IBL Contributed Paper Session at MathFest 2013

Quick FYI -- If you are going to MAA MathFest (LINK), consider submitting an abstract and/or attending the IBL BEST Practices session.  MAA IBL MathFest Contributed Paper Session, Summer 2013

-SY

Inverse Pale Blue Dot

A departure from IBL topics for this post... 

I spent part of last week with my family taking a much needed break in Yosemite.  As we made our way back to civilization yesterday, we were rudely confronted with news of the massacre at an elementary school in CT.  My thoughts go out to those who have been affected by this terrible event.  

While in Yosemite, I made an image that I think is relevant.  I have always lived in urban areas so I rarely get to see the stars.  On this trip, one of the evenings was moonless and clear, so I took my camera and tripod to make a "wide field" night photo from Tunnel View.  At the time I was out there taking pictures, I thought of "A Pale Blue Dot" by Carl Sagan, and how this experience was the inverse of the Pale Blue Dot image.  Instead of the Voyager capturing an image of the earth as a single pixel, I was on the pale blue dot capturing a mass of stars and galaxies that make even a great national park infinitesimally small by comparison.

"There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we've ever known." -- Carl Sagan

Indeed it is all our responsibility to deal more kindly with one another.

Learning is Nonlinear

Learning is nonlinear.  

When a college student is learning to prove a sequence converges in Real Analysis or when a first grader is constructing the notion of grouping powers of ten, there is a period of mental construction that goes on where the ideas, language, problem solving, all that have to be put together coherently.  This process is highly nonlinear.  Certainly it isn't a matter of just transferring information from us to them.  Learning to do math is much more than say the simple act of memorizing your friend's phone number.  Learning a new idea takes time, energy, experimenting, mulling, messing around, all that.  Learning to creatively solve nonroutine problems requires even more of this kind of time.

What does knowing "learning is nonlinear" tell us about teaching?  One thing we can take away is that traditional courses tend to be set up linearly.  Section 1 on Monday, section 2 on Tuesday,... The factory model or industrial revolution genetics of our system become apparent here.  We line them up in rows, bolt on the parts one-by-one, and off they go.  Job well done, right?   

Linear teaching models are dangerously attractive, because they are easier to plan, follow the linearity of books, and institutions can be manufactured along these lines for less cost.  It's efficient on a cost basis, from the standpoint of human capital, because it requires little investment in faculty development and associated resources for teaching (e.g. technology, tools, materials, training, support).

Linear teaching is however inefficient in time allocation within class with regards to learning.  Students struggle on different things at different times.  Basic stuff is easily learned (even outside of class), and linear teaching may inadvertently allocate too much class time to basic material.

Further linear teaching may not allocate enough class time on harder material, which then conflicts with nonlinear learning.  When students need time and support to make it over the next tough section, the class moves on, ideas are half-baked, and misconceptions are formed or allowed to linger.  Students' foundation of thinking is not set upon bedrock, but rather a shifting, insecure base.

What is perhaps the most difficult part for a teacher regarding this linear vs. nonlinear issue (especially for traditional instructors) is the blind spot.  The blind spot I'm talking about is the not having a detailed understanding of student thinking.  If you don't know your students, you don't know where they are likely to struggle, other than through "inverse scattering theory," where in the past you've given quizzes or tests on a particular subject and students did poorly on certain items.  Of course it was too late for that batch of students, but at least the next batch will be slightly better off... Such issues are classic symptoms of traditional assessments, where your only learn about your students from tests (i.e. summative assessment).  Before you give an exam, you should be able to rank order your students (assuming a small enough class size), and then give a test that more or less confirms this.  If you can't do this, then you don't know your students well enough.

The ability to see the nonlinearity of learning is closely connected to student-centered teaching.  Giving students a chance to play and explore, gives us insight into their thinking.  (Just talk to them and ask them what they have tried on a particular problem.)   We use insights into their thinking to guide our instruction and do what is more likely to be useful as the next task.  If things are going well, students will just tell you, "I'm stuck on this..."  Great!  Now you know what the next move should be centered on.  

Linear assessment also takes the form of weighted averages used for grades.  If you rely solely on a weighted average of the hw, exams, final, and do not account for nonlinear learning, then this may be harmful to students.  If a student doesn't understand a topic in week 4, but figures it out by the end of the term, then how should the low score on first test be interpreted and used?  This is a topic of debate, and I am not going to say there is one right way to handle this.  But the point here is that you need to think about this issue and do something consistent with your values regarding education and what the point of education is.  Many faculty use a system where the final exam can make up for earlier lower scores at least to some degree.  This kind of policy is then an acknowledgement of nonlinear learning.  Let us now complete the circle by embracing nonlinear models for (student-centered) teaching!

Lecturing is linear.
Learning is nonlinear.
Teaching is nonlinear.

Online and Distance Learning: What We Should be Thinking About in Education

The wave of the future?  I'm not sure.  What is certain is the democratization of knowledge in the google era.  That's a good thing.  Below is a link to one look at a possible future.

Link to NYTimes Article on Massively Open Online Courses

I don't have a well-formed opinion of online courses today, and in a broader perspective of the role of technology in the math classroom.  It's okay for us to not know answers today as a profession, while at the same time it is NOT okay for us to ignore technological advances.

I would say that math classes today mostly look like they did decades ago.  Some of the differences are fashion, color textbooks, and demography.   That could have been said about brick-and-mortor bookstores just when Amazon, et al were getting started.

Another thing that is for sure is an increasingly rapid change in technology and its usefulness across society and in particular education.  Perhaps ironically for the scientific community the transition away from lecture will not be due to the data collected by education researchers.  Rather the changes might be the fact that we have to confront competing against the "best lecturer in the world in my profession."  Imagine the most dynamic, thoughtful, and articulate person in your field.  Then that person gets recorded by Hollywood caliber videographers giving the same lectures as you would.  At that point, then what does your class offer that can't be obtained for free or more conveniently?

I'm not sounding an alarm bell here or trying to stir up fear.  The point is that knowledge transfer is one generation of technological advance (or perhaps less) away from becoming trivial.  Add in improved interaction via video chat, and a few other features and then the experience of the classroom can be essentially recreated.

Coaching team sports, for example, doesn't seem threatened in the same way by technological advances.  This perhaps hints at what educators at the college level need to embrace.  What is it that we provide as an experience that cannot be learned from books or a set of online videos/classes?

IBL Self Check

Measuring yourself against your personal best is one of the best ways to evaluate yourself.  This is done in sports in particular, where the goal is to improve on one's best time or score.  IBL instructors can do this too!

How can we get better as instructors?  Focusing on the skills and practice of teaching is the way forward.  This list of self-assessment questions are meant as a guide or start. It is not definitive.  Ultimately your ability to improve rests upon being able to see what your students are doing and what you are doing that affects students.   If you can identify areas where you can improve, even if you are successful in that area, it gives you clear direction where you can exert your energy as a teacher.

Classroom Context and Teacher Moves:

1. Are student presenting/sharing ideas in class regularly? Can this be done more often in a way that benefits students?
2. What percent of time is devote to student-centered activities?
3. Are students deeply engaged in the tasks you have given them?  Can your problems/tasks be improved?
4. How many times per class period are you being supportive by giving encouragement, positive feedback, coaching, adjusting tasks to meet the needs of your students?
5. Can you give more positive feedback?
6. Can you improve the problems/tasks given to students?  Are the problems too procedural in nature? Are there good concept questions?  Are the problems too difficult?

Nature of student interaction:

1. How satisfied are you of students' solutions?
2. Do your students view you or a book as the mathematical authority? That is, do they frequently require an external (to them) authority to validate answers?
3. Are students able to understand and explain other students' solutions or strategies?
4. Are students mimmicking a book or model (i.e. "google search solution/proof") or are they generating solutions/proofs "from the ground up"?
5. Are a wide range of students regularly participating in class through presentations, questions, discussions?
6. Are there quiet students in class who do no participate very much?  If yes, then how many?
7. Are there any dominant personalities?  Are such personalities hindering the learning of some of the others sometimes?
8. What is the stress/anxiety level of your class on a scale from 1 to 10?

Keep it simple.  Here are some basic tools in a class that you have direct control over:

• Problems and tasks
• Questions/directions you say
• Basic strategies employed: groups, pairs, think-pair-share, class discussions

An example of how you can improve is if you identify that only 4 students answer your questions in class.  Then instead of asking "What is the answer to...?"  You can say, "Work on this, and I'll call on a person/group at random..."  Then you can get more students involved.

If every term you make a few meaningful improvements, then in a few years you can be brilliant!

IBL Instructor Self Assessment: How IBLish is your class?

Assessing your own teaching is significantly important.  A trait of an effective teacher is one, who is reflective and assessing oneself continuously.  While this is not easy to do, it can lead to continual, meaningful growth in the context of a larger teaching assessment program.  For IBL instruction there are at least a couple of ways to split this up.  The first level is to evaluate the course itself and measure how IBLish it is.  The second level is to self-assess one's methods or techniques, which will be posted in subsequent blog post.

How much of the course is IBL?

• Level 0: All teaching is done by lecturing.
• Level 1: In addition to lectures, other presentations modes are used such as videos and the use of worksheets for the purpose of practicing rote skills.  For example, the instructor shows students how to take a derivative of a trig function, and then provide some more problems similar to the shown example.
• Level 2: The instructor lectures for most of the time, but intersperses some interactive engagement, where students are asked questions and given mathematical tasks that require thinking and making sense, such as "Think Pair Share".  Interactive engagement may take up a few minutes to anywhere up to approximately a third of class time, which may vary day-to-day or be based on weeks (e.g. lecture MW, problems on F). A key feature is that lectures remain a significant component of the teaching system.  The instructor is the primary mathematical authority and validator of correctness.
• Level 3: The instructor lectures for roughly 1/3 to 1/2 of class time.  Students do a variety of activities that focus on understanding of core ideas, problem-solving, generating ideas, evaluating arguments.  The primary student-centered activities are higher-level tasks, such a problem-solving, categorizing, and understanding concepts.  The instructor is not the only authority on the subject matter, and students share responsibility in validating mathematical facts.
• Level 4: All teaching is done via student-centered activities.  Students are either presenting proofs/solutions, working in groups on problem-solving tasks.   In this case, the instructor does talk for part of the time, often in the form of setting the stage for a new unit, facilitating discussions, or summarizing or pointing out important facets of a proof/solution after it has been approved by the class.  The instructor and students primarily work together to form a consensus about the validity of solutions. Logic, reason, and mathematics are used to validate solutions.

In the next post, I'll make a list of questions for the purpose of self-evaluation.  

The "Education Pendulum" is Really a Ball at the Bottom of a Hill

Angie Hodge sent a nice article to me a week ago.  (Thanks Angie!)  On October 11th, Marion Brady published in the Washington Post How long one teacher took to become great.

The article hits on several topics.  It hits on the notion of the dashing stereotype of "good teacher," how effective teaching is when the instruction is student-centered, open, and collaborative, and then onto the difficulty of quantifying what good teaching really is.

Also of note from the article, and the point of this post, is about changing the system:

Because, when it comes to change, you can’t do just one thing. Switching from passive to active learning — which is what that 1960s effort was all about — had, at the very least, implications for classroom furniture, textbook use, length of class period, student interaction, teacher understanding, learner-teacher relationships, methods of evaluation, administrator attitudes, parental and public expectations, bureaucratic forms and procedures. -- Marion Brady

I hear about the pendulum in education frequently from various teachers, parents, educators, random people.  Education goes in one direction, and then there's a switch back the other way.  It feels like that when you're in the middle of it and observe from behind the desk or podium.  From a broader perspective, we are actually NOT on an pendulum.  What I think is a more appropriate model for our failed attempts at changing the system is that we are trying to roll a ball up a hill.  Before we get the ball all the way to the top, however, we give up (one way or another) and the ball rolls back down to the bottom of the hill.  It's definitely a periodic phenomena just like a pendulum, but there's a difference.

We've been talking about education reform for a long, long time.  W. Colburn wrote in 1830

By the old system the learner was presented with a rule, which told [the student] how to perform certain operations on figures, and when they were done [the student] would have the proper result. But no reason was given for a single step... And when [the learner] had got through and obtained the result, [the student] understood neither what it was nor the use of it. Neither did [the student] know that it was the proper result, but was obliged to rely wholly on the book, or more frequently on the teacher. As [the student] began in the dark, so [the student] continued; and the results of [the student's] calculations seemed to be obtained by some magical operation rather than by the inductions of reason. -- W. Colburn, 1830

John Dewey in 1899:

"I may have exaggerated somewhat in order to make plain the typical points of the old education: its passivity of attitude, its mechanical massing of children, its uniformity of curriculum and method. It may be summed up by stating that the centre of gravity is outside the child. It is in the teacher, the textbook, anywhere and everywhere you please except in the immediate instincts and activities of the child himself. On that basis there is not much to be said about the life of the child.  A good deal might be said about the studying of the child, but the school is not the place where the child lives. Now the change which is coming into our education is the shifting of the centre of gravity. It is a change, a revolution, not unlike that introduced by Copernicus when the astronomical centre shifted from the earth to the sun.  In this case the child becomes the sun about which the appliances of education revolve; he is the centre about which they are organized."

Then there was the New Math, there's IMP, CMP, CPM, reform calculus, etc.  The curriculum out there is good, meaningful, and amendable to IBL or hybrid IBL.  We've tried to change the system, but have failed in the past, for the reasons that we try one or two changes, but not all the necessary changes.    So we try a few good ideas, but not enough and the ball rolls back down the hill.

Hope vs. Despair:  The pendulum also represents at least to some degree a sense of futility in the enterprise of systemic change.  For sure, if we continue the one-thing-at-a-time approach we keep going back and forth.  Roll the ball a bit, and it rolls back.  However, if we actually view the ball and the hill model for what it is, then perhaps there's a chance that we'll muster the courage to get enough force behind the ball and move it over the hill. It means tackling more than one thing at a time, and doing a lot of hard work on several fronts.  

Easy? No.
Doable? Yes.

Classroom: For instructors there are implications to your everyday life.  Just changing from lecture to students doing group work or presentations at the board isn't a full switch to IBL.  The interaction of IBL content, managing student interactions with each other and with the material, assessment, coverage, getting students to buy-in, etc.  There are several components that need to be addressed for effective IBL instruction.  While this may seem daunting, all the skills are learnable and doable.  Taking your time or taking small steps is a reasonable strategy, but if you change to little or only in minimal ways, the gains will also be minimal.

The upshot for the classroom instructor is to make teaching changes that are systemic changes.  Doing cute activities are nice and useful, but it's better of the course has at least some minor systemic changes.  If you structure your course so that you regularly incorporate inquiry, plan content and instruction for inquiry, and assess (at least minimally) inquiry practices, then those changes, even if small, add up to a lot.

Teaching is system.  Roll the personal ball up to the next level, and you change your the system.  That helps all of us change the big system.

Upward and onward!

Umami: Pleasant, Savory Taste

Umami is a loan word from Japanese, which means "pleasant, savory taste."  Examples of Umami are when you drink a soda on a hot, sweaty afternoon, and you say "Ahhh!"  Or perhaps you walking in nature and reach a favorite spot to breath in the fresh air, see the wonderful sights, and take it all in.  Ahhhhh!

The notion of umami applies to teaching.  Instead of Ahhhh that was savory, it's more like an Aha!  "That makes sense!"
"Beautiful idea!"
"Great solution -- I never would have thought of that!"

In math courses where students primarily memorize rote skills and learn procedures developed by someone else, it is hard to provide an opportunity for students to have an Aha moment.  The ideas are dull and foreign.  The emphasis is on getting to answers quickly, and there's little time for unpacking deep concepts and enjoying the experience of thinking about beautiful ideas.

Thinking about or discovering for oneself beautiful ideas can result in Umami.  This is another way to look at authenticity of teaching.  By authenticity I mean that students are doing "real mathematics" and are not just memorizing say Gauss-Jordan elimination.  When students do authentic mathematics that allows them to grow intellectually, that's a good thing.  And if the students buy-in and find the experience of discovery an aesthetically pleasing and fulfilling one, then I believe this is a highpoint in these students' intellectual development.  In IBL classes, the opportunity exists to provide regularly authentic math tasks that engage students in figuring out how things work and why things work.  Doing math, then becomes an enriching, pleasant, savory experience.

Umami.

Jo Boaler on IBL

First, we're not switching the name of this blog to "The EBL Blog" -- I'd have to change the url, my business cards,...  :)   

But seriously, in this short clip, Jo Boaler and her students make the case for why inquiry.  Jo Boaler is a professor at Stanford University and has written the excellent book, "What's Math Got to Do With It?"

Ted Mahavier Talk: Legacy of R. L. Moore Conference, Washington DC, 2011

Ted Mahavier opens the 2011 Legacy of R. L. Moore Conference with this inspiring talk.  Ted Mahavier is a professor of Mathematics at Lamar University, and one of the most experienced and dedicated supports of Inquiry-Based Learning.

Does Teacher Personality Matter?

Yes, but only if you're like Darth Vader or Lord Voldemort.  Both of these personality types tend to have issues with restraint, power, and creating a safe (learning) environment :)

Personality matters less than what some may think.  It's very easy to think about a charismatic professor like Ed Burger getting high marks and saying, "Well he's such a wonderful personality.  How could I ever be that?"  First of all, we shouldn't be comparing ourselves based on perceived personality attributes.  Second, we all have different teaching environments, so it's really not about us as personalities and about how we help our students overcome their challenges.

It's really about the teaching skills.  It's really about the teaching practice.  

Noticing big personalities is rather obvious and easy to latch onto.  What is hard to see is how courses are setup and what skills and practices are brought to the table, how the "little stuff" is handled on a day-to-day basis. While being charming may be minimally helpful, when I talk to faculty about why their students are not (yet) buying in to their IBL classes, almost always we can identify something technical that is the root cause.

A starter list of questions to ask yourself to help you reflect on your teaching:

• Have you been marketing IBL effectively and regularly? That is, do students know their role, your role, and how things will operate daily?
• Beyond saying "It's okay to be stuck," what activities have you employed to encourage and support the process of being stuck?
• Are you giving regular, positive feedback? How often?  Have you given positive feedback of some form to every student presentation or comment?
• When the class is silent or not talking enough, do you employ various small group activities to engage your students?
• Are the problems too hard or too easy?
• Have you dealt with the quite/shy students in your class, by finding activities to specifically engage them and get them involved more and more over time?
• What are you formative assessment strategies?
• Are you still relying primarily on traditional exams for summative and formative assessment?
• Do you know how your students are doing before the first exam?  If no, then you've missed out on a lot of data gathering.
• Do you listen to your students to see if they know the answer, or are you also listening to figure out where they are?  Put another way, do you know how your students think and what their specific strengths and weaknesses are?
• Do you spend more than one-third of IBL class time talking?
• Are you still looked upon as the mathematical authority (i.e. answer validator)?
• What grade would you give yourself for restraint and knowing when to step in?
• If a student makes a mistake or is stuck, do you have several strategies for dealing with this situation?  What cues do you use to know when to step in?
• Are students having fun in class doing math?

This list of questions is meant to help you reflect on your practice.  As you consider some of the questions above, it may make you think "Oh, I could do more of..."   Rather than think about who you are as a personality, which really does not matter, you can instead focus on the skills and practice that actually make a difference in learning.

And the data speaks to this as well.  The work by Hake, et al has been well documented and accepted in Science Education.  There's famous graph that speaks volumes:

In the red group are the traditional instructors and the green are the interactive engagement instructors.  Included in both groups are novice instructors, experienced instructors, award winning instructors, and those who have not been rated highly be students.  What matters isn't the personality or the perceived ability (i.e. popularity) of the instructor.  What matters are the skills and practice employed by the instructor.

A more useful self-assessment tool is in the works. Stay tuned!

IBL Miniworkshop at JMM 2013

Matthew Jones, Carol Schumacher and I will be conducting an IBL Miniworkshop at the Joint Meetings in San Diego in January.  Please spread the word!  This is a great way to get introduced to IBL.  If you have colleagues who are interested in IBL, this miniworkshop is a great way for them to get started.  More details are coming!

Thursday January 10, 2013, 1:00 p.m.-2:20 p.m.
MAA Workshop
An introduction to inquiry-based learning. 
Room 1B, Upper Level, San Diego Convention Center
Presenters: 
Stan Yoshinobu, California Polytechnic State University---San Luis Obispo 
Matthew Jones, California State Unviersity, Dominguez Hills 
Carol Schumacher, Kenyon College

Link to the Joint Meetings Website

Teaching Tips and Nuts and Bolts Issues Review

Some useful teaching tips compiled in this post.


Think Pair Share
http://theiblblog.blogspot.com/2011/08/classroom-strategy-think-pair-share.html

Marketing IBL to your students
http://theiblblog.blogspot.com/2012/08/marketing-ibl-to-your-students.html

Teaching Tips: Chaos
http://theiblblog.blogspot.com/2012/02/teaching-tips-chaos.html

Getting Students to Ask Good Questions
http://theiblblog.blogspot.com/2011/11/nuts-and-bolts-getting-students-to-ask.html

Small Groups
http://theiblblog.blogspot.com/2011/10/nuts-and-bolts-small-groups.html

Homework Templates
http://theiblblog.blogspot.com/2011/09/nuts-and-bolts-homework-templates.html

Insights from Ed Burger and Mike Starbird
http://theiblblog.blogspot.com/2012/09/an-insight-from-burger-and-starbird.html

"Euclidean Geometry: A Guided Inquiry Approach" by David Clark

A quick post on a busy Monday... FYI -- David M. Clark, SUNY New Paltz, has recently published an IBL Euclidean Geometry book.  It is flexible enough to be used at many levels.  It has been successfully used in high school, college, and in a masters program in Math Education, with emphasis on inservice teachers.  It is appropriate for HS teachers, college faculty who teach Euclidean Geometry, and faculty who teach in a masters in Math Teaching or masters in Math Ed.

Here's the link to book on the publisher's website:
Euclidean Geometry: A Guided Inquiry Approach

So That About Does It for Lecture

The title for this blog post is a bit tongue in cheek.  Certainly data by itself isn't going to change people.  Change requires actions by people.  But on a data level, there is an increasingly clear picture about what works in education.  This post is about yet another body of work that validates collaborative learning and IBL methods.

All the vectors are pointing in the same direction.  Namely, IBL is more effective than lecture.

Effectiveness of STEM Small Group Learning

Dr. Sema Kalaian et al have been conducting a large, NSF-funded meta-analysis to determine whether or not small group learning in classrooms is more effective than the traditional lecture-based instruction in promoting higher achievement and attitudes toward STEM subjects as well as persistence in STEM college classrooms.  Their work produced rigorous, scientific evidence that establishes a "medium" effect size of 0.37 in favor of small group learning.

As I mentioned this before -- all the vectors are pointing in the same direction.  Whether one looks at elementary school math ed studies, secondary math ed studies, college STEM, other subjects,... all the evidence points towards active learning.  

Students who DO do better.  Students who engage deeply in the subject and are allowed opportunities to collaborate have better learning outcomes.

It's starting to evolve into a moral issue, when looked upon from the seat of an Education researcher.  The evidence in favor of active learning is heading towards the equivalent level of "smoking increases the risks for cancer."  At some point we need to start asking fundamental questions about what we are doing as profession and how we can evolve as a profession appropriately.

If you have thought about the notion that IBL is a style or fad, then I encourage you to think otherwise. Those of us who support and encourage the use of IBL (in whatever flavor) do so, because the evidence from research and our experiences from the classroom compels us to do so. It's a professional choice based on evidence, reason, and the sheer joy of seeing students grow into their potential.

When you're ready to get started with IBL or go to the next step?  Contact us at AIBL. We're here to help!

Dr. Sandra Laursen, UC Boulder, "What Has Ally Learned? Outcomes for Students and Teachers of IBL Mathematics Courses"

This video is of Sandra Laursen's talk from the 2011 Legacy of R. L. Moore Conference in Washington, DC.  If you have wondered about the scientific evidence regarding IBL math vs traditional lecture at the college level, this talk provides strong evidence from a variety of data sources.  I like to describe the work in this video as "all the vectors are pointing in the same direction."  What I mean by this is that they have collected a wide range of data sets, and have found a consistent story that points in the direction that IBL teaching produces better learning outcomes for students.

One of the striking results from the study is that women in IBL courses have some of the biggest gains compared to their peers in non-IBL courses.  IBL courses level the playing field, and could play a role in eliminating the gender gap in STEM! Indeed the gender gap may be perpetuated by traditional instruction.  This is important stuff!!

Dr. Sandra Laursen, University of Colorado, Boulder.

An Insight from Burger and Starbird

I'm in the process of reading "The 5 Elements of Effective Thinking" by Ed Burger and Mike Starbird. I'll start with a quick insight that can help your teaching right now.

Consider the two ways to approach the usual situation where a teacher tries to see if the class understands what is going on.

"Are there any questions?"

vs. 

"Talk to your neighbor for sixty seconds and come up with two questions."

Asking "Are there any questions?" is no longer as useful a line of approach or teaching technique as we'd like it to be.  How often have we been met with silence?  The problem with silence is the lack of data about student understanding.  If you want to know you need them to produce something (an answer, an idea, an example, a question,...).  And you should want to know what your students are thinking or not thinking.  Knowing exactly where your students are at is vital to teaching, just as making a proper diagnosis using evidence is necessary to medical doctors.

Another way to look at this is the following.  If a certain teaching strategy doesn't elicit the response you need, then find another approach.  If you do not give your students a chance to opt out of thinking of a question, then they are going to be more engaged.  Moreover, students will learn how to ask questions and also to seek to find new questions.  Thus, setting the stage for questions has benefits far beyond checking for understanding.  Questioning becomes part of the intellectual life.

Response to "Is Algebra Necessary?"

A month ago, Andrew Hacker published an Op-Ed piece in the NY Times, called "Is Algebra Necessary?"

Here’s the short version of my response:  Yes, Algebra and algebraic reasoning is necessary.  But, the existence of people who write pieces like “Is Algebra Necessary?” is in actuality a symptom of the larger failing of our educational system. 

Many of my colleagues scoffed at Hacker's piece.  It is sad indeed to see this printed in the NY Times.  But that made me think of the question, "How did we get here?"  One thing to note is that lots of different issues are mingled in Hacker's piece and the subsequent responses.

Hacker’s piece is as if it was like something out of the past held in a time capsule.  If he has been reading Math Ed articles during the last couple of decades, it would have been clear that changes were needed in instruction, curriculum, and learning cultures, and that there were innovations and implementation efforts underway.

We should view Hacker as [Hacker].  That is, [Hacker] is an equivalence class of people with his somewhat naive views and opinions about Math Education.  The existence of “Is Algebra Necessary?” could be a symptom of larger failings or shortcomings in the way we teach and learn Mathematics in the U.S.  Further, mathematicians have not done a good job of communicating what math really is.

Let’s take a step back and look at the situation broadly.  The feelings of is dissatisfaction with math education in the United States (in this case Algebra) is unfortunately widespread and has been prevalent for a long time.  There exists deep seeded nonavailing attitudes and beliefs about math.  A nonavailing belief or attitude is one that does not support or inhibits the learning of mathematics.

One of the unintended consequences of traditional math educational is the cultivation of nonavailing beliefs about math.  Krista Muis, Simon Frasier University (Link) conducted a meta-study, which resulted in collecting a list of nonavailing beliefs, including the 15 selected below.  This data is unwelcome indeed.

1. Memorizing facts and formulas and practicing procedures are sufficient to learn mathematics.
2. Mathematics textbook problems can only be solved using the methods described in the textbook.
3. Teachers and textbooks are the mathematical authorities.
4. School mathematics is driven by rules and memorization, and is driven by procedures rather than concepts.
5. If a problem takes longer than 5-10 minutes, then there is something wrong with the student or the problem.
6. The goal of mathematics is to obtain one correct answer and do it quickly.
7. The teacher is the only source of determining whether an answer is correct or incorrect.
8. Students’ role in the classroom is to receive knowledge by paying attention in class and to demonstrate it has been received by producing right answers.
9. The teacher’s role is to transmit knowledge and verify that it has been transmitted.
10. Only geniuses have what it takes to be good at mathematics.
11. Students prefer to have only one way of solving a problem, because it is less to memorize.
12. The processes of formal mathematics have little or nothing to do with discovery or invention.
13. Students who understand mathematics can solve assigned problems in 5 minutes or less.
14. One succeeds in school mathematics by performing the tasks, to the letter, as described by the teacher.
15. The various components of mathematics are unrelated.

Nonavailing beliefs affect perceptions of what mathematics is, contributes to “I hated Math in school!” comments, and could lead policy makers to inappropriate conclusions.  Students who believe that math should not be understood, but merely memorized engage in the study of mathematics in a completely different way than a mathematician does.  This massive gap explains to some degree the differences between those who can and those who can't.

Curiously, some of the major problems we face as a society have the commonality that they are complex, long-term, and gradual.  Understanding something as complex as the educational system in the U.S., with all its intricacies, failures, successes, and fluidity is a big task.  It is difficult, even for well educated people.  A fundamental difficulty when evaluating education is that it is extremely hard to know where one’s knowledge ends and one’s ignorance begins.  We’ve all been to school.  We know what it’s like, and so of course we know how it should be done.  I mean, I've been to elementary school, right, so I should know how it ought to work.  Of course, we wouldn't say that about going to the hospital or dentist.  Ultimately this leads to implicit, perhaps unconscious oversimplification of the problem.

Technical knowledge in education is far beyond the general public.  Most people are blind to many key issues in K-12 math, which are very technical in nature.  How technical?  Here’s an example from elementary school:  What is the difference between quotative and partitive division, and which model is more appropriate for developing and understanding of division of fractions?  This is an important issue to deal with in upper elementary math. Most people I know, however,  do not know what I am talking about if I ask them this question.  Teaching math is hard, technical stuff.  Designing student-centered, meaningful math curriculum and lesson plans is a big, complex task.  Yet, everyone has an opinion about how math should be taught and what policies should be put in place, without knowing major bodies of knowledge that are critical in the development of children’s mathematical thinking.

A main point here is that people who should not be making policy recommendations are doing just that.  As countries have technocrats working in central banks to guide monetary policy, countries should have "education technocrats," who can do the hard, technical work of guiding education policy, curriculum develop, etc.  Education is hard stuff, and it’s well beyond the knowledge and skill set of the general public, even the well-educated sector of the public.

Another issue is that teaching and learning, broadly speaking (and excluding the star teachers out there), is failing on some significant levels.  U.S. students perform poorly on international comparisons.  Our students generally don’t like math.  High school dropout rates are too high.  Many students are developing nonavailing beliefs, and they do poorly on problem solving, proof writing, reading or using the mathematical language, etc.  This makes [Hacker] revolt against mathematics education as constructed today.

“Is Algebra Necessary?” should then be viewed in part as an alert to mathematicians. The notion that the captain goes down with the ship seems appropriate here.  Mathematicians are the leaders, naturally, of mathematics, and we have a tremendously influential and important role in mathematics education.  Thus, whether you disagree with [Hacker] isn't really the point.  There exists a large, unhappy group of people who do not like math, primarily due to traditional paradigms of education.  Sure some members of this class are successful, but they missed the point of mathematics primarily due to how classes are/were taught.  And it is duly noted that the cardinality of [Hacker] is much larger than the cardinality of [Mathematicians].  To provide more emphasis to this point the theme that mathematics education in the U.S. is not up to par is very critically evident in the PCAST document that we released last month, where one of the messages is the frustration and disappointment of the math community in the slow uptake of modern, student-centered pedagogies.  We have been put on notice in more than one way recently by those outside of mathematics.  (See David Bressoud's notes on  PCAST HERE and the MAA response HERE.)

What can you (mathematician or math teacher) do specifically? The easiest way is to start using more empirically validated, student-centered teaching methods in your classroom right now, attend MAA and NCTM conferences that have sessions on innovations in teaching, and sign up for workshops that provide rich experiences for transforming your teaching.   Additionally you could work with your School of Ed in outreach programs to local K-12 schools or with a regional Math Teacher Circle group.

What else?  These types of blog posts are dancing around the issue of the point of school.  If you continue the line of reasoning, "Is Algebra Necessary?" to "What's the point of Algebra?" to eventually "What's the Point of School?", then you start to get somewhere.   (See Guy Claxton's book "What's the Point of School?" and Mike Starbird's talk from June 2012.)

Where do we stand?  We are in the Era of Implementation.  It is clear that we have plenty of innovations in the U.S. about effective teaching.  Some high performing school systems, such as the Finnish system (See Pasi Sahlberg), actually obtain much of their innovative techniques from American researchers.   Simultaneously we have is an educational system that doesn’t provide enough access and support to implement these teaching innovations.  Thus, the challenge we face as a profession is in widespread adoption of empirically validated teaching methods.  The question is whether we will rise to the challenge and do what is necessary to transform our system.

Game on.

Learn by Making Mistakes: Diana Laufenberg

Diana Laufenberg is a high school teacher.  Her insights into learning are straight to the point. Let students explore and make mistakes.  Great stuff in 10 minutes!

Here's Diana Laufenberg's TED talk.

http://www.ted.com/talks/diana_laufenberg_3_ways_to_teach.html

AIBL's Mission

The purpose of this post is to outline the AIBL mission.

The overarching AIBL mission is to support and sustain a growing community of IBL instructors at a national level.  AIBL supports more than just particular IBL courses or a narrowly defined mode of instruction.  AIBL supports individual instructors at all levels of experience.  Indeed, transforming math education will not be achieved without a strong, community working together towards common goals. 

Community: Reform based primarily on new tests or books, without considering larger goals (e.g. effective thinking and advanced problem solving) and environment (e.g. learning culture) are good, necessary steps, but not sufficient.  AIBL's mission, to develop an IBL community, addresses issues of sustainability and continuous professional growth.  Knowing about something is a far cry from being able to do it.  In professions as complex and perhaps daunting as education, a community of support is essential.

Big Tent Philosophy:  AIBL is founded upon the principle of inclusiveness.  There exists a lingering belief that instruction falls into two, distinct categories.  Lecture is category 1, and the Moore Method is category 2.  This is incomplete, and AIBL supports instructors interested in a wide range of inquiry-based methods.  While it is our belief that full IBL courses have the most potential for transformative experiences for students, it is understood that environment, instructor experiences, student body, and other factors affect teaching decisions.  We understand from experience that instructor change is a process that takes time, and encourage instructors to take a responsible, long-term approach to changes in practices.  Go your pace, while at the same time don't wait too long (for your students' sake).  Whatever the situation, instructors who aspire to involve their students in rich mathematical tasks and allow ample opportunities for student collaboration (broadly defined), are welcome and encouraged to participate in AIBL activities.

AIBL is here for the long haul. AIBL is here to help!  Get involved, and get your students doing Mathematics!

Marketing IBL to Your Students

One of the common issues new IBL instructors face is student buy-in.  This is a real concern for all instructors.  In this post, I outline the issue broadly, and then provide some tips for how to ensure that your students get with the program in their minds and in their hearts.

First, let's talk about the issue broadly.  One important thing to remember is that Math is culture.  There exists default expectations about what a math class is and the roles for students and instructors.  These default expectations are often unconscious -- we don't think about them.  When you meet someone new or talk with your boss, you probably do not realize all of unconscious things you do (or don't do) in these interactions.  Likewise, in a math class students have certain expectations that are almost always aligned to traditional instruction.  Students expect instructors to show, and their job is to follow dutifully and write down notes and perform these tasks on exams.

IBL classes are aligned differently, of course.  Students are asked to solve problems they do not know the answers to, to take risks, to make mistakes, and to engage in "fruitful struggle."  These are all very different from normal expectations (as of today -- hopefully that will change).

Tools for making sure your students are on board are

1. clearly defining students' role in the class
2. providing a clear rationale for IBL (regularly)
3. creating a safe and successful classroom environment.

Students need to know what their job is in an IBL class, and it is the instructor's job to make this clear.  Students must know what they are supposed to do (solve problems, write math proofs/solutions, communicate effectively,...).  Students need to know the instructor's role (provide appropriate tasks, coaching, mentoring, adjusting the challenges as needed, moderating discussions,...)

Why IBL?  Well there are lots of reasons.  Research shows it's better for students.  We are now in the era where information about anything is available on your cell phone, and one can run Wolfram Alpha on a cell phone, too!  In other words, all lower-order thinking levels (as per Bloom's Taxonomy) are now nearly worthless due to advances in technology.  Effective thinking is now where it's at.  Thus, IBL is the way forward for students.  This line of reasoning addresses items 1 and 2 above.  What about 3?


The heart is the heart of the matter.

Telling people, "Medicine is good for you!" isn't sufficient.  Students need to know that the instructor is their advocate for learning.  Students need to see themselves as successful mathematicians (where they may never have thought this before in their lives).  Thus it is important for students to struggle, but struggle within reason.  It is suggested that IBL units start off at a basic level, where all students in the class can achieve some success.  Then the problems should ramp up in difficulty as appropriate for your students.  When in doubt, include more easy problems than less, especially at the beginning of the course and at the beginning of new material.  The worst case scenario is that you spend a few extra minutes on them or just skip them entirely in class, and assign them as homework.  There is no cost to including more problems.

In this sense, establishing a safe and successful classroom environment is asymmetrical.  Erring on the side of being "tougher" is fraught with perils.  First, you are going again previous experiences and the traditional classroom culture.  Second, many students have negative attitudes about math.  Third, telling students that are stuck to "just keep going" can lead to the perception that the instructor is not helpful, and thus not teaching.  Struggle is good if the students feel that the struggle leads somewhere.  This type of scenario is often the case for students writing negative comments on course evaluations.  Students may in their minds understand that IBL is good for them, but they experienced too much frustration to truly enjoy the experience in their hearts.  In other words, the experience was not an aesthetic experience.

I also note that saying that you told them is not enough.  You need to know if the students feel it in their hearts.  Look at them and see if they are enjoying the math and interacting positively.

Of course, there is good reason for students to struggle and perhaps not solve a problem.  Such experiences are fruitful on many, many levels of learning.  BUT this is something that should happen down the road, once students are off and running, enjoying math and doing math successfully in a positive and supportive learning environment.  Training for a marathon has similarities to teaching an IBL course.  You don't coach a new runner with hard interval training on day one followed by long 20-mile tempo runs.   Athletes train by exerting an appropriate training load and recovering.  Then they repeat and then move on to new things gradually.  Math is no different.  Tasks should match students' experiences and abilities and grow with them.

If you are not positive, how can your students be positive?  If you never smile, why would your students smile back at you? Be positive!  It's important to let your students know that they are working hard and progressing.  I thank my students for their participation, and I try as best I can to make classes a supportive environment.  Pointing out the good parts of solutions, ideas, and efforts should be a part of daily practice.

Start easy. Establish the learning culture from day 1.  Build on positive class experiences to challenge students to do more and more.


General IBL Points
Some points you can use as a base for discussing IBL classes with your students.  This is a list of talking points to help you find your own way of conveying the message that IBL is good for the mind.

1. IBL is a student-centered method of teaching similar to the Socratic method. It requires more work for me (the instructor), but it's better for you.  Research shows that students who are actively engaged learn better.  While you may not be used to it, I'll do my best to make sure you are comfortable with it and will be successful in this class.
2. One goal of IBL is to help students learn to think independently, and become a successful problem solver.  In other words, a goal of IBL is to help you get better at thinking effectively.  That's really what we will work on.  And you can't learn to think effectively, if someone does all the thinking for you...
3. IBL emphasizes the process of problem solving and theorem proving rather than the memorization of facts.
4. IBL is not experimental.  It has been employed successfully since the days of Socrates.
5. The reason why books and other outside resources are not allowed is because we will discover the ideas ourselves.  We will collectively work on the tasks and come up with our own ideas.
6. It’s OK to be stuck.  Being stuck is a noble state of mind.  It means you are just about to learn something new!
7. It’s OK to be frustrated.  You’re doing fine -- try to slow down and enjoy the process.  We'll get it eventually.
8. Being stuck is natural.  Whatever you do, don’t give up.  If you're stuck, there has to be question in there that you can ask.  
9. What's the best way to learn to play the piano?  Should you just watch videos of pianists?  Do you need to do something else besides watch someone else play?

Teaching is more than content delivery.  Addressing the learning challenges as part of the course is a good thing to do, and acknowledging where students are coming from and building a bridge for them to cross is a core component of effective IBL teaching.

IBL Instructor Perspectives: Ron Taylor, Berry College

Ron Taylor is an Associate Professor of Mathematics at Berry College.
rtaylor@berry.edu


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

Including some adjuncting and teaching in grad school, I have been teaching for 18 years.  The last dozen of these years have been spent in a tenure track position at Berry College, a small liberal arts college in northwest Georgia.  Before I started using IBL I would have classified myself as an interactive lecturer.  By this I mean that I would spend a lot of class time doing the talking, but a significant portion of that talking was in the form of asking questions.  I didn't often talk for a whole class period, but I was the one in charge of what was going on.  I did occasionally use group work, or have students come to the front of the class to write solutions, or parts of solutions, on the board, but I always had a standard textbook and a sheaf of notes that laid out the direction that class was going to go each day.  This approach more or less mimicked the classes I had as a student, though there was some contrast.  All but one of my classes all through college and graduate school were lecture style classes, but there wasn't much in the way of group work in class or having students writing on the board.


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

I had heard of the Moore method before I started using it, but I can't place my finger on where I first heard about it.  I had one Moore style class in graduate school, a point set topology class, and I think that the professor may have mentioned that he was using the Moore method, but I don't remember him telling us much about how the class was going to go or why he was choosing to do it that way.  So I spent the semester thinking that he "wasn't teaching us anything" and didn't enjoy the class, even though in hindsight I feel like I learned a lot of topology that semester.  Then in the spring of 2003 several students asked me to teach a directed study class in number theory and I thought that this would be a good place to try out this method.  We had a textbook, but we spent class time working through problems and talking our way through the proofs in the book.  It wasn't as much of a Moore method class as it could have been, but it wasn't bad for a first go without really knowing what I was doing.  (In addition to being a rank novice at using IBL I didn't really know a lot of number theory either.)  At the time I don't know that I made the connection between what I was doing in the number theory class and what I had experienced in the topology class, but the students seemed to be engaged every day and so I thought that it was at least a partial success.  During that semester I was invited to the Legacy of RL Moore Conference by virtue of being a Project NExT Fellow.  It sounded interesting and so I went.  I felt like I learned some stuff about teaching while I was there and I decided that I would try to ramp up the use of IBL (as I then envisioned it) in my classes with a concomitant decrease in the amount of time I was lecturing.  I've been back to the Legacy Conference ever since, co-chairing the planning committee for the past three years, and I feel as though it is paying off for my students.  Now my goals with IBL are two-fold.  One is to expand the amount of time I use IBL in class and the second is to improve the IBL stuff I have been doing.  I wish my topology professor had explained what he was doing so that I could have had a better experience that semester and maybe started teaching this way sooner rather than later.  I may also have gone on and taken some more topology.

In addition to the payoff for the students in my classes, there has been payoff for students at Berry in other classes.  In 2005 several of my colleagues and I applied for a startup grant from the Educational Advancement Foundation because we had discovered that several of us were trying to move away from lecturing.  We started with a group of six faculty in 4 departments which has grown to a group of almost twenty in more than half a dozen departments.  Since we started this IBL group on campus, other campuses have tried to replicate this process and several of us have won teaching awards based in part on our use of IBL techniques in our classes.


3. What was one of your best IBL experiences?

The single best hour of class I've ever had was in a real analysis class several years ago.  A student was presenting something that she had started the class before, but had to cut short because we ran out of time.  I suspect that she was happy about this because I don't think she had the full answer the first time.  Anyway, she got up to do her presentation and when she got through she asked if there were any questions and about half of the small class had the same question.  They all seemed to think that her proof wasn't quite right and they were asking about one particular part.  But the other half of the class, including myself and a philosophy professor who was sitting in the class, thought that she was right and we couldn't see where they were finding fault with what she had done.  The rest of the class period was spent with the two groups trying to figure out where the misunderstanding was and in the end it turned out that she, and those of us who were saying that she was right, were all making a jump in our heads that the other students wanted to see explicitly written down.  (I wish I could recall the particular detail, but it is lost to the winds of time.)  In the end it wasn't really a big sticking point in the proof, but some of the students weren't making a connection that others of us were making.  They learned a lot that day about how to ask good questions and I learned a lot about how to get out of the way and let the students make sure that things are right. [Editor's note:  This example is a great one!  Please also consider keeping a teaching diary to write down these gems.]

The best whole semester of class I have had may have been a knot theory class I taught where we used a combination of the Moore method and POGIL worksheets.  We didn't "cover" as much material as I had in the past, but we went much deeper into every topic that came up in the class.  The students, who had been immersed in a culture of active learning since they had been at Berry, were really good about asking lots of questions and generating discussion in class almost every day.  When one of them would get up to present the others weren't shy about asking questions if there was something amiss.  And they had learned to do this in a way so that no one ever felt put down when there was a question, so the presenter was able to go with the flow and worry about the correctness of the mathematics or helping a classmate understand something, rather than worry about being wrong in front of peers.  We did some challenging stuff in that class, and some of them worked harder than they had ever worked in a math class before.  But in the end they all seemed to have a really good time and to learn a lot.

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

Find something that sounds interesting (Moore method, POGIL, Just-In-Time Teaching, Think-Pair-Share, etc) and give it a try.  If it doesn't work as well as you had hoped, then tweak it and try again.  Listen to feedback from the students about what worked and what didn't wok so well.  Don't let them talk you into abandoning the methodology, but let them help you make the methods and materials you use better.  Above all else, let your students know that you believe in their abilities and give them the chance to rise to the occasions that you present to them.  Of course, you also have to remember that having the chance to rise to an occasion also gives someone the possibility of falling flat, so let students know that it's ok to make mistakes so long as they are willing to learn from them.

Also, try to find like-minded colleagues who will help you navigate through the hard times that you will invariably encounter.  In addition to the mentoring available from AIBL, you may find people at your local institution.  The culture of inquiry we have at Berry wouldn't be nearly the same if it weren't for the diverse group of people who are using strategies from a wide spectrum of active learning techniques.  This has been extremely beneficial to all of us in terms of having people to talk to when things aren't going as well as we would like.

Announcement: AIBL YouTube Channel

AIBL has a new YouTube channel!   The channel is http://www.youtube.com/user/AcademyIBL  Subscribe and stay connected!

The first AIBL video is a presentation by Michael Starbird, University of Texas.

"What about the high-achieving students?"

The short answer is, no worries!

First, let's look at the big picture.  The major problems in Math Ed are not at the top end of the achievement distribution, where students are successful no matter what pedagogy is employed.  This isn't the first priority when one is thoughtful about the landscape of issues that we face in our system.  This isn't to say that we should ignore opportunities here or the students.  It's just that the way to handle high-achieving students is clear and straightforward.  Let's dispatch this one quickly...

General effective pedagogy includes keeping all your students, no matter the level, engaged in activities appropriate to their level.  High-achieving students are a pleasure to work with.  These students like math, they are motivated, and want to learn more!  So there's no barrier here to deal with other than laziness.  All these students need are good problems and some feedback.

The main ideas to keep high-achieving students engaged are listed below.

• Problem sets should include problems challenging to your high-achieving students.
• High-achieving students can also be given extra problem sets where they submit proofs in writing.  These problems would not be presented in class normally.
• Additional /supplemental articles or chapters can also be assigned.

So there you go.  Give them problems at their level, and then get out of the way.  Offer feedback and support as necessary.  What's the trajectory?  Junior/senior students should be able to do graduate-level work after about a year of full IBL in a particular subject (e.g. a year of Real Analysis, Topology or Abstract Algebra). 

Additionally, high-achieving students can become peer mentors and teachers.  They can model how to write a good proof, offer constructive feedback in ways that are often better received than from the instructor, and can be deployed to help struggling students, via small group work.  

When you have high-achieve, socially skilled students, you have a great, great asset that can positively affect the learning culture of your classroom.  Good teaching practices tell us to keep all students engaged appropriately, and then one can redirect all this talent and ingenuity to increase the level of the whole class. 

IBL Instructor Perspectives: Jackie Jensen Vallin, Slippery Rock University

Jackie Jensen Vallin is a faculty member at Slippery Rock University in PA.  Jackie has been the co-organizer for the Legacy of R. L. Moore conference, and has been actively involved in mentoring new IBL instructors, the MAA, and Project NExT.

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

Counting my teaching experiences in graduate school, I have been teaching for 15 years.  In the early days, including when I finished graduate school and had to write my first “Teaching Statement,” I was much more of a lecture-based instructor. I even said in that statement “I am more of a sage on the stage than a guide on the side.”  This was probably a little misleading since I really had a very interactive style with my students – knowing them all by name and calling on them to further the conversation in class. However, it was pretty unusual to do anything in my classroom except lecture and a few group-work style worksheets.


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

The spring of my first year of “real teaching” (in a faculty position), I attended the Legacy of RL Moore Conference in Austin, TX.  There were a lot of people talking about the Moore Method (and, appropriately enough) refusing to define that method.  But I met some great educators and was intrigued by the idea of putting the responsibility for learning more firmly on my students by guiding them to the answers without pretending that my lecturing them would be enough to get them to understand.


I began simply after that – introducing more worksheets into my classes, doing more group work, and implementing presentation days during calculus. It wasn’t until a couple of years later that I taught a class in which the students were really responsible for presenting almost all of the material. And even then I didn’t do that in all of my classes, but only in upper level courses (intro to proofs and abstract algebra, in particular).  It was another couple of years before I integrated IBL into all of my classes, and now I teach everything (from Math as a Liberal Art to Math for Future Teachers to Intro to Proofs) with as many student-centered activities as I can.



3. What was one of your best IBL experiences?

My best IBL experiences actually happen in lower level courses – students take responsibility for their own learning.  This means that frequently students who thought that they were bad at math get to master that fear, the material, and convince themselves that they are not “bad at math.”  This happened this semester in my Financial Mathematics (for non-majors) courses – students were shown a couple of examples of how to do problems, and then completed problems on their own, in groups, sharing answers at the end of the period.  I walked around the room, checking work, answering questions (by asking more questions) and making sure everyone was on task.  Since this course is based entirely on applying formulas to word problem scenarios, the only way for students to learn to read word problems is by making them *do* word problems.  And they are succeeding!  What a great semester!


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

Start in a way that you feel comfortable – if that means turning over presentation to students one day per week and lecturing the other days, then do that.  Don’t let anyone else tell you how *you* have to do IBL.  Find your own method.  And ask lots of people for ideas, suggestions about notes or activities, or if you feel stuck. I have gotten great pieces of advice from many different people – I steal the parts that work for me and discard the pieces that don’t to have found my own method of IBL.  Everyone is willing to help and talk about teaching – we all care a great deal about our students, or we wouldn’t spend so much time developing good course notes!


Note:  This is great advice -- "find a level that is comfortable for you."  If you want to be "scripted" at the beginning that is also fine, and lets you concentrate on the day-to-day teaching aspects.  Most importantly, if you need help, let me know!

John Dewey + Moneyball = A Key Insight to Change

A quote from John Dewey

"I may have exaggerated somewhat in order to make plain the typical points of the old education: its passivity of attitude, its mechanical massing of children, its uniformity of curriculum and method. It may be summed up by stating that the centre of gravity is outside the child. It is in the teacher, the textbook, anywhere and everywhere you please except in the immediate instincts and activities of the child himself. On that basis there is not much to be said about the life of the child.  A good deal might be said about the studying of the child, but the school is not the place where the child lives. Now the change which is coming into our education is the shifting of the centre of gravity. It is a change, a revolution, not unlike that introduced by Copernicus when the astronomical centre shifted from the earth to the sun.  In this case the child becomes the sun about which the appliances of education revolve; he is the centre about which they are organized."  http://bit.ly/JvR3cG

This passage is from "The School and Society," originally published more than 100 years ago (in 1899).  It is relevant today, sadly.  Are students at the center of instruction in math classes?  Mostly no.  Math teachers predominantly lecture at students, and the great change that Dewey saw has not yet come about, though many teachers have made the shift.  A question one can ask is "Why have things not changed significantly in all these years?"  Sure the books have colors, and we have technology beyond our grandparents' wildest dreams.  But when you look beyond mere surface beauty, you can see that the heart of it is still the teacher telling, and the students following.

One major issue behind the lack of change is data.  More specifically an issue that persists is in assessing teaching and learning, and more pointedly how this data might change our fundamental beliefs (or axioms) about teaching and learning.  What we assess and how we assess it determines our evaluation of student ability and achievement.  Herein lies one of our fundamental issues.  Lack of good assessments can lead us to continue doing what we have been doing.

To get some insights, let's look outside of education to provide a backdrop for analyzing our own system.  One of the unique aspects of baseball is the wealth of statistical information that has been available for generations upon generations of players. It's one of the reasons why baseball is such a wonderfully interesting sport to be a fan of.

Earned Run Average (or ERA) is one of the traditional measures of a pitcher's ability.   A lower ERA is considered better, since the pitcher gives up fewer runs per 9 innings.  The problem with ERA is that it is a noisy and flawed measurement system of pitching effectiveness.  It depends on factors not under control of the pitcher, such as the quality of the defense supporting the pitcher and the effects of stadiums on balls batted in play.  Some pitchers are overvalued and some are undervalued in terms of their contributions to team wins, if ERA is weighted too heavily as a measure of ability.  Voros McCraken conducted some groundbreaking analysis, establishing the concept itself and subsequently methods to measure pitchers that are "defense independent."  This story among others is chronicled in "Moneyball" by Michael Lewis.  But Voros didn't expect baseball teams to rejoice when learning about his findings.  He knew better.

"The problem with major league baseball... is that it is a self-populating institution.  Knowledge is institutionalized.  The people involved with baseball who aren't players are ex-players... They aren't equipped to evaluate their own systems.  They don't have mechanisms to let in the good and get rid of the bad." (Voros McCraken)

This is a striking insight!  Voros essentially identifies why baseball resisted modern statistical methods that could help.  Baseball is not set up as an institution to evaluate how it evaluates players.  Baseball people normally did not have the knowledge, ability or willingness to entertain ideas developed by people like Voros, who is a baseball outsider.

What is the implication for us in the teaching profession?  It should be stated that major league baseball and education are not very similar as institutions.  That said, we have some similarities and we can draw conclusions about our shortcomings from baseball's own struggles.  Indeed teaching is also a self-populating institution. Students who do well in the current system are the ones who end up become teachers or professors.  Some future math teachers state things like, "The reason why I like math is because there's always one right answer, and there's a simple, straightforward structure to all problems."  They are good at memorizing rote skills, the rote skills appear on tests, they get good grades, they are labeled as good in math (which may or may not be true), and then they model themselves after their favorite teacher.  Thus the cycle perpetuates.

Colleges and universities have in their mission the goal of seeking truth and knowledge.  When it comes to teaching, however, discussions among faculty often are about style, "what my students like...," and about delivery of information.  The focus is usually not on learning and what students are doing.   A major point is that we do not use the scientific method to evaluate teaching, just as major league baseball didn't use any scientific methods to validate their player valuation systems.

Consequently we have several metric problems.  Are the usual metrics like skills-based tests and student evaluations the right ones?  Clearly the answer is no.  Let's consider the typical calculus sequence with a thousand-plus page texts.  In the typical chapter on optimization in calculus books, the authors usually highlight in a colored box the steps for how to find relative extrema.  What this tells many (but not all) students is that they should memorize the recipe and regurgitate it on an exam.  That's how one can get a good grade after all.   These students will not walk away with a conceptual understanding of the subject, and probably will forget what they have memorized once the term is over.  In short, their education is unintentionally of a lower quality than what we want.  Mathematics is reduced to applying recipes that many students do not understand or even care to understand.

If you don't believe this can happen, here's data from Physics by Professor Eric Mazur, Harvard University, presenting at the University of Waterloo.   (It's 1 hour long, but worth it!)  At Harvard, 40% of the students in freshmen physics who did well on the procedures had inadequate understanding of basic concepts.

The result of traditional assessments is that many students who are traditionally given good grades have major gaps in understanding of basic concepts.  Students think they know it, but maintain "Aristotelean understanding of Physics" rather than a Newtonian one.  Their education amounts to very little, even at a sublime places like Harvard.

Now let's consider traditional teaching assessments (i.e. student evaluations), and consider data from Physics, based on the Force Concept Inventory (FCI).  All of the red data points are from traditional instructors who lecture.  Represented in these data points are teachers who are highly rated and lowly rated on student evaluations -- the red dots contain some star teachers and the teachers on the "oh bummer" list.  And they all do about the same on the FCI within statistical significance!  Students gain on average about 23% of what is possible in the pre-post test design.  Student evaluations are like ERA.  They are a noisy, flawed metric.  Actually student evaluations are worse than ERA.  ERA has some value in aggregate (whole team ERA), and outliers tend to have outlier ERAs.   In contrast, the highly-rated, award winning instructors are doing no better than Dr. Boring or Professor Snoozer.

The green data points represent faculty who use Interactive Engagement in their classrooms.  One of the main trends is that there is very little overlap between the reds and the greens.   The average green gain is double compared to traditional instruction.  Thus a better way to measure if an instructor is effective is to know what skills and practice he or she utilizes in the classroom.  While crude and incomplete, it at least it tells you whether the instructor is on the red or green distribution.  But these qualities and practices are not usually assessed or measured in teaching evaluations, so there does not exist sufficient data or incentive for the system to embrace change.  We keep on doing the usual, while the traditional assessments tell us things are okay.  And the results keep staying in the "red zone" above.  Education has a bunch of Voros McCrakens, so there is hope.  Baseball has changed, and I believe education will continue to improve for the better.

What about Math?  Calculus Concept Inventory has been rolled out and studies are underway.  Thus there is hope that we will embrace new assessments that tell us what is going on.  Preliminary results suggest similar outcomes to the FCI.  Interactive Engagement and Traditional instruction are different distributions.  I look forward to seeing the published results.  Moreover, a growing body of evidence in research in undergraduate education also suggests that students in traditional courses are not learning what we want them to learn.  (More on this in a future post.)

The MAA's Calculus Study indicates that 80% of college calculus courses are taught in sections of 40 students or less.  Additionally, very few institutions have large lectures for upper-level courses.  Ample opportunities exist for IBL methods to be deployed courses across the nation.

What can an individual instructor do personally?  Looking at data can be demoralizing at times, but one should be optimistic.  In particular, one can turn assessments into valuable tools that guides students and instructors in the right direction.

Assessment is more than grading stuff so that you can assign course grades.  Assessments should be utilized in ways that provide students with regular feedback (formative), instructors with information about their students (formative), and to evaluate demonstrated achievement (summative).  Assessments should provide incentives for the qualities we actually value, including creativity, clarity, exploration, problem-solving ability, and communication.

Ideas for what to assess:

• Student presentations and/or small group work
• Reading or journal assignment
• Math portfolios
• Exams
• Homework

The items above are not revolutionary.  What matters is what we put into them.  Exams can be rote skill based or they can also test for conceptual understanding, application of ideas, and problem solving.  Homework can be made more interesting.

Student presentations and/or small groups are a wonderful way to assess understanding.  When students present their proofs or solutions, it often a rich experience and rich source of information.  You see it all in IBL classes: great ideas, small ideas, half-baked ideas, insightful questions, victories and defeats.  It's a slice of real life, and it's really great.  It's very easy to detect where students are at, and then take action.

Reading assignments can be used to offload (i.e. flip a class) basics to homework, leaving time in class for the harder tasks, where inquiry is useful.  Portfolios are like a CV, and can be used to demonstrate what a student has been able to prove on his or her own.  Additionally portfolios can be used to create a record of the theorems proved by the class. (I'll write more about portfolios in future posts.) 

In IBL courses, one has continuous formative assessment.  Instructors are always analyzing whether students understand an idea or not, by giving students meaningful tasks and then working with them to overcome learning challenges.  If students are stuck, then there is another question or problem that can be posed, and then students are off on another mathematical adventure.  Students are continuously engaged, are monitored, are self-monitoring, get feedback, and so on.  This rich, integrated assessment system of actual learning is a core advantage in IBL teaching.

Returning to Dewey...  If we continue to teach and assess teaching in traditional ways, we will not gather the data and information necessary that support change for individuals and systemwide.  Gathering good data about our students' thinking, which is also a core part of effective teaching, is a key to the way out.  Engage your students, collect good data, share it, publish it.

Upward and onward!

"It ain't what you don't know that gets you into trouble.  It's what you know for sure that just ain't so." - Mark Twain