Teaching Philosophy Thoughts and Statements


Fall 2009, Reflections After 6 Years Teaching at Carroll College:


First and foremost, I feel great gratitude for the privilege of serving here at Carroll College for the past six years.  The time that our young people spend here at Carroll as they pursue higher education is a very special time in their lives, a vital and important time when they are making decisions about who they will become and what directions they will pursue in life, and I consider it a great honor to be given the opportunity to help guide them through this time.


I am also very aware that my teaching is a work in progress.  I know that I have strengths and weaknesses in my teaching and in the coming years I hope to build on my strengths and to address my weaknesses.  I am learning.  I learn something new with every semester that I teach.  I have learned a lot in the time that I have been here, but I have also come to appreciate how much more I have to learn.  I find teaching to be a constant challenge.  I have learned that there is no “right way” to teach something, because every group of students is different.  Sometimes a lesson which works with one group of students will fail miserably with another group of students, and so I simply have to take a deep breath and try another approach.  Thus, I think that perhaps my greatest strength is my flexibility and my ongoing efforts to become a better teacher.


As fundamental principle, I think that what I do in the classroom is far less important than what the students themselves do, in and out of class.  The most important facets of my courses are what I require the students to do, the set of activities and assignments that I give to the students. 


I think that one of the greatest strengths that I bring to the classroom is my enthusiasm and passion for the subjects that I teach.  I believe that a vital component of my role as teacher is to persuade my students that the knowledge and skills we are studying are valuable and worth learning.  If I can convince them that this material is fun, powerful interesting, or useful, then they will make the effort to learn.  So I always make it a strong priority to bring energy and enthusiasm to the classroom.  Before every class, I take a few minutes to clear my head, and bring up a vision of what I want to teach, and what makes this topic in particular so powerful and amazing, what makes this topic worth our effort and energy.  It is almost impossible to learn when you are bored, and it is almost impossible to not learn when you are having fun.  Think about how people effortlessly memorize and analyze every detail of their favorite TV shows and movies.  That mindset, that excited, involved, and enthused mindset is what I strive to cultivate in the classroom, and the easiest way to create it, is to display it myself.  When I am thrilled and excited to be talking about calculus, when I am having fun teaching it, then I can see my students getting exciting and having fun too.  Further, it is vital for the students to understand why they should study this particular material, and to how this topic fits into the broader context, of the course and the outside world.  Thus, repeatedly giving overviews and summaries of the course material helps provide them with the motivation to study and to learn.


During class, I believe that most people learn best when they are not passively observing a lecture, but instead when they are actively participating in the lesson, when they are involved, explaining, solving, talking, trying, working, and struggling.  People learn when they are figuring things out for themselves, rather than expecting others to teach them.  As a result, I have designed my lesson plans as a series of questions, which I ask throughout the class period.  I have found that if I simply ask questions of the class as a whole, waiting for volunteers, then a few students will respond to the questions before others can speak, and so the majority of the class passively watches the dialogue, without actively participating.  Therefore, in all of my classes I ask questions of individual students by name.  This means that one of my primary goals during the first few weeks of class is to learn all student names.  This also allows me to greet students by name when they come into the classroom whenever possible, and begins the process of forming relationships.


On the first day of class, and in my syllabus, I explain this teaching method and the rationale for it.  Initially, students may be surprised and uncomfortable with this process; however I have found that they quickly come to understand that my goal is not to embarrass them, but simply to get them to engage in the material.  Through the teaching process I show that it does not really matter whether each answer is right or wrong.  They find throughout the semester that everyone in the class is right sometimes and wrong sometimes.  What matters is that they are engaged in the discussion, and working to figure things out.


I have found that when used effectively, directed questioning can be a tremendously powerful tool for involving each student in the lesson and for creating a learner-centered classroom.


Even directed questioning only allows me to directly interact with one student at a time, and so whenever possible I use classroom voting, which is a teaching technique where I pose a question to the class, then allow a few minutes for students to work and discuss things in small groups, before every single student in the entire class must vote on the best answer.   Student votes are instantly received and tabulated on a computer which displays a histogram of the results, providing immediate feedback to my students and me.   Then I go around the room, Socratically asking different students to explain their votes, guiding a classroom discussion to figure out the key points.


I brought this teaching technique to my physics class at Carroll College in the fall of 2003, when I started without any clickers at all, just having students vote by pointing to letters printed on the back of their book (A, B, C, etc.)  In spring 2004, we purchased Carroll’s first set of clickers and in the years since I have invested substantial amounts of time and effort to develop the necessary multiple choice questions that have allowed me to use this teaching method in most of the classes that I teach.


I never cease to be amazed at the effectiveness of this technique at producing a more interactive classroom, at getting each and every student to engage, to participate, and to intellectually grapple with the relevant issues.  As a result of using classroom voting, my students report that they have more fun in math class, and I am quite convinced that they learn more as well.


In my first semester of teaching here at Carroll, I found that traditional paper-based homework was not giving many students effective feedback.  Students were repeatedly making similar mistakes again and again.  With traditional homework, often a week would pass from the time that a problem set was assigned and worked by the students until they received their papers back, after being graded and recorded.  This made it difficult for many students to improve, especially those with math-phobias and weaker math skills.  So I began to explore different mechanisms for giving students faster and more effective feedback, and I ultimately discovered the WeBWorK system, a web-based online homework system, developed with NSF support at the University of Rochester, that is freely available to all institutions.  I persuaded the department to purchase a server for this system, and then oversaw the installation and testing of WeBWorK.  During my third year here at Carroll I used this with my courses for the first time and the results have been excellent, so I now use this regularly in almost all of my courses, and it has been adopted by my mathematics colleagues as well. 


WeBWorK has several advantages over paper-based mathematics homework.  First, the system provides students with instant feedback, telling them whether or not their answers to each problem are correct within seconds.  This makes it much more likely that they will be able to identify and correct their mistakes than if they had to turn in written work and wait for it to be graded and handed back several days later.  Further, we have set the system so that students may attempt each problem as many times as necessary, with no penalties for incorrect answer.  As long as they eventually get the right answer before the due date, they’ll get full credit for the problem.  Finally, each student receives a slightly different version of each problem, making it impossible for students to simply copy answers from their peers.  Of course we encourage them to work together so that they can teach and learn from each other.  Instead, because each student receives different numbers, they must explain the process for getting the right answer, which is the important thing. 


The first time that I used WeBWorK, most of the homework problems that I assigned were drawn from sets written by faculty at other institutions.  Over the past years, I have invested considerable amounts of time and effort to programming my own problems into the WeBWorK system, so that the homework assignments are well aligned with my course goals.  I have found that old exams that I have written are often excellent sources for WeBWorK problems.  The addition of these original problems has improved our WeBWorK collection considerably, and in the coming years I will continue to code up new problems, refining and improving these homework assignments.


I have observed that WeBWorK changes the way in which students approach their homework.  With paper-based homework a student might be frustrated with a problem, write something down, give up and just go on.  However with WeBWorK, the students know immediately whether or not each answer is correct, so if the answer is wrong, they try again, and again, and again.  As a result, I have found that they are more likely to come to my office and get help with particular problems before an assignment is due.  A large number of students get in the habit of working on the assignments and getting help until they receive perfect scores.  Thus I can use WeBWork to create a sort of mastery-learning culture in my classes, which can have tremendous positive benefits not only in terms learning the material itself, but also affecting how they approach mathematics and the learning process itself.


I want my students to be able to use mathematics not merely to pass tests, but to deal with real world situations, in their life after Carroll College, and this requires excellent skills at communicating mathematics.  It is not enough to simply be able to perform computations:  They must be able to explain what they have done to others, to persuade others that their calculations are relevant and important.  As a result, in almost every mathematics course that I teach, I require my students to perform open-ended projects in applied mathematics, to write up their work into a formal paper, and then to give an illustrated PowerPoint presentation to the class.


In calculus, I have given out a variety of assignments (see supporting documents), each asking students to create a unique mathematical function to solve some applied problem.  A few examples:

In the assignment statements, I give several limitations to these functions, written in the language of calculus, requiring students to put this knowledge to use.  The key point is that these assignments do not have any one “right” answer, but instead allow for many different approaches, encouraging students to be creative in their function designs, helping them to see mathematical functions as tools that they can use in a variety of contexts.  Students have repeatedly commented to me about how these projects have changed their view about what mathematics is, and how it can be used.


Many of my freshman calculus students have never had to write a mathematics paper before, and so I have to spend time explaining the fundamentals, how the goal is to take a mathematically complex topic, and to explain it with utmost simplicity and clarity.  Students must learn to blend paragraphs, equations, and graphs together into a coherent whole, explaining what they are doing and why, writing a paper to carefully lead the reader through each step of the logic, and to persuade the reader that this particular analysis makes sense.


I teach both of our writing intensive courses in the mathematics major, where I continue this basic idea of having students write up their work on creative, open-ended projects.  In “Math 341:  Probability and Statistics II” I require students to do two papers, giving them very broad requirements:  In the first paper, students must pose a question that could be addressed with statistics, they must gather a set of data, analyze their data, and present their results, in both a formal paper and in a PowerPoint presentation.  In the second paper, the data gathering phase must include conducting a controlled experiment.  I am constantly impressed with how effective these projects are at revealing student misconceptions, and illuminating weaknesses in their understanding.  Students are very good at using external cues to work their way through cut-and-dried, closed-ended textbook problems, even when they don’t really understand the meaning of their calculations.  However in an open-ended setting, it becomes incredibly obvious what they get and what they do not get, allowing me to help them deal with these challenges.


In “Math 342:  Numerical Computing and Visualization” I have created a series of projects deliberately designed to present many of the ambiguities and complexities that appear any time we use mathematics in the real world.  These projects asked students to attempt to extrapolate the future price of heating oil, to develop a mathematical method for managing a deer population within a 50,000-acre reserve, and to use numerical calculus to analyze the data taken by astronomers studying distant stars.  Often it is almost impossible to create a mathematics assignment that that is beneficial to all, so that the strongest students are sufficiently challenged, and the weaker students do not get stuck and frustrated.  I have found that open-ended creative writing assignments like these are remarkably effective at stretching the strongest students without leaving the weaker students behind.  The open-ended nature of the projects means that students can work on them at a variety of different levels:  Some students choose very ambitious approaches, while others take more basic approaches, but no one gets bored, and no one is lost.


In order to help our students become life-long learners, I think it is important that we encourage our students to pursue learning activities beyond the regular demands of their required courses.  Thus, for the past six years I have put considerable time and effort into recruiting students into the mathematical contest in modeling, and to preparing and supporting them in this competition.


The mathematical contest in modeling is a yearly competition where undergraduate students work in teams of three at colleges and universities all over the world, and spend a weekend in February (96 hours) trying to develop a mathematical model of some real world system, using this model to solve an applied problem, and then writing a paper describing their work.  I participated in this contest myself as an undergraduate, I prepared and supported teams when I was in graduate school, and since joining the faculty here, I have worked to prepare and support our teams, in collaboration with Mark Parker, Holly Zullo, and Phil Rose.  Before the competition, we hold a series of five or six practice sessions where we teach fundamental ideas in mathematical modeling, paper writing, teamwork, and time management, in order to help them get the most out of this experience.


In 2009, a record 33 Carroll students volunteered to participate in this competition.  It is important to note that here at Carroll, this contest is offered on a purely extracurricular basis.  Students receive no extra credit or other compensation for doing this (and neither do the faculty).  Each of these students volunteered to complete in this 96 hour math competition purely for the fun of it, which I think says a lot about the sort of learning environment that we have successfully created in our mathematics program.


I believe that if we can help students to get in the habit of pursuing activities for no other reason than for the joy of learning, then we set them on the road to long-range educational success.  This is one of the reasons that I have chosen to present an ongoing series of evening public lectures on various fun and interesting topics in astronomy and physics.  I strongly encourage my students to attend these presentations, and I regularly see a good number of them there.  Each lecture is accompanied by lots of illustrations and images in a PowerPoint, and is taped by Helena Civic Television for broadcast to the Helena community.  These lectures have been well attended, usually attracting between 50 and 100 people, including families with children, retired people, high school students, and of course a number of our Carroll College students.


Concluding Thoughts


Here at Carroll College, I have learned a lot about teaching, and I think that I have been an effective teacher.  I have learned the importance of explaining my teaching methods to the students, so that they understand why we may be doing things differently than they have experienced before in other classes.  I have learned how vital it is to put emphasis on why we are studying this particular material, why it is important, to provide an overview and a context.  I have focused my energies on bringing enthusiasm, innovation, and a student-centered orientation to my courses, and I have been privileged to see my students go on to successfully apply the material that we studied, in other courses, and in the world beyond.


Spring 2004:  What do I hope to accomplish as a teacher here at Carroll College?  What are my goals?  Why do I teach?  What am I trying to do?


  1. Goal number one is to educate the students in my classes.  I want them to learn the material of my courses in a permanent, long-lasting way.  I want them to be able to apply these ideas of mathematics and physics in real-world settings.  I want them to learn not just the specific methods and techniques, but the overarching ideas of quantitative and scientific thought, to learn to view the world through a mathematical lens, and understand the power that comes from measuring things with specific numbers.  I will do anything and everything in the classroom if it results in my students learning more.  I view teaching as a sort of optimization problem:  Given the limited amount of time and energy that I can devote to any class, how can I use these resources most effectively to get the maximum possible learning out of my students as a whole.  A part of this is making my classes fun and entertaining, to joke with students and be silly with them.  If I devote 5% of my efforts to making the students relaxed and involved, then I get more learning out of the class than with my other 95% than if I devoted 100% of time and energy to serious teaching.
  2. I want to inspire students, to change their lives, so that they discover life paths that they had never considered before.  This may seem arrogant and ambitious, but I’ve seen it happen.  Here’s a true story:  I had a close friend who was intelligent, talented, and gifted in more ways that I can count, academically and artistically.  She graduated at the top of her High School class, but seriously considered whether she wanted to go to college at all.  At the time her career plans were in the arts, and she wasn’t sure whether college would help bring her success.  She attended two different colleges, changed majors, and although she got wonderful grades, higher education didn’t seem to be giving her what she needed.  Then, to fulfill a requirement she took an intro humanities course.  As usual she worked hard and diligently, but something different happened in this class.  This time, the professor noticed.  He encouraged her and inspired her.  He complimented her papers and offered her the opportunity to present her work at an undergraduate conference.  By the end of the course she had found her calling, her passion, her life’s work.  Today she is fast becoming one of the world’s leaders in her field.  All of this has flowed out of one intro class, taken to fulfill a requirement.  All this began with the kind supporting words of a wonderful teacher.
  3. I want to revolutionize the practice of college education.  Too often, the practice of college teaching is viewed as an art, an unquantifiable craft, to be practiced and mastered by each generation of teachers.  I believe that by quantifying the results of our teaching, we can empirically test new and revolutionary teaching methods.  Measurement and quantification opens the door to real progress, to allow us to discover new and more effective ways of using classroom time, so that the next generation of teachers can be genuinely better and more successful in teaching than we are today.  I believe that we can quantitatively demonstrate that the traditional “sage on the stage” lecture, no matter how interesting, carefully designed, or beautifully delivered is a less efficient method of teaching than techniques which force every student in the class to be active.  I don’t know the best ways to teach, but I believe that we can reach closer to them through a program of systematic quantitative teaching experiments.
  4. My most selfish goal:  To have fun!  I love the art and the practice of teaching.  I relish the privilege of getting to know the students in my classes, to talk with them, to visit, to joke, to develop relationships and most importantly to become friends.  I have heard it said that “To be a good college teacher you have to remember that you’re their professor, not their friend!”  My reply, is:  “I should be so lucky!”  Just because I have to be their leader and evaluator doesn’t preclude friendship.  Almost all friendships have limitations and contexts:  So what!  Connecting with students, learning a bit about them, becoming a friend, these are some of the great honors, the great joys that come with being a teacher.  As a teacher I love creatively planning lessons and assignments.  I love being the center of a class’ attention, telling jokes, and being a showman about things.  I love sitting back and asking Socratic questions until the student suddenly realizes the answer to their own question.  Teaching a thrill!




Fall 2003:  Excerpts from some e-mails that I wrote to a colleague as part of a discussion on teaching.


Does more experience make you a better teacher?


I really have very mixed feelings about getting more teaching experience.  I mean, yes, it would be nice to feel more comfortable – to have more of a background to know when to worry and when to just relax.  And yes, to a certain extent teaching is a skill, a craft, and like any skill you improve with practice.  But when I look over the teachers that I had at Eastern and CU, there was certainly not any one-to-one correlation between more experience and better teaching.  Often the best teachers I had were not the most experienced ones.  My best teachers injected a real energy, enthusiasm, a passion for the subject.  I think that experience can make you a competent teacher, but not a great one.  Experience can teach you about organization, structure, common difficulties with the material, et cetera.

If teaching is a craft, a skill, then there is no progress in teaching from generation to generation, and I believe in progress.  I believe that any teacher today should be better than any teacher from fifty years ago, because I believe that we can treat teaching as a science, where innovation and experimentation can allow us to develop new techniques that make us all better teachers.  Experience can only refine the practice of existing teaching methodologies – it can never find new techniques that advance the state of teaching as a whole.

With my teaching I want to accomplish a whole lot more than directly educating the students I will see over the course of my career.  I want to use them as a testing ground for new teaching methods – methods of lecture, the structure of homework, uses of technology, other uses of class time – methods that I can evaluate, and if successful methods that can be adopted by others and advance the state of math/physics teaching as a whole.

If anything I'm actually a little bit afraid of becoming experienced.  To be innovative you have to see things with fresh eyes, seeing new possibilities, new perspectives, new ways of doing things in the classroom that no one has ever done before.  As I become more experienced, I'm afraid that I will become more complacent, less able to see new things, and more traditional in my teaching.  That's not useful if my goal is to revolutionize the teaching of math and physics in this country.


Can there really be progress in teaching?


Teaching can be treated as a science, because we can formulate it into a quantitatively measurable system:  Given the finite time and energy that teacher has to devote to a class, how do they allocate that time and energy in order to maximize the amount of student learning that occurs?  Student learning is a measurable quantity – we do it all the time with assignments and exams.  By testing different teaching methods we can objectively demonstrate which are better.

For instance, until the 19th century, college instruction was usually done with the instructor standing at a podium and reading their notes to the class.  At this point the chalkboard was introduced as a new technology.  It was quickly discovered that if the professor wrote on the board during the lecture then students learned the material better – they both saw and heard the material, so it was more memorable.  This also afforded teachers the ability to work examples of mathematics problems for the students visually – which was discovered to be better than merely describing how the problems could be done in words.  This made any teacher of the blackboard-using generation superior to the pre-blackboard generation:  No matter how experienced the two teachers were, using a blackboard increases the total learning that takes place in the classroom in measurable ways that can be measured with exams.  Thus real progress did take place.


Can learning be objectively measured?


Some college educators might argue it's really not fair to evaluate teaching methods by measuring student learning, because the most important part of learning is immeasurable.  It's unquantifiable, ineffable.  There's no way you can put a number on a student's comprehension of Newton's laws – not a fair or meaningful number anyway.

I think this is tremendously hypocritical for an educator to say this.  A large part of our job is to quantify student learning.  I often grade fifty papers a day, and I give each a numerical score, which I used to quantify how well the student has comprehended the material.  We give them assignments, papers, quizzes, exams, and finally at the end of it all we give them a total grade, with which we attempt to put a number on how well they have mastered the totality of the course.  Either this whole operation is a sham, or learning is quantifiable.  It is inconsistent for us to spend hours every day quantifying student learning, but then when we are asked to take our students work together to quantify our teaching, to cry foul!  Of course it would be nice if we could say that success in the classroom, the teaching process, is completely unquantifiable -- because then that saves us from the difficulty of trying to improve some objective standard, but as one who assigns grades to students this is the epitome of academic hypocrisy.

Grades and tests are imperfect measurements of student learning to be sure.  There are a great many factors which contribute to how well a given student does on a given test (how well they are feeling that day, whether they happened to glance over a particular topic just before the exam) in addition to their real understanding, but that doesn't mean we throw them out.  There is no quantity in the universe that we can measure perfectly, with no noise, no sources of error.  You just make repeated measurements -- lots of homeworks, lots of quizzes, a wide variety of questions on the test -- the more data you have, the more likely that these will together tell you something useful about the student's understanding.  And then to evaluate your own teaching, you average together all the students in the class.  Proper use of statistics can allow us to overcome the problems of imperfect testing.


But aren’t the most important aspects of learning unquantifiable?


Any type of learning on which you grade your students, is the result of a type of teaching that you can quantitatively evaluate.  If you assign grades to your students, then you can use those same methods to evaluate yourself.  You can't quantitatively evaluate every aspect of your teaching, any more than you can quantitatively evaluate every aspect of your students' learning.  But every bit of their learning that you can objectively measure corresponds to your teaching, which you can measure in an identical way.

Not everything good that is accomplished in the classroom is quantifiable.  There are aspects of learning and personal growth that really are impossible to measure by some objective standard.  For these aspects of learning, I agree that teaching will forever be folklore, a craft, an art.  There will be no progress, no innovation.  Perhaps one person believes that moral character is what makes someone a great teacher in these respects.  Another person might believe that it is experience, another enthusiasm, another youth and a fresh perspective.  For those unquantifiable aspects to learning, there will never be any way to arbitrate between these perspectives, no way to evaluate what really makes the best teacher.  In the words of Karl Popper, these ideas are `unfalsifiable.'  I find this depressing.

However!  There are aspects to learning which really are quantifiable.  For these there can be real progress.  We can objectively test different teaching methods and really advance, really find better ways of doing things in the classroom.  That's tremendously exciting!  That's why I know that I can be a better teacher than Isaac Newton, Michael Faraday, or Richard Feynman -- all great glorious titans of my field.  But I can be a better teacher in the objective ways because the psychology of learning has advanced since their day.

But in order to do this, we have to choose to treat teaching as a science.  We have to hold our own teaching up to objective standards, and have the courage to evaluate ourselves against the cold light of the numbers.  It is a frightening prospect, but through this door we can find real progress.  We can have students leave our classes knowing more philosophy or more physics than their peers did a generation ago.


Here is the “Statement of Teaching Philosophy” that I wrote in November 2002, when I was still in graduate school.  I sent this statement as part of my application for the position I received here at Carroll:


            I believe that people learn when they are active, when they are involved, explaining, solving, talking, trying, working, and struggling.  People learn when they are figuring things out for themselves, rather than expecting others to teach them.  As a teacher, I promote active learning in and out of the classroom, and I present my students with open-ended problem solving challenges.  As a student of teaching, I work to learn the best new ideas and methods that other teachers are using, and I approach teaching as a field of serious scientific research, where education experiments can point the way to new methods and techniques.

            So how do you get students to be active?  How do you gently demand that they take responsibility for their own learning?  I have taught six college courses now, in both astronomy and physics at the University of Colorado at Boulder and Front Range Community College, and here are some specific things I like to do:


            I love open-ended assignments, because they encourage creativity and challenge students to think analytically.  One assignment that I have designed begins by having each student write a two page letter to NASA proposing a new space probe to be sent somewhere in our solar system.  Next, everyone receives copies of the proposals from five of their peers.  They must rank the proposals 1-5 and write a one page paper justifying their rankings.  Finally, during class the students form committees of five, all having read the same papers.  Together, they must discuss the proposals and come to a consensus about the final committee rankings. Not only does this take them through an important part of the modern scientific process, it forces them to think critically, to discuss, to argue, to question, and to persuade their peers.

            I have had a lot of fun designing new assignments and projects for my classes, but even better, I’ve learned to be a “good thief.”  I have used the web, journals, and the old fashioned grapevine to find out what new teaching techniques are out there, such as this one from Paul Francis at Australian National University:  First, I pose a mystery to the class (for example, a blackbody spectrum with emission and absorption lines).  Then each group of 3 or 4 students receives a briefing sheet, making them an expert on some topic, giving them one piece to the puzzle (e.g. atomic energy levels or photon theory).  To solve the mystery, each group must verbally trade knowledge with the other groups one by one, and the first group to figure out the entire puzzle wins a prize. This activity motivates students with their own competitive spirit to turn just another lecture into a high stakes race.

            I believe that to become better educators, we should treat teaching as a field of scientific research, and perform classroom experiments to study the effectiveness of different teaching practices.  I conducted my first serious education experiment last summer, while teaching introductory astronomy at the University of Colorado.  I incorporated small group activities into every class, but I wondered whether the composition of these learning groups was important:  Is it better to group students of approximately equal ability, so that they will go at the same pace, and all equally contribute?  Or should the groups be mixed, so that the stronger students can help the others?  I designed a pre-test to assess the strength of the students coming into the class and used it to assign my students to either homogeneous or heterogeneous learning groups.  The students spent about 30 minutes of each class doing assignments in these groups throughout the course.  On the last day of class I gave the same test again, and was able to compare 51 pre/post scores.  The mean improvement in test scores, as a percentage of the possible gain, (post-pre) / (100-pre), was 56.7% for the homogenous groups, and 57.1% for the heterogeneous groups:  No significant difference at all!  The next time I teach this class, I will simply group the students arbitrarily.  In the future I would like to study issues such as the effectiveness of small frequent homework assignments versus more challenging infrequent ones, and compare different ways of balancing class time between lecture, discussion, and in class assignments.

            I have not yet had the opportunity to teach upper division classes.  However for the past three years I have worked with upper division students, acting as “coach” for the University of Colorado’s teams in the Mathematical Contest in Modeling.  In this competition, undergraduate students work in teams of three, at colleges and universities all over the world, spending a weekend in February using mathematical modeling to solve a real world problem, then writing a paper describing their results.  To prepare our students, I organized weekly meetings, where we would discuss strategies, problem solving techniques, useful references, important mathematical ideas, and brainstorm on problems from previous years.  I would like to teach upper division classes in a similar style, designing each class around a series of real-world, open ended problems.  The students will write up their work on these projects in a mathematical narrative, a paper which uses calculations and data to persuasively make an argument.  Good upper division classes must demand higher level thinking skills, by asking students to apply the methods we teach in novel situations, and to evaluate the strengths and weaknesses of work done by others.  The lectures, in such a class, are thus motivated by the problems, becoming a way of giving the students the tools they need.

            In summary, our students learn when they are actively figuring things out, trying to teach themselves, not passively drifting through a lecture, expecting to be taught.  I design my classes not around what I will do, but what the students will do, to let them take command of their own learning, and to teach physics and mathematics as a way of thinking, a way of learning about the world around us.