Brussels Sprouts

I don’t like Brussels sprouts.  I never have.  As a child, my parents, who for some reason actually enjoy Brussels sprouts, served them with dinner a few times each month, which means I had to eat them.  The only way I found them even close to palatable was for them to be doing the backstroke in a sea of butter.  As an adult, I can appreciate that they are “good for me”, but that would require that I actually eat them, and since I now have a say in planning the menu, that won’t be happening, ever.

The reason I mention this is that I gave my honors pre-calculus kids a short survey last week with three questions:

1. Please describe the benefits you have found learning in a student-centered, discussion-based class.
2. Please describe the benefits you have found learning in a student-centered, discussion-based class.
3. Which of the following best describes you:
1. I prefer the student-centered, discussion-based classroom.
2. I prefer a teacher-centered, lecture-based classroom, but I see the benefits of being in a student-centered, discussion-based classroom.
3. I prefer a teacher-centered, lecture-based classroom, and I don’t see the benefits of being in a student-centered, discussion-based classroom.

Knowing from the research that even when students have done better in a student-centered classroom they still overwhelmingly prefer a teacher-centered classroom, I was expecting about a 20-80 split in favor of the more traditional classroom.  What I got was more than I could have ever imagined.

First, the kids clearly gave their responses a lot of thought.  The depth of their statements showed a good deal of meaningful reflection was given to the assignment, even though it wasn’t for a grade.  Among the responses were these:

• “There's an ability to learn things from the ground up, to build up understanding rather than just memorize rules or formulas. When you get a firm grasp on the concept from the beginning, you're able to learn it in a more natural way than the usual "memorize this" method. In groups where everyone is at an even level, there's a feeling of investigation when you try a problem and figure it out as a team. On a more personal level, you feel satisfaction when you figure something out the hard way.”
• “It took awhile for me to realize putting whatever thoughts I had about the problem onto paper was more beneficial than just leaving it blank. I would leave questions I didn't know blank in previous math courses and then the teacher would show me how to do them. But now putting whatever thoughts I had down it allowed my peers to help guide me through the problems with whatever work I did have.”
• “Originally (like many others), I disliked the setup of the class as well as the Harkness Discussions. However, I soon realized that there were an abundance of hidden benefits in participating in the said discussions, like: -More engaging class; having your classmates teach you feels more "personal" and "on your level". -Less intimidation; some students find teachers intimidating but this method takes that away--as students can ask each other now. -Better pacing; if you don't completely understand a question, your group can wait and explain it to you while if it were in a classroom-based setting, the teacher would probably have to go on with the schedule and you might have to seek out the teacher on your own time to understand that problem.”
• “Initially I had problems talking in the groups and realizing errors. I was stubborn and wanted to have something right instead of understanding it.”
• “I feel that having to explain my thought process and articulate what I do not understand helps me to better understand the skills that I am supposed to master. Also, when I have to work through problems before actually discussing them, I can best observe areas that require improvement, and then, the frustration of initial failure all the more clearly brings the means for success into my mind.”  
• “I at first was rather timid in asking questions of my peers, for fear of appearing inept due to my want of mathematical understanding. This, obviously, hindered my understanding, yet I was soon compelled to overcome this barrier. At times, I have also struggled to see the bigger skill set and focused too much on individual practice exercises, and not, necessarily, their broader scope. Now that my attention has been drawn to the main topic numbering system, this is much more clear.”
• “By learning in a student-centered class you are able to not only use your opinions and ideas over the question but also how other students approach the problem and are able to answer the question. This is very great to you as you aren't limited to the teacher's perspective and their teaching/solving method. Also being able to discuss with a deeper discussion and allows for a better understanding of the material as long as you are engaged and asking questions/answering the questions while the discussion is going on. Also having the students be the ones teaching, they understand you the best and are able to help you with maybe how you are making a mistake and get in your mindset to correct it and explain it to you in a manner which helps you learn the best.”
• “You are very in charge of your own learning. Independently you have to check yourself on a daily basis to make sure you are caught up and understand everything you're learning.”
• “Student centered classes make sure that the students are all contributing and helps promote leadership among the group because if an individual knows something about a problem that others don't they tend to take the lead and guide the other students so that everyone understands and are on the same page. Also, by explaining a certain question, it ensures that the student truly understands the problem. It helps boost communication skills.”
• “It is harder to learn the material yourself than to be given to you. It takes a lot more time to do the homework because you have to do more problem solving and look things up on the computer. You get frustrated with yourself at first if you can't figure something out, but you have to learn to just give it your best then collaborate with your classmates the next day to solve your issue.”
• “I feel like I'm actually able to discover new things and truly understand them when I figure it out either by myself or through group work rather than being told exactly how to do something. It's easy to remember that one "aha" moment than trying to recall something from your notes.”
• “In the beginning it felt strange to be in a classroom setting where the teacher wasn't in charge of everything. I felt frustrated at times when I couldn't figure problems out. Eventually, I learned to be more open with my classmates, because i realized that everyone is here to help everybody else.”

I received page after page of feedback along these lines.  Beyond the benefit to the students to do this kind of reflection, it was incredibly helpful to me to hear how they are feeling now that have been at this for half a school year now.  And it was obvious to me that they get it.  They get the point and purpose of running the class through Harkness, including that it’s not just about the math.  More importantly to me was this: out of the 90 kids that responded (I have more students than this, but they didn’t all take the survey), 43% said that they prefer the student-centered classroom, and 54% said that even though they don’t prefer it, they see the benefits.  Only 2 kids said they don’t see the benefits, even though they listed out a few benefits in their response to the first question.  Regardless, this was way better than the 20-80 ratio I had anticipated.

So, the big picture: having tried Brussels sprouts, 43% of the kids actually like them, and 54%, despite not preferring them, admit that the sprouts are good for them.  

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.