## Mathematics

**MATHEMATICS EDUCATION
FACULTY AND STUDENT RESEARCH
**

**Current Faculty Research:
**

**Natasha Speer**

**Mathematicians’ Knowledge for Teaching Calculus***Funded by The Spencer Foundation*As a result of research on elementary school mathematics teachers, there is now broader recognition that effective teaching relies on knowledge of more than just mathematical facts and ideas. Kinds of knowledge that are key to effective teaching include (a) how students think about particular mathematical ideas and (b) mathematics that is “specialized” for the work of teaching (e.g., evaluating validity of student-generated solutions). These developments have prompted and enabled creation of courses and materials to enhance teachers’ “mathematical knowledge for teaching.” Unfortunately, the same cannot be said of developments in

*college-level*mathematics. Although there have been advances (primarily focused on graduate student teaching assistants), similar changes in views of what knowledge is used in and necessary for teaching have not occurred. Reasons are likely numerous and complex but the case of elementary mathematics demonstrated that research-driven change is possible. However, a key ingredient is missing: We lack an

*existence proof*that experienced, expert teachers of

*college mathematics*possess and make use of knowledge of student thinking and teaching-specific mathematics. In this project we are gathering evidence to provide that missing ingredient and raise awareness in the college mathematics community that experienced instructors have a wealth of knowledge (of various kinds) that they use in teaching. This is being accomplished by gathering teaching-related task-based interview data from around the country who have been recognized for their expertise in teaching. Armed with such an existence proof, the college mathematics education community may be better able to offer, and have supported, programs that help novice instructors develop such knowledge, thus enriching learning opportunities they provide for students.

*Improving the Preparation of Graduate Students to Teach Undergraduate Mathematics*

*Funded by The National Science Foundation, DUE Award #1432381*A significant number of today’s mathematics graduate students will enter the workforce as college faculty members for which teaching undergraduate students will comprise a significant portion of their professional responsibilities. Although some resources are available for the professional development (PD) of graduate students or novice college mathematics instructors, mathematics departments wishing to develop or expand such a program to enhance undergraduate teaching and learning find it challenging to obtain instructional materials and/or appropriate faculty expertise on which to draw. There are informal networks of faculty who work in this arena, but the resources that do exist often are inaccessible or difficult to locate for those new to the teaching workforce. In this project, we are creating an infrastructure, housed and sustained by the Mathematical Association of America (MAA), to enhance the mathematics community’s ability to provide high quality, teaching-related PD to graduate students. As a result, the project helps to: (i) provide better access to the existing resources; (ii) develop new resources and opportunities for access; and (iii) create lasting versions of the existing informal networks of faculty practitioners and researchers. Three primary audiences are involved: 1) Providers – experienced faculty who provide PD opportunities to graduate students teaching or preparing to teach undergraduate mathematics; 2) Scholars – faculty and graduate students who develop PD materials and programs and/or conduct research on the teaching of undergraduate mathematics; and 3) Teaching Assistants (TAs) – the graduate students themselves who seek professional development. Key components of the infrastructure include: 1) a multimedia suite of resources for Providers and mechanisms for building it; 2) a professional community of practice bringing together Providers and Scholars; 3) workshops and webinars for Providers and Scholars; and 4) distance delivery of PD for graduate students who have no face-to-face opportunity at their local institution.

*Mathematics Capstone Course Resources: Preparing Secondary School Mathematics Teachers*

*Funded by The National Science Foundation, DUE Award #1504551*The preparation of the highest quality teachers and learners of mathematics is a national imperative. As undergraduates, future secondary school teachers take many courses from professors with advanced degrees in mathematics. However, university faculty members often do not possess deep knowledge of the kinds of teaching demands faced by new school teachers. There is a notable need for instructional materials that deliberately and explicitly connect undergraduate mathematics content to the knowledge needed for teaching secondary mathematics. This project is leveraging the existence of Capstone Courses for future teachers found in many universities, which bring together ideas from across the college mathematics experience, as well as recent developments in policy and school accountability, to create and refine activities for use by university faculty in mathematics pre-service teachers Capstone Courses. We are developing Capstone Course modules which provide access to advanced mathematical ideas along with opportunities to unpack mathematics in ways helpful for teaching it in middle and high school settings. The project activities include (1) develop and pilot multi-media instructional modules for advanced pre-service secondary mathematics teacher learning; (2) create guidance for college mathematics faculty for effective use of the modules with target audiences; and (3) gather information from instructors and the pre-service teachers in their courses to inform future module development work.

**Natasha Speer**

Associate Professor of Mathematics Education

Neville Hall 234

581-3937

**Past Faculty Research:**

Natasha Speer

- Graduate Student and Faculty Involvement in, and Knowledge of, Mathematics and Science Teacher Preparation
- Video cases for novice college mathematics instructor development
- Teaching Inquiry-Oriented Differential Equations

**
Graduate Student and Faculty Involvement in, and Knowledge of, Mathematics and Science Teacher Preparation
**There is currently a great deal of attention being focused on the preparation of elementary and secondary school teachers, especially in the areas of mathematics and the sciences. To become a teacher, a person must study the content they will be teaching and instructional methods appropriate for the students they will teach. Associated with The National Science Foundation funded Center for Integration of Research, Teaching, and Learning (CIRTL, www.cirtl.net), this work focuses on the roles that graduate students and faculty play in the preparation of school teachers. One goal of the project is to document the different activities and projects that graduate students and faculty are participating in, or have participated in at some time in the past. The other goal is to find out what aspects of the teacher preparation process graduate students and faculty are most familiar with and which aspects they are less familiar with. This information will be used to inform the development of materials and programs to help graduate students and faculty learn more about teacher preparation and the roles they might play in the process.

**
Video cases for novice college mathematics instructor development
**The goal of this project is to develop a set of video case studies for novice teachers (e.g., graduate teaching assistants (TAs), post-docs, new faculty, etc.) to use as preparation for the teaching of college mathematic courses. The project is funded by the Department of Education, via the Fund for Improvement of Postsecondary Education (FIPSE). Based on similar work at K-12 levels, the materials will provide opportunities for novice teachers to examine and learn about instructional practices and student thinking/learning. Materials will also be developed to connect the video cases to research on the teaching and learning of college mathematics. The video cases may be used as a core for college mathematics professional development programs or as a part of an already-established program. PI: Shandy Hauk, University of Northern Colorado, co-PI: David Kung, St. Mary’s College of Maryland.

**
Teaching Inquiry-Oriented Differential Equations
**Very little is known about the factors that shape the instructional practices of college teachers of mathematics. The goal of this project is to examine connections between mathematicians’ knowledge/beliefs and their teaching practices as they interact with students in undergraduate differential equations classes. In particular, the focus of the research is on the influences of pedagogical content knowledge, specialized content knowledge, content knowledge, and beliefs on the ways that teachers orchestrate whole class discussions.

** **

**CURRENT STUDENT RESEARCH**

**PAST STUDENT RESEARCH**

**Student Research:**

**Adam Barker-Hoyt –**Using Spatial Temporal Rejoining to Understand Connection between Music Education as a Focus Variable toward Mathematical Proof and Justification**William Hall, Jr.**– Symbology of Integral Notation and Student Misconceptions in Definite, and Indefinite Integration**John Stahley**– Students’ Qualitative Understanding of the First and Second Derivative of a Function

**Using Spatial Temporal Rejoining to Understand Connection between Music Education as a Focus Variable toward Mathematical Proof and Justification**

By: *Adam Barker-Hoyt*

This project investigates connections between mathematics achievement and music education via understanding persistence.

Symbology of Integral Notation and Student Misconceptions in Definite, and Indefinite Integration

By: *William Hall, Jr.*

There has been considerable interest throughout the Mathematics education community on how students understand limit concepts, Riemann sums, and accumulation as they pertain to integration. However, there has been little to no research on students’ understanding of the anti-derivative and its relationship to the definite integral. A preliminary study has shown that many students tend to view integration, both definite and indefinite, as area concepts. The indefinite integral is seen as a precursor to the definite integral and many students are neglecting the fundamental distinctions between the two concepts.

Students’ Qualitative Understanding of the First and Second Derivative of a Function

By: *Randy Stahley*

This thesis research is in cooperation with the Dept. of Mathematics and Statistics NEASC study. This research examines how students, who recently completed MAT 126 (Calculus I), qualitatively understand how to create the first and second derivative of a given graphical function. Exploring this non-arithmetic side of calculus differentiation will allow the mathematics instruction community to better understand how students learn, explain, and retain this process. With this information, calculus instructors and curriculum developers can gouge out misconceptions and concentrate on weaker areas of graphical understanding.