 2 is monotonic on any interval but it is not one-to-one and has no inverse on any interval. Theorem 2. Let A such that A is an interval. Suppose f(x) is a continuous function from A to only if f(x) is strictly monotonic.

. Then f(x) has an inverse if and

Proof. The ⇒ direction follows immediately from the previous Theorem. Suppose f(x) is strictly monotonic. By the previous theorem, f (x) has an inverse. For the ⇐ direction, we will prove this by contradiction. We will assume that f(x) is continuous on A, f(x) is not strictly monotonic, and f(x) is one-to-one, which is equivalent to f(x) having an inverse. If for all a, b, and c in A with a < b < c, f(a) < f(b) < f(c) then f(x) would be strictly increasing. Similarly, if for all a, b, and c in A, f(a) > f(b) > f(c) then f(x) would be strictly decreasing. Thus, there must exist a, b, and c in A with a < b < c where one of the two following conditions occurs: (i) f(a) < f(b) and f (b) > f(c) or (ii) f(a) > f(b) and f(b) < f(c). We will prove the theorem in case (i) and leave case (ii) as a homework exercise. Since f(x) is not one-to-one, f(a) ≠ f(c). Suppose f(a) > f(c). (The case for f(a) < f(c) is similar). Note that f(c) < f(a) < f(b). Note also that f(x) is continuous and defined on the interval [b, c]. (This is a consequence of A being an interval). By the intermediate value theorem, there is a d in the interval [b, c] such that f(d) = f(a). But since d is in A, this contradicts that f is one-to-one. Activity 5. Stepping down. Secondary mathematics If we want an inverse for a continuous real-valued function f, but f is not one-to-one, then by convention, we seek to find the largest interval A in the domain of f on which: (i) f is strictly monotonic, (ii) A contains 0, and (iii) A contains at least one positive number. We then restrict the domain of f to A and construct the inverse of the restriction. Q5a. Sketch the graph of f(x) = sin⁡(x) and g(x) = arcsin⁡(x). In what sense are f(x) and g(x) inverse functions? Is it always the case that f(x) = y implies that g(y) = x. Q5b. Consider the graph of h(x) below.

(a) Find the interval A that represents the conventional domain on which h(x) has an inverse function. (b) If we restricted h to the following domains, would the restriction have an inverse function? (i) [0, 0.1] (ii) [1, 3] Activity 6. Advanced mathematics This theorem and example were presented to students: Theorem 3. Suppose we have a continuous real-valued function f(x) that is not one-to-one. We find a partial inverse f−1(x) by restricting the domain of f(x) to an interval A such that f(x) is strictly monotonic on A. When solving f(x) = a, applying a partial inverse function f−1(x) to both sides of the equation will give one solution to the equation but not necessarily all solutions. The solution that is found, x = f−1(a) will be the unique solution on A. Proof. We first show that x = f−1(a) is a solution. We then show that if x is in A and is a solution to f(x) = a, then x = f−1(a). Since f−1(x) is an inverse of f(x) on A, we have that f(f−1(x)) = f−1(f(x)) = x for all x in A. If x = f−1(a), f(x) = f(f−1(a)) = a, so x = f−1(a) is a solution. If x is a solution to f(x) = a, f−1(f(x)) = f−1(a), which implies x = f−1(a). Example. cos 2x = .5. Illustrate this with a graph of cos 2x. The largest conventional interval to restrict f(x) = cos 2x to is [0, π/2]. To find where f(x) = .5 on the domain, arccos (cos 2x) = arccos(.5) 19

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2x = π/3 x = π/6 To find another solution, take advantage of the symmetry of the x-axis. x = −π/6 To find all solutions, use periodicity of f(x) (NOT cos x), which has period π. x = π/6 + 2πk, −π/6 + 2πk Activity 7. Stepping down. Secondary mathematics Q7. Consider again:

sin(2x ) = 0.7 a. b. c. d. e.

If f(x) = sin⁡(2x), from the graph of f(x), how can we restrict the domain of f(x) so that f(x) has an inverse function? What is the solution to f(x) = 0.7 on this domain? How can we use the periodicity of f(x) to find some of the remaining solutions? How can we use the symmetry and periodicity and of f(x) to find the remaining solutions? On the graph below, we labeled eight different solutions as A, B, C, D, E, F, G, and H. Which solutions were found in step b above? Which solutions were found in step c? Which solutions were found in step d?

Appendix. Pre-test The pre-test was identical Activity 1. Appendix. Post-test 1. You are grading the following student homework assignment. Give feedback to the student on his work. 1. Solve the following equation: cos⁡(3x) + 1 = 1.4 cos⁡(3x) + 1 = 1.4 cos⁡(3x) = 0.4 arccos⁡(cos⁡(3x)) = arccos⁡(0.4) 3x = 1.159 x = 0.386 + 2πk

2. You are introducing the inverse sine function to the student. Discuss how you would explain this function to the student? Be specific.

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