"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...?"