What is inquiry-based learning (IBL)? Why use IBL? How can you incorporate more IBL into the classes that you teach? In this talk, we will address all of these questions, as well as discuss a few different examples of what an IBL classroom might look like in practice.
This talk was given at the Mathematics Instructional Colloquium at the University of Arizona.
What, Why, & How?
Mathematics Instructional Colloquium
University of Arizona
Dana C. Ernst
Northern Arizona University
Email: [email protected]
Twitter: @danaernst & @IBLMath
Thanks to Stan Yoshinobu for providing some of the content.
• Ph.D. from University of Colorado.
• Areas of research:
‣ Combinatorics of Coxeter groups and diagram algebras,
‣ A little math education.
• Currently an assistant professor at NAU (dream job!).
• Spent 4 years at Plymouth State University prior to NAU.
• Number of IBL classes I had as a student: 0
• Taught first IBL class in Fall of 2009.
What is inquiry-based learning (IBL)?
• According to the Academy of Inquiry-Based Learning:
‣ IBL is a teaching method that engages students in
‣ Students are given tasks requiring them to solve
problems, conjecture, experiment, explore, create, &
‣ Rather than showing facts and/or algorithms, the
instructor guides students via well-crafted problems.
• Students are responsible for guiding acquisition of
• Often involves very little lecturing, and typically involves
• Sometimes called Modified Moore Method, after R.L.
Continually ask yourself the following question:
Guiding Principle of IBL
Where do I draw the line
between content I must impart
to my students versus content
they can produce independently?
Our main objective
How do we get here?
One minute version of why IBL
• Our system needs an upgrade.
• Unintended negative outcomes via traditional methods.
• Research suggests IBL outcomes are better.
• When I started teaching, I mimicked the experiences I had
as a student (i.e., I lectured).
• By most metrics, I was a successful teacher (e.g., high
evaluations, several awards). Why change?
• Inspired by a Project NExT Workshop run by Carol
Schumacher (Kenyon College), I decided to give IBL a try.
• For 3 consecutive semesters, I taught an intro to proof
course at Plymouth State University.
• 1st two iterations taught via lecture-based approach.
• 3rd time taught using IBL with emphasis on collaboration.
• When I taught an abstract algebra course containing
students from both styles, anecdotal evidence suggested
students taught via IBL were stronger proof-writers & more
independent as learners.
My first IBL class
• 4-5 million freshmen in HS.
• 75% HS graduation rate.
• 1.2 million bachelors degrees annually.
• Less than 1% of BA/BS are in math.
• Only 400-500 U.S. citizens/residents earn a Ph.D. annually.
• Education is a self-populating institution!
You are peculiar!!!
We need to renormalize.
What is happening in STEM education?
• There exists a growing body of evidence suggesting
students are dissatisfied with learning experiences in
• Math Education Research suggests that college students
have difficulty with:
‣ Solving non-routine problems,
‣ Packing/Unpacking mathematical statements,
(Schoenfeld 1988, Muis 2004, Selden and Selden 1995,
1999, 2003, Dreyfus 2001, Sowder and Harel 2003,
Weber 2001, 2003, Weber and Alcock 2004, Tall 1994)
• About half of STEM majors
switch to non-STEM.
• Top 4 reasons for switching
are teaching related.
• Good ones leave, too.
• Loss of interest.
• Non-STEM majors offer
• Curriculum overload.
• Poor teaching.
• Weed-out culture.
Talking About Leaving
The good news
Evidence from the math ed literature suggests that active,
learner-centered instruction leads to improved conceptual
understanding, problem solving, proof writing, retention,
habits of mind, and attitudes about math.
(Boaler 1998, Kwon et al 2005, Rassmussen et al 2006,
Smith 2006, Chappell 2006, Larsen et al 2011, etc.)
The Colorado study
• Sandra Laursen, CU Boulder.
• Statistically significant advantages for students in IBL vs
Hake, Am. Journal Physics, 1998
1. Deep engagement in rich mathematics
2. Opportunities to collaborate.
• “That’s how I learned, and it worked for me...”
‣ But you are peculiar!
• “I like inspiring lectures.”
‣ Inspiration is necessary, but not sufficient.
• “I’m afraid the students won’t like it.”
‣ My kids like Gummi Bears, but that doesn’t mean they
are good for them.
‣ If I lecture, then I dictate pace.
‣ If I write something on the board, then there is a good
chance that it will be done correctly.
The main obstacle:
Most students do not enjoy direct
instruction, but it is what they are used to.
They expect to be passive, & they have had
14+ years of experience to develop deep-
rooted beliefs about how STEM classes
should operate & what is expected of them.
It is vital that student expectations & roles are clearly
established. To help reset these expectations students need
to have a clear understanding of what IBL is & why they
• Most students do not come equipped with the skills and
interests that we have.
• What are the secondary goals of the course? How are these
• Students need to know what their role is.
Students are asked to solve problems
they do not know the answers to, to take
risks, to make mistakes, and to engage in
"fruitful struggle." These are all very
different from normal expectations.
• Students need to know what the instructor’s role is.
• Expectations & goals need to be reiterated throughout the
• Students need to know that it is ok to be stuck and that
you will support them in this endeavor.
• Use analogies: learning to play an instrument, learning to
ride a bike, etc.
• Tip: Get the students to tell you what the best way is to
acquire the skills necessary for effective thinking!
Content delivery & knowledge
• For content that is to be imparted to students, how will it
‣ Assigned readings?
‣ Lectures? Pre-planned vs. by request?
‣ Screencasts (Flipped Classroom)?
• For content that students produce, how & when will it be
‣ Task/problem sequences?
‣ In-class vs. homework?
‣ Collaboration vs. independence?
Keys to success
• Effective marketing.
• Return to your guiding principle.
• Adjusting problems/tasks appropriately.
• Patience & trust!
• Build on positive experiences.
• Pick a style that you are comfortable with.
• Adapt, overcome, & improvise.
Let’s take a look at two examples.
A Moore method approach with collaboration
• My experience:
‣ Intro to Proof, Number Theory, Abstract Algebra, & Real
‣ 10-30 students.
• Encourage collaboration.
• Typical grade determination:
The big picture
Category Weight Notes
Homework 25% Mix of Daily & Weekly Homework
Presentations & Participation 30% Students present problems from Daily Homework
3 Exams 45% Typically take-home exams
Data to support
• 5-10 “tasks” (e.g., exercises, proofs of theorems) are
assigned each class meeting (Daily Homework). Due at
beginning of next class.
• Students are responsible for digesting new material
outside of class.
• Nearly all class time devoted to students presenting
proposed solutions/proofs to assigned exercises.
• Students (usually) volunteer to present.
• My job:
‣ Facilitate discussion
‣ Keep us on track
‣ Mr. Super Positive
‣ Cross my arms and say, “hmmm”
• Students may request mini-lectures or screencasts.
More on student presentations
• Must present at least 2x prior to each exam in order to
receive a passing grade for Presentation category. (Is this
• I take notes during presentation & add to spreadsheet:
‣ Who & what problem
‣ Exercise or proof
‣ Miscellaneous notes
‣ Score 1-4
4 Completely correct and clear proof or solution. Yay!
3 Solution/Proof has minor technical flaws or is lacking some details.
2 A partial explanation or proof is provided but a significant gap still exists.
1 Minimal progress has been made.
• These problems form the backbone of the class.
• Problems from task sequence are assigned based on where
we ended previous class.
• Felt tip pens!!!
‣ Credit: Clark Dollard (Metro State
‣ Each student grabs a felt tip pen
on way into class
‣ Students use pens to annotate
homework in light of presentation
& related discussion
‣ No penalty for use of pen
• Graded on ✔-system. What did they have done before
Advantages of the felt tip pens
• I know what happened before class versus during class.
• Students mark up their work in ways they never did
• Students have (mostly) correct work by the end of class
(pedantic details & logical structure).
• Students have a record of what happened in class together
with their homework.
• When students look back at their notes they see their
comments about what they were thinking & they see
• Students love the felt tip pen approach. Numerous positive
comments about how useful this is.
• Grading of the Daily Homework is fast!
• On week n+1, students choose 2 *-problems (subset of
proofs) from Daily Homework from week n.
• Proofs must be typed (LaTeX preferred) & well-written.
• Email PDF using my naming convention.
• Files stored on Dropbox. Use iPad to annotate PDFs &
then email back to student.
Weekly Homework (continued)
• Submitted on a non-class day.
• Students forced to reflect on previous week’s work by
reviewing their notes from Daily Homework.
• Incorporates multiple rounds of revision.
• Graded harshly on 1-4 scale (credit: Ted Mahavier):
4 This is correct and well-written mathematics!
This is a good piece of work, yet there are some mathematical errors or
some writing errors that need addressing.
2 There is some good intuition here, but there is at least one serious flaw.
1 I don't understand this, but I see that you have worked on it.
An IBL-lite approach
• 20-50 students (NAU has larger class sizes than PSU).
• Class meets 4 days per week.
• 4 midterm exams and a cumulative final.
• 3-4 Daily Homework assignments per week (WeBWorK).
• Homework worth 15% of overall grade.
• 1 Weekly Homework assignment per week. Covers main
topics from the previous week. More challenging than Daily
• 3 class meetings are devoted to introducing new material,
either via lecture or IBL-style worksheets.
• 1 class meeting devoted to students presenting problems
from Weekly Homework. Students annotate with felt-tip
• Presenters are not graded, but 5% of grade is for
The big picture
• Academy of Inquiry-Based Learning
• Journal of Inquiry-Based Learning in Mathematics
• Small grants from AIBL
• IBL Workshop at JMM 2012