Technological , Pedagogical , and Content Knowledge ( TPACK ) and Beliefs of Preservice Secondary Mathematics Teachers : Examining the Relationships

The purpose of this paper is to examine the relationships between preservice secondary mathematics teachers’ beliefs and knowledge regarding teaching mathematics with technology. By conducting three semi-structured interviews, I investigated four preservice secondary mathematics teachers’ technological, pedagogical, and content knowledge (TPACK) and their beliefs about the nature of mathematics, learning and teaching mathematics, and technology use in the mathematics classroom. The findings of this study suggest that preservice teachers with constructivist-oriented or studentcentered beliefs about the nature of mathematics, learning mathematics, and technology use displayed higher levels of mathematical knowledge, pedagogical content knowledge, and technological content knowledge, respectively, than preservice teachers with traditional or teacher-centered beliefs about mathematics, learning, and technology use.


INTRODUCTION
With the increased need for technology integration in mathematics classroom, the role of teachers has become more important.According to the National Council of Teachers of Mathematics (2000, p. 26), "The teacher plays several important roles in a technology-rich classroom, making decisions that affect students' learning in important ways.Initially, the teacher must decide if, when, and how technology will be used."Mathematics teachers are expected to make prudent decisions when integrating technology into their teaching.Many teacher education and professional development programs offer technology courses for preservice and current teachers.These programs encourage student-centered technology uses that "support inquiry, collaboration, or re-configured relationships among students and teachers" (Culp et al., 2005, p. 302) and technology uses that enable students to engage in higher levels of thinking with less cognitive load by providing visualization and representation of problems (Jonassen, 2003).However, teachers have tended to use technology to display lesson content or support their existing practices rather than to implement inquiry-based, collaborative, or problem-solving activities and projects (Culp et al., 2005;Ottenbreit-Leftwich et al., 2010).According to the "Speak Up 2007" survey of Project Tomorrow (2008), 51% of the responding teachers primarily assign homework or drill-and-practice work using computers as a way to facilitate student learning.Project Tomorrow (2011) also compared the results from two "Speak Up" surveys (2008,2010) to show how technology use in the classroom has changed over time.Although some relatively sophisticated uses of technology (e.g., conducting investigations and creating graphic organizers) were significantly higher in 2010 than in 2008, the majority of teachers' technology uses were still limited to providing homework and practice.Moreover, the percentage of assigning homework and practice work increased from 2008.
According to Ertmer (2005), teachers' beliefs are one of the major barriers to the integration of technology into their teaching.Since technology integration has been encouraged and funded, barriers caused by external constraints (e.g., access and support) have been resolved in most schools (Ertmer et al., 2012).Thus, preservice or current teachers' beliefs may constitute a more deeply ingrained barrier to student-centered technology use than Kim / Preservice Teachers' Beliefs and TPACK 2 / 24 external limitations (Ertmer, 2005;Ottenbreit-Leftwich et al., 2010).Although the goals of mathematics teacher education include aligning preservice teachers' beliefs with student-centered learning and developing their knowledge for the effective incorporation of technology into mathematics teaching, little research has examined both preservice mathematics teachers' beliefs and their technological pedagogical content knowledge (TPACK).The purpose of this study is to investigate preservice secondary mathematics teachers' beliefs and TPACK and how they relate to each other.

LITERATURE REVIEW
In the literature, there are diverse terms that many researchers have used to define types of teaching or learning.In general, there are two contrasting sets of adjectival terms: constructivist/student-centered and traditional/teacher-centered.First, "constructivist" is a derivative of "constructivism," which refers to a learning theory.In this perspective, learners are viewed as creators of their own understanding by combining what they already believe to be true based on past experiences with what they learn from new experiences (Richardson, 1997).In addition, knowledge is viewed as a product of an individual's construction of the experiential world.Thus, mathematics is viewed as a human creation that is continually expanding."Student-centered" aligns with "constructivist" in that students are the main agents of their own learning.Student-centered approaches tend to emphasize interactive activities in which students can address unique learning interests and needs to deepen their understanding (Hannafin & Land, 1997).
"Traditional" beliefs or approaches are based on the idea that teaching is mainly the transmission of knowledge from teacher to student and that learning is the passive reception of transmitted knowledge.In traditional classrooms, teachers have authority and can control students' learning activities.From this perspective, mathematics is viewed as a collection of facts, rules, and skills that is fixed, absolute, certain, and applicable (Raymond, 1997)."Teacher-centered" approaches are closely related to "traditional" approaches in that knowledge is primarily transmitted by the teacher through telling.Teacher-centered approaches tend to focus more on content knowledge than on student thinking or processing and place "control for learning in the hands of the teacher" (Brown, 2003, p. 50).

Beliefs and Knowledge about Teaching with Technology
Although teachers' beliefs have been considered a crucial research topic in mathematics education since the 1980s (Pehkonen & Pietilä, 2003), there are various definitions of "beliefs" in the literature.Furinghetti and Pekhonen (2002) derived two types of knowledge from the definitions of beliefs proposed by 18 mathematics educators: objective and subjective.Objective knowledge is accepted by the mathematics community such that individuals are able to approach this type of knowledge and construct "their own conceptions of mathematical concepts and procedures, i.e. they construct some pieces of their subjective knowledge" (p.53), whereas subjective knowledge is informal, personal, and private knowledge that is not necessarily made public and evaluated by other people.Pehkonen and Pietilä (2003) considered beliefs as subjective knowledge that is personal, experience based, and tacit.In this study, I view knowledge as objective knowledge (i.e., formal, verifiable, or justifiable) and beliefs as subjective knowledge (i.e., individual psychological understanding about the world based on personal experience).
Belief systems.Green (1971) suggested three dimensions of belief systems: the relationships between beliefs, the degree of strength of beliefs, and the characteristics of clustering beliefs.The first dimension concerns a "quasilogical" structure of belief systems.According to Green, belief systems have a particular order: primary and then derivative.This order cannot be considered logical, however, because beliefs are arranged according to the logic in individuals' belief systems.The second dimension, the "psychological strength" of beliefs, is viewed as either central or peripheral.Central beliefs are the most strongly held, whereas peripheral beliefs can be more easily challenged

Contribution of this paper to the literature
• Mathematics teacher educators should provide learning and teaching experiences with student-centered approaches and positive experiences with technology and training to integrate technology into mathematics instruction.
• To develop preservice teachers' technological pedagogical content knowledge, mathematics teacher educators should focus on developing preservice teachers' content knowledge and pedagogical content knowledge even if neither type of knowledge includes technology.
• This study provides a foundation for further investigation of how preservice mathematics teachers' beliefs and technological pedagogical content knowledge are related and how this relationship impacts their integration of technology into their teaching.

EURASIA J Math Sci and Tech Ed
3 / 24 and changed.The third dimension is related to the claim that "beliefs are held in clusters, more or less in isolation from other clusters and protected from any relationship with other sets of beliefs" (p.48).This dimension implies that it is possible to simultaneously hold conflicting core beliefs that reside within different belief clusters.
In another perspective on belief systems as sensible systems, Leatham (2006) argued that teachers develop beliefs in ways that make sense to them.This perspective is informed by Thagard's (2000) coherence theory of justification: "To justify a belief … we do not have to build up from an indubitable foundation; rather we merely have to adjust our whole set of beliefs … until we reach a coherent state" (p.5).Beliefs are viable within a belief system when they make sense in the context of an individual's other beliefs.Thus, when contradictory beliefs in different clusters are revealed, the individual must address and rectify the conflict because sensible belief systems do not allow for overt contradictions (Singletary, 2012).
Nevertheless, teachers can hold conflicting beliefs at the same time because their beliefs "are connected to different contexts, certainty and consciousness" (Drageset, 2010, p. 32).Such concurrence of contradictory beliefs can be explained by Green's (1971) third dimension of belief systems.Drageset noted that a belief can be simultaneously derivative and psychologically central or simultaneously primary and psychologically peripheral.For example, beliefs lacking psychological strength may not influence teachers' decisions or actions even if they are primary beliefs.In this study, my view of preservice teachers' belief systems is conceptualized based on Green's (1971) structure of belief systems.This provided insight into how preservice teachers' beliefs are organized and related to their other beliefs; it also explains conflicting beliefs and inconsistencies between beliefs and behavior.

Preservice teachers' beliefs about technology use.
Preservice teachers bring "highly idealistic, loosely formulated, deeply seated, and traditional" entering beliefs about teaching and learning into their teacher education programs (Richardson, 2003, p. 6).These beliefs strongly affect how they interpret courses and classroom practices, how they translate and use their knowledge, and how they determine their later practices as teachers (Pajares, 1992, p. 310).Preservice teachers' beliefs and attitudes are also crucial factors in the integration of technology into their future teaching (Kay & Knaack, 2005).Some researchers have reported that preservice teachers tend to have limited or teacher-centered beliefs about technology use.Messina and Tabone (2015) indicated that most preservice teachers viewed technology as devices assisting teaching (e.g., providing teaching aids or creating materials to teach) rather than enhancing student collaboration, creativity, and active involvement.In Turner and Chauvot's (1995) study, preservice secondary mathematics teachers believed that successfully exploring a mathematics topic with technology requires that students already have knowledge about the topic; the teachers stated they would use technology with their students after they had taught the students how to perform or calculate by hand.
Other researchers have indicated that preservice teachers have shown confidence and intention to use technology for student-centered learning.Preservice elementary teachers in Choy et al.'s (2009) study had positive intentions to incorporate technology into their future teaching to facilitate student-centered learning.During their student teaching, however, they tended to use technology to prepare handouts, record grades and attendance, convey information, or gain students' attention rather than to facilitate student-centered learning.Preservice teachers may also choose not to use technology in their student teaching even if they are competent in its use (Crompton, 2015).This decision can be due to their beliefs or previous experiences with technology in learning or teaching.If they have few or negative experiences with technology, they may not use it in their teaching.External barriers such as lack of time or support from the teacher community and the schools (Amado & Carreira, 2006) as well as preservice teachers' lack of knowledge on how to teach mathematics (Choy et al., 2009) can also be major barriers to technology integration.Technology, pedagogy, and content knowledge (TPACK).Many studies have provided evidence that preservice teachers' knowledge about teaching with technology is another factor that strongly influences their integration of technology into mathematics instruction (e.g., Choy et al., 2009;Pamuk, 2012).The effective integration of technology into instruction requires teachers' intertwined and specialized knowledge.To identify and understand what knowledge teachers need to incorporate technology into their teaching, Mishra and Koehler (2006) developed the technological pedagogical content knowledge (originally TPCK) framework which is now known as technology, pedagogy, and content knowledge (TPACK) framework, building on Shulman's (1986) construct of pedagogical content knowledge (PCK).The TPACK framework consists of three main knowledge domains-content, pedagogical, and technological knowledge (CK, PK, and TK)-and the intersections between and among them: PCK, technological content knowledge (TCK), technological pedagogical knowledge (TPK), and TPACK (also called TPCK; see Figure 1 and Table 1).In this study "TPACK framework" refers to the whole Technology, Pedagogy, and Content Knowledge framework, "TPACK components" refers to the knowledge components that comprise the TPACK framework, and "TPACK" or "TPCK" refers to a specific type of knowledge that intersects with all three: content, pedagogical, and technological knowledge.Table 1 provides summarized descriptions of the TPACK framework components introduced by Mishra and Koehler (2006) and discussed by Abbitt (2011b).

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Niess (2005) also used the term TPACK, but viewed it as extended PCK, referring to technology-enhanced PCK.Niess (2015, p. 22) developed four different aspects that consist of teachers' TPACK, with five levels of development for each aspect: (1) an overarching conception of what it means to teach with technology, (2) knowledge of students' thinking and understanding of specific topics with technologies, (3) knowledge of curricular materials that incorporate technologies, and (4) knowledge of instructional strategies and representations for teaching subject matter with technologies.In her description of aspects and levels of teachers' TPACK, teachers' beliefs play an important role.Teachers develop their TPACK through "a constructive and iterative process" of confronting, reflecting on, and carefully revising their various experiences for teaching mathematics with appropriate technologies based on their beliefs and knowledge (Niess, 2015, p. 24).Therefore, teachers' beliefs and TPACK are closely related, and TPACK should be considered with beliefs.
Preservice teachers' TPACK.Preservice teachers in Choy et al.'s (2009) study were aware of the benefits of technology use and were not reluctant to integrate technology into their teaching.However, they lacked pedagogical knowledge and skills in planning to integrate technology into their lessons, which may have influenced their use of technology in teaching.Similarly, in Pamuk's (2012) study, preservice middle school or high school teachers demonstrated a lack of TPACK and had a difficult time developing intertwined knowledge such as TPK or PCK.Although they had well-grounded technology backgrounds, they displayed limited TPK.Pamuk suggested that the preservice teachers' lack of pedagogical experience may cause their limited TPK and argued that developing their PCK is important and should be a priority in technology integration.Table 1.Descriptions of the TPACK Framework Components (Mishra & Koehler, 2006;Abbitt, 2011b)

Components of TPACK Description
Content Knowledge (CK) Knowledge of the actual subject matter to be learned or taught, including central concepts, theories, and organizing or connecting ideas.

Pedagogical Knowledge (PK)
Knowledge of the processes and practices or methods of teaching and learning, including classroom management, development and implementation of lesson plans, and student assessment.
Technology Knowledge (TK) Knowledge of the standard and advanced technologies, including the skills to install, remove, and operate particular technologies.Pedagogical Content Knowledge (PCK) Knowledge of pedagogy that is applicable and appropriate to teaching specific content.Technological Content Knowledge (TCK) Knowledge of the manner in which technology and content relate to, influence, and constrain each other.

Technological Pedagogical Knowledge (TPK)
Knowledge of the capability of various technologies including affordances and constraints that influence pedagogical designs and strategies in a teaching and learning setting.

Technological Pedagogical Content Knowledge (TPACK or TPCK)
Knowledge of the interaction among content, pedagogical, and technological knowledge that requires an interweaving of specialized knowledge for teaching with technology.

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Relationship between Beliefs and Knowledge about Teaching with Technology Crompton (2015, p. 243) stated that "TPACK cannot be considered as a body of knowledge that exists independently of teachers' beliefs."In Abbitt's (2011a) study on preservice teachers in early childhood education, preservice teachers' beliefs and TPACK are positively correlated, and their TPACK may be predictive of their selfefficacy beliefs about technology integration.Abbitt suggested that efforts to improve teachers' TPACK, especially technology-related knowledge (TPK, TCK, and TPCK), may result in enhanced self-efficacy beliefs.Similarly, Mudzimiri (2010) found a progression in preservice secondary mathematics teachers' beliefs about technology use while their TPK, TCK, and TPACK were improved through a technology-intensive mathematical modeling and methods courses.Smith et al. (2016) investigated the relationships between preservice middle school mathematics teachers' beliefs and their TPACK.Preservice teachers' TPCK levels were the lowest among the TPACK components, and preservice teachers with sophisticated and student-centered beliefs about mathematics, learning and teaching, and technology use displayed higher CK, PCK, and TPCK, respectively, than preservice teachers with traditional or teacher-centered beliefs.Thus, preservice middle school mathematics teachers' beliefs about mathematics, pedagogy, and technology use are aligned with their levels of TPACK components.
Other researchers have found a discrepancy between teachers' beliefs and TPACK.So and Kim (2009) indicated that preservice elementary and secondary teachers recognized the advantages of student-centered learning approaches and saw the benefits of integrating technology into teaching and learning.Preservice teachers, however, faced difficulties in applying their pedagogical beliefs or understanding about problem-based learning in creating tasks and problems, integrating information and communications technology tools, and identifying their role in lesson design artifacts.Chai et al. (2013) stated that teachers displayed low levels of technology-related knowledge (TK, TPK, TCK, and TPACK) and needed robust knowledge regarding technology integration even though they possessed highly constructivist-oriented pedagogical beliefs.
Given the mixed results and few studies that have considered both beliefs and TPACK in the context of mathematics, more research on the relationships between preservice or current mathematics teachers' beliefs and TPACK is needed to clarify and understand teachers' beliefs and TPACK for their integration of technology into mathematics teaching.Most researchers have focused only on preservice teachers' pedagogical or technologyrelated beliefs even though many researchers acknowledged the importance of beliefs regarding the nature of mathematics (e.g., Ernest, 1989b;Raymond, 1997).This study is guided by the following research question: How do preservice secondary mathematics teachers' beliefs (i.e., beliefs about the nature of mathematics, learning and teaching mathematics, and technology use in the mathematics classroom) relate to their levels of TPACK components?

METHODOLOGY
The review of the literature revealed that many studies used self-report surveys to measure preservice or inservice teachers' pedagogical beliefs (e.g., Abbitt, 2011a;Chai et al., 2013).However, these and Likert-type questionnaires have been well documented as inadequate to accurately capture participants' responses (e.g., Best & Kahn, 2006;Gure, 2015;Kothari, 2004).Because individual items may be open to interpretation and numerical responses do not provide detailed information about beliefs, it is difficult to accurately measure participants' beliefs using self-report surveys or Likert-type questionnaires.In addition to beliefs, self-report surveys may not accurately reveal participants' knowledge.Among the research reviewed, many studies used self-report surveys to measure participants' knowledge (e.g., Abbitt, 2011a;Chai et al., 2013;Choy et al., 2009).However, this self-report measure, one of the frequently-used methods, has limitations.Survey items do not provide fundamental questions or statements related to specific content knowledge.Instruments of measurement need to be customized to certain content knowledge.Moreover, we cannot say the self-reporting system assesses or measures teachers' actual TPACK (Abbitt, 2011b) because a self-report is based on teachers' subjective, not objective, thoughts or selfjudgments.Thus, for the current study, a qualitative research methodology, specifically a multiple-case study, was selected in order to accurately identify, describe, and examine individual preservice secondary mathematics teacher's beliefs and TPACK components (CK, PCK, TCK, and TPCK), and to understand how they are related to each other.Yin (1984) defined a case study as "an empirical inquiry that investigates a contemporary phenomenon within its real-life context; when the boundaries between phenomenon and context are not clearly evident; and in which multiple sources of evidence are used" (p.23).In addition, a multiple-case study methodology allows a researcher to construct contextualized experiences and systemic analysis processes (Stake, 2006).Therefore, a multiple-case study methodology was the most appropriate method to identify, describe, and examine individual participants' beliefs and TPACK, and to find possible relationships between their beliefs and TPACK from acrosscase analysis (Creswell, 2013).For a multiple-case study, moreover, Stake (2006) stated that 4 to 10 cases are enough to provide substantial information on the interaction between the cases without overwhelming amounts of differences, thereby restricting comparisons.

Participants and Context
The participants who volunteered for this study were four undergraduate preservice teachers enrolled in a mathematics teacher education program at a university in the southern region of the United States.The participants, Hank, Grace, Abby, and Kate (all names are pseudonyms), were enrolled in a secondary mathematics pedagogy course focused on learning and teaching geometry, probability, and sequences and series.One of the course goals was to develop preservice teachers' knowledge about technology in mathematics teaching and learning and how technology influences student thinking and conceptual understanding.The course instructor regularly provided activities facilitating the preservice teachers' use of technology to explore mathematical concepts, including dynamic geometry environments (DGEs) such as Geometer's Sketchpad 5 (GSP; Jackiw, 2009).The participants were familiar with using technology, such as computers, calculators, the Smart Board, or GSP and with solving and explaining geometry problems.I collected data after the course was over so that (a) participants had been exposed to relevant geometric topics, and (b) there was no implied connection between participating in the study and the course grade.

Data Collection
I collected various types of data from three semi-structured interviews (beliefs, task-based, and performance interviews).Each of all three interviews lasted approximately one hour and were video-and audio-recorded.
A beliefs interview.First, to infer participants' beliefs, I used Smith et al.'s (2016) semi-structured beliefs interview protocol consisting of four categories: the nature of mathematics, learning mathematics, teaching mathematics, and technology use in mathematics class.Based on the works of Raymond (1997) and Zakaria and Musiran (2010), which investigated preservice or current teachers' teaching practices and beliefs about mathematics and pedagogy, Smith et al. (2016) developed the interview questions for the first three categories.The questions for beliefs about technology use in mathematics class were based on Landry's (2010) study on the instrument for measuring middle school mathematics teachers' TPACK.The beliefs interview protocol has 74 questions in total, including the follow-up questions (see Appendix I).
A task-based interview.I used Hollebrands and Smith's (2010) task-based interview protocol, which contains four tasks designed to assess participants' TPACK components within geometry topics (see Appendix II).In the task-based interview, students' DGE use was addressed, and the participants were asked to create activities or tasks using GSP to help the students acquire a deeper understanding of the concepts or to remedy the students' difficulties or misconceptions.I examined the participants' geometric concepts and their understanding of what the students understand and how the students think in specific technological pedagogical mathematical contexts.During this interview, each participant was given a laptop with GSP, the pedagogy course textbook, a compass, a protractor, a ruler, blank paper, pencils, and markers.
A performance interview.Finally, I designed a performance interview in which participants were asked to describe and demonstrate how they would teach the polygon exterior angle sum theorem (the exterior angle theorem: The sum of the measures of the exterior angles of a convex polygon is 360) using GSP.This interview aimed to measure participants' knowledge not covered in the task-based interview and its rubric, to find evidence to support the task-based interview results, and to investigate participants' instructional design and planning process or decision-making based on their pedagogical reasoning and their ability to teach mathematics using GSP (Harris et al., 2010).The theorem was chosen based on the pedagogy course instructor's recommendation.In the pedagogy course, the participants learned various geometric topics.The exterior angle theorem was one topic the participants did not experience in the pedagogy course.In addition, there were diverse possible strategies or ways to teach the theorem.Thus, the exterior angle theorem was appropriate to differentiate the participants' levels of TPACK components.Before the performance interview, I provided information about what participants would do in the interview.The participants were allowed to prepare teaching materials (e.g., pre-constructed GSP materials or worksheets) in advance or could develop such materials during the interview.During this interview, each participant was given a laptop with GSP, blank paper, and pencils.In the task-based and performance interviews, videos of the participants' work on their computers were recorded using a screen capture software program.I also collected any electronic files and artifacts created by the participants during the interviews.

Data Analysis
To analyze the beliefs interview data and assign codes, I used Ernest's (1989a) classification of beliefs and Goos et al.'s (2003) perspectives of technology use (see Table 2).Ernest's belief classification provided insights into mathematics teachers' views and a useful category to conceptualize teachers' beliefs about the nature of mathematics, learning, and teaching mathematics (Smith et al., 2016;Kaiser & Vollstedt, 2007).Goos et al.'s categories regarding how to use technology in mathematics class have been used to investigate how students or EURASIA J Math Sci and Tech Ed 7 / 24 teachers interact with technology in their classrooms (e.g., Morton, 2013;Nzuki, 2010) and what preservice teachers think about technology use in mathematics classrooms (e.g., Smith et al., 2016).After coding, I wrote each participant's narrative about his/her beliefs.Then, by providing each participant with their narrative, I performed a member check (Creswell, 2013).All participants agreed that I accurately captured and described their beliefs.
I analyzed the task-based interview using Hollebrands and Smith's (2010) scoring rubric.The rubric was designed to assess participants' knowledge about the geometry, pedagogy, and technology required to complete the tasks (see Appendix III).The rubric only scores content-based TPACK components: CK, PCK, TCK, and TPCK.Based on the participants' work in each of the four tasks, I assigned one of four levels (emergent, beginner, intermediate, advanced) for each of the four TPACK components (see Table 3).
To assign an overall level of knowledge to each of the TPACK components, I converted the four levels, emergent, beginner, intermediate, and advanced to numerical values, 1, 2, 3, and 4, respectively; the average was then computed.When the average was a whole number, I assigned the level of the corresponding whole number.If the average was a decimal, I assigned a whole number that is close to the decimal value.For example, when the average was 2.75, I considered it as a 3, and assigned the Intermediate level.Whenever the average was exactly in between two whole numbers, such as 2.5, I examined the participant's work across the four tasks of the TPACK components and assigned the level that most consistently represented their knowledge.For instance, when determining Grace's overall level of CK, the average was 3.5.I assigned the intermediate level because Grace lacked basic mathematical knowledge across four tasks.
Finally, I analyzed the performance interview based on three categories-content, pedagogy, and technologywith four levels (emergent, beginner, intermediate, advanced).In the content category, I focused on whether the participants had knowledge about mathematical concepts related to the theorem, a deductive or inductive proof, or connections between mathematical ideas or concepts (e.g., definitions of mathematical figures, the sum of interior angles of a triangle, the sum of interior angles of an n-sided polygon, and parallel line postulates).In the pedagogy category, I focused on what strategies the participants used to teach the theorem to their imaginary students and whether they could anticipate their imaginary students' thinking or potential difficulties.I considered whether the participants allowed their imaginary students to explore many examples to find the theorem themselves or directly provided the theorem, how they led their imaginary students to come up with theorem proofs, or what type of learning environment they provided (e.g., individual learning or collaborative learning).In the technology category, I attended to the participants' knowledge about technology, especially GSP.I observed whether the participants knew how to use GSP to implement certain tasks (e.g., constructing polygons, measuring and marking angles, and using diverse tools of GSP).I also focused on how they used technology to support their pedagogical strategies.I considered whether they used the dragging feature of GSP to have their imaginary students explore many cases and make conjectures based on those cases, or whether they used the benefits of GSP to provide their imaginary students with diverse ways to explore the theorem or helped the imaginary students intuitively understand the theorem (e.g., using parallel line postulates and a circle to show that the sum of exterior angles is 360 degrees).
For the participants' final levels of TPACK components, the results of both the task-based and performance interviews were integrated.While there are four TPACK components (CK, PCK, TCK, and TPCK) that I focused on in this study, the data from the performance interview were analyzed based on three categories (Content, Pedagogy, and Technology).Basically, I used the performance interview data to support results of the task-based interview and to determine participants' final levels of TPACK components.Thus, I considered a participant's content category level in the performance interview when assigning the participant's final level of CK.A participant's pedagogy category level in the performance interview was taken into account when assigning the participant's final level of PCK and TPCK, and the technology category level in the performance interview was considered when assigning final level of TCK and TPCK.

FINDINGS
After classifying each participant's beliefs and examining their levels of TPACK components and performance, I found that each participant's belief classification (see Table 4) and set of TPACK components (see Table 5) and performance levels (see Table 6) were unique.Table 7 shows that participants' beliefs classification and the final levels of TPACK components by combining the results from the task-based and performance interviews.In addition, there seemed to be possible relationships between certain belief categories and levels of TPACK components across data from all three interviews.In this paper, I focused my analysis on the potential relationships between participants' beliefs and TPACK components.In the beliefs interview, some participants seemed to have different beliefs simultaneously, so I was unable to assign a single code.For instance, Grace had both instrumentalist and Platonist views of the nature of mathematics.She said that the more she learned math, the more she realized that she needed to conceptually understand and find the connection between concepts rather than just calculating and following procedures.She explained that students should understand how they solve a problem and why they solve it that way: "Once students understand the reason of learning, they can figure out the relationship between mathematical concepts."She viewed mathematics as connected knowledge or concepts, which is aligned with a Platonist view of mathematics.But she simultaneously believed that mathematics is a set of formulas and calculations and that quick problem solving is important, which is consistent with an instrumentalist view.She said, "Physics is most like math because it requires critical thinking … and diagrams for visual learners.Physics also has many formulas, rules, and calculations."When I asked how teachers can develop students' critical thinking, Grace said, "Use specific examples … I can say [use] a quick way to solve problems or like mental math".She considered critical thinking as a way to solve problems or calculate quickly.

Content Knowledge and Beliefs about the Nature of Mathematics
The participants who held instrumentalist and/or Platonist views of mathematics displayed lower levels of CK, while the participant who held a problem-solving view displayed a higher level of CK.Hank, like Grace, had both instrumentalist and Platonist views of mathematics.He said, "I define mathematics as a manipulation of numbers.… There are formulas … but also an understanding of how to use these numbers."He likened mathematics mastery to building a house: "With math, you need a really good foundation … then you can build up."Abby held an instrumentalist view.She described mathematics as a puzzle that seems to be mere a set of mathematical problems: I think geometry for one … When you're finding missing angles … like finding a missing puzzle piece, that's one aspect of geometry.But then algebra, you probably don't see it as much, but you're solving for x.X is the missing puzzle piece you're always trying to solve for.
When I asked what subject is most like mathematics, Abby said, "Science and math always go hand in hand with solving problems."She did not consider the process of problem solving nor did she focus on how students construct their own solutions but instead emphasized finding the answer to the problem.Grace, Hank, and Abby displayed an intermediate level of CK across the task-based and performance interviews.For example, Grace was able to explain the properties of reflection and rotation and the process of how to perform reflections and rotations, but she seemed to have a misconception about the center of rotation (i.e., only the vertices of a figure can be the center of rotation).Although Hank found that the location of the intersection of perpendicular bisectors (circumcenter) changes, he believed that it "depends on the length of the sides rather than the angles."Abby already knew the Pythagorean Theorem and that all angles of a rectangle and square are 90 degrees, but she was unable to use this knowledge to prove why the diagonals of a rectangle and square are congruent.She displayed an advanced level of CK in the task-based interview but was unable to demonstrate her knowledge about the exterior angle theorem and its proof in the performance interview.Given evidence from both the task-based and performance interviews, Abby seemed to have an intermediate level of CK overall.
Kate held a problem-solving view of mathematics.She said mathematics "can be described as a language because of the way it can explain things."When asked what other subjects are most like mathematics, she answered, "I Kim / Preservice Teachers' Beliefs and TPACK 10 / 24 think writing proofs and all that stuff, it's more like a logical flow, so I'd say it's most like English class."She believed that doing mathematics is similar to creating something that explains the world with logical reasoning.Kate displayed advanced CK in both task-based and performance interviews.She correctly stated not only the properties of different quadrilaterals, rotation, reflection, and circumcenter, but also the inclusion relation among quadrilaterals.She was able to prove that rectangles and squares have congruent diagonals using the Pythagorean theorem and correctly demonstrate two different proofs of the exterior angle theorem (i.e., algebraic and geometric proofs) in the task-based and performance interviews.Therefore, participants with traditional beliefs about mathematics seemed to have fragmentary mathematical knowledge and were unable to connect or use what they already knew to develop theorem proofs.On the other hand, the participant with constructivist-oriented beliefs about mathematics had strong mathematical knowledge and was able to use her knowledge and reasoning to prove the theorems and fulfill various tasks in both task-based and performance interviews.

Pedagogical Content Knowledge and Beliefs about Learning Mathematics
The participants who had passive and/or active views of learning mathematics displayed lower PCK, while the participant who held only an active view displayed higher PCK.Grace, Hank, and Abby held passive and/or active views of learning mathematics.For example, describing how she learned about mod in a mathematics content course in college, Grace said she had to practice doing mod to be able to calculate and solve mod problems.When I asked how students learn mathematics, Grace described the procedures of problem solving: "You see the problem … you look at what is given and what is asked.Find a connection between the hypothesis … then you work until [you find] the solution."She viewed learning mathematics as mastering skills or procedures by repeating problem solving, which is a passive view.Hank held both passive and active views.He said that students learn through "doing again and again, repetition."At the same time, he explained that mathematics can be taught conceptually through diverse approaches including the use of technology.He added,

Let the kids struggle a little bit. That's always when I learn the most. … I would just struggle and have no clue what to do, and then all of a sudden I would get it and I would learn it.
Abby said that students "learn math through their own ways," adding that learning "is your own thing.It's not something that someone can make you do.Or they can help you, but they can't make it come to you."She recognized that learning is an individual process, not a passive reception of knowledge from teachers.She also believed that students learn mathematics through "a trial and error process" saying, "The [learning] process for them [kids] is just trying and trying and trying and until they get it."She emphasized the need for repetition to get the right answer.The participants' passive view of learning mathematics seemed to be aligned with their intermediate PCK levels.For instance, Grace could show whether students' claims were correct, but she did not know what questions to ask or what activities to use to help students develop their mathematical understanding beyond correcting their misconceptions.She also focused more on conveying the proof of the exterior theorem than on providing opportunities to explore examples.Hank was sometimes unable to identify students' geometric thinking levels, and some of his tasks did not lead students to discover properties or mathematical concepts.Without any hints or leading questions about the sum of exterior angles, Hank would directly ask students whether the sum of a triangle's exterior angles was 360 degrees, which is the core idea that students need to discover themselves.Abby did not correctly interpret students' thinking because of her misconceptions about rotation.Although she knew about the distance property of the circumcenter, she did not ask meaningful questions or provide tasks to help students find it.She was also unable to lead students to develop the deductive proof of the exterior angle theorem.
Kate held an active view of learning mathematics.She emphasized students' active learning and believed that students can learn mathematics through problem solving, sharing, and discussing their ideas with peers.She explained the learning process as follows: The students should get a sort of problem, they should try to think about it and figure out the best that they can and if possible, that should be a social thing … They could come together and talk about what they found and say, "Okay, this is what we found, this is why it's true" … So, I think it's important for them to do the figuring out and then come together to define something or say why something works.
In the task-based and performance interviews, Kate displayed advanced PCK.She was able to develop appropriate tasks or activities to help students explore diverse cases, discover properties, and fully understand mathematical concepts.In the performance interview, besides the algebraic calculations, she provided a geometric way using parallel line postulates to show that the sum of exterior angles of a pentagon is 360 degrees.Kate's beliefs about learning mathematics were aligned with what she would do in her teaching.She said that the students would EURASIA J Math Sci and Tech Ed 11 / 24 participate in individual exploration and then, in a class discussion, share and examine their mathematical thinking and ways to prove the exterior angle theorem based on their individual exploration.Thus, participants who believed that students learn mathematics by mastering skills and repeating the same procedures seemed to have limited pedagogical knowledge.On the other hand, the participant who viewed learning mathematics as an active construction of knowledge could provide meaningful questions or tasks and diverse ways to deepen the students' understanding of the content.

Technological Content Knowledge and Beliefs about Technology Use
The participant with a servant view displayed a lower TCK level, while the participants with a partner view displayed higher TCK levels.I categorized Grace's beliefs about technology as a servant view in which technologies are quick and accurate tools to replace mental calculations or pen and paper works.Grace believed that using calculators hinders students' mental math and did not want students to be dependent on technology.She also described a teacher-centered view of technology use: Sometimes students can use the Smart Board but with the teacher observing them.I don't think students are allowed to write on the Smart Board by themselves.… I use the Smart Board as a saving tool, for saving my lecture and lesson.
Grace viewed technology as a tool to save works or communicate with students or their parents rather than as a tool for developing students' mathematical understanding.Across the task-based and performance interviews, Grace displayed a beginner level of TCK.She seemed to feel more comfortable with writing on paper than using GSP when thinking about mathematical concepts or demonstrating her mathematical knowledge.She correctly drew a reflection line on the paper when she performed reflection of a triangle, but when using GSP, she did not know that a reflection line was needed to reflect the figure.She initially did not know how to construct the reflection line and the extended line of the side of the figure using GSP.Because she believed that technology enables students to work quickly, she said she would not use GSP for the exterior angle theorem when dealing with polygons that had many sides, since she thought it would be time-consuming work.
Hank, Abby, and Kate all expressed a partner view of technology.Hank believed that technology allows students' mathematical thinking to "grow exponentially" and that students can deepen their conceptual understanding by manipulating and "playing around" with figures using technology, especially GSP.Abby believed that technology could change the main agent of learning in the classroom from the teacher to the students, as technology allows students "to explore and do what they want and talk to each other about it."Kate stated that the different aspects provided by technology, including the visual aspect, were an advantage of using technology in learning and teaching mathematics.She added:

It [technology] also helps them to figure out what's going on so they can look at it. "I can change this one parameter. What happens to the rest of it?" That's important because it kind of gives them the background for why something is true or what something does, and that kind of thinking is what goes into writing proofs.
Hank, Abby, and Kate displayed advanced TCK levels and were willing to use GSP during the task-based and performance interviews.They were able to construct various mathematical figures using GSP (e.g., quadrilaterals, pentagons, and extended lines and exterior angles of figures) and use diverse affordances of GSP (e.g., dragging, labeling, measuring, calculating, and rotating features) to create tasks or activities in which the students could explore examples and make their own mathematical conclusions.Therefore, the participant who viewed technology as a fast and accurate tool to amplify what one can do by hand had limited knowledge about how to use GSP to construct or represent mathematical figures or concepts.On the other hand, the participants who believed that technology could facilitate students' mathematical understanding by providing opportunities to explore different perspectives were familiar with GSP and knew basic skills to operate GSP and its diverse affordances and constraints.

DISCUSSION
In this study, the preservice secondary mathematics teachers had some overall features.Traditional views were more predominant in their descriptions of the nature of mathematics (instrumentalist and Platonist views) and learning mathematics (passive view) than in teaching mathematics (facilitator view) and technology use (partner view).Second, the preservice teachers' levels of CK and PCK were lower (intermediate level) than their levels of TCK (advanced level).Third, similar to the results of Smith et al.'s (2016) study, preservice teachers displayed the lowest levels of TPCK among the TPACK components.

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My analysis of preservice secondary mathematics teachers suggests possible relationships between preservice teachers' beliefs and TPACK components.This study found that the more sophisticated the beliefs about mathematics, learning, and technology, the higher the levels of CK, PCK, and TCK, respectively.These results are similar to the findings of Smith et al.'s (2016) study. Both Smith et al.'s (2016) and my study indicate that preservice mathematics teachers with constructivist-oriented beliefs about mathematics and learning displayed higher CK and PCK than preservice teachers with traditional beliefs about mathematics and learning.However, in terms of the relationship between preservice teachers' beliefs and knowledge about technology use, the results from this study and that of Smith et al. (2016) were different.In Smith et al.'s study, the preservice middle-school teachers' beliefs about technology use were more closely related to their TPCK levels than their TCK levels.

Beliefs and Knowledge about Technology
In this study, preservice teachers' beliefs and knowledge about technology use seem to have a strong influence on their TPCK level.The preservice teacher with a limited view of technology use and lack of TCK (Grace) demonstrated lower TPCK than the other participants (Hank and Abby), who had more student-centered views of technology use and higher TCK, even though they had similar beliefs and knowledge about mathematics and pedagogy.Moreover, a limited view of technology use and low TCK seem to be associated with preservice teachers' limited experiences with technology in mathematics classes, as discussed by Crompton (2015).Unlike the other participants, Grace rarely used technology in her middle school and high school mathematics classes.In her college mathematics classes, she used technology only for displaying content and communicating with professors, not for exploring mathematics concepts.Her lack of robust knowledge and experiences with technology in studentcentered learning approaches may influence her level of TPCK.This result is consistent with the finding of Chai et al. (2013) that a lack of technology-related knowledge is associated with low TPCK.

Beliefs and Knowledge about Mathematics and Pedagogy
Preservice teachers' beliefs and knowledge about mathematics and pedagogy seem to affect their TPCK levels.Although preservice teachers (Hank and Abby) held student-centered views of technology use and displayed advanced TCK, their TPCK might not be high if they had more traditional beliefs about mathematics and learning and low CK and PCK.The acquisition of technology-related knowledge does not always ensure successful technology integration (Polly et al., 2010).Consistent with the results of Choy et al. (2009), preservice teachers had positive attitudes toward technology, expressed a willingness to use technology in their future teaching, and showed good technological knowledge, but they did not have appropriate knowledge to ask meaningful questions or create tasks that facilitate students' conceptual learning.Kim et al. (2013) had similar findings: Teachers with more student-centered pedagogical beliefs tended to integrate technology more seamlessly into their teaching than those with more teacher-centered pedagogical beliefs.

Importance of Beliefs and TPACK
As indicated by both other studies and the current study's findings, preservice teachers with similar beliefs and knowledge about mathematics and pedagogy may use technology differently to teach mathematics due to limited beliefs and knowledge about technology use.Or, although preservice teachers have strong technical knowledge and positive attitudes toward technology use in mathematics classes, they may not know how to use technology effectively to teach mathematics.Preservice teachers need to improve their knowledge of mathematics, how students think about and learn mathematics with/without technology, and how to use technology to teach mathematics.Just having knowledge, however, would not be enough.Preservice teachers also need to view mathematics as a continually expanding field in which students can construct their own mathematics through active engagement, teachers can facilitate students' conceptual learning, and technology can support studentcentered approaches.Thus, preservice teachers should develop all areas of content, pedagogy, and technology in their beliefs and knowledge to be able to use technology effectively to teach mathematics.

IMPLICATIONS
This study provides insights into what aspects mathematics teacher educators (MTEs) should consider to cultivate teachers who effectively teach mathematics using technology.In this study, preservice teachers' beliefs about the nature of mathematics and learning mathematics were more traditional and inflexible than their beliefs about teaching mathematics and technology use.Their beliefs about teaching and technology use may be more amenable to change because they have had less experience with teaching and technology.Thus, MTEs should place more emphasis on developing preservice teachers' student-centered beliefs about teaching and technology.

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In addition, the preservice teachers displayed lower CK and PCK than TCK.At times, they could not provide appropriate questions or tasks for students due to their lack of CK, which may have caused their low PCK.They knew how to use GSP only for themselves, not for mathematics instruction.To develop preservice teachers' TPCK, MTEs should focus on developing preservice teachers' CK and PCK even if neither type of knowledge includes technology.MTEs should provide learning and teaching experiences with student-centered approaches and positive experiences with technology and training to integrate technology into mathematics instruction.Having preservice teachers work in a constructivist or student-centered environment could affect their beliefs as well (Liljedahl et al., 2007).

STUDY LIMITATIONS
It could be argued that the small sample of four preservice teachers limits the generalizability of the findings of this study.However, although different results could be obtained from a larger sample, the current study focused on detailed analysis of the preservice teachers' interviews and discussion of their beliefs and use of technology.This analysis and observation of how preservice teachers utilized technology to perform various tasks and to teach a specific mathematical concept enabled the researcher to more deeply understand the relationships between preservice teachers' beliefs and TPACK.
Another limitation is that the task-based interview's rubric was founded on student-centered principles.Thus, the rubric is likely biased toward participants who hold student-centered beliefs, and those participants may achieve higher levels of TPACK components than participants with teacher-centered beliefs.Some participants in this study displayed low levels of some TPACK components even though they held student-centered beliefs.Thus, the relationships between the participants' beliefs and TPACK components were not predetermined.
More studies with larger samples and in more diverse mathematical contexts are needed to extend this line of research.However, the case studies discussed in this paper can provide an important foundation for further investigation of how preservice mathematics teachers' beliefs and TPACK are related and how this impacts their integration of technology into their teaching.

Table 2 .
Classifications of Mathematics Teachers' Beliefs

Table 3 .
Preservice Teachers' Levels of TPACK Components in the Task-based Interview

Table 4 .
Classification of Participants' Beliefs

Table 5 .
Participants' Levels of TPACK Components in the Task-based Interview

Table 6 .
Participants' Levels of the Performance Interview

Table 7 .
Participants' Beliefs Classification and Final Levels of TPACK Components