TECHNOLOGY IN MATH PD 1 Developing Technological ...

Running head: TECHNOLOGY IN MATH PD

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Developing Technological Pedagogical and Content Knowledge (TPACK) through Professional Development Focused on Technology-Rich Mathematics Tasks Drew Polly University of North Carolina at Charlotte Chandra Orrill University of Massachusetts at Dartmouth

Note: The research reported here was supported by the United States National Science Foundation (NSF) under Grant No. 9876611. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF.

TECHNOLOGY IN MATH PD

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This study provides findings from a professional development program for middle grades (Grades 4-8) teachers, which focused on developing teachers’ knowledge of mathematics, pedagogy, and technology skills through the exploration of technology-rich mathematical tasks. Using the Technological Pedagogical and Content Knowledge (TPACK) framework (Mishra & Koehler, 2005), data from interviews and open-ended surveys were analyzed to examine teachers’ perspectives of their learning because of participating in the program. Inductive data analyses indicated that participants reported gaining knowledge about technology, and about how technology could support their teaching of mathematics. However, very few participants reported a clear sense about how to use technology while teaching mathematics. Implications for designing and examining the influence of technology-rich professional development focused on content are discussed. Keywords: professional development, technology integration, mathematics, TPACK

TECHNOLOGY IN MATH PD Developing Technological Pedagogical and Content Knowledge (TPACK) through Professional Development Focused on Technology-Rich Mathematics Tasks Students who engage in technology-rich mathematical tasks that also address high-level thinking skills have significantly outperformed their peers on mathematics assessments (e.g., Grandgenett, 2008; Polly, 2008; Wenglinsky, 1998). Technology-rich mathematical tasks are tasks that address higher-order thinking skills, and use technology to model, simulate, or represent mathematical situations. These tasks provide opportunities for students to analyze information, make and test conjectures, and communicate about the mathematics (Grandgenett, 2008; Higgins & Parsons, 2010; Polly, McGee, & Martin, 2010; Zbiek, Heid, Blume, & Dick, 2007). Despite evidence about the potential of technology-rich mathematical tasks to impact student learning, these activities are rarely used in classrooms (Lawless & Pellegrino, 2007; Polly & Hannafin, 2011). In fact, past research on professional development indicates that teachers are only likely to integrate technology and reform-based pedagogies in their classrooms after extensive follow-up, classroom-based support (Davis, Preston, & Sahin, 2009; Polly & Hannafin, 2010; Wei, Darling-Hammond, Andree, Richardson, & Orphanos, 2009). Worse yet, teachers remain unprepared to integrate technology-rich tasks that address higher-order thinking skills (Polly & Hannafin, 2011; Zbiek et al., 2007). Even when teachers attempt to use technology in support of open-ended activities, they often adapt those activities to fit a more didactic, skills-based approach to instruction (e.g., Lawless & Pellegrino, 2007; Oliver & Hannafin, 2001; Polly & Hannafin, 2011). One likely contributing factor to teacher technology use is the manner in which teachers learn to use technology (Davis et al., 2009; Lawless & Pellegrino, 2007; Roschelle, Pea, Hoadley, Gordin, & Means, 2001). Too often, teachers are

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introduced to educational technology in one-size-fits-all experiences that isolate technology from content (Schrum, 1999). Another barrier to implementing technology-rich tasks that focus on higher-order thinking skills is teachers’ beliefs that these types of activities take up too much time, are ineffective, or require too much preparation (Ertmer, 2005; Ertmer & OttenbreitLeftwich, 2010; Polly & Hannafin, 2011). Lastly, school culture, including pressure from colleagues and administrators, also inhibits the use of technology-rich tasks (Ertmer & Ottenbreit-Leftwich, 2010). While teachers report having more access to and more knowledge of technology-rich activities, teachers still feel they do not have enough equipment, time, and support to teach effectively with technology (Project Tomorrow, 2008). Professional development programs must provide opportunities for teachers to simultaneously develop knowledge and skills of technology, mathematics content, and pedagogy (Mishra & Koehler, 2006), while also addressing teachers’ beliefs about how technology supports the teaching and learning of mathematics (Ertmer, 2005). This study examined the influence of a professional development program on teacher-participants’ technology skills and beliefs about integrating technology into their mathematics classrooms. Theoretical Framework Learner-Centered Professional Development We grounded this study in a theoretical perspective that assumes the program of study, InterMath, embodies research-based principles referred to as Learner-Centered Professional Development (LCPD; see National Partnership for Educational Accountability in Teaching [NPEAT], 2000; Polly & Hannafin, 2010). The LCPD principles, grounded in constructivist epistemologies, assert that teachers should have access to learning opportunities that develop both their cognitive and metacognitive abilities. That is, LCPD experiences simultaneously

TECHNOLOGY IN MATH PD support teachers’ development of content knowledge, refinement of self-regulation skills to guide their learning, and development of the skills and understandings necessary to attend to student thinking once back in their classrooms (Polly & Hannafin, 2010). In short, LCPD provides teachers with the opportunities to take responsibility for their own learning and focuses teacher learning on meeting the needs of the students in their classrooms. Technological Pedagogical and Content Knowledge (TPACK) The second aspect of this study’s theoretical framework is Technological Pedagogical and Content Knowledge (TPACK; see Figure 1). TPACK has been advanced as a way of addressing the intersecting knowledge base of content, educational technologies, and effective pedagogies (Grandgenett, 2008; Mishra & Koehler, 2006; Niess, 2005). The TPACK framework is based on the premise that a teacher’s learning and classroom practices could focus on any of the regions in Figure 1, but for technology to be effectively integrated, he or she must have a deep understanding of and employ pedagogies in the central region, where technology, pedagogy, and content intersect (Mishra & Koehler, 2006). To this end, professional development should address aspects of all three domains of knowledge. LCPD programs provide rich environments that can address the teachers’ TPACK needs. In LCPD programs, teachers deepen their content and pedagogical knowledge by engaging in the kinds of activities they are expected to teach. The LCPD framework assumes that teachers own their learning, and during content-specific professional development that addresses technology integration, learning may focus on any combination of technology, pedagogies, or content (NPEAT, 2000; Orrill & the InterMath Team, 2006).

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Research Questions The purpose of this study was to examine the broad question, What happens when the learning of technology is situated in the context of a mathematics professional development program? Specific interests focused on the following subquestions: •

What did teachers report learning about technology from their InterMath experience?

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How did teachers view technology in support of their own learning?

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How did teachers think technology could enhance their students’ mathematics learning?

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How did teachers think they could use technology in their classrooms?

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What barriers did teachers report as influencing how they use the technology in their own classrooms? Method

We examined interview data from 29 participants from five different InterMath courses. As part of the evaluation of the InterMath program, both interview and survey data were collected. Based on the research questions, interview data was the primary data source for this study. Context of an LCPD Program: InterMath. InterMath, a United States National Science Foundation-funded professional development program, is the focus of this study. InterMath aims to develop all aspects of teachers’ TPACK through an LCPD design. During courses, participants use technology (e.g., dynamic geometry software, spreadsheets, and graphing software) to explore mathematical tasks that align to mathematics content covered in middle grades classrooms. Similar to Swan’s

TECHNOLOGY IN MATH PD (2007) work, mathematical tasks are used to create opportunities to develop knowledge of both content and reform-based mathematics pedagogies. In Table 1, we discuss how InterMath addresses the various aspects of TPACK. InterMath embraces all components of TPACK in its approach to professional development, but the extent to which teacher-participants focus on each component varies, primarily based on their interests and professional needs (Doering & Veletsianos, 2007; Orrill & the InterMath Team, 2006). There are five InterMath courses: one course integrates all four mathematics strands (i.e., number concepts, algebra, geometry, and data analysis), while the other four courses each align to one of the aforementioned individual strands. Each course was designed to be taught over a full semester (45 contact hours) to middle grades (Grades 4-8) teachers. These courses were offered to two school districts in the southeastern United States: (a) a midsize urban district and (b) a small suburban district. Additional courses were also offered at the university where InterMath was developed. Typical InterMath session. A typical InterMath session begins with a facilitator selecting an open-ended task from the InterMath website (http://intermath.coe.uga.edu/) and then guiding participants through it. This demonstration-and-discussion task relies heavily on one or more technologies so the instructor can (a) help participants develop a coherent understanding of the mathematics involved, (b) support participants in understanding when and how technology can be used as an appropriate instructional tool in mathematics, and (c) allow a representation of the mathematical concept to support teachers in communicating about their mathematical understanding. Discussions during the introductory task(s) are often very rich because participants bring very different experiences to the course (Orrill & the InterMath Team, 2006; Polly, 2006). While InterMath was designed for middle grades teachers, participants often range

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TECHNOLOGY IN MATH PD from high school teachers to persons who do not teach math but are seeking mathematics licensure. The use of the technology often enhances the discussion by allowing participants with less robust mathematical knowledge to see concrete examples of concepts and for those with higher understandings to make connections between mathematical concepts (Polly, 2006). After the class discussion, participants engage in their own exploration of one or more investigations related to the mathematical focus for the class session. As a deliverable for the course, participants are responsible for completing at least 10 written explanations of how they approached a task. These explanations describe how the teachers arrived at their answers, how they used technology, and any other observations that teachers made about the task or the mathematics embodied in the task. These write-ups are one of the key components of the professional development, as they provide participants with an opportunity to engage in mathematics as a communications-based discipline and to reflect on their own thinking and their own problem-solving strategies. Further, participants are encouraged to design and implement technology-rich mathematical tasks in their own classrooms. At the time of this study, about half of the typical InterMath course was comprised of mathematics teachers, while the other half included teachers who worked in other areas and did not have access to mathematics classrooms. Participants Participants in this study came from five different InterMath courses. The first two courses included all four mathematical strands (i.e., number concepts, algebra, geometry, and data analysis), while the final three courses focused on one strand each: one number sense, one algebra, and one geometry (see Table 2). The courses lasted between 1 and 15 weeks and engaged the teachers in 40 to 45 hours of activities. The instructors included college mathematics and mathematics education faculty, as well as mathematics education doctoral

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TECHNOLOGY IN MATH PD students with prior experience teaching mathematics at the high school and college levels. The 29 participants included 14 middle grades teachers (Grades 5-8), 5 elementary school teachers (Grades K-5), 8 secondary mathematics teachers (Grades 9-12), and 2 participants who were not teachers. Some participants were current teachers who taught mathematics, while others were certified teachers who were pursuing an endorsement in middle grades mathematics. Twenty-seven of the participants were female and two were male. All participants came from three school districts in the southeastern United States. Procedures In the smaller courses, all participants were asked to participate in the research interviews during one of the final two course meetings. (See Appendix A for interview protocol.) However, in larger classes (i.e., 15 or more participants), approximately one third of the participants were invited to participate in the interviews. The participants from these classes were selected at random in an effort to obtain varied perspectives about the InterMath experience. Throughout this paper, we identify participants using a coding scheme that relates them to the course in which they participated. We gave each participant a unique identifier (e.g., A2 is always the second person interviewed in course A, as defined in Table 2). One person was enrolled in two courses simultaneously; therefore, she has been coded as CD to signify the two courses in which she was enrolled. Each interview was audio taped and transcribed verbatim. Participants in the 15-week courses were interviewed at the halfway point and at the end of the InterMath course. In shorter courses, participants were interviewed only after completing the course. The interview data were combined into a single dataset. They were then coded and sorted into emergent categories (Patton, 2002). Once critical strands of interest were identified,

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the research team considered the data in each of the categories and re-sorted it into finer subcategories. These subcategories were then used to support or refute the assertions that had emerged throughout the data analysis. Results In response to the broad nature of our research question (i.e., What happens when the learning of technology is situated in a context that is focused on the learning of content?), the analysis of interview data led to numerous themes in the dataset. Our findings, therefore, focused on the five subquestions: •

What did teachers report learning about technology from their InterMath experience?

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How did teachers view technology in support of their own learning?

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How did teachers think technology could enhance their students’ mathematics learning?

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How did teachers think they could use technology in their classrooms?

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What barriers did teachers report as influencing how they use the technology in their own classrooms?

A brief summary of findings for each of the aforementioned questions are presented in the following sections, are summarized in Table 3, and are elaborated below. Teachers’ Reports of Learning about Technology According to the interviews, InterMath participants believed they learned a variety of technology-related skills (Technological Knowledge), independent of learning mathematics, during their courses. Previous research indicated that teachers’ reports of their learning during InterMath fell into three categories: (a) how to use technology, (b) how technology helped them

TECHNOLOGY IN MATH PD learn mathematics content, and (c) how to use technology to solve mathematical investigations (Author, 2006). In this study, participants reported general learning in terms of becoming less “computer illiterate” and learning more about how to use computers. Others, however, honed in on specific skills they acquired in the context of solving investigations during InterMath. One participant (D1), when asked about what she learned, responded: “Using the graphing calculator software…I had to use it where I had to change the x and y minimums and maximums. It used a lot of algebra.” The interviews also uncovered some commonality in the learning of technology for participants. Most participants began InterMath with very little experience with any of the technologies in the course. While some participants had used Microsoft Excel, most were novices with other technologies such as web design software (e.g., Microsoft Front Page or Microsoft Word), Geometer’s SketchPad (GSP; Key Curriculum Press, 1993), and Graphing Calculator (Pacific Tech, 1994) software. Further, the novelty of using computers caused some participants to feel uneasy at first. For example, one participant (E1) reported feeling uncomfortable for the first two days of this course because she was using a different type of computer: “It was terrible at first. I haven’t used a Macintosh in 10 years, so the first 2 days were a struggle just figuring out the basics.” However, based on our interview data, participants only felt this kind of tension for approximately the first half of the course. Then, they seemed to hit a turning point and became more comfortable and proficient with both the computers and the software. In a postcourse interview, A1 said, “In the beginning there were all kinds of technical glitches and that was a

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TECHNOLOGY IN MATH PD little frustrating. Partially it was me and partially it was the program as a whole. Once I worked through the speed bumps it was alright.” Participants attributed their successes with technology to the opportunity to receive help from their peers and their instructors. B2 commented,

I would like to have done more how to use this Excel, basic formulas. If B1 hadn't been there, I would have done awful. But he helped me, showed me how to do dollar signs, the equal signs, find the average and the sums, and do that. And that made a significant difference for me. Same with Geometer’s Sketchpad. He showed me how to, you know, make the constructions. And I needed that basic background. And if I had had that, I think the rest of it would have been easier - the problem solving.

Views on How Technology Supported Learning Several teachers in the InterMath courses reported that the technology really helped their learning because it allowed them to represent problems in different ways (E2) and “it’s a lot better when you can see things” (D2). D2 went on to say that the InterMath use of technology was “not just drill and practice,” and it made the teachers think through the problems and how they might represent the problems graphically. Based on these kinds of statements, it seems likely that the teachers were starting to value technology and to understand there are different ways to use technology—to calculate, organize data, graph, and model situations. Some of these purposes for integrating technology are more effective than others for supporting meaningful learning.

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TECHNOLOGY IN MATH PD GSP, one of the software packages used to support geometry learning in InterMath, was cited often as being effective. GSP supports learners in visualizing the mathematics by allowing them to construct objects and test conjectures. For example, a user may use perpendicular lines to construct a rectangle, and then manipulate the figure to test whether the rectangle changes such that its sides are no longer parallel and perpendicular. The teachers almost universally came into InterMath having heard of GSP, but had either limited or no experience using it. When asked about the technology in InterMath, one teacher said the program had made a “big impression” on her because it allowed her to understand some things “a little more fully than [I] had ever understood them before” (E3). This same teacher even attributed an “epiphany” about parabolas to this program. She said she knew that the definition of parabola was a “set of points that are a distance from a fixed point” but had never been able to “see it” prior to using GSP. Interestingly, the teachers reported that the technology not only helped them learn mathematics, but also, for at least one teacher, that it made InterMath a more enjoyable professional development experience than others in which she had participated (C1). This teacher asserted the hands-on approach offered by the technology was an easier way to learn than listening to a lecture and taking notes, which characterized her other experiences. She said she had not had much success in traditional mathematics courses in which she listened to a lecture on how to do mathematics, and then tried to solve practice problems. Based on her interview, this teacher learned mathematics in InterMath and enjoyed it because of the technology. While this sentiment was from only one teacher, her experience was not unique. Data from other interviews support the idea that participants began the course concerned they would not succeed, but found themselves not only learning mathematics, but also enjoying it and developing understandings they previously did not have.

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TECHNOLOGY IN MATH PD Views on How Technology Could Enhance Students’ Learning Teachers varied in how and why they believed that technology could enhance their students’ mathematical learning. Some teachers recognized that technology does not do the math for the students, but rather that the students must understand the math in order to use the technology correctly (CD1, C2). One teacher reported, “You have to know the math in order to use [the spreadsheet]. You can’t just use it without knowing math, which is the same thing with the calculator (CD1).” This was a critical finding cited by many participants; many reported that technology only helps students learn mathematics when they are able to use the technology to set up the task. Previous research showed that teachers tended to be ill-equipped to use technology to set up tasks, and instead relied on didactic models of teaching (Cognition and Technology Group at Vanderbilt, 1997; Hannafin, Burruss, & Little, 2000; Oliver & Hannafin, 2001). Without recognizing that the role of technology is not to do the math for the students, teachers will be unlikely to implement technology in ways that are aligned with the standards. Several teachers, based on their own learning experiences, suggested that being able to investigate a mathematics problem with technology could help students build and deepen their own understanding (C3, D3, and E1). One teacher gave an example of using GSP with her students to first define various math definitions, and then to discuss constructing versus drawing shapes. She reported, “They really get a sense of the properties of shapes by doing this, and dig deeper in their understanding… ” (E1). While InterMath courses do not require teachers to integrate technology in their own classrooms, teachers who attempted to integrate technology became advocates. For example, B2 mentioned, “…some of my students that were the weakest students were very good with GSP and got a great deal of success from it.” One teacher brought up the point that technology can be used as a motivational vehicle to

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TECHNOLOGY IN MATH PD engage students in mathematics. C3 stated, “[…math is a subject students] don’t feel confident in. This is a way to emerge those students. And really it piques their interest into it and [what] they can do with it.” While technology can assist students in developing a meaningful understanding of mathematics, most respondents did not focus on the motivational aspects as those too often fade over time (Gillespie, 1991). Views on How Technology can be Used in Classrooms Several teachers described how they could use technology in the classroom, but not necessarily how they did use it with their students during the courses. However, in their descriptions, there were many shared responses to the roles that technology can play in a classroom with the most common role being that of a demonstration tool (A2, D2). A few teachers expanded on the advantages they saw in using technology as a demonstration tool. Specifically, C2 explained, “Having the graphing calculator was helpful as a student, so you can just see the graph there. It’s quicker.” Similarly, A2 recalled using graphing software to help illustrate linear equations in her class and mentioned, “that’s really nice to be able to have instead of try[ing] to draw it myself on the overhead” (A2). In these responses, it is clear teachers were viewing technology as a teaching tool rather than as a learning tool. Across the interviews, there were two cases in which teachers gave specific examples of how they already used technology in their classrooms. One teacher (CD1) who teaches a careers class and was already familiar with Excel told how she used Excel to solve a problem in which students determined how long it would take to pay off credit card debt if only paying the minimum monthly charge. In recalling this lesson, CD1 said, “Luckily, I had a computer right there so I threw it on an Excel spreadsheet.” The other teacher (A2) required her 7th-grade students to each pick three problems to investigate and create write-ups. She then posted their

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TECHNOLOGY IN MATH PD write-ups on her InterMath homepage. This was the only teacher who specifically drew from InterMath and the only teacher who seemed to allow students to use technology as a learning tool rather than reserving it as a teaching tool. One other teacher (E4) reported that she planned to use technology in hands-on ways with her students after the course was over. While it is understandable that teachers would not draw from the InterMath problem set because the problems were, for the most part, too complex for middle grades students, teachers focused too narrowly on technology as a teaching and demonstration tool—both in practice and in discussing how they might use it. When the use of the technology in the InterMath courses was mostly hands on, it was being used as a learning tool teachers were using to engage in their own knowledge construction. This finding warrants future exploration into how teachers believe that students learn. Perceptions of Barriers to Using Technology in Mathematics Classrooms The data analysis results created interest in the finding that few teachers used technology at all in their classrooms and, when they did, teachers relied on technology as a demonstration tool. Based on the interview data, teachers still felt constrained by their access to technology and the time it took to use technology. Surprisingly, teachers in the InterMath courses frequently reported limited access to computers for instructional use, a lack of useful computer software, a lack of technical support, and their own lack of computer literacy skills. The major constraint teachers felt in using technology in their classrooms came from the lack of access to computers or computer labs. Teachers reported a lack of access because the computer labs were shared with teachers in other content areas that required report writing, and those areas were given higher priority by the administration in their schools. Others talked about getting the lab once or twice a term “if you’re lucky” (A1) and having to schedule times for

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TECHNOLOGY IN MATH PD using the lab 3 months in advance (E2). One teacher mentioned that she did not have access to a computer lab during “math time” and that she was told she would be losing several computers in her classroom without replacement because they were too outdated to be maintained by the technology department (C1). Despite having ample access to hardware, teachers often reported limited access to highquality software. A common issue was the overprevalence of drill-and-practice software. One teacher (C2) commented that she hated that so much money was spent on drill-and-practice software and D2 reported that she “begged and pleaded” for the graphing software used regularly in InterMath courses but was denied. Another teacher (D1) also talked about graphing software; she called it “nice to see, but something you can’t have in our schools.” Her school had handheld graphing calculators, and she could not make a connection between using the computerbased graphing software and the hand-held graphing calculators. Clearly, raising teacher appreciation of uses of technology is only one step in changing the way mathematics is taught with technology. Another barrier brought up by several teachers was their own limited computer literacy. Teacher A3 said she felt “somewhat computer literate” before the InterMath course but laughingly said she later found out she was illiterate and had a lot of learning to do with the technology. When identifying major obstacles for teaching with technology, many teachers spoke of students not having “as much experience with computers as they need” (D1), students being “computer illiterate” at younger ages (C3), and students being at different levels with technology literacy (CD1). In other words, teachers felt that both they and their students had to have a certain level of technology literacy in order to implement technology-based approaches in their classrooms.

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Implementing technology in the classroom is still not seen as an easy task and is often reported as being too time consuming. A4 reported technology as being the “extra stuff” that would get done if there was any “extra time.” There were also teachers who still did not believe in the value of using technology for mathematics due to it making students “handicapped because they do not always have to use their brains (A5).” Clearly, for these teachers, InterMath did not provide the kind of experiences that lead to meaningful change in student learning experiences. Discussion In this study, researchers examined what happens when the learning of technology is situated in a context that also focuses on learning content and pedagogy. Several findings warrant further discussion. A Variety of Foci during Professional Development This particular professional development setting incorporated the learning of technology and the learning of content in a way that each individual teacher could choose to focus more on technology, on mathematics, or to focus on learning mathematics through the use of technology. Because the professional development was designed around the perspectives of LCPD and TPACK, teachers controlled the focus of their professional development; thus, some teachers reported focusing solely on technology, while others focused on how technology could be used to enhance the teaching and learning of mathematics. In line with prior research (Peterson, Putnam, Vredevgood, & Reineke, 1992; Polly, 2006; Polly & Hannafin, 2011), teachers in LCPD programs focused on concepts and issues that are of greater interest and most connected to their daily work.

TECHNOLOGY IN MATH PD Focus on Learning Technology The results indicated that many of the teachers perceived of InterMath as focusing on helping them develop their understanding of and ability to use technology rather than as a joint technology–mathematics program. Many reported during interviews that their major focus was learning how to use a spreadsheet, Geometer’s Sketchpad, and the graphing calculator software. As a result, many participants finished the InterMath courses feeling they learned a lot about technology, but not much about mathematics content or pedagogy. This perception could possibly be explained by the freedom embedded in each course. Participants completed one task together at the beginning of course sessions before choosing tasks to then explore on their own or with a partner. As a result, many participants chose tasks with mathematics concepts that were familiar and comfortable to them. Hence, participants had to learn technology to explore the tasks, but in many cases worked on concepts that they already knew or that were familiar to them. In order for participants’ content knowledge to be extended, teachers needed to be challenged in an environment that felt safe and comfortable (Polly & Hannafin, 2010; Swan, 2007). Engaging in Mathematics Content Teacher-participants valued the hands-on use of the technology for engaging in mathematics for their own understanding, and experienced the mathematical power that technology can contribute to the investigation of mathematical tasks compared to nontechnological methods. Further, at least some of the teachers reported an understanding that mathematics teaching and learning could be enhanced through the use of technologies beyond simple calculations.

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TECHNOLOGY IN MATH PD Those teacher-participants who reported learning about how technology could support their mathematics learning, as well as their teaching of mathematics to middle grades students, represent the ideal result of this program. Technology represents an instructional tool that enhances opportunities to more effectively teach and learn mathematics concepts through the representations, models, and dynamic nature of the technology (Earle, 2002; Zbiek et al., 2007). Meeting the Goals of InterMath InterMath has five main goals related to technology integration: (a) teachers learn how to use technology, (b) teachers recognize that technology is valuable as a mathematics learning tool, (c) teachers can use the tools as demonstration instruments in their classrooms, (d) teachers value the mathematics tools for student learning, and (e) teachers are able to implement the technology in their classrooms as learning tools rather than instructional tools. Based on the TPACK framework, the fifth goal is the desired outcome. The data suggest that InterMath participants made progress in learning how to use technology. All of the teachers had considerably more confidence at the end than they had at the beginning of the course in their abilities to use the tools. Similarly, many of the teachers provided evidence that they saw technology as being important in their own learning of mathematics. Moving to the third goal, using technology as a demonstration tool, we start to see a significant difference among the teachers. Some reported using the technologies as a demonstration tool, others reported understanding that it could be used as a demonstration tool, and others, however, did not report any viable or successful way to use the technology in their classrooms. Finally, only one teacher provided her students with an opportunity to explore technology-rich mathematical tasks in her classroom (CD1). She reported using technology to

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TECHNOLOGY IN MATH PD guide her students through solving mathematics problems. It seems that the approach of modeling good uses of technology (e.g., demonstration and hands-on engagement) conveys the message that technology is important for teaching and learning mathematics and helps the teachers build a model for how it might be used in their classrooms. However, the InterMath participants were largely unable to make fine distinctions in the ways in which the technologies could maximize learning, citing only integration ideas that involved teachers modeling and showing students’ representations of tasks. Consistent with prior research, participants in technology-rich professional development need extensive classroom-based support in order to see effective carryover from the knowledge and skills learned in workshops to their classrooms (Lawless & Pellegrino, 2007). The InterMath approach of focusing a teacher’s attention on the content knowledge development facet of TPACK contributed to this outcome. After all, the teachers were able to build a model of learning with technology as they participated as learners, but those who were less facile with the technology initially were unable to move beyond seeing the role of technology for themselves as learners. This is consistent with the TPACK model as outlined above. Limitations of the Study While the findings in this study show potential for a model to develop teachers’ capacities for technology integration in mathematics, this study has limitations. Based on the nature of our research question and subquestions, the data focused solely on teacher-participants’ self-reports and did not attempt to gather more objective measures such as assessments of mathematics content knowledge, work samples to examine pedagogical content knowledge, or classroom observations to examine teachers’ enactment of TPACK. Future studies should

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address these limitations by collecting a wider variety of data sources, including possibly following participants into classrooms and examining how InterMath influences teachers’ instructional practices and, ultimately, how it influences student learning outcomes. Concluding Thoughts Professional development programs, such as InterMath, that embody learner-centered principles and simultaneously develop teachers’ knowledge of content, pedagogy, and technology, show promise to influence teachers’ learning and instructional practices. However, further research is needed to understand some aspects of technology learning in this context. For example, what aspects of learning mathematics are particularly enhanced when the teachers use technology? What do the teachers learn about using technology as a result of the kinds of mathematical problems in which they engage? And, what are the teachers actually learning about teaching with technology? Clearly, there are a number of research questions that could be raised about the effectiveness of modeling technology use in the context of content knowledge development in a discipline that would help us better understand the extent to which this is a viable model for promoting high-quality technology integration both as a teaching tool and as a learning tool. This study also highlights implications for middle grades teachers in Grades 4 through 8. The InterMath website (http://intermath.coe.uga.edu) provided teachers with technology-rich mathematical tasks that they completed during professional development to develop their Technological, Pedagogical, and Content Knowledge (TPACK). The tasks on InterMath can also be used by teachers to use with middle grades students. The website includes hundreds of tasks, an interactive dictionary, and other resources that can support both teachers and students in their work.

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Cognition and Technology Group at Vanderbilt (CGTV) (1997). The Jasper project: Lessons in curriculum, instruction, assessment, and professional development. Mahwah, NJ: Lawrence Erlbaum Associates. Davis, N., Preston, C., & Sahin, I. (2009). ICT teacher training: Evidence for multilevel evaluation from a national initiative. British Journal of Educational Technology, 40(1), 135-148. Doering, A., & Veletsianos, G. (2007). Multi-scaffolding learning environment: An analysis of scaffolding and its impact on cognitive load and problem-solving ability. Journal of Educational Computing Research, 37(2), 107-129. Earle, R. S. (2002). The integration of instructional technology into public education: Promises and challenges. Educational Technology, 42(1), 5-13. Ertmer, P. A. (2005). Teacher pedagogical beliefs: The final frontier in our quest for technology integration? Educational Technology Research and Development, 53(4), 25-39. Ertmer, P.A., & Ottenbreit-Leftwich, A. (2010). Teacher technology change: How knowledge, confidence, beliefs, and culture intersect. Journal of Research on Technology in Education, 42(3), 255-284. Gillespie, R. (1991) Manufacturing knowledge: A history of the Hawthorne experiments. Cambridge : Cambridge University Press. Grandgenett, N.F. (2008). Perhaps a matter of imagination: Technological pedagogical content knowledge in mathematics education. In M. Koehler & P. Mishra, (Eds), The Handbook of Technological Pedagogical Content Knowledge for Teaching. New York: Routledge.

TECHNOLOGY IN MATH PD Key Curriculum Press (1993). Geometer’s sketchpad. [Computer program]. Cambridge, MA: MIT Press. Lawless, K. A., & Pellegrino, J. W. (2007). Professional development in integrating Technology into teaching and learning: Knowns, unknowns, and ways to pursue better questions and Answers. Review of Educational Research, 77(4), 575. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A new framework for teacher knowledge. Teachers College Record, 108(6), 1017-1054. National Partnership for Excellence and Accountability in Teaching (NPEAT) (2000a). Improving professional development: Research based standards. Washington, DC: NPEAT. National Partnership for Excellence and Accountability in Teaching (NPEAT) (2000b). Revisioning professional development: What learner-centered professional development looks like. Oxford, OH: NAEP. Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge. Teaching and Teacher Education, 21, 509-523 Oliver, K. & Hannafin, M. J. (2001). Developing and refining mental models in open-ended learning environments: A case study. Educational Technology Research and Development, 49 (4), 5-33. Orrill, C. H. & the InterMath Team (2006). What learner-centered professional development looks like: The pilot studies of the InterMath professional development project. The Mathematics Educator, 16(1), 4-13. (Special issue on InterMath project). Pacific Tech (1994). Graphing calculator. [Computer program]. Berkeley, CA.

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TECHNOLOGY IN MATH PD Patton, M.Q. (2002). Qualitative research and evaluation methods. 3rd edition. Thousand Oaks, CA: Sage. Peterson, P.L., Putnam, R.T., Vredevoogd, J., & Reineke, J.W. (1992). Profiles of practice: Elementary school teachers’ views of their mathematics teaching. In W.G. Secada (Ed.), Special issue of the International Journal of Educational Research: 17 (5), 471–488. Polly, D. (2006). Participants’ focus in a learner-centered technology-rich mathematics professional development program. The Mathematics Educator, 16(1), 14-21. Polly, D. (2008). Modeling the influence of calculator use and teacher effects on first grade students’ mathematics achievement. Journal of Technology in Mathematics and Science Teaching, 27(3), 245-263. Polly, D. & Hannafin, M.J. (2010). Reexamining technology’s role in learner-centered professional development. Educational Technology Research and Development, 58(5), 557-571. Polly, D. & Hannafin, M. J. (2011). Examining how learner-centered professional development influences teachers’ espoused and enacted practices. Journal of Educational Research, 104, 120-130. Project Tomorrow. (2008). 21st century learners deserve a 21st century education. Selected National Findings of the Speak Up 2007 Survey. Retrieved from: http://www.tomorrow.org/speakup/speakup_congress_2007.html Roschelle, J., Pea, R., Hoadley, C., Gordin, D., & Means, B. (2001). Changing how and what children learn in schools with computer-based technologies. The Future of Children, 10(2), 76-101.

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TECHNOLOGY IN MATH PD Schrum, L. (1999). Technology professional development for teachers. Educational Technology Research and Development, 47(4), 83-90. Swan, M. (2007). The impact of task-based professional development on teachers' practices and beliefs: A design research study. Journal of Mathematics Teacher Education, 10(4-6), 217-237. Wenglinsky, H. (1998). Does it compute? The relationship between educational technology and student achievement in mathematics. Educational Testing Service Policy Information Center. Zbiek, R.M., Heid, M.K., Blume, G., & Dick T.P. (2007). Research on technology in mathematics education: The perspective of constructs. In F.K. Lester (Ed.) Second Handbook of Research on Mathematics Teaching and Learning (pp. 1169-1208). Charlotte, NC: Information Age Publishing.

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TECHNOLOGY IN MATH PD Table 1 How InterMath Addresses TPACK TPACK Components Technological Knowledge (TK) Pedagogical Knowledge (PK) Content Knowledge (CK) Technological Content Knowledge (TCK) Technological Pedagogical Knowledge (TPK) Pedagogical Content Knowledge (PCK) Technological Pedagogical and Content Knowledge (TPACK)

How InterMath addresses it Teachers use technologies (e.g., spreadsheets, graphing software, geometry modeling software). Teachers participate in the processes of exploring a task and discussing the content in the task. Teachers explore content by completing complex mathematical tasks. Teachers use mathematics-specific technologies to explore mathematical tasks. Teachers use technologies to explore content, model a situation, and organize information. Teachers explore mathematical tasks and participate in discussions about the approaches to solving tasks and embedded mathematical concepts. Teachers use technology to explore mathematical tasks, and discuss the approaches to solving tasks and embedded mathematical concepts.

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Table 2 Description of Courses Code A B C

Course

Instructor

Duration

All strands All Strands Number Sense

Mathematics Educator (InterMath PI) Mathematician (InterMath Personnel) Mathematics Educator (Grad Student) Mathematics Educator (Grad Student) Mathematics Educator (InterMath PI)

15 weeks 15 weeks 15 weeks 15 weeks

D

Algebra

E

Geometry

1 week

Course Participants

Research Participants

23**

8

7*

4

5

5

4

4

23**

8

*Three participants withdrew before interview data were collected. **In courses with numerous participants (n > 20), approximately one-third of the participants were randomly selected to participate in the research study.

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Table 3 Summary of Findings Research Question What did these teachers report learning about technology from their InterMath experience? How did these teachers view technology in support of their own learning? How did these teachers think technology could enhance their students’ mathematics learning? How did these teachers think they could use technology in their classrooms? What barriers did these teachers report as influencing how they use the technology in their own classrooms?

Summary of Findings Participants reported learning: 1) how to use technology, 2) how technology helped them learn mathematics content and 3) how to use technology to solve mathematical investigations. Participants discussed that technology “helped” their learning because it allowed them to represent problems in different ways and create visualizations (graphs, geometric figures) of tasks. Participants recognized that students needed to understand the math in order to use the technology correctly. Technology was viewed as a way to extend or deepen their understanding by allowing them to create multiple representations of tasks. Participants reported that technology was a tool for teachers to create graphs and pictures to help students learn. Two participants reported that students could use technologies, such as Excel or the graphing software, to explore mathematical concepts. Participants reported three barriers: 1) lack of access to technology, 2) time needed to use technology with students, and 3) need for more confidence using technologies themselves.

TECHNOLOGY IN MATH PD Figure 1. Framework of technological pedagogical content knowledge (TPACK). From Mishra and Koehler (2006).

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31 Appendix A: Interview Protocol

What has your experience been like in the InterMath course? What have you learned about technology? What have you learned about using technology to teach mathematics? How can technology help your students learn mathematics? How could you use technology in your own classroom? What barriers prevent you from integrating technology in your own classroom?

TECHNOLOGY IN MATH PD

32 Authors

Dr. Drew Polly is an Assistant Professor in the Department of Reading and Elementary Education at UNC Charlotte. His work focuses on supporting teachers’ enactment of learnercentered tasks and teachers’ use of standards-based pedagogies in mathematics classrooms. Correspondence concerning this article should be addressed to Drew Polly, Dept. of Reading and Elementary Education, COED 367, 9201 University City Blvd., Charlotte, NC, 28223. Email: [email protected]

Dr. Chandra Orrill is an Assistant Professor at the University of Massachusetts at Dartmouth and a Research Associate at the Kaput Center for Research and Innovation in STEM Education. Her research is focused on how teachers understand mathematics and the ways professional development can support that understanding.