PBL for Engagement and Empowerment in the Mathematics Classroom

 Well, the end of my transition year with this new curriculum for my colleagues is almost done.  We have realized that we wrote way more problems than we needed in order to get through our old geometry curriculum content-wise.  It is an interesting thing, though, to have to go through and cut problems that you really loved when they were written.

 Anyway, something else that comes to mind at this point in the year is how far the students may or may not have come in their work with problem solving.  Clearly, the stronger mathematics students basically just eat up these problems.  They are confident in their skills and ideas, so being part of the conversation is not difficult.  However, the students who might have come into this course with the preconceived or realistic notions that they are pretty weak mathematics students are much more of a concern.  The concerns range from how attentive they can be during a problem discussion to how they can possibly be metacognitive enough to write a journal entry.  All of these issues surfaced throughout the year with different students and I thought I’d share a few with you.

One student, I’ll call John, came into this course with a barely passing grade in Algebra I.  In fact, there was serious rethinking of him moving on at all into this course.  The year began for him as a studet who was barely engaged in the problems, but because of his positive attitude, persevered.  John had severe retention and learning issues, which a few months into the year were diagnosed by a professional.  Clearly, there was no way this student could remember to solve a linear equation from Algebra I, if he couldn’t recall the properties of parallelograms from the previous problem, two days ago.  This was clearly an extreme case, but worth noting.

John never stopped working and half-way through the year finally was linked with a tutor who specialized in his special needs.  John started meeting with his english teacher to help in writing his journal entries, and finally something clicked.  The entries became much more connected to class problems and ideas.  The work in class kept him engaged and sometimes even volunteering to share a solution that he knew was wrong.  The growth in this student since September has been tremendous, even if grade-wise he’s still in the C- range.

Another success story is a girl I’ll call Sheila.  This is a student who came into the course with very little confidence in her ability to speak her mind(with respect to math) or have good ideas.  She was a B-/C+ student in Algebra I and definitely retained some of her skills from that course.  She started the year mainly complaining blatantly in class about how much she did not enjoy the pedagogical framework of the course (my summarizing of course), and frequently stopped class to comment about how a problem could’ve been more easily stated or how unfair it was that we would expect her to draw her own diagram.  I allowed this to happen a number of times since I wanted to create a sense of freedom of expression in the classroom.  However, at mid-year, I made it clear to her that this type of sharing with the class will no longer be tolerated.  Interestingly, once that happened there was a change in her work and attitude.

Now, I’m no psychologist, so I can’t claim to understand what made Sheila change the way she worked, but all of a sudden so much of this course mattered to her.  It came across in her contributions to class, which were often surface in her understanding, but sometimes showed great insight.  She once described the reasoning behind the Triangle Inequality brilliantly with a piece of her hair.  At the end of the fall semester, she had a C-, and most recently earned an A- on a problem set.  Again, I can’t exactly put my finger on what has changed for this girl, but I can hypothesize.

It might be that Sheila finally stopped complaining enough to see that she did have important things to say in class besides her complaining.  It might also be that once she did that, she actually enjoyed being a productive part of a solution process with her peers.  I sometimes catch her explaining something to another student and think that her confidence in her abilities has grown tremendously.

The more important question that follows is twofold 1) are these changes because of the coursework or are they just the normal intellectual maturity that happens to teen-agers and 2)what will happen next year when they move on to a more traditional course in Algebra II/Trig.  It is the second question that has my colleagues most troubled, and for now we will have to just wait and see.  Although our head of school, would be happy to have us writing another year’s worth of curriculum this summer, we really feel that we need to edit and really solidify the year we have right now.  Hopefully, in the future, we’ll be able to bring this experience to other courses as well.

As we have completed our first week back to school, I feel the need to write something else, but I’m not quite sure what to discuss. I thought this entry could be just a summary of what we are doing right now.

The geometry teachers met to discuss a bunch of problems that we were planning to do very soon. Many of them included more difficult proof problems. Interestingly, we found that for some of them, we had all done them differently. I love when this happens, because it shows how differently we, as adults, all think. It helps to keep us in perspective about the students’ work. It’s so easy from our perspective, to know where a problem is going and what its point it. Can you imagine how the kids feel? It must be totally disconcerting to think you are going in one direction and then come to class and realizing your thought process was off track. It can be very discouraging.

That’s why attitude plays such a big part in this type of curriculum. A few of my students are definitely getting the hang of coming to class and being open to being wrong. They do not see it as a bad thing anymore, but as their learning experience enhanced. Yesterday in my class, we had a little time at the end of the period and I had them all start on one problem that was very abstract. The first part of this problem gave them four general points A(0,0), Q(a,b) and S (c,d) (where a>c and b<d). These three points were supposed to be three vertices of a parallelogram. They were asked to find the coordinates of R, the vertex not shown. I had them think alone for a little while and then I walked around the table and asked what they got. It was an interesting discussion and I thought it worked well, because now everyone had that part of the problem done and correct and could go back to their homework later and feel that they had accomplished at least a part of it. As the bell rang, one student said, “I really liked that” and she meant that it helped her feel better about her homework when we started discussing the problems in class. I think I might consider being much more deliberate about this in the future.

Another thing that worked really well this week was doing a Think/Pair/Share exercise. This is where I give them a problem and first have them think alone for 3-5 minutes. Pair up with another student (randomly by some silly rule that I come up with) and then the pairs share with the whole class. I believe I had everyone talking in the class at least once during this exercise, which is generally my rule of thumb for a good discussion.

I am encouraged by the positive feedback that I have received from not only my department colleagues, but colleagues in other deisciplines as well. I was approached by a teacher in the science department who said that they were considering teaching Biology with this method in the future and could we meet to discuss this. The Algebra I teacher is considering moving in a direction where there is much more problem-based learning going on. I feel proud that others are finding value in what we are doing and that we had the courage and deliberateness to create such a good curriculum that it has impressed others with its value.

Of course, no curriculum is perfect and my fellow geometry teachers and I are already looking to the summer to plan our work on our revisions of this curriculum…but that’s another entry!

I’ve been meaning to write this entry for a while, but of course, as a teacher with faith in her students, wanted to keep giving kids a chance. One of the biggest issues we had this fall with transitioning to this new curriculum was how to deal with students who were resistant to this new method of teaching. Initially some of the comments were centered around trying to understand why this change was made. Clearly, even with, what, by this time in the year, was probably hours of discussion, these students would not see the power of this pedagogical decision for many reasons. There were some students who simply needed to readjust their defintion of success in mathematics. Some students really had a learning style that did not fit with this type of teaching. Overall, though, many of these students have come around by this time and have seen the benefits for them, either in their own growth or at least in some way, in their grades.

It’s the tough cookies that are still getting to me. This is what I call the 3 or 4 students out of the 5 sections that we are teaching who are just being tough. The typical characterization of this student is a student who is actually quite bright, but has chosen in their academic career to just not work and get by on their smarts with OK grades. They have also kept up an attitude that has given them a certain reputation with their classmates in order to show them all they this individual simply doesn’t really care and can’t be bothered working for this class. We’ve all seen it and dealt with it in many classes in our careers.

However, in this course, it is different. The tough cookie is really a hard one to handle. The problem being that problem-based learning really requires a great deal of investment on the part of the student. When this does not happen, learning is impeded and often stifled. PBL needs the student’s investment for many reasons. For one, there is an assumed amount of effort on the learner’s part in struggling with the problem on a nightly basis. Struggling in a good way of course, where they simply jot down ideas and formulas that might lead them, with further discussion, to an answer. Without the attempts on the learner’s part, there is no regular practice of independent problem solving skills, which I believe is a necessity. Further, without regular practice of independent problem solving skills, it is difficult for the learner to track their own progress.

One of my tough cookies, I’ll call Tara, is a talented athlete and pretty bright girl, but it is difficult for her to admit to me that she enjoys solving problems. She has good retention of her algebra skills from last year, but she continues to keep up this “I-really-don’t-care-what-you-think-of-me” attitude in order to perpetuate her “too cool for school” reputation in and out of class. Tara is a great example of a tough cookie, who on the inside is really intrigued by problems and knows she can do it, but has not made the important investment in order to see progress.

I am going to continue watching Tara throughout this year and see how she reacts to certain ways that I interact with her. The other day she came to class with no work written on a problem except for copying the diagram that was given in the problem booklet. Our conversation went like this:

me: “You know that you are supposed to write something for every homework problem.”

Tara:”I did”

me:”You just copied the diagram from the booklet – no new ideas or information from you. You should have at least labeled the diagram with values you knew or label something x for the variable you were trying to figure out.”

Tara: “Of course, I knew that, you could just ask me.”

me: “I really don’t have time in class to ask every student what they meant to write on their homework when there’s nothing written there.”

(I walk away to check the next student’s homework)

Tara: (under breath) “Jesus Christ”

me: (touching her head softly) “Yeah, we’re both pretty demanding”

I often try to use humor in situations where students are frustrated. I know that Tara’s reaction is not an example of what she really thinks of the class. It is merely her way of venting her own frustration with herself. I wish she would come and talk to me outside of class, but I know that’s way to much to expect from her right now. I will keep working on her and see what happens. Until then, we can lead a horse to water, but cannot make them drink. But maybe there’s still hope for her yet…

The constant hum of November is over and the bright rush of the holiday season of December are upon us at my school. The group of geometry teachers that I work with are all settled in for their “long winter’s nap”. Well, OK, not really, but that’s something of what it feels like. We have passed the “three month” period of teaching with problem-based learning and it seems that most students, but not all, have caught the fever. One of my colleagues, whose class was still clearly retaining a negative feeling toward the class and how it was taught decided to do something drastic one day. She went into class, after a day before where the students had actually been rude to each other, and asked each student to go around the table and say one positive thing about this course. Was she taking a risk? Most definitely, but for this teacher it paid off. The students all had different things to say:

“I like that it’s OK to be wrong in this class.”

“I like that we can share our ideas and don’t have to have the homework all right every day”

“I like that my opinion matters”

These were just some of the comments that she shared with me. I saw a weight lift from my colleague when she told me this and I feel like it was a turning point not only for the students, but possibly for her as well. I have the utmost respect for the two teachers who had faith in my ideas and theories around PBL and who were brave enough to take a chance on this curriculum with me. They both are pioneers in this method at our school and it has taken a toll on them this semester. But, no matter how down they got about the classes, they still believed that what they are doing is in the best interest of the students. We clearly have a strong team of dedicated teachers working on this project and I feel very lucky to have them.

So, as the kids say in the car, “Are we there yet?”. Have things settled and we won’t have issues with students for the rest of the year? Are we at our best in our teaching with PBL? Do we sit and rest on our laurels? Well, first we take a break and relax. Then we meet at the beginning of the new year and start again.