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Does the choice of observation instrument make a difference in the feedback and ratings that teachers receive? This study explores how lessons are rated differentially across various observation instruments. To investigate this question, ten randomly selected mathematics lessons were rated using six different observation instruments. Overall scores varied little across instruments. Our analyses indicate that differences in scores can be attributed to what we call instrumental occlusion, instrumental emphasis, and element density. This article concludes with implications for the selection and use of observation instruments in school settings.

In this article, the authors (a mathematics educator and a literacy educator) offer a framework to engage students with mathematics and children's literature through a read-aloud experience in which teacher and students engage with the text while problem solving. The authors describe the narrative-first approach as a strategy during which the teacher starts with a piece of children's literature and reconstructs it through a mathematical lens, enacting the narrative-first approach in five sequential steps. During the first step, the teacher reads the story aloud to students; following that, the teacher asks connecting questions relating the story to the mathematical concept to be studied. For the third step, once connections have taken place, the teacher then poses the mathematical problem. For the fourth step, the students complete the initial problem; they broaden their thinking with an extension problem, which challenges them further. The fifth and last step is a reflective discussion during which teacher and students reflect on mathematical learning. The authors aim to adapt this framework to integrate literature and mathematics with children ages 6-9, taught in a virtual setting.

A transition from "doer" to "teacher" for prospective teachers requires them to reorient from thinking about how they do mathematics to engaging with students and their work, understanding student representations, and planning instruction accordingly. To scaffold a transition, we developed a five-step mathematics as teacher heuristic (MATH) model. The study investigated the impact of MATH on the development of teacher candidates' mathematical knowledge for teaching (MKT) and their pedagogical knowledge. Twenty-two preservice secondary mathematics teachers enrolled in a methods course participated in the study. Findings of the study showed that teacher candidates' MKT was engaged as a result of analysis of the student work. While some teacher candidates based subsequent instructional planning work on what they noticed in the student work, others had gaps between what they noticed and their further planning. Teacher candidates' work samples with noticing and noticing with gaps are shared in the results section.

Too often, high school algebra students encounter topics in a discrete, disconnected fashion. In the authors' first years of teaching, they found students to be adept at solving problems they were currently studying; for instance, they could solve quadratics during a quadratics unit. Too often, however, students struggled with that same content once they moved onto new topics in the same course. For instance, the quadratic formula, a routine algorithm in October, was quickly forgotten by spring, when exponential functions were the focus. As the authors gained confidence and experience as teachers, they retooled their instruction to encourage students to make connections across topics throughout the school year. In this article, they share an investigation that exemplifies this style of practice, a rich task designed for second-year algebra students, one that connects polynomial and exponential functions in a way that fosters inquiry, discovery, and problem-solving tenacity. More specifically, they explore the Quadratic to a Quadratic (Q2Q) task, a low-threshold, high-ceiling task that encourages students to use a variety of problem-solving strategies as they find solutions to a nonroutine algebraic equation.

In this article, the authors share activities designed to help students make a connection between the complex roots of X[superscript 2] - 2x + 5 = 0 and the graph of y = X[superscript 2] - 2x + 5. We piloted these activities in three precalculus classes. The activities involved both concrete and technological representations of the mathematical situation. Students manipulated the concrete representation, whereas one of the authors manipulated a GeoGebra representation. In so doing, the authoss believe that students were using these two representations to model and interpret a mathematical phenomenon, as suggested in "Principles and Standards for School Mathematics" (NCTM 2000), as well as making connections between the two representations.

Mathematical knowledge for teaching is a complex web of knowledge domains. In this article, we share findings from an 18-month professional development project that aimed to improve middle school mathematics teachers' mathematical knowledge for teaching (MKT) of proportional reasoning by focusing on the critical analysis of mathematical tasks and student work. Although multiple studies have shown that professional development can contribute to teachers' MKT globally, little is known about how this knowledge grows and how specific domains of MKT can be targeted through professional development. Findings in this study show how professional development positively influenced participants' knowledge of content and teaching and knowledge of content and students, two domains of MKT, through teachers' twinned analyses of tasks and student work in proportional reasoning.

Generating mathematical conjectures is an important mathematical habit of mind for preservice teachers to develop so that they can, in turn, help their students to develop this skill. In this paper we present a classroom episode in which preservice mathematics teachers experience conjecturing in the context of a rich, open-ended task. They also reflect, throughout the task, on how they might nurture the generation and exploration of mathematical conjectures with their own students.

In this article, the authors present the use of digital tape diagrams--implemented using GeoGebra (2001) dynamic mathematics software--as a support to static diagrams to pose and solve proportional reasoning tasks. They show how dynamic sketches can allow students to explore standard tasks in proportional reasoning in ways that emphasise visualisation of the task and support conceptual understanding.

Middle school students need opportunities to craft informal mathematical arguments beyond justification of steps in an algorithm. Pentominoes provide an excellent vehicle for such activity. In this article, the authors describe an exploration with pentominoes that engaged a group of ninth-grade students in proportional reasoning, dilation, and spatial visualization while encouraging them to engage mathematical argumentation. (A bibliography is included.)

Sustaining engagement with a mathematics task is not a novel suggestion for effective mathematics teaching. "Principles and Standards for School Mathematics" (2000) specified that "students need to know that a challenging problem will take some time and that perseverance is an important aspect of the problem-solving process and of doing mathematics" (NCTM 2000, p. 186). Yet there is often an expectation in mathematics class that solutions should come quickly, and teachers work against a norm that has equated fast responses with proficiency in mathematics. The Common Core's Standards for Mathematical Practice (SMP) (CCSSI 2010) have brought renewed attention to the importance of perseverance in mathematical problem solving. In this article, we offer a set of classroom practices that foster students' perseverance in problem solving.

The Common Core State Standards (CCSSI 2010) and NCTM's (2014) "Principles to Actions" agree that "for meaningful learning of mathematics, tools and technology must be indispensable features of the classroom . . . that support students in exploring mathematics as well as in making sense of concepts and procedures and engaging in mathematical reasoning" (NCTM 2014, p. 78). With these tools come novel forms of mathematical representations. The National Council of Teachers of Mathematics confirms the dichotomy by defining representation as referring "both to process and to product--in other words, to the act of capturing a mathematical concept or relationship in some form and to the form itself" (NCTM 2000, p. 67). Thus, by working with external rep representations, students access, assimilate, accommodate, and build internal representations. With advances in technology, the forms and capabilities of external representations are opening new windows both to do and to study mathematics so that students' internal representations will evolve. The intent of this article is to provide teachers with a more nuanced explanation of the role technology plays in helping students mediate between external and internal representations of linear functions. Teachers may (re)consider the questions that they might ask when creating an activity using technology or engaging in formative assessment during an activity using technology. In this article, the author first shares an activity in which students use TI-Nspire to study linear functions. Then she shares some of the major results of a study that she completed and its implications for the teaching and learning environment.

This study is based on a Task Design and Analysis activity from a year-long professional development program. The activity was designed to increase teacher growth in several areas, including knowledge of mathematics, understanding of students' cognitive activity, knowledge of good questions, and ability to develop and improve high quality tasks. The study collected data from 30 classroom teachers. Results include teachers' initial understanding of the task, teachers' interpretations and awareness of the strategies their students used to solve the task, and the pattern of revised tasks. The process and findings of the study will expand teacher educators' understanding of structured approaches to link an inservice PD with teachers' work context as well as develop a better analytic framework within which teachers and teacher educators analyze student work.

The article opens with a Geoboard Triangle Quest in this form: "How many noncongruent triangles can be constructed on a 4 × 4 geoboard? How do you know? Justify your answer with significant supporting work." The use of advanced digital technologies as tools for problem solving receives much attention in the methods classes if the authors. They anticipated that some students would use technology to solve the task. The authors looked forward to the interplay between those approaches and hands-on approaches mirroring those adopted by Allen's students. Ultimately, the students attempted the task in many ways, using both digital and analog techniques. Interestingly, different solution strategies yielded different numbers of triangles. Although Allen's students tentatively concluded that 72 triangles were possible (2013, p. 118), they did not provide a proof, and none of the students arrived at this answer. In this article, the authors discuss how they came to what they consider a definitive solution and, on the way, address a challenge that arises in many inquiry-based classrooms: When confronted with multiple, conflicting results, how does one determine which--if any--answers are correct?

In mathematics, students should represent, model, and work with such real-world situations as those found in the physical world, the public policy realm, and society (CCSSI 2010). Additionally, students need to make decisions and be flexible enough to improve their decisions after analyzing realistic situations. The LEGO® Pets activity does just that. Modeling with the Common Core's Standards for Mathematical Practice is a process that begins with identifying variables, analyzing relationships, and formulating a model, then moves into interpreting, validating, and reporting on the results (CCSSI 2010). To bring this modeling process to life in classrooms, teachers need sources of authentic problems. But how can teachers determine the authenticity of a problem that will allow students to model and analyze a situation mathematically? Herrington and Oliver (2000) and Lombardi (2007) have developed frameworks for authentic learning situations. They indicate that authentic problems share a number of characteristics. In particular, they (1) have clear connections to the real world; (2) require significant investments of time and intellect; (3) use information from multiple sources and often require taking more than one perspective; (4) allow for student collaboration; and (5) require not only finding a solution but also interpreting that solution in the context of the problem. In this article, the authors describe how they developed the LEGO® Pets activity (activity sheet included at the end of the article) in which students model a real-world problem situation, first concretely and then abstractly. They taught the activity in three sixth-grade classes in an affluent suburban district as well as in one sixth-grade and one seventh-grade class in a much less affluent suburban district. Approximately thirty students were in each class. The students in all five classes were motivated, interested, and able to work through all the questions posed. Sometimes they needed no scaffolding to answer the questions; at other times, minimal scaffolding; and a few times, extensive scaffolding.

In a recent collaboration with an area high school teacher, the authors were asked to develop an introductory sinusoidal curves lesson for a group of second-year algebra students. Because the topic was abstract and unfamiliar to these tenth graders, they looked for hands-on lessons to support their learning. One lesson that they found, which they refer to as the Spaghetti Sine Curves activity, had students use uncooked spaghetti to measure and generate line segments to create sine curves on a coordinate system (see, e.g., Dadmehr 2006; Coleman n.d.; and Sturdivant 2002). Using a modified version of the lesson, the authors were surprised to discover how problematic the use of hands-on materials was for students. Many struggled to break the spaghetti strands precisely. Others found taping the spaghetti to the paper both tedious and time-consuming. In the original lesson, the use of hands-on materials actually interfered with student learning. Over the course of several brainstorming sessions, the authors revised the original Spaghetti Sine Curves activity, incorporating technology into the lesson. Here they share their revised Spaghetti Sine Curves lesson, highlighting the use of GeoGebra applets to enhance student learning. As they share this lesson, they illustrate the important role that technology may play in supporting effective use of physical materials.

Amazon, Walmart, and other large-scale retailers owe their success partly to efficient inventory management. For such firms, holding too little inventory risks losing sales, whereas holding idle inventory wastes money. Therefore profits hinge on the inventory level chosen. In this activity, students investigate a simplified inventory-control problem. Within this context, students develop tables, graphs, and algebraic representations to reach a decision. This activity has been successfully completed with students in both first- and second-year algebra.

Latest standards-based reform documents from mathematics and science education organizations and communities bring forward one important issue--higher-order thinking skills, namely reasoning. In this article, we share our experiences and our approach to developing an integrated course for mathematics and science prospective teachers centered on fostering reasoning with the use of technology. This study took place in an integrated mathematics and science methods course. The prospective teachers were career changers with STEM (science, technology, engineering, and mathematics) backgrounds. A sociocultural theoretical approach framed this study, wherein learning is evident in the constructed and enacted plans to engage STEM students in reasoning. The analysis of prospective teachers' reflections and integrated lesson plans suggested that the structure of the course scaffold with minilesson study helped prospective teachers extend their view of reasoning and consider reasoning when planning lessons in the future.

The article discusses lessons learned through integrating children's literature with mathematics through the lens of researcher C. Deck's "Power of Yet."

A transition from "doer" to "teacher" for prospective teachers requires them to reorient from thinking about how they do mathematics to engaging with students and their work, understanding student representations, and planning instruction accordingly. To scaffold a transition, we developed a fivestep mathematics as teacher heuristic (MATH) model. The study investigated the impact of MATH on the development of teacher candidates' mathematical knowledge for teaching (MKT) and their pedagogical knowledge. Twenty-two preservice secondary mathematics teachers enrolled in a methods course participated in the study. Findings of the study showed that teacher candidates' MKT was engaged as a result of analysis of the student work. While some teacher candidates based subsequent instructional planning work on what they noticed in the student work, others had gaps between what they noticed and their further planning. Teacher candidates' work samples with noticing and noticing with gaps are shared in the results section. [ABSTRACT FROM AUTHOR]

In the following paper, the authors share the Trout Pond Exploration, an activity designed to engage students in sequential reasoning. The authors revise the exploration to include student access to advanced digital technologies, namely spreadsheets. Using these tools, students follow their natural inclination to solve the task using recursive methods. [ABSTRACT FROM AUTHOR]

The article discusses the role of dynamic geometry software application GeoGebra in supporting effective use of physical materials like spaghetti to study frequency curve and amplitude of the sine function. It informs that the traditional method which uses spaghetti solely to measure each segment separately for different angles was a time consuming task for teacher and student. It cites that the software helps students to understand periodic nature of the sine function.

This study investigates the effectiveness of the Ohio High School Transformation Initiative using several years of state data on attendance, graduation, dropout rates, and performance index scores. Results indicate differences between small schools or small learning communities, and traditional schools that have similar school profiles to small schools in: graduation rates by year; attendance rates among the traditional schools over the years; and the performance index scores. Small schools' performance index scores gradually increased, but traditional schools with similar school profiles outperformed small schools. There was no difference in the dropout rates. Creating a sense of community, rather than simply transforming the school structure, seems to be the key to improving high school education. [ABSTRACT FROM AUTHOR]

There is a dearth of research on the mechanisms for preservice teachers development of the pedagogical knowledge necessary for effective use of such technologies. We explored the emergent Technological Pedagogical and Content Knowledge (TPACK) (Niess 2005, 2006, 2007) of a group of secondary mathematics preservice teachers in a methods course as they designed and implemented technology-rich teaching materials in field settings. Participant surveys and collected assignments were analyzed through the lens of the TPACK framework. The data were also analyzed to examine the trajectory of the participants' beliefs about the appropriate role of advanced digital technologies in mathematics. The results indicate that the participants' understanding of technology shifted from viewing technology as a tool for reinforcement into viewing technology as a tool for developing student understanding. Collected data supports the notion that preservice teacher TPACK development is closely related to a shift in identity from learners of mathematics to teachers of mathematics. In a class where advanced digital technologies were used extensively as a catalyst for promoting inquiry-based learning, preservice teachers retained a great deal of skepticism about the appropriateness of using technology in concept development roles, despite their confidence that they can incorporate technology into their future teaching. [ABSTRACT FROM AUTHOR]

Handheld graphing technologies have fundamentally been used in the teaching and learning of many secondary school and college mathematics concepts. To observe the effects of these new kinds of technologies on students' learning of mathematics, their teachers need to decide whether it is reasonable to do, learn, and teach mathematics with a calculator, and moreover, if graphing calculators are worth using in their classrooms as an instructional tool. With this aim in mind, this study used two conceptual frameworks—the possible roles of advanced calculators in mathematics teaching and learning, as well as technological, pedagogical, and content knowledge—to investigate the views of preservice secondary mathematics teachers on the use of calculators in mathematics instruction. Open-ended survey questionnaire and group interviews were the main data collection methods for this study. Results revealed preservice teachers' views on the use of handheld technology in mathematics instruction and the roles that they assigned to the calculators such as a visualization tool, a transformational tool, a computational tool, or a discovery tool. Their beliefs about advantages and disadvantages of using graphing calculators and when and why they would prefer to use graphing calculators were also discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]