Revelation and Revolution

What is the point of an assignment?  Any assignment...homework, project, essay, test: what is the point of the assignment?

For a teacher, the answer to this question would probably vary depending on the assignment.  Homework may be for the sake of practicing a particular skill.  An essay may be for the sake of demonstrating a thorough understanding of a historical event.  The list goes on.  However, from the standpoint of the students, the point of any assignment is essentially the same: to finish the assignment and get as many points as possible.  I had a revelation this week: this attitude about the purpose of assignments is one of the reasons students, parents, and other teachers have difficulty understanding discovery-based learning in general and Harkness in particular.  For most people, the purpose of the assignment is to complete the assignment, and the presumption made is that the students already have a reasonable if not a complete understanding of the underlying material.  However, the point of most assignments in a more discovery-based approach is to learn the material by doing the assignment.  The presumption made is that the students have an understanding of the background material necessary to discover the material, and the point of the assignment is not to show what they already know, but rather to use what they already know to learn something new.

The confusion on the part of the students, parents, and other teachers is understandable, especially when it comes to mathematics.  The textbooks are written with the idea that the material will be explicitly shown to the student (by the teacher, by reading the textbook, etc.) and that the student will do the exercises to practice what they were shown.  Even when it comes to word problems, the textbooks explicity show the students what to do, and the exercises are about practicing.  Problem solving, creativity, critical thinking, students making connections for themselves...none of these are the goal of the mathematics textbooks.  And since the textbooks have not really changed very much in decades, none of these were the goal of the mathematics education most of us received.  Since many teachers teach the way they were taught (for the record: up until two years ago, that statement would have described me), it's easy to see why so little has changed in mathematics education.

Contrast this with what happens in some upper level mathematics books...for instance, an abstract algebra book.  The written portion of the book is, in general, not repleat with examples that the students are expected to mimic in the homework.  Instead, the section contains the background and foundational material, and the students are expected to creatively use that information to make connections and discover more of the material.  In other words, the homework isn't for practice and it isn't for demonstrating a command of the content of the course.  Instead, it is geared toward extending the material presented either by the professor or by the textbook.  This is why so many students who "have always been good at math" stuggle so mightily when they reach these kinds of math courses.  I believe it is also why they struggle in college science classes that require them to do the same thing.  To be clear, most college calculus courses don't require this, and therefore neither does the AP Calculus test.  That being said, many students then struggle in the subsequent courses that creatively use the calculus skills they supposedly know in ways they have not been shown...so much so that some colleges are beginning to rethink the way calculus is being taught.

Here's the problem for those of us who teach high school: we are now being asked to prepare the students to be creative, make connections, and think critically on standardized assessments to solve problems the likes of which the students have not necessarily seen before.  The basic skills being tested are not the issue, since they are (in mathematics, at least) the same skills that we have expected the students to learn for decades.  However, instead of asking the kids to regurgitate what they have seen, they are asking them to use the skills in ways that, while reasonable, are ways they have not encountered.

If we don't ask the kids to be creative as part of their daily experience of mathematics, how can we expect them to be successful on the new standardized assessments or in the (college) courses that proceed from ours?  If all they are used to doing is following the prescription laid out by the textbook or by the teacher, how can we expect them to be successful when they are presented with an exercise (or, in "real life", with a situation) that does not rely on any previously-seen prescription?  It is going to take a fundamental shift in the method of delivery from the textbooks and the teachers to  prepare the students.  It is going to take a fundamental shift in the attitude of the students, parents, and other teachers about the purpose of assignments, both in and out of class.  As I said above, the current attitude is completely understandable, so we need to explain the rationale behind these assignments.  We also need to adjust the way we grade these assignments...which should naturally lead to a conversation about formative assessment.  We need to help other teachers understand why teaching the way we were taught isn't going to be enough if we want our students successful.

Put simply, we need to help everyone involved see why we are including more discovery-based teaching and why we are expecting the students to gain a deep understanding of the material that goes beyond the surface-level comprehension that was required of us in school.  Simply knowing what to do is no longer enough; the students need to understand the "why" of the material, and they need to understand it well enough to be creative with it.  We need them to see that, for the most part, completing the assignment is no longer the point; what the student can learn by doing the assignment is.  Much as I enjoy teaching through Harkness, I have become convinced that the students need me to teach through Harkness.  And I need to help the students, parents, and other teachers understand this as well.

I have no illusions about what this means.  This is calling for a revolution.

Let the revolution begin.

Technology

I like technology.  I really do.  Calculators, computers, tablets, cell phones...facebook, twitter, class websites...I use them all, and I like using them all.  They make a lot of what I am able to do in and out of the classroom as a teacher possible.  From exploring mathematics to answering questions and giving homework hints to keeping up with former students (not to mention hearing from people I have never met in response to this blog), so much of this simply wouldn't be possible without technology.

And yet, if you were to look around my classroom on any given day, you would often wonder if technology was even present.  The kids have their calculators, and every once in a while they'll reach for them. If they need a definition they may reach for their phones and look up the term in question.  But most of the time the focus of the in-class discussions is on the concepts and the content, trying to understand the mathematics rather than mindlessly plugging in numbers or relying on calculator-produced graphs.

Which got me thinking this week: with the huge push from seemingly everywhere to incorporate more technology into our lessons, are we using technology for the sake of looking modern, or are we doing so to actually improve the quality of the learning that is happening?  As we encourage the kids to make use of the technology, are we also teaching them to discern when the technology is being helpful and when it is actually getting in the way?

And then there is the bigger question: for as connected as we are and with all of the forms of communication we have at our disposal, how are the kids at one-on-one communication?  While there are plenty of kids who are fine when it come to having a face-to-face conversation, for others it is a definite weakness.  Most of the kids are comfortable having a texting conversation, but when it comes to actually talking with another person the percentage drops drastically.  And by encouraging the use of technology (for instance, discussing a reading assignment on a wiki instead of in class), are we inadvertently denying our kids an opportunity to work on and develop a skill they lack?

Just some thoughts from the week as I was sending out homework hints on twitter, looking around my classroom watching the kids have face-to-face conversations with one another every day, reading the self-assessments they submitted online, and wishing former students "happy birthday" on facebook.

The Advantages of a Large Class

Thursday and Friday of this past week, the band students in my classes (and there tend to be quite a few of them) were on a trip to Indianapolis, which for two of my classes took the number of students down to 15 or less.  So, rather than sitting at separate tables, those classes sat as one group and discussed the exercises.  Since it was only one group, I didn’t need to circulate and sat at the “table” with them the entire bell both days.  The atmosphere was, as you might expect, completely different in those two classes.  The most striking thing to me was the way that most of the kids presented the material to me instead of the group.  There was an almost constant look of, “I’m doing this correctly…right?” in their eyes as they kept turning toward me about every five seconds as they presented their solutions.  It took a lot of reminders to get them to refocus on presenting to each other, interacting with each other, questioning each other, etc.  Friday was definitely better than Thursday, but even then there still wasn’t the level of interaction that normally pervades the classroom.  These two classes are my morning classes, so it made for quite a contrast to the afternoon, where things were running normally, with three or four tables going at once and me checking in visually and verbally with each group rather than sitting with them the entire bell. 

From this, it occurred to me that there may actually be an advantage to having enough students in the classroom to force the move to more than one group.  Specifically, the students are forced to become more self-reliant.  If I’m constantly at the table with them, it becomes really easy to fall back into the old habit of relying on the teacher for everything from moral support to verification of the correctness of answers.  (I avoided asking questions and/or agreeing with their answers until the end of each presentation, allowing plenty of time for the other students to catch any mistakes and ask any questions, but the request from their glances was there almost constantly.)  If, however, there is an understanding that I need to keep track of other tables and that I can’t be with any group the entire time, then the responsibility of running the conversation, catching the mistakes, asking the questions, and thoroughly investigating an exercise falls to the students by default.  And normally the kids rise to this responsibility, especially by this point in the year.  Despite the fact that we just began the new trimester on Tuesday and that these kids have not been together in my classroom before (they’ve had the first half of the course so they know how things are supposed to run, but it’s a new distribution of the students each trimester), the afternoon classes are already running smoothly, and I’m sure the morning classes will be next Monday with the return of the band kids.  However, the fact that the instinct to look to me rather than to each other is still there is something that I found disconcerting to say the least.

So, one more thing to add to the to-do list: make sure the kids understand that they can and should rely on themselves and on one another, not only in my class but in others as well.  I’m there to double-check and refine their results, but I’m not the driving force behind the conversations.  I thought I was already doing this, but clearly I need to do a better job.  Hmmm…

Cause or Effect

In every discipline, there are some pieces of information that simply need to be memorized.  Each subject has its foundational definitions and fundamental facts: names and dates in history, proper grammar and syntax in language, basic arithmetic in mathematics...all of these need to be fairly automatic to be successful in understanding the content of the subject at hand.  However, I've been giving this some thought lately, especially with final exams this past week, and the question I have is this: should memorization be part of the cause of understanding the content of a subject, or should it instead be the result of understanding the content? 

For example, is it necessary to sit down with flash cards and memorize the important names and dates as part of the process of understanding historical events, or, instead, should the memorization of the names and dates be the natural result of understanding the events?  Does memorizing the formula for the area of a rectangle actually help me understand what it is I'm finding, or instead, does understanding what it is I'm finding result in the memorization of the formula?

The reason this came to mind this week is I noticed that some of the kids who had, up to the point of the exam, earned As on the tests were lightly reviewing for the exam, whereas others with As were studying rather furiously.  Those who were lightly reviewing were participating in the discussion during the in-class review in a very relaxed, almost nonchalant way, and through their contributions it showed that they have a solid understanding of the material.  On the other hand, those who were studying furiously seemed to be attempting to re-memorize the material, and their conributions to the in-class review discussion were more in the form of double-checking their facts rather than demonstrating their knowledge. 


My conclusion after watching this was that both directions (memorize to understand vs. understand to memorize) are viable methods of achieving success on tests.  Both groups of kids had As on the tests, so both, from a purely data-driven perspective, were doing well in the class.  But looking more deeply into that success, it became obvious to me that only some kids were being truly successful when it came to actually learning the material.  And those were the kids whose memorization of the material was the result of understanding it, and not the other way around.

I wonder how prevalent this is in every class, regardless of the subject matter.  I wonder how much I have missed during my teaching career, thinking that if the kids were doing well on the tests (perhaps even the AP tests when I taught those classes) they were understanding the material.  And I'm grateful to have found a means through which I can at least partially discern which kids are truly understanding the material and which ones, despite their grades, are not. 

Just one more reason to use and promote discussion-based learning.