Inquiry-based instruction is one effective teaching approach useful in developing these desired student outcomes. Though many definitions of inquiry are used, we view inquiry-based instruction as a student-centered approach, in which students explore mathematical ideas and think critically about the content prior to receiving or developing their own explanations.<\/p>\n
In contrast to an inquiry-based approach, most people have experienced mathematics as a set of procedures to be learned, practiced, and automated. Within their own mathematical experiences teachers have rarely been asked to challenge their mathematical understandings; to question why things work the way they do; and to appreciate the beauty, coherence, and value of mathematics.<\/p>\n
These types of experiences can have a direct and profound impact upon teachers\u2019 beliefs, conceptions, attitudes, and practices (Fennema & Franke, 1992; Pajares, 1992; Thompson, 1992). If new and different educational outcomes are to be achieved, teachers must be encouraged to try new and different approaches of instruction. A considerable body of research however, has demonstrated that transformation of teachers\u2019 practices is a difficult and complex process (Thompson, 1992).<\/p>\n
In addition, teachers are expected to use technology to encourage students to explore mathematics and to foster deeper learning (NCTM, 2000; CCSSO & GSA, 2010). Koehler and Mishra (2009) suggested that technology should be used from the outset of any professional development efforts designed to transform teacher practices. Teachers may then experience mathematics and learn mathematics in ways similar to the way their students will experience and learn mathematics, leading to a more coherent pedagogical approach. However, asking teachers to simultaneously develop their content knowledge, transform their practice to one that is inquiry based, and incorporate new technology into their instruction may prove to be overwhelming.<\/p>\n
This paper describes a professional development project, Inquiry in Motion, and our attempts to effect transformative changes in middle grades teachers\u2019 mathematics practices. This report describes the participation of 6 middle school mathematics teachers.<\/p>\n
Our goal for the project was to develop and support teachers\u2019 use of technology in teaching mathematics through inquiry. In the first 2 years of the project, we undertook two different approaches in reaching this goal. Both approaches challenged teachers to examine their current content and pedagogical knowledge and the ways in which their knowledge bases interacted as they taught.\u00a0 The first approach developed teachers\u2019 knowledge bases prior to introducing and considering ways that technology could support their instruction and student learning. The second approach immediately introduced technology as a tool for inquiry-based instruction. In our examination of these two approaches, we sought to answer the following questions:<\/p>\n
- \n
- Prior to training and sustained support on the use of color-graphing calculators, what are middle school teachers\u2019 beliefs about and attitudes toward incorporating graphing calculators into instruction?<\/li>\n
- How do middle school teachers\u2019 beliefs and attitudes about incorporating graphing calculators into instruction change as they participate in a professional development program composed of intensive summer training and the availability of multiple support structures throughout the school year?<\/li>\n
- Was the strategy of infusing technology contemporaneously with inquiry-based instruction or the strategy of introducing them separately was more effective in producing desired change?<\/li>\n<\/ul>\n
Literature Review and Theoretical Frame<\/strong><\/p>\n
This article reports on a portion of a larger professional development project called Inquiry in Motion (Marshall, Horton, & Smart, 2009; Marshall, Smart, & Horton, 2010; Marshall & Horton, 2011), which has worked with middle grades mathematics and science teachers to develop and assess content-embedded inquiry pedagogy. The project was designed on principles and research on both inquiry instruction and effective professional development.<\/p>\n
For this study, we expanded this frame to include other research bases, including teachers\u2019 beliefs and conceptions about mathematics and incorporating technology into instruction and the emerging literature on technological pedagogical content knowledge. This expanded framework guided the formation of our research questions and the professional development supports we provided teachers.<\/p>\n
Inquiry Instruction<\/strong><\/p>\n
We take our view of inquiry-based instruction from the National Science Education Standards<\/em>, in which inquiry-based instruction refers to \u201cthe development of understanding through investigation, i.e. asking questions, determining appropriate methods, gathering data, and formulating and communicating logical arguments\u201d (National Research Council, 1996, p. 105). Although this definition was written for science education, it is germane to the pedagogical goals of mathematics educators. A mathematical approach to effective inquiry is the union of the five Process Standards espoused by the NCTM (2000): problem solving, communication, representation, connections and reasoning with content-based objectives. The key facet to an inquiry approach is that students must have opportunity to explore the concepts at hand before receiving a formal explanation of them.<\/p>\n
An inquiry-based approach to teaching science has been shown in the literature to be moderately effective in helping students to learn (e.g., Haury, 2002; Shymansky, Kyle & Alport, 1983; Wise & Okey, 1983). Within the Inquiry in Motion project we have seen moderate gains in student performance after Year 1 of teacher participation and significant student gains in science content and processes after Year 2 (Marshall & Horton, 2011).<\/p>\n
From a mathematical education standpoint, standards-based teaching approaches have been shown to increase students\u2019 problem solving skills while not significantly influencing (positively or negatively) students\u2019 procedural knowledge. Boaler\u2019s (1998) work is one such example of this result.<\/p>\n
Design of Effective Professional Development<\/p>\n
The professional development for this project aligns with the work of Loucks-Horsley, Love, Stiles, Mundry, and Hewson (2003): \u201cThere is no prescription for which designs are right for which situations\u201d (p. 7).\u00a0 Rather, several things must be considered:\u00a0 teachers\u2019 and students\u2019 knowledge and beliefs, the nature of mathematics teaching and learning, state and national standards, current curricula and assessments, and organizational structures in place to support change.\u00a0 Understanding that these items play a crucial role in the design and delivery of professional development is vital.\u00a0 Further, the role of technology in mathematics was considered in the design of this project.<\/p>\n
Neufeld and Roper (2003) suggested that professional learning experiences should be grounded in inquiry, reflection, and experimentation; collaborative; ongoing and supported by modeling; connected to teachers\u2019 current work with students; and supported by a school\u2019s current organizational structure. Additionally, professional developers may assume several roles during professional development (Killion & Harrison, 2006). For the purposes of this project, the researchers served as coaches, mentors, consultants, collaborators, and coordinators.<\/p>\n
Teachers\u2019 Beliefs and Conceptions of Mathematics<\/strong><\/p>\n
Much of the earlier research done in the area of teacher beliefs was summarized and synthesized by Thompson (1992). Thompson discussed the differences between beliefs and knowledge. This distinction is important, because these two concepts are often intertwined; teachers often treat their beliefs as knowledge. Further, beliefs can be held with varying levels of conviction and the understanding that not all people may believe the same thing. In contrast, knowledge has the connotation of being universally accepted and is judged solely as being right or wrong.<\/p>\n
Researchers have classified and explained the beliefs held by mathematics teachers on the nature of mathematics in several ways (Ernest, 1988; Lerman, 1990; Pajares, 1992; Thompson, 1992). Ernest\u2019s (1988) three different conceptions of mathematics (the problem solving view, the Platonist view, and the instrumentalist view) are connected to the four main models of mathematics teaching offered by Kuhs and Ball (1986), which classify mathematical teaching approaches based upon research on learning and teaching as well as philosophies of education and mathematics.<\/p>\n
1) Learner-focused: mathematics teaching that focuses on the learner\u2019s personal construction of mathematical knowledge; 2) Content focused with an emphasis on conceptual understanding: mathematics teaching that is driven by the content itself but emphasizes conceptual understanding; 3) Content-focused with an emphasis on performance: mathematics teaching that emphasizes student performance and mastery of mathematical rules and procedures; and 4) Classroom-focused: mathematics based on knowledge about effective classrooms. (p. 2)<\/p><\/blockquote>\n
The learner-focused view is most closely associated with an inquiry approach to teaching mathematics. The role of the teacher in this approach is that of a facilitator and a poser of questions and explorations. The second view takes the content as the focus of the classroom and encourages the students to develop understandings of ideas and processes; the lesson is driven not by student inquiry but by a scope and sequence of mathematical content. In the third mode, the teacher is expected to \u201cdemonstrate, explain, and define the material, presenting it in an expository style\u201d (Thompson, 1992,<\/strong> p. 136),<\/strong> and to be successful, students must demonstrate their knowledge of and proficiency with mathematical rules and algorithms. The classroom-focused model is not based upon any particular learning theory, but rather on the notion that students learn best when lessons are well structured and organized.<\/p>\n
Based on multiple observations of each teacher and on teacher interviews prior to the professional development experience described in this study, the third mode, with its emphasis on performance, was by far the most dominant mode of instruction. The primary goal of the professional development was to help teachers transform their practice to a learner-focused model.<\/p>\n
Use of Technology in Teaching<\/strong><\/p>\n
The NCTM and the Association of Mathematics Teacher Educators (AMTE) have called for mathematics teachers to use technology to support their instruction, enhance student learning, foster mathematical discourse, and facilitate discovery and conceptual understanding (NCTM, 2000; AMTE, 2008).\u00a0 AMTE (2009) further stated that for teachers to use technology effectively they must have a vast understanding of the content and connections that exist within the content, a rich and thorough understanding of how students learn mathematics, and a pedagogical base that encompasses the use of technology in their instruction. This view influenced us most in assisting the teachers in integrating technology into their instruction.<\/p>\n
Burrill and colleagues (2002) performed a meta-analysis of research articles concerning the relationships between teacher beliefs about technology and about mathematics and mathematics education. Their work revealed a positive relationship between teachers\u2019 beliefs about mathematics and their beliefs about the use of handheld graphing technology. For example, teachers with non-rule-based beliefs about mathematics viewed technology as integral to their instruction and were more likely to allow students more freedom in the way they chose to use the technology.<\/p>\n
Teachers who had rule-based beliefs about mathematics were less likely to view technology as a means of improving or enhancing their instruction, focused on students\u2019 affective reactions to calculator use, and were more likely to control the ways students used the technology (Burrill et al., 2002).<\/p>\n
In their review of the literature related to teachers\u2019 beliefs about technology integration, Zbiek and Hollebrands (2008) categorized teachers\u2019 beliefs into three types: personal concerns, management concerns, and technology concerns. Teachers\u2019 personal concerns deal with logistics, planning, and control and are related to how teachers view themselves as content experts and authorities in the classroom.<\/p>\n
Management concerns relate to classroom management issues and the student learning that does or does not occur as a result.\u00a0 Teachers who have a rule-based approach to teaching mathematics have a structured approach to classroom management and to integrating graphing calculators into their instruction (Tharp, Fitzsimmons, & Ayers, 1997).\u00a0 These teachers tend to view the calculator as a computational tool, use it as such, and fail to acknowledge the ways in which the tool can be used to help students explore topics. The technology promotes structure rather than learning.<\/p>\n
Technology concerns are identified as those relating to using the technology effectively during instruction.\u00a0 Taken together, these concerns related to broadening beliefs about using technology in the classroom initially present a challenge. These concerns were integral to our professional development efforts during our initial efforts to help teachers incorporate technology into their instruction. Other classifications of technology use (e.g., Beaudin & Bowers, 1997) influenced our thinking but did not have as profound an influence upon our design and data collection.<\/p>\n
Zbiek and Hollebrands\u2019 (2008) review of the research related to technology use indicated that \u201cthe ways in which technology is integrated into teachers\u2019 classrooms is influenced by their conceptions of technology, mathematics, learning, and teaching.\u201d\u00a0 Consequently, in designing our professional development, we paid close attention to teachers\u2019 conceptions and made conscious efforts to perturb the teachers in ways that would help them experience the power of technology to deepen learning.<\/p>\n
Chen (2008) found that in many cases a disconnect exists between teachers\u2019 beliefs and practice.\u00a0 Chen cited three reasons for this disconnect:\u00a0 external factors such as administrative support, teachers\u2019 conflicting beliefs, and teachers\u2019 incomplete theoretical frameworks concerning teaching mathematics with technology. For our project, we had full support and encouragement from the administration, we worked to confront and influence teachers\u2019 beliefs, and we discussed and provided experiences that were intended to aid them in infusing technology into their practice.<\/p>\n
Further, there is a relationship between teachers\u2019 beliefs about the effective use of technology and their beliefs\u2019 about how mathematics should be taught. Turner and Chauvot (1995) suggested that teachers who view themselves as the content authority in the classroom believe students must completely understand mathematical procedures and concepts before being allowed to use technology.\u00a0 Nevertheless, other research has demonstrated that when teachers have opportunities to use a new technology as learners their conceptions of technology are broadened and their beliefs about technology integration expand (Drier, 2001).\u00a0 Consequently, we posited that supporting teachers and providing them with opportunities to use technology as learners would positively impact their instructional practice.<\/p>\n
Technological Pedagogical Content Knowledge<\/strong><\/p>\n
Technology can be a useful tool for facilitating inquiry-based mathematics instruction, and through experience learning with technology teachers can gain knowledge and insight into using technology in their teaching. Scholars in the field of education developed a description of the knowledge required by teachers to incorporate technology effectively into their instruction, calling it technological pedagogical content knowledge (e.g., see Koehler & Mishra, 2005; the concept is often more recently referred to as technology, pedagogy, and content knowledge or TPACK). Technological pedagogical content knowledge consists of (a) the knowledge of teaching content with technology, (b) the knowledge of instructional decisions and representations for teaching content with technology, (c) and the knowledge of students\u2019 learning with technology (Niess, 2008).<\/p>\n
Grandgenett (2008) recommended that teacher education programs should integrate technology, content, and pedagogy within their coursework, prepare teachers to gain a disposition to experiment with new technologies, and offer teaching methods and mathematics content courses supporting the development of TPACK.\u00a0 This framework guided our work with teachers and, specifically, the questions we asked about the ways in which the professional development for the mathematics teachers should unfold.<\/p>\n
Methods<\/strong><\/p>\n
Setting<\/strong><\/p>\n
The interventions and study were conducted at a middle school in a relatively small (approximately 27,000) urban district.\u00a0 Two middle schools were involved in the first year of the study, focusing on inquiry-based instruction in mathematics.\u00a0 Only one school was selected for adding the technological component for the second year. This was conscious decision was made primarily due to limitation of resources. The school that was selected for the technological component had designated itself a School of Inquiry and Innovation. The school had an enrollment of just under 600, with 54% White, 32% African American, 8% Hispanic, and 5% mixed race; 61% on free lunch and 8% on reduced lunch; and 15% classified as having special needs.<\/p>\n
Participants<\/strong><\/p>\n
During the first year of the study, 4 of 6 mathematics teachers from the school participated in the Inquiry in Motion effort. During the second year, all 6 mathematics teachers participated. The two mathematics teachers who joined the effort the second year had prior commitments during the first year of the study; they had expressed an interest but were unable to participate.<\/p>\n
Of the 6 teachers, all were female with a minimum of 3 years of experience. Five were elementary certified and had been grandfathered in by the South Carolina State Department of Education to obtain their middle school certificate; the other was originally certified to teach secondary mathematics.\u00a0 As such, though the teacher with the secondary background had extensive course work in mathematics, none of the teachers had special training or preparation for teaching middle school mathematics.\u00a0 Teacher information is provided in Table 1.<\/p>\n
Table 1<\/strong>
\nParticipating Teachers\u00a0<\/strong><\/p>\n
![\"Figure](\"https:\/\/citejournal.org\/wp-content\/uploads\/2016\/04\/v13i4Math1Fig1.jpg\")
Figure 1.<\/strong>Total initial response types on the beliefs survey.<\/em><\/p>\n![\"Figure](\"https:\/\/citejournal.org\/wp-content\/uploads\/2016\/04\/v13i4Math1Fig2.jpg\")
Figure 2.<\/strong> Total initial response types of Candice, Abby, and Beth on the beliefs survey.<\/em><\/p>\n![\"Figure](\"https:\/\/citejournal.org\/wp-content\/uploads\/2016\/04\/v13i4Math1Fig3.jpg\")
<\/p>\n