Editing

I read a the book Five Elements of Effective Thinking over the summer, and while lots of things in it struck me as stuff I need to remember to mention to my students, one thing has so universally prompted a “wow, that’s so obvious, how did I miss it” response that it’s quickly becoming something that I mention in pretty much every conversation I have about problem solving.

So here’s the question: which one is easier, writing a first draft or editing a first draft?  Without exception, everyone has responded with “editing” as being the easier task.  Editing whatever is there, discerning what is good and what is not, what works and what doesn't, has been seen as the easier thing to do.  This leads my follow-up question: if editing is easier, then why not just get the first draft out of the way, regardless of how bad it may be.  That way you can get to the task of editing, keeping the stuff that was good in the brainstorm first draft, and working with the stuff that wasn't to make it better.

Relating this to a math class: when it comes to problem solving, why not take the same approach?  When tackling a problem, try something…anything.  Get your thoughts down on paper, and then start sifting through what’s there to see what is worth keeping and what needs to be “edited”.  Just brainstorm some ideas about how to attack the problem, not worrying about forcing any “algebra” or “geometry” into the process, but just working through how to solve the exercise.  Once the idea about how to solve the problem comes into view, then put the equations and/or the pictures into the solution to communicate your ideas to others in the common languages of algebra and geometry.

For so long we have shown the kids how to solve the problems that they often don’t even consider brainstorming ideas about how to do it, and instead they go looking for “the formula” of “the example” that relieves them of any real thinking, which is a shame.  The analog to this would be to have a kid writing an essay for an English class to forget about writing any sort of a rough draft, and instead asking them to simply use a template with lots of almost complete sentences that have a few blanks to fill in.  That’s not how we teach kids to write an essay, and it shouldn't be how we teach kids how to problem solve.  They learn to write by writing, discussing, and editing.  The same holds true when they learn to problem solve. 

Just one more reason to run a discussion-based classroom.

Grammar

Well, summer officially ended on Thursday as we went back to school.  This year I have my normal load of honors pre-calculus classes along with one section of algebra 1.  All the classes are off to a good start, but the first “real” discussions won’t happen until Monday, so we’ll see how things go.  Yes, this includes the algebra 1 classes.  While I’m not running my section in as independent and discovery-driven a way as we do with the honors pre-calc kids, I am still infusing a lot of discussion into a relatively small amount of lecturing.  In particular, the emphasis is going to be on the applications as opposed to being on the mechanics of algebra.  That doesn’t mean that we’re not going to work on the mechanics, because we are.  Obviously, it’s a little difficult to do a basic algebra problem without the mechanics.  However, I thought of/realized something over the summer about the mechanics of high school mathematics that seems to have struck a chord with everyone to whom I have mentioned it.

Grammar is important in English class.  No one disagrees with this.  Grammar is important and it needs to be emphasized.  However, proper grammar is not the point of English class.  The point of English class is to improve the communication skills of the students, in terms of their ability to both take in and interpret information and to share information with others.  Proper grammar is a point of focus and an important aspect of attaining this goal, but it is not the actual goal.

Now, let’s look at a typical algebra 1 class.  Are the mechanics of algebra important?  Absolutely.  We really can’t do much without them.  However, the mechanics of algebra are the “grammar” of the subject.  The point of algebra 1 (or of any high school math class, in my opinion) is to improve the problem-solving skills of the students.  Solid mechanics can certainly help the students reach this goal.  But if all the students can do is push the symbols around while having no idea about how to use the mechanics to solve a problem, then we haven’t really done much in terms of realizing the actual objective.  For that matter, the mechanics of algebra are not the only means available to the students to solve a problem.  Geometry and statistics play a vital role in helping the students become well-rounded problem solvers.  Sadly, I experienced several conversations in different settings over the summer where a person solving a problem got to the correct answer without algebra and described the process they used as “not really involving any math” precisely because there was little to no algebra involved.  Some used well-drawn pictures and a healthy dose of geometry, some used data tables and graphs, but since there was a lack of creating an equation and pushing the symbols around, the conclusion was that there wasn’t really any math going on.

AUGH!

So, in addition to incorporating a healthy amount of discussion into my algebra 1 class, my goal for the year is to get the students in all of my classes to see everything they are doing as they attempt to solve an exercise as “doing math”.  I want the kids to realize that drawing a picture, creating a table, making and testing a conjecture, making a quick calculation, and yes, writing and solving an equation are all “doing math”.  All are valuable tools to have at their disposal in order to reach the goal of improving their problem-solving skills.