The purpose of this study was to investigate a function machine in the form of a Vending Machine applet as a means to motivate preservice teachers to examine their own understanding of the function concept. The applet was designed to purposefully problematize common misconceptions associated with the algebraic nature of typical function machines. Findings indicate that the preservice teachers for which the applet provoked a dilemma elaborated on or transformed their understandings related to the definition of function.
As technology becomes more prevalent in the mathematics classroom, teachers will need to be able to effectively evaluate technological tools to use with students. In this study, the authors examined secondary mathematics teachers’ evaluation of online dynamic geometry tools. The analysis focused on the teachers’ noticing of technology; specifically, what features within the tools mathematics teachers attended to, how they interpreted these features, and in what ways they responded. Findings indicated that secondary mathematics teachers attended mostly to mathematical features of the tools and considered the tools’ ability to focus on student engagement and student thinking to be very important, as well as the ease of implementation of the tool. The secondary mathematics teachers tended to begin their evaluation by determining how the tools work and attending to its appearance and then moved toward examining the mathematical features and how they related to student thinking.
Dynamic geometry software can help teachers highlight mathematical relationships in ways not possible with static diagrams. However, these opportunities are mediated by teachers’ abilities to construct sketches that focus users’ attention on the desired variant or invariant relationships. The study described in this paper looked at two cohorts of preservice secondary mathematics teachers and their attempts to build dynamic geometry sketches that highlighted the trigonometric relationship between the angle and slope of a line on the coordinate plane. The authors identified common challenges in the construction of these sketches and present examples for readers to interact with that highlight these issues. They then discuss ways that mathematics teacher educators can help beginning teachers understand common pitfalls in the building of dynamic geometry sketches, which can cause sketches not to operate as intended.
A student, Stuart, related perimeter to pixels and the professor, Beth, moved back and forth between reserved believing and reserved doubting and doubting teacher actions (Elbow, 1986; Harkness & Noblitt, 2017) while assessing the merit of his conjecture in the moment. Video allowed the researchers to rewatch the episode multiple times after the moment and to attempt to believe (Elbow, 1986; 2006), or find merit or strength, in Stuart’s conjecture and then explore the mathematics that he suggested. Within this paper the researchers “restory” (Creswell, 2012) chronologically what transpired in the moment in the classroom, their later conversations, and their after-the-moment mathematical explorations of Stuart’s conjecture. Video can, perhaps, allow teacher educators to help preservice teachers and classroom teachers notice and reflect on missed opportunities for believing. Video also has the potential to empower teachers to explore the mathematics suggested by students after the moment and then use what they learn in future lessons.
This study examines whether preservice teachers’ experiences with video analyses during teacher preparation have long-lasting effects on their practices once they enter the profession. Specifically, the authors examined whether teachers who had opportunities to analyze student thinking and learning during teacher preparation continued to do so when they reflected on their teaching effectiveness as full-time teachers. A group of elementary school teachers who attended a video-enhanced mathematics methods course were compared to a control group at the end of their first year of full-time teaching. Teachers were asked to assess two lessons they had just taught by describing lesson learning goals and providing a rating of lesson effectiveness and a rationale for their evaluation. Teachers who attended the video-enhanced course during teacher preparation outperformed their counterparts in both the quality of evidence they drew upon and their attention to individual or subgroups of learners. Study limitations and future directions are discussed.
In this article the authors described their exploration of a particular design element they labeled “video in the middle.” As part of the video in the middle design, the viewing of carefully selected video clips from teachers’ classrooms is sandwiched between pre- and postviewing activities that are expected to support teachers’ engagement in and learning from the video. These three elements (prevideo, video, and postvideo), taken together, comprise a videocase. Videocases can then be further sequenced to create a specified professional development (PD) curriculum. Purposeful selection of each video clip allows for coherence between the prevideo, video viewing, and postvideo activities, which in turn, supports the link between a given videocase and identified teacher learning goals. Incorporating a video in the middle design within a video-based mathematics PD environment can promote a detailed and focused examination of complex mathematical content, the relationship between pedagogical decisions and practices, and an unpacking of students’ mathematical thinking. It is essential to underscore the major role that facilitators play in video-based PD, and that the effective application of the video in the middle design is, in large part, dependent on skillful facilitation. The video in the middle design can be useful across different content areas and teacher education settings.
The purpose of this paper is to present a multitechnology-enabled lesson used with secondary preservice mathematics teachers to develop their technological pedagogical statistical knowledge. This lesson engages preservice teachers in a statistics lesson aimed at developing their reasoning about the measurement units of data using TinkerPlots and then engages them in reasoning about students’ approaches to the task. A description of the lesson, preservice teachers’ approaches, and how they reasoned about sixth graders’ strategies are included. The authors further discuss the affordances of the specific technologies used in creating the learning opportunities for these preservice teachers and implications for teacher education.
As computational thinking (CT) is increasing in focus in K-12 education, it is important to consider how teacher education programs may better prepare teacher candidates (TCs). Previous studies have found that TCs do not always have a firm understanding of what CT involves, and they might not have clear ideas about how to develop CT in their future classrooms. In this context, the authors developed a course for elementary school TCs focusing on CT in mathematics education. The course integrated CT in the context of mathematics activities to help TCs develop both a conceptual understanding of mathematics and mathematics teaching with CT. The paper presents a case study analysis of TCs’ online discussions and reflection assignments of the course, as well as themes in their learning about and attitudes toward CT in mathematics teaching and learning.
In preparing future elementary educators in mathematics, helping them overcome their anxieties of mathematics and teaching mathematics is paramount. This study examined how different instructional practices (in-class lecture, flipped learning with teacher-created videos, flipped classroom with Khan Academy videos) compared in improving students’ mathematics anxiety and anxiety about teaching mathematics. Results suggest that, while all three methods improved students’ anxieties related to mathematics, flipped learning with teacher-created videos significantly had the greatest decreases in mathematics anxiety and anxiety about teaching mathematics. Survey responses and class interviews also suggested that flipped learning with teacher–created videos better aligned with course content and activities, thus helping students feel prepared and more confident before entering the classroom.
Online teacher professional development is becoming more prevalent as the ability to harness technology to bring teachers and resources together becomes easier. Research is needed, however, to determine the effectiveness of models and to share practices that increase teacher knowledge of content and pedagogy. This study examines how a hybrid professional development model impacted secondary teachers’ implementation of handheld graphing technology through an analysis of the participants’ perceived growth in skill with the technology and their perceived ability to provide support to other teachers using the same technology. Participant surveys as well as follow-up observations and interviews of selected participants indicated an increase in handheld graphing technology use prompted by participation in the professional development workshop.
This article presents an overview of the ways technology is presented in textbooks written for mathematics content courses for prospective elementary teachers. Six popular textbooks comprising a total of more than 5,000 pages were examined, and 1,055 distinct references to technology were identified. These references are coded according to location within the textbook, role of technology, and type of technology. The treatment of technology varied across the textbooks in the sample. The number of references to technology ranged from 71 to 451. Two textbooks mentioned technology on less than 10% of the pages, while one mentioned technology on over one fourth of the pages. For each textbook, the majority of references were to mathematical action technologies. Across the sample, calculators, websites, and e-manipulatives were most frequently mentioned. Examples of textbook activities that may influence the development of technological pedagogical content knowledge in prospective elementary teachers are provided. Recommendations are made for future directions in curriculum development and research to address the challenge of preparing teachers to effectively teach mathematics in the digital age.
Prospective elementary teachers at three universities engaged in online modules called the Virtual Field Experience, created by the Math Forum. The prospective teachers learned about problem solving and mentoring elementary students in composing solutions and explanations to nonroutine challenge problems. Finally, through an asynchronous online environment, the prospective teachers mentored elementary students. The researchers assessed the prospective teachers’ solutions and explanations to problems at the beginning of the semester, at the middle of the semester after completing the training in mentoring, and again at the end of the semester after the mentoring was completed. The researchers observed improvements in the prospective teachers’ abilities to write explanations to problems. Specifically, growth was seen in prospective teachers’ communication of their explanations and their ability to construct viable arguments and critique the reasoning of others (Common Core State Standards Initiative, 2010, Standard for Mathematical Practice 3), and attend to precision (Standard for Mathematical Practice 6).
In a course emphasizing interactive technology, 19 students, including 18 mathematics education majors, mostly in their first year, reinvented the definition of limit of a sequence while working in small cooperative groups. The class spent four sessions of 75 minutes each on a cyclical process of guided reinvention of the definition of limit of a sequence for a particular value, L = 5. Tentative definitions were tested systematically against a well-chosen set of examples of sequences that converged, or not, to 5. Students shared their definitions and the problems they were having with their definitions with their peers through whole class presentations and public postings on a course electronic forum. Student presenters received feedback from their peers both in person and through the forum. The approximation, error, error bound framework was used to help structure students’ thinking. The use of interactive examples with epsilon bands and movable N values, in which students could zoom in to adjust the value of epsilon or zoom out to find a value of N, proved especially helpful in the process. The changes in their tentative definitions show the difficulties students had as well as the learning that occurred.
Teaching mathematics for understanding requires listening to each student’s mathematical thinking, best elicited in a one-on-one interview. Interviews are difficult to enact in a teacher’s busy schedule, however. In this study, the authors utilize smartphone technology to help mathematics teachers interview a student in a virtual one-on-one setting. Free from physical constraints and preconceived biases, teachers can concentrate on building questioning, listening, and responding skills when noticing student mathematical thinking. Teachers engaged in four communication types when working with students through this technology: clarification, verification, and either extension or redirection.
This paper explores student interactions from the Virtual Math Teams-With-GeoGebra Project, a computer-supported collaborative learning environment that allows individuals to interact, collaborate, and discuss user-created dynamic mathematics objects. Previous studies of virtual math teams have focused on the coconstruction of a joint problem space and the ways collaborative meaning making can be accomplished in the online environment. Instead, this study explored the development of the students’ argumentation practices. The researchers used Toulman’s (1969) model to analyze and explain the structure of the online interactions and the argumentative practices that become normative among students. In particular, the researchers found that the students made increasingly detailed and mathematical descriptions of the data, developed more abstract warrants, and increasingly acted as if giving reasons was normative in the discussion.
A test project at the University of Puerto Rico in Mayagüez used GeoGebra applets to promote the concept of multirepresentational fluency among high school mathematics preservice teachers. For this study, this fluency was defined as simultaneous awareness of all representations associated with a mathematical concept, as measured by the ability to pass seamlessly among verbal, geometric, symbolic, and numerical representations of the same mathematical object. The preservice teachers in this study attended a seminar where they were introduced to the underlying concepts and the pedagogical advantages of multirepresentational fluency. For select topics, this idea was reinforced with interactive GeoGebra applets that allowed preservice teachers to alter a parameter and simultaneously view how it changes all four associated representations simultaneously. A qualitative study found that this approach appeared to (a) promote the use of multirepresentational fluency in problem solving approaches used among preservice teachers, (b) change preservice teachers’ perceptions of what it means for a student to understand a concept, and (c) change the nature of evaluations that preservice teachers felt were appropriate for high school students.
While digital environments can offer convenient, viable options for preservice and inservice teachers to engage in or continue their studies, little is known about teachers’ experiences with and perceptions of various existing online learning spaces. This paper describes an initial investigation using data from a group of preservice and in-service mathematics teachers who interacted by posting their reflections regarding online learning spaces to an asynchronous, electronic discussion board. Inductive qualitative techniques were incorporated to determine which online learning spaces study participants had experienced as well as their perceptions of each. Results are reported in light of possible implications for teacher education, including specific suggestions for additional study of online learning spaces.
In this paper the authors explored the question of collective understanding in online mathematics education settings and presented a brief overview of traditional methods for documenting norms and collective mathematical practices. A method for documenting collective development was proposed that builds on existing methods and frameworks yet is sensitive to the particularities of interaction in an online setting. This study used data from recent projects to further describe and highlight the steps of the proposed method for analyzing collective development in an online setting and to ground discussions of research and practice.
The collision between a growing, inexperienced teaching force and students’ algebra struggles should be one of great concern. A collaboration of four public and private universities in Oregon restructured mathematics methods courses for preservice teacher candidates by using the affordances of technology to counteract this loss of experience. Over time, veteran mathematics teachers develop extensive knowledge of how students engage with concepts. Preservice teachers, on the other hand, do not have the same experience that they can rely upon to anticipate important moments in the learning of their students. To address preservice teachers’ lack of experience with student thinking the Algebraic Thinking Project synthesized 859 articles of research into multiple technology-based resources: (a) Encyclopedia of Algebraic Thinking, (b) Student Thinking Video Database, (c) Formative Assessment Database and Class Response System, and (d) Virtual Manipulatives. The technology is used in coursework to influence preservice teachers’ dispositions toward and understanding of students’ algebraic thinking.
The authors present examples of analysis of online discourse and interactions among prospective middle-grades and secondary mathematics teachers in a technology methods course. The online group met synchronously using Elluminate Live! to study data analysis and probability with dynamic technology tools. Analysis of class sessions included broad lesson maps, which captured instructional decisions, big ideas related to content, use of technology, and general discourse. Critical episodes, where prospective teachers seemed to address common misconceptions and develop their own understandings about data analysis and probability, were identified and analyzed further. Trends related to design and management and discourse in the synchronous, online environment are reported, along with implications for further work with online technology methods courses.
This study sought to identify components of an asynchronous online teacher professional development program, Prime Online, that potentially affected participants’ mathematical knowledge for teaching (MKT). Twenty-three third- through fifth-grade general education and special education teachers completed a yearlong online teacher professional development program focused on improving MKT, instructional practices for all learners (particularly those with disabilities), and practitioner inquiry. Latent growth modeling and focus group data indicated growth in participants’ content knowledge and initial growth in knowledge of students from pretest to midtest, with a decline at the end of the program. Module components are described to highlight the online teacher professional development program structure and specific activities that potentially supported participants’ growth. Mathematical modeling, engaging with practitioner-focused journals and websites, developer-constructed materials, classroom implementation, and reflection and discussion provided participants with the opportunities for professional development resulting in increased MKT.
This paper builds on Grossman’s notion of approximations of practice as scaled-down opportunities for preservice teachers to learn to teach by doing. The authors propose the use of media rich, collaborative web-authoring tools for preservice teachers to create, complete, or edit scenarios in which they practice particular activities of teaching, such as explaining a mathematics concept or reviewing students’ work. The ways these environments can be used to fit the notion of approximations of practice are described, along with the authors’ experience using the web-based software Depict (in the LessonSketch platform) in the teaching of secondary mathematics methods. This use of multimedia scenarios combines the advantages of visual and video-based approaches to the study of practice with those approaches that ask the preservice teachers to create scenarios (e.g., lesson plays). The value of integrating this storyboarding web software in a larger environment where scenarios can be created collaboratively, annotated, and commented on in forums is presented.
Teacher professional development and course work using asynchronous online environments seems promising, yet little is known about how mathematics teacher educators (MTEs) develop practices for such spaces. Research has shown that views of learning impact design of online learning spaces, enabling and constraining particular student action. More remains to be examined about the steps being taken to make sense of MTEs’ practices to support learning. In this paper, facets of MTEs’ struggle to design an asynchronous online environment and enact a practice aligned with a view of learning are explored.
The use of instructional technology in secondary mathematics education has proliferated in the last decade, and students’ mathematical thinking and reasoning has received more attention during this time as well. However, few studies have investigated the role of instructional technology in supporting students’ mathematical thinking. In this study, the implementation of 63 mathematical tasks was documented in three secondary and one middle school mathematics classroom, and the Mathematical Tasks Framework (Stein & Smith, 1998) was used to correlate the cognitive demand of mathematical tasks with the use of technology as an amplifier or reorganizer of students’ mental activity (Pea, 1985, 1987). Results indicate that the use of technology generally aligned with teachers’ current practice in terms of the distribution of low- and high-level tasks enacted in their classrooms. However, the use of technology as a reorganizer of students’ thinking was strongly correlated with these teachers’ attempts to engage their students with high-level tasks. The distinction between using technology as an amplifier or a reorganizer is refined and extended through its application at the grain size of mathematical tasks, and implications for mathematics teacher education are discussed.
Over 2 years a small group of middle school mathematics teachers’ beliefs, attitudes, and practices were investigated in order to transform their practice to an inquiry-based, technology-rich model. Research has suggested that technology and pedagogical innovations should be introduced together, so that teachers learn mathematics in the way their students will. This claim is examined within the study. Results indicate that all of the teachers expressed interest and had positive attitudes about incorporating the calculators as they moved to an inquiry-driven model, and these attitudes continued to improve over the course of the project. However, teachers were divided about when to introduce the calculators. Some were adamant that they would have been too overwhelmed had the technology been introduced during the first year when they were trying to deepen their own content knowledge and implement inquiry. One teacher, for whom all aspects were integrated from the outset, believed the technology provided the motivation to transform her practice. Results suggest that teachers’ backgrounds, their depth of knowledge, and their familiarity and comfort with integrating technology into their instruction should inform professional development design.
This article describes how a free, web-based intelligent tutoring system, (ASSISTment), was used to create online error analysis items for preservice elementary and secondary mathematics teachers. The online error analysis items challenged preservice teachers to analyze, diagnose, and provide targeted instructional remediation intended to help mock students overcome common error patterns and misconceptions. A short description of how the ASSISTment system was used to support follow-up in-class discussions among preservice teachers is provided, as well as suggestions for producing similar online error analysis items in other content areas. Directions for accessing all of the mathematics error analysis problem sets currently available in the ASSISTment system, sample error analysis items and responses, and a rubric for implementing these assignments in mathematics methods classes to support preservice teachers are included at the conclusion of the article.
The integration of technology, pedagogy, and content in the teaching of secondary mathematics was explored among 280 secondary mathematics teachers in the State of New South Wales, Australia. The study adopted the technological pedagogical content knowledge (TPCK) model through the administration of a 30-item instrument called TPCK-M. The instrument consisted of three major theoretically based constructs: technological content knowledge (TCK), technological pedagogical knowledge (TPK) and technological pedagogical content knowledge (TPCK). Results indicated that PowerPoint and Excel constitute the two TCK modal technological capabilities while TPK scores revealed teachers’ lower capacity to deal with the general information and communications technologies goals across the curriculum, such as creating digital assessment formats. TPCK-M scores seem to suggest a healthy standard in teachers’ technological skills across a variety of mathematics education goals. However, the magnitude of such influence in practice needs to be further ascertained, given that the study identified a number of instructional, curricular, and organizational factors seriously inhibiting the integration of technology into teaching and learning. In general, to take advantage of more novel learning technologies, teachers need to be trained in working with online tools (webquests, wikis), mobile learning, and interactive whiteboards and in authoring digital learning resources.
This paper focuses on how preservice primary teachers can be supported to embrace digital learning technologies (DLTs) in their teaching of mathematics. The nature of the instruction and the assessment in the final mathematics unit of the bachelor of education program were changed. Despite being tagged as “tech-savvy,” preservice students use digital technologies primarily for social networking and information retrieval. These uses of digital technologies do not guarantee any facility for their utilization as learning technologies, which may result in early career teachers being unprepared to enact the effective use of expensive equipment in schools. The provision of a communal constructivism environment supported student learning as they met the challenges of creating interactive digital applications to teach a mathematical concept to their peers. This paper is likely to be of interest to mathematics educators who are trying to steer preservice teachers away from “worksheet maths” as well as other preservice teacher educators who wish to incorporate digital technologies into their content and methodology units.
This study examined a random stratified sample (n = 62) of prospective teachers’ work across eight institutions on three tasks that utilized dynamic statistical software. The authors considered how teachers utilized their statistical knowledge and technological statistical knowledge to engage in cycles of investigation. This paper characterizes their problem solving and the ways they represented and explored data and discusses how teachers’ work with representations seems to inform their problem solving. Recommendations are included for ways mathematics teacher educators can engage teachers in developing their knowledge for doing and teaching statistics with technology.
Middle school teachers’ use of digital curricula incorporating dynamic technology has been found to support student learning of complex algebraic concepts. This article reports on pilot research involving collaboration among faculty from a public university’s college of education, educational researchers from a nonprofit research organization, and school district leadership from a large, urban school district. The purpose of this paper is to document a series of inquiry-based professional development sessions provided to middle school teachers on the implementation of a digitally based mathematics replacement unit emphasizing algebraic concepts. The professional development experiences allowed the participating teachers to implement the digital unit successfully using a variety of instructional approaches.
This study highlights ways in which generative activities may be coupled with network-based technologies in the context of teacher preparation to enhance preservice teachers’ cognizance of how their own experience as students provides a blueprint for the learning environments they may need to generate in their future classrooms. In this study, the design of generative learning environments is used as a framework for developing an activity for students to explore modeling by interpolation and function approximation in the classroom. The research question explored whether the implementation of a generative activity on function interpolation can lead to a qualitatively different mathematical space of solutions when used in a calculus class when compared to its use in the context of a class on learning theories in science, technology, engineering, and mathematics (STEM) education. Participating students included preservice STEM teachers and students in a first-year calculus course. In order to determine possible qualitative differences in mathematical activity between the two classroom contexts (calculus class or learning theories in STEM education class), the authors focused on the evidence of student individual and collective thinking from three different groups, as documented in their corresponding generated public spaces and explored and characterized each group by its respective generated mathematical spaces of solutions.
This paper reports on a study of 22 preservice teachers enrolled in a first-semester mathematics teaching methods course. Course activities included participation in two separate field experiences in neighboring school districts. The methods class placed considerable emphasis on the use of advanced digital technologies in the teaching and learning of mathematics, with particularly extensive use of the TI-Nspire. The purpose of the study was to examine preservice teachers’ evolving relationships with advanced digital technologies in their teaching, examined through the lens of their technological pedagogical content knowledge (Koehler & Mishra, 2005; Niess 2005, 2006, 2007), and to examine the interplay between their field placements and the quality of their use of advanced digital technologies in inquiry-based lessons. The principal conclusion of the study is that there seems to be a crucial, perhaps decisive effect that modeling of exemplary practice in the field placement has on candidate attitudes regarding the use of advanced digital technologies in their teaching. There is evidence that the pre-service teachers’ experiences in the classroom primed them for the possibilities of technology use but it takes the experiencing of exemplary practice to convince them of the benefits of working to incorporate technology in their own teaching.
Teacher preparation for the 21st century deserves a front-end approach to addressing the use of technology in the learning environment. To study the effect of instructing with technology, pedagogy, and content knowledge (TPACK), teachers were asked to apply pedagogical, mathematical, and cognitive fidelity to technology used in an instructional unit they were designing. Initial results indicated that teachers were conflicted by a conceptual approach to technology use. Through clarifying and defining pedagogy, mathematics, and cognitive fidelity within the TPACK framework, teachers became more aware of the misuse of instructional technology, what attributes of technology lead to conceptual development, and integration of meaningful technology into instructional units. TPACK, with fidelity carefully defined, creates a research-based model by adding the qualifying features needed to maximize the potential of technology in the classroom. The purpose of this study is to look at the knowledge structures of TPACK and examine them in designing instruction units.
With funding from NSF, the Prime the Pipeline Project (P3) is responding to the need to strengthen the science, technology, engineering, and mathematics (STEM) pipeline from high school to college by developing and evaluating the scientific village strategy and the culture it creates. The scientific village, a community of high school students, teachers as learners, undergraduate students as mentors, and university scientists as leaders, collaborate to solve challenging long-term problems/projects that develop villagers’ expertise with STEM concepts/skills and give them a taste of the work of STEM professionals. Data were collected from both a group of intervention students and a matched control group to address the research question, “Does participation in P3 increase students’ interest in and success with the study of mathematics and science in high school?” Data were collected through surveys and interviews to address the question, “Does participation in P3 change teachers’ instructional practice and expectations for student performance?” Results showed that intervention students completed significantly more and more advanced courses in science and mathematics in high school, and their GPAs were significantly higher than their matched controls. Surveys of students’ postsecondary plans and intended college majors confirmed increased interest in STEM or business fields.
This paper will present results from an action research study that incorporated two different technologies for the purpose of students valuing and communicating a geometric proof to their peers. The author followed the Mathematics TPACK (Technology, Pedagogy and Content Knowledge) Framework (Association of Mathematics Educators, 2009) to structure an action research project in her classroom. Preservice elementary education majors used The Geometer’s Sketchpad (Jackiw, 2006) to illustrate a problem solution and its supporting proof based on the mathematics of similar or congruent triangles. The students then used a freeware screen capture program, Jing, to produce an audio supported screencast. Once the screencasts were completed, the students engaged in the process of self- and peer review. The results of the study indicate that the self- and peer review process allowed the students a chance to reflect accurately on the essential elements of a proof. Students reported that the experience of reviewing their peers’ screencasts had a stronger influence on developing their skills at writing and evaluating geometric proofs compared to the process of creating their own screencast proofs.
Because technological pedagogical content knowledge is becoming an increasingly important construct in the field of teacher education, there is a need for assessment mechanisms that capture teachers’ development of this portion of the knowledge base for teaching. The paper describes a proposal drawing on qualitative data produced during lesson study cycles to assess teachers’ development of technological pedagogical content knowledge. The specific qualitative data sources include teachers’ written lesson plans, university faculty members’ reviews of lessons, transcripts and videos of implemented lessons, and recordings and transcripts of debriefing sessions about implemented lessons. Using these data sources, inferences about teachers’ technological pedagogical content knowledge are drawn and validated. An example of the implementation of this lesson study technological pedagogical content knowledge (LS-TPACK) assessment model is provided. The example includes inferences drawn about high school teachers’ technological pedagogical content knowledge in the context of two lesson study cycles that involved teaching systems of equations with graphing calculators. Reflections on the strengths and weaknesses of the LS-TPACK model are included from a qualitative perspective, as well as from a psychometric perspective.
This study investigated the comparative efficiency of Web-based instruction (WBI) and traditional teaching methods on preservice teachers’ fraction knowledge. Students’ knowledge of fractions was measured using a Fraction Knowledge Test. The test consisted of 32 items and was administered as pre- and posttests to a total of 42 preservice teachers in two intact classes at the same university. One of the classes was randomly assigned as the experimental group (n = 21) and was given WBI. The other class was assigned as a control group (n = 21) and was given traditional instruction. Analysis of covariance was used to determine treatment effects on students’ knowledge of fractions when the pretest result was used as a covariate. The analysis of results showed a statistically significant difference between the experimental and the control groups’ posttest mean scores in favor of the experimental group.
This article describes experiences from a professional development project designed to prepare in-service eighth-grade mathematics teachers to develop, explore, and advance technological pedagogical content knowledge (TPCK) in the teaching and learning of Algebra I. This article describes the process of the participating teachers’ mathematical activities and teaching and learning tasks, each of which required a TPCK framework. Sessions were organized to transform content through strategies that integrate technology with the teachers’ content and pedagogical knowledge. Content of the professional development sessions ranged from analyzing algebraic learning activities to examining appropriate uses of technology in the teaching and learning of algebra. Teachers participated in 60 hours of summer sessions and 60 hours of academic year sessions. Results revealed the need to provide teachers with opportunities to develop and explore an integration of technological, pedagogical, and content knowledge in the teaching and learning of algebra.
What knowledge is needed to teach mathematics with digital technologies? The overarching construct, called technology, pedagogy, and content knowledge (TPACK), has been proposed as the interconnection and intersection of technology, pedagogy, and content knowledge. Mathematics Teacher TPACK Standards offer guidelines for thinking about this construct. A Mathematics Teacher Development Model describes the development of TPACK toward meeting these standards. The standards and model provide structured detail to further the work of various groups. The proposals may guide teachers, researchers, teacher educators, professional development consultants, and school administrators in the development and evaluation of professional development activities, mathematics education programs, and school mathematics programs.
Several organizations have highlighted the importance of preparing teachers to teach students mathematics using appropriate technology (e.g., Association of Mathematics Teacher Educators, 2006; International Society for Technology in Education, 2008). This article provides examples from teacher education materials that were developed using an approach that integrally develops teachers’ understandings of content, technology, and pedagogy to prepare them to teach data analysis and probablity topics using specific technology tools.
This study examined teachers’ uses of virtual manipulatives across grades K-8 after participating in a professional development institute in which manipulatives and technology were the major resources used throughout all of the activities. Researchers analyzed 95 lesson summaries in which classroom teachers described their uses of virtual manipulatives during school mathematics instruction. The findings indicated that the content in a majority of the lessons focused on two National Council of Teachers of Mathematics (2000a) standards: Number & Operations and Geometry. Virtual geoboards, pattern blocks, base-10 blocks, and tangrams were the applets used most often by teachers. The ways teachers used the virtual manipulatives most frequently focused on investigation and skill solidification. It was common for teachers to use the virtual manipulatives alone or to use physical manipulatives first, followed by virtual manipulatives. One important finding of this study was that teachers used the virtual manipulatives during the main portion of their lessons when students were learning mathematics content. These results represent an initial exploration of teachers’ current use of virtual manipulatives in K-8 classrooms.
This paper reports the results of a doctoral research pilot study that paired a researcher with an experienced classroom teacher for a 12-week time span with the goal of effectively integrating the use of Geometer’s Sketchpad (GSP) into the classroom teacher’s practice. Using a teacher development experiment, the researcher created an apprenticeship model to foster the transmission of the knowledge to the classroom teacher required to successfully teach with Geometer’s Sketchpad. Specific results indicate a positive change in the facilitation of mathematical communication and inquiry-based instruction in the classroom teacher’s practice as well as sustained use of GSP beyond the time span of the pilot study. General results include the development of the constructs of technological knowledge (TK), technological pedagogical knowledge (TPK) and technological pedagogical content knowledge (TPCK).
This article describes a technology integration course planning assignment that was developed to enhance preservice teachers’ technological pedagogical content knowledge (TPCK). This assignment required preservice teachers work with peers to integrate various technological tools (e.g., graphing calculators, web-based mathematics applets, etc) in a secondary level mathematics course (e.g., Algebra 2). A description of the context and the course in which this assignment is given is provided and lessons learned from several years of implementation are discussed.
The purpose of this study was to investigate effective uses of digital ink technology in an elementary mathematics methods course. A survey methodology was used in the study to examine the participants’ perceptions toward this technology for teaching and learning. All of the items on the survey produced response means between 5.0 and 6.0, with a median standard deviation of 1.095, on a 7-point Likert-type scale. The findings indicate positive perceptions regarding the benefits of the use of digital ink technology.
The use of videotaped episodes of elementary mathematics classrooms for professional development is not new. However, without appropriate support, preservice teachers may find it difficult to hone in on the underlying features of the targeted practices displayed in the swift-moving action of the classroom being observed. The focus in this study is to investigate the benefits of including scaffolding supports directly into the software that facilitates the viewing of the videotape episodes to enhance preservice teachers’ understanding of the teaching of mathematics. The data indicate that the preservice teachers who used the software product, MathStore, were able to develop significant insight into specific aspects of the teaching and learning process.
Two teachers participating in an online study group provided the foci for in-depth case studies. Transcripts of conversations they had with colleagues about issues related to reform-oriented pedagogy were analyzed from both acquisition and participation perspectives on learning. Both teachers exhibited mainly marginal changes to their pedagogical reasoning structures and were generally resistant to adopting ideas posed during online debates. At the same time, the text-based environment provided a setting for both participants to structure their emerging thoughts about changes to their existing pedagogical reasoning structures. It also served as a forum for them to identify gaps in their personal knowledge and to obtain further professional development to address them. The methodology and theoretical perspective employed in the report provide a foundation for further research on teachers’ learning in online environments.
In this paper are discussed recent efforts to provide preservice mathematics teachers with opportunities to connect elementary teaching methods and content with the content and methods of secondary school mathematics. Through an in-depth exploration of the game, Shut the Box, preservice elementary and secondary mathematics teachers thoughtfully analyzed and manipulated computer-generated output, developed and tested their own conjectures, and collaboratively answered questions involving theoretical probabilities across courses and content levels. Through their collaboration, the preservice teachers gained a better appreciation of mathematics content and pedagogical strategies that lie beyond the grades they will likely teach, as they reconsidered the importance of content and pedagogical knowledge at every level of mathematics instruction. These interactions are considered in this document through a discussion of the mathematical underpinnings of the popular board game.
The challenge for mathematics teacher educators is to identify teacher preparation and professional development programs that lead toward the development of technology pedagogical content knowledge (TPCK). TPCK is an important body of knowledge for teaching mathematics that must be developed in the coursework in teaching and learning, as well as within the coursework directed at developing mathematical knowledge. Preparing teachers to teach mathematics is highlighted by its complexities. What technologies are adequate tools for learning mathematics? What about teacher attitudes and beliefs about teaching mathematics with technology? What are the barriers? These questions and more frame the challenge for the development of a research agenda for mathematics education that is directed toward assuring that all teachers and teacher candidates have opportunities to acquire the knowledge and experiences needed to incorporate technology in the context of teaching and learning mathematics.
This article shares an approach to teaching mathematics teacher education courses incorporating asynchronous online discussions. Specifically, this research is guided by the following research questions: (a) How would online discussions contribute or hinder teachers’ learning in mathematics methods courses? and (b) What pedagogical strategies need to be considered when incorporating online threaded discussion? The analysis of data collected provides the basis for conclusions and recommendations for educators who are interested in integrating online discussions into mathematics methods classrooms.
This study investigated the influence of a mentor-supported model of technology training on mathematics teachers’ attitudes and use of technology in the classroom. The treatment included six training sessions, informal focus groups, and mentor-provided support.
The results indicated that mathematics teachers participating in mentor-supported professional development increased the amount and level of technology use in their practice. Teachers had a desire to learn about technology and understood it was important. Levels of accommodation, interest, comfort and confidence related to the use of technology improved. Teachers continued to be concerned with barriers such as lack of release time for training, planning and collaboration, and a need for ongoing support. It was also found that when teachers perceived there was not enough time for training or a lack of technological resources they did not make an effort to become technologically proficient.
Recommendations include providing teachers additional support when implementing new strategies and allowing more release time for training, planning, and collaboration. Recommendations for future research include investigating further the effectiveness of peer teachers and mentor teachers as trainers; ways to best change teachers’ perceptions and attitudes about technology; and ways teachers best learn to integrate technology into practice.
The preparation of preservice teachers to use technology is one of the most critical issues facing teacher education programs. In response to the growing need for technological literacy, the University of Northern Colorado created a second methods course, Tools and Technology of Secondary Mathematics. The goals of the course include (a) providing students with the opportunity to learn specific technological resources in mathematical contexts, (b) focusing student attention on how and when to use technology appropriately in mathematics classrooms, and (c) giving opportunities for students to apply their knowledge of technology and its uses in the teaching and learning of mathematics. Three example activities are presented to illustrate these instructional goals of the course.
The potential to use mathematics software to enhance student thinking and development is discussed and a taxonomy of software categories is outlined in this paper. Briefly, there are five categories of tool-based mathematics software that can be used fruitfully in a mathematics curriculum: (a) review and practice, (b) general, (c) specific, (d) environment, and (e) communication. A description of the affordances and constraints of the five types of software and how each facilitates different aspects of student learning clarifies the ways in which diverse off-the-shelf offerings can be used to address the goals of mathematics instruction, from building basic skills to exploring mathematical applications in the real world.
This article focuses on three key factors that a survey of literature indicated impact the teaching and learning of mathematics with graphing calculators: access to graphing calculators, the place of graphing calculators in the mathematics curriculum, and the connection between graphing calculators and pedagogical practice. Access to graphing calculators is associated with student achievement gains and a wide array of problem-solving approaches. The research suggests students’ achievement is positively affected when they use curricula designed with graphing calculators as a primary tool. Studies of teachers’ use and privileging of graphing calculators illustrate the impact professionals have on students’ mathematical knowledge and calculator expertise. Implications of these research findings for preservice and in-service teacher education are summarized.
Graphing calculators have been used in the mathematics classroom for speed, to leap hurdles, to make connections among representations, and to permit realism through the use of authentic data. In this study, a graphing calculator tutorial provided on the Casio FX1.0 and FX2.0 PLUS models was found to serve a fifth purpose, improving manipulative skills. Specifically, after using the tutorial, students in a beginning college algebra course scored significantly higher on a test on solving linear equations. Results concerning a change in attitudes were tentative, although they suggest that the tutorial also may contribute to improved attitudes.
This paper outlines the efforts of two mathematics teacher educators in their use of online videos to expose their elementary preservice teachers to examples of reform teaching, as espoused by the National Council of Teachers of Mathematics. The online videos provide an excellent source for reflection, and each author shares their different avenues to encourage both discussion and reflection about the practices seen on the videos. Actual student comments about videos they have viewed reveal the motivating and enlightening nature of this delivery method. While several websites provide access to online videos, this paper highlights PBS Mathline (http://www.pbs.org/teachersource/mathline/lessonplans/search_k-2.shtm).