Ed News: Middle Class Squeeze

While not directly IBL related, this affects all of us at the college level.  The rising cost of college education is a growing concern.  UC Berkeley has announced that it will begin subsidizing the cost of education for middle class families.  That is, families with incomes between \$80,000 and \$140,000 per year will receive a reduction in tuition.  The annual cost of attending to a UC school is now up to $32,000 per year for California residents.

The original CA master plan intended to provide free college education for qualified students in CA.  That dream is over for good.  More importantly, the cost of education has risen to the point where even middle class families struggle to afford paying for public colleges.  This is not good if our best and brightest students are not able to attend our best public colleges in our state.  It means we ultimately are squandering chunks of our society's talent pool.  Not good.  More on this later.

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?

Nuts and Bolts: Assessment of Presentations

IBL courses deserve assessments that match what students are asked to do in class.  In most full IBL courses, presentations form a significant portion of the grade.  Grading breakdowns could be something like this:

30% Presentations
30% Final Exam
20% Midterm
10% Written work
10% Portfolio

In this post I'll focus on presentations, and discuss the other items in future posts.  Student presentations are one of the key components of an IBL course.  Students should present their work to their classmates to be vetted for correctness and clarity.  This process by itself is immensely valuable.  In fact, it lies at the heart of IBL.  Students work on hard problems at home.  They bring back their findings, get feedback and repeat, until a problem is solved.

Assessing presentations can take on many forms.  One way is to use a point scale.  Here's a rubric which can be adjusted to suit the style of an instructor.
10 Completely correct and clear
8 or 9 minor technical issues, but the proof is correct
5, 6 or 7 Proof is incorrect, has a significant gap(s)

Points don't tell the whole story.  Instructors should have in mind the overall qualities they want from students.  Below is a general guideline that can be adapted to match your own criteria and your institution's.  Students should be encouraged and rewarded for their intellectual contributions to the class.  Generally students should present regularly and also participate meaningfully in discussions in groups and in class.  Students who are able to prove major theorems and present regularly in class should earn an A for presentations.  They are capable of doing original (to them) mathematics.   Students who present regularly, but none of the harder problems typically earn a B for presentations.  Students who show up to class regularly and present only occasionally earn a C for presentations.  Students who miss a significant number of classes and/or do not present more than once typically would earn a D or F for their presentation grade.

*Some instructors also include some bonuses for creativity and ingenuity.  When a student does something that you have not seen before that shows real creative thinking, it should be rewarded in some way.  I take notes in class and write comments and jot down the creative idea.

The main message is that students have an incentive to

• Show up to class having worked on problems
• Participate and discuss mathematical ideas
• Prove theorems on their own
• Contribute to their own and their classmate's intellectual development

Are these not the qualities we want from our students?  Aligning incentives in positive ways to what society values in people is a something we should strive for.  Assessment is not merely a way to determine grades -- it's also a tool to encourage students to be young mathematicians!

Nuts and Bolts: Getting Students to Ask Good Questions

By Matthew G. Jones, Cal State Dominguez Hills <mjones@csudh.edu>

One of the keys to making IBL work is to make sure that students really engage with the topic. There are lots of ways to do this, but I will focus on two ways to get a whole class discussion going. The first of these ways is to use question starters. I have done this in two ways. In the first case, I simply write a question on each of three or four index cards, and distribute them around the classroom while another student is presenting a solution. The index cards have a single question type, such as, "How did you know where to start?" "Could you explain how you went from ... to ...?" "Can you explain what the question is asking?" "Was this the first thing you tried?" The students are told that if they are given a card, they must pose a question, and they can use the one on the card if they wish. This way, those few students are thinking of a question to ask while the presenter is working. The presenter also knows to expect questions, and the class understands that discussion of the solution is the norm, rather than passive silence. The other way I have seeded questions is to hand out a sheet to the entire classroom, and to tell them that you will call on a few students to pose a question, and that they can use the handout to help them formulate a question.

In either scenario, if I call on a student who claims to have no questions, then I will ask the student to paraphrase a specific part of the presentation, such as, "Could you explain what you think the presenter means by this line?" or "What is the goal of this part of the solution?"

The second way to get a whole class discussion going is to let a presenter complete his/her solution, and then to give the student observers 2 minutes to discuss the solution with a neighboring student in the room. Sometimes I will ask a pointed question to prompt the discussion, such as, "What kind of proof did the presenter use, and why do you think that was his/her choice?" and sometimes it is left open. Then, I open the whole class discussion by calling on students and asking, "What did you and your partner discuss?" This kind of question diffuses the pressure for students to report themselves as confused, because they often give replies like, "We were trying to figure out..." or "We weren't sure about..." The main idea is that students are more comfortable reporting on their actions in a partner discussion than identifying their personal confusion or misunderstanding.

Carol Schumacher, Kenyon College

This article originally appeared on the AIBL User Experiences website in 2010.  It has been reposted here on the IBL Blog, the new home of AIBL User Experiences.



By Carol Schumacher, Kenyon College
Inquiry-based learning (IBL) has been transformative for me as a student and as a teacher.  I was an undergraduate at Hendrix College in Conway, AR, where I was fortunate enough to have all of my upper level courses taught with an inquiry approach.  I had always liked math and had been good at it, enough to be pretty sure from an early age that I wanted to major in math in college.  But it was my first IBL course that really made me a mathematician.
For me, as a student, the inquiry-based approach was exciting and empowering.  For the first time, I had a real sense of “ownership” for the mathematics I was doing.  I was able to see that mathematics is not only a set of techniques and ideas that I could master, but a powerful way of thinking.  Moreover, my appreciation for the beauty of mathematics increased, and I came to crave the “high” that I got from solving a problem or proving a theorem on my own.  So the inquiry approach also made me thirst for more mathematics in a way previous courses had not.

When I went to graduate school, many of my fellow students “knew” more mathematics than I did.  But this was really no problem, because I knew how to prove theorems.  When a topic came up that I had never studied, I never had a problem learning it on my own.  And I was pretty much fearless in the face of the problems that were set before me.  At one point, I was flabbergasted to find out that some of my fellow graduate students just scoured the library for solutions to the assigned problems.  This approach would never have occurred to me.  My IBL training showed me that mathematics is something you do, rather than something you read about in books.

When I began teaching, I knew that I would use an inquiry-based approach in all of my upper level theory courses.  So you may wonder in what sense it has been “transformative” for my teaching.  There are a number of ways.  Inquiry-based teaching is perforce centered on students’ learning.   (The main question in our preparation as IBL teachers is not “what am I going to do in this class,” but “how shall I craft these materials so that the students can make headway in the mathematics?”)  This insight from inquiry-based teaching, has infused all of my classes.  I have improved as a teacher by thinking less about teaching and more about learning.  Because it is not what I do, but what happens to my students that is important.

I certainly do not teach all of my courses with a strict inquiry-based approach, but I do believe that engaging students actively in the business of doing mathematics is the key to fostering deep learning.  This is just as true for non-majors taking a liberal arts math course or a calculus course as it is for a graduate school bound math major.  And I have found that I can adapt inquiry ideas for use in classes of all kinds and at all levels.  Even in courses that are more traditionally “teacher centered,” I intersperse periods of lecture and discussion with opportunities for students to work through some ideas on their own.  And I am entrepreneurial in looking for ways of turning the class over to my students in the form of active learning and exploratory exercises.  I have found that when I can devise a good way to substitute something that the students do for something that I have previously tried to do for them, it is always a good idea. The learning is more profound and the knowledge longer lasting.  (I was told by a recent graduate who is now a graduate student that he remembers the content of my courses best, because he developed the subject himself.)

Inquiry-based learning is, also, transformative for many students.  An IBL course can completely change the outlook of mathematically talented students who have never before found it particularly interesting.  They come to see that mathematics is not just as a set of techniques for building the new and improved mouse traps of the 21st century, but is a powerful tool for understanding the world around them.  They learn to appreciate this tool and they learn to wield it by making it their own.  What more could we want?

Links to some IBL textbooks written by Carol Schumacher:
Chapter Zero
Closer and Closer

Reminder: AIBL Small Grants Program

A quick reminder... The AIBL Small Grants program deadline is coming up soon.  You still have time to apply and receive support for your efforts.

Click Here for more information.

See What They Can Do

My good friend and colleague, G. Edgar Parker, Professor of Mathematics, James Madison University, has written a wonderful story for The IBL Blog.  Thank you, Ed!


See What They Can Do

The bedrock for what became the basis for my teaching philosophy was forged when I was an undergraduate at Guilford College.  I was in college for all of the wrong reasons; I didn’t like school, but I knew my parents expected me to go to college and I was in no hurry to find gainful employment unless I could make a living playing baseball.  I was at Guilford for all the wrong reasons; I didn’t want to follow my older brother Elwood for another four years (the expectations associated with coming behind a straight-A student who was also a two-sport varsity athlete, all-conference in one sport and team captain in the other, even if they were imaginary, seemed real to me), but I went anyway because I had seen the baseball teams that my two older brothers played on there and knew that I could pitch better than anybody I had seen pitch in those six years (and besides, it had snowed on my campus visit to the school I visited that I liked the most and it was the week before the regular baseball season’s schedule back home was to start).

During the advising session before my first semester, fall of 1965, I jettisoned any ideas of following the subject that intrigued me most, Biology, because I found out that, at Guilford, you went to labs as a priority over practice and made up labs if you missed them for games.  My advisor, J. R. Boyd, placed me in two mathematics courses, Calculus I and Mr. Boyd’s Linear Point-Set Theory.  Interestingly enough, it was calculus that scared me.  I had actually heard of that and didn’t think I was prepared.  I should have guessed that Mr. Boyd was cut from different cloth; when I protested that I had only had four weeks of trigonometry in high school, he had “reassured” me by telling me, “Don’t worry, if you need to know more, you’ll learn it.”.

I was so naïve that I just assumed that the way point-set was being taught might be something that people did in college, so I just tried to solve the problems and hung on.  Since Mr. Boyd never said anything negative, for all I knew, I was doing okay.  I was getting some problems, or at least Mr. Boyd seemed satisfied with what he wrote on the board when I described my arguments.  Since I didn’t know the difference between a hard problem and an easy problem, I was trying them in the sequence they came in the notebook.  And then I ran into the problem that, looking back, could have been stated as “the continuum is not denumerable”, but was stated in a way that tempted a student to find a map from the natural numbers onto the numbers.  I thought I had an argument, but Hal Phillips, whom I considered to be the best student in the class, chose the problem when his turn came.  As Hal presented, I saw his ideas (which I had discovered myself) crash one by one and finally doom the argument on which I had worked so long and hard.  I was, at that time, still pretty meek in public and very reticent to call attention to myself, but the ardor of the moment overwhelmed me and I burst out in class in frustration (Mr. Boyd had made no suggestion as to how to remedy the now-evident flaws in the argument), “Mr. Boyd, why don’t you just show us how to do it?”.    Without missing a beat, he turned to me and said, “Mr. Parker, why should I limit you to what I know?”.  It was an opportunity to learn a lesson about teaching, but at the time, I was just relieved at the way Mr.
Boyd handled the situation and didn’t add to the embarrassment I had created by calling attention to myself.  Besides, I still wanted to pitch in the Big Leagues and education was just a diversion to enable me to continue to play the game I loved.  Education was not my intended vocation.

Fast forward to 1977.  I had finished my PhD in mathematics at Emory University, had courses given by Moore method from David Ford, Bill Mahavier, John Neuberger, Phil Tonne, and Mary Frances Neff, and chosen academe as my vocational home.  I recognized by now that Moore method is what allowed me to blossom creatively and provided me with the tools to learn other persons’ mathematics as well.  Still possessing, at that time, some sense of humility, I concluded that Moore method should provide the same growth potential for others it gave to me and decided to use it as the core for teaching the mathematics courses in the major at Pan American University that I gave.  Over the next seven years, for each such course, I painstakingly constructed a problem sequence with the self-assurance that, anybody who could do the problems in sequence would have to be able to see how to do the next one in the sequence because the connections were so obvious to me.  And each time, as often as not, the students leaped right past my lemmas to the important problems and found ways I hadn’t thought of to do the problems or ways that I considered “less natural”, or patiently proved my lemmas, but then showed me what the lemmas should have been by proving the theorem without using the lemmas.

Fast forward to the summer of 1988.  I had been at James Madison University since 1984 and I now understood how to follow the students’ ideas in the major courses, but still thought, for some reason, that non-majors had to be led to the fountain before they could drink.  That summer I was assigned yet another section of Mathematics 103: The Nature of Mathematics, the lowest numbered mathematics course in the catalogue and a course that I had taught every semester since it was introduced in 1985 (I had developed such a course for a less well-prepared clientele at Pan American, so, naturally, even though I had voted against our department offering such a course for General Studies at James Madison since I thought a school of our pretension should make calculus the core requirement, I was picked to give one of the initial offerings and was blessed with it on a continuing basis.) .  On a lark, I had the following conversation with myself: “The students are in this course because all they want is a mathematics credit.  So no one will fail the course.  But I will not tell them that and I will teach them as if they were majors.”  So I designed a course in which I “taught” them rigor by having them justify, on the basis of the field axioms applied to the words number, +, and $\times$, some of the fundamental techniques from high school algebra, and then put them to work on proving, as theorems, the field axioms stated for the ordered pairs of numbers with addition and multiplication defined so as to make the structure the complex numbers (without having i a part of the notation, I hoped nobody would recognize that they may have studied this algebra already and I was correct in this guess).  The only modification of Moore method I made was that the class was split into eleven groups, each with two or three students, and each group was responsible for proving a single theorem (chosen by lot), with all students responsible for certifying the correctness of the arguments and for reproducing and using what they certified to be correct.  The class got them all!  It was not that the students populating this class could not do the mathematics, it was my not believing they could do it that never gave them a chance to do it.  The summer of 1988 may have been an act of God.  I have used this problem set many times since, and no other class has gotten them all.  But every class has gotten some of them, and most classes have gotten the existence of reciprocals, the problem I consider the most difficult in the problem set.

Dr. Parker, why should you limit your students to the way you think about things, and why should you impose limitations on them?  


The longer I teach, the more I wonder why I didn’t ask myself this question earlier.

G. Edgar Parker, James Madison University

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.

Nuts and Bolts: Small Groups

The topic of this post is group size in collaborative learning environments.  There are differing opinions about this, and I will not go into any data regarding small group sizes.  Rather I'll talk about what I do, and you can use this as your starting point.

Size:  I like groups of size 2 or 3.  Clearly 2 is the smallest group size.  Once groups get to size 4 or greater, students can hide.  This is something I want to avoid.

Mix 'em up -- generally it is a good idea to move students around.  One can use the Random() command in excel to create a random list of numbers, and then sort students on that column.  Then take bunches of 2 or 3 students to form the groups.

Random shuffles aren't always the best, however.  For instance, there may be certain personalities you want to keep away from each other.  In this case, one has to do some group engineering.  Experience tells me that as you get closer to the boundary of a classroom (i.e. the walls), there exists a greater probability of personalities, especially if groups were allowed to form naturally.  I also look for students who are quiet.   Then I make a list of personalities and quiet students.  Seed each group with exactly one from this list, and then fill the groups with the other students.

The number of people in the group is but one aspect of heathy groups.  Students can get noisy, chatty, or a person can start dominating discussions.  It is important to be an active coach and mentor.  An instructor might be called upon to manage personalities and guide through gentle questions/directions like, "Can I count on you guys to work as a team today?" or "Let's stay in mathland (to the student texting)."

It can also be useful to talk about what it means to be a good group member.  A good group member shows up to class ready and on time, listens carefully to their partners, is supportive, and offers appropriate levels of help at the right time (i.e. no blurting out answers and having a genuine respect for learning).

Short version:

• Find a good size (2 or 3, in my opinion) and mix of students.  
• Coach and mentor group dynamics until they work as teams.

"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!

AIBL Small Grants Program: (For Faculty at U.S. Institution Only)

The Academy of Inquiry Based Learning has small grants for mathematics faculty working at U.S. institutions.  Click on this Link for descriptions of the categories, eligibility, and application process.  The deadline for applications is November 8th.

Video: Ken Robinson "Do Schools Kill Creativity?" (2006)

When I start a new academic year, I like to revisit sources that help me re-center as I plan.  It's like being an athlete and doing drills and base training to get the body ready for an upcoming season.  I revisit why we teach as a check to keep me from straying from the real reasons why we love our jobs as teachers.  We are also privileged as math experts to have the capacity to see the wonder and beauty that lies far beneath what nearly everyone else considers mathematics.

What we are presented with every academic year are opportunities.  Opportunities to use Mathematics (or whatever subject you teach) as a vehicle to provide transformative experiences for students.  We very easily can get bogged down in the details of covering chapters 1-8, making sure all the t's are crossed and techniques are sharply executed.  But what we are really after is, if you think for a moment, is helping students become powerful, creative learners.   

Ten or fifteen years later, what do you want your students to have retained from your class?  My guess is that knowing how to compute $\frac{d}{dx}\arctan(x)$ isn't highest on the list.

Nuts and Bolts: Homework Templates

One big idea I learned in college English and Literature courses was the notion of writing and rewriting drafts.  Constructing an essay is a process -- you read, you think of some idea you want to argue for, you construct an outline, you write a draft, and then you rewrite and reprocess until a finished product (or the deadline) arrives.

Of course you learn this in any subject.  To master any discipline require a long, long process of thinking, working, reworking,... Discipline and focus lead over time to an increase in skill and understanding.

All too often in math courses, the content is broken down into bite size pieces.  That is, the material is overly preprocessed for "easy" consumption.  Students follow.  Teachers get high marks.  Life is good.  But we know that's not good enough.  Certainly this doesn't develop the kinds of work ethic, habits of mind, and problem-solving ability we value.

One issue that needs to be dealt with is the specific process students use in their daily practice.  Do they just try something and either (a) get it or (b) give up after getting stuck?  This is probably the case for far too many students.  Poor process and practice leads to poor results, and diminished intellectual develop in the long run.   To shift the nature of the practice is not an easy task, and what I propose is not *the* solution -- it's just a start to one.

How does it work?
In my classes, I require students to use a homework template.  It's just a word document with the name of the course at the top and two little prompts.  The first is the statement of the problem or task.  The second is the solution/work/whatever the student has.   Students are instructed to do scratch work on separate paper, and most importantly analyze their work and transfer it to a final version onto the template.  One problem per page, unless the problems are really short.  Students are given the option of including their scratch work to the problem.

This system forces the drafting process that is much more explicit in the humanities and is easy to pull off in math classes.

A sample Template

Does it work? Yes.  The quality of student work improves considerably.  Instead of getting scratch-work quality written work, students are now required to take their homework more seriously.  Students also comment that they feel proud of their work, and why not?  If you think of an original idea, and present it well, that's something you should be proud of.  The homework has good content, it looks good, feel professional,... a job well done!

Does it solve all problems in the math process "pipeline"?  No. Templates and requirements of drafting do not address all issues.  Teaching is a complex system, and addressing students' process of doing homework is but one part of the picture (though a very big one).  It is, however, a very welcome step in the right direction.

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.

Mistakes Are Good!

One of the messages in an IBL that goes along with "Being stuck is okay!" is "Mistakes are Good!"

Mistakes are generally stigmatized in U.S. Math Education.  When a students does something wrong, it is unusual if the student thinks of the mistake or error as an opportunity for greater, perhaps even profound insight.

There exists many reasons why we build prototypes or practice in a batting cage or simply use scratch paper.  We need to see how things work.  We need to practice and fail, so that we can learn to do what is right.  In short, practice and experimentation.

One cannot grow without experimenting or trying things.  It would be nice if our students could all have the disposition to say things like "Let's see if the idea works for a special case..." or "Let's see if we can check our thinking..."

If students fundamentally believe that mistakes are bad, then the very nature of their interaction with mathematics is limited.  Over time this leads to poor self image and then ultimately poor habits of mind and work ethic.  That would be the nail in the coffin.

One of the ways to get students over the negative image of making mistakes is to provide opportunities for students to experiment, and to allow for mistakes to play a central role in the learning process.  In fact, in an IBL class students make *great* mistakes.  They say or do things in ways that maximize their learning.  As an instructor I no longer make these mistakes, because I already know the material.  First time learners of a subject reveal, through their mistakes, what they know and what they don't know.  This is where the learning zone is, and this is where one can create magical learning experience!

Student: <Writes or says something that is incorrect>
Teacher: "Oh, did you just say/write... Well I'm really glad you brought this up.  How many of your were thinking about this the same way?  Good!  Let's rewrite this as a question, and then investigate it further to get to the bottom of this."

A Good, But Not Good Enough Idea

In the NYTimes, Sol Garfunkel and David Mumford published a piece titled, How to Fix Our Math Education.  In this column, Garfunkel and Mumford claim that the curriculum should be changed to include more realistic and practical situations that are meaningful to students.  This I agree with 100%.

The problem with their piece is that the curriculum (AKA the specific content in courses) is only one of many issues.  Their analysis is limited, and here's why.

There are numerous examples from the K-12 system, where schools have adopted a research-based, NSF funded curriculum like IMP or CPM.  Many times these programs are shut down by some exogenous force.  For example parents, who have good intentions in their hearts have organized and shut programs down. They say things like, "Well that's not how I learned it...," never mind that the world has changed and students are being prepared for new challenges, not challenges from the 19th century.   Or a new superintendent wipes out the programs, or a new principal comes in and moves teachers to new classes or grade levels, or budget cuts,..., the list could go on and on.  I note that none of these people do this intentionally.  People in education and parents are well-intentioned.  They mean well.  It's just that we don't know better as a society, and look for simple fixes for long-term, multi-layered problems.  Insufficient.

The problems we face now are complex, and my sense is that they are on the order of magnitude to problems related to ecosystem sustainability.  Our educational system is vast and complex, with outdated doctrines interfacing with modern challenges, and a currently fashionable and misguided desire to apply business models to non-business systems.  So when someone says, "Just fix X, and all will be good," I know that it just won't.

Content (i.e. textbooks or materials) is just one small slice of the education pie.   In addition to content is

• Instruction (and all this huge category entails)
• Student attitudes, beliefs, habits of mind, experience with math, and their natural cognitive development mathematically.  (Example: the developmentally appropriate order of topics is sometimes different than the logical order in a subject.)
• External issues to the classroom like tenure and promotion for research (but not teaching), pacing plans, poverty, etc.
• Systemic issues like standardized testing (K-12) or standardized courses (calculus)
• The factory model mindset held by most people regarding education.  We value certain kinds of intelligence.  Math and reading are at the top....Art, Dance, Music, craft, and manual skills are at the bottom, which leaves some students feeling marginalized for being born with the "wrong" abilities.    
• We educate students in disciplines in ways such that they are not connected to each other, even though they often are connected and of more value when studied as a whole.
• Community and culture.  Most people think math is "2+2 = 4."  That there is only one answer.  That there is only one way to get to that singleton answer.  That math is immutable and the same across all time.  Learning math means you know how to calculate.  Additionally, Parents, even well educated ones, can destroy a positive change, thinking that they are doing a good thing.  I do not doubt parents' love of their children or desire for good education. It's just that we as a society are generally naive about what math is.  What we assume to be true about math is actually false.  So based on false assumptions we act unwittingly against our own interests. Mathematicians bear some of this responsibility, as we are not good ambassadors of our wonderful subject. 

Viewed even from this slightly broader perspective, changing the content without thinking about the education ecosystem is limited.

So how does IBL come into play?  At the college-level, a math instructor, who uses IBL, can control enough of the ecosystem within the classroom to construct a little, temporary greenhouse.  This little greenhouse has enough of the conditions necessary to foster intellectual growth and provide opportunities for students to undertake the effort required for transformative changes.  Considering how to scale these little greenhouses up is one way to realize the enormity of the challenge in education reform, and emphasizes that changing a textbook or the curriculum is like upgrading the rake or the shovel to a new, more effective one.  While it's a necessary step in the right direction, it is insufficient.

Respect the Struggle

In IBL classes, being stuck is a critically important part of the experience for students.  One of the greatest lessons a student can learn in school is how to manage being stuck.

One of the issues we face, particularly in the U.S., is that mistakes are stigmatized, and this context makes teaching problem solving more difficult.  When a student gets a problem wrong it is far too often interpreted as a criticism of their mathematical ability.  Yet, in contrast to this problem-solving ability and creative thinking are highly regarded attributes in all areas of life.

What can we do as teachers?  Students must know that

It's okay to be stuck!

In fact, being stuck is a noble state.  It's when we are stuck that we learn to learn. It's when we are stuck that we construct new ideas, and discard or improve upon ones that are not good enough for our current situation.  Being stuck is good!

One facet of effective teaching is respecting the struggle.  What this means is to allow students to struggle and think, in such a way that they are not stressed out, rushed, or feel that making mistakes is "bad."  Ensuring that students feel that they are allowed to explore, think, experiment, and build ideas that may not work is critical to building a positive learning environment.

Giving away answers or letting students flounder excessively are two ways we can get off track.  As a teacher one should monitor students and manage the struggle so that students are challenged, making progress (over time), and not overly frustrated.

How do you know if you have a positive learning environment in your classroom?  Ask yourself if your students feel it's okay to be stuck.

"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.

IBL User Experience: Kyle Peterson

Kyle Peterson is a Professor of Mathematics at DePaul University.


Let $n$ be an integer greater than one. Since $n$ and its successor,
$n+1$, are relatively prime, their product, call it $n_2 = n(n+1)$,
has at least two distinct prime factors. By similar reasoning, the
product of $n_2$ and its successor, say $n_3 = n_2(n_2+1)$, has at
least three distinct prime factors. Continuing in this way, we can
construct, for any positive integer $i$, a number with at least $i$
distinct prime factors. Hence the set of primes is not finite.

Not a bad little argument, eh? It's my favorite proof of the
infinitude of the primes, and I first read it on the homework turned
in by one of my students, whom I'll call Sara. If you are a regular
reader of American Mathematical Monthly, you may remember this
argument from a note by Filip Saidak in the December 2006 issue, two
months after Sara handed in her homework.

Sara was an average student, certainly not the ``best" of the class. I
had expected her, like many of her classmates, to construct an
argument similar to Euclid's classic contradiction argument, since
that's what my carefully chosen sequence of problems pointed to. (Or
so I thought.) I remember being so floored by what I read in Sara's
paper that I initially thought there must be some error. It was so
different from what I was expecting to see! But no, it was correct,
and I immediately ran across the hall to share it with a friend.

This is the potential of an IBL class. Given only the necessary
preliminaries, along with some time and space to think, Sara had come
up with something truly novel. It was just a month or so into my first
semester of IBL teaching, and, like the hack golfer who hits a
hole-in-one, I was hooked. I had never had a student so thoroughly
surprise me, and I wanted to experience more of those surprises.

I've been teaching IBL for about five years now, and while a gem of
that magnitude is rare, I find smaller surprises happen on a regular
basis in my IBL courses. For one thing, just because a student comes
up with a proof, the way they arrive at the proof often takes its own,
fascinating path. Other sorts of surprises include the time a student
who doesn't seem to be paying attention pipes up to point out a flaw
the rest of the class missed. Or when a student who would rather die
than go to the board in the first few weeks leaps out of her chair to
go to the board in the last few weeks. People laughing and smiling...
in a math class! Rather than: What do I have to talk about today?, I
walk into the room thinking: What will the kids show me today?

My favorite overheard conversation last year:

Student 1: ``So I think we have a bijection. Do we have a bijection?"
Student 2: ``I don't know...(mumble, mumble)... Wait! Yes!"
Student 1: ``We do?"
Student 2: ``Yes!"
Student 1: ``Yes!"
-Slap!- (the students give each other a high five)

Not every IBL course will necessarily produce a revelation like Sara's
proof, but they all produce everyday miracles that will delight and
surprise you. Using a lecture-only format pretty much guarantees that
your students won't surprise you, since they will be doing
their best to mimic what you do. To paraphrase something I've often
heard Ed Parker say: Why should we limit our students to what we know?

Classroom Strategy: Think Pair Share

One of the most effective and easiest ways to get students involved in your classroom is to use Think Pair Share.  Harvard Professor and Physicist, Eric Mazur, has been one of the strongest proponents of using peer instruction.

How does Think Pair Share work in a mathematics classroom?  It goes like this...

• Pose a question or task to the class, such as "Give an example of..." or "Which ones of these, if any, is an example of...?"
• (Think) Let students think about the question/task individually for about a minute (or whatever appropriate time) 
• (Pair) Ask students to explain their solution/idea/thoughts to one person sitting next to them.
• (Share) Involve the entire class in a discussion of the question/task.  A good way to start off is to walk around the class while the pairs are discussing and ask one or two pairs to share their ideas.
• (Recycle) If necessary, a class may not arrive at a consensus.  In such cases the class can re-enter the pair phase and work with their partner to sort out the details.

Below are some examples chosen from elementary Number Theory.  But these ideas can be easily adapted to any math course from Calculus to Math for Elementary Teaching to Real Analysis.

Example1:  (Starter question)

• State the definition of $n|k$, where $n,k$ are integers.
• Question: In your own words, interpret what $n|k$ means.  Write a few sentences.
• Share with your neighbor your sentences and revise if necessary.
• Pick a few groups to share their work.

Example 2:

• Task: Determine which of the following statements is true.  
1. If $n$ is even, then $2|n$.
2. If $n$ is even, then $n|2$.
• Think for yourself which one is correct.
• Convince your neighbor of your answer.
• Class discussion. Recycle if there is confusion or lack of consensus.  

Example 3:

• Question: If $n|a$ and $n|b$, then $n|(a+b)$.
• Think of some strategies you could use to prove this theorem
• Discuss your strategies with your neighbor.  Write questions, if you have any.
• Pick a few groups to share their strategies and/or questions
• Make a list of the ideas, and let students continue to ask questions.  Then one can move on to the next task, leaving the proof as a homework problem that will be shared later.  Another option is to let students come up with a sketch of a proof in class and clean it up at home to be turned in/presented the next time.

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.