Creativity

So here's a question from the exercises that were assigned during the past week:

You have often wondered how tall the water tower near Pine Hill Park is, so one day you take your protractor and head for the park.  Looking at the top of the tower, you measure the angle of elevation to be 30°.  After moving 108 feet closer to the tower, you again measure the angle of elevation to the top of the tower and find it to be 50°.  What is the height of the tower?

A pretty standard exercise, I admit.  Here's the thing: the kids came up with at least four different ways to determine the height of the tower based on this information.  Some used a system of equations involving tan(30) and tan(50).  Some used the law of sines.  Some used the fact that the "large" triangle is a 30-60-90 triangle.  Some used the Pythagorean Theorem.  Many used some combination of the above, but not necessarily the same combination.  Most arrived at the correct answer.

Among others, this is one of the things I love best about teaching through discovery and discussion.  The kids came into the class knowing basic right-triangle trig, and they had used special right triangles before.  Through the course of the exercises, we have emphasized both of these topics, along with discovering the law of sines.  That sets us up to work on this exercise.  If I had done a similar problem in class as an example to "show them how" the problem is to be done, the next day I would have had classes full of kids who all did this problem exactly the same way I did it.  No critical thinking, no real problem solving, no creativity, just mindless imitation.  Instead, the kids had to make sense of the exercise themselves, use the "tools" available to them, and solve the exercise.  Working through things this way sets them up to handle a wide variety of similar exercises that don't necessarily ask them for the height of the tower, but that could instead ask them for the distance from the base of the tower, the angle of elevation, etc., whereas me giving them as example and having them imitate it sets them up to do an identical exercise with the different numbers and nothing more.  To ask them to find the angle of elevation would require a separate example, and asking them to find the distance from the tower would require a third, none of which build or even encourages critical thinking, problem solving, or creativity.  Instead, it promotes memorization, and little to nothing else...memorization which, by the way, fades away quickly.  And if what the kids memorize is inaccurate, then fixing it is difficult.  I have found that it is much easier to correct a reasonable but flawed guess about which a kid isn't completely sure than it is to correct a flawed bit of memorization about which a kid is absolutely certain.

Sadly, I've seen teachers essentially train kids to do exercises only and exactly the way that they are shown in class and then get upset that the kids aren't able to generalize the process on the test.  Worse, they don't make the connection that if we don't regularly ask the kids to understand the material well enough that they can explain it and be creative with it, then they won't be able to put the pieces together and generalize the process once every 2-3 weeks on a test.  Of course, rock bottom is the teachers whose test questions are only ever the same exercises as those that were specifically shown as examples in class, just with different numbers.  This kind of teaching may have cut it in the past, but it won't now, especially with end-of-course exams that require creativity and critical thinking coming soon to our classrooms.

If we want the kids to be creative, problem-solving, critical thinkers, then we need to ask them to be creative, problem-solving, critical thinkers every day.  We need to encourage them to make a conjecture as to how a problem can be solved, attempt to solve the problem based on that conjecture, and right or wrong learn from the experience.  This requires us to create an atmosphere in our classrooms where it's safe to make a mistake and then try something else.  It requires us to see things the way they do and help them find a way that the material makes sense to them.  It requires us to accept different but mathematically valid methods of solving an exercise.  

It's easier to just ask the kids to do things the way we do them.  It's easier to grade a set of papers that all look the same.  However, good teaching doesn't focus on the teacher.  Good teaching focuses on the learning being done by the students.  It isn't easy.  But if we don't want the kids to be creative, problem-solving, critical thinkers because it makes things difficult for us, then we need to get out of the classroom.

Physical Space

During the last week, as we prepared for the second test of the trimester, it occurred to me just how profoundly the physical space in which a class is held can influence the way in which the class is run.  I am fortunate enough to have tables that seat two students apiece as well as an entire long wall (20+ feet) on which is a white board complete with sliding panels.  As the room changed this week from the usual “groups” configuration to “stadium” style for the review to the more traditional rows for the test, I was first grateful to have the ability to change the room so easily for all of these different needs.  However, it got me thinking about what I would do differently if I were to design the room “from the ground up”.

First, I would include more board space.  Yes, I know I already have more board space than most.  However, to comfortably run four groups of 7-8 students, I would need at least one more wall worth of white boards.  This is probably more indigenous to a math class, but since that’s what I teach, it’s one thing that I would definitely include.

Second, I would get rid of the rectangular tables and replace them with something that would allow for better eye contact and give more of a community feel to each group.  I've seen large trapezoid-shaped tables and I think that’s probably what I would opt for, but I’m not certain.  A little more research would be needed before making a final decision.  I would also equip each of the groups of tables with a tablet computer for looking up any information they may need during class.

Third, I would get rid of my teacher desk and replace it with stand-up desk that can accommodate a desktop computer (or maybe a laptop/tablet combo), along with a tall chair.  I rarely sit at my desk during the day, and quite honestly it takes up valuable space that could be used to give the kids a little more room.  However, during class I am constantly going back-and-forth to my lectern to make notes about what I’m observing and the feedback I am getting from the kids about how well they understand the material.  A stand-up desk would make a lot more sense for me.

Now don’t get me wrong.  I am acutely aware that I already have far more in my classroom than most, and I’m extremely grateful for it all.  But, if we’re really going to ask the kids to do more collaborative work in class, then we need to be aware of the needs we have of the physical space itself to make it happen, and in most instances, the traditional classroom with one white board and 30 individual student desks just won’t cut it.

Feedback

As the year has progressed, and the more comfortable I have become over the last two years with using discussions in my classroom, I have struggled with trying to keep track of everything.  I have become firmly convinced that the most important thing I do in the classroom is pay attention to the kids: find out what they know, what they don't, and determine what is preventing them from making the progress they need to make.  Admittedly, the opportunity to do all of this is a huge advantage of running a Harkness classroom (or a "flipped" classroom, for that matter) as opposed to a "traditional" one.  In the past, the amount of information I was able to get from the kids on a daily basis was extremely limited, mainly because I was the one doing most of the talking.  Now, I have the opportunity to get a good feel for where each and every one of the kids is every day.  However, trying to write down notes in enough detail that I can remember and then act on what I am observing has proven to be a huge undertaking.

My solution so far has been to make a spreadsheet on my tablet and carry it with me from table to table.  The spreadsheet has the names of the students down the first column, and the skills they are to be learning during the course of the trimester across the top row.  During class, I mark an "x" in the appropriate cell when one of the kids presents a solution on the board or when they make a solid summary statement about one of the concepts to the group.  After each test, I go through the spreadsheet and mark a "t" in the appropriate cell if the student earned at least a "B" on the exercise that was on the test that relates to a particular skill.  All of this has been very helpful, and some patterns are certainly emerging in terms of who goes to the board to do only review exercises, who rarely goes to the board at all, and who has a reasonable mastery of all of the skills to the point that they are willing  to go to the board at any time.  It's not perfect, because I still feel I miss a lot of what happens at the other tables when I get deeply involved in the discussion at one table, but it's certainly better than what I was able to do in the past, when most if not all of the feedback I got from the kids came in the form of a formal quiz or test.

This leads me to one of the other big revelations I've had recently.  There has been a lot of talk at our school recently on the topic of "feedback", mostly in the form of discussing the work of John Hattie.  Initially, my reaction was, "OK. how can I give the kids more information about what they're doing right, about where they still need some extra practice,..." etc.  However, it occurred to me that the only way for me to give feedback to the kids is if I'm getting feedback from them.  Trying to give them feedback every day requires that I get feedback from them every day.  This kind of interaction, the give-and-take in both directions necessary to allow the two-way feedback happen, is precisely what was missing from my classroom in the past.  This interaction is not only promoted by a discussion-based classroom, it is the very heart of it.  Whether the classroom is running on a Harkness model (which I would classify as discovery-based discussion), a true Direct Instruction model (brief lecture followed by lots of closely monitored in-class practice in the form of exercises or activities done in small groups), or a flipped model (somewhere between Harkness and DI), the important part of the time in class is when the students and teacher are getting and giving feedback to one another.

In short, the most important part of any classroom should be the real interaction between the teacher and the students.  Discoveries can be made anywhere, and lectures can be delivered by online videos, but the frequent feedback necessary for the teacher to know where the students are in their learning and for the students to receive affirmation of or correction to their progress and conclusions can only happen during real dialogue, and this happens best in a discussion-based classroom.

Is it easy to keep up with everything?  No, but it's certainly worth the effort.  And I'll definitely take this over the only-way dissemination of knowledge that used to take place in my classroom.