## Monday, November 25, 2013

### IBL is Fun!

Something that doesn't get talked about enough is how much fun it is to teach via IBL, when things are working well.  At workshops and conferences we focus on the benefits to students and the benefits to learning.  We bring out research papers from the education research literature, and discuss the benefits of active, student-centered instruction.  All these are appropriate and clearly the right thing to do, since there is evidence from literature on K-12 math, K-12 science, K-college non-STEM subjects, research on how people learn, and undergraduate STEM fields.  The evidence from a variety of sources, across subjects, through a long time span, and across education levels points in the same direction.  That's the argument from the scientific point of view.  But is it fun?  The answer is undoubtedly yes!

One aspects of IBL teaching that gets overlooked in our efforts to be careful and scientific is that at a basic level IBL courses are fun teach (when it's working well).  You get to be the mentor in the middle.  Students, who are initially tentative start to open up.  They do more, they show more, and then there are those special moments when you see students realize that they can be players in the game.  They enter the fray, respond, and contribute.

One of the great experiences as a teacher is working with students as a team with the goals of becoming better, smarter, and more knowledgable.  That's a wonderful process to be a part of.  Working with the students,... coaching, mentoring, celebrating the successes, getting through the brick walls, and seeing the development over time is highly rewarding.   The growth mindset is something you can see, and when the growth-mindset light bulb turns on, and you were there to be a part of it… that's magic! That's what makes it worth all those nameless late Tuesdays nights, grading, mulling, thinking of what to do next...  We might ask what this is all for?  And then you see the results, stand back, and smile.

IBL is fun!

Note: We'll have more posts in the future showcasing the transformative experiences students have had.  These experiences have not be capture in research papers yet, and we are collecting videos of students who can share their experiences in IBL classes.  These videos hit on "IBL is fun" and how rewarding it is professionally to be able to create a classroom culture, where such things are possible.

## Monday, November 18, 2013

### 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.

## Friday, November 8, 2013

### A Characteristic of IBL Teaching: Mediating the Interaction Between Students and Mathematics

A question that comes up frequently is, "What are some of the main features of IBL teaching?"  One of the main characteristics of IBL teaching that I'd like to highlight in this post is mediating the interaction between students and mathematics.

Education is full of monikers for active, student-centered teaching.  We say things like, "Guide on the side," "Mentor in the middle," "Coach," etc.  These are nice ways to think about the nature of IBL teaching, and it's important to unpack what these things specifically mean.

In IBL Math there exists a particular role for instructors, which is to set-up and mediate the problem-solving process.  Students are engaged in exploring the mathematical landscape via well-chosen, logically ordered sequences of problems.  As students engage in their explorations, the instructor works to support the class so that students' interactions with the mathematics results in learning.

Here are some ways IBL instructors mediate the interaction between students and mathematics.  Instructors
1. design/adapt appropriate curriculum.  The problem sequences must be matched to the goals of the course, must be logically consistent and coherent, and meet the needs of the specific students in the class.
2. keep students going without getting overly stuck.  That is, struggle is good if the struggle is fruitful.  Being stuck is part of the learning process, and students come in with a "tolerance threshold" for being stuck.  It's important to stay within these tolerances so that students stay in the game.  One positive outcome of an IBL course is that tolerance for being stuck can increase. It's something that can be improved upon.
3. provide scaffolding.  This idea is related to the previous point is being able to control the "being stuck phases" by offering "bread crumbs" (hints, lemmas, sub problems, deploying group work, etc.) in just the right type and size to keep the cognitive level of the task high, while simultaneously avoiding student shutdown due to being overly stuck.  This is where calibrating to specific classes comes into play, and making good choices depends on having good data about students.  Listening, observing, and interacting regularly with students forms the foundation of this piece.  With good data in hand, it's often clear what scaffolding is needed.  "Oh, students are having trouble thinking about supremum.  Students may need to work on a couple of problems about upper bounds…"
4. set the class structure and environment for positive discourse.  Class presentations are of no value if the class environment is not setup appropriately.  IBL instructors set the roles, expectations, and procedures of class discourse, which is used to share and validate ideas.  The ethos of the class must include inquiry, discovery, mutual respect for students and learning, and valuing "mistakes" as important discoveries.
5. provide structure of the material being covered, so that students know where they are in the mathematical landscape.  This is an excellent place to inject small lectures and/or organizational tasks, where students are not necessarily solving new problems, but organizing the information they have recently studied.
The central focus is on the math and how students are working on the math.  The first major component is creating problems that guide students through the mathematical landscape.  Deploying these tasks determines the nature of the interaction students have with the mathematics.   As students work on solving problems, help or direction is needed.  The instructor provides enough assistance to keep students in the learning zone, without taking away all of the fun and exploration.  When students make progress and discoveries, the instructor provides a forum for the ideas to be shared, vetted, and learned by all.

This is the notion of mediating the interaction between students and the mathematics. It is a central component of effective IBL teaching, and a healthy perspective for instructors to embrace.

Related Posts: