Thursday, October 27, 2016

MAA Books for Active Learning

Some people like to write their own notes for their active learning classroom. However, many resources that we've already shared on this blog, like the Art of Mathematics and the Journal of Inquiry Based Learning, provide great textbooks and resources you can use. The MAA also has several books that are useful for an Active Learning Approach.

Number Theory Through Inquiry”, by David Marshall, Edward Odell, and Michael Starbird

Distilling Ideas: An Introduction to Mathematical Thinking  by Michael Starbird and Brian Katz

Exploratory Examples for Real Analysis”, Joanne E. Snow & Kirk E. Weller

Explorations in Complex Analysis”, Michael Brillslyper, Michael J. Dorff, Jane M. McDougal, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, & Kenneth Stephenson

“A TeXas-style introduction to proof”, by Patrick Rault and Ron Taylor.

Coming soon. Preliminary copies available through the authors.

Wednesday, October 26, 2016

Faculty development - how to learn about active learning

Participating in faculty development is one key way instructors can learn how to apply active learning approaches. Our group has been evaluating workshops on Inquiry-Based Learning (IBL) to learn what makes them effective in changing instructors' teaching practices - results show 60-80% of instructors reported using IBL in the year following the workshops (See here & here for more information). One key feature is that these workshops help participants learn how to implement IBL in a variety of contexts from large, introductory courses to small, upper-level courses. Building upon the results of these evaluations, the Academy of Inquiry-Based Learning and E&ER have recently begun a new NSF-funded project, PRODUCT , to increase capacity for training greater numbers of instructors to use IBL. Over 5 years, they will train new workshop facilitators, who will offer a dozen intensive workshops as well as shorter workshops. They are great hands-on opportunities for those interested in active learning to learn from skilled colleagues how to implement IBL in their courses and develop plans for an IBL course that fits your own context.

(Shared by Chuck Hayward, Ethnography & Evaluation Research, UC Boulder)

Active Learning and Self-Awareness

One of the best "side effects" of active learning is that students develop metacognition - self-awareness of their own learning and problem-solving approaches. Metacognition supports mathematical problem-solving because students with good metacognitive skills can better analyze and adjust their own problem-solving approaches or abandon ones that are not fruitful. Thus metacognition transfers to other settings - it's a lifelong learning habit. A metacognitive classroom has a rich mathematical culture where students are behaving like mathematicians, examining claims and asking for justifications. Alan Schoenfeld demonstrates this link in his classic article on metacognition and problem solving and suggests four ways to foster metacognition in your own classroom. 

(Shared by Sandra Laursen, UC Boulder). 

Friday, October 21, 2016

Beyond Lecture: Techniques to Improve Student Proof-Writing Across the Curriculum

This book has some ideas on how to provide active learning experiences to our students.  The e-book version is available for free to MAA members. One of the articles, by Patrick Rault, focuses on Teaching Proofs via Inquiry-Based Learning.

A TeXas Style Introduction to Proof

This new textbook serves to help instructors teaching a college level Introduction to Mathematical Proof course in transitioning to an active classroom.

Citation:    Taylor, Ron; Rault, Patrick X.  A TeXas Style Introduction to Proof.  Mathematical Association of America (MAA) Textbook Series.  To appear.  

The Greater Upstate New York Inquiry-Based Learning (UNY IBL) consortium

The Upstate New York Inquiry-Based Learning Consortium is a group of professors in the greater upstate New York region which have been meeting share their enthusiasms, frustrations, and triumphs related to the use of Inquiry-Based Learning in their mathematics classes.  We provide support and training workshops to college mathematics faculty and secondary school mathematics teachers in the use of active learning in their classes.

(Shared by Patrick Rault.)

Calculus Concept Inventory (Epstein)

While this paper and the tool it uses have been critiqued, this paper makes a compelling case that not all active learning classrooms are equal and that we need to strive for environments in which students articulate their thinking to get feedback in quick and meaningful cycles. While the paper is intended for educators, having students read it can be helpful for building student buy-in for active learning, especially if they feel that they would be learning more by listening.

(Shared by Brian Katz.)

Inquiry-Oriented course materials

"Teaching Abstract Algebra for Understanding" (now "Inquiry-Oriented Abstract Algebra"), "Inquiry-Oriented Differential Equations", and "Inquiry-Oriented Linear Algebra" are each sets of course materials based on extensive research build from Freudenthal's "Realistic Mathematics Education" paradigm. After requesting access, instructors can see polished course materials that includes discussion of goals and design concerns, examples of student work, and likely student conceptions along the way. Website:

(Shared by Brian Katz.) 

PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies)

PRIMUS is a pedagogically focused journal for mathematicians. The articles discuss math problems or teaching interventions, and they are written with the goal of allowing readers to adapt ideas for their own context. Many of the papers in PRIMUS can help mathematician readers apply research coming out of the Mathematics Education (especially RUME) community without having to develop the skills to read full research-level papers. Special issues on inquiry or active learning will be particularly fruitful, including the upcoming special issue on "Teaching Inquiry".

(Shared by Brian Katz.)

Journal of Inquiry-Based Learning in Mathematics

JIBLM contains peer-vetted classroom resources for inquiry-based learning. Most of these materials are for full courses or larger modules, but the site is moving to include smaller-scale resources. This is also a nice repository where you could share materials that you have developed.

(Shared by Brian Katz.)


Differential (really difference) equations, such as the famous SIR model from epidemiology, can be coded into spreadsheet technologies, but they require some technical expertise, and it can be hard for students to make sense of the model because some columns will end up representing populations while others are constants or changes. VenSim allows a graphical layout that helps separate the kinds of variables at play. Moreover, it allows users to see the impact of changing parameters dynamically, which facilitates the kind of exploration that would be extremely difficult symbolically. It's October, so you might consider adapting SIR to be about a zombie epidemic, asking students to evaluate different plans to fight the scourge.

(Shared by Brian Katz.)

How College Works

Book by Chambliss and Takacs, How College Works, Harvard University Press: Cambridge MA, 2014.

A look at what makes college "successful" (or not) for students, based on many different notions of "success".  A major discovery is that early connections with a good professor and/or mentor make a bigger difference than we think.

(Shared by Annalisa Crannell.)

Make it Stick: The science of successful learning

Book by Brown, Roediger, and McDaniel, Make it Stick: the science of successful learning, The Belknap Press of Harvard University Press: London, England, 2014.

This book describes how to use quizzing, distributed practice, interleaving to increase student learning (or rather, to "interrupt forgetting"). Highly readable, both for teachers and for students.

(Shared by Annalisa Crannell.)

Wednesday, October 19, 2016

Ethnography Research and Inquiry Based Learning

There is much research showing that IBL is effective at teaching students mathematics. One of the most striking results is that the students' attitudes towards mathematics can change significantly after taking an IBL course, especially among students who are women. In the forefront of this research are the faculty of the Ethnography & Evaluation Research group at UC Boulder, led by Sandra Laursen.

Some of their articles on this topic are:

-Laursen, S., Hassi, M.-L., Kogan, M., Hunter, A.-B., & Weston, T. (2011) Evaluation of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics. [Report prepared for the Educational Advancement Foundation and the IBL Mathematics Centers.] Boulder, CO: Ethnography & Evaluation Research, University of Colorado, Boulder.

-Laursen, S.L., Hassi, M., Kogan, M., & Weston, T. (2014), Benefits for Women and Men of Inquiry-Based Learning in College Mathematics: A Multi-Institution Study, Journal for Research in Mathematics Education, Vol. 45, No. 4 (July 2014), pp. 406-418.

-Kogan, M. & Laursen, S.L. (2014), Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics, Innovative Higher Education, Volume 39, Issue 3, pp. 183--199.

The complete set of their research can be found in

Discovering the Art of Mathematics - Newsletter

In order to support a growing sense of community among faculty interested in active learning and inquiry-based approaches, the Discovering the Art of the Mathematics project publishes a monthly email newsletter, which typically introduces a new blog about our experiences, tools, and questions arising from teaching in an inquiry-based way.  

Faculty and teachers can sign up at

(Shared by Volker Ecke.)

Discovering the Art of Mathematics - Videos

Active learning classrooms require different pedagogical tools and teacher moves than a traditional lecture classroom.  Video vignettes of active learning classrooms can support faculty in identifying such teacher moves, analyzing their impact on students, and becoming curious about  practicing some of these moves in their own classrooms.   The media library on our project web site contains over 60 classroom videos, of about 2-15 minutes in length, exhibiting a wide range of classroom situations, teacher moves, and student responses.

These videos are typically part of a specific section of our online e-book "Discovering the Art of Teaching and LearningMathematics Using Inquiry, where we reflect on the pedagogy visible in these videos.

(Shared by Volker Ecke.)

Discovering the Art of Mathematics - Workshops for Faculty

Curriculum materials and pedagogy books are a vital part of professional development for faculty.  Often, however, they are not sufficient to faculty in the shift from traditional teaching methods to more active learning approaches. The community of a small workshop that allows faculty to be students in an inquiry-based classroom, voice their hesitations, practice new teacher moves and discuss inquiry-based techniques has a much bigger impact on faculty. The project “Discovering the Art of Mathematics” offers 2-3 day traveling workshops with the goal of establishing a local inquiry community that will stay connected and support each other in trying new techniques. See for a description of past and future workshops.

(Shared by Volker Ecke.)

Discovering the Art of Mathematics - Faculty resources and membership

As our project has grown, we have learned that successfully using our materials depends on more than just the books.  Much of what has made our approach successful has come from observing our students and the various ways in which they may approach a problem.  These insights, along with solutions to selected problems and possible extensions have been included in a teacher’s edition for each book.  These editions are only available for instructors so to support faculty who use our materials we have created a faculty login.  Faculty members may request a free account to access the teacher edition for each book. Our account request form can be found at  A member of the Art of Mathematics team will review the application and respond shortly thereafter.

Discovering the Art of Mathematics - Textbooks

Active learning requires that students be provided with challenging, engaging, open, meaningful mathematical tasks. Traditional textbooks are not often organized around such active, student-centered investigations.

The Library at Discovering the Art of Mathematics is a collection of 11 freely available texts developed as inquiry-based alternatives to traditional textbooks.  The primary audience of these texts are mathematical for liberal arts students.  Each of the themed texts is built around deep mathematical topics and provides inquiry-based materials which can be used as content for a semester-long course.
More generally, the volumes in this library provide materials that enables teachers to experiment with inquiry-based learning in their classrooms.  Chapters and sections of the texts can be used as supplements, replacement units or enrichment opportunities suitable not only for college classrooms but in high schools and math circles.
All materials in the library can be copied and freely distributed for educational purposes.

Volumes in the library are:  Geometry, Games & Puzzles, Music, Knot Theory, Calculus, Proof, Dance, Patterns, Infinities, Number Theory and Art & Sculpture.

(Shared by Volker Ecke.)

Discovering the Art of Mathematics

"Our experience and expertise with the pedagogy of Inquiry-Based Learning extends across the full K-16 spectrum of mathematicslearning. We all routinely teach the full range of college-level mathematics courses using inquiry with strong success: lower-level undergraduate mathematics courses including Calculus I-III, upper-level math major courses focusing on proofs, and content courses for future elementary and secondary teachers. We also work with K-12 teachers in bringing IBL techniques into their classrooms.

In our experience, support in the following dimensions can support faculty wishing to shift from traditional to active learning approaches:  providing them with challenging, meaningful, engaging investigations and an extended set of pedagogical practices, giving them personal insight into the student experience in an active learning classroom, and opportunities to explore, analyze, develop, and practice their own way of facilitating this new ecosystem of learning. A community of reflective colleagues will be invaluable for sustaining deep and lasting change." - Volker Ecke

The website for "Discovering the Art of Mathematics", has all of these resources. Among them:
- Textbooks.
- Teacher books and faculty login.
- Faculty Workshops.
- Videos of IBL in action.
- Newsletter & blog.

The Academy of Inquiry Based Learning

The AIBL website is a hub for resources on Inquiry Based Learning in mathematics. AIBL President Stan Yoshinobu writes the IBL Blog to push out useful information for the IBL community. Yoshinobu says: "we offer NSF-funded IBL Workshops in the summer (for the next four years). We are working to confirm dates for summer 2017, when we will have three workshops in different locations."

Brian Katz says "This website contains resources for instructors who are running active-learning classrooms, including course materials for instructors, student testimonials about their experiences, opportunities for professional development, and connections to a community of peers doing similar work."

You can also join the AIBL Facebook group. Katz says "This Facebook group can help you connect with other instructors doing similar work. The group gets posts about resources for instructors, suggestions for classroom materials, and brain-storming techniques for addressing teaching challenges."

Below are some photos from the AIBL Workshops from 2014 and 2015, respectively (courtesy of Stan Yoshinobu).

On Teaching and Learning Mathematics

The AMS blog "On Teaching and Learning Mathematics" has published numerous articles about active learning and related topics.  For a concise overview of the articles published by the blog, see these two "year in review" articles: and

Specifically, there is a five-part series on active learning from Fall 2015.

Editor Ben Braun says: "This five-part series on active learning from the AMS blog "On Teaching and Learning Mathematics" provides an overview of issues related to active learning in postsecondary mathematics education, including example tasks, self-assessment of active learning methods, example environments and structures that support active learning, balancing direct instruction with active learning, and more."