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!

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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!