Search

Research suggests that tasks that use technology as a reorganizer (technology is used to produce dynamic representations) are linked with the development of students' conceptual knowledge, yet many secondary mathematics textbooks in the USA predominantly include tasks that use technology as an amplifier (technology is used to produce static images). Thus, if teachers wish to incorporate tasks that use technology as a reorganizer, they must locate these resources elsewhere or construct them themselves. This study took place in a technology, pedagogy, content course where each prospective teacher (PT) engaged in at least one component of an ideation, rehearsal/refinement, enactment, and reflection intervention, where they were asked to adapt traditional textbook lessons to promote students' rich conceptual understandings of function transformations via technology as reorganizer tasks. A total of 15 PTs agreed to participate, 8 seeking a grade K-8 certification and 7 seeking a grade 6-12 certification. On the initial lesson plan involving linear function transformations, the majority of PTs (13 out of 15) used technology as an amplifier, but on the final lesson plan involving absolute value function transformations, 13 PTs used technology as a reorganizer. There were also increases in the incidence of lesson elements that promoted moderate or rich forms of conceptual knowledge between the initial and final lesson plans. We discuss the implications of these results and introduce readers to dynamic conceptual components as manipulable bridges between different mathematical representations that hold the potential to develop students' richer forms of conceptual knowledge.

An important component of the Common Core State Standards for Mathematics (CCSSM), used by the majority of states in the U.S., has the eight standards for mathematical practice (SMPs). While surveys have investigated teachers' perceptions of the CCSSM few have investigated middle school mathematics teachers' (MSMTs') (grades 6-8) perceptions of the SMPs. Similar to other countries, teachers in the United States frequently use mathematics textbooks. Two different types of textbooks have been in use in the United States, conventional and standards-based. The latter were designed on the basis of earlier standards documents including versions of the SMPs. As a result of these changes to the educational system in the U.S., we sought to characterize MSMTs' perceptions of the SMPs and investigate if these perceptions differed by the type of textbook teachers used. We found that MSMTs struggled in naming the SMPs, misinterpreted the modeling SMP, and conflated components of reform-oriented instruction with the SMPs. MSMTs using a standards-based textbook were more likely to view their textbooks as aligned with the SMPs, x[superscript 2](3) = 7.708, p = 0.026 and to view the SMPs as an instructional philosophy for the CCSSM, Fisher's Exact Test = 3.881, p = 0.05. The implications of these results are discussed.

In this article, the author presents a set of design principles that were used to construct and refine a lesson involving the TI-Nspire handheld graphing calculator through multiple enactments in high school algebra classrooms to assist students in constructing a deeper understanding of systems of linear inequalities, a high school algebra content standard in the Common Core State Standards for Mathematics (CCSSM) (NGA Center and CCSSO 2010). These principles are rooted in years of crafting and implementing technology-based lessons in secondary school mathematics classrooms. Furthermore, they are research-based and aligned with the Common Core. These principles could be used to develop a technology-rich lesson as is done in this article or could be used as a framework to assist in revising a found resource. In this article, the author argues that using these principles to construct or adapt existing technological resources offers students opportunities to develop deeper mathematical understandings and nurture their capacity to investigate mathematics. This article also highlights the power of planning a lesson with others and reflecting on lesson implementation either alone or with others.

Eight middle school mathematics teachers' perceptions and uses of curriculum materials and the Common Core State Standards for Mathematics (CCSSM) were investigated. Adapting a noticing framework and models of dialogic instruction and direct instruction, teachers' noticing practices with curriculum materials and the CCSSM when planning, enacting, and reflecting on lessons were examined. Teachers who were committed to implementing the CCSSM and who were using one of two substantively different curriculum programs were purposefully selected. Data sources included multiple forms of interviews and classroom observations. The teaching evidenced three distinct noticing patterns. These patterns indicated that teachers' curriculum materials were associated with how teachers perceived and enacted the CCSSM. Teaching with a curriculum program that was designed as a thinking device prioritized the Standards for Mathematical Practice of CCSSM evidenced noticing that was consistent with dialogic instruction. Teaching with a curriculum program that was designed as a delivery mechanism prioritized the Content Standards of CCSSM and evidenced noticing consistent with direct instruction. Findings indicated that the designated curriculum and contributed to differing interpretations of CCSSM and served as a lens for noticing. However, a dialogic curriculum program was not sufficient to support dialogic approaches in practice. One pattern showed teachers planning dialogic lessons, but the lesson enactments were not consistent with teachers' plans, with evidence that the teachers were not aware that their practices differed from dialogic approaches. Implications for research and practice are discussed.

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.

This study examines Middle School Mathematics Teachers' (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers' use of district-adopted and non-district-adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers' planning practices, and perceived rigor of the CCSSM. The data suggest that teachers' use of digital resources located online was disconnected from their district-adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform-oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one-third of MSMTs wanted more CCSSM professional development.

In this study, U.S. middle school teachers' perceptions of Common Core State Standards for Mathematics (CCSSM), CCSSM-related assessments, teacher evaluation processes, and resources for implementing CCSSM were investigated. Using a mixed methods design, a national sample of 366 teachers was surveyed, and 24 teachers were interviewed. Findings indicated that teachers viewed CCSSM as including new content for their grade level. Teachers also reported using multiple curriculum resources to align with CCSSM and indicated that new assessments would serve as a proxy for CCSSM. Implications for rapidly changing policy, curriculum, assessment, instruction, and professional development related to CCSSM are discussed.

The symbolic manipulation capabilities of computer algebra systems, which we refer to as CAS-S, are now becoming instantiated within secondary mathematics textbooks in the United States for the first time. While a number of research studies have examined how teachers use this technology in their classrooms, one of the most important factors in how this technology is used in the classroom is how it is embedded within curricular resources such as textbooks. This study introduces readers to an analytical framework for examining CAS-S within textbooks and presents the results of its application to three secondary U.S. mathematics textbook units involving polynomial functions. The framework consists of two components: application of CAS-S and reflection on CAS-S uses. The analyses identified differences among the three textbook units in pedagogical intent and task connectedness involving CAS-S. The majority of CAS-S tasks were coded as involving low procedural complexity and there were few instances in which the technology was used in the construction of proofs. Textbook developers asked students to reflect on the visible CAS-S result as opposed to the invisible process leading to those results. The implications of these results as well as the potential transformative role of CAS-S are discussed.

This study investigated how a group of 10 prospective secondary mathematics teachers (PST) read, evaluated, and adapted a textbook lesson involving the symbolic manipulation capabilities of computer algebra systems (CASS). PST read the entire lesson and tended to focus on the organizing question at the beginning of the student lesson and the CAS-S sections of the lesson. PST frequently evaluated the lesson with respect to the teacher and whether lesson elements would promote student understanding. Only one PST evaluated the lesson with respect to the student and how the student might interpret lesson elements. There were five categories of curricular adaptations exhibited by PST: following, additive, mix, reductive, and adaptive. In general, the adaptations made to the lesson by PST were productive. These productive adaptations included asking students to make predictions before using CAS-S and understanding the hidden procedures used by the technology.

Using technology to explore the coefficients of a quadratic equation can lead to an unexpected result. This article describes an investigation that involves sliders and dynamically linked representations. It guides students to notice the effect that the parameter "a" has on the graphical representation of a quadratic function in the form f(x) = ax[superscript 2] + bx + c, and understand why this effect occurs. Students engaging in this activity develop a deeper understanding of quadratic functions. This investigation uses dynamic sliders connected to quadratic function parameters and the symbolic manipulation capabilities of a computer algebra system to understand this complexity. In the process, students develop a deeper understanding of the quadratic function concept and the connections among its graphical, standard, and vertex forms. Knowledge that is rich in connections is more likely to be retained by students (Bransford Brown, and Cocking 2000), and the explorations into the effect of parameters described here helps prepare students for more advanced mathematics topics. In addition, this article embodies a strategic use of technology as advocated in the Common Core State Standards for Mathematics Standards for Mathematical Practice.

Students' struggles in learning reasoning and proof (RP) in mathematics are well known. Despite the instantiation of RP opportunities in mathematics textbooks in the USA and the important role that textbooks play in teachers' instructional decisions, little research has been conducted on analyzing student textbook tasks and narrative sections of mathematics textbooks in the USA. One polynomial functions unit located within a reform-oriented, conventional, and hybrid secondary mathematics textbook was examined for RP instances. RP tasks comprised 4, 9, and 22% of the student tasks in the conventional, hybrid, and reform-oriented textbook units, respectively. The textbook exposition of all three textbook units promoted the presentation of proof building blocks (e.g., definitions) with no occurrences of conjecture development or testing. Overall, only 19% of the learning objectives across the three textbook units were validated. The conventional and hybrid textbook units had the potential to promote an authoritative proof scheme in students while the reform-oriented textbook unit had the potential to promote an empirical proof scheme in students.

This study examined the instances of a representational toolkit (RT) in one reform-oriented algebra textbook developed in the United States. The majority of RT uses were active (66%) as opposed to passive and connected (63%) in that the technology results were used in some other manner by either the textbook authors or students as they solved problems. Symbolic representations were most prevalent in the textbook while tabular representations appeared least frequently. Although the textbook involved a variety of different functions the parameter experimentation feature of the technology was rarely used. Comparisons were also made with a previous edition of the textbook to determine the impact of computer algebra systems (CAS) on the curriculum. These results suggest that CAS was used to make the textbook more investigatory, include new content to a limited extent, demonstrate solutions to example problems, verify mathematical theorems, and enable student work with more sophisticated problems. (Contains 2 figures and 3 tables.)

This study examines the influence of reading and planning from two differently organized mathematics textbooks on prospective high school mathematics teachers' pedagogical content knowledge and content knowledge of exponential functions. The teachers completed a pretest and two posttests. On the pretest, the teachers possessed an incomplete understanding of content and pedagogical content knowledge related to exponential functions. The teachers' understanding of how to translate from table to closed-form and recursive equations grew as a result of their use of the "Mathematics: Modeling Our World" textbook, while the "Discovering Algebra" textbook appeared to be more beneficial in terms of pedagogical content knowledge. Teachers read from the student lessons in both textbooks, but read differently from the sections of both textbooks intended for the teacher. They focused more on the purpose of the "Mathematics: Modeling Our World" lesson and more on the places where students might experience difficulties in the margins of the "Discovering Algebra" lesson. The teachers' learning was influenced by their own personal characteristics (e.g., previous textbook experiences) as well as textbook qualities (e.g., organization).

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.

This study examined achievement patterns of middle school students enrolled in Standards-based curricula, in particular those curricula that were funded from a solicitation of proposals through the National Science Foundation (NSF) in the early 1990s (NSF RFP 91-100). Approximately 1400 middle-grades students who had used either the Connected Mathematics Project (CMP) or the MATHThematics (STEM or MT) program for at least 3 years were assessed on two widely used tests, the Stanford Achievement Test, Ninth Edition (Stanford 9) and the New Standards Reference Exam in Mathematics (NSRE). Hierarchical Linear Modeling (HLM) was used to analyze subtest results following methods described by Raudenbush and Bryk (2002). When "Standards"-based students' achievement patterns are analyzed, traditional topics were learned. Students' achievement levels on the Open Ended and Problem Solving subtests were greater than those on the Procedures subtest. This finding is consistent with results documented in many of the studies reported in Senk and Thompson (2003), and other sources. (Contains 6 tables and 4 figures.)

This article describes how student-generated terminology for the y-intercept evolved within one Standards-based classroom. It also discusses the teacher's role in this evolution as well as students' understanding of this terminology within different function representations, and it presents ways teachers can help students develop mathematically precise definitions. (Contains 2 figures and 2 tables.)

Research on student achievement within the University of Chicago School Mathematics Project (UCSMP) and Core-Plus Mathematics Project (CPMP) at the secondary level is beginning to accumulate, however, much less is known about how prepared these students are for post-secondary education. Therefore this study involving students within one tracked school district used multiple linear regression to examine the role of differential experience within secondary Standards-based mathematics programs, gender, and prior mathematics achievement on college algebra and calculus readiness placement test scores. Results show that there are no significant differences between students who had completed three and four years of the CPMP curriculum. UCSMP students with four or five years of experience significantly outperformed CPMP students on both assessments. Prior achievement was a significant predictor of student achievement on both examinations. Male students outperformed female students on the algebra placement exam. Students who had studied from both CPMP and UCSMP significantly outperformed students who had studied from CPMP for four years on the calculus readiness examination.

One classroom using two units from a "Standards"-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher's interpretation of different activities and his assumptions about the transparency of different representations, as well as students' past experiences left the student-generated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself. (Contains 10 footnotes, 4 figures, and 7 tables. Appended to this article is the Sample Paper and Pencil Test Items.)

The current study examined the mathematical achievement of high school students enrolled for 3 years in one of three NSF funded "Standards"-based curricula (IMP, CMIC, MMOW). The focus was on traditional topics in mathematics as measured by subtests of a standardized achievement test and a criterion-referenced test of mathematics achievement. Students generally scored at or above the national mean on the achievement subtests. Hierarchical linear modeling results showed that prior mathematics knowledge was a significant but modest predictor of achievement, student SES had a moderate effect, and increasing concentrations of African American students in a classroom were associated with a stronger effect of attendance on achievement. No differences on the standardized achievement subtests emerged among the "Standards"-based curricula studied once background variables were taken into account. The two suburban districts providing data for the criterion-referenced test achieved well above the national norm. (Contains 10 tables and 1 footnote.)

Conceptual understanding in mathematics within the area of functions involves the ability to translate among different representations, table, graph, symbolic, or real-world situation of a function. Students' procedural knowledge for solving equations may become separated from their conceptual knowledge, and if these connections can be maintained students are more likely to remember skills longer or be able to reconstruct them if they are forgotten.

In this study, U.S. middle school teachers’ perceptions of Common Core State Standards for Mathematics (CCSSM), CCSSM-related assessments, teacher evaluation processes, and resources for implementing CCSSM were investigated. Using a mixed methods design, a national sample of 366 teachers was surveyed, and 24 teachers were interviewed. Findings indicated that teachers viewed CCSSM as including new content for their grade level. Teachers also reported using multiple curriculum resources to align with CCSSM and indicated that new assessments would serve as a proxy for CCSSM. Implications for rapidly changing policy, curriculum, assessment, instruction, and professional development related to CCSSM are discussed. [ABSTRACT FROM AUTHOR]