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.