Active vs. "Active"

Not all active classrooms or activities are forms of IBL teaching.  The nature and quality of active learning experiences can differ widely among instructors.  In fact, two people saying they implement active, student-centered instruction can be as different as it gets, where one is essentially a traditional instructor and the other on the full IBL side.  For instance using handouts/worksheets in class can be implemented in a vast array of different ways from low engagement in memorizable procedures to true mathematical investigations and discoveries.  How active learning is implemented matters greatly.

We can highlight the differences between truly engaging active classes from those that have the veneer of active learning in several ways.  In this post, how instructors ask questions is highlighted as one area where differences can be seen.  The quality of the questioning by an instructor says much about the active vs "active."  Below are two examples, set in a calculus context, but the specific context can easily be from any math course.

Example 1:  Instructor A asks students, "What is the derivative of $f(x)=x^{3}$ at $x=2$?"  Pause. A student volunteers, "12"  "Good!"

In the first example, Instructor A asks a simple question that has a numerical answer.  If the answer is correctly stated by someone, the instructor may feel that the students can do the problem.  An untrained observer would think all is well and that this is an example of active learning.  The teacher asked a question and received the correct answer from a student.  All good, right?  Well, not so fast.  I think we can do better.  This is essentially a closed process.  Once the answer is given, it's time to move.  Instructor A may ask, "Are there any questions?" but that question has lost much of its value in math classes today.

Example 2:  Instructor B asks students, "Take a moment to find the derivative of $f(x)=x^{3}$ at $x=2$, and check with your neighbor… [Instructor B selects a student using some method to distribute evenly who gets called on].  Linda, would you share how you did this example?… Thank you!"

In the second example, Instructor B asks a student to explain how she went about the problem, hence encouraging a discussion of the important aspects of a computational problem.  The focus is on the process, and the answer is a component of the discussion.  Based on the response, Instructor B can restate Linda's ideas, ask follow-up questions, ask another student if he/she agrees, and so on.  It's a discussion that is open-ended.

These two examples are based on a basic, computational task.  Greater distinctions in questioning and ensuing discussions exist when topics become more sophisticated, which highlights the importance of asking good questions.

Let's go back to example 1 to see what else we can learn.  Asking a question to the class and waiting for volunteers usually leads to the situation where only a handful of students regularly contribute to the discussion.  Hence a related trap to conventional questioning strategies is that it may only encourage a few students to participate.

Another potential trap for instructor A is reluctance to give harder questions in class or at least discuss them in class, due to the fact that many questions do not have simple, succinct answers.  If instructor A seeks quick answers (for the sake of time), then it limits the level of the tasks presented to students.  While issues like time and coverage play a role, it is a limited framework for learning if the hard material is covered by the instructor and the students are left with the easy material here and there.

Let's push this idea further and consider a more involved task related to Calculus 1.  The task is "Find the equation of the tangent like to $f(x) = \sin (x)$ at $x =\frac{\pi}{4}$."  It is likely that some (many?) calculus 1 students would get stuck on some aspect of the problem, at least for a short period of time.  Such a task would not be given to students as a quick answer question, since it involves several steps, so Instructor A is more likely to demonstrate it, because "it's too hard" or "takes too long."

We know from math education research that the twin pillars are (a) deep engagement in rich mathematical tasks, and (b) opportunities for students to collaborate in some form.  Collaboration is defined broadly, where collaboration can mean anything from pairs to small groups to specialized whole group discussions (i.e. pure Moore Method presentations.)   Instructor A achieves neither of these goals through the style of questioning employed.

Engaging students requires a different skill set compared to making clear, lucid presentations, and starts with effective questioning.  Here are some examples.

• Could you explain how you got the answer?
• Could you explain how you thought about the problem?
• Talk to a neighbor for a moment and come up with two comments or questions.
• Could you explain Linda's strategy to your neighbor/the class?
• Consider alternative strategy ABC, and determine if it will work on the previous problem?  Think for a moment, and feel free to discuss it with someone...  Now I'm going ask a few students to share their thoughts…

How we ask questions is critically important.  Questions are moments in a class when students are invited to be participants.  Starting with engaging, open questions is a positive first step towards full IBL instruction, and using engaging questions regularly provides opportunities for new IBL instructors to practice critically important questioning and listening/observational skills.  Even if you are just dipping your toes in the IBL waters, you can try these techniques right now with low risk and minimal investment.

Instructors can self assess quickly, by reflecting on what kinds of questions they ask and how these questions are implemented.  Such self assessment is useful and can lead to big changes in the positive direction.