Marilyn Carlson - US grants

This research will extend current theories describing how students'
acquire an understanding of the function concept, the central concept of
the undergraduate mathematics curriculum. The model that emerges will
provide a framework for the development of two research instruments to
broadly assess students' function knowledge. The Precalculus Concept
Assessment Instrument (PCA) and the Calculus Concept Assessment Instrument
(CCA) will provide a consistent means of quantifying students' concept
development for two large-enrollment undergraduate courses.
Investigations of students' problem solving behavior, using
interviews, case studies and observations will attempt to identify major
beliefs and behaviors that both promote and present obstacles to students'
mathematical success and continued mathematical study. The results of
these studies will form the basis for the development of a framework and
instrument to broadly define and assess undergraduate students' beliefs
about knowing and learning mathematics.
The emergent instruments will provide objective means for examining
patterns in students' function understandings and mathematical beliefs.
The resultant investigations will provide insights for guiding the
development of precalclulus and calculus level curriculum modules. During
the course of this project model workshops will also be offered to
disseminate the newly developed instruments and curricular materials.

Mathematical Sciences (21)
This project develops a prototype course in scientific programming that is aimed at majors in the mathematical sciences (as distinct from engineering and computer science) and provides a comprehensive introduction to software development tools for large codes in scientific computing. The undergraduate computer science curriculum typically includes one or more courses in which students work on a large software project, often as part of a team, that requires them to develop, document, and respect functional interfaces and to compile and link the final product from multiple files and libraries. In contrast, majors in mathematics, physics, and similar disciplines usually do not learn such skills in the introductory programming and numerical analysis courses that typically are part of the curriculum. As a result, majors in the mathematical sciences often are at a disadvantage, relative to their peers in computer science, when they seek employment after completing their baccalaureate degrees or when they continue on to graduate study in fields that involve the development of large scientific codes. The course provides mathematical science majors with experience in developing and prototyping software for applications that involve large data sets or large amounts of CPU time.

This is a proposal for a 3-year longitudinal study of developmental trajectories in middle school students' learning about rational numbers. The goals of the proposed work include:
1. Uncovering patterns and mechanisms of development in students' understanding of rational numbers and proportional reasoning;
2. Integrating the current piecemeal body of research on rational number into a coherent developmental model by examining how understanding of rational number sub-constructs evolve concurrently and interactively;
3. Developing insight into the ways in which classroom instruction, especially teacher questioning techniques and tasks, impact students' ability to think about, represent, and communicate their understanding of rational number concepts and operations as it develops over time; and
4. Generating transportable models of rational number development that can be factored into teacher pre- and in-service staff development to promote quality instructional practices in the future.

Broad Impact:
The conduct of the work will affect the lives of 160 middle grade students of diverse backgrounds throughout the life of the project. The results of the project will influence pre-service and in-service teaching nationally. Finally, the study will reframe math education issues longitudinally and deliver high quality software products I support of student learning.

Panel Summary:

Intellectual Merit:
This is a proposal for a 3-year longitudinal study of developmental trajectories in middle school students' learning about rational numbers. The panel believes the aims of the project to be very worthwhile, with the longitudinal focus on this age group making a unique contribution to existing research on the understanding of rational numbers. Reviewers were also enthusiastic about detailed modeling of individual students' knowledge and understanding being combined with detailed, rich, contextual examination of learning opportunities, discourse, and inscriptions in classrooms, especially in multiple classrooms over several years. The research team is strong and presents an organized plan for accomplishing the proposed research.

This five-year project, entitled "Developing a Professional Learning Community Model for Secondary Precalculus Teachers: A Model for Teacher Professional Growth," will produce a model for a Professional Learning Community (PLC) for pre-calculus teachers in secondary schools. It will generate research knowledge from cycles of (1) defining, (2) studying and (3) refining the model and its components. The project leadership team will then be able to describe the support structure, group processes and tools needed to assist secondary pre-calculus teachers in providing high quality instruction for their students.

PLC tools will be developed and refined to facilitate Teachers' "reflection-on-students'" thinking and reasoning relative to the major concepts of pre-calculus mathematics. The activities of the PLCs will also support teachers' continued conceptual and mathematical development, while promoting reflections on the effectiveness of their classroom practices.

The leadership team expects that the teachers' rich reservoir of new knowledge and understandings will be observable in the ways in which they orchestrate learning experiences for their students. They anticipate qualitative improvements in their classroom practices (perceivable in their questions and questioning patterns, the design and substance of their tasks, the conceptual focus of their exams), resulting in improved learning in their students. Implementation of this model will produce a set of research-developed tools to assist individual schools in devising and implementing a plan to support secondary teacher professional growth.

This Nanotechnology in Undergraduate Education (NUE) award to Arizona State University supports Dr. Ray W. Carpenter, Center for Solid State Science, along with colleagues Prof. Marilyn Carlson (applied mathematics and Science, Technology, Engineering and Mathematics (STEM) teaching methods), Prof. Jeff Drucker (Physics), Prof. Stephan Goodnick (Electrical Engineering), Prof. Vincent Pizziconi (Bioengineering), Dr. Andrew Chizmeshya (Physics), Dr. Michael McKelvy (Chemistry), Prof. B. L. Ramakrishna (Chemistry and Plant Biology), and Dr. Renu Sharma (Chemistry)to teach undergraduates, at three levels, the abstract concepts and properties dependence on length scales of nanoscience and engineering by leveraging existing cutting edge nanoscience and engineering research projects to produce teaching modules for undergraduate classes and opportunities for senior thesis projects. The levels of students who will be targeted in this program are: first year honors students, second year students who have completed calculus, and advanced undergraduates participating in senior projects.

The proposal for this award was received in response to the Nanoscale Science and Engineering Education announcement, NSF 03-44, category NUE and was jointly funded by the Division of Engineering Education and Centers (EEC) in the Directorate for Engineering (ENG), the Division of Materials Research (DMR) and the Division of Mathematical Sciences (DMS) both in the Directorate for Mathematical and Physical Sciences (MPS).

Project Pathways targets mathematics and science learning and achievement in grades 9-12. The project will produce a research-based and tested model to support secondary mathematics and science teachers. Core partners include four school districts (Chandler, Mesa, Tempe, and Tolleson) and the Center for Research on Education in Science, Mathematics, Engineering, and Technology (CRESMET) at Arizona State University (ASU). The Maricopa Community College District and Intel are collaborators with ASU in delivering the project's research-based services and products to these districts. The demographics of the partner school districts mirror those of Arizona, where 45% of students are persons of color and the Hispanic population is expanding rapidly.

Pathways will produce tools and knowledge to guide secondary mathematics and science teachers in promoting conceptual learning and STEM behaviors that the literature deems essential for continued STEM learning and course-taking. These key behaviors include competence and flexibility in scientific inquiry, mathematical problem solving, and engineering design. A central concept is that of function, which research identifies as a unifying concept of secondary mathematics and science. As the core strategy of the Pathways model, teams of engineers, mathematicians and scientists will partner with master teachers and STEM education faculty to generate instructional sequences for both teachers and students. The instructional materials will take the form of modules for secondary mathematics and science teachers, with companion modules for secondary STEM classrooms and companion tools for secondary STEM learning communities. The professional learning community tools will support teachers in adapting their new knowledge and instructional approaches to their own classrooms by engaging them in deep reflections on their instruction and their students' learning. Pathways materials will be easily adaptable in any learning environment. In this project, however, they will be packaged for use in four courses in an ASU graduate degree program for inservice teachers, delivered on-site in the teachers' schools. To better assist Arizona's many bilingual students, Pathways will adapt student modules to an innovative, research-developed English language learner technology platform. Other Pathways strategies include activities (such as science fairs for students and a regional conference for high school guidance counselors) to encourage all students to take challenging mathematics and science courses and to consider science-based careers.

Teams of STEM education faculty and graduate students will research the effectiveness of the courses and learning communities on teachers' understanding of mathematics and science concepts and their understanding of the process by which foundational STEM concepts and behaviors develop in students. They will also investigate the process by which teachers shift their classroom practices to promote improved STEM learning in their students. Graduate research assistants will be recruited to participate in the Pathways project, preparing many future faculty for careers as STEM education researchers. Pathways will establish new patterns of information-sharing and collaboration among STEM scientists and educators, community college faculty, secondary administrators and teachers, and industry partners. The project aims to narrow the majority/minority achievement gap, encourage students to take challenging STEM courses, increase high school student STEM learning and achievement, and improve the pass rates in ASU's introductory calculus, physics and biology courses.

The Center for Nanotechnology in Society at Arizona State University (CNS-ASU) helps ensure "that advances in nanotechnology bring about improvements in the quality of life for all Americans" (PL 108-153). The Center's vision is that research into the societal aspects of nanoscale science and engineering (NSE), carried out in close collaboration with NSE scientists and combined with public engagement, will improve deliberation and decision making about NSE. CNS-ASU builds the capacity to address the societal implications of NSE by creating a broad institutional network, instituting a coherent research program, promoting innovative educational opportunities, and engaging in meaningful participation and outreach activities, especially with under-represented communities. Its goal is nothing less than charting a path toward new ways of organizing the production of knowledge and developing and testing new processes of anticipatory governance to meet the emerging promises and challenges of NSE.

CNS-ASU joins Arizona State University with the University of Wisconsin - Madison, the Georgia Institute of Technology, North Carolina State University, Rutgers, The State University of New Jersey, and other universities, individuals, and groups in the academic and private sector, as well as the International Nanotechnology and Society Network (www.nanoandsociety.org) that ASU is developing. At ASU, the project's two guiding organizations are the Consortium for Science, Policy, and Outcomes (www.cspo.org), which provides an institutional home for science and technology policy scholarship and engagement, and the Biodesign Institute (www.biodesign.org), which provides a substrate of NSE research and a test bed for interdisciplinary collaboration.

CNS-ASU will implement a program of research and engagement called "real-time technology assessment" (RTTA), which consists of four methods of inquiry: mapping the research dynamics of the NSE enterprise and its anticipated societal outcomes; monitoring the changing values of the public and of researchers regarding NSE; engaging researchers and various publics in deliberative and participatory forums; and reflexively assessing the impact of the information and experiences generated by its activities on the values held and choices made by the NSE researchers in its network. Through RTTA, CNS-ASU will probe the hypothesis that trajectories of NSE innovation can be steered toward socially desirable goals, and away from undesirable ones, by introducing a greater capacity for reflexiveness - that is, social learning that can expand the range of conscious choice - into knowledge-producing institutions. It organizes the research around two broad NSE-in-society themes: freedom, privacy, and security; and human identity, enhancement, and biology.

The Center's educational and training plan includes innovations at the undergraduate, graduate, and postdoctoral level that encourage interdisciplinary opportunities among NSE students and social science and humanities students. Partnerships with proven programs, including the Hispanic Research Center (www.asu.edu/clas/hrc) and the Center for Ubiquitous Computing (http://cubic.asu.edu), ensure recruitment and retention of students from under-represented groups. A collaboration with the Center for Research on Education in Science, Mathematics, Engineering, and Technology (http://cresmet.asu.edu), CNS-ASU generates training modules for high school teachers in NSE-in-society.

Designed as a "boundary organization" at the interface of science and society, CNS-ASU provides an operational model for a new way to organize research through improved contextual awareness, which can signal emerging problems, enable anticipatory governance, and guide trajectories of NSE knowledge and innovation toward socially desirable outcomes, and away from undesirable ones. In pursuit of this broadest impact, CNS-ASU trains a cadre of interdisciplinary researchers to engage the complex societal implications of NSE; catalyzes more diverse, comprehensive, and adventurous interactions among a wide variety of publics potentially interested in and affected by NSE; and creates new levels of awareness about NSE-in-society among decision makers ranging from consumers to scientists to high level policy makers.

The Summer Certification Institute in Secondary Mathematics (SCISM) program at Arizona State University is a parallel certification sequence that is enabling 30 upper-level students to take the courses and fieldwork required to become secondary mathematics teachers while they are completing a bachelor's degree in mathematics or engineering. SCISM students complete their education requirements in the senior academic year and the two summers before and after the senior year. They graduate with a bachelor's degree in mathematics or engineering and 15 graduate credits toward the M.Ed.

Two distinctive features of SCISM are an undergraduate teaching practicum and the use of professional learning communities (PLCs). In the practicum, SCISM candidates are mentored in teaching a section of College Algebra in ASU undergraduate classrooms. The candidates participate in teacher professional learning communities that are linked to the practicum. In their PLCs, SCISM candidates dig deeply into the content they are teaching and support one another in developing effective lesson plans and methods of instruction. The PLCs continue into the new teacher induction period, serving as a framework for supporting the teachers during their novitiate as high school math instructors.

SCISM students gain their field experience and student teaching credits in high-need Arizona school districts. They are required to teach mathematics for three years in a high-need district upon completing the SCISM program. CRESMET, ASU's Center for Research on Education in Science, Mathematics, Engineering and Technology, is sponsoring SCISM in collaboration with the College of Education, the Department of Mathematics, and the Ira A. Fulton School of Engineering. SCISM is expanding on a pilot effort that CRESMET launched in 2005 with eight students and start-up funding from the Boeing Company. SCISM draws inspiration from the UTeach model established at the University of Texas at Austin and innovates on the UTeach model by combining a college classroom-teaching model with the use of professional learning communities.

This study to be conducted under the auspices of the Mathematical Association of America (MAA) undertakes a national investigation of Calculus I to identify the factors that contribute to student success, to understand how these factors are leveraged within successful programs and to use the publications, committees, and public fora of the MAA to disseminate the information and help mathematics departments build on its insights.

The study uses both quantitative and qualitative methods. During the study's first phase, a large-scale national survey of Calculus I instruction will be conducted through mathematics departments. The study builds on the Conference Board on the Mathematical Sciences (CBMS) studies to link the survey data with student success. Using the epidemiological model developed by Sadler and Tai, the PIs will examine factors such as instructor attributes, departmental focus, classroom variables, and out of class expectations as likely explanatory factors. A web-based student survey will also be conducted. Data will be studied through HLM analyses and institutions, course-level characteristics, and student characteristics will all be examined.

Phase II of the project will develop a preliminary theoretical framework for institutional models that produce successful calculus students. This informs the selection of eight cases to be conducted in different kinds of institutions. The cases will be analyzed using the approaches of pattern matching, explanation building, and cross-case syntheses.

The study is significant because the number of calculus students has been steadily decreasing in spite of the fact that more high school students are taking higher level mathematics courses. The PIs state that many students leave STEM fields as undergraduates because of poor calculus instruction. Identifying what works in college calculus instruction and why it works and widely disseminating the information will improve student success in calculus courses.

The Pathways to Calculus: Disseminating and Scaling a Professional Development Model for Precalculus Level Instruction Phase II MSP involves four Core Partners; Arizona State University, as the lead, and the Mesa, Chandler, and Scottsdale school districts. Supporting partners are: Brigham Young University, the University of Northern Colorado, the University of Georgia, Northern Arizona University, and Scottsdale Community College.

This Phase II project builds on the work of Project Pathways, a targeted MSP that identified attributes of professional development for secondary mathematics and science teachers that resulted in substantive and sustained improvements in student learning, as documented by student performance on district exams, state exams and research-based tools. The Phase II project leverages the research-based processes and tools that emerged in Phase I research to be highly effective for shifting teachers' instruction to be more inquiry-based and conceptually oriented. Phase II builds on Phase I findings in five broad categories that are critical for supporting mathematics teachers to realize significant shifts in their students' learning of key ideas of mathematics. These are teachers': 1) knowledge of the mathematics they teach; 2) beliefs about what constitutes effective mathematics learning and teaching; 3) ability to engage in reflection on student thinking and learning in relation to their teaching; 4) use of curricular support materials that promote inquiry-based and conceptually oriented instruction; and 5) participation in Pathways Professional Learning Communities (PLCs).

Phase I resulted in the development of the Pathways Precalculus Professional Development Model (P3DM), which includes in-class student activities with detailed teacher notes, computer animations and assessments. This supported Precalculus teachers in making instructional transitions that realized significant gains in student learning. Phase II extends this work by scaling the P3DM in three ways: 1) implementation of the P3DM at the community college and university levels, 2) implementation of the model in larger classes; 3) engagement of school administrators, including department chairs, to support all teachers in Core Partner school districts (11 schools) in adopting P3DM in Precalculus. Phase II studies the process of scaling P3DM in each of the three ways, and continues to examine how the P3DM experience affects teachers' instruction in other Precalculus level courses such as algebra II, college algebra and trigonometry.

The Phase II Research Agenda addresses the following:
1. What institutional factors of a school inhibit or support quality implementation of Pathways Precalculus materials? What external resources are needed to mitigate inhibiting factors and capitalize on supportive factors?
2. What is the typical developmental trajectory of teachers understanding and taking ownership of Pathways Precalculus materials? What support do teachers need to enable this development?
3. What is the typical developmental trajectory of teachers' understanding of the core mathematical ideas in the Pathways Precalculus materials and ability to accurately assess students' reasoning about these ideas in the classroom setting? What support do teachers need to understand these concepts and effectively assess student reasoning?
4. What administrative support enables teachers to effectively implement Pathways Precalculus materials?

A minimum of 150 high school mathematics teachers will be involved in addition to university and community college instructors and faculty. The products of this research will contribute knowledge and tools for scaling up the P3DM in the Precalculus strand (precalculus, college algebra, trigonometry and high school algebra II) of mathematics. The investigations will also produce insights about the factors that contribute to a mathematics department's transformation to support students in developing the capacity and confidence to solve novel problems and construct deeper and more connected understanding of the central ideas of a course.

This project focuses on mathematical knowledge for teaching at the secondary level, building upon and extending existing work for elementary mathematics. The project is developing two related assessment instruments: one for assessing teachers' mathematical knowledge for teaching secondary mathematics (MKTsm) and one for assessing the quality of secondary mathematics instruction (IQAsm). Used simultaneously, these instruments are capable of linking secondary teachers' mathematical knowledge with their instruction. The first instrument draws from prior research on teachers' and students' understanding of key mathematical ideas. The second extends the Instructional Quality Assessment (IQA), an existing assessment that focuses on the quality of elementary and middle school mathematics, so that it has a clearer focus on secondary mathematics and on the quality of mathematical ideas that a teacher's instruction fosters. The instruments are being constructed, validated, and tested in three overlapping phases. Development of the MKTsm assessment draws on existing research on teachers' and students' understandings of foundational ideas in algebra and precalculus. The IQAsm extension draws from existing IQA validation studies, from the co-PIs' recent research on transforming secondary teachers' classroom mathematical practices, and from what is learned through the development process in designing the MKTsm instrument. The MKTsm and IQAsm instruments are being piloted in three projects of the Math and Science Partnership (MSP) program, to ensure that the instruments are broadly valuable to MSP projects and mathematics education in general. The instruments are being distributed online, and measures are being taken to ensure valid and appropriate usage by other researchers, or by administrators who might use the instruments for purposes that could include evaluation. In addition to the instruments, the project is developing training videos, scoring guidelines, and an overview for administrators, and also is creating a process by which the publisher will require certification of professional development for use of the instruments when they become commercialized.

This project is developing a new research-based instrument for placement of students in algebra and precalculus and disseminating documented descriptions and sample items of this new instrument. The work is based on earlier research and testing that produced a Calculus Concepts Readiness test (CCR) and a companion Precalculus Concepts Assessment (PCA). In particular, researchers considered students' conceptual understanding of key concepts required for success in subsequent mathematics courses, in future careers, and as informed citizens. The purpose of this follow-on development and dissemination effort is twofold: i) to influence and improve college mathematics placement tests and ii) to inform and shape instructional practice in algebra and precalculus courses. The intellectual merit of this project rests in its recognition that while there is an adequate research base to inform more effective instructional practice, teachers at all levels lack ready access to assessment tools that reinforce teaching for improving students' deep conceptual understanding. Indeed, most placement tests currently in use focus on procedural fluency, thus reinforcing traditional instructional approaches. The new placement test becomes part of a coherent suite of instruments that serve to improve teaching and learning throughout the entire algebra-precalculus-calculus pathway. The project is exerting its broader impacts in several ways. Firstly, the reasoning and conceptual taxonomies of the new instrument, as well as sample items to illustrate the conceptual reasoning coverage, are being made publicly available and used throughout the dissemination phase of the project to inform professional development efforts not only for college and university faculty teaching lower-division mathematics courses, but also for K-12 teachers. Secondly, the project is being led by the Mathematical Association of America (MAA), the largest professional organization in the world with a focus on undergraduate mathematics. The organization's online and print publications, and sessions at national and regional meetings provide diverse and far-reaching channels for dissemination. In addition, existing relationships with other national organizations seeking to raise the quality of mathematics instruction gives the MAA a unique opportunity to develop, pilot, and rigorously document the reliability and validity of placement tests. Finally, the results of this project are contributing to improved articulation between secondary and post-secondary mathematics and informing pre-service and in-service teacher education in mathematics.

Graduate students in mathematics are often called on to teach sections of introductory courses ranging from college algebra, to precalculus, to calculus. In this project a team of mathematicians and mathematics education researchers are developing a sequence of courses and associated activities that support Ph.D. students in mathematics to adopt a scientific approach both to their teaching and to evaluating the effectiveness of the underlying curriculum. The intellectual merit of this effort lies in its use of previously developed research-based student and teacher materials that have been documented to transform precalculus and calculus teaching and learning. Based on this prior work the investigating team is creating a model education certificate program for mathematics graduate students that is envisioned to transform their teaching practice. Concurrent with these teaching experiences, students complete a sequence of courses and seminars in which they read select publications, review materials, and design and conduct small-scale research studies of student learning. These combined program components lay the groundwork for producing mathematics Ph.D. students who are prepared to take leadership roles in improving the teaching and learning of undergraduate mathematics. The project is exercising its broader impact both by engaging the faculty to change the culture and departmental infrastructure in the PI's own institution, and by serving as model for other departments of mathematics who can incorporate the education certification program into their graduate offerings, thus supporting pre-professional faculty development of their students.