By Estrella Johnson, Assistant Professor of Mathematics Education at Virginia Tech University, Karen Keene, Associate Professor of Mathematics Education at North Carolina State University, and Christy Andrews-Larson, Assistant Professor of Mathematics Education at Florida State University
Making
By Estrella Johnson, Assistant Professor of Mathematics Education at Virginia Tech University, Karen Keene, Associate Professor of Mathematics Education at North Carolina State University, and Christy Andrews-Larson, Assistant Professor of Mathematics Education at Florida State University
Making fundamental changes to the way you teach is a difficult task. However, with a growing number of students leaving STEM majors, instructors’ dissatisfaction with student learning outcomes, and research indicating positive avenues for improving undergraduate mathematics instruction, some instructors are ready and eager to try something new. In this post, we describe some promising research-based curricular materials, briefly identify specific challenges associated with implementing these materials, and describe a recently funded NSF project aimed at addressing those challenges.
Teaching Inquiry-Oriented Mathematics: Establishing Supports (TIMES) is an NSF-funded project (NFS Awards: #143195, #1431641, #1431393) designed to study how we can support undergraduate instructors as they implement changes in their instruction. A pilot is currently being conducted with a small group of instructors. In the next two years, approximately 35 math instructors will be named TIMES fellows and will participate in the project as they change their teaching of differential equations, linear algebra, or abstract algebra. As project leaders, we will study how to best support these instructors, as well as how their instructional change affects student learning. More details about the project follow later in this blog post.
The curricula we utilize in the project are each examples of inquiry-oriented instructional materials. Inquiry-oriented instruction is a specific type of student-centered instruction. Not surprisingly, different communities characterize inquiry in slightly different ways. In the inquiry-oriented approach we describe here, we adopt Rasmussen and Kwon’s (2007) characterization of inquiry, which applies to both student activity and to instructor activity. In this approach, students learn new mathematics by: engaging in cognitively demanding tasks that prompt exploration of important mathematical relationships and concepts; engaging in mathematical discussions; developing and testing conjectures; and explaining and justifying their thinking. Student inquiry serves two primary functions: (1) it enables students to learn new mathematics through engagement in genuine exploration and argumentation, and (2) it serves to empower learners to see themselves as capable of reinventing important mathematical ideas.
The goal of instructor inquiry into student thinking goes beyond merely assessing student’s answers as correct or incorrect. Instead, instructor inquiry seeks to reveal students’ intuitive and informal ways of reasoning, especially those that can serve as building blocks for more formal ways of reasoning. In order to support students, instructors routinely inquire into how their students are thinking about the concepts and procedures being developed. As instructors inquire into students’ emerging ideas, they facilitate and support the growth of students’ self-generated mathematical ideas and representations toward more formal or conventional ones. The instructor’s role is to guide and direct the mathematical activity of the students as they work on tasks by listening to students and using their reasoning to support the development of new conceptions. Additionally, instructors provide connections between students’ informal reasoning and more formal mathematics.
With an inquiry-oriented instructional approach, instructors use mathematically rich task sequences, small group work, and whole class discussions in order to elicit student thinking, build on student thinking, develop a shared understanding, and introduce formal language and notation.
Curricular Materials for Undergraduate Mathematics Education
The TIMES project is organized around three sets of post-calculus, research-based, inquiry-oriented curricular materials.
· Inquiry-Oriented Abstract Algebra (IOAA), developed by Sean Larsen under the NSF grant Teaching Abstract Algebra for Understanding (#0737299), http://www.web.pdx.edu/~slarsen/TAAFU/ (User:AMSBlog; Password:teacher). These materials are designed for an introductory group theory course and include units on groups and subgroups, isomorphisms, and quotient groups. Supplementary materials for rings/fields are available upon request.
· Inquiry-Oriented Linear Algebra (IOLA), developed by Megan Wawro, Michelle Zandieh, Chris Rasmussen, and colleagues under NSF grant numbers 0634074/0634099 and 1245673/1245796/1246083, http://iola.math.vt.edu (must request login & password). These materials are designed for an introductory linear algebra course and include four units on span, linear dependence and independence, transformations, and eigenvalues, eigenvectors, and change of basis. Tasks for determinants and systems are also available upon request.
· Inquiry-Oriented Differential Equations (IODE), developed by Chris Rasmussen and colleagues under NSF grant number 9875388, website coming soon. These materials are designed for a first course in differential equations and include the following topics: solving ODEs; numerical, analytic and graphical solution methods; solutions and spaces of solutions; linear systems; linearization; qualitative analysis of both ODEs and linear systems of ODEs; and structures of solution spaces.
For each of these three curricular innovations, the student materials have been developed through iterative stages of research and design supported by grants from the NSF. In the early stages of these respective projects, the developers carried out small-scale teaching experiments focused on uncovering students’ ways of reasoning and developing tasks that evoke and leverage productive ways of reasoning. Instructional tasks then went through additional cycles of implementing, testing, and refining over a series of whole class teaching experiments. In the last stages of research and design