Tuesday, January 12, 2016

Math Autobiography

Want to know one of my favorite assignments that I have ever given my students? Want to know learn a lot of useful information about your students in a short amount of time?

I know it sounds too good to be true, but this one simple assignment could change how you teach your classes and how well you know your audience.

Last year after helping lead an inquiry-based learning (IBL) workshop in Portland, I decided to try something new in my own classrooms. One of the other instructors, I wish I could recall who, said he/she started off every class with a math autobiography. That's all I had from my notes, but it sounded interesting so I did some research on it before school started last fall.

I went home, Googled “Math Autobiography” and found this assignment (MS Word document) online written by Kathy Wellborn from Fort Lewis College.

Math Autobiography 

Purpose of the Assignment 
As your instructor, I want to get to know you as a person and as a student of mathematics. This will help me better meet your needs. It also helps our department as we work to improve our services to students.

Your autobiography should address the four sections listed below. I’ve listed some questions to help guide you, but please don’t just go through and answer each question separately. The questions are just to help get you thinking. Remember the purpose of the paper. Write about the things that will give me a picture of you. The key to writing a good piece is to give lots of detail. See the example below:

Not enough detail: I hated math in fourth grade, but it got better in sixth grade.

Good detail: I hated math in fourth grade because I had trouble learning my multiplication tables. I was really slow at doing problems, and I was always the last one to finish the timed tests. It was really embarrassing. …

Section 1: Introduction 

  • How would you describe yourself? 
  • Where are you from? How did you decide to attend Fort Lewis? 
  • What is your educational background? Did you just graduate from high school? Have you been out of school for a few years? If so, what have you been doing since then?  
  • General interests: favorite subjects in school, favorite activities or hobbies. 
Section 2: Experience with Math

  • What math classes have you taken and when?
  • What have your experiences in math classes been like? 
  • How do you feel about math? 
  • In what ways have you used math outside of school? 
Section 3: Learning Styles and Habits (specifically for math) 

  •  Do you learn best from reading, listening or doing? 
  • Do you prefer to work alone or in groups? 
  • What do you do when you get “stuck”? 
  • Do you ask for help? From whom? 
  • Describe some of your study habits. For example: Do you take notes? Are they helpful? Are you organized? Do you procrastinate? Do you read the text? 
Section 4: The Future 

  • What are your expectations for this course? 
  • What are your responsibilities as a student in this course? What do you expect from your instructor? 
  • What are your educational and life goals? 
  • How does this course fit into your educational goals? 
It was fantastic! Students took it way more seriously than I could have imagined. Some wrote pages and all wrote enough to get to know them. It made me realize that we don't give our students opportunities to share their math baggage/backgrounds/etc. with us often enough.

Students shared everything from horror stories about being shamed in math courses to their excitement about math. Some let you know what they have heard about your class and even fears they may have such as a fear of presenting or working with others. This is good to know in an IBL course, since my students are all expected to work with others and also present problems. I even learned about people who were taking the course "for fun" and others who were in it for the second or third time. It really helped me get to know my students perceived strengths and weaknesses.

The first time I did this assignment, I was surprised that almost all of my students said they preferred to work alone and considered themselves introverts. This meant that I had to help them learn how to work together and intentionally help them see the benefits of this type of learning.

I kept these autobiographies and looked back on them as the semester passed. I want to create a similar end of course writing assignment and would be happy to share it when I do. If you already have one, please share it.

I'll let you try this assignment in your own classrooms and see what you think. Please share your comments on what you learned from this assignment.

Monday, October 12, 2015

Many Voices Are Greater Than One

Photo credit Stan Yoshinobu
It turns out that talking to yourself can be a sign of intelligence and lower stress. The down side is that you don't tend to hear a lot of new ideas.  Perhaps it is best to work out ideas by yourself and, once you get stuck, find new ideas by talking to others. Though this is a simple way to describe how discovery happens in a community setting, and describe expectations for students in our classes, it has not always been a common approach to teaching. It was certainly not how we were taught to do mathematics. However, what helped many mathematicians to learn and love mathematics might not work for our students. If you could learn by just being told things, then children would never have to be told twice to brush their teeth or wash their hands. If you have ever seen a tablet or phone screen after an eight year old has used it, you know that the lesson on washing hands doesn’t stick. How do we get mathematics to stick in our students' minds like grime on an eight year old’s hands? In order to make lasting knowledge, you need to get in there, explore, and experiment with everything you see, much like a child at play.

It is a shame that so many students never get their hands dirty working with math, but rather most view mathematics as an adversary. A simple measure of this is to ask them “When was the last time math really made sense?” Even good students may have progressed without making sense of any of the ideas that make math so interesting to us. They have had excellent training in finding answers, when faculty think they should have been sense-making instead.

Photo credit Stan Yoshinobu
How do we bridge the gap between where we want our students to be and where they are now, while accounting for their lack of preparation in mathematical thinking? The good news is that you can guide your students to build a solid foundation and connect to the most important ideas in education. Key to this approach are the multitude of voices that need to be included in the conversation. If the goal of a class is for students to develop a deep understanding and appreciation of mathematics, then the students' voices are even more necessary than their instructor's. And lasting changes in the lives and minds of our students are exactly what we are trying to achieve as educators. Outside voices are also necessary: the many other like- minded educators, including past, present, and future, and in all disciplines, have to collaborate to decide how to address the ever-changing needs of our students. Faculty at many institutions are already implementing active methods of engaging students and fostering lasting learning.

For the last 10 years, the Academy of Inquiry Based Learning has run a workshop to aid mathematics educators in implementing inquiry based learning (IBL). This summer, about 40 faculty, including all three authors, attended a four day workshop in beautiful San Luis Obispo. At the workshop, we worked with experienced faculty to develop materials, and individualized approaches to implementing IBL in our classrooms. This was an immersive experience into the practice and methods of “big tent IBL”, which is a broad and inclusive version of this active learning philosophy. IBL is being implemented in all manner of classes, not just upper level proof-based mathematics courses. In fact, the largest groups of people at the workshop were working on implementing IBL into the sequence of mathematics classes for elementary education majors and intermediate and college algebra.

The four day workshop also introduced us to the community of faculty across the country that want nothing more than to help others get the most out of the time spent with their students. It was just enough time to get energized and get to know a lot of great people, as well start thinking about the myriad of ways to implement IBL. Unfortunately, this was not nearly enough time to fully figure out how we would carry out these ideas in our upcoming classes. The workshop wisely includes a year of follow up mentoring with the organizers. As a way to expand on these mentoring conversations, several of us decided to write about our early experiences at A Novice IBL Blog. Already, blogging has been a wonderful reflective exercise in connecting what we are trying in the classroom to the changes we want for students. Writing posts provides an opportunity to struggle with new pedagogical tools, and learn how to effectively communicate in a new forum. This struggle has helped us each appreciate the difficulties that our students face when grappling with new material.

Blogging has also helped us appreciate how difficult it can be to do new things. Especially when the new thing is a fundamental change in how you approach your job. Just as when our students are learning, it’s important to have many voices for ideas and support. Our blog is about support, not just for its authors, but for anyone trying IBL or anything new in the classroom (and not just at the post-secondary level). So, come visit our blog, participate in the discussions and learn about inquiry based learning along with us.

-Liza Cope, David Failing, and Nick Long

Liza Cope is an Assistant Professor at Delta State University and can be contacted at lcope@deltastate.edu.

David Failing is an Assistant Professor at Quincy University and can be contacted at failida@quincy.edu.

Nick Long is an Associate Professor at Stephen F. Austin State University and can be contacted at longne@sfasu.edu.

Thursday, September 24, 2015

IBL Conference: A New Kid’s Perspective

A guest post by Katie Wanek, University of Nebraska at Omaha

The UNO group after the panel presentation.
This past June, I was given the opportunity to attend the 18th Annual Legacy of R.L. Moore Inquiry-Based Learning Conference in Austin, Texas, along with three of my fellow undergraduates from the University of Nebraska at Omaha (UNO). I was one of only 6 attendees who were still working towards their undergraduate degree, as a majority of the conference attendees were teachers and professors at the collegiate level. Needless to say, I felt very much like the new kid in school where everyone else is already on chapter 8 and I am only on chapter 2. However, although I was much younger than most of the people I interacted with, and much less experienced, everyone I met was not just willing, but seemed excited to share their knowledge. I felt welcomed to share my own ideas and thoughts in sessions, and I was encouraged to actively participate. As it was my second time attending the conference (I attended the conference in Denver last year), I made a lot of comparisons between the two conferences while I was there. I found that there were three main characteristics that stayed constant between the two:
  1. Inspiring
  2. Enlightening, and 
  3. Empowering
Inspiring. I left the conference each time inspired by both the session topics and the people I met. Whether it was an idea for a Math Teachers’ Circle, a lesson idea for my future classroom, a method to engage and encourage students, or an activity to try for outreach events, the sessions were always informative, entertaining, and thought-provoking.

The people I met at the conference also inspired me. One of my favorite activities at both conferences was the Round-Table Discussion at which I sat with a table of other people passionate about secondary math education. We discussed roadblocks to implementing IBL in a secondary classroom and ways to work around them. People freely shared resources they found to be helpful and we all exchanged e-mails. Within the first week after this past conference, there were multiple emails sent from the other table members with links to sites and Google documents. These people inspired me to be not just an average teacher, but a great teacher who actively searches for ways to help her students and make her classroom and teaching better.

Enlightening. I learned a lot about inquiry-based learning and the various techniques that people use to implement it in their classrooms at both conferences. Every year, I learn more and more about inquiry-based learning and what it actually means. I’ve learned that there are tons of different ways to use IBL in classrooms – flipped classrooms, projects, in-class activities, etc.

There are also multiple ways to encourage IBL in outreach activities such as with puzzle competitions. One of my favorite sessions at the 2015 Conference was the “IBL and Mathematical Puzzlehunt Competitions” by Steven Clontz and PJ Couch. They talked about puzzlehunt competitions that they have adopted at their respective universities and how puzzles are a great way to spark interest in math.

I learned so much at the conferences that my brain was exhausted by the time I got back to Omaha. There was so much to soak in and in a short amount of time.

Empowering. Both conferences were empowering. I was empowered with knowledge to implement IBL in my future classrooms. I was empowered with new networks of people to turn to for ideas, techniques, and encouragement. I was empowered with ways to sell IBL to students, parents, and administrators and the data to back-up my claims. Most of all, I was empowered with a can-do attitude and the thought that this is something that is possible and will help students.

Although the two conferences had a lot in common, especially with the three main themes I detailed, there were some differences as well. It felt like the 2014 conference had more parallel sessions and there seemed to be a larger focus on hands-on sessions and ready-made ideas to take back to the classroom. The 2015 conference focused more on how to help students be confident in their mathematical ability and how to encourage them to not give up and to keep going. Both of these differences tied back to their respective themes. The 2014 conference theme was “Engaging with Inquiry-Based Learning” and the 2015 conference theme was “Empowering with Inquiry-Based Learning”. When connecting the differences to the themes, the differences make sense. Regardless, both conferences had a lot to offer and I was left with a stronger desire to implement inquiry-based learning into my future classroom after both conferences.

Altogether, the conference was an incredible experience and I highly encourage everyone to attend the conference, especially pre-service secondary math teachers who are interested in the idea of inquiry-based learning. For the pre-service teachers and other undergraduates planning on attending a future IBL Conference, my suggestion would be to go in with an open mind, talk to as many people as you can, and participate as actively as you can. It is a wonderful conference that is not only inspiring, enlightening, and empowering, it also focuses on student learning and how to help students deeply understand material. And, whether we are pre-service teachers, new teachers, or teachers who have been in the field for years, anything that will assist student learning is something that we can all get behind.

About the author: Katie Wanek is a senior at the University of Nebraska at Omaha where she is a double degree student in Math and Education. She has been the Math Club president for the past three years and also works as Dr. Angie Hodge’s undergraduate assistant. Katie will graduate in May 2016 after which she aims to find a position teaching middle school math where she hopes to incorporate active learning into her classroom.

Wednesday, July 1, 2015

Knowing What to Do is not Doing

A guest post by Bob Klein, Ohio University

In Summer 2014, I attended the Academy of Inquiry-Based Learning Workshop in Portland that took place prior to MAA MathFest. While I had always come out relatively well in teaching evaluations, I also have been trying to improve my teaching. For the IBL workshop I redesigned my two fall courses (a capstone math content course for secondary education math majors, and a geometry content course for middle school teachers). Later that semester I redesigned my history of mathematics class to follow an inquiry-based learning (IBL) format. Overall, students were more engaged and I was doing far less “delivery” and much more “data collection” and “decision-making.” When students were working at the board or discussing, my role shifted to listening/observing/evaluating so that I could guide discussion. For a while, I’ve adhered to the mantras in my teaching: “Be less helpful” (DM) and “Never Say Anything a Kid Can Say” (SR). But aside from my work in math circles, it hasn’t been until this academic year that I’ve been able to realize those in the classroom.

A side effect of this is that I both thought MORE of my students’ abilities and LESS of their mastery at the same time. MORE because they showed me that they could embrace a learning environment that put more on them—this was a matter of willingness to adapt/change what took place in the classroom. But also LESS because I had a much more detailed picture of what they knew and what they were able to do at any given point. The content capstone course for secondary math teachers-to-be contained mostly seniors who have completed college geometry, three semesters of calculus, number theory, discrete mathematics, algebra, statistics, and an elective. It may just be this lot, but I’m worried about their abilities as they move forward next year to teaching. Worried especially that we may have promoted them through “the system” of courses without an accurate read on their mastery of the content of those courses. That said, I think I caused those students to be worried about their mastery of content because I asked them to PERFORM that content in rather public ways and subject to peer criticism. This is cause for hope and also a good basis for me to make further refinements to the running of the curriculum.

The other course I redesigned for fall was a geometry content course for future middle school teachers. The sophomore level course is significantly different from the capstone course and I think students here were challenged and pushed though in different ways from the capstone course. I was able to push the geometry students to see mathematics itself (as a practice) differently. They began the course trying to convince me that the “(I do)→(we do)→(you do)” formula of exercises (not problems) was what math was all about. By the end of the semester they were asking each other, “How would you defend that?” and digging deeper. I think I probably needed two semesters with them to achieve my stated goals for the course so I might need to be more realistic or more specific about the syllabus goals.

History of mathematics was the hardest to redesign for me. The course enrolls mostly middle-level education majors though secondary math education majors and a few STEM majors also enroll. As such, the group has a wide range of backgrounds and it cannot be a course that focuses in large part on proofs (without significant support) though we do cover some demonstrative mathematics. To adopt an IBL approach, I carefully chose a focus area (in this case Egyptian Mathematics) and we spent a long time engaging in problems and discussions centered on Egyptian techniques. We then spent some time working on Mesopotamian, Greek, and European mathematics (as well as other topics). Having a “big focus” for the course (almost an area of expertise for the students) made IBL techniques of problems and board work something that challenged the disparate backgrounds of the students equally and stimulated discussion about the deep ideas behind mathematics, including the assumptions, leanings, and contextual influences thereof.

All in all, though, this was very eye-opening for me. As an educational researcher (rather than a research mathematician), I KNEW what I was supposed to be doing, but the IBL workshop and community gave me the support and courage necessary to try it and to stick with it, especially in those first few weeks when I was still in the mode of ‘selling the approach’ to the students. The mass of existence proofs for the model, backed up with the leadership team’s smiles and insistence that this was a flexible set of practices rather than orthodoxy, were especially helpful.

One upshot of this work has been my nomination and eventual awarding of the Presidential Teacher Award at Ohio University. I was selected as one of only two faculty on my campus to receive the award after a thorough process involving a teaching portfolio, multiple observations of my teaching throughout the year, and interviews by the selection committee of me and my chair. It was truly humbling and nourishes my belief that the IBL approach is making a difference that is visible to students and beyond. I’m eager to engage in further discussions about approaches that worked for others, and welcome others to contact me at the address below.

The reflections above are from Bob Klein, PhD, Associate Professor and Undergraduate Chair, Department of Mathematics, Ohio University, Athens, Ohio USA. kleinr@ohio.edu


[1] Meyer, Dan. http://blog.mrmeyer.com/2009/asilomar-4-be-less-helpful/.

[2] Reinhart, Steven. (2000). Never Say Anything a Kid Can Say. Mathematics Teaching in the Middle School. Vol 5, No 8, pp. 54-57.

Thursday, April 9, 2015

Calculating Community: The Running Equation

How does one do something they perceive to be so difficult that they never thought they would even consider doing it? How does one go from being a non-athlete to running 100 miles at elevation? And, how does this have anything to do with mathematics? I challenge you to read this blog post and replace “running” with “mathematics” for most people.

In my free time, I run. I run a lot. In fact, my new favorite race to run is the 50-miler, but I have completed three 100-mile events. One of these 100-mile events I completed in under 24 hours.

Was I born a runner? No, in fact I was a scrawny girl with chicken legs who would do anything to get out of gym class. Running used to make me cry. It was hard and it hurt. But, I was curious. Why did so many people run and say they loved running? In January 2009, I set out on a mission to find out what this running hype was all about. I trained for my first half marathon (13.1 miles) diligently for 4 months. I followed a training program to the nth degree, ran my race, and found my new hobby. The next day I signed up for a full marathon, 26.2 miles. I again trained and still liked running after completing it. Everyone I talked to told me that after running it I would never want to do another one, but instead I was looking for my next race shortly after I finished my first one. I didn’t want to tell anyone, but I wasn’t all that tired by the end of the race. I was hungry, but not tired. What was going on here?

I had thought I hated running because it was difficult, but I did my homework and ended up liking the feeling of success. With each step of the way I thirsted for more, even though others told me it would be “too hard.”

For a few years, I was able to fill this thirst with trying to run faster, but then I moved to Omaha. Here is where I found the ultra running community – the Ph.D. program of running.

26.2 miles was enough “difficulty” for me for years and I even thought it was insanity to run anything over that distance. Who does that for fun?!?! Not only is an ultra marathon anything over 26.2 miles, but they are often ran on dirt trails following flags on a course that may or may not be well marked. However, they do feed you multiple times along the way and I like food. Think of an aid station as a semester break in grad school. It means you have survived another semester. You get to rest, eat, recover, and forget the pain for a while. Then you go back for more with each leg of the journey providing new challenges and opportunities for growth.

How does one survive such madness? My answer to surviving both mathematics classes/programs and ultra running events is one and the same: community. When people are brought together to complete a task that at the time seems impossible, there’s something special that develops. Here are a few tips I have for creating that community that comes with embarking on a hard task.

The buddy system: Find people to train/work with who inspire you to do more than you think you can. This does not always mean finding people who are better than you. I would say that this should people of varying abilities. For instance, I have three training partners who come to mind when I think of the types of people who help me go beyond my “limits.”

First, there is my friend, Steve Stender. He is fast. He is very fast. He’s also the person I have ran more miles with than anyone I know. By being a friend and getting to know me as a person and as a runner, I value what he has to say. He shows me on a regular basis that I can go faster than I think I can and longer than I think I can with good company. He also has flat out told me to “go out and win something.” This being told to a non-athlete is something I am still not even sure if I believe. However, I have won races now and don’t think I would have even tried without Steve. He would be the friend encouraging you to get an A+ on an exam instead of just passing it.

Second, there is my friend, Kim Moore. She is tough as nails. She challenges me to go beyond my comfort zone. Somehow this lady has gotten me to run outside in the cold all winter and to run in the dark. I hate the cold and am scared of the dark. She would be the friend who encourages you to take the hard instructor or the difficult class. These things make us stronger, even though they may be painful at the time.

The third “type” of running friend I have is similar to my friend, Eric Schelker. He is someone who I mentor/train. He couldn’t run two miles when we met a year and a half ago, but now he is training for his first marathon. There’s something about helping someone and knowing they are looking up to you that makes you not want to quit. This also helps me want to do well and be a good role model. This would be the friend who you have to “teach” while working together, but in return you then understand the material better.

Regular meetings: This one is rather simple. Once you have a core group of people to train/work with you, meet regularly. This helps develop friendships and holds you accountable for doing your training/work. Don’t forget to make these meetings fun! Our running group meeting at a local bar on Tuesdays that has 50 cent tacos. We run 3-10 miles and hang out afterwards eating cheap tacos. My calculus classes meet daily before class to finish homework in a casual setting.

Talk to strangers: Whether there are newcomers in your group or you are seeking out a group, don’t be afraid to make new friends. Everyone feels awkward talking to new people, but you may just meet a life long friend or future colleague by talking to strangers.

Cave into peer pressure: Before you know it you will be running crazy distances (with these people who were once strangers) or taking crazy hard math classes. Sometimes peer pressure is a good thing!!

Impossible goal: Lastly, don’t be afraid to set a seemingly impossible goal for yourself. Break that goal into baby steps and conquer it with the help of your new friends.

For more details watch my TEDx talk at: https://www.youtube.com/watch?v=W6JG9-MWCKA

Tuesday, January 27, 2015

Setting the Stage

Whenever I’m teaching via inquiry-based learning (IBL), it is important to get student buy-in. I often refer to this as “marketing IBL”. My typical approach to marketing involves having a dialogue with my students, where I ask them leading questions in the hope that at the end of our discussion the students will have told me that something like IBL is exactly what we should be doing.

In the past, I would just wing it on day one and it’s been different every time. However, I’ve had lots of people ask me to describe exactly what I do and I also thought it would be a good exercise for me to sit down and think carefully about the activity. So, in the fall of 2014, I created some slides to guide the activity, which I am now calling "Setting the Stage". Since then I have shortened the activity and made some improvements. The current version of the activity is inspired by TJ Hitchman, Mike Starbird, and Brian Katz.

The main idea is that I want to get students thinking about why we there and what we should really be striving to get out of the course. In addition, it helps students understand why I take an IBL approach in my classes. Below is an outline of the the activity.

Directions to the Students

  • Get in groups of size 3–4.
  • Group members should introduce themselves.
  • For each of the questions that follow, I will ask you to:
    1. Think about a possible answer on your own.
    2. Discuss your answers with the rest of your group.
    3. Share a summary of each group’s discussion.


  1. What are the goals of a university education?
  2. How does a person learn something new?
  3. What do you reasonably expect to remember from your courses in 20 years?
  4. What is the value of making mistakes in the learning process?
  5. How do we create a safe environment where risk taking is encouraged and productive failure is valued?
Each time I’ve run the activity, the responses are slightly different. The responses to the first two questions are usually what you would expect. Question 3 always generates great discussions. The idea of "productive failure" naturally arises when discussing question 4 and I provide them with this language sometime while discussing this question. Listening to the students’ responses to question 4 is awesome. It’s really nice to get the students establishing the necessary culture of the class without me having to tell them what to do.
After we are done discussing the 5 questions, I elaborate on the importance of productive failure and inform that I will often tag things in class with the hashtag #pf in an attempt to emphasize its value. I also provide them with the following quote from Mike Starbird:
“Any creative endeavor is built on the ash heap of failure.”
I wrap up the activity by conveying some claims I make about education and stating some of my goals as a teacher.


  • An education must prepare a student to ask and explore questions in contexts that do not yet exist. That is, we need individuals capable of tackling problems they have never encountered and to ask questions no one has yet thought of.
  • If we really want students to be independent, inquisitive, & persistent, then we need to provide them with the means to acquire these skills.

Lofty Goals

  • Transition students from consumers to producers!
  • I want to provide the opportunity for a transformative experience.
  • I want to change my students’ lives!

Below is the Spring 2015 version of the slides that accompany the activity.

You can always find the current version of the LaTeX source at my GitHub repo located here. Note that I’m using the beamer m theme for the slides, which require the Mozilla Fira fonts by default. Feel fee to steal, modify, and improve. And please let me know if do.

Note: This post originally appeared on Dana’s personal blog.

Thursday, January 8, 2015

The Twin Pillars of IBL

By Dana C. Ernst

As regular readers of this blog know, I am passionate about inquiry-based learning (IBL) and the Moore method for teaching mathematics. This educational paradigm has had a profound impact on my life as a teacher. Actually, scratch that. It has had a profound impact on my life!

When I started teaching, I mimicked experiences I had as a student. I tried to emulate my favorite teachers. Because it was all I really knew, I lectured. And this seemed to work out just fine. By standard metrics, I was an excellent teacher. Glowing student and peer evaluations, as well as reoccurring teaching awards, indicated that I was effectively doing my job. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to: (i) heavily depend on me to be successful, and (ii) retain only some of what I had taught them. In the words of Dylan Retsek:

"Things my students claim that I taught them masterfully, they don’t know."

Inspired by a desire to address these concerns, I began transitioning away from direct-instruction towards a more student-centered approach. The goals and philosophy behind IBL resonate deeply with my ideals, which is why I have embraced this paradigm.

While there is variation in practice, IBL courses typically provide students with experiences that differ from traditional lecture-based courses. In many mathematics classrooms, doing mathematics means following the rules dictated by the teacher and knowing mathematics means remembering and applying these rules. However, an IBL approach challenges students to create/discover mathematics. According to the Academy of Inquiry-Based Learning, IBL is a method of teaching that engages students in sense-making activities. Students are given tasks requiring them to conjecture, experiment, explore, and solve problems. Rather than showing facts or a clear, smooth path to a solution, the instructor guides students via well-crafted problems through an adventure in mathematical discovery.

I believe that there are two essential elements to IBL. First, students should (as much as possible) be responsible for guiding the acquisition of knowledge, including the pace at which this happens, and second, they should be responsible for validating the ideas presented. That is, students should not be looking to the instructor as the sole authority. In most IBL courses, student-led presentations and small group work form the backbone of the course. In general, the majority of class time is spent on these types of student-centered activities, which provide ample opportunity to discuss and critique ideas that arise from a group problem or student-presented solution.

One guiding principle of IBL is the following question:

Where do I draw the line between content I must impart to my students versus the content they can produce independently?

While instructors might give a mini-lecture to introduce or summarize the day's work, the instructor's main role is not lecturing, but rather to foster a safe environment, facilitate discussion, and redirect as necessary. In an IBL course, instructor and students have joint responsibility for the depth and progress of the course.

A research group from the University of Colorado Boulder lead by Sandra Laursen has conducted a comprehensive study of student outcomes in IBL undergraduate mathematics courses while linking these outcomes to students’ and instructors’ experiences of IBL. This quasi-experimental, longitudinal study examined over 100 courses at four different campuses over a period that spanned two years. The courses examined self-identified as IBL versus non-IBL. Classroom observation was used to verify that IBL classes were indeed different from those designated as non-IBL sections of the same course. The following is a list of characteristics that the IBL sections shared:

  • Students solve challenging problems alone or in groups; share solutions; analyze, critique & refine their solutions 
  • Class time is used for these student-centered activities 
  • Students play a leadership role 
  • Activities change often 
  • Course is driven by a carefully built sequence of problems or proofs, rather than a textbook
  • Pace is set by students' progress through this sequence 
  • Course goals usually emphasize thinking skills & communication; content “coverage” is less central 
  • Instructor serves as "guide on the side" not "sage on the stage"—manager, monitor, summarizer, cheerleader

On average over 60% of IBL class time was spent on student-centered activities including student-led presentations, discussion, and small-group work. In contrast, in non-IBL courses, 87% of class time was devoted to students' listening to an instructor talk. In addition, the IBL sections were rated more highly for a supportive classroom environment and students conveyed that engaging in meaningful mathematical tasks while collaborating was as essential to their learning.

Below is a brief summary of some of the outcomes of Laursen et al.'s work:

  • After an IBL or comparative course, IBL students reported higher learning gains than their non-IBL peers, across cognitive, affective, and collaborative domains of learning. 
  • IBL students’ attitudes and beliefs changed pre- to post-course in ways that are known to be more supportive of learning, compared to students who took the non-IBL sections. 
  • In later courses, students who had taken an IBL course earned grades as good or better than those of students who took non-IBL sections, despite having "covered" less material. 
  • On a research-based test of students' ability to evaluate proofs, IBL students showed evidence of greater skill in recognizing valid and invalid arguments and of the use of more expert-like reasoning in making such evaluations. The volunteer sample consisted of only high-ability students; no instructors gave the test to all students during class time. 
  • Non-IBL courses show a marked gender gap: across the board, women reported lower learning gains and less supportive attitudes than did men (effect size 0.4-0.5). Women’s confidence and sense of mastery of mathematics, and their interest in continued study of math, was lower. This difference appears to be primarily affective, not due to real differences in women’s mathematical preparation or achievement. 
  • This gender gap was erased in IBL classes: women’s learning gains were equal to men’s, and their confidence and intent to persist similar. IBL approaches leveled the playing field for women, fixing a course that is problematic for women yet with no harm to men. 
  • When sorted by prior achievement, the grades of most students (IBL and non-IBL alike) dropped in subsequent courses as course work became more difficult. But grades of initially low-achieving students who had taken the IBL course rose 0.3-0.4 grade points, unlike their low-achieving, non-IBL peers, and unlike their higher-achieving classmates.

This work determined that there are two "twin pillars" of IBL that may explain the student outcomes, namely (i) deep engagement in rich mathematics and (ii) opportunities to collaborate. Here, deep engagement refers to individual and group effort in tackling meaningful problem-solving tasks that are not merely "busy work." Collaboration is a key component as students learn from explaining their ideas and trying to understand others. According to Laursen et al.:

"The twin pillars reinforced each other: after struggling with a problem individually, students were well prepared to contribute meaningfully during class, and interested in the solutions that others proposed. Collaboration in turn motivated them to complete the individual work. It also made class enjoyable, encouraged clear thinking, and built communication skills."

IBL is not a magic bullet, but the experiences that I have had watching students transform into independent learners is why I am so passionate about it. I want students to have life-changing experiences! Learning the content of mathematics is just a bonus.

One of my principal goals is to make my students independent of me. I want them to experience the unmistakable feeling that comes when one really understands something thoroughly. In the words of Carol Schumacher:

"When one can distinguish between really knowing something and merely knowing about something, that individual will be on his/her way to becoming an independent learner."

I’m not terribly picky about the particular flavor of IBL or active learning one chooses to employ, but it is becoming increasingly clear to me that if we want to produce life-long independent learners, then the twin pillars need to form the foundation for the pedagogical approach we choose to take.


[1]  Laursen, S. L., Hassi, M.-L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study. Journal of Research in Mathematics Education, 45(4), 406-418.

[2]  Kogan, M., & Laursen, S. L. (2014). Assessing long-term effects of inquiry-based learning: A case study from college mathematics. Innovative Higher Education, 39(3), 183-199. DOI 10.1007/s10755-013-9269-9.

[3]  Laursen, S. L. (2013). From innovation to implementation: Multi-institution pedagogical reform in undergraduate mathematics. In D. King, B. Loch, L. Rylands (Eds.), Proceedings of the 9th DELTA conference on the teaching and learning of undergraduate mathematics and statistics, Kiama, New South Wales, Australia, 24-29 November 2013. Sydney: University of Western Sydney, School of Computing, Engineering and Mathematics, on behalf of the International Delta Steering Committee.

[4]  Assessment & Evaluation Center for Inquiry-Based Learning in Mathematics (2011). Evaluation  of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics. (Report to the Educational Advancement Foundation and the IBL Mathematics Centers) Boulder, CO: University of Colorado, Ethnography & Evaluation Research.

Further resources can be found here.