Easy Questions

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

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

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

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

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

Book Club

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

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

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

Comfortable

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

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

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

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