*This blog post is an Q&A session conducted via email with Dr. Karen Keene and Dr. Justin Dunmyre. They are sharing information about the TIMES project. Thank you Karen and Justin!*

**0. Please tell us about yourselves.**

**Karen Keene**has her Ph.D. in Mathematics Education from Purdue University. Karen was introduced to active learning in undergraduate mathematics education while she was a graduate student involved in the creation of the Inquiry-Oriented Differential Equations materials. She has been serving as a project leader on the TIMES project where inquiry-oriented instruction, one form of active learning since 2013. She is currently an Associate Professor of Math Education at North Carolina State University and is currently serving as a rotating Program Officer for the National Science Foundation.

**Justin Dunmyre**has his Ph.D. in Mathematics from the University of Pittsburgh, and is a Brown ’13 Project NExT fellow. He is currently an Associate Professor and Chair of Mathematics at Frostburg State University. Through Project NExT, Justin got interested in active learning, and subsequently participated in the IBL Workshop. This transformative experience led him to wonder what IBL would look like in his discipline (differential equations) and almost as soon as he had that thought he got an email through the IBL mailing list about this exciting TIMES project!

**1. We'd like to learn about the TIMES project. What is the main idea behind this effort?**

The TIMES project began as a collaboration of second-generation authors of varied inquiry-oriented (IO) classroom materials. By second-generation we mean that Michelle Zandieh, Sean Larsen, and Chris Rasmussen wrote the original IO materials for Linear Algebra, Abstract Algebra, and Ordinary Differential Equations, respectively. The TIMES Principal Investigators, Christy Andrews-Larsons, Estrella Johnson, and Karen Keene were each graduate students of these original authors and launched the TIMES project to study how they might support other faculty in using these materials. That’s why TIMES actually stands for Teaching Inquiry Mathematics: Establishing Supports. These supports were three fold: providing the curricular material, a 3-day summer workshop on using the materials, and weekly online working groups. The three supported curricula were IOLA (Inquiry-Oriented Linear Algebra) and IOAA (Inquiry-Oriented Abstract Algebra formerly known as TAAFU - Teaching Abstract Algebra For Understanding) and IODE (Inquiry-Oriented Differential Equations).

Justin was in one of the first cohorts of TIMES fellows for IODE, and became involved in running the online working groups and became a coauthor on the materials, along with Nick Fortune whose dissertation research was supported by TIMES project.

**2. What classes do you have materials for?**

A first course in Linear Algebra, an abstract-algebra course focused on groups, and a first course in differential equations.

**3. What is a typical day like in an IO class?**

A typical day is centered around the guided reinvention of particular mathematical concept(s). The tasks are based on the principles of the instructional design theory of Realistic Mathematics Education (RME), of which one of the tenets is that the material must be experientially real for the students. The Inquiry Oriented materials are grounded in contexts that the students can initially understand and reason about, maybe from a less sophisticated viewpoint, even if they’ve never had that specific experience before. For example, in IODE, one tasks early in the materials is focused on population growth of owls in a forest, which students may not know a lot about, but can understand. As in any IBL classroom, students are constructing the mathematics for themselves, and taking ownership of that mathematics. The students work through a series of tasks, often encountering the the tasks for the first time in class. Therefore, we rely on small group work for an initial translation from the context to concepts relating to the learning outcomes of the class.

The instructor facilitates this guided reinvention by primarily using four instructional components (Kuster, Johnson, Keene, Andrews-Larson, 2018) , not all of which may happen on the same day. These four instructional components are general instructional goals: eliciting student thinking, building on student thinking, building a shared understanding of the mathematics reinvented in the classroom, and connecting the students’ mathematics to formal mathematics.

In any given day in our classes, one would see the students sitting in small groups, working on tasks. One would also hear whole class discussions where the instructor is soliciting student input, re-voicing, and, reshaping it, innocuously, to guide the conversation with an eye on the instructor's mathematical agenda. Occasionally, the students would make presentations of their ideas, but this would not always happen. The whole class discussion and small group discussions do happen every day. Finally, when the class has finished an idea, or perhaps developed a need for notation to express their ideas, the instructor connects their work with formal language and notation.

**4. What are some of your best moments as a teacher in an IO class?**

**Justin**: There are so many! One of my favorites came from the first time I taught our optional unit on bifurcation theory. This unit starts with a task where students are challenged to model the introduction of a parameter that represents harvesting of fish from an otherwise logistic model. After settling on a simple shift of the form dP/dt = 0.2P(1-P/25) - k, where k is the harvesting parameter, students are asked to come up with a one page report to explain to the owners of the fish hatchery the ramifications of varied choices in k. The one page report is the trick! By requiring students to use space efficiently, they can actually invent the bifurcation diagram for themselves. What really surprised me was how many different forms this bifurcation diagram can take. I’ve seen students use spreadsheets that show if dP/dt is positive in green or negative in red (the bifurcation diagram then emerges as the change from green to red), carefully stacked phase lines, analytically drawn bifurcation curves using the quadratic formulas, and more. When the students present these ideas to one another, they realize they’re all saying the same thing, and absorbing insights from other groups result in very deep understanding of this sophisticated concept. The first time I taught this unit I was giddy, I couldn’t believe that these bifurcation diagrams were emerging before my eyes, completely invented by my students! We wrote about this task sequence in a PRIMUS paper here (Rasmussen, Dunmyre, Fortune & Keene, 2019).

The “salty tank” problem is practically a rite of passage for students in my differential equations classes. It’s developed its own legend here on campus, because the discussions are so robust, and it is the first time that students are really asked to develop their own equation. The students really marvel at how they can have an 50 minute long debate over a prompt as simple as “A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute.” This problem is the seat of another favorite moment of mine. Ideas were flying all around the room, what does it mean to be well-mixed, what should the input term look like, what should the output term look like? A student said something to her group, but she didn’t want to cut into the whole class discussion. Her group thought it was important though, so one of her more outgoing group members interrupted the discussion and he said “I think we should all hear what Sarah has to say.” Of course, it was a critical insight that helped reframe the conversation in a productive direction! But the act of one student elevating the status of another student, that was a powerful moment that will always stick with me.

**Karen**: When I taught this course to math and physics majors together, my favorite times were when we talked about the difference between instantaneous rate of change and rate of change over a "very very small time interval". The conversations were always spirited and deep, with the Physics Majors declaring it doesn't matter if there is a difference and the Math majors wanted it to matter and try to understand what a "limit" really is. Of course, I was always rooting for the Math Majors, but it didn't really matter, as it was a situation where the students were engaged in thinking deeply about the math and taking the authority of learning on themselves. Of course, ultimately, we had to agree to disagree and the physics majors usually could go along with the idea of instantaneous rate of change as that being the foundation of a differential equation.

I can think of other times that after we had small group discussions, two groups would present their ideas about a particular task- and they were not the same. When that happened, I would encourage each side to state their case. Then I would send the students into small groups to continue the discussion and decide what they thought. This might go on for much of a class. I know it took up a lot of time, but it was worth it--- they were not waiting on me to tell them- but making their own mathematical judgments. Most of the time, it all seemed to move the agenda forward. I do remember one time that the whole class agreed on something that I knew was mathematically wrong. I made the decision (there was no test or assignment the next day) to let it stand. On the next day, I brought of the decision with a question that led them to believe they were wrong-- and all was forgiven!

**5. If you could give some advice to math instructors thinking about using active learning, who have not tried yet, what would you say to them?**

**Justin**: There is evidence in the research that supports your decision to try active learning, so you should proceed with confidence. For the IO curriculum, it is extremely exciting to be a sort of curator of the conversation. You don’t know what students are going to say, and you get the exhilaration of thinking on your feet to fit their ideas into your agenda. So, my advice is: this is hard work! Be kind to yourself when you just can’t reshape their ideas the first time. We have found that, although active learning is our main mode of instruction, there is a “time to tell.”

**Karen**: Take it a step at a time— some instructors might go all out first time around, but trying one or two days or tasks and seeing how it works in your classroom is just fine. Ultimately, students will be more engaged and take ownership of the mathematics they learn in your new active learning classroom.

**6. How can readers learn more about the TIMES project and get involved?**

The TIMES NSF grant has essentially run its course, so we are no longer running workshops. You can find the course materials, including instructor’s notes, at these websites:

IODE: iode.wordpress.ncsu.edu

IOAA: www.web.pdx.edu/~slarsen/TAAFU/home.php

IOLA: iola.math.vt.edu

When you begin to investigate these materials, please don’t hesitate to contact us; we are more than happy to help! With sufficient interest, we may even run informal online working groups.

**References**:

George Kuster, Estrella Johnson, Karen Keene & Christine Andrews-Larson

(2018) Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles, PRIMUS,

28:1, 13-30, DOI: 10.1080/10511970.2017.1338807

Chris Rasmussen, Justin Dunmyre, Nicholas Fortune & Karen Keene (2019) Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram, PRIMUS, DOI: 10.1080/10511970.2018.1472160