Placing the Blame

A good bit of the teaching that occurs in a discussion-based classroom happens behind the scenes.  The preparation of the questions, especially when it comes to making sure they are worded well and scaffolded appropriately, is time-consuming and difficult, but getting it right is crucial if the kids are to learn the material.  Any teacher who has written a project for a class knows what I’m talking about.  The wording of the description of the project is quite possibly the most important part of the project, and unfortunately you don’t find out how well, or how poorly, the directions are worded until after the kids have the directions in their hands.  At times, it’s the questions that come immediately after you hand out the information sheet that clues you in to exactly what needs to be reworded.  At other times, it’s not until after the projects are turned in that you realize you didn’t get anything close to what you thought you were requesting, and you certainly didn’t get the information about how well the kids understand the material.

In a discussion-based classroom, every exercise you assign has the potential for this to happen.  Fortunately, in the third year of using the problem sets we wrote for honors pre-calculus, most of the bugs have been worked out (though we found two typos in the last week alone), but there is still the matter of writing the questions for the individual exercise sets (we’re not calling them “tests” this year, and several kids have mentioned that just the removal of the word “test” does lessen the anxiety…credit to Carmel Schettino for the idea), and since we write a new set of exercises for these every year, the process never really ends. 

The difficulty this year has been writing the discussion exercises for my algebra 1 class.  Some of the worksheets have worked as expected, and I couldn’t be more pleased with the way the kids have tried the exercises for homework and discussed them the next day in class.  In particular, I love the fact that these are non-honors freshmen, and yet they have taken to the discussions just as well as the honors juniors in my pre-calculus classes.  A few of the worksheets have not worked out as well, and while it is tempting to just give in and show the kids what I meant, I have come to understand the importance of having the kids struggle with the material and make sense of it for themselves.  I have also come to understand that if the kids didn’t get it, it’s probably because I didn’t ask the correct questions, or at least didn’t ask them in the right way, and as such the appropriate response on my part is to search for and ask the right questions.  During my career, I have heard many teachers place all of the fault for not understanding the material on the kids, unwilling to take a critical look at the way they presented the material.  Lecture-based teachers say that they have explained the material as well as could possibly be done, and the rest is on the kids.  Discovery-based teachers say that the directions in the activity are absolutely clear, and the rest is on the kids.  While I agree that there are a few kids who aren’t learning the material because they are actively refusing to learn it, now that I am in the habit of critiquing everything I do for my classes, I have become far more aware of the fact that the lack of learning is more than likely my fault, and it’s my responsibility to fix it. 

Most of the kids are trying to succeed in our classrooms.  Most of the kids are preparing for and actively participating in class.  If they’re not understanding the material, our first port of call needs to be to ask ourselves what we could have done better, period.

Every Once in a While...

I cannot recommend this highly enough: have your kids bring in questions for you to answer in class. 

As I mentioned in the last post, I asked my kids to do this, and this week they responded with some really great questions, most of which I had never seen or tried to solve before.

I gave each problem about 5 minutes of time in class, just thinking through it out loud, trying an idea or two to see where it would lead. For a few of the problems the initial idea was fruitful and I was able to solve on the spot in class.  For others, the initial idea was off the mark and I needed to start over and try something else.  I think the kids enjoyed watching me squirm a bit too much on these, but in a way the whole point was for them to see me struggle, be wrong, try again, make a silly arithmetic mistake, go back to an idea that had been previously discarded, and so on.  And for a couple of the problems, I needed to take it home and work it through that evening, returning with an answer the following day.

I wasn’t able to take a problem every day in every class, just because of other things going on (a tornado drill on one day, for example), but I was able to do so a couple times in each class, and by the looks of it they got the point.  A few kids told me outright that they got a lot out of the discussion around these questions.  Also, there was a little more “I’m not sure if I’m right, but I’ll put what I’ve got on the board” by the end of the week.  By seeing me be imperfect right there in front of them, there was a little more understanding that I’m not looking for them to be perfect, but instead I’m looking for them to learn from their imperfections.  And by the end of the week, one of the first statements made in each class as the kids were walking in was, “I’ve got a questions for you to try today,” so if nothing else I know that the kids are enjoying the discussion and that we have at least a couple more weeks’ worth of exercises ready. 

One of the students was intrigued enough by one of the problems that we had a short conversation on Twitter about it that evening, which for me was one of the highlights of the week: a student doing math for the sake of solving the puzzle and for the learning, and more importantly not for the sake of the grade.  It also got me thinking about possibly putting a student-posed problem of the week online for the kids to discuss, just for the sake of having the discussion.  The difficulty, of course, is that communicating mathematics online is cumbersome, at best, but it may be worth the try to see what creative ways the kids come up with to overcome the difficulty.

Another thought that crossed my mind was that this idea would have merit in other disciplines.  Having students bring in a poem for the teacher to analyze in English class, an article for the teacher to translate in Spanish class, or a document for the teacher to explore in history class would be a great exercise for both the teacher and the students, as would the resulting discussions.

So, nothing but a positive update to last week’s post.  Nice to have this happen every once in a while.

Leading by Example

It occurred to me this week that for as much as we tell the kids to not be afraid of making mistakes (so long as they learn from them) and convince them that problem solving is actually a pretty messy process, we don’t dare go into a class without having meticulously worked the exercises ourselves so that any struggle on our part is hidden from view.  The kids never get to see us actually do any real problem solving.  They never get to see us actually work our way through a problem we haven’t seen before.  We may think we’re demonstrating the process at times, but truth be told we never actually model the process for them.

That ends tomorrow in my classroom.

I have asked the students in my honors pre-calculus classes to bring in a math problem for me to solve.  Any math problem.  I gave them fair warning that if they wanted to see me work the problem all the way through then they would need to make sure the problem was “reasonable”.  However, I also told them that regardless of what they brought in, there would be merit in me making an attempt at it, as they would get to see how to begin to attack a problem when they have no idea where to start.

Yes, I may have just set myself up to face plant onto the tile of my classroom.  Yes, they may bring in something that I will have no idea where to begin.  Yes, they might just get to see me make a mistake or five.  And that’s the point.  If I’m asking them to get comfortable with making mistakes in front of the class, then I need to show them that I’m comfortable making mistakes in front of them.  If I’m asking them to at least make an attempt on a problem, then I need to do the same right there in front of them.  If I’m asking them to admit that they’re human, then I need to admit the same.

Honestly, I’m interested to see what they bring in.  I’m excited for the challenge.  I’m looking forward to displaying my love for the subject and for the “puzzle” that, for me, is what mathematics is all about.  And if I mess up, that’s ok.  The kids will be able to learn at least as much from that, and possibly more, than if I am able to solve the problem on the first try.

I’ll report how things turned out next week.

Stage Two

I’m now a few weeks into the next phase of infusing discussion-based learning into my classroom. Specifically, I’m teaching a section of college-prep algebra 1 this year, and so far, I’ve not needed to lecture much at all.  Every few days or so I will summarize what we’ve been doing, or if there is a topic that I know from experience kids find confusing (like function notation) then I will spend a few minutes dealing specifically with that topic.  Other than that, I have been creating worksheets for each section, much in the same way I did a few years ago when we did the “test run” with Harkness in the conics unit of honors pre-calculus.  Put simply, I look at the section, determine how I would lecture on the section, and then I create a list of questions that (hopefully) lead the kids through the material.  Instead of putting examples on the board and going through them, I take a little more care in scaffolding the examples, creating them in such a way that the kids can at least begin if not complete the examples themselves, discuss their results in a group, and through this process learn the material.

So far, things are going well.  Half the class earned an “A” on the first “individual exercise set” (it sounds less threatening than “test”), and for the most part those that didn’t made simple arithmetic mistakes as opposed to making fundamental algebra mistakes.  Moving into the second unit, things are going equally well, and the kids are getting a lot out of the discussions.  A few haven’t “bought in” yet, and while I’m still trying to get them on board, I don’t believe that the class being run through discussion is the reason for their lack of effort.  Rather, they appear to be genuinely disengaged from school in general, which is a far taller wall in my way when it comes to reaching them.  Even an activity we did last week (“So, how many standard-sized Post-It notes would it take to cover the walls of the classroom, except for the white boards?”) didn’t catch them.  The rest of the class was up, making different measurements they thought would be useful, and worked hard on the exercise for 30+ minutes.  The other few just sat at their desk, waiting for the rest of the group to give them the answer.  Talking with them, encouraging them, trying to get them to participate…nothing worked.  It’s really sort of sad to see that the natural curiosity that fills kids when they are young has been essentially removed from these kids.  What it is that deadened the natural curiosity can vary, and part of my role is to help them through that, whatever “that” is.  Still sad, though.

Anyway, in general I’m happy with the results so far and am optimistic that things will continue going well.  If nothing else, these first few weeks of the year have convinced me that discussion-based learning isn’t just for honors kids, nor is it just for upper-level material; it is working well with college-prep freshmen.  Hopefully, I will find a way to get the few holdouts to join the rest of us in learning some math in a slightly more relaxed manner than they’re used to.

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.

The Fundamental Question

This is going to seem like a silly, and potentially obvious question, but I've done a lot of thinking over the last month, and I’m becoming convinced that the way someone answers this question defines their entire outlook regarding mathematics education, including what content should be included, how it should be taught, and how it should be tested.  The question, and I believe it to be the most important question in mathematics education today, is this: what is mathematics?

If your answer to this question centers around memorizing times tables, learning the basic mechanics of algebra, or memorizing geometry formulas for area and volume, then in my experience you probably lean toward a lecture-based method of delivery in the classroom and test students using problems that strongly resemble those that were in the assigned homework.  While not explicitly teaching to the test, you do expect that the problems on standardized tests will resemble those found in the textbook from which you are teaching, and that, if you so decided, you could teach to the test and your students would succeed in passing these tests.

On the other hand, if your answer to this question centers around discovering patterns, regardless of where they may be found, then in my experience you probably lean toward a more constructivist approach in the classroom and test students using exercises that require them to use the material covered in class in ways that may or may not have been done in class.  Teaching to the test is an impossibility for you, since the point of the test is to see how well the students can use the material in new ways…in short, to see how well they can problem solve. Standardized tests are just one more opportunity for your students to problem solve, and as such the preparation for them is a regular, ongoing part of the course, not a separate entity on which to focus.

The answer to this question also seems to relate to how you feel about the Common Core standards and the tests that are due to accompany their implementation.  The “mechanics” people tend to not agree with CC, while the “patterns” people tend to see it as not quite so big a deal. (Note: I’m not talking about the part where the test scores are part of the teacher evaluations…that’s an entirely different discussion.)

Now, to be clear, I’m not saying that the “mechanics” people don’t try to teach the kids to problem solve.  Nor am I saying that the “patterns” people don’t understand the value of being fluid with the fundamentals.  What I am saying is that for every mathematics educator I’ve met, one seems to take center stage, while the other is either a follow-up (as the problems solving seems to be for the mechanics folks) or a part of the process included along the way (as it seems to be for the patterns folks).  But the conversation we are currently having in math education seems to be missing a discussion about the fundamental question that underlies all of our beliefs.

So, how would you answer the question?

(Admittedly, this is rhetorical, but I’d love to hear some of the answers.)