Connecting Secondary and Tertiary Mathematics: Abstract Algebra and Inverse

Eileen Murray Matthew Wright Debasmita Basu Montclair State University Montclair State University Montclair State University

This study explores how practicing teachers make connections between secondary and tertiary mathematics. Using three frameworks for teacher knowledge of mathematics, coupled with key developmental understandings (KDUs) (Simon, 2006) as related to teacher knowledge (Murray & Wasserman, 2016), we observe how a professional development workshop focused abstract algebra impacts teachers’ understanding and teaching of secondary mathematics. Key Words: mathematical connections, professional development, teacher knowledge

Within the mathematics and mathematics education communities, ongoing consideration has been given to the knowledge secondary mathematics teachers require to provide effective instruction. At the focus of this debate is what mathematical content knowledge secondary teachers must have in order to communicate mathematics to their students, evaluate student reasoning, and make informed curricular and instructional decisions. Many believe that mathematics teachers should have a solid base of mathematical knowledge and mindfulness of how tertiary mathematics is connected to secondary mathematics (Papick, 2011). But some have shown how more mathematics preparation does not necessarily improve instruction (Darling-Hammond, 2000; Monk, 1994). Thus, questions endure about what connections are between secondary and tertiary mathematics are important and how knowledge of these connections may impact classroom practice. While there are many who support the notion that mathematics preparation for secondary teachers must involve knowledge of vertical connections, less is known about how inclusion of courses such as abstract algebra in teacher preparation programs may bring this about.

Framework

In order to explore the nature of vertical connections in mathematics, we draw on three areas of teachers’ knowledge of mathematics research: Mathematical Knowledge for Teaching (MKT) (e.g., Ball, Thames, & Phelps, 2008), Advanced Mathematical Thinking (AMT) (e.g., Zazkis & Leiken, 2010), and Knowledge of Algebra for Teaching (KAT) (e.g., McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012). These frameworks, along with the research on key developmental understandings (KDUs) (Simon, 2006) as related to teacher knowledge (Murray & Wasserman, 2016), provide a way to think about how understanding of tertiary content may impact teachers’ understanding of the teaching and learning of secondary mathematics.

Building on Shulman’s seminal work on teacher knowledge (1986, 1987), Ball and colleagues (2008) conceptualized the domains of MKT according to elementary mathematics. Although there exist many challenges in translating the definition of these domains in a secondary context (Baldinger), our current work utilizes MKT constructs to better understand the nature of knowledge needed for secondary teaching. In particular, we consider the development of horizon content knowledge (HCK) as it pertains to secondary mathematics through exposure to tertiary mathematics. According to Ball, Thames, & Phelps (2008) HCK is “an awareness of how mathematical topics are related over the span of mathematics in the curriculum” (p. 403), but this construct varies considerably from the elementary to secondary level (Howell, Lai, &

Phelps, 2008). For the purpose of the current work, we use the definition of HCK as rendered by Jacobson et. al., (2013), as “an orientation to and familiarity with the discipline (or disciplines) that contribute to the teaching of the school subject at hand, providing teachers with a sense for how the content being taught is situated in and connected to the broader disciplinary territory” (p. 4). We use this definition to consider how instruction in abstract algebra coupled with secondary tasks prompting teachers to apply this knowledge may develop an awareness of connections between abstract algebra and secondary mathematics. We believe that knowledge of specialized content such as abstract algebra may provide secondary teachers with a better understanding of the mathematical horizon as it pertains to their teaching of secondary algebra.

Even with more students taking algebra, the preparation of algebra teachers is still not well researched (Stein et. al., 2011 in McCrory et. al., 2012). This lack of research on teacher preparation and shortcomings of existing frameworks serves as motivation for the development of KAT. This framework contains three domains of knowledge believed to be essential to the teaching of secondary algebra: school algebra, advanced mathematics, and algebra for teaching (McCrory, et. al., 2012). Knowledge of advanced mathematics is pertinent as this is the knowledge that provides teachers with “some perspective on the trajectory and growth of mathematical ideas beyond school algebra” (McCrory et. al., 2012, p. 597), much like HCK. This knowledge is of special significance because many students experience difficulty transitioning from high school to college mathematics and many teachers perceive the undergraduate mathematics that they themselves learned as immaterial with regard to their teaching practice. For these reason, and because secondary teachers are typically required to graduate with an undergraduate degree in mathematics, we also draw upon AMT, defined as “knowledge of the subject matter acquired in mathematics courses taken as part of a degree from a university or college” (Zazkis & Leiken, 2010, p. 264).

Finally, in order to better understand how exposure to abstract algebra content may impact secondary teachers’ knowledge of teaching mathematics, we consider mechanisms by which teachers develop awareness of connections and how this awareness may impact instruction. To accomplish this, we utilize the construct of KDUs (Simon, 2006). The essential characteristics of KDUs to teachers and teaching are that a KDU must involve a conceptual advance on the part of the teacher, and that without the knowledge, teachers must build their understanding through activities and reflection rather than explanation or demonstration. We posit that through awareness of connections between secondary and tertiary mathematics, it is possible for secondary teachers to develop KDUs, thereby furthering their understanding of and ability to teach secondary mathematics.

Our current work explores how in-service teachers make connections between secondary mathematics and abstract algebra. This work builds on the results of a smaller pilot study, which revealed an interesting change in participants’ understanding of various mathematics concepts, including inverse. In the previous and current work, frame participants’ understanding of inverse using the APOS framework (Asiala, Brown, DeVries, Dubinsky, Mathews, & Thomas, 1997). The APOS framework conceptualizes an individual’s understanding of mathematical content according to four levels - action, process, object, and schema. At the action level, inverses are used algorithmically to perform mathematical tasks such as solving equations, e.g., multiply both sides of an equation by the multiplicative inverse. As a process, inverses are viewed as both an operation and mathematical property of equality. At the object level, inverses are understood as elements within a set defined with respect to a binary operation. Finally, at the schema level, a comprehensive understanding of inverse is attained and the operational/elemental duality of

inverse is understood as well as its utility as a mathematical property of equality (Wright, Murray, & Basu, 2016).

In our pilot, we found that some participants initially discussed inverses through an action or process-level of understanding. After engagement in activities focused on the algebraic structures of groups, rings and fields, participants began to consider inverses as objects. We used this finding to further explore how an extended professional development workshop highlighting the connections between abstract algebra and secondary mathematics may not only change teachers' ideas about the inverse concept, but also influence their thinking about mathematics teaching and learning. The research questions for this study are: (1) How does understanding of tertiary mathematics change teachers’ knowledge and teaching of secondary mathematics concepts? (2) How does exposure to and instruction in tertiary content, specifically abstract algebra, change the way teachers understand and teach the concept of inverse?

Methods

To answer these research questions, we conducted a four-day professional development (PD) workshop with four in-service teachers (Conor, Dylan, Orlaith, and Aidan) from an urban high school in northern New Jersey. The workshop consisted of four three-hour sessions held on consecutive days prior to the beginning of the school year.

Data Collection During the workshop, participants worked through three packets of activities that included

scripting tasks (Zazkis, 2013) (e.g., extend an imaginary interaction between a teacher and students in a form of dialogue, including explanations and/or examples), secondary content activities (e.g., describing the mathematical properties used to solve a multistep linear equation), and tertiary content activities. The tertiary activities focused on abstract algebra content including algebraic structures (groups, rings, and fields) and formal definitions of binary operations, inverse relation and function. The first two packets prompted participants to consider mathematical properties used when solving equations in a secondary classroom and how these properties relate to algebraic structures. The third packet focused on connecting the content on algebraic structures to functions. The purpose of these activities was to unpack how discussions about inverses through an abstract algebra lens might help participants reconsider functions as objects rather than actions or processes. Additionally, we sought to challenge conventional thinking about what a function is and common problems distinguishing between additive, multiplicative, and compositional inverses for functions.

Participants engaged in these tasks individually and as a group. One researcher engaged participants in group discussions, which were audio and video taped. We collected all written artifacts containing participants’ responses and reflections for future analysis. Future data to be collected is classroom observations and participant interviews that will allow us to further explore how exposure to and instruction in abstract algebra content impacts instructional practice.

Data Analysis In our initial analysis, each researchers independently isolated episodes from the video

recordings that highlight teachers’ understanding of inverse and identity. Once the significance of these episodes was mutually ratified, we transcribed and analyzed the audio using an initial or “open” coding method, searching for words or phrases that showed evidence of participants’

making connections between secondary and tertiary mathematics. Open coding was the preferred method for our initial analysis as it allows for the development of tentative codes that may lead to further inquiry, thereby allowing the study to take direction naturally (Saldana, 2009).

Preliminary Results

We focus our results on the fourth day of the workshop to explore how abstract algebra impacts secondary teachers’ understanding of inverse. Similar to our pilot, we found participants moved from an action/process level toward an object level of understanding of inverse. To highlight how the participants’ understanding evolved, we divide our results into three sections: understandings of the concept of inverse and inverse function as action/process, the transition in understanding towards object, and impact on instructional practice.

Initial Understanding of Inverse At the beginning of the session, participants read and responded to a scripting task capturing

the conversation between a teacher and three students regarding the inverse of the function f(x) = x2. Students in the task provided three answers for the inverse of this function: 1/x2, -x2, and √x. After considering this scenario, Dylan, commented, “Only one of them [1/x2, -x2, or √x] is the inverse of the function f(x) = x2, but they all have different thinking. For example, x2 and -x2. When you combine them it is 0 but does that mean that that’s an inverse?”

Conor viewed the inconsistency of responses as a lack of “conciseness in mathematical vocabulary” rather than “just students’ misconceptions”. Based on this discussion, we see participants beginning to understand why there are inconsistencies in students’ understandings of inverse functions. In trying to unpack this ambiguity, participants exhibited an understanding of inverse at the action/process level. For example, when participants were asked to communicate their understanding of inverse, Conor referred to multiplication and division as inverse operations, while Dylan asserted, “my definition of the inverse of a function is something that undoes another function.” In providing this definition, Dylan overlooks the binary operation with respect to which the inverse is defined. He references an algorithmic approach to “undoing” without considering the operation that is being undone or the product of this undoing, arriving at the identity, thereby evidencing action-level understanding.

Transition in Inverse Concept Understanding During the discussion of abstract algebra in connection with the inverse and identity function,

participants transition from action/process to object level understanding. In the introductory scripting task described above where participants realize the ambiguity in their conception of inverse, they discuss the importance of clearly defining the term inverse. Orlaith states, “What does it mean that we are asking them for the inverse? So define the term inverse. I think that's where we would have to start.” She also notes how inverse is different for functions, but expresses her inability to provide a specific definition for this concept. Through this reflection, we posit that Orlaith’s HCK might not include the understanding of functional inverse. Specifically, she seems to be aware of how her inability to provide an accurate definition of a function’s inverse is indicative of a gap in her knowledge of the broader mathematics territory of inverses in algebraic structures. We interpret this as evidence of a transition in her understanding of the concept of inverse.

We also observe transitioning through participants’ written responses to reflection questions posed at the end of the four-day workshop. In particular, Conor writes, “You think about them in

terms of other definitions/properties that contain the words inverse or identity.” In addition, Aidan reflects on his transitioning knowledge by stating, “It is important to emphasize what operations we are working with.” Through these reflections, we claim that participants have developed a deeper awareness of inverses as either a mathematical property (process level) or as an element within a set of objects that must be defined with respect to an explicit mathematical operation (object level). Furthermore, we posit that this advance in conceptual understanding may signify the development of a KDU with regard to the higher-level mathematical underpinnings of inverse.

Impact on Instructional Practice At the conclusion of the workshop, participates seemed more aware of the connection

between tertiary and secondary mathematics than they had been at the beginning. This awareness prompts teachers to reconsider their own practice in different ways. For example, during one conversation, Dylan mentions how he will integrate newfound understanding of inverse, particularly in regard to defining inverses with respect to their identity under composition, into this year’s lessons. Upon realizing that the identity of a function under composition is the function “y=x” Dylan states, “I never taught it like that” and “I’m one-hundred percent using that this year!” When asked to reflect upon future instruction, he further states, “I will be lesson planning completely different. Our first unit is all about solving and inverse operations. I must change how I approach my introduction throughout this unit because I want my students to not have misconceptions as they approach higher levels.” In so stating, Dylan is considering how this new knowledge impacts his teaching of secondary mathematics as it may help prepare his students for higher-level mathematics, showing an awareness of the mathematical horizon.

Applications and Implications

Based on our pilot findings in which participants’ understanding of the mathematical concept of inverse evolved from action/process to object understanding, we created an extended PD workshop to push teachers’ thinking about inverse, identity, and solving equations. This extended workshop allowed us to present classroom scenarios and scripting tasks and to delve more deeply into abstract algebra content.

In the current study, participants’ understanding of inverse evolved in a similar way as our pilot participants. That is, participants began to think about inverses as objects within an algebraic structure, rather than as an action or process. The difference between the current work and the pilot came from the participant’s connection to their classroom practice. Similar to the pilot, the participants reported a newfound appreciation for being precise and consistent with their language in the classroom, especially around definition of identity, inverse, and binary operations. In the current work, the participants went even further by connecting their experiences in undergraduate mathematics classes to their teaching. As Conor stated, “I left it all in the floor in college, but it’s so important for Algebra II as well!”

As we continue our data collection and analysis, we will use classroom observations and interviews to verify that the teachers’ reports of how this new understanding of content might impact instruction. The implications of this work are a first step in helping us understand how knowledge of mathematics is related to being an effective secondary mathematics teacher. We hope to engage mathematicians and mathematics teacher educators in discussions about how this data might provide confirming or disconfirming evidence for teachers’ reports on instructional impact of the knowledge of connections between secondary and tertiary mathematics.

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