Friday, November 14, 2014

Instructor Voices Part 4: "What Makes a Good IBL Instructor?" (Video)

Here's the fourth episode in the Instructor Voices series on the AIBL channel.   This time just a quick take on what a handful of IBL instructors think about important attributes or skills of an effective teacher.

I'll add on to their good comments by saying that these aren't a complete list.  What goes into becoming a solid IBL teacher isn't the size of a sound bite.  If you're interested in taking the next steps, please consider attending the IBL Workshop in July 2015. Registration opens in January.

Careful listening, providing appropriate structure for exchanging ideas, restraint, patience, understanding the learning process, and compassionate joy.

Friday, November 7, 2014

Instructor Voices Part 3: Importance of Failure (Video)

In an earlier posts, I wrote about de-stigmatizing mistake.  Some of us are now  doing things (when appropriate) in our classes now to raise mistakes to their proper place in the learning (Productive Failure).  This post is a continuation of this theme.

Here's another short video of IBL instructors sharing their thoughts on the value of mistakes or getting stuck.  Thanks again to the interviewees for sharing their beautiful insights!

Tuesday, November 4, 2014

Instructor Voices Part 2: "Benefits to Students" (Video)

What is IBL's biggest benefit to students?  A groups of IBL instructors give us their perspectives.

Thursday, October 30, 2014

Instructor Voices Part 1: Your Most Memorable IBL Experience (Video)

At the RLM and IBL Conference last June, we asked a handful of IBL instructors to tell us about one of their most memorable IBL experience.  Here they are in their own words.

What is one of your most memorable IBL experience?

Wednesday, October 8, 2014

Other Blogs Strategies for Grouping Students, Student Resistance, Teaching Math Through Immersion,...

Some useful links from recent posts in the iBL blogosphere:

Strategies for Grouping Students by Chriss von Renesse is a great post on how to use grouping to develop an environment to increase student success.

Matthew Jones has a nice post on strategies for dealing with student resistance.  The shift from traditional, passive learning to active engagement can be hard.  Students can say, "I don't learn this way."  Matt has a nice post on how to handle these situations.

AMS blogs have really been interesting to follow.  Here's a nice post by Priscilla Bremser on Teaching Mathematics Through Immersion.

Sunday, September 28, 2014

Dynamics of Misinformation in Education

A post about a broader topic, not just about IBL...

Misconceptions about IBL (or CCSS or other education efforts in general) are in part due to what I call "selective reporting."  When there's news about evidence that points towards using active, student-centered methods of teaching, it rarely receives much fanfare, so most people miss it.  On the other hand, if school district or academic institution is struggling to implement effective teaching, then the story makes it to mainstream news, creating a bias.  Put another way, the problem is that the full picture isn't being reported in a reasonable way, so the takeaway message is distorted one.

Here's an analogy that gets across the depth of the difference.  In photography, cropping is a way to change or enhance an image.  What we exclude can significantly alter the impact of an image.  

Picture #1.  The subject is a boy who appears to be pensive or annoyed.  Presented by itself the image leaves the viewer with a very specific interpretation.  It's definitely not about playful, youthful themes.

Picture #2:  What was really going on?  The subject was engaged in a variety of silly poses and only pretended to be annoyed in the image above.  Taking a look at the frames below shows the broader story, and a completely different perspective of what is happening.

In a similar way, news is reported in bits and pieces about education that leaves the broader story out of the picture.  An oversimplification of education is presented below just to get the point across.  The diagram is not how I actually think about the education system, but I think it's good enough for these purposes for a "back-of-the-envelop calculation."  

Let's assume the main groups of topics in education reform are listed in the diagram below.

What happens is that the media employs selective reporting, where the emphasis is on the problems with say implementation or highlighting a small subgroup's opinions disproportionately more than their earned merit (other stuff).  So to the non-expert it's easy to make incorrect/limited conclusions.

For instance with respect to CCSS, the media emphasis has been on implementation struggles.  The public then could be swayed to think the entire CCSS idea is flawed, as opposed to seeing the problem for what it is -- early struggles with the transition.  Reporting rarely (if ever?) asks natural follow-up questions or provides a broader view to put the issues into context.  It's especially unfortunate, because education is a complex, long-term issue.  Therefore, context is fundamentally important to understanding what is going on in education, and context and framing is just what is being excluded by media reporting.

The math profession isn't entirely free of this.  The AMS published somewhat recently a Doceamus article in the AMS Notices, where a narrow study on "worked examples" was extrapolated to imply that constructivist and minimally-guided approaches were invalid.   This article made the rounds, but the article by Freeman et al, based on a metaanalysis of 225 research articles from STEM disciplines appeared to get less fanfare, despite being categorically a vastly stronger body of work in scale, quality, and value to society.

The dynamics of misinformation is subtle, because it's hard to know about things you don't know about already.  Undoing misconceptions is harder than informing people right the first time, so it's a problem that can snowball and lead to unproductive or harmful resolution.

The misinformation gap is here and is real.

Monday, September 15, 2014

Is Teaching = Art? Is Reform = New Curricula?

Founds this entertaining and well-written essay by Peter Taylor via my twitter feed recently.  I first want to say that I agree almost entirely with what Peter Taylor has to say, especially the call for doing more "artistic" math and letting go of doing it all for the students.  It's a good read and I recommend it.

There are two points in Peter Taylor's essay, however, that I find incomplete, which ultimately affects how we go about transforming our educational system.  I'd like to expand on those here and encourage others to think about these two points further.

Is teaching = art?  It's been debated, but in my opinion the answer is no.  Effective teaching has aspects of art (more artisan like) that require creativity, but teaching is teaching.  It is its own thing, and making analogies to other disciplines can help us make sense of it, but taken too far diminishes the unique activities and mindset required in the teaching profession. While teaching is a creative endeavor, it also requires mentoring, organization, managing young learners, learning outcomes, learning goals, assessment, etc. 

Style matters to an extent in teaching, but learning outcomes matter more.  Much more.  I can't just teach my style and not think about the impact on my students.  I'm not saying Peter Taylor says this.  He doesn't.  But that's the way "teaching is art" gets interpreted by some people.  Artists have a freedom of expression and freedom of intention.  Their work can be interpreted in many ways by viewers, who see things through their own lenses and experiences.  Teaching on the other hand is very intentional work with real outcomes that matter to young people.

Math is beautiful to mathematicians, but math is not art, just because it is beautiful.  Yes it requires similar kinds of dedication, creativity, and expressing our ideas in clever ways.  But that doesn't make math an art form, unless we really relax the definition of art. 

Teaching $\neq$ art.  

Now this all sounds like academic banter.  I bring this issue up, because I think it matters in the real world.  The danger in thinking that teaching is art is that some people mistakenly take that to mean it's primarily about personal teaching style, which can disconnect the act of teaching from learning.

Teaching as a profession is more akin to medical practice.  Medicine uses science and problem solving in creative ways with the goal of improving health.  Likewise, teaching can be studied, it relies on knowledge of the academic disciplines, with the goal of improving learning and thinking.   We can study teaching, find better ways to teach through scientific research, and so on.   It's better to then to think of teaching as a unique profession.

Teaching is also fundamentally a system and a cultural activity, which brings us to the next point regarding curricula and reform.   Many reformers in the past have been tempted by the Sirens of "Reform = Curriculum Change."  Many reform efforts have crashed on the rocks of model courses and innovative curricula as complete solutions to reform.    I emphasize that good curricula is absolutely necessary for authentic reform.  Updated curricula alone, however, is not sufficient.   If all it took was good curricula to make our teaching system change, then reformed would have occurred already decades ago.   So why is curricula insufficient?

Math classes have their own distinct culture and history.  When students and teachers walk into math class, there is an expectation of what is going to happen and what math is.   The teacher shows.  Students mimic.  Students practice quietly to get the same answer as the teacher in the same way as the teacher.  Then it's on to the next topic.  That's the standard culture, where the assumptions for our (K-12) system have roots back in the industrial revolution, when the focus was on learning algorithms and facts for the factory workers, and a few elite would rise up to run the empire (by attending college).  The challenge is to shift this culture to a new model, and that has been a challenge that has resisted change for centuries.  The entire culture of teaching and learning that has to be shifted, and that is why changing curricula is insufficient.  Necessary, yes.  But insufficient.

What we somehow need to figure out is to change our teaching culture and teaching system.  It's a big, complex problem, where the core subproblem is implementation of active, student-centered teaching on a broad scale.

Do we have full answers to the big problem?  No, not yet. 

Can we do this?  Yes.  But my guess is that it's going to take creative problem solving, coordinated efforts by large, organized groups of people, and about a generation of time.   Hopefully we can bet on exponential growth, and see changes sooner. 

I have had the fortune of seeing Picasso's Guernica in person that Peter Taylor mentions. It's absolutely a great work of art with meaning and power that transcends time and culture.   All subjects should inspire people to equivalent stature.  Picasso once said that all children artists, and that the problem is to remain an artist when one grows up.  Likewise I believe all children are mathematicians.  The problem is to remain a mathematician when one grows up!

Upward and onward!

Thursday, September 11, 2014

Learning to Ride a Bike

More light-hearted posts here.  Thanks Paul Harper for starting this thread... Paul Harper recently posted this 7-minute video about active learning, using the context of his daughter learning to ride a bike.

In IBL land a central idea is creating a sequence of problems, and letting students experience the learning process through guided discovery.  The problems are spaced so that students have opportunities for authentic ownership of the mathematics and intellectual (and personal) growth.  In the spirit of showing our kids learning to ride, here's an image of my son, Hutch, working on a lemma (i.e. riding a balance bike with no pedals).  After he learned how to balance himself, he got on a bike and rode off.  All he needed to learn was how to use the brakes.  And you didn't need to tell him he got the right answer.  Q.E.D.

Monday, August 18, 2014

Julian Fleron, Phillip Hotchkiss "Inquiry-Based Learning to Engage and Empower the Disfranchised"

Julian Fleron and Phillip Hotchkiss, Westfield State University presented at the 17th Annual RLM and IBL Conference Co-Hosted by MAA, EAF and AIBL.  The conference was held in Denver Colorado, June 2014, and their talk has just been posted on the AIBL YouTube Channel.   This talk shows some practical ways to get going with students who typically have math anxiety and transform the course into a fun, engaging learning environment.

"Inquiry-Based Learning to Engage and Empower the Disfranchised"

I also recommend checking out their website on teaching courses for Liberal Arts Students at Discovering the Art of Mathematics.

Friday, August 15, 2014

IBL Poster Session at MathFest 2014

Some images from Portland.  The IBL poster session was held in the exhibit hall, and the turnout was steady and engaged.  Thank you Angie Hodge and Dana Ernst for organizing this session, and an especially big thank you to all those who presented posters!

Poster sessions are a great way to involve people in discussions on a particular theme.  Attendees can interact with presenters in a conversation.  I like both poster sessions and presentation, and perhaps there will be ways to mix the two strategies into a "mixed media" format in the future?  Just a thought.

Some images from the IBL Best Practices Poster Session, MathFest, Portland OR 2014.

Sunday, August 10, 2014

AIBL Booth at MathFest 2014

What is AIBL?  "AIBL is the organizational front to an existing community" says TJ Hitchman, University of Northern Iowa.   At MathFest 2014, the IBL community organized a booth in the  exhibit hall.

The co-organizers of the booth are Angie Hodge, University of Nebraska Omaha, and Dana Ernst, Northern Arizona University.  They asked IBLers to hold "IBL Office Hours."  These wonderful volunteers spent part of their busy conference schedule at the booth to talk to attendees interested in learning more about IBL.  A big thank you to...

  • Angie and Dana,
  • TJ Hitchman
  • Victor Piercey
  • Brian Katz
  • Elizabeth Thoren
  • Ron Taylor
  • William Lindsey
  • Melissa Lindsey
  • Susan Crook
  • Natalie La Rosa

In the age of top-down, centralized command-and-control reform efforts, based on model courses, external incentives, and penalties for non-compliance, we take a fundamentally different approach.  We take the bottom-up view, where working with individuals and supporting them to solve their own specific implementation challenges is the core.  Through intensive workshops, mentoring, small grants, and visiting speakers, we help individuals and small groups grow their IBL skills and practices, and cultivate a culture of learning at their institutions.  Over time we believe that this community will be more durable, more sustainable, and ultimately impact more students' lives.

One community. Infinite possibilities!

Monday, June 30, 2014

Productive Failure (#PF)

This spring quarter I taught Math 423, a course for future secondary math teachers.  This course is often called a "capstone" course and is intended as an advanced look into the secondary curriculum.  It's a hybrid course in that it is a math course, but it also has the goal of transitioning math majors from being a student to being ready to enter a credential program.  Put more simply, it's a transition course from being a math student to being a math teacher.

One of the main themes of the course was productive failure.  In an earlier post De-stigmatizing Mistakes, I wrote about how Ed Burger makes productive failure part of the course.  So I did the same for Math 423!  Five percent of the grade was based on sharing productive failure.  Students were required to share at least twice during the quarter a mistake that they learned from.  These mistakes could be natural or could be intentional (as in a strategy like trial and error).  

The results were better than I had anticipated!  Students felt as future teachers they needed to learn this lesson about the value of productive failure.  They felt a sense that everyone makes mistakes and that we can all learn from them and others can learn from them, if we share our newfound insights.  We de-stigmatized mistakes in our little segment of society, and it felt right and good.

One student writes in an portfolio assignment:
One of the biggest themes that I will carry not only into my teaching career but in my life is the idea of productive failure. Failure is given the stigma of being negative and until I came to this class I believed that. After going through this class my thoughts on failure have completely changed. I never thought of failure as a device that can enhance learning and ideas. Every day, watching everyone present their productive failure I noticed how no matter how small the failure was someone learned something. It not only taught us how not to do something, but the right way to think about certain problems and common misconceptions that can help you better adjust your lessons as a teacher. Failure is a part of life and should be embraced and not chastised. By
giving failure in learning such a negative connotation you can inhibit students from good learning habits and for a love of school. I believe that failure should be considered productive and embraced in classrooms all around the world.  -- Jordy Adamski, Cal Poly Math Graduate
Experiences like this are some of the major reasons why I spent so much of my time thinking about improving teaching and improving the system.  When it works, it's wonderful!    When we say the classes are more fun to teach and students get more out of it, it's hard to communicate the impact.  Some might think that the C student moves up to a B, but that captures little of the real transformation that occurs in some the hearts and minds of our students.  Changing one's entire outlook on mistakes and how that might impact that student's math teaching practices in the future is a tremendous change!

An important point to mention is that productive failure fits naturally into an IBL framework.  Productive failure can easily be included in the course grade, since IBL courses already have the active, student-centered dynamic that can easily accommodate short student presentations on productive failure.  On the other hand, a lecture-based course normally does not have the comfort level and student buy-in that would allow students to open up and expose themselves by sharing their latest and greatest mistakes.   Hence, it is emphasized that the teaching system used is fundamental and that adding low-cost, high-impact strategies, like productive failure, should be done within a broader framework that supports it.

Throughout the term we used hashtags.  We labeled productive failure with #PF, which made class more fun and also elevated productive failure to it's rightful, dignified place in the learning process.  #PF showed up all over the place throughout the course, and I hope it finds it way into your classes, too.


The #PF Crew

Sunday, June 22, 2014

Legacy of R. L. Moore and IBL Conference 2014

Denver, CO hosted the 17th annual Legacy of R. L. Moore and IBL Conference, co-hosted by the Educational Advancement Foundation, The Mathematical Association of America, and the Academy of Inquiry Based Learning.  The theme: Engaging in IBL.  More than 80 presenters, over 200 participants over 2 days.

If you missed the conference or were not able to attend a parallel session for one reason or another, videos of all of the sessions will be available by the fall of 2014 on the AIBL YouTube Channel.

Many thanks to Harry Lucas, Jr., Norma Flores, Albert Lewis, Fain Brock, Judy Diaz and everyone else at EAF for their tireless efforts to setup the conference.    Thank you to Angie Hodge and TJ Hitchman and the conference organizing committee for putting together a wonderful program.

A big thank you to all participants!  Your contributions made it a special event!

Some images from the conference (more to follow)...

Saturday, May 17, 2014

Data Points Toward Active Learning

NSF recently posted a summary findings from a study by Freeman et al (Link to NSF News Release "Enough with the Lecturing")    The original article appears in the Proceedings of the National Academy of the Sciences and is a meta-analysis of research from STEM fields.

 I have been saying for a few years that "All the vectors are pointing in the same direction."   The data continues to accumulate and the preponderance on evidence suggests that we should be engaging our students actively in high-quality tasks.

No guilt or shame.  We're all in this together, and I believe everyone who works hard at teaching has good intentions and the best interests of students at heart.  The perspective I like to take is one that is used in medicine.  When new techniques or treatments are shown through evidence to provide better care for patients, then the medical community adopts those new practices.  Likewise we can do the same in teaching.  We study, we collect data, we learn from our efforts, and we put those things into our classrooms. It's the logical thing to do.

Implementation is where the biggest challenge is.  Implementing IBL methods is challenging and takes a significant effort initially.   The estimates we have available are that it takes about 100 hours for a new instructor to get started, and several hundred additional hours to build the necessary expertise.  This is based on my experiences running weeklong IBL workshops and organizing the AIBL Mentor Program.   But the skills learned stick, and once instructors get acclimated to IBL methods, they tend to continue using them.   At the moment professional organizations including (but not limited to) the MAA and AIBL are the platforms for implementing these changes, as they focus on instructors and supporting them through the process of developing teaching skills and practices.

The evidence grows each day in support of active, student-centered instruction. If you feel the call to take action, come join the Academy of Inquiry Based Learning and get started!

Upward and onward!

Thursday, May 8, 2014

Student Testimonial: Alfred and Diana

I am happy to share an interview of Alfred and Diana, two math majors at CSU Monterey Bay.  This interview was filmed in April 2014, by Kaylene Wakeman, AIBL and Cal Poly.  Their instructor is Professor Rachel Esselstein, Department of Mathematics CSU Monterey Bay.

Transformation is a term we use in the IBL community.  The word, transformation, truly is appropriate.  It's hard to convey the experience of teaching via IBL through data or talking about it in a presentation. The experience of working with students, seeing them grow, believe in themselves more and more each day, discovering that they can be movers and doers.  It's special.  This interview captures some of that magic.

Tuesday, April 8, 2014

Frenkel, Bressoud, Brights Spots, and the Implementation Era

David Bressoud recently wrote a post Age is Not the Problem in his MAA Blog Launchings.  There are several topics in his post in his response to Edward Frenkel's Op-Ed piece in the LATimes.

One topic I want to emphasize is that teaching and implementation aspect of the pieces.  There is this sense that the education community needs to wake up and get its act together, and this notion comes up implicitly in Frenckel's piece (and he may or may not have a strong opinion about this topic or have intended something by it).   Readers of his piece, however, may pick up on it, so I think this is a good opportunity to highlight the bright spots.

There are literally thousands of us who are working on implementing high quality, inquiry-based, student-centered methods of instruction that incorporates what we have learned from education research and experience.  Many of us in the community have heard the calls for change and are doing something about this.   There are people who are devoting their careers to address the majors issues in mathematics education.  We have bright spots to celebrate and to embrace as pillars for building real, long-term solutions.

Pointing fingers at "bad teachers" or "bad textbooks" or whatever else is one way to deal with education system issues, but it lacks a constructive outcome.  Truly great nations or communities go much, much further.  They look honestly at the problems, they evaluate and think about the evidence available, and they forge alliances and build systems that provide opportunities for the stakeholders to make good decisions and to do the long, hard job of building solutions to complex, long-term problems.  It's a big job to change a cultural activity like teaching.

Hence, this is the implementation era!  Implementation is a major if not the major challenge we face in education.  We have enough good ideas about how to teach effectively now.  It is worthwhile now to expand efforts and get these methods into our classrooms.  Are these methods perfect? No.  Do we need to do more work on improving our methods? Yes.  But we know enough that it's time to move so we can make differences in the lives of students today.  That's the implementation challenge!  Good ideas are on our shelves.  Lots of good ideas!  Now how do we get those good ideas into the classrooms implemented at a high level across the nation, globe?

If you're interested in engaging in this kind of work, please join the IBL community and AIBL (or NCTM or whatever  appropriate professional society for your area).   AIBL's mission is to help math instructors implement at a high level what we have learned.  We don't just talk about the issues, we implement them in our classrooms right now.   We can do more than point fingers and lament and complain.  We can take action and be movers and changers, and you're all invited.

Upward and onward!

Find AIBL at
Dana Ernst has started a G+ community

Saturday, March 22, 2014

Learning Zone Analysis Part 2: Evaluating Math Content

This is part 2 (of 3) of the Learning Zone Analysis (LZA) idea.  LZA Part 1 discusses how one can choose on a macro level the teaching methodology best suited to the specific goal.  In this post, I'll discuss how I use LZA to take apart a unit and use it to guide how I might construct problems.

Let's return to the integers unit for grade 6 (IBL Integers Unit).  The actual context doesn't really matter so much as the framework presented here, and as before I want to capture a wider audience.

We start with a traditional rote explanation of subtracting a number in traditional math settings.  I sometimes see subtracting an integer as (a) change the sign ($-(-1) = +1$), (b) remember to move right on the number line.  I personally experienced (a), when I was a student.  I was told when you see two minuses, you change it to a plus.   So I learned how to get to an answer, without having to learn the concepts that make things work.

I think it's easy to say that most of us agree that merely doing the computation in the ways presented above are a limited and not ultimately beneficial to students without a broader understanding of integers.  The IBL Integers Unit uses a context, includes a model for thinking, introduces zero pairs and mathematical equivalence, and requires students to write a justification why subtracting a negative is equivalent to adding.

Let's break things down...

LZA of the Traditional Integers content
  1. Computing how to subtract integers, skills practice
  2. A connection to the number line, but perhaps without conceptual grounding
LZA of the IBL Integers Unit
  1. Computing how to subtract integers, skills practice
  2. Context for problem solving
  3. Modeling numbers and equivalence (zero pair)
  4. Problem solving
  5. Argumentation and justification
It's immediately obvious the difference in the list.  One misconception in the general public is that the new teaching vs. old teaching is about style and that they are assumed to have the same goals and achieve the same ends.  It's clear that the goals are different, and that one is more sophisticated than the other.  Moreover, both instructors can say, "I covered integers."  The nature of the coverage is vastly different, and while one got through it faster, I'd like to say, "So what?"  What real math was learned if all we achieved is answer getting.

Once again cultivating dispositions is done more appropriately in the IBL setting than the traditional setting.  One can risk saying that a big missing piece in the general discussion about education reform is the difference in what the point of education is.  It makes me wonder if unacknowledged differences in "education axioms" may be a significant contributor to the friction in public discourse.

Another point worth mentioning is that there is an interaction between the method of teaching, teaching philosophy, and the content.  When we think of students as explorers and doers of mathematics, then we are more inclined to present to them tasks that are a different nature than if our view of teaching is focused on skills acquisition (or passing standardized tests).  So teaching isn't just a method.  It's a system.   What we value is important in education, our methods, how we assess, what we assess, our beliefs about what students are capable (and not capable of doing), and the goals of education all feed into what happens in the classroom.

One can argue that it is the case that one can lecture on concepts and conceptual understanding.  So the traditional content can be expanded to some degree.  I point out that the teacher explaining a concept is not equivalent to students actually demonstrating their conceptual understanding through a presentation or written work.  How content is covered and how students engage in it are important, intertwined factors.

A highly useful application of LZA is to use it when you're teaching out of a textbook.  An instructor can look at a section and make a quick list of the content and dispositions that students are likely to engage in.  Then using this list, an instructor will know the strengths and weaknesses of a unit, and fill the "gaps" appropriately.  Knowing students are good/not good at certain dispositions can also add valuable data for the instructor to consider.  When we say, "My students normally are not good at explaining/solving...," then there exists a set of tasks or problems that should be deployed.

Short story: Get your content.  Use LZA.  List what's there and not there.  Adjust.  Win!

Upward and onward!

Tuesday, March 11, 2014

MAA PREP IBL Workshops Summer 2014

Hello IBL Community!

This is a quick reminder that AIBL is offering two IBL Workshops under the MAA PREP umbrella.  Information about our workshops is available at  These workshops are for college math instructors, and early-career faculty are especially encouraged to register.

Registration for the workshop is handled by MAA through their registration portal

We hope to see you this summer!

Wednesday, February 26, 2014

IBL Best Practices Poster Session, MathFest 2014

Hello IBLers!  Please consider presenting a poster at the IBL Best Practices Poster Session, at MathFest  2014.  Poster sessions are a great way to interact with people directly who are interested in similar courses or ideas.   Please join us, share your ideas, and contribute to the IBL community!

This poster session is co-organized by
Angie Hodge, University of Nebraska Omaha, AIBL Special Projects Coordinator
Dana Ernst, Northern Arizona University, AIBL Special Projects Coordinator
Stan Yoshinobu, Cal Poly San Luis Obispo, Director of AIBL

Friday, February 21, 2014

Learning Zone Analysis Part 1: Dispositions and Skills

How do you know when to use a specific teaching method or technique?  This is a question that all teachers deal with, and I believe that a general tool for sorting some of this out can be very helpful.  One idea I have been working on is a framework called "Learning Zone Analysis" or LZA for short.   In this post, I'll discuss one aspect of LZA, which is useful for deciding when to use active learning and when one can get away with a mini lecture or flipping a topic outside of class.

Zone 1 contains dispositions.  Dispositions include (but not limited to) problem-solving ability, learning to read and write proofs, positive attitudes about mathematics, being willing to experiment, searching for counterexamples, advanced techniques, communicating ideas, utilizing effective practices in the study of mathematics.  

Zone 2 contains basic skills, factual knowledge, connecting Math to other subjects (or other disciplines within Math), motivation, organizing information or a unit of work that students have just presented proofs on, etc.

LZA can be represented in a diagram:

For Zone 1, it can be argued that it is most appropriate to use active, student-centered methods, such as IBL.  Zone 1 is about dispositions, habits of mind, and cultivating higher-level skills.  Such dispositions must be developed by students for themselves through sustained practice and reflection in a supportive environment.  Dispositions cannot be learned by listening to others, and this is fundamentally why actively solving challenging problems is necessary.

Zone 2 can be effectively and efficiently covered through lectures or mini lectures.  Learning about where your office is shouldn't be a problem-solving experience.  Similarly, students could learn that Fourier Series can be applied to signal processing on their own, but it's much more motivating and useful if the instructor presents a succinct, clear exposition of the connections, providing value and motivation.  Further I can envision setting the context of a unit, what students are responsible for learning outside of class, students' roles, and and should be done via direct instruction.

Motivation actually exists in both zone 1 and zone 2.  In zone 2, the instructor can give explicit motivation for mathematical concepts.  A different kind of motivation can be addressed by the instructor in the form of encouragement and praise.  Encouragement and praise should be regular and clearly positive.

Motivation in Zone 1 is tacit.  It is through individual successes over long time periods that students become ever more confident and motivated to learn mathematics.   It is also arguable the the motivation from being successful at solving hard math problems is more authentic and long lasting compared to pep talks.   Motivation from mentoring or coaching and from success are both necessary.

How does this all work in the practical world?  For a specific topic, list the goals of the lesson(s) into zone 1 and zone 2.  Then select IBL or teacher-centered to cover each zone.  A rule of thumb is 75% IBL and 25% teacher-centered is a good place to start, with variation class-to-class to suit the specific mathematical landscape and how students are getting on with the material.

There exist other ways to use LZA.  We could evaluate lessons or curricula to see how much higher-level thinking vs. factual or skills knowledge is present.  LZA can also be used in class observations to measure how much of the visible work is in zone 1 or 2, and the relative effectiveness of lecture vs. IBL.  More on these other uses in future posts.

Friday, February 7, 2014

Student Testimonial: Nora Ortega

Nora Ortega is a math major in the teaching option at Cal Poly San Luis Obispo.  Nora has taken several IBL Math courses, including Intro to Proofs (Math 248), Euclidean Geometry (Math 442), and Modern Geometry (Math 443).  Nora intends to become a high school math teacher.

One of my favorite parts of this video starts at around 6:15, when Nora is asked about the impact her experiences in IBL classes have had on her intended career choice.  Nora discusses how seeing her instructors take a risk has left an impression upon her to do more.


Saturday, February 1, 2014

IBL Workshops in 2014

AIBL is offering two IBL Workshops in summer 2014 for professors and instructors of undergraduate mathematics courses.  The 4-day workshops are built around hands-on, interactive sessions focused on the skills, practices, and concepts necessary for successful implementation of IBL methods.  Participants are supported for one calendar year via a follow-up mentoring program, and invited into in the IBL community.

Workshop 1 will be hosted at Kenyon College, Gambier OH, June 23-26, 2014.
Workshop 2 is a pre-MathFest workshop in Portland, OR, August 3-6, 2014.

More information is available on the IBL Workshop website.

Both workshops have identical programs and are designated MAA PREP workshops.  The workshops are funded with generous support from the National Science Foundation, The Educational Advancement Foundation, and the Academy of Inquiry Based Learning.

Tuesday, January 28, 2014

Quick Start Guide to IBL Teaching

Disclaimer:  This is just a quick start guide, not a manual, and getting started in IBL is hard work. Please visit the IBL community to learn more at

Step 1: Determine the overarching goals of the course, and use these big goals to help with decisions you have to make.  For example, when teaching Real Analysis 1, one of the big goals of the course is to understand deeply the fundamental concept of convergence. 

Step 2: Find/Adapt a sequence of problems.  Some course materials are available from  Another option is to build your own set of problems, using a textbook as a guide for how to sequence the problems. 

Step 3: Understand your role as an instructor.  IBL instructors select the math problems, make assignments, select presenters, moderate and facilitate discussions, use group work appropriately, mentor, pose sub problems or special cases when the class is stuck, ensure all students are deeply engaged.

Step 4: Marketing what and why IBL to students and colleagues.  Marketing is used in the best sense of the word here.  Students and colleagues need expectations reset, especially if IBL is not commonly used at your institution.  Student buy-in is critical, especially at the beginning of the term.  Because teaching and learning are cultural activities, there exists a set of default, often unconscious assumptions of what math is, what teaching math is, and what students are “supposed” to do in a math class.  Based on the “distance from IBL” your students are, use the appropriate amount of regular, ongoing marketing and sign posting of tasks.

Step 5: Pick a rubric for grading presentations and homework.  Here’s one example.
            4 points = correct
            3 points = mostly correct, except for technical or clarity issues
            2 points = there exists a logical flaw
            1 points = went to the board

Allowing students to pause and return later once is a regularly practiced by IBLers.   Bonus points for productive failure are awarded by some IBL instructors.  A score equal to a 1 is rare.  Other rubrics exist. This is shared as a common choice.

Step 6: Keep a course diary with notes from class and thoughts as you prepare, grade, and reflect on your teaching.

Step 7: Use a spreadsheet with names in the rows and problem numbers across the top.  Use this spreadsheet to keep track of who has presented.   Separately, keep another spreadsheet with the names in the rows and days of the term across the top. This second sheet is used to keep track of participation, and helps you make data-driven decisions regarding who to call on and how to build groups that work effectively.

Wednesday, January 22, 2014

Destigmatizing Mistakes

Ed Burger, President of Southwestern University and mathematician, presented in Mike Starbird's session on Mathematics and Effective Thinking.  One of Ed Burger's main points is that we must raise mistakes to their proper level of dignity and value in the learning process.

Mistakes are stigmatized in mathematics.  Timed tests, answer-math curricula, questions from instructors that start with "What is the answer…?" all feed the perception that we are to do math right the first time, and any deviation from the correct path may be discouraged or labeled as wrong or inefficient.

While the quickest path might allow us to get through the material in class quicker in time, silent and lasting consequences sometimes take root.  Being reluctant to experiment and think for oneself is one of these consequences, resulting in underdeveloped problem-solving skills, reliance upon external validation of solutions, and for some students a damaging self-image in math that is exposed by the sentence, "I'm no good at word problems."

What does Ed Burger do about this?
  1. 5% of course grades are based on the quality of student failure.  In order to get an A in the class students must fail productively.
  2. Use specific activities to teach intentional mistake making as a positive strategy.  An example of this strategy is to give a problem to students to work on.  Their first task is to intentionally do the problem incorrectly and then share their mistake with a neighbor.  Often learning why a strategy is wrong leads to deeper insights, and getting used to making mistakes forms a habit of mind in the spirit of experimentation.  
  3. A related strategy is for the instructor to present an incorrect proof/solution, and then have the students discuss what insights we can learn from it.
Productive mistakes and experimentation are necessary ingredients of curiosity and creativity.  A person cannot develop dispositions to seek new ideas and create new ways of thinking without being willing to make mistakes and experiment.  Instructors can provide frequent, engaging in-class activities that dispel negative connotations of mistakes, and simultaneously elevate them to their rightful place as a necessary component in the process of learning.

Wednesday, January 8, 2014

Joint Mathematics Meetings IBL Sessions and Events

A list of IBL related talks and events are listed on the AIBL JMM 2014 page.  Follow this link IBL Sessions and Events at JMM2014   I look forward to seeing some of you in Baltimore next week!

Update:  See also the post by Math Ed Matters (Ernst + Hodge)

Friday, January 3, 2014

AIBL Special Projects Coordinators

SY,  Dana Ernst, and Angie Hodge
photo by Kirk Tuck

Dana Ernst, Angie Hodge, and I clearly have different, complementary skills.  As you can see, I don't like to talk much.  I must be a quiet, bookish type.  Dana is always riding his bike, so he can't hear what you say, and Angie would rather not see another ultra marathon advertisement, since she might sign up for it even though her schedule is full.

Dana and Angie are AIBL Special Project Coordinators.  They are working with AIBL to help disseminate IBL methods and support new IBLers, they are co-bloggers on Math Ed Matters, and they super supportive!  I am looking forward to working with them again in 2014!

Please visit our website to learn more about The Academy of Inquiry Based Learning

Wednesday, January 1, 2014

Happy New Year and Aim High!

Best wishes to all for a happy and successful 2014!  I'll share an oft used and famous quote, because I think it pertinent in our profession.
"The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low, and achieving our mark." -- Michelangelo
The arc of learning is long, but bends towards the bold and hard working.