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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.

Successful teacher preparation programs provide learning experiences that help teacher candidates make the shift from "student" to "teacher." In this paper we present research on the implementation of a process for providing candidates such experiences. Utilizing the Mathematics as Teacher Heuristic (MATH) process, prospective high school mathematics teachers explore rich problems by solving the task, analyzing samples of student work, designing a solution key, and modifying the task. We use their engagement in these explorations and reflections on the process to analyse the development of candidates' Content Knowledge and Pedagogical Content Knowledge.

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. (Contains 5 figures.)

The use of technology to create multiple representations of a concept has become one of the significant instructional environments that the National Council of Teachers of Mathematics (2000) suggests strongly for mathematics teachers to consider. One example of this type of environment is educational software with linked multiple representations. An activity for both linked and semi-linked versions of multi-representational software which was used in a dissertation study is presented along with two ninth grade algebra students' responses in order to provide an example of possible uses and effects of semi-linked and linked computer software in mathematics classrooms. It was also aimed to make connections between practice and research. The conclusion of this study was that semi-linked representations could be as effective as linked representations and that there was a role for each in different situations, at different levels, and with different mathematical concepts. (Contains 3 figures and 3 tables.)

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.

OBJECTIVE: The current study aimed to examine the psychiatric symptoms that can be seen in fibromyalgia (FM) patients, their attitudes toward seeking psychological help, and their concerns about stigma. Besides, it was investigated whether the stigma concerns that they may experience about receiving psychiatric treatment constitute an obstacle for patients to receive psychiatric treatment. SUBJECTS AND METHODS: This cross-sectional descriptive study was conducted between February and July 2020. Various seeking help were measured with Attitude Towards Seeking Psychological Help Scale-Short Form (ATSPPHSF), Self-Stigma in the Process of Seeking Psychological Help Scale (SSPSPHS), Intention to Seek Psychological Help Inventory (ISPHI), and Social Stigma Due to Seeking Psychological Help Scale (SSDSPHS). FM symptoms of patients were measured with The Symptom Screening Questionnaire, Revised 90 Items (SCL-90-R). Quality-of-life parameters were measured with Fibromyalgia Impact Questionnaire (FIQ). RESULTS: Fibromyalgia patients had higher somatization (p=0.001), psychotism (p=0.045) and phobic anxiety (p=0.015) scores than controls. The ATSPPH-SF (p=0.002) and SSPSPHS (p=0.043) scale scores of the FM patients were higher than the controls. There was a significant positive correlation between FIQ and SSPSPHS (r=0.288, p=0.043) and SCL-90 overall (r=0.602, p<0.001) and all subscales scores. Patients with high active psychotic symptom levels had higher FM exposure scale scores and SCL-90 overall scores than those with low active psychotic symptom levels (p<0.001). CONCLUSIONS: The findings of this study showed that fibromyalgia patients have more somatization symptoms than healthy individuals, and as psychiatric symptoms increase in these individuals, their level of being affected by FM increases. [ABSTRACT FROM AUTHOR]

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.

This experiment was carried out to investigate the effects of reduced dietary calcium (Ca) and non-phytatephosphorus (NPP) levels at 2:1 constant ratio on performance, carcase parameters and bone mineralization of Ross-308 broilers. A total of 11400 one-day-old chicks were randomly allocated to five dietary treatments with twelve replicates and were fed starter diets including either recommended (0.96 and 0.48 % Ca and NPP respectivelycontrol treatment, T1) or a 6.25% lower level of both Ca and NPP according to breeder's recommendations during starter period (SP). After SP, T1 continued to be provided through diets containing 0.87 and 0.78 % Ca for grower (GP) and finisher periods (FP), respectively, while 6.25 % reduced group was divided into 4 dietary treatments regarding extent of decrease in Ca and NPP levels and provided by diets containing 0.79;0.65 (T2), 0.79;0.60 (T3), 0.75;0.65 (T4), and 0.75;0.60 (T5) % Ca for GP and FP, respectively. After SP, moderate and even substantial reduction of Ca and NPP significantly decreased feed intake (FI) (quadratic, P<0.05), but obtained similar body weight gain (BWG) and FCR compared to the control (P>0.05). On the other hand, reduced dietary Ca through T1 to T5 significantly decreased the total Ca and NPP intake of broilers (linear and quadratic, P<0.01), and accompanied to significant linear and quadratic relationship between Ca intake and FCR and BWG respectively. It can be concluded that reducing Ca level down to 0.75 and 0.60 % in GP and FP, respectively are possible without compromising the growth performance and bone mineralization of modern broilers. [ABSTRACT FROM AUTHOR]