Mathematical Content, Pedagogy, and Technology ...

Mathematical Content, Pedagogy, and Technology: What It Can Mean to Practicing Teachers Beth Bos Texas State University Kathryn S. Lee Texas State University ABSTRACT: The purpose of this study is to look at how practicing teachers integrate technology-based instruction involving the study of number concepts, geometry, and statistic and probability, during their masters program. Though results show significant improvement, there is a need for district support in the form of one-to-one technology for all students if technology is to become a seamless student tool. The program’s courses produced a positive attitude about technology, pedagogy, and content (mathematics) knowledge (TPACK) that lasted a year after the program was completed. (KEY WORDS: mathematics, interactive learning environments, technology, and elementary education)

Introduction Technology has a natural drawing power for today’s youth. It stimulates their interest, curiosity, and creativity. To harness this energy and direct it towards learning mathematics teachers embrace information computer technology (ICT) as a formative power. Teachers need experiences in using technology to go deeper into the mathematics, to direct student interest toward exploring mathematics to problem-solve, reason and prove, strengthen communication and transform a student’s view of mathematics. The following study takes a longitudinal look at the results of a professional development program focused on elementary teachers and their development of a deeper understanding of mathematics through the use of technology and the design of instructionally sound mathematics lessons. Literature Review According to the TIMSS 2011 International Results in Mathematics American schoolchildren continue to lag behind major global competitors in mathematics exams (Mullis, Martin, Foy, & Arora, 2012). Despite U.S. progress on some of those tests, students in Singapore, South Korea, Japan and Finland, among others nations, outperformed U.S. fourth- and eighth-grade students on the 2011 TIMSS. The nation's fourth-graders made some progress on the mathematics exam since it was last given in 2007, but U.S. scores on the other exams were statistically unchanged. Asian students have long dominated the math and science exams, dating as far back as 1995, and the current results show they not only earned higher scores, but also a larger number of them performed at the highest levels. In fourth-grade math, for example, 43% of students in Singapore scored "advanced," compared with 13% of their U.S. counterparts. In eighth-grade math, 47% of Korean students scored at the top level as compared to 7% of U.S. students (Mullis, Martin, & Arora, 2012). The lack of preparation and interest in mathematics has major consequences in higher education. Only about one-third of bachelor’s degrees earned in the United States are in a science, technology, engineering or mathematics (STEM) field, compared with approximately 53% of first university degrees earned in China and 63% of those earned in Japan. More than one-half of the science and engineering graduate students in U.S. universities are from outside the United States (National Science Board, 2012). Two meta-analyses on computer technologies in mathematics education showed encouraging results and credit part of their success to pragmatic constructivist teaching approaches (i.e., problem-based, inquiry-oriented and situated cognition) and suggest that appropriately designed computer technologies can play a role in supporting and encouraging students as they learn (Hattie, 2009; Li & Ma, 2010). Project Tomorrow, a nonprofit education organization, evaluated use of mobile phones with internet access in a program during the 2009-2010 school year and found that not only did mathematics scores increase, but teachers changed their manner of instruction and students were more engaged than before (Project Tomorrow, 2012). Technology can bring together the NCTM’s process standards, or best practices, with the use of today’s engaging technologies. In a recent study it was found that high achievement in mathematics was associated with (a) high levels of mathematics confidence, (b) strongly positive levels of affective engagement and behavioral engagement, (c) high confidence in using technology, and (d) a strongly positive attitude to learning mathematics with technology. Low levels of mathematics achievement were associated with (a) low levels of mathematics confidence, (b) strongly negative levels of affective engagement and behavioral engagement, (c) low confidence in using technology, and (d) a negative attitude to learning mathematics with technology (Barkatsas, Kasimatis, & Vasilis Gialamas, 2009). In some cases, the use of virtual manipulatives in mathematics has been shown to increase student understanding despite their achievements levels. There are multiple affordances within each virtual manipulative application (applet), as noticed in another study using virtual

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manipulatives, and one or more of these affordances may be more influential and beneficial for one achievement group while another affordance (within the same virtual manipulative applet) may be more influential and beneficial for another achievement group (Moyer-Packenham & Suh, 2012). “Integration of technology is more about the pedagogy of good instruction and less about knowledge of how the technology works” (Earl, 2002, p. 8). We know that adequate pedagogical integration of digital technologies is a critical factor for instructional success. Technology will not reach its potential in maximizing teaching and learning without pedagogical integration (Conlon & Simpson, 2003; Cuban, Kirkpatrick, & Peck, 2001; Niess, 2007). Additionally, in order to pedagogically integrate a technology, teachers must first perceive and understand the affordances of the specific technology and then relate the affordances to their instructional goals during lesson planning (Angeli & Valanides, 2009). The challenge for mathematics teachers is to leverage technology affordances of digital tools in their classroom. Leveraging begins with cognitively integrating these affordances with teachers’ knowledge of specific mathematical tasks and instructional guidance. Technology affordances that teachers construct or activate are important for planning the use of technology in class within a problem-based instructional learning model. Problem-based instruction creates an atmosphere for reasoning and critical thinking and teamed with technology can be very powerful (Donnelly, 2010). Conceptual Model TPACK, a conceptual model used to help teachers understand the relationship between technology, pedagogy and content (mathematical) knowledge, assists in holistically viewing the relationships involved in integrating technology into learning and instruction (Mishra and Koehler, 2006). Many researchers, beginning with Koehler and Mishra (2005), advocate that one way to learn about the complexities of teaching with technology is to engage in the design process (Koehler, et al., 2011). As Koehler et al. (2011) explain, “through the design process, learners must constantly work at the nexus of content (what to teach), pedagogy (how to teach it), and technology (using what tools)” (p. 151). Theoretical Framework In the past few decades, a constructivist discourse has emerged as a powerful model for explaining how knowledge is produced in the world, as well as how students learn. For constructivists like Kincheloe (2000) and Thayer-Bacon (1999), knowledge about the world does not simply exist out there, waiting to be discovered, but is rather constructed by human beings in their interaction with the world. "The angle from which an entity is seen, the values of the researcher that shape the questions he or she asks about it, and what the researcher considers important are all factors in the construction of knowledge about the phenomenon in question" (Kincheloe, 2000, p. 342). Thayer-Bacon (1999) invokes a quilting bee metaphor to highlight the fact that people are socially and culturally embedded, rather than isolated individuals constructing knowledge. To assert that knowledge is constructed, rather than discovered, implies that it is neither independent of human knowing nor value free. Indeed, constructivists believe that what is deemed knowledge is always informed by a particular perspective and shaped by various implicit value judgments (Gordon, 2009). According to Windschitl (1999), constructivism is based on the assertion that learners actively create, interpret, and reorganize knowledge in individual ways. "These fluid intellectual transformations," he maintains, "occur when students reconcile formal instructional experiences with their existing knowledge, with the cultural and social contexts in which ideas occur, and with a host of other influences that serve to mediate understanding" (p.752). Methods A quasi-experimental design was used for the modified TPACK Survey by Schmidt, Baran, Thompson, Koehler, Shin, and Mishra (2009), and a qualitative method, Lyublinskaya and Tournaki’s (2011) TPACK Levels Rubric, was used to assess use of technology as found in the teachers’ lesson plans. The samples were carefully selected to represent kindergarten through sixth grade teachers who had at least three years of teaching experience and had students classified as at-risk of dropping out of school as determined by Texas Education Agency (TEA) criteria (Public Education Information Management System [PEIMS], 2011-2012). By design the study represented a wide range of elementary teachers (age, nationality, type of school environments) who all taught at-risk students. The study addressed the following research questions: 1) What effects are noticed about teachers’ attitude toward the use of technological, pedagogical, and content (mathematical) knowledge and the integration of technology before, directly following participation in a mathematics content class enhanced with technology, and a year after the courses were completed? H1: Teachers’ attitude toward the integration of technology using TPACK will improve over the period of treatment and will be maintained over an extended period of time. 2) What effects emerged in comparing teachers’ lesson plans over a series of three semesters using Lyublinskaya and Tournaki’s (2011) TPACK Levels Rubric? H2: Lesson plans will show advancement in the TPACK Levels and improved use of best practices.

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Participants The population studied included 45 practicing teachers enrolled in a professional development program with an emphasis on elementary mathematics. The teachers ranged in age from 22 to 60 years old and taught in five different school districts having at least 50% or more at-risk students as identified by TEA. The cohort’s ethnicity was 3% Asian, 10% Black, 27% Hispanic, and 60% White; and gender 7% male, 93% female. Teachers taught in urban, suburban and rural areas of Central Texas. Treatment The study’s focus centered on three semesters where number theory, geometry, and probability and statistics were taught using various forms of computer related technology. Students were taught about mathematical fidelity and analyzed elementary applets for both mathematical and pedagogical fidelity (Bos, 2011). Did the application appear to be procedural or did it increase their awareness of the patterning and logic involved? Did the application only serve as motivational or did it engage the participant as learners and doers of the mathematics through exploring rich word problems? Was it used to replace pencil and paper tasks or was it used to get students to think and reflect on past mathematical knowledge structures? Is the application related to a mathematical concept and does the action on the object make sense of the mathematics? After the application was examined teachers were asked to find applications they could use in their classroom to strengthen the teaching of mathematics. This proved to be more difficult than they thought and participants came to the conclusion that using technology effectively in teaching mathematics was more difficult than they realized. Instructional time was spent on posing questions with more than one right method to solve. The interaction centered on participants’ dialogue as they rationalized their thinking, often arguing over strategies and solutions. The instructor did not offer solutions and instead posed more questions to be considered. Beyond rich problems based on mathematical content the teachers worked in cooperative groups to develop rich problems and tasks for their students. Participants in the course operated in assigned groups to arrive at what their students would perceive as a relevant problem. Participants would encapsulate their problem into one driving question related to their students’ environment to be resolved through a multidisciplined approach. With the driving question at the center, participants brainstormed the various cognitive avenues students might take and the information their students would need to solve the problem. A wiki was used as a platform for the participants to collaborate on lesson plans and use as a presentation tool for their instructional unit that included their driving question, concept map, project calendar, lesson plans, assessment, and resources as illustrated at www.ci5303.pbworks.com. Over the course of the second semester on geometry, GeoGebra, graphing calculators, Google Sketch Up, and Patty Paper were used to explore basic constructions, transformations, rotations, dilations, and three dimensional shape manipulations. The teachers explored Van Hiele’s (1999) stages of development (visualization, analysis, informal deduction, deduction, rigor) in terms of understanding where students are and how to move them to the deduction stage or possibly the rigor stage, the highest level of thought. Problem-based units were to be designed with a geometry theme around a real life problem. Units ranged from designing quilts and miniature golf courses to planning an eco-safe playground. In the third semester the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12 Curriculum Framework guided probability and statistics instruction (Franklin et al., 2007). The teachers learned to help their students develop an understanding of data, number relationships, probability, and graphs with TinkerPlots® Dynamic Data™ Exploration software. TinkerPlots is designed for students to learn from visual representations of data, and its drag-and-drop interface makes it easy to learn. The teachers planned lessons from measuring climate change to designing a successful school carnival. At the end of the semester, the teachers presented their instructional units and peer-evaluated their units using the International Society for Technology in Education’s (2008) National Educational Technology Standards for Teachers (NETS-T) as a guide. Data Sources The TPACK Survey by Schmidt, Baran, Thompson, Koehler, Shin, and Mishra (2009) was used to obtain data. The self-reported survey was designed for pre-service teachers and has been used by both pre-service and in-service teachers. Though the survey claims to determine technology knowledge, pedagogical knowledge, and content knowledge, because the data is self-reported the researchers have used it as indicating participant attitude toward the indicators. In the original test all core subject areas are represented. Because our focus was only on mathematical knowledge content the other subject areas’ content were omitted. The TPACK Survey was administered during the teachers’ first semester of the program, their last semester of the 36-hour mathematics specialist program, and one year after completion of the program. Lyublinskaya and Tournaki’s (2011) TPACK Levels Rubric was developed based on the TPACK framework for technology integration in the classroom where teachers progress through five progressive levels in each of four components of TPACK as identified by Niess, van Zee, & Gillow-Wilese (2010). The developers organized the rubric as a matrix where

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each cell represented a specific TPACK level (one of the four components of TPACK). Thus, each row of the rubric represented a specific component of TPACK and each column of the rubric represented a specific level of TPACK. For each cell of the matrix Lyublinskaya and Tournaki developed two specific performance indicators that were consistent with qualitative descriptors developed by Niess, van Zee, & Gillow-Wilese (2010) and the principles for a practical application of technology developed by Dick and Burrill (2009). The relationship between TPACK components and the TPACK levels rubric is highlighted in Table 1. Table 1. TPACK Components and TPACK Levels Rubric TPACK Components Technology knowledge (TK) Content knowledge (CK) Technological content knowledge (TCK) Pedagogical knowledge (PK)

Component Descriptor Understanding of technology tools

TPACK Levels Rubric

What is known about a specific subject (mathematics) – Number Concepts, Geometry, Probability and Statistics? What is known about the affordances to represent or enhance content? Teaching methods and processes. (i.e. problem-based, inquiry-oriented, concept attainment, and situated cognition)

Pedagogical content knowledge (PCK)

Pedagogy specific to a particular subject area.

Technology pedagogical knowledge (TPK) Technology Pedagogical Content Knowledge

Understanding how technology supports particular teaching approach

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Knowledge of instructional strategies and representations for teaching and learning subject matter topics with technologies • Knowledge of curriculum and curricular materials that integrate technology in learning and teaching mathematics; An overarching conception about the purposes for incorporating technology in teaching subject matter topics. Knowledge of students’ understandings, thinking, and learning in subject matter topics with technology

The purpose of the TPACK Levels Rubric is to assess teachers’ TPACK level based on qualitative data collected from teachers, such as lesson plans. The instrument is not intended for direct data collection. The following scoring procedure is applied when using the rubric. The possible range of scores for each component is 0 – 5, where the component score can be an integer (both performance indicators are met) or half-integer (one out of two performance indicators are met). The score is assigned for each component independently. The TPACK Levels Rubric, used in this study for evaluating lesson plans, was tested for reliability and validity. Content validity was addressed by employing two TPACK experts. The experts were both researchers who were involved in the initial development of the TPACK conceptual framework for mathematics educators. They reviewed the rubric and provided written comments in response to three specific free-response questions about the rubric. The developers revised some of the rubric’s items according to the experts’ comments. In order to test for inter-rater reliability, two different experts in the field used the revised rubric to score the 45 documents (13 lesson plans with supplemental TI-Nspire documents, 13 narratives of lesson presentations during professional development, and 19 narratives of classroom teaching observations). Each expert was provided with specific instructions and explanations on using the rubric. Both experts found the rubric to be easy to use with all artifacts provided to them for scoring. The range of correlations between the scores of the two experts on the same components was from r = 0.613 to r = 0.679 p < .01. Correlations that examined whether there was a relationship among the four components of the rubric for each expert were also found statistically significant, i.e., the range of correlations for Expert 1 was from r = .85 to r = .94 p