Easy Questions

One of the great lies we tell ourselves is that easy questions are not worth answering.  Regardless of whether we find easy questions to be useless because we can answer them quickly or because we don't see how the answers can possibly lead to anything important, we have convinced ourselves that easy questions are not worth answering.

Sadly, this is not what we instill in our students.  By and large, the kids are firmly convinced that easy questions are the only ones worth answering.  In fact, in the face of anything that resembles a difficult problem, the kids tend to give up and wait for someone to show them the answer, regardless of whether or not the answer is accompanied by an explanation.  This is the common experience of pretty much any high school teacher.  It is not, however, the experience of anyone who has regular contact with younger children. And by "younger", I'm not talking about much younger than high school age; my 6th grader is a shining example of this.  These children are never satisfied with only knowing an answer, and anyone who tries to simply give them an answer to a question without explaining why the answer is true is in for it.

I'm not going to attempt to delve into why it is or when it is that this thirst for an answer to the question "Why?" is killed off.  Instead, this has to do with an observation I made in my classes this week.  In one of my classes we are in the middle of the material on trigonometry.  The way we organized this part of the course was to spend the first unit on sine and cosine exclusively, from the basics inside a right triangle to the graphs to the identities.  We then repeat the process with the tangent and secant functions in the second unit and the cotangent and cosecant functions in the third, while still holding on to the material from the previous units.  This term, the kids did reasonably well with the first unit and notably worse in the second unit.  I sort of saw this coming because in discussing the exercises in the first unit the kids were running through the "easy" exercises without really delving into the "why" of them.  Since the main topics were sine and cosine, topics with which they were already quite familiar from the last several math courses they have taken, they were able to still do well on the first test.  However, coming into my course they are not as familiar with the tangent and secant functions, so when they weren't bothering to slow down and discuss the "why" of the easy questions, I sort of knew that they would struggle with the "more difficult" questions late in the unit, which they did.  However, rather than putting forth the effort to answer the more difficult questions, and rather than going back and looking at the easy questions a second time to see what they may have missed, the kids tried to fall back to their old habit of memorizing everything for the test.  This, of course, did not serve them well, since the tests are designed to make evident whether or not they actually understand the material.

On the flip side of this, the first unit also includes the material about ellipses, and the second unit contains the material about hyperbolas.  Overwhelmingly, the kids did well on the questions about these topics on the tests.  In my opinion, the reason they did so well is because they did not come in to the course feeling completely comfortable with these topics, and spent the time necessary with each exercise, not seeing any of the questions as easy, and really trying to understand the content.  In doing so, they did not resort to simply memorizing the material, but focused on understanding it instead.  In other words, none of the questions was easy, so from the beginning they were focused, as opposed to what happened with some of the trigonometry questions where the easy questions were effectively ignored and the important content from them was missed.  On the positive side, in neither instance did the kids turn and run from the more difficult questions, nor did they wait for me to simply give them the answers.  Both of these are very good things.

The moral of the story is this: there are no easy questions.  There are foundational questions, the answers to and the content of which are more readily understood at a surface level, but when it comes to really understanding something, there are no easy questions.  The obvious must be discussed and thoroughly understood for there to even be the possibility of grasping the more difficult material.  The kids will get another opportunity at this in the third unit, and I plan to explicitly tell them to slow down and make sure they really understand the answers to the "easy" questions...questions which technically don't exist.

Book Club

The atmosphere of a classroom can make all the difference when it comes to whether or not a student will learn anything in that classroom.  For example, if the teacher lectures all the time and the student finds this to be boring, then chances are the student will not be able to learn from that teacher.  Or, if  the student finds the teacher or the content of the course to be intimidating, then the student will probably not learn anything from that teacher.  In instances where the student has difficulty learning the material in the classroom but still cares enough to try to pass the class, they tend to find ways to compensate for this, such as reading the textbook, doing a bit of research on the internet, or hiring a tutor.  On the positive side these can help the student be successful in the class.  However, it also gives the teacher the feeling that what they are doing in the classroom is actually working, when, in fact, the kids are learning despite what is happening in the classroom.

The reason I bring this up is that there was an episode during the past week that let me know the atmosphere in my classroom is conducive to student learning.  It came from one of the current students, who came in to class on Friday morning and said he feels like the class is a lot like a "math book club".  "We go home, do some work, and come in, sit back, and talk about the math.  It's so relaxed...I really like it."  I love this description of the class.  It's a great description of the class: the prospect of learning the material without lectures is intimidating at first, but eventually the students get comfortable with the system and with each other, and together they figure out that the material isn't really that intimidating either.

So, when asked, I now have a new way to describe my class.  I'm not teaching precalculus through Harkness.  I'm running a precalculus book club.

Comfortable

It only took a year and a half, and it was bound to happen eventually, but this week I became acutely aware of the fact that I'm finally feeling really comfortable with Harkness.  Things have settled down, not just for this trimester, but with the course, to the point that I can relax a bit and thoroughly enjoy each class.

Up to this point, there have been several things that have prevented the comfort from taking hold.

1. Creating the exercises and then tweaking them constantly as we were running the course had been the main thing preventing it.  In creating the course essentially "from scratch", and with no real template to guide us, we were constantly looking for the flaws in the questions and for ways to make both the individual exercises and their sequencing better.  We're still doing this, but the changes are becoming fewer and further between.  
2. Learning to shut up and let the kids take the lead was a huge hurdle.  And while there are still a few times every day where I step in a bit too early, for the most part I've learned to relax and let the kids play the exercise through until they arrive at an answer (correct or incorrect) or ask for help.  
3. Writing the tests was another source of tension.  Each of our tests has six open-ended questions the likes of which, for the most part, aren't found in textbooks.  The creativity involved in writing questions that synthesize several topics and are accessible to the students given the time constraints of a test is not something that happens a couple days before a test.  We have gotten better at this in the course of the last year and a half, but initially we were sort of grasping for good problems.
4. The kids were making the transition as much as we were.  This was especially true last year, whereas this year the kids had "fair warning" from last year's group. If were honest about it there is a transition period for the students in any class, but normally the transition is eased by the teacher.  This year, we could help with the transition far more than we could last year.

Personally I think this level of comfort has made things better for the students as much if not more than it has for me.  Much like anything else in the classroom, the demeanor of the teacher sets the tone, and this is no exception.  Hopefully, this will manifest itself in the test scores and level of understanding of the students.  We'll see...

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...

Method or Philosophy?

Next Wednesday, I will be making a presentation to the faculty of our high school on the Harkness Method, how I've been using it in my classroom, and what it has taken to implement it, with the hope being that it piques the interest of some of the other teachers to the point that they will pursue including more discussion-based methods of instruction in their classrooms.  However, the more work I've done for the presentation, the more I've begun to ask the following question: is Harkness really a method, or is it a philosophy?

The reason for the question is this: it seems to me that the real push behind Harkness is the discussions, allowing the students to dig deep into the material during the discussions and coming to a thorough understanding of the material through the discussions.  And even though the students need to prepare for the discussions, the actual shape of the preparation is not really defined.  The preparation my students need to do looks essentially the same as it does in the math classes at Exeter, with the homework exercises scaffolded so the students discover the material through them.  But is this the same way the preparation looks, or even should look, in the other disciplines?  For that matter, is this the way the preparation should look in other mathematics classrooms?

For example, there is the "flipped classroom" model of instruction, where the students watch a video lecture to prepare for class and then do the exercises based on the lecture during class.  Done correctly, meaning that the in-class work is dominated by discussion and problem solving, I can see this as fitting well with the Harkness philosophy.  I can also see the team-based method of instruction, a project-based method with a strong emphasis on in-class discussions, as really being just another form of Harkness.

It should be clear that I'm leaning toward Harkness as being a philosophy with the following components:

• the in-class work is dominated by student-led discussions
• student preparation for the discussions is vital
• the teacher acts as a guide, both in terms of designing the course so that student discovery of the material is possible and in terms of  helping the students learn from their mistakes

Exeter says that Harkness is flexible to the individual style of the teacher.  Until I began to view Harkness as a philosophy, I didn't really understand what that means.  With this view, I'm hoping to be better able to help other teachers help their students by implementing the philosophy in their classroom.  Updates to follow...

Spreading the Word

With the classes running relatively smoothly this year, and with the course exercises in need of far less revision than they were last year, I have had a bit more time to try to “spread the word” about running a discussion-based classroom to the other teachers in my building.  As you can imagine, there has been a great deal of variation in the reactions and responses. 

Some teachers are very interested, seeing the discussion-based classroom as a method that fits well with an emphasis on problem solving.  While interested in trying it out, there is a hesitation that seems to be based on one recurring difficulty: “We were never shown how to teach this way.”  And I completely agree with them.  For all of the talk about problem-based learning, project-based learning, and other such ideas that are really just repackaged and expanded versions of ideas that have been floating around for a couple decades, there have not been substantive changes made in the way teachers are being prepared for the classroom.  Admittedly, I have not been in a “college of education” classroom in a while, but there are currently several student teachers in our building and the first round of observations has begun.  Based on the way they are being evaluated, it’s easy to conclude that things haven’t changed.  The things for which the observers are looking make it evident that there is an assumption the teacher is at the front of the room leading the class the vast majority of the time, and if that’s what the evaluation form is looking for, then it only makes sense that the training the student teachers are receiving puts them at the front of the classroom the vast majority of the time as well.  While I would hope that an ambitious student teacher would be rewarded for doing the kinds of cooperative learning and discussion-based activities researchers are telling us are best, someone going strictly by the forms could easily count any and all of that against the student teacher.  

That being said, “we were never shown how to teach this way” shouldn’t stop us from doing what is best for the kids.  Changing my classroom over to a Harkness classroom has taken a lot of time, effort, and energy, most of it very much behind the scenes.  The classroom management portion of running a discussion-based classroom for 30 students was not something I found in a book or article, but instead relied on me figuring it out.  This isn’t a matter of bragging...it’s simply a matter of fact.  My point is that if we truly believe that running a discussion-based classroom the best way to teach the kids, then we who are currently in the field should be willing to do whatever it takes to make it happen.  To their credit, those who are interested in doing so in my school are actively seeking out the information, and the administrators in my district have been and continue to be incredibly supportive.  It is truly a blessing to be able to call these wonderful people my colleagues.  In addition to this, we should expect the schools of education to be showing those who are about to enter the field how to actually implement the best practices touted by the research.  

Put simply, the time for the chasm that has existed for decades in education between theory and practice to disappear is long overdue.  But those of us who are willing to do the hard work and make the necessary changes in the classroom, both in the schools and in the colleges of education, are few and far between.  And the same is true of administrators at both levels as well.

This, in my opinion, is one of the fundamental problems with our education system as it currently exists. 

In Their Own Words...

Each week, I have the students complete a self-assessment, in which I ask them to be honest with me regarding their effort for the week with the homework and in the discussions, the group dynamic, and anything else they feel the need to share.  Here is a representative sample of the comments I received in the "anything else" section this week:

"I feel like from this class, I've understood so many more math concepts on a deeper level than I did previously. I think the past couple of years it was mainly focused on just applying equations or certain steps, but with this method, I can actually think through the problem logically without having to rely on just the equations."

"Throughout the class I have improved my math skills by observing other students' work and comparing it to mine. This helped me to solve problems overall, which before this class seemed to be a weakness for me rather than a developing strength. I am having a great experience in learning math in a new way that brings together problem solving, communication, and collaboration."

"I find that each person tackles a problem differently, or is able to complete different questions in a homework assignment. I think it's cool that each person can contribute in a different way at school because of how they interpreted the homework assignments."

"I think the class structure is VERY well thought out, and it's working very well for me. I retain the concepts much better this way, and the class is actually fun."

"Everyone is positive and supportive of each other, it's always a good atmosphere."

We're in good shape so far.

Patience is a Virtue

Overall this was a great week.  Courtesy of a minor surgery, I was not at school on Tuesday or Wednesday, and the plans I left for the sub were essentially "the kids know what to do".    One of my sons is a sophomore at the high school, and when he went into my room before school on Tuesday to get his books from a table in the corner, he said "hi" to the sub and mentioned who he was and that he would be in between each bell  to get his books, since using my room as his locker is far more convenient than using his actual locker.  The sub called him over and asked, "Is this really the entire lesson plan?"  My son quickly read it over, and responded with, "Yep, that's what happens in here every day."  The report from the sub was that things went really well...no difficulties, and particular mention that the lesson plans were very complete, with enough for the kids to do the entire bell.  Specifically, Tuesday the kids discussed the assigned exercises from the long weekend, and Wednesday they did ten review exercises for the first test, which was given on Friday.  Thursday I took questions, just in case there was any lingering confusion from the two previous days (and, happily, there were very few questions), and gave them a few more review questions to work on, so I could make sure they were ready for the test.  Having graded the tests, and with the mean being a B+ and the median being an A-, I'm happy with the results.  For the most part, the mistakes were careless ones rather than fundamental misunderstandings, and the only kids down close to the D range were those who have not been participating fully in the discussion...in other words, there were very few surprises.

However, one thing that I noticed as we went through the review on Thursday was that while the students were feeling fairly comfortable with the material that was going to be on the test, they were not comfortable with the idea that not everything we had been covering was going to be on the test.  And I admit, there are a few "loose ends" from the exercises that were not on the test, the reason being that the topics being taught through those exercises have not been fully covered yet.  We haven't completed all of the exercises that cover those topics, so the test questions on those topics will have to wait.  This is something the kids aren't used to.  What they expect is that the test covers all of the material since the previous test, and after the test we "reset" and begin preparing for the next test over the new material, which will include little to no reference to the old material.  They are not used to being patient when it comes to learning the new material, mainly because we tend to shove a bunch of stuff at them all at once, neatly compartmentalized into chapters on exponents and logarithms, or on basic trigonometry, or on the conic sections, rather than allowing them to build their knowledge gradually over all of these topics at once, which is what I'm asking them to do.  The reasoning is simple: in future courses and in future applications, the expectation will be that they have relatively rapid recall of any and all of the topics covered in my course.  Calculus does not work exclusively with only one type of function as derivatives are taught, and neither do the applications of physics.  And yet, we often teach the different types of functions separately.

In talking with other teachers, and in particular with those who cling to the idea that lecture-memorize-mimic is the best way for the kids to learn, I've come to realize that this lack of patience is also one of the reasons why they cling to lecturing.  In order to get the kids to receive a large amount of material on a particular topic in a short amount of time (for example, learning everything there is to know about graphing trigonometric functions in under two weeks), lecturing makes sense since there simply isn't time to allow the students to patiently and gradually understand the material.  So, in addition to helping the kids gain some patience in learning the new material, I've come to realize that if we are going to get other teachers on-board with changing their classrooms over to a student-centered, discussion-based, discovery method of delivery, we first need to get them to understand that it's not about covering one topic in-depth for two weeks and then moving on, but rather it's about slowly and patiently building all of the topics throughout the term (be it quarter, trimester, or semester).  I'm not expecting this to be an easy sell, but I firmly believe it to be worth the effort.  After 22 years of teaching through lecture-memorize-mimic, I've come to see the enormous benefits of patiently presenting the material through discovery and discussion.  I only hope others will come to see the benefits as well.

...and we're back!

After a summer of roller coasters, little league baseball, revising worksheets, and the conference at Exeter, the new school year began on Wednesday.  The luxury of knowing that the students can successfully self-start and run the class has given way to beginning the process again with a new group of students and hoping the kids I had last year will continue to use the principles of Harkness as they study in their new classes which are not being taught through Harkness.  After three days, things are looking good.

First, while I tend to see my former students a lot since a good number of them come back to me for a college recommendation letter, several have come back specifically to tell me that they found the summer work for AP Calculus "really easy"...something that has never happened before, with the common response being a comment about how much they had forgotten over the summer.  Others stopped by specifically to tell me they miss my class because of the way it was taught.  One in particular came back to say that, when asked her favorite math class this year in another class, she said honors pre-calculus precisely because it was taught through Harkness.  While I'm sure that there are some students who are relieved by the fact that lectures have returned, I know that there are also plenty who are not.  To each of the students who came back, I gave the same advice: use the principles of Harkness as much as you can, both in and out of class.  They know the value of talking things through, of pursuing and achieving a deep knowledge of the material by discussing it. Hopefully, the fact that they miss the way the class was taught will translate into valuing discussing the new material in calculus when they get the chance.

Second, the new students have taken to the discussions very quickly.  Some were told beforehand by friends how much they loved the way the class was run, while others came in with no idea what was about to hit them.  And there were a few students who, after the discussion on Thursday, went to their counselor and switched out of honors pre-calculus.  This happens every year, so I know it's not something exclusive to teaching through Harkness, and it always makes me a bit sad that the students don't even give themselves a chance.  The rest of the students have quickly fallen into good, productive, respectful discussions.  It has been a pleasant surprise how quickly this has happened, but I suspect that the fact that the honors algebra 2 teachers began implementing more discovery learning in their classes last year is at least partially responsible for it.  A few of the students have inquired about what the tests will look like...an understandable concern, but one that I hope will fade as they get more comfortable with the discussions, and as they learn that the problem-solving practice they are getting every day is precisely the skill necessary for success on the tests.

So overall, a good start to the new year.  Hopefully, the success of Harkness will be evident in the AP Calculus courses to the point that the other math teachers (and maybe others?) will begin implementing more discussion-based methods in their classrooms.  Only time will tell.

After Exeter

Now that the conference at Exeter has come to an end, a few reflections on the week:

(1) The week-long courses were great, not only in terms of content but in also in terms of modeling the Harkness philosophy.  From a content standpoint, it was nice to be doing math again rather than teaching it.  From a pedagogy standpoint, it was great to see some of the nuances in action, nuances which can be difficult to detect when you're in the heat of the classroom.  And putting both the content and the pedagogy together, it was fun but aggravating to be the one who is frustrated rather than being the one providing the frustration for others.

(2) The camaraderie among the conference participants was equally great.  There was a seriousness of purpose that led to an almost instant bond with the people with whom I had the pleasure of working.

(3) No, the Exeter materials are not appropriate for every class.  The materials were written for the caliber of kid that attends Exeter, and as such they are not right for every classroom in the country.  Likewise, a private school has more independence when it comes to curriculum, and because of that the materials do not necessarily translate to every classroom in the country.  That being said, I am now more convinced than ever that the Harkness method is appropriate for the vast majority of the classrooms in the country.  The only thing standing in the way is the commitment by the teachers to write the worksheets appropriate for their classroom (both in terms of the level of the kids and the curriculum) and the support of the administrators.

(4) New England is delightful.  From the people of the town of Exeter that I met walking around town to the people in Boston who asked if we needed help maneuvering the subway and train, as well as recommending a restaurant during our two-hour layover (including an approximately 25-year-old blind girl...no kidding), the experience with the people was great.

(5) Finally, I need to increase the awareness of Harkness to the rest of my district and to the districts around me.  The method was never mentioned in any of my undergraduate education nor in any of the professional development since then.  And yet, it's the method of choice at one of (if not the) best high schools in the country.  More people need to know about it, and those of us who spent the past week at Exeter are just the ones to do it.

Time Travel

Day two at Exeter, the first day of "real" classes and Harkness discussions, was wonderful and enlightening on many levels.  I signed up for two content-related classes, knowing that this would mean having homework each night since the classes were going to be taught via Harkness.  The homework has been fun, the discussions today were good, and much as I witnessed in my classroom on the few days when there were only 12-14 students present, the depth and for the most part the focus of the conversations were solid.  There was also a panel discussion about how to run things in a classroom of 25+ students, which was not so much a "how-to" manual as a sharing of ideas...all good stuff.

However, as I was sitting in the math class today as a student, I wondered what things would have been like had this been the way I was taught in high school.  Would I have the same appreciation for it then as I do now?  Would I have actually enjoyed some of the math courses (I loved some but loathed others)?  For that matter, what would it have been like if my college courses were run this way?  Would I have been enthusiastically in favor of such a method, or would I have simply been trying to get through the material regardless of how well I understood it?  It's tough to tell from this vantage point, of course, but one thing is for certain: I need to make sure I instill in my students an appreciation for Harkness and for the Socratic Seminars they experience in their English classes and for the POGILs they experience in their science classes.  Looking back, I wish this were the way I had been taught.  I wish I had been taught not to fear mistakes but to acknowledge them and embrace the lessons learned from them.  I hope that my kids last year got at least a small sense of this, and I intend to make sure that the kids in the years to come do.  And I need to have this attitude myself moving forward.  There were a lot of things that we did right last year...that really worked well and that are worth continuing.  But being here I'm gaining some of the insights into Harkness that are difficult to see from the outside, especially from hundreds of miles away.  I need to actively look for other potentially better ways of doing things, discuss these ways with other people, and learn from the experience.  It is becoming clear to me that, viewed properly, life is a Harkness table, and it's high time we told the kids (and ourselves) to pull up a chair.

Welcome to Exeter

After a Saturday filled with planes, trains, and subways, the math conference at Exeter began in earnest today, and so far it has certainly lived up  to its billing.  The people running the conference are amazing (in terms of both friendliness and organization), the food is wonderful (in terms of taste, abundance and diversity), and the introductory sessions to the week-long courses are what I expected, namely, some homework and some discussion.  The discussion should be better tomorrow...it lagged a bit today, and I now understand how the kids felt on the first day of honors pre-calc last school year.

This is also my first trip in New England, and thus far it's great.  The people in Boston and here in Exeter and very warm and helpful, and the weather, while really hot and humid during the day, has been very pleasant in the evenings.  There is also an acute sense of history about almost everything you see, especially the buildings, which something you don't get to experience as much in Ohio, since most of the history at home involves the geography rather than the architecture.

My hope is to give a short update each day this week, though my math homework may make that difficult (it's been a long time since I've used "I have homework" as an honest excuse for anything), so we'll see how it goes.

Grading

Now that the school year is over, this blog will probably become less Harkness-focused (though that will certainly still be the main theme of many of the posts) and include other topics that hopefully will be of interest.

Today is the first such post.  Over the summer, some of the teachers at my high school have been asked to begin to rethink the grading practices both in our classrooms and in the district.  Having read the book provided by the district that will serve as the starting point for our discussion in the fall, the push seems to be toward eliminating any scores that do not directly evaluate how well a student has learned the material for the course.  The goal is to have the grade be an accurate reflection of how well the student has mastered the material, and I agree that if we're honest about it our current grading practices do not do this.  Among other things, this would mean not giving credit for homework, since homework is intended to be practice and not actual evaluation.  It would also mean eliminating extra credit, since it distorts the accuracy of the grade.

In my mind this raises two immediate issues:

(1) Using this sort of a standards-based grading system, the traditional method of assigning letters to certain percentages simply doesn't make sense.  For example, let's say that there are five standards the student is to meet during a certain marking period.  Throughout the term, the student has demonstrated (through in-class discussions, project presentations, tests, etc.) that they have mastered one of the standards, are proficient in three of the standards, and not proficient on the remaining standard.  To what letter grade does this correspond?  If we base the letter on mastery, the student is at 20% mastery - a really low F on a traditional scale.  If we base it on proficiency, the student is at 80% - a B- on a traditional scale, and very different than the F just mentioned.  Or should the letter grade be some combination of the two?  These are good questions to which I don't currently have an answer.  However, this makes me very curious to see how the grading is done at Exeter - definitely a question I'll be asking during the conference in two weeks.

(2)  Any change in the way the letter grades are assigned is going to be met with a lot of questions from the students, the parents, and the community, let alone something as drastic as this.  We will need to educate each of these groups regarding not only the new system itself (if we put such a thing in place), but also the rationale and research behind it.

If anyone reading this has comments, suggestions, or ideas, please send them along.  I would love to have a good discussion here, and to include the discussion here in the conversation we will be having in my district in the fall.

Lessons Learned

With the school year now completed and my classroom cleared, and having thoroughly enjoyed the first morning of not waking up at 4am, it's probably time to reflect on the year while things are still relatively fresh in my mind.

First, exams went well this week, with the median grade being a B+, which is higher than in years past and higher than the median score at the end of second trimester.  As I admitted in an earlier post, I understand that we're talking about a limited amount of data here, but the reduction in the number of questions during the review for the exams and the increase in the scores on the exams themselves certainly point to Harkness being a better method of delivering the material to the students, at least in terms of their retention of the material.

So, what were the key points I feel I need to keep in mind for next year, and what would I tell someone who is thinking of changing their classroom over to a Harkness classroom?

1. The size of the group is important.  If I get additional white boards, I plan to reduce the number of students in each group from ten down to seven or eight.  Twelve to fourteen is great if the teacher can sit at the table with the group the entire time, but with 28-30 kids in the class that's simply not possible.  I was able to experience 12-14 in a class during the weeks of the AP tests, groups of 7-8 during third trimester since a few of my classes had 22-23 students, and groups of 10-11 the rest of the time.  The additional board space will allow me to go with four groups next year and break the classes of 30 into what I believe to be the ideal size of 7-8 students.  The larger groups tended to either split into two smaller groups, or to have one or two students not participating in the discussion as much or as well as they should.  The teacher at the table with the students the entire time prevents this from happening with a group of 12-14, but without that option the groups of 7-8 students seemed to work the best.
2. The questions on the worksheets are important.  The exercises in the textbooks are not geared for discovery and discussion, so relying on them is not a good idea.  Even the discovery activities provided by the textbook companies are not sufficient to get the job done, though they might be a good place for someone to start using Harkness in their classroom.  The questions need to be written for the level of the students in the class, stretching them to make the connections and discoveries, but not so far that they break and give up.  Were I to teach a "regular" pre- calculus class, I would need to rework the sheets I have, because they were written for an honors level "audience".
3. Harkness is not a one-day event, nor is it a once-in-a-while method.  It is a philosophy that needs to be infused into every aspect of the instruction and learning.  From the daily work to the review, and next year probably even to the quizzes, Harkness must be at the heart of it all.  Reverting back to feeding the students the information and mechanics even for a day or part of a day will undo a lot of what has been accomplished.  I learned this early in the year and never forgot it the rest of the way out.
4. Harkness is better, but it's not easier.  From making (and revising, and revising) the worksheets to constantly evaluating the students' work during class to grading lots of correct solutions that have been obtained by very different but mathematically sound methods, Harkness is more mentally taxing than traditional lecturing.  I saw that last spring during the ""test run" so I knew what I was getting in to, but having seen the results, I can enthusiastically say that it has absolutely been worth it.

And finally, what reasons would I give for making the switch?  Ultimately, it comes down to a decision: do we want to teach the kids to solve problems, or do we want to teach them to problem solve? What I mean by that is this: do we want to equip them to successfully complete specific types of exercises, or do we want them to be able to use a set of skills and successfully complete whatever is thrown at them that uses these skills?  Do we want them to be completely confident in their abilities when it comes to a limited set of exercises, or do we want them to have enough confidence to be willing to tackle new exercises, the likes of which they may or may not have seen before?  Lecturing prepares the students to solve problems; Harkness prepares them to problem solve.  For nearly 22 years I thought I was teaching the kids to problem solve. Now, I know that I was actually only teaching them to solve problems.  Lesson learned.

Final Week

From my standpoint, class this week went about as well as it could have, and was a great way to end the course.  The two days of school next week are exam days, so at this point the discussion portion of the course is finished, and the positive results of running the course with a Harkness philosophy were on full display.

On Tuesday, the students took the test for the final unit of the course.  For various reasons, the other honors pre-calculus teacher and I decided to make this a group test.  The "rules" for the test were:

• the groups were slightly smaller, with 6 or 7 students in each group
• this was not an open book nor an open note test
• the students were to discuss the exercises as close to usual as possible
• the test was worth 12 points, as usual, but 2 of the points came from participation in the discussion (I went from group to group noting the participation like I do every day)
• each group turned in one test; however, each student was required to contribute to the writing of the solutions

I love stumbling upon a great idea, and boy was this one of them.  I have never seen a group test go as well as this.  There is always the concern with any group work, especially when the grade is significant, that one person will do all of the work and the others will just latch on to that one person.  That didn't happen.  All of the students participated and contributed, pretty much equally, to each of the five questions on the test, and I had no reservations assigning the grades earned by each of the groups.  In fact, the whole thing went so well that we are considering incorporating group quizzes into the course next year.  There still is, and always will be, a need for individual assessments, but considering the group-discussion  nature of the course, it really only makes sense that some of the formal assessments be group assignments.  Probably should have realized this earlier in the year.

As the students left class, I gave them a set of twelve exercises to prepare as a means of studying for the final exam, and on Wednesday and Thursday they discussed the exercises.  On Friday, I was gone at graduation rehearsal, so I prepared an additional twelve exercises for them to work on without the usual preparation at home.  These three days of discussion were the review for the exam...a far cry from the usual review for the exam which tended to consist of me going through the review exercises in detail.  The fact that they were able to get through most of the exercises on Friday without me even being in the classroom is a huge testament to the progress the students have made this year in terms of taking charge of their own education.  And as I sit here writing this on Sunday night, I have received two emails, both of which were requests for answers so they could check the work they had done.  Again, quite the change.

All in all, a great way to end the course.  Hopefully, the grading of the exams over the next couple days will be just as enjoyable.

Finishing Strong

It's been fun this week watching the kids work with the new material.  Struggle is probably the more accurate word.  We have been working with rotating conics, writing conics in parametric and polar forms, and finding powers and roots of complex numbers by using DeMoivre's theorem.  None of these topics is necessarily difficult, but all of them are new twists on old material, and the rotations in particular are long and tedious, requiring a fair amount of patience.  And with the inconsistencies in the schedule during this unit, the connections to the old material haven't come as easily.  Couple that with the fact that this unit has a greater percentage of new material than the others and it's made for more struggle than usual for the students.  However, that did not deter the vast majority of them.  Instead of backing down, they dug in.  Admittedly, I have honors level students, and it irritates them when they don't understand something, and yes, I understand that Harkness allows me to use that to my advantage.  However, the fact that they redoubled their efforts to come to grips with several bits of new material has been really fun to watch.

Even more, it has been great to see the progress the kids have made during the course of the year in terms of their willingness to take on the challenge themselves, coming to me for help as their second or third and not their first line of attack.  Most have truly taken responsibility for their own learning, and have learned to work together and rely on one another.  Most are willing to admit when they are lost...quite the accomplishment for an honors student.  The fact that they are not letting this deter them is another accomplishment, since often an honors student, who is so used to things coming easily to them, can actually give up faster than other students when confronted with an academic challenge.  Unfortunately, those who have not reached this point tend to slow down the conversation, trying to force an answer and an explanation out of themselves rather than working to discover or create a solution with the group.  And there are still a few kids that are not preparing for class; however, the other students did not allow that to slow down the discussions, essentially working around the fact that there was a bit of "dead weight" in the group, still including them in the conversation but not allowing them to dominate the group either with questions or with an explanation intended to cover up the fact that they are unprepared.  The good side of this is that the number of unprepared kids dropped drastically this week, and it showed in the discussions.  Which means that, for the most part, the kids are back on the gas and finishing strong, just like I had hoped they would at the end of last week.

Next week will go quickly: Memorial Day Monday, test Tuesday, exam review and a couple end-of-the-year disruptions the rest of the week, and then final exams the following Monday and Tuesday.  Which means that technically I have completed the year without lecturing.  Never would have thought it possible...

Willing to Try

There's not much to report this week.  We're all recovering from the two weeks of AP tests and trying to get back in to something that at least resembles normal for the remaining two weeks.  We took the fourth test of the trimester this week, and there were no major surprises, especially with the disruptions caused by the AP tests.  There has been a slight increase in the number of students who are coming to class unprepared, but again, nothing out of the ordinary for the end of the year.  So overall, a somewhat low-key week.

One thing that did hit me during the week was the struggle the kids were having with the review exercises we planted with the exercises that cover the current material. The second half of the course is dominated by trigonometry, with the conic sections thrown in for good measure.  The review questions, however, have included analyzing polynomial and rational functions and simple finding the area under a curve problems (without the shortcuts of calculus...it's all Riemann sums), for example.  These were things that the kids knew how to do close to the end of the first half of the course, but without consistent practice have forgotten.  If we're honest, we often forget how to do a good bit of the mathematics we've learned if we don't practice it very often, so it's not really a surprise that the kids have lost some of these  skills over the course of the intervening months.  I've thrown in similar types of review exercises near the end of the second half of the course in years past, and while the struggles were similar, there are a couple key differences this year: (1) a lot - as opposed to very few - of the kids were able to start the problems, and had the right idea but were shaky on the details; and (2) when someone who did remember how to do the exercise went to the board, the other students remembered the material much more easily than in years past.  My hope is that this carries over to next year when they encounter some of the algebraic gymnastics necessary to be successful in calculus, that while they may initially be shaky on the details of the manipulation necessary in an exercise, they will remember the details more easily with a slight nudge in the right direction.

More importantly from my vantage point, the kids are willing to dig into the exercises.  Even if they have forgotten how to do them, or in the case of the current material are unsure how to begin them, they at least try something.  And in that attempt, they at least have a place to start the discussion the following day.  In that attempt, they are trying to draw on the mathematics they have covered not just in my course but in previous courses.  In that attempt, they are showing a willingness to actively engage the exercise, understanding that even if they aren't successful at home, they are increasing their chances of understanding the point of the exercise when it is discussed in class.  And in that attempt, they are showing me they are ready not only for calculus next year, but for college the following year.

This is also why it is really obvious who has done the homework and who hasn't.  The kids who have done the preparation more easily participate in the discussion, whereas the kids who have not, even though they try to participate and try to do the problem at the board, well, there's just something about the way they do it that screams, "This is the first time I've even looked at this problem!"  Hopefully the kids who have let off the gas pedal the last couple days will get back to normal by Monday and finish the year as strong as they began it.  We'll find out soon enough...

Flexibility

It's been a relatively abnormal week in terms of the schedule.  Monday I was out for religious reasons (any Orthodox Christian knows that this is "recovery week" from Pascha), and since I teach the honors-level pre-calculus, lots of my kids were out a day or two the rest of the week taking AP exams.  In fact, yesterday my first bell class was down to 12 students.  So, a few observations:

• The report from the sub on Monday was glowing.  The kids who were in class (the AP Chem test was Monday, so numbers were down) came in, got to work, kept track of the participation...all without needing a prompt from the sub.  Looks like they've taken ownership of the course and responsibility for their learning.
• Switching the groups every day as I do is an advantage during weeks such as this.  Since the kids are used to being with a different set of kids each day, they easily reorganized themselves into reasonably-sized groups when the number of absences made it necessary, and the conversations continued as usual.
• Along those lines, one of the students from my 3rd bell class was going to have to miss several days this week because of the AP tests.  Since she does not have a 5th bell class, she asked if she could join my 5th bell on the days she would miss 3rd bell.  Of course I said yes (seriously, a kid is asking to not miss class...what's not to love?).  Her experience, in her own words: 

"This is the beauty of this class. I was able to come into a completely different class than the one I'm normally in and have meaningful and interesting discussions. The format of the class encourages a kind of intimacy that doesn't usually exist in this school. I could sit down at a table with people and have a conversation, when I was probably avoiding eye contact in the hall three minutes prior. It's fantastic."

• Harkness for 12 is very different than Harkness for 30.  On the one hand, with only 12 students in the class we had one group instead of the usual three, which allowed me to be at that one table the entire class.  I saw and heard every comment, suggestion, and inquiry, and was able to keep the tally of the discussion points myself rather than relying on the students to double-check my count.  On the other hand, with me at the table with them the entire time, they were looking for my input more than normal.  So, 12 allows me to have more direct contact with all of the kids the entire class period, but 30 allows for (or more accurately, requires) more ownership on the part of the kids.
• The kids were great about making sure the kids who were absent were brought up to speed the next day, without me prompting them to do so.  They made sure that no one missed the opportunity to discuss any of the exercises.  Granted, the initial part of these discussion was much smoother than usual, since some of the kids had already discussed them once and could therefore more easily explain them to the students who were absent, but still, the team-oriented aspect of it was great to see.

Conclusion: Harkness is far more flexible than a standard class, and is able to handle schedule changes far more easily.  With this much disruption during the week, finding a way to make sure all of the kids got all of the information used to take some serious creativity.  Delivering a lecture more than once for the sake of the absent kids while figuring our what to do with the kids who didn't miss the lecture the first time made things interesting to say the least.  Having the kids who were in class the first time explain the new material to the kids show weren't there was one option, but for kids who aren't used to explaining the material on a regular basis this was difficult at best.  Meanwhile, the Harkness classroom just keeps rolling along without any major changes and, in all honesty, making more progress than the regular class does with the same disruptions.

So overall, a good week of learning both for the kids and for me...just the way it should be.

Listening

It's May, and even though we have five weeks left in the school year, there is always an atmosphere that arrives in May that makes it seems as though tomorrow is the last day of school when in reality it is five weeks away.  Prom is now over, AP tests begin next week, and so the struggle to stay focused continues.  On the good side, the overwhelming majority of the students are putting forth the effort, preparing for class, and responding positively to the requests I've made over the last couple weeks.  And the "poster class" is still doing pretty much everything right.  However, the other classes, while certainly better, are just not quite as on track as the poster class.  So, this week I tried to pinpoint exactly what it is that the poster class does better than the others, and I think I've figured it out: they listen.

By that, I don't just mean that they hear one another.  No, they really listen to one another.  They seriously consider the suggestions of each and every person in the class, trying to find the possibility of a solution in each proposal.  They respond to the questions and confusion others are experiencing, adjusting their explanations in the hopes that the new description of the solution will hit home.  They takes turns, rarely having more than one person in the group speaking at the same time.  They politely make corrections to the proposed solutions of others in the group.  There is a level of respect for the individual, in particular for the value of both their contributions and their needs.  And in all of this, the conversations do not feel forced or stilted...they are very natural, complete with a reasonable but unobtrusive dose of joking around.

The question, of course, is: what has gone right with this class that has not in the other three classes?  The answer to this still eludes me, because from my perspective I started each of the classes the same way at the beginning of the trimester, but have needed to give reminders to the other classes throughout the term that the poster class simply hasn't needed...if it ain't broke, don't fix it is coming to mind.  I really need to continue sifting through things to try to figure this out.

Finally, there is a very real test of Harkness coming on Monday: I won't be at school.  Pascha is this Sunday, so in addition to those church services we will be back at church on Monday morning.  The test will be this: the plans I left for the sub state, quite simply, that the class is run through student-centered, student-run discussions, that the kids know what to do, and that they will take care of the class for the day, including keeping track of the participation points.  I told the kids this at the end of class on Friday, so they know the expectations for Monday.  We'll find out in Tuesday how much progress they were able to make and how much they have taken responsibility for their own education.  In other words, we'll see how much Harkness has become part of the way they approach learning.  I'll let you know the results next week...

Different Classes

It has always been interesting to me as a teacher how different one class of kids can be from another.  You're the same teacher, and the kids are all taking the same course so supposedly all of the kids have at least close to the same level of ability, and yet certain classes do really well while others just don't live up to their potential.

Case and point from this week: one of my classes is, as I told them, the "poster class" for Harkness in a mathematics class.  They came in prepared every day, had good, deep, respectful discussions, and made the most progress through the current set of exercises.  They also had the best average on the last test.  Two of the other classes are doing reasonably well.  They are coming in prepared, for the most part, and are taking the class seriously, but they are a bit less focused during the discussions, and are not as in tune with the idea that the discussions are about the individual and about the group.  There are a few disrespectful tendencies that need to be broken, such as moving on to the next exercises when someone at the table doesn't completely understand the current one.  I need to refocus them on the idea that if someone at the table isn't understanding their explanation, then their explanation needs just as much help as the other student's understanding does (there are teachers I know who have this same difficulty, automatically blaming the kids for doing poorly without reflecting at all on the delivery of the material they are providing).  Then there is my remaining class.  They have been the primary offenders when it comes to not being prepared for class; for example, one day this week, about one-third of the class had not done the homework.  Calling them on it helps the next day, but it doesn't "stick" the way it should and a couple days later they start to fall back into their bad habits.  Of course, as I mentioned last week, lack of preparation leads to slow,  difficult discussions, and because of that they made the least amount of progress this week on the current exercises.  They also had the lowest average on the last test - all to no surprise.

I do not, however, see this as a weakness of the Harkness method.  Quite the opposite.  Harkness makes it more obvious when a kid isn't prepared for class, when as kid doesn't really understand the material, when a kid is trying to "phone it in", etc.  It is also more obvious when a kid really is putting forth the effort, when a kid has a solid, deep understanding of the material, when a kid is "invested" in the course, etc.  Because of this, I believe I have a more complete picture of the students individually than I ever have in the past.  By the end of class each and every day I have a good feel for where they are individually, both in terms of understanding the content and of how hard they are working, much more so than I have even been able to have in the past.  So...weakness?  No way.  I just need to figure out how to get the kids in the other three classes to have the focus, work ethic, and respect for the individual that the "poster class" has.  I need to step back and uncover what it is that I'm doing that has led the other classes in a different direction than the poster class.  Granted, it could just be the make up of the kids in the other classes, but I need to make absolutely sure it's not me.  Unfortunately, I don't think it's something drastic.  It seems to be something very subtle.  Hmmm....

Staying Focused

Every year, the students (and the teachers, if we're honest about it) reach a point where staying focused and finishing the school year strong becomes close to impossible.  The weather slowly gets nicer, and the warmth and freedom of summer begin to dominate our thoughts.  "Homework? How can I possibly do homework?  Look how nice it is outside! I'll do my homework after I go for a walk...a long, slow walk." (As I write this on Saturday morning, the temperature outside is 37°, after being in the 80s earlier this week...welcome to Ohio.)

The problem is: not doing the homework this year in my class means not learning the material.  This was in the spotlight this week for me on two occasions.  First, early in the week, the discussions were lagging, and a quick check around the classroom gave me the information I needed: a lot of the kids had not done their homework, i.e., had not struggled with the exercises at home and prepared for the discussion.  This has happened several times in the last couple weeks, but I decided to wait and see if the kids would pull themselves out of it.  They didn't, and I called them on it, gently but firmly reminding them that not preparing for the discussion affects not only their understanding of the material, but everyone else's, since the poor discussions that result from the lack of preparation negatively impact even those students who have done the homework.  From that point forward, the kids did the homework, the discussions improved, and as I gave them the positive feedback, specifically mentioning that the discussions were much better than they have been recently, I saw a lot of nods around the room from both the kids who had never stopped preparing for class and those who had finally returned to doing so. 

Then yesterday I had a discussion with one of the English teachers who had thrown an open question out to her teacher friends on Facebook, asking how we assigned and used homework...in other words, what was our "homework philosophy".  I gave this some thought before discussing it with her, and it made me realize just how profound the difference from last year to this really is in my personal "homework philosophy".  Last year, homework was for practice of material and skills learned (or at least material and skills seen performed by the teacher) in class.  Homework was not for learning; homework was for doing the repetition necessary to memorize the algorithm, regardless of whether or not any true understanding of the material and skills existed.  This year, homework is for preparation.  It is for beginning to cultivate the questions about the material that will need to be answered, and discovering (or recovering) the skills, definitions, etc., necessary to find the answers.  Homework is for discovering and learning what you can on your own, and preparing your questions for the material that just doesn't quite click.

Through this reflection, I realized that in the past a lot of kids were able to get away with not doing the homework, or not taking it seriously, or just copying it from someone else.  The classroom was the only place where new material was presented, so putting off the practice required to memorize the material for the test until absolutely necessary wasn't a problem.  Everything needed for success on the test was delivered in class, so as long as the student paid attention or took notes, the timing of the practice (that day, or the day before the test, or whenever...) wasn't as important.  This is why cramming for a test could, potentially, be a successful strategy.  If the student can memorize well, then the day before the test is the perfect time to "learn" the material.  Of course, some kids need more repetition than others, so for some kids the daily practice was very important.  For others, however, not so much.  Which then, of course, brings forth the question: if a kid doesn't need the repetition, why make them do it?

This simply isn't true in a Harkness classroom.  The questions being asked on the test demand a solid understanding of the material to the point that some creativity is required.  Memorization doesn't come from sitting down and memorizing all of the material; rather, it comes from struggling with and working through and making sense of it, so that comprehension, rather than just pure memorization, takes place.  Both the discovery at home and the discussions in class are important parts of this process, as my students saw during the past week.  Personal preparation is important not only for the individual's understanding of the material, but also for that of the others in the class.  This makes the homework an integral part of the course rather than just practice (necessary or not) of the material covered in class.

Now admittedly, there are still a couple kids who possess solid problem-solving skills and can still contribute to the class (though not as completely or meaningfully had they done the homework) without doing the preparation at home.  However, a few of the exercises this week stumped them in class, and I firmly believe that had they done the preparation at home they would have successfully done the exercise.  Hopefully, this, coupled with the fact that they are perilously close to falling out of the A range, will inspire them to do the homework.  We'll see...

Reviewing Revisited

I mentioned a few months ago that I was struggling with how to help the kids study/review for the tests.  While I have not come up with the ultimate answer and solution, I've been trying something in the class this trimester I think is a bit more in line with the overall Harkness philosophy.

Each unit in this part of the course has 48 exercises, and we have planned on each of the five units taking 10 days total.  That gives us eight exercises per day for six days, three days for review, and one day for the test.  The three days of review may seem to be a bit much, but the plan gives us the necessary flexibility in case the discussion of a few of the exercises gets more in-depth and therefore takes longer than average to complete.  For the review, the plan going into the unit is:

• day 1: tie up loose ends form the exercises on the worksheets, revisiting the exercises on the worksheets that summarize or apply the skills and material on which we are currently working
• day 2: the students look through the textbook and bring in exercises from the sections that relate to the material on which we are currently working (I tell them which sections, they bring in the exercises)
• day 3: the students work through a set of exercises that contain applications of the current material, but that do not contain any new material; they receive and work through these exercises in class, so these exercises are not prepared and discussed as happens with the regular exercises

So far, the review itself has been much better, but the results on the tests (two so far) have been about the same, with no significant increase or decrease in the scores.  On the positive side, this is giving the students responsibility for the review, and keeps the focus on them working through the material with me doing the constant analysis and assessment of their work to make sure they are gaining a real, deep understanding the material.  And this method of review is certainly more in keeping with the Harkness philosophy.  On the negative side, I had hoped that the scores would increase, showing the review to be more effective.  But at least it is not having any detrimental effects.  As such, I take more student-centered and student-driven with the same test scores to be a win.  Not a win-win, but a win.  Hopefully by the end of the trimester, the second win will arrive.

What Else Could I Ask For?

The first week back from Spring Break is always one of the slowest and most difficult of the school year, not only because waking up early has returned to the daily routine, but also because we don't have another day off until Memorial Day - a full two months from now.  I gave the kids the normal amount of homework to do over the break- one worksheet.  On Monday at the beginning of first period, I said good morning, but not much else, and after the required five minutes of complaining about how tired they were, the students began to discuss the worksheet, picking up where we left off before break.  My other classes went pretty much the same way.  There wasn't a huge need to go back and review the material from the first worksheet of the new unit which we began that previous Friday, and as we made our way through the week it became evident that the kids pretty much have a handle on the material we're covering in this unit.  The classes are self-starting at this point, and at times the students go to the board to start in on the exercises before the bell rings to officially begin class.  I've guided the discussions more by asking questions (sometimes as simple as, "Are you sure about that?") than by explicitly pointing out any mistakes, and I really don't fix the mistakes at this point, since the kids make the corrections themselves, most of the time without me needing to point out the error.  This didn't happen as quickly during the previous trimesters (though it did eventually happen), and there are still certain bells on certain days that need me to get them on track at the beginning of the period, or to get them back on track during the period, but these days are becoming fewer and fewer in number as the term goes on.  In fact, this week, out of a total of 20 periods, all of them self-started, and only two needed to me to help them refocus.

Seriously: What more could I ask for?

The kids have bought in to Harkness.  They have taken responsibility for their own learning.  They have started to make connections with the previous material.  They have started to get deep into some of the exercises, not stopping at the level of just answering the questions, but really trying to see the important ideas underlying the exercises.  There are times when I need to help them extend their findings, to nudge them toward a deeper understanding of a particular piece of the material, but again here these times are becoming fewer and fewer in number.  The time it takes to nudge them is a lot less than the time it would have taken had I been lecturing to them, and most times I don't need to finish my sentence or two before at least one of the kids makes the connection, at which point I get out of the way and let the student finish nudging the rest of their group.

Seriously: What more could I ask for?

I don't know what I did right this trimester that has allowed us to get to this point so quickly this term.  Perhaps it's actually nothing I did, but rather it's just the particular mix of kids in each of the sections I'm teaching, or some other thing I'm not recognizing and over which I have no control.  But I intend to enjoy the remaining eight weeks for all they are worth, looking for what went right so it can be replicated as much as possible next autumn and beyond.

Along those lines, I'm fortunate enough to be going to Exeter this summer for their annual math conference, hoping to fill in some of the gaps in my understanding of Harkness which I'm sure are there.  I'm also hoping to connect with other public school teachers who are trying to implement Harkness in their classrooms, giving and taking through a week of discussion, with the aim being not only to do things better next year, but to look for ways to inspire others to join us in bringing out the best in the students and better prepare them for what lies beyond high school.

I guess that's what else I could ask for.

S.P.I.D.E.R.

It is the first weekend of spring break, and having administered the first test of the new trimester last Thursday, I'm honestly feeling really good about the way things are going this trimester.  No, not everyone is the class earned an "A" on the test, but from the standpoint of the daily discussions, the students seem to have a solid feel for what is expected and how to get the most out of the them.  The mistakes on the test were more along the lines of not reading directions or making other sorts of careless mistakes, as opposed to not having any idea what is going on.  And in general, the students who did not participate as much in the discussions were the ones with the lower scores (although I seem to have a couple kids with test anxiety - those who are great during the discussions and just can't seem to put it together on the test).

Beyond the fact that the students are more experienced with Harkness at this point in the year, I did something subtle late last week in my classroom that I think the kids have noticed.  On the side wall of the room, I put the following "guidelines" for the Harkness discussions - one each on six separate pieces of paper, with the font size large enough to be read from most parts of the room:

Synergetic - the discussion is team-oriented and one in which each member participates equally

Practiced - the concepts and skills learned during the discussion are practiced by the students as they review the material and as they prepare for future discussions

Independent - students run the discussion, questions and analyze possible solutions, and summarize results

Developed - the discussion gets deep and builds on itself and on previous discussions; individual questions, worksheets, and discussions do not stand in isolation, but are inter-related

Explored - more than simply a discussion, it is a discussion-based exploration of a topic through the exercises on the worksheets

Reflective - students evaluate their participation in the discussions and look for ways to improve their contribution to the class in order to improve their understanding of the material

The "S.P.I.D.E.R." acronym is stolen from Alexis Wiggins (http://modelsbydesign.wordpress.com/2012/09/16/goodbye-harkness-hello-spider-web/), though I tweaked it slightly for the purposes of teaching math and of being in a classroom with 30 students instead of 15.  

I plan on giving the students a handout with the expectations on it when we return from the break, not so much to inform them of what is expected - as I said above, they really seem to have a good feel for this - but more to quickly refocus them and get them back into "school mode".

Beyond that, there's not really much to report this week.  Things are good, and hopefully they will remain that way.