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.

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.

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

The Role of Inspiration

One of the most frequent things I hear from my fellow colleagues is "I like lecture.  I love listening to lucid presentations."  I do to.  In fact, great lectures are wonderful experiences, and many instructors want to inspire students just as they have been inspired.  This is a noble sentiment, and shows that instructors care deeply about their students.  In this post I discuss the role of inspiration and lectures and their limitations.

When we attend performances by musicians it is an aesthetic experience.  We are living in the moment.  If we are lucky enough to attend a great performance, we can be swept away by the beauty and passion of the event.  Art, music, great scholarship are indeed high points of society.  

Yo-Yo Ma plays "The Swan"

Herein lies the complexity of teaching and some of the pitfalls of the profession.  We can mistake beauty for effectiveness, and this is very easy to do.  Just as Sirens luring sailors with their enchanting songs, beauty, lucidly, and inspiration can lure a teacher to lose sight of the point of school and effective teaching.  

I make my point by analogy.  Going to see musicians like Yo-Yo Ma perform are an inspirational experiences.  It is a necessary but not sufficient condition for studying to become a musician.  If you want to be a musician, you have to play. Play lots.  Play often, and reflect.

Similarly learning mathematics requires one to see the subject as meaningful.  But one also has to do the hard work to build the mind.  Listening to a lucid, inspirational math talk can be enlightening.   BUT lectures do not build capacity to do mathematics with nearly all listeners, especially novices.  Mathematicians already have the capacity to think mathematically -- we can learn what we need from a lecture much of the time.  For students who are not yet advanced in mathematical thinking, they cannot take away the same lessons from the very same lecture. 

If you want your novice cellist to learn to play, you can't just show a vid of Yo-Yo Ma, and say "There you go.  Now go home and practice hard."  It's not that simple.  Telling isn't teaching.  Likewise really understanding calculus, deciphering nested quantified statements in theoretical math courses, and learning to build differential equation to model a physical situation requires more than just following a recipe, processed by the professional mathematician (i.e. the equivalent of Yo-Yo Ma).

Students need a supportive environment, well-matched problems, time to be stuck, and opportunities to figure things out for themselves.  It is this long, arduous, and rewarding process that unlocks potential.  Good musicians know this.  They don't just listen to someone else play.  They also practice with intent with teachers, with collaborators, and with new music to keep their minds and hearts fresh.  They experiment.  They learn new skills, they interpret pieces in their own way.  They do.

To put it simply, lectures and IBL methods should be used for different purposes.  Lectures can be used to inspire.  IBL methods should be used to build students' abilities and capacities to do.

The myth that "Teaching is Art" is one that lets us rationalize "aesthetics = effectiveness."  I am not criticizing aesthetics, by the way.  Beautiful ideas are why we are here.  But there is a difference between showing students something beautiful and helping students become young mathematicians.  Horses for courses, as they say across the pond.  Or use the right tool for the job over here in the U.S. 

What's the right mix of lecture and IBL?  We don't know exactly.  Data suggest a small percentage of the time should be lecture, and that as instructors talk more, students report less learning gains.  Many of the most effective and experienced IBL instructors use IBL daily, and intersperse mini lectures as needed or at opportune times.  Examples are (a) when the students have completed a body of work, (b) to showcase for students how certain ideas or techniques can be further used, (c) to summarize big ideas, (d) enculturation, (e) exposing students to things that there is not sufficient time for, (f) summarizing student strategies and proof techniques from the week,...

"Science is 1% inspiration and 99% perspiration." -- Albert Einstein

Thinking About Thinking

One of the main reasons why IBL produces superior learning outcomes compared to non-IBL teaching methods is metacognition.  Metacognition described simplistically is thinking about one's own thinking.  Professional academics do this as a habit of mind.  We ask ourselves, "What is my approach to this?"  or "The way I'm thinking about this is..."  It's one of the reasons why we are peculiar.

Students sometimes (often) do not think about their thinking.  Most have not had experiences in school that support this.  This is easy to think about.  If all you do is follow rules and procedures to compute algorithms you don't have to think about your own thinking.  All you need to do is follow the thinking of someone else.  My intuition about why passive learning fails for most people is that unless you are predisposed to independent thinking, there is little in traditional education that can transform one towards independent, critical thinking.   Monitoring one's own thinking is part of the sophisticated set of thinking processes that distinguish experts and novices.  And it can be trained!

Where this comes into play in IBL math classes boils down to this:  students in IBL math classes must explain their reasoning on a regular basis.  I claim that the process of justifying answers engages the metacognitive process.  It goes something like this:

Student: "I believe the statement is false."
Instructor: "Can you tell us why?
Student: "Well... I looked at these examples and then I thought that this one here doesn't satisfy the second condition."
Instructor: "Very interesting.  What do the rest of you think?..." <discussion ensues>

The support of metacognition in IBL classes is much richer than what is presented in the vignette above.  Students are stuck on problems.  They are proving theorems from first principles, and are asked to write proofs outside of class.  Moreoever, students are required to present their proofs to the entire class for peer review.  Students cannot get through class without having to think about their thinking.

Some questions to get you to think about your students' thinking about their thinking:

• Do your students think about their thinking and how do you know?
• If you are not sure students are thinking deeply about their thinking, what can you do in class and via assignments to encourage this?
• What mathematical tasks and class setup could you use to support thinking about thinking?

IBL Levels (with apologies to Van Hiele)

In this post I present the IBL levels, a la the Van Hiele levels, which sincere apologies to Van Hiele. These are based on my observations of IBL instructors over the past decade.

A very important point is that the IBL levels presented here are simplistic (reductionist) and are intended as a form of guidance to instructors.  The IBL levels indicate the direction in which you ought to go and provide a framework for evaluating your own teaching.

Level 0: non-IBL, teacher-centered instruction.


Level 1: some student engagement via occasional "active lecture" methods like
Think-Pair-Share, concept questions, or students working out examples.  The instructor and textbook remain as the predominant mathematical authorities, where students seek confirmation from the instructor or textbook to check if their answers are correct.


Level 2: a significant percentage of class time (50% or more) is used to engage students in solving problems, discussing solutions, and peer reviewing work.  The problem solving activities are focused on problems that students do not know the answer to or are not shown the strategy or solution method.  Students are supported and encouraged to make their own conclusions, think for themselves, and share their insights.  Students have a predilection for determining the correctness of solutions without always seeking external validation from an instructor or book.  Students have frequent opportunities to engage in rich mathematical task.


Level 3: full IBL.

While level 3 is a lofty goal to set for oneself for teaching all courses, level 2 is more easily attainable, especially for courses like freshman calculus.  Levels 2 and 3 are then goals that can be achieved by math instructors.  In particular, level 2 does not require wholesale changes in curriculum, and is thus appropriate for various contexts and environments.  It should be noted, however, that level 2 does not have the same potential as level 3 for transformative experiences.  There are always tradeoffs.

For novice IBL instructors, you do not have to wait for the "right course" to begin your journey in IBL instruction.  Level 1 can be done in any class, and allows you to test the waters and build up your own IBL teaching skills and understanding of IBL methodology.  Making the transition to level 2 would be much easier, and it sets the foundation for making the transition to full IBL instruction.

If you have been hesitating, start with level one and try out Think Pair Share!

Another look at the 'Coverage Issue'

One of the most common complaints or worries that instructors make is that there is so much material to cover.  I agree.  Many people agree.  Courses are jammed packed with far too many topics.  In some cases, this is warranted, and I am the first to agree that there exists a vast amount that students need to learn.  No one in their right might wants to teach a course with very little content.  The debate then is about the mix of course content.  This mix should be balanced with concepts, problem solving, opportunities for real growth, and of course contain an appropriate number of topics.

The coverage issue become problematic when we sacrifice understanding for the sake of getting through the long list of topics.  I'm going to use the words of college students to convey the core ideas. Keep in mind that these students were some of the best in their high schools, have taken at least precalculus, and most of them have had calculus.  Several have had some upper level courses.

• "I feel as if this was the first math class ever that I actually took the time to understand what exactly a graph was saying.  Before graphs were just lines, or parabolas, that I plotted..."
• "I'm astounded by the fact that I only did 5 actual math problems in my middle and high school careers.  I drilled procedures, and didn't really learn anything from it."
• "The idea of teaching students how to spit out answers to problems is not effective in learning.  As a learner, I would like to understand why we do math a certain way and not that it is 'just done this way.'"
• "I had never known why we 'completed the square.' I understood that completing the square produced an equivalent expression or function because whatever was added was also subtracted, but it always seemed so complicated.  For the first time [using algebra tiles] I really had a clear understanding of what was going on when I completed the square."

I re-emphasize that these are college students, who are eager to learn.  They have done algebra for years, and have not ever been allowed to explore why very basic things make sense.  They do not report learning as much as they wanted to or could have from earlier courses.  We know those courses are all "jam packed" with content from top to bottom, and that there is "no time to do anything else." Yet students are not reporting that they learned what they need to learn, which begs the question, "What is this all for then?"

Instructors often do not have much control over how much material a course must cover.  In this way, there exists a systemic issue or crisis.  Our system measures coverage in antiquated ways (i.e. a list of topics) and does not include covering "practice standards," such as problem-solving ability, communication, understanding concepts, and being able explain ideas.

To IBL instructors this poses a Teaching Minimax problem -- Minimize as much as possible unnecessary content and maximize the amount of time available for students to explore why.

More on this in future posts...

IBL Instructor Perspectives: Professor Dana Ernst

A Q&A session with Professor Dana Ernst, Department of Mathematics, Plymouth State University.  Dana  is an award-winning, highly energetic math professor, who has recently learned to teach using IBL.  In this blog post, we ask him a handful of questions related to his IBL teaching.

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

I have been teaching college-level mathematics, starting as a graduate student, since the spring of 1998.  My classes have always been interactive, but initially they were predominately lecture-based. Probably like most teachers, I modeled my teaching style after my favorite teachers.  I was aware of the Moore method and IBL, but I had never experienced this paradigm as a student.  By most metrics, my approach in the classroom seemed to be working.  My teaching evaluations have been consistently high and I have received several teaching awards. However, prior to implementing IBL, I was suspicious that I could get so much more out of my teaching.  More precisely, I was suspicious that I could provide my students with the opportunity to get so much more out of my classes.

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

It was not until I sat through a Project NExT workshop at the 2008 MathFest led by Carol Schumacher that I began to consider using IBL.  Carol's workshop was about implementing a modified-Moore method approach in an undergraduate real analysis course.  I do not remember the details of the workshop, but by the end, I was inspired to give IBL a shot.  In the spring of 2009, despite having no prior experience or formal training, I decided to teach my very first IBL course.  The course I chose is called Logic, Proof, and Axiomatic Systems, which is meant to be the introduction to proof course at Plymouth State University.  Perhaps surprisingly (since it was my first go), the course was a huge success and I was immediately sold on the potential impact that IBL can have on a student's learning and character development.  I have loved teaching since the day I started, but nothing compared to the joy of watching students truly learn mathematics, and often completely independent of me.  I had taught the same course two semesters in a row using a mostly lecture-based approach, and I had thought that the previous two iterations went very well.  However, the IBL version was a vast improvement.  I have since had students from all three variations in upper-level proof-based courses and the students from the IBL version are much more independent and, in general, better proof-writers.

3. What was one of your best IBL experiences?

There are so many!  Here is one event that illustrates why I am hooked on IBL.  During the Fall 2011 semester, I was chosen for jury duty, which required me to miss six days of classes.  For all but two of these days, I was able to find a faculty member to cover my classes.  For the two days that I did not have faculty coverage, I convinced a graduate student in education to cover my IBL abstract algebra course.  This student had taken my IBL introduction to proof course, so he had some IBL experience, but he had never had an abstract algebra course.  On the days the graduate student covered for me, the class ran as usual and the students were highly productive.  They didn't need me!  The students were so proud of what they achieved while I was gone, they sent me pictures of the work that was presented on the board.  The graduate student that covered for me indicated that all he had to do was jot down who came to the board to present.

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

In order for IBL to be successful, the students have to buy into it.  To pull this off, instructor need to do some marketing at the beginning of semester.  The right amount of marketing varies from class to class and semester to semester.  For classes filled with students with prior IBL experience, I don't have to convince them of the benefits of IBL.  These students are generally ready to dive in and get started.   For classes consisting of students that are new to IBL, it is important to explicitly spell out the format, expectations, and goals of the course.  One approach I take on the first day is to ask the students what skills a college student, specifically a math major, should have upon graduating and how best to acquire these skills. Through some Socratic questioning, I am able to get them to tell me that we should be doing something like IBL. I'm not trying to trick them, but rather give them some ownership in the philosophy behind the structure of the course.

Even if the students are sold on IBL, you still have to be willing to adapt, overcome, and improvise.  Issues will come up that you couldn't have predicted.  Building a community of trust will make any challenges a lot easier to deal with.  I believe that the two most important qualities of an IBL instructor, heck any instructor, are patience and being "Mr./Mrs. Friggin' Positive." Lastly, I would like to echo something Ed Parker has said.  Sit back, shut up, and "see what they can do".


[Added by Stan Yoshinobu]  Summary: 

1. Go to a workshop if you can.
2. We can all improve our teaching!
3. We are not just teaching facts, we are developing appropriate habits of mind, such as independence
4. Marketing to students (i.e. getting them on board with an IBL class) at the beginning of term is one of the keys to success.
5. Be flexible! Be patient! Be super positive!!

Thanks Dana!

A Peek into the (Near) Future

Apple Inc. introduced iTunes 10.5.3, which includes interactive textbooks.  This is almost surely the wave of the future, no matter what teachers and instructors might think of technology.  Just as the CCSS represents a great opportunity, so does advancing technology.  The Math community should experiment and investigate the options.

For those who say, "Run what brung ya," see Kodak.  Kodak, once a giant, the inventor of the digital camera sensor is now headed for bankruptcy.  The point isn't just about our individual preferences as instructors.  We also have to consider what our students need to compete in the 21st century.

Do we have to have answers today?  No.

Should we explore and study this? Absolutely yes!

Are/will international competitors be exploring technology and education? Absolutely yes.

Architecture and Education Part 1

Last week I was in Boston, MA for the Joint Mathematics Meetings.  I have never been to Boston before, and I was excited about the trip.  After a long travel day and a couple of days of meetings, I needed to get outside, feel the sun, breath fresh air.  I needed to see a bit of Boston before getting back on a plane headed to the west coast.

Before I left for Boston, my wife told me to go and visit the Boston Public Library.  She said it's a special place.  This then leads to the point of this post -- the influence of architecture on education.  I'll do this through photos and captioning.  Part 1 is about a big idea.  Part 2 will be about our little 'ol classrooms and how they can be set up in simple ways to encourage students to interact.

Outside on the steps

A statue gazing at a sphere deep in thought

Inscriptions of the names of the giants, upon whose shoulders we stand 

Entering the building reveals old-world architectural themes.  The patterns above you as you walk in lead your eyes up.

Then as you walk past the entrance, stairs lead you slowly up to a new chamber with art up high, lions honoring the fallen, and arches that again lead the eye upwards.

The stairs for you to walk slowly, and make several 90 degree turns.  These 90 degree turns force you physically to change directions as well as encourage you to prepare the mind for what is to come.

When you reach the top of the stairs, you arrive at a hallway.  The black iron doors on the right side are the entrance to the great hall.  Two more 90 degree turns to go.

Finally one arrives in the great hall.  Arriving in this special place is truly inspiring.  You are encouraged to learn in  a space like this.  Great minds have studied here in the past, are studying here now, and will study here in the future.

Some takeaways (at least for me).  Environment matters.  Indeed, if architecture and decorations didn't matter, we would all live and work in boxes with no paint, no art, no human influence.  But that's not the case.  Building designs affect how we interact with people and how we go about our work.  Living and working in well-designed spaces matters.

See you next time!

Guy Claxton on Skills Versus Dispositions

I stumbled across this just now.  In a brief 8 minutes, Professor Claxton discusses learning power and the difference between learning antiquated skills and having transformative experience.

Why Students Leave STEM Majors

A recent (11/3) article appeared in The New York Times called "Why Science Majors Change Their Minds"

Quoted from the article:

But as Mr. Moniz sat in his mechanics class in 2009, he realized he had already had enough. “I was trying to memorize equations, and engineering’s all about the application, which they really didn’t teach too well,” he says. “It was just like, ‘Do these practice problems, then you’re on your own.’ ” And as he looked ahead at the curriculum, he did not see much relief on the horizon.

This article is a micro version of the book "Talking About Leaving: Why Undergraduates Leave the Sciences" by Seymour and Hewitt.  It should be on your bookshelf if you are an educator in a STEM field.  The top four reasons why students leave STEM fields in undergraduate education are teaching and learning related.

The Top Four Reasons why undergraduates leave STEM majors:

1. Lack or loss of interest
2. Non STEM majors offer better education/learning experience
3. Poor teaching by STEM faculty
4. Curriculum overload, fast pace overwhelming.

Traditional explanation or "justification" of this is that we are "weeding out" the weak or morally inferior students.  The data presented by Seymour and Hewitt says otherwise.  One cannot predict based on academic measures such as GPA which students will switch out of STEM majors.

In short, we waste talent and turn away many strong, capable students.  In our rush to "get through all the material" our system has to a large degree lost sight of the big prizes.  We can, however, regain this by focusing on the learner and the learning experience.  We can be coaches and mentors rather than gate keepers, and departments across the nation should reconsider the definition of success.

What should success really mean for a Math program?

The Teaching Axiom of Choice

I was reminded of this quote by Bertrand Russell, while watching Ken Robinson's TED talks.

"Is a man what he seems to the astronomer, a tiny lump of impure carbon and water crawling impotently on a small and unimportant planet? Or is he what he appears to Hamlet? Is he perhaps both as once?” -- Bertrand Russell.

The power of imagination is built quite strongly into the fabric of our very existence.  One of the traits that separates us from other animals is that we read poetry and listen to Miles Davis.  Other animals communicate and make sounds, but they don't write plays or compose music.

One of strongest and most consistent beliefs among the IBL teachers is that creativity can be taught.  This is quite evident at the Legacy of R.L. Moore conferences -- IBL practitioners have this unending supply of belief in their students, and not assuming what a student can or cannot do.

I have internalized this for myself as the Teaching Axiom of Choice, which is stated as "Every student has the capacity to learn Mathematics."  Our educational system (in the U.S.) has a tendency to label students at a young age.  Mathematical ability it thought of by some as a fixed, unchanging attribute that cannot be altered significantly by effort.

What this does to teachers is set expectations, which then in turn affects teaching decisions. It goes like this.  If you are a teacher who believe in fixed attributes of students and are given a track of students who are "low level," then you are likely to assign "easy" tasks and not challenge students.  Effort is to be minimized and the goal is to just get some of these students through the material so they can pass and get on to the next level.

It is unlikely that problem solving, deep engagement in rich mathematics, and a developing other higher-level thinking would be part of the class.  So then the labels lead ultimately to a dry, barren math experience (though unintentional -- no one means to do harm).

It's a choice.  We can, though not always easily, believe in our students.  Give them our best, and work hard on our teaching methods to provide the best possible chance for learning.  I am the first to realize that the system plays a strong role in what we are allowed to do in the classroom.  With that said, the Teaching Axiom of Choice then is a precondition to providing transformative experiences for students.  If you do not believe in students (or at least willing to suspend belief), then you are unlikely to give your students the kinds of tasks and structured freedom to let them compose, to dream, and to be creative.

"I now view math more as an art..."

At the beginning of the term of a course for future elementary school teachers, I ask my students to write a math autobiography.  Normally this is done in the first course in the sequence for future elementary teachers.  This time I have done it in the third course.  One of my colleagues, who taught the first two courses via guided inquiry is on leave this term.

As I read through the math autobiographies, I can see those who have had negative experiences noticing a change of heart.  The seeds of hope have been planted, and they see themselves liking math more than they used to.

Most of the negative attitudes about math appear to develop or at least surface somewhere between upper elementary school and HS Geometry.  This is a pattern in the biographies, so I won't try to explain it. I'll just report it.  Why this is the case is not the point of this post.  What I'd like highlight is one student's poignant statement:

I now view math more as an a art because of the diversity in how to get one answer.  Growing up I was only taught one way to solve a problem, so I did not even think about the other methods that may have been easier for me personally.  It is so important to teach children today that there is not always one correct answer and not one method to get to that answer.

Beautiful!

Let us focus on what we can do for students in our classes, and let us be reminded that students have the capacity to change, if given the opportunity to do better.  There is a way to make it happen!

Farewell, Lecture?

Eric Mazur, Harvard University, is a well known physicist, who has championed active learning teaching methods in Physic.

Here's the abstract.

This article presents a method for teaching in large introductory required courses that is different from the traditional lecturing. The responsibility for gathering information now rests on the shoulders of the students. Class time is devoted to discussions, peer interactions, and time to assimilate and think. Instead of teaching by telling, we use teaching by questioning students. Research shows that learning gains nearly triple with this approach. Students not only perform better on a variety of conceptual assessments, but also improve their traditional problem- solving skills.

 http://www.sciencemag.org/content/323/5910/50.full

Eric Mazur's talk "Confessions of a Converted Lecturer" is highly compelling.

"We Are In the Business of Transforming Lives!" -- Mike Starbird

As the fall term approaches, I hear as clear as ever Mike Starbird's words, "We are in the business of transforming lives!"   It's important to remind ourselves that we are not just teaching algebra or calculus.  We are using mathematics as a vehicle to help students find their mathematical talent.  We are nurturers of talent, not machines created show students how to plug in numbers or solve for $x$.

When I think of the fall term and my classes, I also think about how I can help create more opportunities for students to transform themselves, their self image in math, and how they go about learning and doing math.  Before diving into all of the details of building up a syllabus and deciding how much this or that, the beginning of the academic year is an opportunity for teachers to revisit the reasons why we teach and what the point of education is.

Cheers from SLO!

"Thank you for treating us like professionals"

This week I had the pleasure of working with San Luis Coastal Unified School District math teachers.  My colleagues Linda Patton, Marian Robbins, and Elsa Medina did a fantastic job of running sessions specially designed for K-12 math teachers.  The teachers were inspired to think, to problem solve, to work together, and to think of ways to get their students to do some real mathematics.  It was a great 3-day workshop, and I am looking forward to the follow-up days in the coming year.

On the last day of the workshop, one of the teachers said, "Thank you for treating us like professionals."  The teachers were kind, energetic, wonderful to work with, and highly appreciative of the support we provided.  They accomplished a great amount in a short time.  But this workshop experience, where teachers feel like professionals and are treated as such, is not the norm.  It is a strange thing that such a circumstance could even exist in the U.S.  The fact that society has evolved to make teachers feel marginalized, despite their importance to society and our future, is astonishing.

In future posts, I'll dive into some of the ideas of why this issue is really a signal for a larger set of problems in education reform.  That is, how society treats, supports, and values teachers in society leads to key insights.

For college faculty, what all this probably means is that the (relatively) easy part of the whole enterprise is to teach future teachers the math (or fill-in your subject).  

The Colorado Study: The Vectors Are All Pointing in the Same Direction

Sandra Laursen, Marja-Liisa Hassi, Marina Kogan, Anne-Barrie Hunter, at the University of Colorado Ethnography & Evaluation Research and Tim Weston, ATLAS Assessment and Research Center University of Colorado Boulder have conducted a large, mixed-method study of IBL at 4 research universities in the U.S.  This is the largest study of its kind, and the results are striking.  (Link to the study)

What did they learn?

• LESS instructor talk time, results in BETTER outcomes.
• Women in IBL classes reported as high or higher gains than their male classmates across all cognitive, affective and social gains areas (3.2.3). But women in non-IBL classes reported statistically much lower gains than their male classmates in several important domains: understanding concepts, thinking and problem-solving, confidence, and positive attitude toward mathematics.
• Among students who entered with low math GPAs (<2.5), IBL students generally earned better grades in later classes than did their non-IBL peers (6.4.1).
• Attitudinal changes were modestly positive in IBL groups, and mixed and somewhat negative in non-IBL groups.  Overall, IBL math courses tended to promote slightly more sophisticated and expert-like views of mathematics and more interactive approaches to learning. In contrast, traditional mathematics courses appeared to weaken students’ confidence and enjoyment, and did not help them to develop expert-like views or skillful practices for studying college mathematics.

The overall results of the Colorado study points in the same direction as the bulk of the results from the Math Education literature.  When students are (a) deeply engaged in high quality mathematical tasks requiring critical thinking and reasoning, and (b) have some form of collaboration, then student outcome are statistically significantly better compared to students in a non-IBL setting.  (Collaboration is broadly defined here.  Collaboration is not only group work, but includes activities such as class discussion and student presentations to the whole class.  In this last instance, the class is peer-reviewing the presenter's work.  This is collaboration.)

When the Colorado study is combined with the literature from Math Education, then we start to put together a rather clear picture.  We have known already that students have historically had poor attitudes and beliefs about Math from K-college.  Students have beliefs such as "all problems can be solved in 5-minutes or less" and "it's the form of the answer that important, not the quality of the process or content of the proof."

We also know that students, even college students, have limited ability in problem solving and proof.  Students are rarely exposed to the kinds of experiences necessary to develop problem solving, the critical thinking and reasoning for proof, and other higher-level thinking strategies.  High stakes testing and the drive for further standardization has made it more difficult for students to develop the kinds of attitudes and thinking skills needed to learn math beyond rote skill.  

In light of this, the Colorado study shows that IBL methods (broadly defined) is a glimmer of hope.  When students are given a chance to think for themselves and are properly supported by the instructor and their peers, that students are capable of rising up and fulfill their potential.

The data speaks -- all the vectors are pointing in towards IBL.

Why IBL? The Road to Present Day IBL

By Amélie G. Schinck

(Originally posted on the AIBL website)

Inquiry-Based Learning (IBL) is not a recent or passing movement in mathematics education. IBL is based on a wide body of research and has a long track record of success. The following is an outline of IBL’s theoretical background and empirical grounding.

At the university level, IBL is also known as the Modified Moore Method (MMM), named after professor R. L. Moore of the University of Texas. In the majority of undergraduate mathematics classrooms across the nation, “doing mathematics means following the rules laid down by the teacher; knowing mathematics means remembering and applying the correct rules when the teacher asks a question; and mathematical truth is determined when the answer is ratified by the teacher” (Lampert, 1988, p.437). Moore aimed to challenge students’ assumptions about what it is to do, know, and understand mathematics. Beginning in the 1920’s, and continuing for half a century, Moore taught collegiate mathematics through inquiry, challenging his students to think like mathematicians. Moore believed: “That student is taught the best who is told the least” (Parker, 2005, p.vii). Through a sequence of carefully crafted problems and theorems, Moore would have students pose conjectures, construct their own proofs, justify their reasoning to their peers at the board, and assess the validity of proposed solutions and proofs. Textbooks were generally not used. Lectures were kept to a minimum. Collaboration between classmates was strictly prohibited. For a biography of R. L. Moore, and an account of the origin and impact of the Moore Method, see Zitarelli (2004) and Whyburn (1970). For more information on R. L. Moore, also see http://legacyrlmoore.org.

The Modified Moore Method is a less strict version of Moore’s approach to the teaching and learning of mathematics. For instance, MMM courses may make use of an IBL inspired textbook. Varying degrees of importance can be placed on formal examinations. Student collaboration is sometimes encouraged, with solutions to problems shared during small-group and/or whole-group discussions. For descriptions of different modifications and their rationale, see Chalice (1995), Mahavier (1999), and Padraig & McLoughlin, (2008). For some examples of IBL textbooks, see Burger & Starbird (2005), Hale (2003), Schumacher (1995) and Starbird, Marshall & Odell (2007). For refereed, IBL classroom tested course notes for university level mathematics classes, visit the website for the Journal of Inquiry-Based Learning in Mathematics (www.jiblm.org).

Students are thus engaged in the creation of mathematics, allowing them to see mathematics as a part of human activity, not apart from it. MMM courses are in direct contrast to the traditional lecture-based approach to the teaching of mathematics. Reporting on his use of MMM, Chalice (1995) stated:

While using this method, I have been able to cover as much material (and in few cases more material) as in the usual lecture-style course. More importantly, with the Modified Moore Method, the students and I have covered that material in a far more enlivening, enjoyable, and intellectually stimulating way (p.317).

An inquiry-based approach was recommended by the National Science Foundation in their 1996 report of a year-long review of the state of undergraduate Science, Mathematics, Engineeringand Technology (SME&T) education in the United States entitled Shaping the Future (NSF, 1996). In this report, the researchers stated that it is imperative that:

All students have access to supportive, excellent undergraduate education in science, mathematics, engineering, and technology, and all students learn these subjects by direct experience with the methods and processes of inquiry (NSF, 1996, p.6).

The IBL movement found in undergraduate mathematics, and supported by the Academy of Inquiry-Based Learning (AIBL), is in line with, and a natural extension of, the reform efforts in grades K-12. Recommendations by the National Council of Teachers of Mathematics (NCTM) for the past three decades (NCTM, 1980, 1989, 2000) have consistently included a call for a focus on teaching problem solving by teachers, positioning problem solving ability as the overarching goal of mathematics education. These recommendations are founded on the notion that the learning of mathematics is an active, social process in which students construct new ideas or concepts based on their current knowledge. Student understanding is connected to open- ended questions and an inductive teaching style. Principles and Standards for School Mathematics (NCTM, 2000) emphasizes the need for teachers to create a culture of learning in their classroom in which students learn with understanding and construct conceptual mathematical meaning through a problem-solving approach:

Problem solving means engaging in a task for which the solution is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understanding. Solving problems is not only a goal of learning mathematics but also a major means of doing so. (NCTM, 2000, p.51)

Discussing the importance of fostering Reasoning and Proof in Grades 9-12, Principles and Standards for School Mathematics (NCTM, 2000) states:

As in other grades, teachers of mathematics in high school should strive to create a climate of discussing, questioning, and listening in their classes. Teachers should expect their students to seek, formulate, and critique explanations so that classes become communities of inquiry (p.346).
To sustain and support the recommended focus on problem solving, active learning and inquiry in grades K-12, undergraduate mathematics education must also change, especially in the area of teacher preparation (NSF, 1996).

As mathematics education researchers turn their attention to IBL, evidence is mounting that this approach to the teaching of mathematics is ideal for the teaching of proof (e.g. Smith, 2005; Dhaler, 2008). Despite the emphasis on proof in higher level undergraduate mathematics courses, research on students’ conception of proof consistently shows that most struggle with appreciating, understanding and producing mathematical proof (Dreyfus, 1999; Harel & Sowder, 1998; Jones, 2000; Selden & Selden, 1987, 2003; Weber, 2001). Many mathematics educators argue that students’ (mis)conceptions about proof are the inevitable result of the traditional, lecture-based approach to the teaching of proof (Dreyfus, 1999; Harel & Sowder, 1998; Jones, 2000; NCRTL, 1993; Shoenfeld, 1988; Silver, 1994; Smith, 2005). In his article Why Johnny can’t prove, Dreyfus (1999) noted that “the ability to prove depends on forms of knowledge to which students are rarely if ever exposed” (p.85). Dreyfus (1999) concluded that a classroom environment in which students are required to explain and justify their reasoning is key to helping students transition from a computational view of mathematics to a view that conceives of mathematics as a field of intricately related structures.

Smith (2005) reports on the results of an exploratory study of the perceptions of mathematical proof and strategies for constructing proof of undergraduate students enrolled in lecture-based and problem-based (MMM) “transition to proof/number theory” course. Smith (2005) found evidence that the problem-based approach provided students with more opportunities to make sense of the proof construction process in a personally meaningful way than the lecture-based approach. Smith (2005) noted marked differences between the two groups of students. For instance, students in the lecture-based course focused on the form of the proof rather than on its meaning, and were reluctant to work concrete examples. On the other hand, students in the MMM course emphasized meaning over surface features, introduced notation in the sense-making process, conjured up previous proof strategies on the basis of the concept under study, and made use of concrete examples to gain insight into the main idea. Based on these results and the preliminary analysis of other collected data, Smith (2005) hypothesized that classroom communities of inquiry (such as MMM) encourage students to produce proofs by making global or intuitive observations about the mathematical concepts and transform these observations into formal, deductive reasoning.

In a dissertation study on the effects of the Modified Moore Method on college students’ concept of proof, Dhaler (2008) found that MMM had a positive effect on student’s conceptualization of mathematical proof, as well as on self-confidence in their abilities, their appreciation of the relevance of proof, and their ability to be independent thinkers.

An inquiry approach to teaching has also been shown to have a positive effect on students’ acquisition and retention of conceptual understanding. At the K-12 level, Boaler (1998), for instance, showed that students who learned mathematics in an open, project-based approach developed superior conceptual understanding to their counterparts who had learned the same subject matter through a traditional, textbook approach. A central part of Boaler’s study was to compare students’ capacity to use their mathematical knowledge in new and unusual situations. Boaler (1998) found that students who had been taught in the traditional way “did not think it was appropriate to try to think about what to do; they thought they had to remember a rule or method they had used in a situation that was similar” (p.47). On the other hand, students who had been taught in an open approach could use mathematics in novel situations as they had developed the belief that mathematics required active, flexible thought. Furthermore, they had gained the capability to adapt strategies and methods depending on the situation. Though the project-based approach described in Boaler (1998) is not identical to the inquiry-based learning approach sponsored by AIBL, the implications of Boaler’s study remain ; conceptual understanding is improved when students learn mathematics by engaging in inquiry.

At the undergraduate level, Rasmussen & Kwon (2007) provides a summary of two quantitative studies that assessed the effectiveness on student learning of an inquiry-based approach to the teaching of differential equations (as part of the Inquiry Oriented Differential Equations (IO-DE) project.) Rasmussen, Kwon, Allen, Marrongelle, and Burtch (2006) compared students that had taken inquiry-oriented differential equations (IO-DE) classes versus students that had been taught using a traditional approach. Rasmussen et al. (2006) found that although the two groups did not show a significant difference in procedural fluency (i.e. routine problems), the IO-DE group scored significantly higher on conceptual problems.

In a follow-up study one year later, Kwon, Rasmussen, and Allen (2005) compared the retention effect on procedural and conceptual understanding between the traditional and IO-DE group. The data showed no significant difference between the two groups in procedural fluency. However, the IO-DE group showed a significant positive difference compared to their traditional counterpart on conceptual understanding.

The IO-DE project described above uses an adaptation of Realistic Mathematics Education (RME), an inquiry approach to the teaching of K-12 mathematics in the Netherlands. RME is based on curriculum developed at the Freudenthal Institute. Through the posing of true problematic situations (not simply “word problems”), RME encourages student investigation and inquiry. Students’ construction and representation of mathematical concepts such as number sense is valued. The book series Young Mathematicians at Work by Fosnot and Dolk outline the translation of the Dutch approach to the teaching of mathematics to numerous American urban classrooms.

The famous Swiss psychologist Jean Piaget stated: “to understand is to invent”, highlighting the active nature of the learner. The above discussion provides an outline of the theoretical foundation on which Inquiry-Based Learning rests. Furthermore, it provides a summary of the mounting evidence that students who are given the opportunity to learn mathematics through inquiry develop deeper procedural and conceptual understanding of mathematics.

References

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41-62.

Burger, E., & Starbird, M. (2005). The Heart of Mathematics: An Invitation to Effective Thinking, Emeryville,CA: Key College Publishing.

Chalice, D. R. (1997). How to teach a class by the Modified Moore Method. The American Mathematical Monthly, 102(4), 317-321.

Dhaler, Y. Y. (2008) The effect of a Modified Moore Method on conceptualization of proof among college student. Dissertation Abstracts International Section A: Humanities and Social Sciences, 68(11-A), 4591.

Dreyfus, T. (1999). Why Johnny can’t prove, Educational Studies in Mathematics, 38, 85-109. Hale, M. (2003). Essentials of mathematics: Introduction to theory, proof, and the

professional culture. Washington, DC: MAA.

Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education III (Vol. 7, pp. 234-283). Providence, RI: American Mathematical Society.

Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53-60.

Kwon, O.N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics, 105, 227-239.

Lampert, M. (1988). The teacher’s role in reinventing the meaning of mathematical knowing in the classroom, in Proceedings of the PME-NA, pp. 433-480.

Mahavier, W.S. (1999). What Is The Moore Method? Primus, 9(4), 339-354. Martin, G., & Harel, G. (1989). Proof frames of preservice elementary teachers, Journal for

Research in Mathematics Education, 20(1), 41-51. National Center for Research on Teacher Learning (1993). Findings on Learning to Teach, Lansing, MI: NCRTL.

National Science Foundation (1996). Shaping the future: New Expectations for Undergraduate Education in Science, Mathematics, Engineering, and Technology, Advisory Committee to the NSF Directorate for Education and Human Resources. Accessible at www.nsf.gov (file nsf96139)

National Council of Teachers of Mathematics (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA.

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics.Reston, VA.

Parker, J. (2005), R. L. Moore: Mathematician and Teacher, Washington DC.: Mathematical Association of America.

Padraig, M, & McLoughlin, M. (2008). Inquiry Based Learning: A Modified Moore Method Approach To Encourage Student Research. Paper presented at the 11th Annual Legacy of R. L. Moore Conference, Austin, TX.

Rasmussen, C., & Kwon, O.N. (2007). An inquiry-oriented approach to undergraduate mathematics. Journal of Mathematical Behavior, 26, 189-194.

Rasmussen, C., Kwon, O. N., Allen, K., Marrongelle, K., & Burtch, M. (2006) Capitalizing on advances in mathematics and K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. Asia Pacific Education Review, 7(1), 85-93.

Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well- taught” mathematics courses. Educational Psychologist, 23, 145-166.

Schumacher, C. (1995). Chapter Zero; Fundamental notions of abstract algebra, Addison- Wesley Publishing Co.

Selden, A., & Selden, J. (1987). Errors and misconceptions in college level theorem proving. In J. D. Novak (Ed)., Proceedings of the second international seminar on misconceptions and educational strategies in science and mathematics (Vol III, pp. 457-470). Ithaca, NY: Cornell University.

Selden, A., & Selden, J. (2003). Validation of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34, 4- 36.

Smith, J. C. (2005). A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses. Journal of Mathematical Behavior, 25, 73-90.

Starbird, M., Marshall, D., & Odell, E. (2007). Number theory through inquiry. DC: MAA textbooks.

Weber, K. (2001). Student difficulty in constructing proof: The need for strategic knowledge, Educational Studies in Mathematics, 48(1), 101-119.

Whyburn, L.S. (1970). Student oriented teaching – The Moore Method. The American Mathematical Monthly, 77, 351-359.

Zitarelli, D. E. (2004). The origin and early impact of the Moore Method. The American Mathematical Monthly, 111(6), 465-486.