# Teaching Inquiry-oriented Mathematics: Establishing Supports (TIMES)

TIMES is a collaborative NSF grant with principle investigators Estrella Johnson at Virginia Tech, Christy Andrews-Larson at Florida State University, and Karen Keene at North Carolina State University. The central goal of this project is to study how we can support undergraduate instructors, as they implement changes in their instruction. We have a three-pronged instructional support model, consisting of:

*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).*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.*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 will 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).

## Inquiry-oriented instructional approach

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:

- Generate student ways of reasoning
- Build on student thinking
- Develop a shared understanding
- Connect to standard mathematical language and notation

## The Curriculum Materials

The TIMES project is organized around three sets of post-calculus, research-based, inquiry-oriented curricular materials:

**Inquiry-Oriented Abstract Algebra (IOAA)**

- Topics include: groups, subgroups, isomorphisms, quotient groups, homomorphisms
- Intended to be used for a junior/senior introductory abstract algebra course
- Materials currently available: Student materials (task statements) and instructor support materials (learning goals, examples of student work, implementation notes). For more information (and materials), visit: https://taafu.org/ioaa/ (User:guest; Password:guest)
- History: Developed under NSF grant number 0737299 (PI: Larsen). Key personnel: Johnson.
- Evidence of efficacy: Larsen, Johnson, & Bartlo, 2013

**Inquiry-Oriented Linear Algebra (IOLA)**

- Topics include: span, linear dependence and independence; transformations; eigenvalues, eigenvectors, and change of basis (tasks for determinants and systems also available upon request)
- Materials intended for use in an introductory linear algebra course
- Materials currently available: Student materials (task statements) and instructor support materials (learning goals, examples of student work, implementation notes). For more information (and materials) visit: http://iola.math.vt.edu (login & password required)
- History: Developed under NSF grant numbers 0634074/0634099 (PIs: Zandieh & Rasmussen) and 1245673/1245796/1246083 (Wawro, Zandieh & Rasmussen). Key personnel: Andrews-Larson.
- Evidence of efficacy: data analysis is in progess

**Inquiry-Oriented Differential Equations (IODE)**

- Topics include: 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; structures of solution spaces
- The materials are meant for a first course in differential equations.
- Materials currently available: http://iode.wordpress.ncsu.edu
- History: Developed under NSF grant number 9875388 (PI: Rasmussen). Key personnel: Keene.
- For more information, contact kakeene@ncsu.edu or chris.rasmussen@sdsu.edu
- Evidence of efficacy: Kwon, Rasmussen, & Allen, 2005

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, 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 project is designed to help instructors overcome such difficulties.

## Grant Activities

Summer 2014: Pilot summer workshop as a mini-course at MathFest

Spring 2015: Pilot online working groups for IOAA and IODE

Summer 2015: Summer workshop after MathFest, DC, August 9th - 11th

Fall 2015: Online working groups for IOAA, IODE, and IOLA

Winter 2016: Minicourse at JMM, Seattle WA, January 6th and 8th.

Summer 2016: Summer workshop, June 22nd – 24th, Raleigh NC.

Fall 2016: Online working groups for IOAA, IODE, and IOLA

Summer 2017: Summer workshop for IODE and IOLA (for highschool teachers), tentatively planned for June 28th - 30th, Arlington VA.

Fall 2017: Online working group for IODE

## Becoming a TIMES Fellow

We are no longer accepting TIMES applicants. If you are interested in learning more about the project or the curriculum materials, please contact Christy Andrews-Larson for linear algebra, Karen Keene for differential equations, or Estrella Johnson for abstract algebra.