Current Results

We've reached the end of the first semester, and overall the results from both honors pre-calculus and algebra 1 are positive and promising.

With the honors pre-calculus kids, we found a way to "incentivize selflessness"(which still sounds like an oxymoron to me).  We made a deal with the students that if everyone in their particular section of the course earned at least a B, then everyone would get a grade bump of "one sign", meaning B becomes B+, B+ becomes A-, and so on.  And while the kids weren't able to cash in on the deal, the overwhelming majority of them saw their grades jump by over half a letter grade on both the final IES and on the semester exam.  My hope is that they have come to realize what I knew all along: the more they help one another and the more they look out for one another, the more they will learn themselves and the better they will be able to communicate their understanding both verbally and in writing.  Granted, I'm going to explicitly mention this when we return in January, but I'm confident that my words will simply be reinforcing what they already know.  And just in case, we plan to keep the incentive in place for the entirety of second semester.

With the algebra 1 kids, giving the kids feedback rather than numbers on their papers and allowing them to make corrections to their work went very well, as the kids did remarkably well on the semester exam.  At our school, we have common tests and exams for the required courses, and the norm on the semester exam is a C average, with a few Fs and a couple As.  My kids were able to achieve a B average with plenty of As (including a 100%) and no Fs.  For the most part they took my feedback to heart and  made to necessary corrections to their misconceptions. In addition to this, they worked well with and for one another during the last few weeks of the semester merely because they understood that doing so was helpful to everyone involved.  There was no incentive as there was in honors pre-calculus, and yet the selflessness was there all the same. Perhaps the replacement of numbers with feedback served as a model or perhaps it took a bit of the competitive edge off of things, but regardless I'm hoping to be able to incorporate some of this into the honors pre-calculus classes next semester.

So, a solid first semester for both classes, and a successful (albeit exhausting) infusion of discussion-based learning into algebra 1.  On we go...

Questioning Grades

I'm quickly coming to the conclusion that assigning grades is detrimental to learning.

Let's start at the beginning: why do we assign grades in the first place?  Looking back on my career as a teacher, in an earlier time I probably would have answered this with something along the lines of "to report how well the student has done in class this term" or "to determine whether or not the student is ready for the next course".  And I would imagine that many teachers would probably say something similar.  However, looking at things the way I do now, where "how does this help the student learn" is the central question that needs to be asked regarding everything I do as a teacher, I'm having trouble figuring out how grades promote learning.

That doesn't mean I shouldn't give tests or homework or correct the work of my students, because I most certainly should.  But the part of any of those that has the potential to help my students learn the material is the feedback that I give them, not the number I put at the top of the page.  In fact, the number at the top of the page is, in general, a distraction, since the kids more often than not look at the number and nothing else.  The comments I write are, in many cases, ignored by most of the students.  Sadly, I believe this is why many teachers don't even bother to give comments on assignments, choosing instead to simply mark how many points the student lost on a particular exercise and nothing else.  Equally sad is that the emails I receive from parents are normally not concerned about how well their child is learning the material, but rather what the kid can do to raise their grade.  The grade is everything to everyone involved in the process.  Learning, while nice, is ultimately not the goal.  And looking at all of this, the conclusion is quickly reached: the grades are getting in the way of the learning.

So, in algebra 1, I've started not putting numbers at the top of most of the papers I "grade", putting only comments for improvement and allowing the student to make corrections to the assignment, corrections which include an explanation of the correct answer.  This seems to be helping the students actually focus on learning the material, and the stress the normally accompanies the assignments isn't there, since there they know a number isn't going to be placed on the paper.  To put it more bluntly, they know I'm not going to be judging them; instead, they are beginning to see that my role is to help them learn the material and that this corrective feedback is a step toward that goal.  I'm also "grading" a lot more of their practice work.  But again here, it's not about the number; rather, it's about giving the kid helpful feedback so they can hold onto what they are doing well and correct what misconceptions they have.  For the sake of mentioning it, this feedback also happens during the  in-class discussions in algebra 1, and is, in many ways, the discussions are the exclusive way that feedback is given to the students in honors pre-calculus.

Of course, eventually I still need to put a number in the gradebook.  My hope is that the number will be more reflective of how much they have actually learned, rather than a confusing conglomerate of effort, compliance, and assessments which may or may not indicate how much the student has learned by the end of the semester.  Hopefully, as is currently the case, most the students will continue to do the work for the sake of the feedback and for the sake of actually learning the material rather than for the sake of the grade, a grade which will take care of itself so long as they actually take the feedback seriously and learn the material.  We'll find out soon enough.

All For One

So it's been an up-and-down couple of weeks.

The up: Three days before the last individual exercise set, or IES, (i.e., test) in both algebra 1 and honors pre-calculus, I gave a  practice set under "real" conditions. For algebra 1 I graded the papers, returning them the the next day for the kids to discuss.  The following day was our normal review, which I refer to as me being on the "firing line" because the kids fire questions at me. The next day was the IES, and of 28 kids in the class, 14 got some sort of an A.  This was a solid improvement over the previous IES, so of course we're doing this again with the current unit. With honors pre-calculus, the homework for the evening of the practice IES was to grade someone else's paper (think peer editing from English class), and the next day the kids discussed the papers with their partner.  These were quite possibly the best discussions of the year. There was a gentle but firm honesty and a seriousness of purpose that was really great to see. The following day I was on the firing line, and the day after that was the IES. The grades improved slightly,  but not as dramatically as they did with the algebra 1 kids. Definitely going to do this again, mainly because I do think it helped, but also because something else is going on that I believe is at the root of the lack of improvement in the honors pre-calculus classes, which leads me to...

The down: Normally by this point in the term, the classes are on a slow, steady climb with regard to both the grades and the quality of the daily discussions. There's a certain comfortable hum that takes over the classroom, indicating to me that things are going as they should, that the kids are in it for each other, helping one another learn the material.  This year, however, there seems to be one thing preventing this from happening. The word that comes to mind is "selfishness", but that seems to me to be a bit harsh, even if it's honest.  You see, some of the kids are able to do the homework exercises "on the fly", and they do so during the discussions in class...and by do the exercises, I mean mechanically push the symbols around and get an answer, not explain the material with any kind of clarity or depth.  While this does them some good, it does not benefit the other students, at least not as much as it could. If the students who understand the material would do the homework, they could work through their difficulties at home instead of in front of their group, and be better prepared to help the other students in the class as well as have a more complete understanding of the material themselves.  Their motivation,  however, is to get their participation points, not participate in the discussion for the everyone's benefit.  Sadly, this should come as no surprise.

You see, I've come to the conclusion that we actually teach the kids to be selfish.  We teach them to not share information with one another, to not help one another, at least not when it comes to academics.  The entire thing is an individual competition to see who can get the best grades, who can earn the best score in class, who can "win", for lack of a better term.  For as much as we may want the process to be about everyone succeeding, and by succeeding I mean learning, we are currently in a system that in many ways prevents that from happening.  What we need to promote and cultivate is a culture that values learning over grades. This is what I try to do every day in my classroom.  Many of the students have "bought in", and are doing well in the class, not only in terms of learning the material and in terms of their grades, but also in terms of learning how to learn, which if asked is what I would say is *the* point of education. Those who should be the "top students", those I mentioned above who can do the problems on the fly, are not making any progress because they are content with what they already know, content that they can "win" and get an A (though for several of them this is currently in question), unconcerned that they can mechanically "do the math" without truly understanding it...a fact that is evident from the fact that they struggle to explain the material to the others in the class.

So the question I'm wrestling with is how to incentivise selflessness.  How do I get the top kids to stop the chase for the grade and instead focus on improving their explanation of the material, which would not only benefit the other students, but would also increase their own understanding?  I don't have any answers...yet.  But, just as the students shouldn't stop looking for solutions to their problems, neither will I.

Testing vs. Assessment

I have been told over the last couple years that what I am doing with the honors pre-calc kids can’t possibly be done with college-prep-level students, that it must be nice to have the opportunity to work with the honors kids where discussion-based learning is possible, and that I’ve lost touch with the reality of most classrooms.  I have essentially dismissed all of this, firmly believing that discussion-based learning is the way to go regardless of the “level” of the student.  There was no doubt in my mind that if we don’t prejudge the ability of the students to take responsibility for their own education and give them the opportunity to do so, great things will happen.  

This year, I have five honors pre-calculus classes and one section of algebra 1, which has provided me with the opportunity to run a college-prep-level class by discussions.  With as much work as we have done over the last couple years for the honors pre-calculus class, the preparation in terms of  exercises and so on has been minimal, and instead we have been able to focus almost exclusively on the kids.  However, I have not taught algebra 1 for about six years, long before I converted my classroom to a discussion-based learning environment.   So, the preparation work for that class has been extensive, writing exercises that lend themselves to discovery and discussion (which are sorely lacking in the textbooks).  In addition, of course, there has been the usual struggle of getting the kids to take responsibility for their learning, which is something that the honors pre-calc kids struggle with as well.  That being said, the algebra 1 kids have been, for the most part, far more willing to take chances, make mistakes, and learn from the experience than the honors kids.  As such, I have had the opportunity to watch them in the early stages of solving linear equations, finding an equation of a line, and multiplying polynomials, and then help correct the misconceptions in a much more individualized way.  This informal daily assessment is crucial for any student learning anything, and a discussion-based classroom is the perfect place to allow this to happen.

Two days before the most recent test, I gave the kids a practice test in class, running that day in exactly the same manner as I do the actual test day.  I graded the practice test that evening, which allowed me to see where the misconceptions remained and where the common mistakes were being made.  The next day in class we discussed the practice test exercises, and the kids got to use the practice test as a means of preparing for the real thing. Was the practice test taken for a grade?  Nope.  Did the kids take it seriously?  Yep, most of them did.  They understood the benefits of practicing well for the test, and realized that the only way to correct the mistakes was to make and become aware of them.  

Then came the actual test.  On a test with which students have historically struggled, fourteen of twenty-seven students earned some version of an “A”.  Most of the rest earned a “B”.  The test was the only grade that actually went in the gradebook.  However, the success most of the students had was set up by the informal daily assessment and the formal practice assessment, the more personalized, detailed feedback the students received, and the seriousness with which the students took the feedback they were receiving.  

In other words, testing and assessment are not the same thing.  Assessment is what happens as we prepare the kids to take a test, or at least it should be.  And in my experience, a discussion-based classroom allows the necessary assessment and feedback to happen in a way that a lecture-based classroom never could, regardless of the “level” of the kid.

A Growth Mindset

How do you react when a student says, "I can't do this. I'm just not good at math and I never will be.  It's just who I am."?

Hopefully, you have a talk with the student, calmly discussing with them that with focus and by doing the required work, they can learn the material.  We shouldn't believe that they can't learn the math, and we shouldn't let them buy into that lie, either.  If we don't believe that every kid can be successful in our class, we shouldn't be teaching, period. We expect kids to be able to achieve a certain level of competency in every subject, and we don't accept the excuse that they just aren't good at any specific subject.  Yes, they may be more comfortable with some subjects than others, but that doesn't matter. There is a level of success that we expect from every kid, regardless.

Now, how often do we allow ourselves to get away with telling ourselves we can only teach a certain way.  We balk at the idea of teaching any way other than that with which we are completely comfortable.  We convince ourselves that the only way we can possibly be effective in the classroom is to continue what we have been doing, regardless of current research that clearly says we should be doing things differently, hiding behind the idea that "I can't teach that way. I'm just not good at it and I never will be.  It's just who I am."

This is the mindset of many teachers I know. It was me just a few years ago.  There is a current push to instill a growth mindset in the kids and yet we have a fixed mindset when it comes to our abilities as teachers.  This needs to stop.  If the research says we need to teach in a way that encourages discussion, promotes creative problem solving, and requires real understanding, then we need to take on a growth mindset, calmly focus on the task at hand, and put in the work necessary to teach that way. If we don't believe we can do this, then we shouldn't be teaching, period.

It is time to stop making excuses and start growing as educators.  We need to stop limiting ourselves (and, in turn our students) and start becoming what the students need us to be.  And it's time for those of us who have already taken on the task to start promoting a growth mindset in our colleagues.  Anything less is a disservice to the profession and the students.

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.)

Looking Backward and Forward

Well, year two of Harkness has come to an end, which makes this a good time to reflect on what has gone well and what could go better.  I’ll start with the positives:

1. I’m basically happy with the problem set we created two summers ago and have been editing ever since.  There is still work to be done, and honestly I don’t think it will ever be “finished”, but I’m happy with where it is and how well the questions have moved the students where we needed them to be.
2. The kids who buy into the system and do what we ask them to do are being successful in lots of ways.  By “doing what we ask them to do”, I mean preparing for class, participating in the discussions, and reflecting on their work every day.  By “succeeding”, I mean learning the material, period.  The evidence for this is not only provided by their grades on the tests and on the exam, but also on their day-to-day work and on their standardized test scores.  As examples of this, three students mentioned to me that their ACT score rose by 5, 7, and 8 points respectively compared with their previous scores, and they attributed the increase to the critical thinking and problem solving skills they gained from my class.  I don’t know how much credit the class actually deserves, since there are most certainly other factors involved, but the kids gave the credit to the class.
3. My management of the class has improved.  I’ve reached a point where I have the paperwork I need to get the daily information I need, and I’m able to do it in a way that allows me to spend more time with the kids at the tables.  Keeping track of everything that goes on simultaneously at 3 or 4 tables was (and is) a huge task, but I’ve finally reached a point where it’s manageable and doesn’t take overshadow the time I get to spend with the kids.
4. Some of the other teachers are beginning to include more discussion-based methods in their classes.  Is it completely Harkness?  No.  Should it be?  That’s up to them.  But at least they’re giving it a try and having some success.

 The negatives:

1. There are still kids who don’t buy into the system.  Some kids want to be told what to do and how to do it (a few said this to me directly in those words).  Others try to get by with only the in-class discussions and none of the preparation.  Still others try to memorize everything instead of truly learning the material.  The one theme that runs through all of these students is that they are focused on the grade and not on learning.  Of course, they’ve been in school for 10+ years now, and breaking them of something that has worked for them in the past isn’t easy.  Somehow I need to impress upon the kids early on next year that my classroom is focused on learning, not on grades.  And I have evidence at this point that if you do what I ask in the way I ask, then not only will you learn the material, but the grades will fall into place as well.
2. There are still plenty of skeptics when it comes to any “non-traditional” method (i.e., anything other than lecturing), and any shortcomings the kids have are placed squarely on the shoulders of the non-traditional methods.  Besides Harkness, some of the methods used by teachers in my building include the flipped classroom model, team-based learning, and competition-based learning.  All of these have received their fair share of criticism, even when the shortcomings noted in the students are just as common if not more so in the students in one of the traditional classrooms.  Somehow, we need to remove the stigma.  A good place to start may be to find a way for the traditional teachers to accept the invitation to observe our classes.

Over the summer, I will be heading back to Exeter for the Greer conference, and this year the other honors pre-calculus teacher will be with me.  During the week, we will have the opportunity to refine some of the questions from our problem sets.  We will also be working on ways to incorporate more discussion-based learning in the algebra 1 classes in our building, since we will both be teaching that course next year as well.  Finally, I’m thrilled that the Exeter Mathematics Institute will be coming to my school for a week to share Harkness with teachers from several schools in the Cincinnati area.  The goal, of course, is to help the kids succeed.  My contention, of course, is that discussion-based methods, whether flipped, team-based, competition-based, Harkness, or otherwise, are the best way to make that happen.  Hopefully we can find a way this summer to get the momentum moving in what I would consider to be the right direction.

Favorite Lesson

I subscribe to several education sites and a popular theme recently has been teachers sharing their favorite lessons.  This got me thinking about whether or not I have a favorite lesson.

A few years ago, it would have been difficult for me to answer this because there would have been several I would have wanted to choose.  “Everything You Need to Know about Logarithms in Less Than 45 Minutes” would have made the short list.  “Crash Day” near the end of the first half of honors pre-calculus when so much material we have covered up to that point goes into proving e^(iπ)+1=0.  Partial fraction decomposition and rotation of conics would have been on the short list as well.  I enjoy the material from these lessons, it’s easy to build a story around them, and the kids were able to follow them (for the most part). 

However, I’m no longer lecturing, and at this point I would have to say that I don’t have a favorite lesson because I don’t have math lessons in my classroom.  Refocusing the course in a Harkness style has meant transferring the class over to a more holistic outlook when it comes to the material.  We don’t focus on individual topics.  Rather, we focus on how the material they already know coming into the course can be extended, utilized, and synthesized.  As such, there aren’t lessons on the individual topics.  Instead, we focus on making a little progress every day with a lot of different topics.  For example, a “normal” day this term could include working with parametric equations and projectile motion, equations of conics, solving trigonometric equations, and solving triangles, all on the same day.  And that’s not an exhaustive list of the material we cover. 

So if pressed to give my favorite lesson, it would have to be the one where I help the kids see that math isn’t a bunch of separate topics, but rather is a unified whole.  In other words, I get to present my favorite lesson every day.

The Goal

I’ve been giving a lot of thought lately to the common core standards and what they are really trying to accomplish.  There are many supporters and many critics, and I find myself somewhere in the middle, mainly because in working with the standards and in trying to prepare for their implementation I have seen the good and the bad.  However, there is one main “theme” that tends to stand out that distinguishes the supporters from the critics, and that is the way the following question is answered: What is the goal of education?

If one sees the goal of education as the transmission of facts or techniques, then the common core standards make no sense whatsoever.  Reading through the standards and looking at some of the workbooks and other materials that have been produced, there is a feeling that the “basic skills” are not being emphasized, whether those basics are knowledge of parts of speech or memorization of arithmetic facts or whatever.  In fact, at a glance, it appears as though these skills are not being valued at all.  As such, the outcry against the common core standards tends to take the form of “just teach them to do the math” or “just teach them to write”.  And in many ways I can understand the complaint. 

However, I believe the complaint comes more from misunderstanding than it does from actual rejection of what the common core is actually trying to do.  You see, the emphasis of the common core is on two things: (1) understand the material well enough to explain it simply; and (2) understand the material well enough to apply it creatively.  This is the goal of education for the common core.  Implied in this is the reality that the students still need to know the basics of writing a sentence and adding fractions.  However, being able to go through the mechanical processes is not enough.  In particular, mechanically getting the correct answer does not show a real understanding of the material.  In many cases, what it shows is the memorization of a process rather than the understanding of a concept, and as such asking a student to explain what they did is important, since in the explanation you will see which of the two (memorization or understanding) the student has actually accomplished.

Unfortunately, the workbook pages I have seen on social media sites complaining about how we need to get rid of the common core seem to be pages from late in the learning process, and therefore are asking the student to go beyond the mechanical process and explain what they did.  The misunderstanding on the part of the critics is that they see these worksheets as trying to teach the students the basics, when what the worksheets are actually trying to do is assess how well the student has understood the material. 

The implications of all of this for the classroom teacher are quite profound.  If all I do is show my students how to perform the standard multiplication algorithm, and then assess them on how well they can perform the algorithm, I will not be checking how well they understand what multiplication actually is nor how well they can use the algorithm in a problem-solving situation.  The same would hold true for teaching a student the parts of speech, or factoring, or many other “basics”.  However, if I emphasize problem-solving in my classroom, and as the need arises show the students the usefulness of a particular algorithm, then the students will still get the basics, but they will do so while developing a deep understanding of the material and creatively applying the basics. 

This is where the discussion-based classroom is at its best.  Students work collaboratively to creatively solve a problem, and in doing so they learn the basics because they need those basics to solve the problem.  This is what the “real world” looks like.  In life, we are rarely confronted with a problem we already know how to solve.  Instead, we are confronted with a situation and we need to go find the tools and develop the skills necessary to solve the problem.  A classroom centered on memorization of the basics does not prepare students for this; a classroom centered on the essential spirit of the common core, i.e., a discussion-based classroom, does, and it does so in a way that does not short-change the basics.

All that being said, the implementation of the testing regimen that is to accompany the common core standards is where this entire process meets with resistance from teachers, even those of us who agree with the intent of the standards.  Asking the kids to be creative and then testing them for conformity is, in my mind anyway, a contradiction, and is a cause for concern.  Yes, some of the exercises are performance-based, but I wonder just how flexible the graders are going to be.  If a kid uses a method that is mathematically valid on an exercise, will the grader have the mathematical prowess to determine the validity of the method?  For that matter, will the rubric constrain the ability of the grader to award credit for a mathematically valid response?    Asking the kids to patiently problem solve and then testing them with a relatively short time constraint is contradictory as well.  Add to these the fact that we are losing several weeks (or more) of regular class time to administer the tests, and you can see where a teacher might be concerned.

So, do I think we need to return to just teaching the basics?  Clearly, the answer is no.  Do I think the current implementation of the standards, and in particular the testing that comes along with them, is in the best interest of the students?  Again, and despite the fact that I agree with the fundamental tenets of the common core, I would have to say no.  There has to be some middle ground, where the basics are included in the bigger picture of problem solving (rather than problem solving being included as an add-on to the teaching of the basics), and where collaborative work and communication skills are the fertile ground in which the creative use of these skills take root and grow.

 

Sounds like Harkness to me.

Baseball Practice

I have been asked a couple times recently why I run my classroom the way I do.  Here is my current response:

It is a beautiful Saturday afternoon and, more importantly, it’s the first day of baseball practice.  The coach, having never met any of the players, quickly introduces himself, and then begins batting practice.  The coach lines up the players behind the backstop while he spends the next hour hitting baseballs being pitched to him by a machine he set up on the mound before practice began.  The boys watch intently as the coach explains and demonstrates how to properly stand, hold the bat, swing the bat, and so on.  At the end of the hour, the coach tells the boys to go home and practice what they have seen.

At practice the following Tuesday evening, the coach asks the players how their practice at home went.  One of the players says that he struggled with his stance, so the coach demonstrates again the proper way to stand in the batter’s box, showing the player by example how it should be done.  Much the same as Saturday, the player watches intently as the coach hits the baseball.

“Do you understand what you were doing wrong?”

“Yep, I think I get it now.”

“Good. Any other questions?”

Silence. 

Then the coach begins fielding practice.  The boys stand behind the backstop and watch intently as the assistant coach hits ball after ball to the head coach who fields ball after ball from the shortstop position.  He shows the players proper technique, how to stand, how to hold and place the glove, and so on, and after an hour of demonstrating what the players are supposed to do, sends them home to practice on their own.

This is repeated over the next several weeks with the coach showing the players how to catch fly balls, how to bunt, how to pitch, how to throw.  He shows them everything they need to know to play baseball. He never sees them play.  They never practice together or play together. The first time the coach actually sees them play baseball is at the first game.  You can imagine the results.

The coach is confused.  He showed them everything they needed to know.  They were very attentive at the practices, and what few questions they had he answered.  How did they not do well?

Absurd as this sounds, this is precisely how many people teach mathematics, performing in front of the class while the students watch, and then sending the students home to practice on their own.  The teacher answers questions by showing the students how to solve the problem at hand.  The teacher never actually sees the students do any mathematics except for the tests.

 

Why do I run my classroom the way I do?  Much the same as a coach spends most of his time watching the players at practice, noting the mistakes they are making and helping them correct those mistakes before the actual game rolls around, I want to see my kids doing math, see the mistakes they are making, and help them correct those mistakes as they are learning the material.  Just as the majority of the time spent at baseball practice involves the players playing baseball or practicing a particular skill, I want the majority of the time in class to involve the kids actually doing math.

Why it took me so long to figure this out is a travesty.  How others don’t see the benefits of running a student-centered classroom is a mystery.

Differentiating

At the end of the last unit, quite a few of the students were struggling with the basics of vectors, as the grades on the last test made evident.  Knowing that some of the material to follow depended on understanding this information, the other honors pre-calculus teacher and I came up with an optional replacement assignment for the students to do.  The descrpition itself was simple: convince me you know what you're doing when it comes  to addition, subtraction, scalar multiplication, finding unit vectors, determining component form from magnitude and direction, and determining magnitude and direction from component form.  How this was to be accomplished we didn't specify.  They could write a short paper, make a poster, make a video...whatever.  The "how" wasn't important.  Just convince me you know what you're doing, and the score you earn will replace the score for the vector exercise on the test.

To say that it went well would be an understatement.  Granted, overall what the kids turned in wasn't necessarily creative, as most turned in explanations of the above topics with an example of each (though one student did write new lyrics to "The Devil Went Down to Georgia" and performed the song in class), but afterwards the students said that they definitely learned a lot from doing the activity, and the subsequent exercises we have done in class have convinced me that this is, in fact, the case.

So, after the most recent test, we have given the students the opportunity to choose one problem from the test and turn in a similar project to convince us they really do understand the material the particular exercise was assessing.  Again, we were specific enough so they know what skill/knowledge we want them to demonstrate, but beyond that the assignment is fairly open-ended.  And I have no doubt that when they turn in the assignment next Wednesday, we will have achieved the same result as before: quality work from which the students learn the material well.

Notice by the way that none of this is focused on "extra credit".  The focus is on understanding the material and demonstrating that understanding, period.  Yes, the assignment helped the kids' grades, and while that may well have been the initial motivation for doing the assignment, I'm hopeful that enough of them got enough out of the first assignment that they will see the true importance of doing the second one.

All of this got me wondering: Why do we have the students complete some sort of in-class assessment wherein the students have to demonstrate their mastery of the material by doing exercises prescribed by the teacher?  If the answer is: because that's how they're going to be assessed on state-mandated standardized tests, or because that's how they're going to be assessed on college entrance exams, or that's the way they're going to be assessed in the next course or in college, or because we've always done it this way, then haven't we missed the point of assessing the students?  These may well appear to be practical reasons, but are they valid?  I understand that there needs to be some sort of assurance that the students did their own work, and for the record we achieved that with the assignment described above since the exercises that have been assigned since then that depended on knowledge of that material have been completed well by the students.  And there are plenty of teachers I know who, for various reasons, would never dream of giving an assignment as open-ended as the one described above.  But seriously, if the point of assessing the students is to give them an opportunity to demonstrate their mastery of the material, then how is the above assignment any different than a short essay in an English or history class?

My current conclusion: we need to give the students more variety when it comes to the opportunities we give them to demonstrate their understanding of the material.  There needs to be accountability, but we need to admit that, in the same way that we differentiate instruction, we need to differentiate assessments.

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.

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.