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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]

Manipulatives have been used in many mathematics classrooms across many age groups with the aim of helping students to understand abstract concepts through concrete, kinesthetic, and visual experiences. In this paper, after we provide a background for the use of physical and virtual manipulatives in teaching and learning of mathematics, we will describe our experiences with a group of students who complete an activity with a physical manipulative which is followed by a virtual manipulative to study the concept of residuals in a linear regression. (Contains 5 figures.)

In this technology-oriented age, teachers face daily decisions regarding the use of advanced digital technologies--graphing calculators, dynamic geometry software, blogs, wikis, podcasts and the like--to enhance student mathematical understanding in their classrooms. In this case study, the authors use the Technological, Pedagogical, and Content Knowledge (TPACK) model in conjunction with a five-stage developmental model, which can be used to describe growth in TPACK to describe the initial attempts of a teacher, Jane, to develop TPACK as she learns and attempts to integrate an advanced teaching technology into her classroom, namely the TI-Nspire graphing calculator. The study tracks her struggles to reconcile some traditional beliefs about how students learn with her desire to be responsive to what she perceives as affordances of advanced digital technologies. Main data collection methods were journal writing, observations, document analysis, and interviews. Using the five-stage developmental model, we saw that this experience helped Jane to move among different stages. This study showed that the TPACK model with the five-stage developmental model can be a beneficial tool for researchers to study teachers' professional growth and is also a valuable tool for teachers to reflect on their own growth. (Contains 8 figures.)

A common activity--having students collect data to measure the radius and circumference of circular objects--can be given a new twist. These data can be viewed virtually using technology. A dynamic geometry environment coupled with the powerful capabilities of a spreadsheet can greatly enhance student learning. The activity in this article addresses these grades 6-8 tenets found in "Curriculum Focal Points," which state that students should be able to: (1) analyze tables to discern relationships; (2) use equations to describe relationships inherent in a table; (3) study areas and circumferences of circles; (4) identify the unit rate of the graph of a proportional relationship as the slope of the line; and (5) translate among verbal, tabular, graphical, and algebraic representations of functions. These suggestions can be explored in a technological environment where teachers and students use TI-Nspire[TM], GeoGebra, or The Geometer's Sketchpad[R]. Students collect data from a dynamic geometric construction and study the relationships among the radius of a circle, its circumference, and its area. Many technologies allow students to create and study dynamic geometric constructions by changing a part of the construction. The effects of that change in other parts of the same construction become clear for students to see. For instance, students can use a graphing calculator such as TI-Nspire with dynamic geometry features to grab and drag the radius of a circle to make it longer or shorter. However, in GeoGebra or Sketchpad, students would need to drag the endpoint of a radius. Students can also use the technology to quickly measure many circles, display these measurements, and observe the effect that changing the radius has on the circumference. These tools can also capture an approximation of the areas of the circles. A TI-Nspire-based activity sheet is included at the end of this article. (Contains 6 figures and 1 table.)

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. (Contains 2 figures and 10 tables.)

"I liked to grab things on the graphics and see it first hand, see what we were doing on the problem. It makes more sense." These words were from a student who completed a series of lessons on lines of best fit. She made this comment after comparing a hands-on approach using a strand of spaghetti with a high-tech approach using a calculator. To give students meaningful experiences while studying regression lines, the authors designed a learning environment that accommodates "Bruner's three forms of representation." Bruner advocated a sequence of learning that runs the gamut from using physical materials to using symbols to teach mathematical ideas effectively. This article presents a regression activity using the TI-Nspire[TM] calculator and spaghetti. (Contains 4 figures.)

Evolving technology has played an important part in a common quadratic-function lesson. Having been mentioned repeatedly in numerous reform documents, a recurring lesson has involved changing the parameters in f(x) = ax[superscript 2] + bx + c and studying the effects on the graph. In both NCTM Yearbooks and NCTM Standards documents, technology is discussed and explored as a way to enrich quadratic function tasks. However, these authors will show how advances in technology have enriched these activities. Using quadratic functions as an example, this article describes how the evolution of technology has altered the expectations of students' understanding and critical thinking. (Contains 9 figures.)

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers' views and beliefs about when and why they prefer three possible uses of CAS--black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

Handheld graphing technologies have fundamentally been utilized in the teaching and learning of many secondary school and college mathematics concepts. In order 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 utilized two conceptual frameworks--the possible roles of advanced calculators in mathematics teaching and learning, as well as technological pedagogical 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 discussed. (Contains 2 tables and 1 figure.)

The purpose of this study was to examine the proportional reasoning skills of Turkish middle school students. It was also aimed at revealing the differences in students' performances in terms of their gender, grade level, and problem types--missing value and numerical comparison. The study was conducted with two sixth and seventh grade classes from a public school in Ankara, Turkey. The data were collected through a proportional reasoning performance instrument consisting of eight ratio-proportional tasks--three missing value and five numerical comparison problems. The results revealed that there were differences in students' performances according to the type of problem and their grade level. Students were more successful in solving missing value problems than comparison problems. Also, it was found that seventh grade students performed better than sixth graders.

This article offers an introductory activity for the limit concept with a geometrical and historical foundation. A connection among Geometry, Measurement and Calculus is highlighted with the help of technology. The geometrical drawing, measurement and graphing capabilities of both TI-89 and Geometer's Sketchpad make it possible for students to experience Archimedes' process for determining circular area. (Contains 1 table and 7 figures.)

Teacher educators and researchers have studied the improvement of teacher education programs so that pre-service teachers may be well prepared. Research has shown that not only teachers' knowledge but also their beliefs have major influence on their approach to teaching. This present research examined the mathematics and science pre-service teachers' beliefs and perceptions in Turkey. Specifically, the study focused on the student-teachers' perspectives on "effective education" at the end of their subject area dominant program before the student teaching and the changes in those beliefs after having teaching experience. Many pre-service teachers described the teacher-centered environment as effective, and those who supported a student-centered environment discussed difficulties they faced in creating it during student and full-time teaching.

Concept mapping has been widely used as one of the most efficient methods of revealing the cognitive structure of an individual on any concept. There are not only different concept mapping techniques but also different ways of analysis. It has been suggested that concept maps provide valuable and rich information which becomes disoriented when they are analyzed only quantitatively. This study has described a qualitative analyzing method of concept maps, developed for investigating the cognitive structures of preservice teachers on some educational concepts. An analysis of the relations between the concepts in concept maps is developed and discussed which was neglected by both qualitative and quantitative analyses in the literature. This analyzing method aims to examine concept maps created by pre-service teachers using "construct-a-map-from-scratch" method centred on multiple key concepts which emphasized the categorization of linkage labels and other connected concepts. (Contains 3 tables and 3 figures.)

The focus of this study was on comparing three groups of Algebra I 9th-year students: one group using linked representation software, the second group using similar software but with semi-linked representations, and the control group in order to examine the effects on students' understanding of linear relationships. Data collection methods included mathematics pre- and posttests, follow-up interviews, computer-based clinical interviews at the end of the treatment, classroom and lab observations, document analysis, and survey. 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 2 figures and 2 tables.)

Presents results from a study that uses questionnaires to ask mathematics teachers' (N=64) opinions about mathematics education computer software. Indicates that respondents overwhelmingly favored discovery and remediation as the purpose of using software in mathematics classrooms. Teachers believed that real world situations and multiple representations were the most important attributes of educational mathematics software. (Contains 11 references.) (Author)

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