Fundamental Differences

Well, today was the last day of school before winter break, which in my classroom means we eat waffles and "celebrate" Undiramahanukwanzrohgurmas (I'll let you try to figure out the combination of holidays and Holy days). It also means it's a good time to reflect on the half year that has passed and the "lessons learned" about the Harkness Method:

(1) Harkness requires the students to be prepared. The daily discussions don't work if the kids aren't prepared. The class doesn't move forward if the kids aren't prepared. And I mean this in a very positive way, because the class shouldn't move forward if the kids aren't actually learning the material. Their preparation or lack of preparation, and their subsequent progress or lack of progress, are more obvious in a Harkness classroom, and that's a very good thing.

(2) Harkness requires the teacher to think of the course as a whole, instead of planning chapter by chapter or topic by topic. Good teachers already know that this should happen, but in a regular classroom, if you fall behind and need to take things day-by-day for a few days, it's ok. Not so in a Harkness classroom. The planning that we had to do over the summer was absolutely crucial, and there is no way that we could be teaching this way without having the overarching vision of the entire course in mind. In a set of e-mails last week, the other honors pre-calculus teacher and I were planning for the bulk of what we need to do in terms of pace and timing when we return in January. This long-term planning is not a luxury...it is essential for Harkness to work.

(3) Harkness requires the students to truly understand the concepts, and not just memorize a few facts and algorithms. The whole point is to have the students learn how to solve problems, and as such the questions on the worksheets and on the tests need to use the skills and concepts in ways which they have not seen before. The basic mechanics must be solid, but just using the mechanics or using them in word problems the likes of which they have seen repeatedly in class and in the homework does not help the kids learn how to problem solve and does not test to see if they are learning how to problem solve. This shift from "the students imitate the teacher and the textbook" to "the students creatively solve problems" is difficult for both the students and the teacher. Students are used to being shown how to do the homework exercises and required to do strictly similar exercises on the tests. Teachers are used to telling the students how to solve the exercises and directly answering any questions. Harkness requires the students to struggle with the material and through the struggle put the pieces together. It requires the teachers to lead the students to the answers through carefully constructed questions on the homework and responding to their questions with questions in the classroom. It is unfamiliar territory in many ways, but the resulting understanding of the material is well worth it.

(4) Harkness takes patience, both from the students and from the teacher. The students need to understand that we are slowly building the material for the entire course during the entire course, so some questions they may have will not be answered for a while (though encouraging them to seek the answers on their own is a great idea). The teachers need to understand that requiring the students to put the material together will take more time than just telling them what they need to know. They also need to fight the instinct to save the kids when they are struggling, and instead lead them through the struggle by solid questioning techniques. Both need to understand that the short track only leads to a surface-level, short-term understanding of the material, whereas the slow but steady patient discovery of the material is both deeper and more permanent.

(5) Harkness is harder than being in a "regular" classroom, both for the teacher and the students. The focus, the creativity, the mental energy, the ability to handle frustration, and the perseverance required in a Harkness classroom are all more difficult. But the rewards are more than worth the extra effort.

A quick story to wrap up the week, and the year: a student whom I had in class first trimester came to me today to tell me that he had retaken the SAT a few weeks ago. Despite the fact that he had forgotten his calculator, he earned a better math score this time. I know it's just one piece of anecdotal evidence, but I see the benefits of Harkness all over his success. The problem-solving skills promoted by a Harkness classroom, along with the fact that we have relegated the calculator to its proper place as a tool to be used as necessary and not a crutch upon which we rely without questioning, deserve at least some of the credit for this success. This student's story was the perfect way to start my break.

Work, Work, Work...

We are now one week away from the beginning of our winter break, and I haven't been this ready for a break in a long time.  Put simply: I'm exhausted.

When I describe the Harkness Method to some people, they mistakenly get the impression that all I do is sit around and listen to the students while they do the work.  And while on the surface that description is true, just sitting around has turned out to be more exhausting than lecturing for several reasons:

1. Lecturing involves preparing a set of notes and, after careful editing and polishing, delivering the information to the students.  Every once in a while, you evaluate the students' work, whether this is homework or a quiz or a test.  It is stressful on your voice for about two weeks at the beginning of the school year, and it can be stressful if you have not taught a particular course in a while, but once things settle in it's really not that bad.  Harkness, on the other hand, involves constantly evaluating students' work in class, and in my classroom, that actually means evaluating the work of at least three students at a time.  Yes, it is easier on my voice (this is the first year in a long time that I have not needed cough drops at the beginning of the year), but mentally, it is constantly taxing. 
2. Lecturing involves being prepared for an essentially predictable set of events.  And after teaching a course a few times, even the questions that the students have becomes essentially predictable, to the point that a good teacher will change the lecture to address the anticipated questions before they are asked.  It also involves doing the homework exercises.  Since you have shown the kids how to do the exercises, the way you do the problem on your sheet will almost always match the correct answers on the kids' sheets, and the kids who don't get the correct answers will either have something that strongly resembles what you have on your sheet, or they will have a blank spot on their sheet.  Quite honestly, if you have to teach the same class more than a couple times a day, this can often get a bit boring.  Harkness, on the other hand, involves preparing the worksheets for the students, designing the questions so they are led to discover the material.  Initially, you think it involves preparing for the questions the students will have or for the mistakes they will make.  However, you quickly realize that you can't predict the many different ways the kids will devise to solve a problem.  So instead, being prepared involves knowing the material you are teaching so well that you can quickly analyze the vast majority of what the kids put together and respond accordingly.  It involves doing the homework exercises so you have the correct answer along with a way to get the answer, but a lot of the time the method the kids use to answer the problem won't match yours.  Even if you teach the same class several times a day (as I do), this doesn't get boring, ever.  The kids in one class normally don't use the same methods as those in another class, so the evaluation of whether or not they are doing an exercise correctly, or are at least going down a path that will lead them to discover  correct method, never ends.  Again, this is way more mentally taxing that lecturing.
3. For many people, lecturing involves following a textbook and assigning problems from a textbook.  Little to no thought goes into the order in which to teach the material because the book provides the order for you.  Little to no thought goes in to choosing which exercises to assign because the teacher's edition of the textbook suggests which exercises are appropriate.  (To be fair, I do know a number of teachers who, despite the fact that they lecture, do not necessarily take the textbook "in order", who make their own worksheets, etc.  However, I also know a lot of teachers and administrators who believe that "the textbook knows best" and would never even entertain the thought of doing something that was not prescribed by the textbook.)  Harkness, on the other hand, involves knowing your material well enough to be able to teach it without a textbook.  It involves thinking through the entire course and planning the delivery of the material in the order and at the appropriate depth for your students.  It involves writing your own worksheets because the exercises in the textbook are inadequate in that they do not lead the students to discovered the material, nor do they encourage the students to do anything other than repeat the processes you have shown them.  

In short, Harkness involves more work, both in terms of preparation and in terms of the day-to-day work in the classroom, which is only fair.  After all, if we're going to ask the kids to start thinking for themselves and not relying on being spoon fed by us, then it only makes sense that we, the teachers, should start thinking for ourselves and stop relying on being spoon fed by the textbook companies.

They Get It

I have been waiting for this week to arrive.  I wasn't sure when it would arrive, but I was confident that it would, and at long last it is here.

I have mentioned before that I have the kids turn in a journal each week.  So far the journals have been pretty straight-forward, telling me about the material we covered, the successes and struggles they are having with the material, and how the groups they are in are working...the usual stuff that I have requested from the students for years.  However, this week the kids in my "part A" class (the first half of the two-trimester honors pre-calculus class) took their first test.  And while there is plenty of room for improvement as far as the grades are concerned, there were two very positive occurrences.  First, the vast majority of the mistakes were either simple arithmetic or not answering the question (i.e., not reading the directions), so from a content standpoint the class is in great shape.  Second, and this is the "event" I have been waiting for, the journals contained passages such as the following:

"This week, the advantages of using the Harkness method became evident for me, especially for our first test. Instead of having to go back and study things that we learned two weeks ago and never talked about again, we built on those things to the point where we were still incorporating some of our preliminary topics into tougher problems. It really benefited me during the test because I didn’t have to go back in my mind to a few weeks ago and think of things that we discussed. Also, instead of having all teachers show us one method to solving a test problem, I was able to use some “shortcuts” that some of my classmates showed me throughout the week(s)." 

"Lo and behold, you somehow pull out [...] an A! A miracle! And I truly believe that it is all because of the Harkness method. In the nights leading up to the test, unlike in past math courses, I was not looking at review problems and having no idea where to start.  Yes, I had a few topics that I needed to reexamine, but I wasn't going into the test stressing that I didn't know that information.  I understood the information with a much higher retention rate."

"I feel much more confident in my ability to succeed in the class.  More specifically with the discussions, I feel that I am contributing my ideas well, and taking a lot out of the processes of my peers.  For example, when we discuss the features of a particular graph, it is always reassuring to have multiple people explain the different aspects of a function - quite frankly, some people are inevitably going to be better at simplifying and furthermore, explaining their process."

What else is there to say?  They get it.  They get the whole point.  They get that it's about understanding the material.  They get that having a discussion makes for a deeper understanding of the material.  They get that it's OK to admit when you're having difficulty with the material and to ask for help.  They get it.  Amen.

Now admittedly, not all of the journals contained such passages.  But the only negative comments in the bunch were about ways to make the discussions even better.  None of them...literally none...were negative to the point of wanting to return to lectures.  Despite the admissions that this does take more work than "the old way", there was the immediate follow-up that it was worth it.

Hopefully this attitude will not only last the rest of the term, but will make it's way into the other sections of the class and have a positive impact beyond this trimester and possibly even beyond this year.  I'm well aware that once a class or a teacher gets a reputation, be it good or bad, it's hard to shake.  And at this point I'm optimistic that Harkness is starting to get a positive reputation.

Not Lecturing is Not the Same as Not Teaching

It's always fun to watch the reaction when I describe how things are being run in my classroom this year.  After describing the work that was done over the summer to carefully put together the worksheets, and how the worksheets are designed to lead the students through the material, and how I go from group to group listening to the conversations and helping guide the discussions as necessary, I still regularly get one of the following two questions: (1) "So you're not teaching them?"; (2) "So you just sit there all day and don't really do anything?"

I've learned to translate the actual meaning of these questions as, "So you're not lecturing?"  The answer to this question, of course, is, "Yes, I'm not lecturing."  However, for those of us who were taught almost exclusively by means of being lectured to, there is this sense that if there is no lecture, there is no teaching going on.  And I used to be very much of the same opinion.  Clearly I have changed my mind and have come to understand that while lecturing is one method of teaching, it does not guarantee that any learning is going on.  I have seen ample evidence of this over the years.  Specifically in math, I have seen both of the following on a far-too-regular basis:

(1) Students (and parents) who complain that questions on a test were "different" than those that were assigned for homework, meaning they expected the questions on the test to be identical to those that were assigned for homework, except with different numbers.  This, of course means that the students didn't actually learn the concepts...they just memorized the steps involved with the problems that were assigned.  This is how many of us got through many of our classes in high school: just memorize what the teacher gives you, and give it back to them on the test. Then, forget it almost immediately and start memorizing the new stuff.  This of course leads to...

(2) Students who forget the material they were taught last week/month/year.  Ask almost any math teacher how they spend the first six weeks of the school year, and you will get pretty much the same response: reviewing.  And if they are feeling particularly honest, they will admit that what is really going on is reteaching, since the kids don't remember very much from their previous math courses, despite earning an A or a B in the course.  The refrain "I've always gotten As and Bs" is far too common, and most of the time it's honest.  However, the grade does not reflect how well the student has learned the material...only how good their short-term memory is.

In short, in my experience, lecturing is good for short-term memorizing, but not for long-term learning.  So how do we actually learn things to the point that we really understand them?  The answer to this is easy if you stop and think about it: learning and understanding rely on struggling with the material yourself, reasoning it through for yourself, discussing the topic with others who are struggling with it as well, discussing the questions you have with an expert, and then practicing a few exercises.  Think back: if you really understood a concept in math or science, there was no need to do a lot of practice.  In fact, if you understood the material really well, the practice got annoying.  On the other hand, if you didn't really understand a topic, then the late exercises from the homework were just as difficult as, if not more so than, the early ones, emphasizing the fact that you hadn't really learned the material, and no amount of extra practice was going to change that.

So, as I sit at the tables in my classroom with my students, listening to them as they discuss their successes and struggles with the assignments, responding to the questions they have (usually with a question in return to help them finish the discovery they began), no, I'm not lecturing.  But they are most certainly learning.  And since I'm the one who put together the worksheets that are guiding their discovery, and since I'm the "expert" to whom they are bringing their questions, then ultimately I must be the one teaching them.  But in addition to teaching them some math, I'm also teaching them how to learn.  Sounds good to me.

Round Two

This week, we began the second trimester, which means I have four new classes.  Three of the four classes are the second half of honors pre-calculus, and most of the students in there I had during the fall term (though not all).  However, the kids that were in class together last trimester are not necessarily together this trimester.  So, in many ways, we started over yesterday, beginning to build the "community" that leads to productive discussions and solid understanding of the material.  On the positive side, regardless of who they had for the first half of the course, all of the students have had twelve weeks of Harkness, so we hit the ground running yesterday, and so far the discussions are going well.  Two of the three bells have had great discussions during these first two days, and I'm really looking forward to working with them this trimester.  The other class has a few students who still want to be spoon-fed.  I understand that it's difficult to make the change from passive to active learning, to go from having things handed to you to having to work for it.  But I also know that having to really struggle with the material makes the learning more permanent.  I'm under no illusion that they could go back to passively learning the material, getting their daily lecture from the other students.  But I hope they will come to understand they are cheating themselves out of a true understanding of the material by not actively doing the work.  I also hope they will come to understand that if they want to help run the business or engineering firm or whatever, then simply imitating someone else is not the way to make it happen.  Rather, they need to be the active problem solvers in the organization, and that is precisely what I'm trying to prepare them to be.

My other class this trimester is one section of the first half of the course.  These students should be going through the culture shock of not being lectured to every day.  However, they have been phenomenal so far.  I'm not sure if it's the particular students I have in the class, if it's the fact that this is my second round using Harkness with this material, or what, but the discussions today, on day two, were great.  The vast majority of the students in the class were ready to go, had done the appropriate prep work, and went deep into the material.  Yes, there were some logistical questions about how to take notes and what was really expected of them, but for the most part the answers to their concerns were, "You've got the right idea...that's what I'm looking for."  It probably also helps that several of the students in the class are on the speech and debate team, so they understand the meaning of prepare for a discussion, be prepared to defend your answers, and so on.  Whatever it is, I hope it continues for the entire trimester, because these first two days have been amazing.

One final comment for this week: the Spanish teacher across the hallway from me implemented some of the ideas of Harkness in her classroom last trimester.  As we were talking today with a couple other teachers, she said that the problem solving and discussion of Harkness seemed to her to be a perfect fit for a math classroom, but that it would be difficult to implement it in other disciplines.  I found this interesting since the usual way the conversation goes on the web is that Harkness is a perfect fit for English and History, but not for math.  It sort of put things into perspective: Harkness can  be a perfect fit for any of the disciplines with the right commitment to it from both the teacher and the students, and I took it as a compliment that the commitment was evident enough to her for her to make the observation of this perfect fit happening in my classroom.  Now if I could only get others to see that Harkness is a good fit for their classroom as well.

Final Exams

Well, we've reached the end of the first trimester, which means we've reached the first round of final exams.  One of the things the other honors pre-calculus teacher and I decided over the summer was that we would not change the format of the final exam or of the review we gave for it.  We did this for the obvious reason: we wanted an objective measure of whether or not the students were doing better this year.  As in the past, we spent two days on the review exercises.  The results were not surprising.

First, the review itself was easier.  The students came in better prepared and had far fewer questions than in the past.  And on the exercises with which they initially had trouble, they were more willing to at least try something than students in the past.  And on the day of the exam there were far less last-minutes questions than there have been in the past, and overall there seemed to be less anxiety about the test (either that, or the students this year are remarkably good actors).

Then came the test itself.  From a grading standpoint, this year was more difficult, but for all the right reasons.  The mistakes that the students made were more "calculation" than they were "content".  Content mistakes are easy to grade, since it is immediately obvious that the student does not know how to solve the problem.  Finding calculation mistakes in an otherwise correct solution, however, is far more difficult and far more time-consuming.  Don't get me wrong, I'm very happy for the change.  Grading the paper of a student who at least knows what they're doing but (like me) has occasional difficulties with 1+1=2 is much more enjoyable than the alternative.  As I've said previously, I would much prefer that the emphasis in education were placed on how much the student has learned rather than on the grade they have earned, and as such when the test indicates that the students have learned the material, the rest is worth it.

Now, admittedly, the mathematician side of me knows that the scores on one set of exams is not enough to definitively indicate that one method of teaching is better than another.  But this was about more than just the numbers.  The way the students were able to review more independently, the variety of methods they used to solve the some of the exercises, and the overall tenor of the classroom over the last few days all have their source in one place: Harkness.  The students are less dependent on me for the initial attempt at a problem and have less anxiety when it comes to solving multi-step exercises, and having seen the progress the students have made in both of these areas as the trimester progressed (and having not seen the same kind of progress in previous years) tells me it's working.  

And despite the initial resistance, the vast majority of the students have acknowledged their progress as well.  In all fairness, there are still a few students who would prefer to be spoon-fed and then regurgitate the method they were shown rather than figuring things out on their own.   After all, mimicking the teacher has worked for them for the last 11 years, and at this point, it's easier for them.  But looking at what will be better for them long-term, I know that mimicking others is not the way to go.  Those who can mimic are plentiful; those who can problem solve are rare.  And those who are willing to make the effort to at least attempt to solve any problem set in front of them are the ones the colleges and employers keep saying they want.  By the looks of it, Harkness-trained students are what they're looking for.  And one trimester in, I'm more than willing to provide.

So far...

This is the last week of the trimester, so in part I have spent the week looking at the worksheets as a whole and reworking them for next trimester.  In our school, the year is broken into three terms, and a standard course is two terms in length.  That means a student could have the two halves of the course during 1st and 2nd term, 2nd and 3rd term, or 1st and 3rd term.  There are advantages and disadvantages to this, but from a teaching standpoint, it means that one period each day I will be teaching the first half of the course again next term, and as such one of the advantages is the immediacy with which we get to reflect on the first half of the course as a whole and make the changes we feel are needed instead of having to wait until next year.

So, things that need to change:

(1)  We need to cut the number of worksheets.  The allotted time in the trimester along with the unforeseen interruptions (students being pulled out of class, field trips that take enough students out of my class to have an impact, etc.) made 37 worksheets simply too many to cover.  Put honestly, there were times, especially toward the end of the trimester, where things just felt rushed, and not being able to take the time to thoroughly discuss the exercises is antithetical to Harkness.  Taking into account Thanksgiving and Winter Break, and that 2nd trimester always has the potential for a random snow day, reducing the number of worksheets is imperative.  In particular, we included a few worksheets at the beginning as a means of introducing Harkness, worksheets that were mainly review.  These are being cut from the course next trimester, along with other changes.

(2) We had an expectation going into the trimester that the students should be able to complete one worksheet every day.  Because of this, we felt that 37 worksheets was a reasonable number.  Knowing now that this led, at times, to a "rushed" atmosphere in the classroom, we need to be clear from the beginning of the trimester that the expectation is for the students to thoroughly discuss each of the exercises and to take notes on the material and methods used to solve them.  Obviously we have a certain amount of material that we need to cover, but based on this trimester, setting the focus on the depth of the discussion is important, and with fewer worksheets, the pace should not be a problem.

(3) In addition, there are a few topics in our curriculum that are labeled with the phrase "if time permits".  In the past we have always had time to get to these topics, so in writing the worksheets over the summer, we felt comfortable spreading the topics throughout the worksheets.  However, because of the discovery involved with and the depth of the discussions resulting from using Harkness, the time that in the past was more than available ran short.  As such, we are re-arranging some of the questions on the worksheets (about 10-15 questions total) to place all of the "if time permits" topics on the last few worksheets.  The last few worksheets will also include some "trimester review" exercises (just as they did this trimester), so hopefully there will not be a break in the flow of the material.

(4) Since the entire process is so new to the students, they need to be reminded often of the "ground rules" for the discussions.  This didn't happen nearly enough this trimester, so my plan is to have the students read through the handout outlining the ground rules every Monday at the beginning of class.  Hopefully it will set the tone for the week and keep the groups focused not only on the math but on the group dynamics.

In preparing for the second half of the course, the lessons we have learned from the last few months have influenced the worksheets we have been making.  In particular, we have set the goal at 30 worksheets, meaning we want to cover the entire course in 30 worksheets.  Hopefully, this will allow for a more relaxed atmosphere and allow the students to thoroughly discuss the exercises, as mentioned above with regard to the changes we are making on the worksheets for the first half of the course.

Finally, as the students are preparing for exams this week, I hope that they will take Harkness home with them as they review for their other classes.  The lessons learned in my classroom from Harkness should be about more than just the mathematics.  It should also be about a way of discussing anything, whether the "anything" is material in other courses, planning a school-wide event, or preparing for a speech and debate tournament.  There are many residual benefits from using Harkness, and even if they don't realize it, I hope the students can take the method with them into the other parts of their lives.  It's funny, but for as much as we as educators have talked for years about preparing our students not just for future classes but for life, my experience is that we've never really done it.  This year, for the first time, I feel like I actually have an opportunity to do so, and it's an opportunity I'm not about to let slip past me.

More Committed Than Ever

As we near the end of the trimester - next week is the last full week, with final exams the following Wednesday and Thursday - it never fails that there are some students who, having come to the realization that the grade they had hoped for is probably out of their reach, begin pass through what are commonly referred to as the "five stages of grief". All five of the stages were on full display in the journals from the students this week. Before I get into this, however, a few disclaimers:

(1) There are only a few students each term that go through this. The vast majority of the students are comfortable with or have an understanding about their grade, and what follows in no way applies to them. In fact, many of the students mentioned in their journals this week that they have learned the importance of consistent, daily effort this trimester in my class courtesy of Harkness. They mentioned that missing a homework assignment or a day of class has never had the kind of negative impact on any other class in the past. In addition, they mentioned that they understood more acutely the need to get the notes from and discuss the material with another student if they had to miss class. One more positive, unintended consequence.

(2) In what follows, every time I mention "grade they want", it means "A". I am teaching nothing but honors pre-calculus this year, and for many of the students "pass=A" while "anything else, including A- = fail". As a teacher, I can't stand this attitude for several reasons, not the least of which is that it takes the focus off of the learning and puts it on the grades. However, as a student, I remember this attitude all too well, so I do feel some empathy. Now then..

Stage 1: denial
These are the students who ask if there is any possible way, if they do well enough on the remaining assignments, to earn the grade they were looking for when the term began. Since we're talking about a math class, my first instinct is often to give them a list of the remaining assignments and tell them to go make the calculation. However, it always seems mean-spirited no matter how I say it, and so I grab my calculator and normally we figure out that if they get a perfect score on all of the remaining assignments, including the exam, that the grade they want is in fact possible. Of course, this is not what they want to hear, and that leads to...

Stage 2: anger
This could just as well be labeled "blaming", since the anger tends to be channeled away from the student and against anything else available, from extracurricular activities taking too much of their time to excessive pressure from their parents to requirements in other classes to how horrible the teacher is...you get the idea. This year, of course, there is an easy obvious target: Harkness. In fact, one of the students who has earned an A- to this point wants to go back to a lecture-mimic because of her low grades...no, I'm not kidding. Note that this is all about the grade and not about how much the students have learned. That theme is continued in...

Stage 3: bargaining
These are the students who are willing to do absolutely any extra assignment you put in front of them, so long as there is a promise that they will receive the grade they want if they complete the assignment. Notice this doesn't say "earn" the grade, nor does it say that they do well on the assignment. The students' part in this bargain is nothing more than completion of the assignment - it's all about effort, and not much more. If I decide to give an extra credit assignment, there are two important guidelines: (1) any student who has not completed all of the "regular assignments" is not eligible for "extra"; and (2) the "extra" assignment is available to all of the students others than those noted in (1), not just the ones who are struggling (very loose definition on that term). Of course, the extra assignment is no more nor less difficult than the other assignments during the trimester, and I actually grade it. To me, it's not about effort...it's about showing me you've learned the material. As I said above, for the most part this is not what the students had in mind when they mentioned extra credit, and this leads to...

Stage 4: depression
This is the one that can have a devastating effect. Lingering at this stage too long can result in not completing the extra assignment, or not putting the effort into the remaining "regular" assignments, rendering the extra one meaningless. These are the students I honestly worry about, not just for the sake of my class and not just for the sake of the other classes they are taking, but overall. I understand the quest for perfection, and I understand the thrill in attaining it, even if it's just one test and not an entire course. However, I also understand that perfection is fleeting, and that when it comes to academics, focusing exclusively on grades and not on learning is far more devastating than an A- on a transcript. I also understand that perfection is rarely a solo performance. Ask a pitcher who has thrown a perfect game how much the other eight players had to do with the achievement, and he'll tell you it's a lot. That being said, these are the students with whom I try to have a conversation, with the intended outcome being...

Stage 5: acceptance
By this I don't mean acceptance of an A-, or a B+, or whatever. I mean accepting that they have made a lot of progress during the trimester, that they have learned a lot of material, and that like the team that loses the Super Bowl or the World Series, there is still a lot to be proud of. This acceptance does not come easily, but it is quite possibly one of the most important lessons they can learn. State championships are rare for a reason. But there is no shame in being knocked out of the playoffs after winning the district championship. There should, instead, be pride in the accomplishment, and an understanding that the district title was earned through the same kind of hard work and dedication, both on and off the field, necessary to win a state championship. Academically, this means there should be pride in giving consistent, complete effort both in and out of the classroom, and an understanding that actually learning something and earning a B+ really is better in the long run than learning little to nothing (other than how to memorize something for two weeks and then forget it) and receiving an A.

So we enter the last week of the first trimester dedicated to having a Harkness classroom...dedicated to real, lasting, meaningful learning. And as I said, the overwhelming majority of the students have figured out that they really are learning and retaining more this year than they have in the past, and that there really is a reason to put forth a consistent, complete effort. These are not empty words, as I have seen the struggle and the success they are having because of it. This year more than ever before I have seen the "light bulb" go on...in fact, it happens pretty much every day in every class, and it has reminded me why I got into teaching in the first place. Or rather, it has introduced me to what is possible if the focus in on the students and their learning, and not just on their grades and how well those grades reflected the degree to which they could mimic the way I do mathematics. There is an ever-growing list of reasons why I am dedicated now more than ever to having a Harkness classroom, but this focus on what is really important is way, way up there.

Too Good...

Just really needed to share what happened in class yesterday.  I'm still not used to how well Harkness works without me giving a detailed lecture.

First, here is the worksheet for yesterday's discussion:

This is the first worksheet of the final unit of the trimester.  No, they have not had anything about derivatives, Riemann sums, or partial fractions in class.  And yet, this is what happened:

Class Discussion - 31 October

Every group was able to estimate the area under the curve in #5, an every group came to the conclusion that in order to write the sum using "sigma notation", a choice needed to be made about "which end of the base" (left or right) to use to calculate the height of the rectangles.  Every group was able to find the partial fraction decomposition in #6.

When the students actually do the prep work and actively participate in the discussions, this is what happens.

The Value of Patience

I have mentioned in the past that part of the preparation for the change from a standard lecture-and-mimic classroom to a Harkness classroom was creating the worksheets for the class, since the textbooks we have (and, in fairness, none of the textbooks I have seen) have the types of questions we need, let alone in the order we wanted.  One reason for this is that we chose to not teach the material one chapter at a time, but rather to teach all of the topics for the course concurrently.  So, for the first half of our pre-calculus sequence, this means that pretty much every worksheet has exercises about polynomials or rational functions, and exercises about exponential and logarithmic functions, and exercises about discrete mathematics (sequences, series, binomial theorem, and the like).  Throw in a review question and a "think-outside-the-box" question (the online materials from Exeter are great for this), and you have a good worksheet.

However, one of the struggles we had was determining the pacing of the material.  We needed to create questions that led the students to discover the new material on their own, that challenge the students, that "stretch" them every day, but that don't make such a huge leap that the students can't put the pieces together.  For example, today's worksheet asked the students to evaluate 7 nCr k for k from 0 to 7 inclusive.  Monday's worksheet asks them to expand (a+b)^7.  Layer by layer, worksheet by worksheet, the material contained in the section on the binomial theorem will be covered (and thus, discovered).  However, at the same time, other exercises on the same worksheets are building the layers of solving polynomial and rational inequalities.  Other exercises are building proof by mathematical induction.  You get the idea.  This is a drastic departure from the "normal" way of delivering the material, where all of the material for the binomial theorem is delivered in a lecture on one day, all the material about proof by induction is on a different day, and the material about polynomials was covered weeks ago.  In doing this, one of the huge advantages is that the students are working with all of the topics, at some level, throughout the trimester.  I'm hoping that this will pay off on the trimester exam, since all of the topics will have been seen recently, and none of the topics will have been essentially ignored for several weeks.

On the other hand, there are times when a question on a worksheet seems to come out of nowhere and seems to not be connected to anything we are doing.  A good example of this is the question about 7 nCr k on today's worksheet.  We have not done any combinatorics or factorials this year, and the question is not immediately put into the context of expanding binomials.  It's just sitting there, asking the kids to make the calculation.  Even at this stage of the trimester (we are 8 of the 12 weeks through the term), when they know they need to be patient and let the worksheets gradually reveal the material to them, the students are a bit irritated by the fact that the some of the material seems to move so slowly.  Or at the very least it gets to them that they have to wait for the "big picture".  From my standpoint, several good things come from this:

(1) the kids are working on more than one topic at a time;
(2) the kids get to see and/or use the basic mechanics of a topic over several days, which gives the mechanics more time to sink in; and
(3) there is an almost constant sense of anticipation, waiting for the pieces on which we have been to finally come together.

In the "instant knowledge" world in which these kids have been raised ("I'll just look it up on Google."), this kind of patience is not easy to for the kids to handle.  But, in the "real world", whether it is scientific research or business strategies, this kind of patience is required.  So, in addition to promoting a better understanding of mathematics and having a better grasp of what problem solving is really all about, Harkness promotes and teaches patience.  Sounds good to me.

Discussion vs. Debate

One of the things I have been watching in my classroom during the past week is the dynamics of the different classes.  Two of my classes are very much into discussing the material, making sure every one understands an exercise before moving to the next one, putting different methods of solution to the same exercise on the board for comparison, and in some cases even putting an incomplete solution on the board so the group can finish it together.  For lack of better terms, these two classes "get it", and I couldn't be happier with the progress they are making.  The students in one of my classes are still very hesitant about putting solutions on the board unless they are absolutely certain they have a correct answer.  This has led on many occasions to one person putting multiple exercises on the board, and rather than having a discussion it becomes a student lecture.  Because of this, on Monday a "new rule" will be put in place: no one is allowed to put more than one solution on the board on a given day.  Finally, there is my remaining class, a group that is very much into debating one another, trying to prove "who's right" and, by implication, "who's wrong".  Unlike the other classes, solving the exercises in this particular class has become more of a competition rather than a collaboration.  Though they make a reasonable amount of progress each day, they are nowhere close to as productive as the other periods.  It often takes then a lot longer to complete an exercise, if for no other reason than neither side will "give in".  This really becomes a problem when both of the students are actually correct, but have used such different methods to solve an exercise that the answers they get look as different as their methods, despite the fact that the answers are actually the same.

The difficulty for me is that, despite the extra time and the difference in "styles", the discussions being had by the debate class are still really good.  The students are getting into the depths of the exercises and as such I don't really want to stop the debates, if for no other reason than debating really does seem to fit the personality of the class.  They come in each day, ready to state their case (or if they are unsure of their answer, to take sides between two people who are sure of theirs), they work through the exercises, and leave at the end of class still liking each other.  Who am I to complain about that?

On the other hand, the students have gotten used to completing one worksheet every day, and there are times when a debate that occurs makes that impossible.  And there is the matter of making sure this group makes close to the same amount of progress during each class period as the other classes.  Several of the students are very uncomfortable with falling behind (and told me so in their weekly journals), so Monday I will need to try to strike the balance between making sure the discussions go just as deep, but that they are doing so in a more efficient manner.

Despite it all, I am seeing the benefits of Harkness.  I am seeing how much better, how much more permanent the students are learning and understanding the material.  And because of that, I'm more than willing to continue to work through the difficulties, and work with the students to make the class everything I know it can be.

Summary and Review

Over the course of the last few weeks, one piece of the "puzzle" has been lost in the discussions, and that is the importance of review.  By this, I don't mean reviewing the material at the end of a unit immediately before a test, because we have been doing that.  I mean reviewing every day, for about five minutes at the end of the period, summarizing the stuff we covered that day, and I'm as guilty of forgetting to make sure this happens as the students are.  My recommendation to them from the beginning has been to summarize each problem once it has been completed, writing a clean set of notes in a separate notebook that lists the important skills, formulas, etc., that were needed to solve the exercise.  Lately, however, this piece is being left out, due to time constraints (shortened bells) or just getting so deeply involved in the discussion that  the end of the class arrives long before any of us are expecting it.  And while I love it when those discussions happen (and they have been happening more frequently), the price to be paid is the students not having a good set of notes from which to study when the test rolls around.  So, the sacrifice of leaving some questions unanswered until the next day is going to have to be made.

I also noticed this week a particular quality of the more effective groups: they all have the student at the board explaining their exercise as they are writing, rather than writing everything on the board and then explaining what they wrote.  The group is more actively engaged in the problem from the beginning, the mistakes get corrected earlier in the process, and students at the table are not just scribbling down what's on the board, but instead are asking questions about what is being written and including the explanation of the process in their notes.  So, I mentioned to the students on Friday (the beginning of the new unit) that this seems to be more effective, and so far...OK, today...the groups were all working in this way, and in that the discussions were more effective.

Finally, I mentioned above that the good, deep discussions are happening more frequently.  One of the reason I believe this is happening is that the material on the worksheets is taking more than just recall, more than just brushing off the dust and getting the algebra 2 material back up to speed.  The "new material" from the beginning of the year is now being used, and the "newer material" is being built on top of it.  Because of this, the discussions are not only covering the introduction of the new material, but are also solidifying the "older new material".  And in that, none of the material is being forgotten...at least not for long.  My hope is that this will pay off at the end of the trimester as we review for the exam.  I guess we'll see.

A Normal Day...

The link below is not intended to show anything other than what a normal 10 minutes of class looks like.  The video was taken this morning.  I didn't "hand pick" this because it shows something over-the-top spectacular.  Instead, I picked it because it is a good example of what normal looks like.  The three conversations, me floating from group to group answering a few questions, the kids presenting and explaining the material...in other words, the new normal.  This is the first day of a new unit, and the problem they are working through is a review exercise from algebra 2.

Class Video - 5 October

Settling In

This week, the "new" has finally begun to feel normal for the students.  Despite the fact that this was homecoming week, complete with all of the interruptions (fun and wonderful as they are), the classes have truly begun to care for themselves.  Courtesy of the predictability of the daily routine, if I don't have any announcements to make at the beginning of class, the classes are pretty much self-starting at this point, which is great.  And courtesy of a bit more new material being woven into the exercises, the discussions are going much deeper, with different students interpreting some of the exercises or definitions in very different ways, which leads not so much to a debate as to a thorough discussion of the material...which is really great to watch and to participate in.  I've been struggling with trying to strike a balance between being too helpful and not being helpful enough.  It's tough to quickly discern whether a discussion that has taken a "wrong turn" will lead to a correct interpretation in the long run or will only lead to confusion.  So I need to constantly be aware of my participation in the discussions, making sure I don't lead them to the method I used (which defeats the purpose of having them solve the problems), but likewise making sure they are being guided to a viable method that, for the most part, they are discovering or creating themselves.

In creating the worksheets over the summer, we kept in mind that since most of the students in the class are juniors, the ACT and SAT are looming large in the near future for them.  In the past we have given the students weekly review sheets containing exercises that, at least in part, served the purpose of reviewing some of the material they had in all likelihood forgotten.  However, this year we have put at least one review exercise on each of the worksheets.  And, as in the past, it always seems to be the geometry problems that cause the most difficulty. For instance, this week we spent at least an hour of class time on one problem...yes, one problem...that we still haven't resolved as a class.  It made for great discussion, though, really forcing the kids to justify their assumptions about the picture that was given in the problem and about the relationships that result from those assumptions.  Honestly, it's the kind of discussion I wish we could have about lots of the exercises and concepts.  To get there, though, means writing the correct kinds of questions that will lead to those discussions.  As such, the worksheets are already being revised, and will probably be in a constant state of revision.

One final thought for this week: it occurred to me this week that despite the fact that we haven't done a lot of basic mechanics practice ("drill and kill"), the kids have gained a remarkable fluency with the basic mechanics of the material.  Factoring, solving exponential and logarithmic equations, and the like, have not been specific "stand alone" questions, but rather have been necessary to solve some of the exercises on the worksheets, and in the process the kids have remembered (or in some cases, finally really learned) and practiced the mechanics.  It truly put my mind at ease on Thursday when the kids were using and explaining the laws of logarithms...all without me saying a word.  The worksheets and discussions are not sacrificing the basics for the "advanced", which makes me all the more convinced that this really is the best way for kids to honestly learn and not just memorize the material.

Video #1

Here is the first of what I hope will be several videos showing how things are run in my classroom.  This clip is short (just under two minutes), but it gives you a feel for the level of the kids and the kinds of conversations they have.  This is one of three tables in my classroom, and while they were having this discussion I was at one of the other tables, so pretty much all of this is student-directed, including one student who is noting who goes to the board and who makes a "helpful" contribution, whether it's a comment, a question, or whatever.  Other than review and test days, this is what the classroom looks like (or at least it's what it should look like) every day.

Classroom Photo

Just a quick "update": this is the set-up of my classroom.  My table is just off the right side of the photo.

Taking Notes

One thing I have noticed during the last week is the difficulty students have taking notes.  Don’t get me wrong, the students are taking notes.  They have two notebooks for my class: one in which they keep all of the work they do at home while working through the exercises on the worksheets, and one in which they place a “clean” set of notes after the daily discussions.  It is in the clean set of notes that I have noticed the difficulties.

You see, in a “normal” math class, the lectures provide information not so much about the skills the students need, but rather examples of specific types of exercises the students will be expected to do on the homework and on the tests.  In fact, a few years ago I took the time to type my lecture notes, complete with examples, and have given them to the students in the past as a means of making sure the notes they had were, for lack of better terms, “complete”.  This kind on note-taking leads to and perpetuates the lie we have told the students (or at least led the students to accept): they are supposed to memorize a specific algorithm for each particular type of exercise.  Because they have been trained to take notes in this way in math class for years now, the students are having a difficult time transitioning to taking notes on the skills they will need to master, the definitions they will need to understand, and the relationships they will need to see.  And in this, they are having difficulty taking the skills learned on one worksheet and using them on the subsequent worksheets.  They are looking for similar problems of the same type rather than exercises that require the same skills.  The frustration with this has begun to set in, and several students are now asking that I give them notes rather than having them rely on one another to discern the important information they are to glean from the daily discussions. 

Of course, the notes I would be giving them are not the notes they actually want.  Any notes I would give them at this point would focus on the skills, definitions, and relationships, whereas they are requesting notes that cover specific examples for them to memorize.  Since part of the point is that there is more than one way to arrive at an answer (not just “my way”), since the focus is supposed to be on understanding the content and applying it in a variety of situations and not on memorizing an algorithm for each “type” of problem, and since they are supposed to be relying more on one another and less on me, I have no intention of giving them notes.

That being said, the students are supposed to “use their resources”, such as the internet, their textbook, etc., and as such I have placed my typed notes online for them to use.  No, the notes do not contain examples and algorithms for the exercises on the worksheets, but they do contain examples of the skills and explanations of the definitions.  The relationships, however, remain theirs to discover.

Finally, as a follow-up, the permission slips are in and the taping will begin this week.  Hopefully I will have something to upload next weekend.

Shutting Up

Seriously, I need to learn to stop talking. I'm finding it amazingly difficult to help the students by giving them leading questions rather than answering their questions directly. So, this week I took a few steps that should encourage me to shut up.

(1) Each day, I had the students in each of the three groups choose a leader for their table. The student leader was responsible for keeping their group focused, for making sure that everyone is participating, and for organizing who goes to the board to do each exercise. The student leader was also responsible for noting who went to the board and who made a meaningful contribution to the conversation about each problem. They didn't need to detail the type of contribution (asking a question, answering a questions, clarifying a remark, etc.)...just note that a substantial contribution was made on the little, easy-to-follow form I created for them.  This has allowed me to slow down and spend longer at each table listening to each of the conversations, which means I can be more patient while the students struggle with a concept. I'm still noting down what I observe at each table, but the "pressure" to get to the other tables more quickly, just in case I miss something, has been relieved.

(2) I outright told the students that I would not be getting involved in the conversations unless (a) they asked me for help directly; or (b) as a group, all ten students came to a conclusion that was incorrect. I was still going around to the three tables, still sitting with the students listening to their conversations, but the expectation that every time I at down at a table the attention was to refocus on me was removed. Honestly, I still find this very difficult. Twnenty-two years of responding to a question, or even a confused glance, with a direct, "here's-how-to-do-it" answer has ingrained a habit that will not go away easily. At least I'm responding to questions with questions rather than with direct answers more often than I used to, and the kids are asking one another first and having a discussion about the difficulties they're experiencing before asking me to participate..so there is something positive to report.

(3) I began to take videos of the class "in action" (yes, I sent out the permission slips to place parts of the videos on the web; no, I don't have them all back yet). Now, I can see and hear how often I interrupt instead of letting the conversation happen, as well as take a more objective view of how well the class is going.

I heard this week from several unsolicited sources (other teachers, parents, friends in the district) that the kids are really enjoying how the class is being run and that they feel they are getting more out of the class because of it. On the other side, one of the students mentioned in his journal this week that he really wants to go back to the "lecture-take notes-regurgitate" way of doing things. Having run the class using Harkness last May, and now for the first few weeks of this year, I have some unfortunate news for him: I'm not going back. Yes, I understand the struggle and frustration involved from his end, and yes, I get that figuring things out rather than being told takes more time and therefore more patience, both on his part and on mine. However, I also get that the end result is better, deeper, more permanent learning. And because of that, I'm not going back.

So, I need to be more patient and I need to shut up. That should be easy enough. (If you can't hear the sarcasm in that statement, you're not paying attention.)

"Are We Learning Anything?"

Each week for many years now, I have had my students write a one-page journal that serves two purposes.  First, it allows them to summarize the week in terms of the mathematics, and to reflect on what has gone well and on what they need to focus.  Second, it allows them to tell me how things are going.  The response in the journals last week was unanimously positive, and this week pretty much continued the theme.  However, there was one comment that appeared in several journals that I honestly loved seeing: "I don't feel like we're learning anything."

Now, I could list off a number of skills that, while they were certainly introduced to the students in previous courses, the students have brought to a level of mastery during these first couple weeks of the year.  I could also list off several brand new skills and concepts of which the students have achieved a level mastery during these first two weeks.  But, in good Harkness fashion, I'm going to let them figure out that they are, in fact, learning a good amount of new material.  Happily, one of the students has already figured that out, stating in his journal that he "really likes the fact that we are learning without it feeling like we're learning".

This is part of the beauty of the Harkness Method.  Like learning to walk, you don't realize just how much progress you've made.  It seems as though you've always been able to do this, and even though you can now walk more confidently and walk on slightly more difficult terrain, it's still just walking so you've not really learned anything new, right?  It's not until  you take a good, long look at it that you realize just how well you're doing and just how far you've come.  In writing our worksheets, we tried our best to imitate what we saw as one of the essential aspects of the worksheets at Exeter: it's not about gaining complete mastery of something new every day.  It's about making a little progress on several things every day.  Over time, little by little, the progress becomes substantial.  But since it's little by little, the students don't realize that they are making progress, let alone making progress on so many different things at once.

The test on Thursday went well for most of the students...about right for the first test of the year.  The  first test sort of marks the end of the "introductory" phase of any class, with the students now seeing how they will be formally assessed.  Many of the students mentioned that now that they have seen a test, they have a better idea how to study and prepare through the next couple weeks as well as how to study the last day or two before the next test.  Specifically, several students said that they wish there were formal notes for the class (i.e., lecture notes, which are not coming...I'll be talking a bit about this at the beginning of class on Monday), while others said they now have a clearer picture of the kinds of notes they need to take during the discussions (i.e., they "get it").

Finally, open house was this week.  A few parents were curious how the students were going to be able to learn the new material "on their own".  Of course, they're not on their own...the careful writing of the worksheets and my participation in the discussions are more subtle than me standing in front of the room, but it doesn't mean they're on their own.  However, it was during these discussions with the parents that I realized one of the necessary shifts is that the students need to get comfortable with the homework not being the practice of new skills they have seen (and through the practice, the skills are solidified) but rather the homework is where the skills they need to discover are cultivated, and that the in-class discussions are where the new material is solidified.  Also, a few parents were very supportive of the "new method".  I heard more than one story of college graduates who are very knowledgeable when it comes to skills and "book information", but who don't know how to creatively use the information they have to solve the problems they face in their job.  Harkness certainly addresses this weakness, and give the students the problem-solving practice they need while delivering the required new material.

So, overall a successful week...nothing to deter me from continuing having a Harkness classroom.  In fact, plenty to convince me that this really is the right thing to do.

The Adventure Begins

I just began my 23rd year of teaching, and in many ways, I'm starting over.

Last fall I was directed to the website of Exeter Academy (www.exeter.edu) by a friend, who told me that I really needed to look at their curriculum, the worksheets that they use (all created in-house) and, most importantly, their method of delivery.  What I found has had a profound impact on my philosophy of education.

Exeter uses what it refers to as the Harkness Method, named for a visionary benefactor who made a substantial donation to the academy with stipulations as to how the money was to be used:

"What I have in mind is [a classroom] where [students] could sit around a table with a teacher who would talk with them and instruct them by a sort of tutorial or conference method, where [each student] would feel encouraged to speak up.  This would be a real revolution in methods." ---Edward Harkness

This method, which I can best describe as a student-directed Socratic Method, is described in greater detail on the Exeter website and relies on several things:

(1) the questions the students are asked must be worded in a way that allows them to discover the material rather than relying on the teacher presenting the material to t he students; this requires a lot of prep time, and in particular, the textbooks do not have questions appropriate for this purpose;

(2)  the teacher must observe and redirect the conversation the students have about the material, rather than leading the conversation; this requires a major adjustment for the teacher, especially one who is used to lecturing;

(3) the students must take responsibility for their learning; in particular, they must do their homework and take an active role in the in-class discussions; this can be a major adjustment for the students, especially those accustomed to passive participation...or no participation.

I wrote the worksheets and  piloted the method in my Honors Pre-Calculus class last spring as we did the unit on conic sections.  The feedback was overwhelmingly positive (a few students still wanted to be spoon-fed, but played along since this was the only way they were going to learn the material), so the other Honors Pre-Calculus teacher and I spent the summer writing worksheets for the first half of the course, and this year we are using the Harkness Method pretty much exclusively in our classrooms.

This is where the folks at Exeter will think we're crazy: we have around 30 students in each of our classes.  The method, as it exists at Exeter, relies on a maximum of 15 students in a class, so that the teacher can hear, note, and respond (if necessary) to everything that is said.  So, we have needed to tweak the method for 30 in a class.  To do this, the students are separated into groups of 10 and seated around three "tables".  I rotate   around to each of the groups...not ideal, but at least there are only three groups, so I can spend a decent amount of time with each of them, especially if one group gets into a particularly engaging discussion.  So far this year, the feedback from the students has been essentially unanimous and positive. We'll see what happens after tomorrow's test...the first test of the trimester.

Anyway, my plan is to update this blog on a weekly basis, both for personal reflection and in the hopes that some others who have more experience with Harkness will be able to give advice, share wisdom, and commiserate.  So, please feel free to ask questions, give feedback, etc. ...that's part of the point.

'Til next time...