"Wow, that's amazing!"

Calculus is one of the greatest intellectual achievements in the history of mankind!  Very few other achievements rise to the level of Calculus.  Indeed, Uri Treisman said in 1992,

"The subject drips with power and beauty.  It rendered thousand-year-old questions immediately transparent.  Calculus is truly amazing.  But, how many students who take the course as freshmen look up and say, `Wow! That's amazing!'?  How often, math faculty members, have your students had that experience?"

These questions, 20 years later are still relevant.  When I talk to students about calculus, some of them like it, many are afraid of calculus, and very few have a grasp of the subject that goes beyond mere surface computations.

I emphasize emphatically that no one wants this to be the outcome.  This isn't something that people intend to happen.  But it is our reality.  Making our courses into "Wow!" experiences on an intellectual level is a challenge that is presented to the mathematics profession.

So here we are with this tremendous gift of having one of the greatest set of ideas of all time in our hands to teach to hundreds of thousands of freshmen every year.  Should we not feel in our hearts, "What a great opportunity!"?  If calculus is just a job, then there's room for improvement.

IBL Levels (with apologies to Van Hiele)

In this post I present the IBL levels, a la the Van Hiele levels, which sincere apologies to Van Hiele. These are based on my observations of IBL instructors over the past decade.

A very important point is that the IBL levels presented here are simplistic (reductionist) and are intended as a form of guidance to instructors.  The IBL levels indicate the direction in which you ought to go and provide a framework for evaluating your own teaching.

Level 0: non-IBL, teacher-centered instruction.


Level 1: some student engagement via occasional "active lecture" methods like
Think-Pair-Share, concept questions, or students working out examples.  The instructor and textbook remain as the predominant mathematical authorities, where students seek confirmation from the instructor or textbook to check if their answers are correct.


Level 2: a significant percentage of class time (50% or more) is used to engage students in solving problems, discussing solutions, and peer reviewing work.  The problem solving activities are focused on problems that students do not know the answer to or are not shown the strategy or solution method.  Students are supported and encouraged to make their own conclusions, think for themselves, and share their insights.  Students have a predilection for determining the correctness of solutions without always seeking external validation from an instructor or book.  Students have frequent opportunities to engage in rich mathematical task.


Level 3: full IBL.

While level 3 is a lofty goal to set for oneself for teaching all courses, level 2 is more easily attainable, especially for courses like freshman calculus.  Levels 2 and 3 are then goals that can be achieved by math instructors.  In particular, level 2 does not require wholesale changes in curriculum, and is thus appropriate for various contexts and environments.  It should be noted, however, that level 2 does not have the same potential as level 3 for transformative experiences.  There are always tradeoffs.

For novice IBL instructors, you do not have to wait for the "right course" to begin your journey in IBL instruction.  Level 1 can be done in any class, and allows you to test the waters and build up your own IBL teaching skills and understanding of IBL methodology.  Making the transition to level 2 would be much easier, and it sets the foundation for making the transition to full IBL instruction.

If you have been hesitating, start with level one and try out Think Pair Share!

Another look at the 'Coverage Issue'

One of the most common complaints or worries that instructors make is that there is so much material to cover.  I agree.  Many people agree.  Courses are jammed packed with far too many topics.  In some cases, this is warranted, and I am the first to agree that there exists a vast amount that students need to learn.  No one in their right might wants to teach a course with very little content.  The debate then is about the mix of course content.  This mix should be balanced with concepts, problem solving, opportunities for real growth, and of course contain an appropriate number of topics.

The coverage issue become problematic when we sacrifice understanding for the sake of getting through the long list of topics.  I'm going to use the words of college students to convey the core ideas. Keep in mind that these students were some of the best in their high schools, have taken at least precalculus, and most of them have had calculus.  Several have had some upper level courses.

• "I feel as if this was the first math class ever that I actually took the time to understand what exactly a graph was saying.  Before graphs were just lines, or parabolas, that I plotted..."
• "I'm astounded by the fact that I only did 5 actual math problems in my middle and high school careers.  I drilled procedures, and didn't really learn anything from it."
• "The idea of teaching students how to spit out answers to problems is not effective in learning.  As a learner, I would like to understand why we do math a certain way and not that it is 'just done this way.'"
• "I had never known why we 'completed the square.' I understood that completing the square produced an equivalent expression or function because whatever was added was also subtracted, but it always seemed so complicated.  For the first time [using algebra tiles] I really had a clear understanding of what was going on when I completed the square."

I re-emphasize that these are college students, who are eager to learn.  They have done algebra for years, and have not ever been allowed to explore why very basic things make sense.  They do not report learning as much as they wanted to or could have from earlier courses.  We know those courses are all "jam packed" with content from top to bottom, and that there is "no time to do anything else." Yet students are not reporting that they learned what they need to learn, which begs the question, "What is this all for then?"

Instructors often do not have much control over how much material a course must cover.  In this way, there exists a systemic issue or crisis.  Our system measures coverage in antiquated ways (i.e. a list of topics) and does not include covering "practice standards," such as problem-solving ability, communication, understanding concepts, and being able explain ideas.

To IBL instructors this poses a Teaching Minimax problem -- Minimize as much as possible unnecessary content and maximize the amount of time available for students to explore why.

More on this in future posts...

"Do You Understand?"

Perhaps the question, "Do you understand?" is the most commonly asked question by teachers.  All instructors have uttered this, but there are better ways to get at the issue.   In IBL classes students must demonstrate their understanding through careful explanations of their solutions to the whole class (and sometimes in small groups).

Asking someone if they understand can be unreliable.  One reason is that students may give the answer that they think the teacher wants to hear rather than say the through.  Another reason is because students may not be able to self-evaluate on the spot the level of their understanding.  Rather than ask if they understand, it might be more useful to ask students to complete another task that would allow them to demonstrate their level of understanding.

Of course asking "Do you understand what the homework assignment is?" is different than "Do you understand convergence?"  The point here is that if you want know if students understand a mathematical concept, then it is best to ask them to demonstrate it to you rather than just ask if they get it.

Some alternatives:

• "Now that Joe solved problem N, let's see if we can take it another step.  Here's a related problem... (write problem on the board)."
• "I would like to know if you understand this concept.  Here is a question that I'd like you to work on and answer in the next few minutes..."
• "Can you come up with conjectures that extend the problem just presented?  What else might be true (or false)?"
• "We have just started a new section, and there have been a couple of presentations on problems related to definition 25.  Here's a question about this definition I'd like you all to answer..."
• "Janis just solved problem 35.  Using a similar strategy to Janis, how would you solve this type of problem...?"

From the Mouths of Mathematicians

"The first things i learned about mathematics was that there is a hell of a difference between, on the one hand, listening to math being talked about by someone else and thinking that you are understanding, and, on the other, thinking about math and understanding it yourself and talking about it to someone else." Sarah Flannery

Reuben Hirsch from his book What Is Mathematics Really?  "Mathematics is learned by computing, by solving problems, and by conversing, more than by reading and listening."

It is fair to say that mathematicians enjoy doing mathematics.  We enjoy solving puzzles, working on ideas, making connections, and all that.  It is the so called "heart of the matter," and one of our goals as instructors is to make this happen for all our students, at whatever level they are at.

IBL Instructor Perspectives: Professor Dana Ernst

A Q&A session with Professor Dana Ernst, Department of Mathematics, Plymouth State University.  Dana  is an award-winning, highly energetic math professor, who has recently learned to teach using IBL.  In this blog post, we ask him a handful of questions related to his IBL teaching.

1. How long have you been teaching and what was your teaching style before you started using IBL?

I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1998.  My classes have always been interactive, but initially they were predominately lecture-based. Probably like most teachers, I modeled my teaching style after my favorite teachers.  I was aware of the Moore method and IBL, but I had never experienced this paradigm as a student.  By most metrics, my approach in the classroom seemed to be working.  My teaching evaluations have been consistently high and I have received several teaching awards. However, prior to implementing IBL, I was suspicious that I could get so much more out of my teaching.  More precisely, I was suspicious that I could provide my students with the opportunity to get so much more out of my classes.

2. How did you learn about IBL and when did you begin using it in your classes?

It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to consider using IBL.  Carol's workshop was about implementing a modified-Moore method approach in an undergraduate real analysis course.  I do not remember the details of the workshop, but by the end, I was inspired to give IBL a shot.  In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course.  The course I chose is called Logic, Proof, and Axiomatic Systems, which is meant to be the introduction to proof course at Plymouth State University.  Perhaps surprisingly (since it was my first go), the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student's learning and character development.  I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me.  I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well.  However, the IBL version was a vast improvement.  I have since had students from all three variations in upper-level proof-based courses and the students from the IBL version are much more independent and, in general, better proof-writers.

3. What was one of your best IBL experiences?

There are so many!  Here is one event that illustrates why I am hooked on IBL.  During the Fall 2011 semester, I was chosen for jury duty, which required me to miss six days of classes.  For all but two of these days, I was able to find a faculty member to cover my classes.  For the two days that I did not have faculty coverage, I convinced a graduate student in education to cover my IBL abstract algebra course.  This student had taken my IBL introduction to proof course, so he had some IBL experience, but he had never had an abstract algebra course.  On the days the graduate student covered for me, the class ran as usual and the students were highly productive.  They didn't need me!  The students were so proud of what they achieved while I was gone, they sent me pictures of the work that was presented on the board.  The graduate student that covered for me indicated that all he had to do was jot down who came to the board to present.

4. What advice do you have for new IBL instructors?

In order for IBL to be successful, the students have to buy into it.  To pull this off, instructor need to do some marketing at the beginning of semester.  The right amount of marketing varies from class to class and semester to semester.  For classes filled with students with prior IBL experience, I don't have to convince them of the benefits of IBL.  These students are generally ready to dive in and get started.   For classes consisting of students that are new to IBL, it is important to explicitly spell out the format, expectations, and goals of the course.  One approach I take on the first day is to ask the students what skills a college student, specifically a math major, should have upon graduating and how best to acquire these skills. Through some Socratic questioning, I am able to get them to tell me that we should be doing something like IBL. I'm not trying to trick them, but rather give them some ownership in the philosophy behind the structure of the course.

Even if the students are sold on IBL, you still have to be willing to adapt, overcome, and improvise.  Issues will come up that you couldn't have predicted.  Building a community of trust will make any challenges a lot easier to deal with.  I believe that the two most important qualities of an IBL instructor, heck any instructor, are patience and being "Mr./Mrs. Friggin' Positive." Lastly, I would like to echo something Ed Parker has said.  Sit back, shut up, and "see what they can do".


[Added by Stan Yoshinobu]  Summary: 

1. Go to a workshop if you can.
2. We can all improve our teaching!
3. We are not just teaching facts, we are developing appropriate habits of mind, such as independence
4. Marketing to students (i.e. getting them on board with an IBL class) at the beginning of term is one of the keys to success.
5. Be flexible! Be patient! Be super positive!!

Thanks Dana!

Architecture and Education Part 2

In part 1 I described an ideal situation where the physical space of a building can encourage learning.  Now I turn to a pragmatic issue -- how to make your classroom physically and mentally a good learning space. Small groups are useful in a wide variety of situations.  Although some IBL instructors forbid collaboration of the group work kind, many instructors use group work highly successfully.  Both style have merit, and what I highlight here are ways to setup a classroom, should you choose to use group work (which I recommend as a good way to get started using IBL methods).

Physically most of our college classrooms are setup with the factory model.  Students are sitting in rows.

In a room like this, the desks can be moved.  In this case you can easily ask students to move their desks into small groups of size two to four.  One suggestion, if possible, is to arrange it so that it is easy for you to walk through the classroom. Creating a boulevard in the middle allows you to walk down the boulevard and get to all the groups (e.g. ask the first two rows, and last two rows to move together) . 

If your room has fixed desks, all is not lost.  You have to ask students to work with neighboring students and get them to turn their bodies at face on another.  It's not ideal, but it works.

Don't be shy about moving people. This is part of IBL instruction -- move students to where they will be successful!  It is better to be mildly intrusive than to allow other factors to inhibit learning.  

Non-physical caveat:  The buildings, setting up groups, desks, these are all physical and can assist with developing a healthy, productive learning environment.  By themselves they are not sufficient of course.  Instructor skill, leadership, facilitation, and coaching are major drivers and can overcome almost any physical boundaries.

Ideally we would have small tables in one part of the room for small group work, and chairs in another part of the room for whole-group discussions and presentations.  I hope that future buildings are designed for class discussions, collaboration in small groups, as well as some private space for those times when students need to think by themselves.  This is doable and low cost.