Philokalia

Philokalia (Gk. φιλοκαλία): love of the beautiful, the good

There is a certain, I would say unfortunate, trend that has occurred in both mathematics and music education during my teaching career that has really begun to bother me recently.  While on the surface is appears as thought they are very different trends, I would say that at their root they are, in fact, the same.  Before I get started, though, full disclosure: I am also certified to teach high school music, and spent the first 10+ years of my career teaching both math and music, as well as directing theatre.  

In music education, the trend has been to make almost everything a competition.  On one side, I understand the need to win competitions, because that makes it easier to justify the existence of the music program at the school.  To be clear, I think that the need to justify the role of the performing arts in education is absurd, but right or not the need is there.  What this can lead  to is a band or choir working on much less music for a much longer time with the goal being to make the performance perfect, since perfection is essentially what it takes to  win competitions.  What can be lost along the way, however, is the inherent aesthetic beauty of the music itself.  The focus changes from moving the audience to impressing the judges.  And in my opinion that is not a good thing.  In many instances there is a need for greater balance between being competition ready and just playing the music.

In mathematics education, there has been a push toward making sure the mathematics being taught is useful.  If the content is not immediately and obviously applicable to the everyday life of the student, then it is not important, or at the very least not as important as that which is applicable.  What gets lost along the way is the inherent beauty of the subject and the joy of just solving the puzzle.  My graduate degrees are in pure mathematics, so I find it fun to take the pieces, discover the patterns, and "play the game".  Don't get me wrong: I understand the need to show the kids that math is useful, and that this usefulness can be a great motivator.  But one of the things we tried to put into the exercises we created as we built the course was a balance between the basic mechanics, the applications, and the "math for the sake of the math".  And what's amazing is how the kids actually appreciate and enjoy the math for the sake of the math.  

In both math and music, we have, in many ways, abandoned the beautiful for the useful.  And while I understand the need for the useful, I firmly believe that we need to rediscover a love of the beautiful.  

Taking Notes

We have reached the end of the first trimester, with final being given last Wednesday and Thursday. At the end of each trimester, as the kids take their exam, I collect their binders and grade them for completeness. The binder is to contain everything: handouts, notes, tests...everything. One thing I have noticed in making the transition to Harkness is that the kids don't really know how to take notes. Before making the change, the kids were ok at taking notes, meaning they were ok at scribbling down what I wrote on the board, but even then they missed a lot of the important information because they normally didn't write anything down unless it was written on the board. And in a Harkness classroom, with so much of the information being delivered in the course of the discussions, the kids have a difficult time discerning what they should be writing down. It's not surprising to me that the kids who do well in the course tend to be the ones who have organized binders and a fairly complete set of notes. I really focused this trimester on what the successful kids were doing as far as their notes were concerned, and will be "imposing" the following on the kids next term:

(1) The homework must be done on a separate sheet of looseleaf paper. Some of ths kids try to do the initial work on the actual worksheets. None of the successful kids do this.

(2) Corrections are to be made on the homework sheet without erasing the original mistake. The successful kids understand that they need to learn from their mistakes, and they tend to make corrections in pen next to the original work so they can remind themselves of what their instinct told them to do originally, and so they can have the self-awareness to avoid making the same mistakes in the future,

(3) A clean set of notes must be made on looseleaf paper during the discussion, especially at the end of the discussion of each exercise. The reason for the looseleaf paper is to allow the kids to see their original work as they write the clean set of notes, but still be able to keep them separate from each other.

(4) A glossary and a formula sheet must be created as we make our way through the trimester and kept at the end of the "clean notes" section of the binder.

For my part, I will need to check the binders with a certain amount of regularity, the hope being that not only will this improve things for the kids in my class, but also in their other classes. Granted it will be up to them to transfer the skills, but my hope is the success I'm anticipating will be contagious. I'm certain I'll write more about this later...

Being Human

It is amusing for me to watch the student office aides when they walk into my classroom with a note for one of the kids in my class, simply because most of the time it's difficult for them to find me.  It's clear from the look on their faces that they don't expect to me to be seated around the table with my students.  Most often they look toward the board first, then toward my desk, and finally toward the lectern, at which point I raise my hand, waving it back and forth to get their attention and inform them of my location.  The same thing happens when another teacher unfamiliar with Harkness comes to my room.  The administrators, however, are not only used to the fact that they won't be able to immediately locate me in my classroom, but they will sit down at one of the tables with the kids when they come in to my classroom for one of their monthly walk-through observations.  

While I never would have expected to be saying this a couple years ago, there's no place I'd rather be in the classroom than seaeted with the students.  When I used to lecture, I tried as much as possible to be a real person. When we went through the homework, I didn't rely on prepared notes or the work I had done in going through the exercises ahead of time.  Instead, I worked through the problem, talking through my thought process, and, yes, occasionally making mistakes.  I didn't try to hide the fact that I make the same sorts of mistakes the kids do...mistakes along the lines of 1x1=2 (which, for the record, is a mistake I made on one of my Ph.D. preliminary exams).  In short, I didn't hide the fact that I am human.  There are some teachers who feel threatened by even the possibility that they might make a mistake in front of their students, and even when they do they try to cover it up rather than simply admitting it and moving on.  They don't make a move at the board without first consulting their notes or the textbook.  They promote an air of invincibility and absolute control that, quite honestly, I have never understood.  In my opinion, this is one of the reasons some teachers are hesitant to implement, or are even hostile to the idea of trying, a discussion-based method of instruction in their classroom.  They cannot bear the thought of not being the infallible master of the domain that is their classroom.  They cannot bear the thought that their students just might figure out that they are human.  

For me, building a solid rapport with my students has always involved making sure the students understand it's not me against them, but rather us working together to help them learn the material.  And one of the things I love most about Harkness is that I get to sit at the table with the students and just by the physical arrangement of the room emphasize that we are in this together.  I love the fact that I get to hang out with the kids a learn some math.  And yes, I've learned plenty of math from the kids over the years, most especially during the last year and a half.  I love the fact that they know I'm not perfect, and that one of the things they learn in my classroom is that it's ok to make mistakes so long as you admit them, learn from them, and try to avoid them in the future.  In the course of the discussions, I find out more about the kids than just their math ability.  I find out about their other classes and the parts of their lives outside the classroom they are willing to share.  I get to hear about marching band rehearsal, soccer practice, youth  group, Catcher in the Rye, and family holiday traditions.  In turn, they get to find out that I'm very sarcastic, that I root for the Buckeyes, that some of the novels they have to read for English class I like but others I don't, that church is important to me, and so on.  In other words, without crossing any inappropriate lines, we get to be real with one another.  We get to be human.  And against the backdrop of being human, we learn some math for 70 minutes a day.  I honestly can't imagine my classroom being any other way.

Writing Good Questions

Over the course of the last two years, one thing that has become abundantly clear is that one of if not the most important parts of successfully implementing Harkness is my classroom has been the packet of exercises we put together for the honors pre-calculus course.  It was a struggle to find just the right balance between not challenging the kids and pushing them too hard.  With every topic, we had to make sure that the problems led the students to the information in such a way that they were stretched just enough to put the next step along the path within their reach without making the stretch so minimal that they didn't see the value in doing the exercise.  Since we wrote the exercises, I have discovered a term for this type of exercise.  It is what Dan Meyer refers to as "perplexing".  His definition is that this kind of question is one with the following qualities: (1) the students understand what is being asked; (2) the students do not currently have the answer; (3) the students believe they have the ability to answer the question.

Within the last week, though, I have stumbled upon one of the reasons the writing of the questions was so difficult: I was never shown how to write good questions.  Not during the earning of my undergraduate degree. Not during any of the professional development at any of the schools at which I have taught in my 23+ years of teaching. Never.  Once this hit me (and why it didn't before this I have no idea), I asked a few of the other math teachers as well as a few of the teachers in other subject areas, and the response was unanimous: no one was ever shown how to write a good question.  And the more I thought about it, the more I realized that this is probably the key obstacle to implementing any kind of meaningful reform in education.  If we, the professionals, struggle to write good, meaningful, well-structured questions, what are our options?  More often than not, the option is to go looking for questions, normally in textbooks from which we aren't currently teaching or from other teachers.  The trouble, of course, is that is we are talking about writing questions that are fundamentally different than any we have asked before (for instance, the questions we needed for our worksheets or the questions that we need to prepare our students for the performance assessments of the common core end-of-course tests).  The "new" textbooks that claim to be designed for the common core tests are not up to the task, and the other teachers are in the same position we are.

This is one of those problems to which I really don't have an easy answer.  For us, writing the exercises was a long, difficult process, and honestly, we are still editing them, searching for better ways to put the scaffold in place so the students can successfully make the climb.  It was also one of the most rewarding things I have ever done in terms of professional development.  Digging that deep into the material revealed new connections and provided new insights into a course that I had taught for over a decade.  And because of that, writing your own questions is, in my opinion, one of the best things you can do to improve as a teacher.  I'm not saying it's easy, and in fact I know from personal experience it's not.  But it's sooo worth it.

Differentiating Instruction

Looking at a classroom that is running in Harkness mode, it is easy to see that many of the best practices we are supposed to implement are happening.  The classroom is student-centered.  The students are highly engaged. The students are taking responsibility for their own learning.  However, one thing that people perceive as missing is differentiation.  "You mean that they do this discussion thing every day?  What about the kids who have trouble learning this way?"

One of the things I used to question a few years ago was how to structure a lecture in such a way that I would be able to reach the variety of kids in my classroom.  And honestly I used to pass off the idea of making that happen as impossible.  Over the course of the week I might have been able to teach one day or one part of one day for each of the kids, but to try to reach each of the kids every day during every lesson was simply not an option.  So, I relegated the idea where I normally relegated all such ideas: into the category of "educational theory that isn't in touch with reality".

However, with Harkness, what I have found is that the differentiation all but takes care of itself.  Early in the trimester, it is fairly common for the kids to ask if it's ok for them to attempt to solve an exercise a certain way.  "Am I allowed to draw a picture for this?"  "I found a formula...can I use it?"  But after a few days of hearing me say, "Yep, if you think a picture will help," or "As long as you can explain where the formula comes from,", the kids begin to figure out that they can use pretty much whatever method they can devise and understand.  Slowly they begin to take responsibility for their own learning, and once that happens they begin to figure out how they learn best.  Does drawing a picture help, or do the equations make more sense?  Or is it some combination of the two...or something else that works best?  Because they aren't being told how to solve the exercises, they tend to head for their comfort zone rather than doing what they're told, and in that the instruction is differentiated.  Granted, the kids who are stuck in memorization mode don't reach this point as quickly since they are spending their time desperately looking for someone to imitate rather than working through the material themselves.  But even in that search these kids tend to find which of the other students attacks the exercises in a way they understand, and in that the instruction is differentiated.

So yes, we do this discussion thing every day.  And it's the most differentiated the instruction has ever been in my classroom.

How to Succeed in a Discussion-Based Classroom

The second test of the trimester happened last Tuesday, and after grading them one thing became abundantly clear: some of the kids have caught on to trying to understand the material, while others are still clinging to trying to memorize the material.  The kids who are trying to understand the material are the ones who are more thorough in their preparation, more active in the discussions, and therefore are doing better on the tests.  Those who are still trying to memorize the material give up easily on their homework, passively take notes in class, and are not doing as well on the tests.

In terms of the homework, let's be clear: the exercises have been scaffolded so that the kids should be able to make the connections between what they already know and what the exercise is asking them to do.  It may take looking up a definition or finding a formula they've forgotten, but the work is within reach.  The kids who are trying to understand the material may not successfully complete all of the exercises, but on those that they get stuck, they write out the questions they have and therefore come prepared to discuss each of the exercises regardless.  The kids who are struggling still seem unwilling to work through an exercise if the path to a solution does not present itself within a few seconds of reading the problem.  During the discussions, these kids tend to mindlessly take notes, trying to copy everything being said and everything written on the board without putting thought  into what it is they are writing.  By contrast, the kids who are trying to understand the material are more deliberate in their note-taking, waiting until the end of the discussion about a particular exercise and writing down only the important, underlying ideas and processes.  When the tests come around, the kids who are used to struggling with the exercises and have focused on learning the ideas and processes are not thrown off by the fact that the exercises on the test are not essentially identical to those on the homework; instead, they have practice with being persistent and with using the knowledge they have in a "new" way.  Those who have been trying to memorize the material tend to quickly get defeated and resort to writing down any formulas and definitions they remember in the hopes of getting some partial credit points.

While I feel for the kids who are still stuck in memorization mode, their struggles have convinced me all the more that running a discussion-based classroom is the right thing to do.  In addition to learning "the basics", the kids who are putting in the effort to learn the material are also learning problem-solving and, more importantly, are learning how to learn.  In other words, they are learning what they should be learning...what they wouldn't be learning if all I was asking them to do was memorize the material for the test.  So the question is how to convince the memorization kids that they need to change, and then help them do it.  It's not an easy sell, since memorization has worked for them for years, and probably still works in a number of their other classes.  Hmmm...

Meanwhile, at The Ohio State University

This weekend was parents' weekend at Ohio State, and I spent a wonderful weekend with my daughter who is a sophomore there.  Among other activities and presentations we attended was one by Dr. Matthew Stoltzfus (or Dr. Fus, as he is known to his students), a chemistry professor who has flipped his general chemistry classes.  Yes, this means he has flipped a class that regularly has over 200 students per section in a large lecture hall.  Yes, it means he doesn't lecture, at least not in the way that any of us would expect.  His class is very interactive and very discussion based, with the general setup of the class being: (1) the students watch the introductory video on the material before class; (2) the students are given a situation or problem to discuss, and Dr. Fus walks around, guiding the discussions but not giving away the answer; (3) the students enter their response electronically, and Dr. Fus, receiving the responses in real time, goes around and gives advice and suggestions to the students; (4) Dr. Fus gives a short explanation of a similar or related situation; (5) the students resume discussing the original problem, with an opportunity to change their answer if they want.  For the most part, the students arrive at the correct answer after the second attempt.  It was really neat to see that the idea of not lecturing for 50 minutes straight while the students take notes has begun to be put into practice at the college level.  I had the opportunity to speak with Dr. Fus briefly after the presentation, which included us answering a simple chemistry question in the manner described above.  He said that there are other professors who are getting away from the traditional lecture and, in different ways, implementing a more interactive, discussion-based method of instruction.

Then at lunch, I had the opportunity to have a discussion with another student at OSU.  Along the way, he asked how my daughter and I had spent the morning, and I told him about the presentation by Dr. Fus. The student said that he has had several professors who have flipped their classroom, some good, some bad.  Responding to my question about what he saw as the difference between the good and the bad, he said that it really seemed to come down to how much the professor interacted with the students.  In his opinion, those who did had a great class, and he added that he felt he got more out of those classes than he would have from a traditional lecture.  However, he said that he felt he would have gotten more out of a traditional class than he did from the classes in which the professor did not interact with the students but instead essentially provided a space for the students to discuss the lecture they had watched.

Needless to say, this was all music to my ears.  Student-centered, discussion-based guided inquiry as the basis for learning in a college classroom dispels the misconception that we at the high school level need to be lecturing since this is what the kids will experience in college.  The mentality among some high school teachers is that high school needs to be the bridge between a school experience where the teacher adjusts to the students, and college where the students must adjust to the professors.  To see a professor as student-focused as Dr. Fus, and to hear from him and from the other student that this is (finally) beginning to happen on college campuses reassures me that we who are (finally) implementing this at the high school level that on the right track.

And it energizes me to continue to spread the word about the Harkness philosophy.  On we go...