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, instructors who were not involved in the development implemented the materials and provided feedback.
Over the course of the last 10+ years, these extensive and ongoing research projects have produced many results, including: instructional sequences comprised of rich problem-solving tasks, instructor support materials, research showing positive conceptual learning gains (e.g., Kwon, Rasmussen, & Allen, 2005; Larsen, Johnson, & Bartlo, 2013), insights into how students think about these concepts (e.g., Larsen, 2009; Wawro, 2014; Keene, 2007) and the identification of specific challenges that instructors face as they implemented these materials. Some of the difficulties experienced by instructors implementing the materials include: making sense of student thinking, planning for and leading productive whole class discussions, and building on students’ solution strategies and contributions (e.g., Johnson & Larsen, 2012; Speer & Wagner, 2009; Wagner, Speer, & Rossa, 2007).
The TIMES grant will allow us to better understand how to support instructors as they work to implement these three inquiry-oriented curricula materials. We have a three-pronged instructional support model, consisting of:
(1) Curricular support materials – These materials, created by the researchers who developed the three curricular innovations, include: student materials (e.g., task sequences, handouts, problem banks) and instructor support materials (e.g., learning goals and rationales for the tasks, examples of student work, implementation notes).
(2) Summer workshops – The summer workshops last 2-3 days and have three main goals, 1) building familiarity with the curricula materials, including an understanding of the learning trajectories of the lessons; and 2) developing an understanding of the intent of the curricula in particular and inquiry-oriented instruction in general.
(3) Online instructor work groups – The online instructor work groups have between 4 and 6 participants, each currently implementing the same curricular materials. Each group meets for one hour a week and works on selected lessons from the curricular materials. For each of the focal lessons, we discuss the mathematics and plan for implementation. Then, after instructors have taught the lesson, the group watches video clips of instruction with a focus on student thinking. The goal is to help instructors develop their ability to interpret and respond to student thinking in ways that support student learning. Every meeting also has time dedicated to address specific and immediate needs of the participants (e.g., difficulty with managing small group work, a particularly challenging task, strategies for getting students to share ideas).
Over the course of this three-year grant, we will offer these supports and investigate their impact. Our research will focus on the relationships and interactions among the supports, the instructors, and their instructional practices. In addition to assessing the impact of the support model, project data will be analyzed to identify aspects of the supports and instruction that have a positive impact on students’ learning.
We hope that this post provided a useful description of the inquiry-oriented instructional approach that can help instructors think about how they might (or already do) incorporate some of these ideas into their teaching. For instance, regardless of how you currently teach, really inquiring into your students’ thinking (not just their answers) can provide you with very valuable insights. We also hope that, after reading this post, you will be encouraged to see that some tangible, practical steps are being taken toward scaling up and supporting inquiry-oriented instruction.
If you are interested in learning more about the curricular materials or this project please visit http://times.math.vt.edu/. If you are interested in learning more about becoming a TIMES fellow, please contact Estrella Johnson (firstname.lastname@example.org) for Abstract Algebra, Christy Andrews-Larsen (email@example.com) for Linear Algebra, or Karen Keene (firstname.lastname@example.org) for Differential Equations. We are the principal investigators on the project and would be glad to hear from you if you are interested in learning more.
Johnson, E. M. S., & Larsen, S. (2012). Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior, 31, 117 – 129.
Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter. The Journal of Mathematical Behavior, 26(3), 230-246.
Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics, 105(5), 227-239.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2), 119-137.
Larsen, S., Johnson, E., & Bartlo, J. (2013). Designing and scaling up an innovation in abstract algebra. The Journal of Mathematical Behavior.
Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics. The Journal of Mathematical Behavior, 26(3), 189-194.
Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. ZDM, 46(3), 389-406.
Speer, N. M., & Wagner, J. F. (2009). Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions. Journal for Research in Mathematics Education, 40(5), 530-562.
Wagner, J. F., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. The Journal of Mathematical Behavior,26(3), 247-266.