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.

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.

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:

IBL Teaching Levels

Asking Good Questions

Sense-Making Continental Divide

AIBL YouTube Channel

Videos from the 2013 Legacy of RLM/IBL Conference have been uploaded to the AIBL YouTube Channel.  You can find all of the sessions from the conference HERE. 

David Pengelley's Closing Plenary Talk

Answer Getting vs. Inquiry

Edited on 10/12

Phil Daro discusses the reasons against Answer-Getting HERE.  Daro's 18-minutes talk gets to an important aspect about teaching.  In this talk, Daro suggests that in the U.S. we (math teachers) focus on helping students get answers.  On the other hand in countries like Japan, teachers focus on the mathematics that can be learned from problems, this simple, fundamental difference leads to vastly different outcomes and perspectives about teaching and learning.  When the focus is on mathematics and not just Answer Getting, then students can engage in doing the kinds of things that mathematicians believe is real mathematics.

Additionally Daro discusses the notion that mistakes and answers are part of the process of learning mathematics.  They are not the ultimate goals.  Answers, while still essential, are only a part of a larger endeavor, and not a signal that there is nothing left to do.

Mistakes should also be valued as useful discoveries.  When we discover a method that does not work, it needs to be fleshed out so that we can be sure that we can learn as much as possible about the related mathematics.  Such a process is not usually part of the standard method of instruction.

IBL instruction is consistent with these ideas.  Instructors in college-level IBL courses use a well-crafted set of problems to provide the context for the learning experiences.  (Course materials for some college-level courses can be found at The Journal of Inquiry Based Learning in Mathematics.)  Students work on these problems without being shown solutions ahead of time.  Part of class time is used by students to present solutions or ideas to the rest of the class, and the audience peer-reviews these ideas.  Logic and reason are used to determine if solutions are correct, and mistakes (discoveries) are used as opportunities for further investigations.   

It's a wonderful notion to view mistakes as important discoveries.  This sets up the framework for class discussions in a positive, scientific setting.  Mistakes then become identified as useful explicitly in the running of the class, and this is where diverging from Answer-Getting becomes fundamentally different than doing mathematics.  If the goal is getting answers, then by definition getting non-answers isn't getting us to our goal.   While we mathematicians view mistakes from a healthy perspective (at least when we are doing math), our views and attitudes are divergent from the way some students look at mathematics.  They "FOIL it" or "Butterfly it" or "Cross Cancel/Multiply."  

A related point is what new IBLers often say.  A common statement is, "I'm surprised that students have so much trouble with these concept questions."  These concept questions may be true-false questions or questions that ask students to apply an idea or method to a slightly novel (to students) problem.   The results are usually discouraging, and college instructors wonder why this is the case.

Daro's talk sheds some light on this phenomenon.  In the segment where students apply the "butterfly algorithm" to add fractions, it is noted that the trick doesn't generalize (easily or obviously to students) to adding three fractions, and U.S. students perform especially poorly on adding $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ compared to their international peers.   Daro suggests that it is because we spend too much time on tricks and on Answer-Getting, often at the expense of doing the underlying mathematics (in this case equivalence, equivalent fractions, and common denominators).  By the time these students get to college, their limited experiences with logic, problem solving, and higher-level thinking in mathematics leaves them underprepared for rigorous thinking.  

What can we do in college?  Focus on mathematical discoveries of all types, and let students inquire together about the meaning of mathematics.  Each new idea is a discovery and we can provide supportive classroom experiences, high-quality tasks, and effective coaching/mentoring to move students towards successful habits of minds and attitudes.

One can start a course by sharing prepared common mistakes and use them as the first experience in learning from mistakes and how the course will view mistakes as important discoveries.  Instructors can state something along the lines of... "What discoveries have you made about this problem? Please work with your partner to write these down and be ready to share your discoveries."  

Quick Post: Building Self-Esteem and Confidence

Just a quick post to share a quote by Randy Pausch, who presented and wrote The Last Lecture.

"There's a lot of talk these days about giving children self-esteem. It's not something you can give; it's something they have to build."

Self-esteem and confidence are built from hard work and success.  The overly simplistic model of IBL teaching is that we pitch (some) problem just outside the grasp of students, and through hard work, support, and guidance, students succeed.  Repeat. Repeat. Repeat...

Success breeds confidence like no other.  Good teaching practice can support fruitful struggle that then leads to new knowledge (for students), ways of thinking, habits of mind, and the oft-elusive quality of confidence.

Being Stuck

Dealing with "Being stuck" is one of the most critical components of IBL teaching.  IBL teaching rests on several factors, such as good content, questioning strategies, setting up a safe and productive environment for learning, having an assessment system that is consistent with the goals and ethos of the course,...    In this post the focus is on handing situations when students are stuck, which can sometimes make or break a course.   A little background first to set the stage.

The short, oversimplified background story is that mistakes are stigmatized (in the U.S.).  Thus struggling in Math is equated by some students as a sign of being dumb or slow or that the teacher isn't doing a proper job.  Traditionally math teachers present nice, clean solutions, and there are very few instances when students can witness the math process that actually is what makes us successful learners.  It can be the case that a student has never experienced or witnessed what mathematicians do regularly.  That is, the process of problem solving, inquiring, experimenting may all be unconnected from Mathematics.

Consequently, the IBL instructor who gets a group of such students must not only deal with the "regular" learning challenges that a math course presents, but also the legacy of underdeveloped/negative attitudes and habits of mind that promote learning.  When these underdeveloped/negative components of the learning process come out is when students get stuck.  Being stuck is both an opportunity and a risk.  It takes courage (at least initially) for students to admit to being stuck and to then engage in problem solving.   The risk to the students and teacher is when students are so frustrated and stuck that they shut down and stop learning.  (In math speak, we want to avoid the boundary.)

What we can do as IBL instructors?

1. Make sure students know and feel that it's okay to be stuck.  "Are you guys stuck?  Great!  It's okay to be stuck!  Let's use being stuck as an opportunity to work on our problem-solving skills... How can we break this problem down to a manageable size?..."
2. Scaffold enough so that the students see your role as their advocate and facilitator in learning. It's better to error on the side or more scaffolding than less, early in the term.  What this means is to provide enough hints/lemmas/basic examples so that students perceive themselves as progressing.  
3. Create a positive, relaxed class environment by using group work and visiting groups to check-in with individuals.
4. Use (more) starter problems.  One of the main roles of an instructor is selecting appropriate tasks.  Giving several starter problems, where all students can get traction is important.  Early in the term this is especially important, and the lower the level of the course the more important it is to have good entries into topics.
5. Summarize, restate, and give alternative solutions here and there to provide the expert insight that students often gain from.  When students have finished a section or unit, that is a wonderful opportunity to highlight all of the wonderful insights, ideas, and strategies that students learned.  It's a way to make explicit progress and achievement, as well as review material.
6. Marketing what IBL is and why it is good for students should be steady and ongoing.  What this means specifically is clearly indicating that the goals of the course include handling being stuck, problem solving, communicating ideas with peers.  These are in addition to the content goals.  "The reason why we are doing these activities is so you get better at..."
7. Consider employing reading assignments and journal writing.  Such assignments can be especially useful in addressing beliefs and attitudes.  Burger and Starbird have a book, "The Five Elements of Effective Thinking," that can be used in any math course as a supplement.  Students can read a couple chapters at a time and write a one-page reflection paper on what they learned and how they might use the ideas in the class.  Getting a second or third opinion in writing is very effective.  It's one thing to get the message from class, and yet another when multiple sources support the same messages, thus providing more opportunities for students to take necessary steps towards successful learning.

If being stuck in class is explicitly a good thing, students' struggles are respected, and class activities are designed to take advantage of these opportunities in a positive spirit, then being stuck can be a positive force in learning!   

Have ideas?  Send the via email or post them in the comments.