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In this article, the authors present a framework for guiding elementary preservice teachers in adapting mathematics word problems to better meet English language learners' (ELLs) needs. They analyze preservice teachers' ELL adaptations implemented in a one-on-one setting. Through qualitative methods, four themes regarding implemented adaptations are identified: language adaptations, mathematical adaptations, tool/visual adaptations, and structural adaptations. The authors conclude that the framework was successful in helping preservice teachers learn about adapting curriculum by interacting with ELLs. Implications for teacher education are discussed.

The article discusses the use of Tinkercad, a free 3D simulation platform, to foster interdisciplinary STEM learning in middle school classrooms through hands-on design and problem-solving activities. Topics include the advantages of interdisciplinary STEM education, strategies for using Tinkercad effectively in teaching, and a detailed classroom activity emphasizing the iterative engineering design process.

Preservice elementary teachers often struggle with prime decomposition and other mathematical topics that correlate with number theory. This paper provides a framework for integrating prime factor tiles into their curriculum with a particular emphasis on prime decomposition. Using this framework, preservice teachers explored and evaluated numbers using prime factor tiles. The results of the exploratory inquiry showed that preservice teachers made some progress in their understanding of prime decomposition after exploring with the tools. However, they struggled with problems requiring the application of prime decomposition. More time to delve into this topic is probably needed in order to observe further gains.

As there has recently been an onslaught of video cases being developed and implemented with preservice teachers, it is important to evaluate how we should use these cases. This research investigates the features elementary preservice teachers consider valuable when using video cases in mathematics education. The researchers used hierarchical cluster analysis to create a dendrogram that displayed statistically significant features. The study revealed two primary clusters--(a) cross-subject lessons emphasizing adaptations and techniques to reach a variety of learners and (b) problem-based lessons with students in groups supported by lesson analysis--as well as (c) a small third cluster of collection and distribution of materials. Ultimately, such findings can be used to guide the design and incorporation of tools for using mathematics-based video cases. (Contains 2 tables and 1 figure.)

Family math night can easily be implemented into mathematics methodology courses providing an opportunity for field-based learning. Preservice teachers were asked to develop and implement an inquiry-based activity at a family math night event held at a local school with personnel, elementary children and their parents in attendance. This action research examines what preservice teachers discovered about how children learn mathematics during their interactions with parents and children. Preservice teachers' individual reflections focusing on children's learning were qualitatively analyzed and clustered. In addition, data from teacher educator's observation notes was used to triangulate findings. Four themes were identified in relation to children's needs for promoting learning in mathematics: (1) activity adjustment (2) engagement through guiding and questioning (3) motivational issues and (4) the use of manipulatives and visuals. The event appeared to offer preservice teachers insight into the elements of activities needed to guide children's learning in mathematics.

Elementary pre-service teachers in an upper-division algebra mathematics content course explored graphing through the use of Calculator-Based Rangers™ (CBRs). This study explores how they used the "Distance Match" feature of the technology to engage in algebraic thinking. Operating CBR technology, pre-service teachers' perceptions regarding how position/time graphs represent algebraic concepts was explored with an emphasis on physical movement. Using an inductive approach for qualitative data, the pre-service teachers' reflections regarding the use of CBRs were analyzed. Specifically, patterns and structures were discovered through coding of the pre-service teachers' raw descriptions of their movements. Using the data sources (a reflection and assignment), the results showed increased understanding and beneficial affordances with the use of CBRs. However, the pre-service teachers failed to connect the graphs thoroughly to numeric representations and described slope only superficially. For example, movement was not contextualized using numbers but rather using verbal descriptions. Implications are provided for the integration of CBRs as tools for enhancing pre-service teachers' knowledge for teaching mathematics with technologies.

This study examined the structure of two preservice teachers' understandings of educational software in mathematics using repertory grid techniques. Specifically, the study focused on how teacher educators can enable preservice teachers to discern the features of mathematics software, and develop pedagogical goals that reflect the affordances and constraints of available tools. Results showed a deepening of knowledge and a differentiation of knowledge following experiences of exploration, evaluation, and comparison of different types of software. Results imply that directing preservice teachers' attention to utilizing the functional characteristics of software for developing plans and for the design of instruction may be fruitful for more effective integration of software in their future teaching. However, participants' preexisting content knowledge and their pedagogical worldview tempered this effect. (Contains 8 figures and 2 tables.)

Recognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern‐based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described. [ABSTRACT FROM AUTHOR]

Using a dog's paw as a basis for numerical representation, sixth grade students explored how to count and regroup using the dog's four digital pads. Teachers can connect these base-4 explorations to the conceptual meaning of place value and regrouping using base-10.

To support students' development of concepts in mathematics, the use of technology is often encouraged (Common Core State Standards Initiative [CCSSI] 2010). Technology can contextualize learning and provide a meaningful setting for mathematical ideas. Most teachers are supportive regarding the use of technology to encourage learning and understanding in mathematics. However, using technology can present many challenges. Particularly, technology training is not as strong as it could be in supporting teachers' growth and ability to create lessons, and most teachers would like support in using technology in the classroom. To address this concern, this article presents a guide for an activity using graphing calculators and Calculator-Based Ranger™ 2 (CBR) as the technology to support understanding of position/time graphs with preservice elementary school teachers. This activity supports a grade 6 standard from the Common Core State Standards for Mathematics (CCSSM) that encourages students to see the relationship between variables in real-world situations. Students should "analyze the relationship between the dependent and independent variables using graphs and tables". This activity allows students to experience how their distance from the CBR (dependent variable) changes the graph as related to time (independent variable). CBRs are sonic motion detectors designed to be used with graphing calculators (specifically the TI-83 Plus and the TI-84) and can collect data on motion, including distance, velocity, and acceleration. Additionally, one of the preservice teachers taught the same lesson individually to a fifth-grade student who sometimes struggles in mathematics. The teacher's reflections are included. A bibliography is included.

Since the 1950s, the understanding of how the base 10 system works has been encouraged through alternative base systems (Price 1995; Woodward 2004). If high school students are given opportunities to learn other base systems and analyze what they denote, we believe that they will better understand the structure of base 10 and its operations (Verschaffel and De Corte 1996). Special pedagogical knowledge is needed to teach numbers and operations (Ball and Bass 2000). In the primary grades, teachers focus on base 10 and often use base 10 blocks to support the development of operations and number structure. Learning other base systems forces the students to deconstruct a system that they do not always structurally conceptualize (Anderson, Reder, and Simon 1996). For example, students must critically analyze regrouping and scrutinize place values and digits (Woodward 2004). The activities presented in this article, for students grades 8-10, focus on two alternative base systems: base 5 and a student-selected base system beyond 11. Polynomial operations are also presented as they relate to alternative base systems. A bibliography is included.

Earth Day is celebrated annually on April 22. Teachers often commemorate Earth Day with their classes by planting trees, discussing important conservation topics (such as recycling or preventing pollution), and encouraging students to take care of planet Earth. To promote observance of Earth Day in an intermediate elementary school classroom, this article presents the integration of sustainability with the hopes of preserving Earth and its resources for as long as possible. This article describes a six-week unit that integrated sustainability and mathematics at the intermediate elementary school level. Students explored a topic and identified solutions that would support preserving resources for the betterment of our planet. The iSTEM (Integrating Science Technology Engineering in Mathematics) authors share ideas and activities that stimulate student interest in the integrated fields STEM in K-grade 6 classrooms.

The purpose of this investigation is to describe an implementation of a modeling task using mock data from an ancient archeological find. Students discover the relationship between the height of a person and his or her stride length. Qualitative data from student discussions document thinking and reasoning.

Video cases have frequently been developed to model educational instances from an authentic elementary classroom setting for preservice teachers in mathematics and science. Preservice teachers' predilections regarding mathematics and science video cases should influence how cases are integrated within the university classroom. Using Personal Construct Theory, constructs were identified, used to create repertory grids, rated and then analyzed using hierarchical cluster analysis. Three primary clusters were discovered along with a minor fourth cluster. When designing or using a video case with preservice teachers, these findings can help determine the content for integration.

Narratives are a wonderful way to allow students to personalize mathematics. In the context of this article, the authors define narratives as short stories, written by students, that contain a mathematical work problem centered on students' lives and interests. These narratives can be supported with art, such as drawings or photographs. When students conceptualize mathematics through these narratives, they are able to connect the relevance of mathematics to their everyday lives. This article describes a framework using student-authored narratives to help students build stronger connections to the application of mathematics. This framework allows middle school students to use their creativity while applying and reinforcing mathematical material. The lesson is appropriate for all levels of middle school learners, including students with special needs and English language learners. Students will create narratives at their own ability levels, making the lesson appropriate for learners with a variety of needs (Hitchcock et al. 2002).

In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance" as one such tool to model quantitative relationships of an algebraic equality. It models an equation by illustrating the meaning of equal because the center of the balance (the "fulcrum") can be correlated to the "equal sign." For the balance to stay steady at its center, values must be equal on both sides. The use of a balance to help students develop algebraic understanding and a deeper meaning of "equal" has been shown to be beneficial to student learning. The authors then recommend "balance tools" available online for students from Pre-K through grade 12 to afford students the opportunity to visualize how their manipulation of numbers or items sways or impacts the balance.

To improve access to STEM curriculum, an activity was planned that presents the opportunity to design and build using gears and other tools. In this challenge, preservice elementary school teachers were asked to mathematically analyze gears and create an amusement park ride that uses gears to spin. Although this lesson was implemented with preservice teachers, it was designed to embed elementary school curriculum within the context of learning content. The Amusement Part Ride challenge offers the chance to explore STEM within the context of future curriculum that elementary school teachers may teach. The lesson can easily be explored with students in intermediate elementary grades as well.

Do manufacturers' claims about water projectile toys measure up? It is one thing to assert that a Super Soaker[R] shoots thirty linear feet, but can it really shoot that far? This lesson gives students an opportunity to test the claims, experience measurement in a meaningful context, and analyze data within that context. Although measurement can be a challenging task to teach and learn (Clements 1999), using water toys makes this investigation topic interesting, exciting, feasible--and fun. By integrating measurement and data analysis, students have the opportunity to test manufacturers' claims. This investigation focuses on the grades 3-5 Standard in which students are expected to "understand measurable attributes of objects and the units, systems, and processes of measurement" as well as "the need[s] for measuring with standard units and becom[ing] familiar with standard units in the customary and metric systems." Expectations for K-grade 2 include understanding "how to measure using nonstandard and standard units" (NCTM 2000). This Super Soaker investigation includes extensions (for grades 5-8) and modifications (for grades 1-4) for other investigations using water projectiles.

Literature is viewed as a natural "thinking" medium to help children experience learning in a contextual setting (Ward 2005). Though, as Draper (2002) contended, mathematics is a subject that children are often unable to recognize as useful in their lives, literature with mathematical components may be able to address this concern. Combining mathematical thinking skills with literature may be a useful means to explore mathematical reasoning and literacy development (Draper 2002; Ruiz, Thornton, and Cuero 2010). Designing problem-based curriculum can be conceptually challenging. One good starting place can be to design problem-based mathematics inquiry using a narrative, storytelling technique. The premise of the approach described in this article is that authoring narrative storybook literature provides an accessible and positive vehicle for teachers and students to learn to express mathematical constructs. A sample book is used to describe the approach in detail and to demonstrate how to create a children's book with a problem-based component embedded within the storyline. In addition to describing the strategy, the authors provide a framework to implement mathematical, storybook narratives in a preservice teacher education class for use in the school setting with students as the mathematical problem solvers. (Contains 2 figures.)

With the ever-changing dynamics of society, teachers are often faced with a classroom of students who have diverse linguistic and academic needs. Research has shown that schools are becoming more linguistically diverse throughout the United States and that English language learners (ELLs) are posing their own sets of challenges for teachers. Teaching mathematics to ELLs can be particularly challenging. Mathematics contains many of the same challenges as other subjects and can be perceived as being less difficult because it is based on the language of numbers. By using deliberate strategies to adjust the phrasing of word problems, teachers can provide a richer mathematics experience for ELLs. The author presents two examples of word problems written in a traditional format and rewritten in an adjusted format reducing linguistic demands while maintaining rigor. The framework can be used by teachers to support their ELLs by modifying word problems to focus more on the mathematics. (Contains 2 figures.)

Research shows that preservice teachers can enhance their growth and development through interactions with children in the school setting as a component of their coursework in education. Additionally, preservice teachers feel they need more interactions with parents in their university preparation. Family Math Night has the potential to address these needs. This article provides a conceptual framework for merging Family Math Night into a mathematics methodology course. First, a method for coordinating the event with local schools is described. Second, the structure for the preservice teachers' assignment is presented as a model for integration. With this structure, preservice teachers are provided with an opportunity to challenge what it means to teach and learn mathematics through field-based experiences. (Contains 1 table.)

Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)

After interacting with a Best Practices video case whole class and individually, preservice teachers were asked to reflect on what features they deemed helpful. In addition, they were asked to reflect on what additional features they would find desirable. The constant comparative method was used to analyze the data (Glaser, 1965). Three features were considered constructive: (1) modeling of teaching techniques (2) supplemental materials to support the video case and (3) observing classroom management. Concerning additional features, they wanted (1) an analysis of the lesson (2) improvements in technology and editing and (3) extended video case selections. The expressed features can guide the creation and use of video cases in preservice teachers' education by providing direction for integration and development.

Video cases are becoming more and more prevalent throughout the United States to assist in the development and growth of preservice teachers' instruction. This research investigates the perspectives of preservice teachers and their naive understandings of the kinds of learning and assistance video cases can provide in their methodology courses. Personal Construct Theory was used to uncover participants' perceptions of their needs in terms of curriculum content of video cases. Hierarchical cluster analysis revealed six distinct clusters which serve as themes for discussing the early needs of preservice teachers and how video cases might be configured to address these perceived needs. Results reveal that preservice teachers' beliefs, though naive, do reflect the research on the actual needs of teachers in their first years of induction and support. Perhaps if preservice teachers' predilections are examined, this data can help guide the future implementation of video cases in their university courses.

This article describes a structure to help preservice teachers get invaluable field experience by aligning theory with practice supported by the integration of elementary school children into their university mathematics methodology course. This course structure allowed preservice teachers to learn about teaching mathematics in a nonthreatening environment without the formal observations and evaluations common in student teaching situations. It also allowed teachers to view lessons and to teach using appropriate framework to produce optimum problem-based learning. In addition, teachers experienced the proper use of manipulatives and discovered how manipulatives support student learning. (Contains 1 figure.)

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.

Teaching methods often promote fact gathering and memorization rather than the use of those facts to solve complex, realistic problems. Students easily memorize basic facts, steps and procedures, repeating them flawlessly from memory when asked. The problem, however, lies in teaching students how to use these facts and procedures creatively in new, unknown problem-solving situations. While they recite this knowledge when prompted by the teacher, the students find it difficult to decide which piece of knowledge to use and how to use it when solving problems in more complex situations. Case methodology has emerged as a powerful tool for creating a bridge between theory and practice in teacher education. Teaching cases have been a keystone of professional training in schools of business, law and medicine for more than a century. Case method instruction provides models of how to think professionally about problems. As such, case studies bring the complexities of classroom activities into focus and allow students to connect concepts and real life. Case-based instruction has been defined as an active-learning pedagogy designed for problem analysis and problem-solving stressing a variety of viewpoints and potential outcomes. According to Shulman (1992) there are no studies or evidence that confirm the effectiveness of this method. Nevertheless this author found that its constitutive elements account for the potency of the case method. flexibility. The use of case-based teaching and learning can be utilized to help preservice teachers connect knowledge and practice. Case-based teaching and how it is used is discussed in this article. [The activities reported in this manuscript were supported, in part, by a grant from the United States Department of Education (#P336B990064).]

Anchored instruction is designed to present problems in a meaningful context to allow for investigations into real life environments. The Jasper Project was created to allow students to investigate mathematical dilemmas using anchored instruction techniques. This study uses case study methods to examine the perceptions that preservice teachers have toward the use of anchored instruction after having the opportunity to investigate and evaluate a Jasper predicament as a group. The guiding research questions were: 1) Through exploration and evaluation, are preservice teachers able to determine significance of using anchored instruction with their future students? And if they see significance, 2) How do preservice teachers envision student learning and mathematical growth using anchored instruction? Four individual cases were evaluated, specifically examining the features of anchored instruction each preservice teacher described. The preservice teachers were optimistic about the utilization of anchored instruction and described features that support student learning and growth.