PBL for Engagement and Empowerment in the Mathematics Classroom

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.

Well, it’s almost Thanksgiving break, and boy do we need it.  I’m probably going to jinx everything by writing this post, but I think it’s important to step back and look at the good things that are going on.  It is very clear to me and my colleagues that the students have come to some sort of milestone in our geometry classes.  Girls who were not excited or engaged with the problems just a short 2 months ago, are now making efforts in class.  I have one student, who when faced with a problem introducing the Triangle Inequality Theorem, ripped a piece of paper from her notebook, folded it in half and then took one half of the paper and folded it again, showing that when you make a triangle out of two halves, (one half becoming two sides), the two put together must be bigger than the third side.  This is a girl who just 3 weeks ago, got back a problem set with a “B” on it and outrightly stated to the class “I hate math.”  Now I look back on that time and see how far she has come, but how do you get the students to see that?

I’ve heard similar stories from my colleagues.  One student said thank you to a teacher by saying, ” This is the first math class where I feel like what I think matters.”  Times like this definitely make us feel good about what we are doing and how much it means to the students.  Clearly, they are not intellectually mature enough to know what to thank us for or to articulate what they are getting out of this class right now.  However, that’s the nice thing about this type of curriculum.  It seems that the skills we are trying to impart to the students just kind of “sneak up” on them and they don’t even realize what they are learning.

So, we may be over the hump, or maybe just over one hump.  We have number of students invested in the process, and although they might not like it yet, they are learning and that’s a pretty darn good feeling.

While doing some research for my latest literature review assignment, I came across a chapter in the Handbook of Research on Mathematics Teaching and Learning (1992) that was written by Alan Schoenfeld entitled “Learning to think mathematically: Problem solving, metacognition and sense making in mathematics.” It is a very good overview of the history of the paradigm shift in mathematics education and research to more of a focus on problem solving as a goal in the mathematics classroom and makes some good comparisons between differing definitions of what problem solving means in the classroom.

Something that I thought was extremely interesting in this chapter was the argument that teaching with the “big picture” of problem solving skills in mind is actually much more difficult for the teacher. This totally resonates with me and my colleagues who are making this shift this year. We are all feeling more tired, more spent at the end of our day. Now, this could be because of a new “block” schedule that we are using this year, but I also think some of it has to do with the way we are teaching.

Schoenfeld quotes another researcher, Burkhardt, who says that teaching problem solving has its unique challenges for classroom teachers. The first of which is that the teacher “must perceive the implications of the students’ different approaches, whether they may be fruitful and, if not, what might make them so.” This is clearly a challenge for teachers without experience with this type of pedagogy. Not only do you need to be free to pursue incorrect solutions, but you need to reserve judgement, find “kind” ways of telling students they are wrong, control the dynamic of discomfort that students have when the teacher is uncertain, to name a few challenges. During the summer, when we did professional development before adopting this new curriculum, we had days of discussions about all of these issues. However, discussion only gets teachers to a certain level. I firmly believe that the best education is the experience of doing it, while having some observation for another opinion.

Secondly, it’s difficult pedagogically. It’s a BIG change from what most teachers are doing currently. There are so many new decisions that the teacher has to make that he/she may not be trained for. The teacher must “decide when to intervene, and what suggestions will help the students while leaving the solution essentially in their hands, and carry this through for each student, or group of students, in the class”. How do you assess for each student individually when they are solving a problem in a group? How do you give them enough practice in order to allow them to be ready for an assessment? I think that our curriculum makes these goals pretty accessible.  I’ll try to assess that this year in some way.

Thirdly, it is difficult personally.  The implications of putting yourself out there and being vulnerable every day are many.  Burkhardt says “the teacher will often be in the position, unusual for mathematics teachers and uncomfortable for many, of not knowing: to work well without knowing all the answers requires experience, confidence and self-awareness.”  I believe that these are just the tip of the iceberg of the personal implications of teaching with PBL.  One of my colleagues and I got in a conversation recently about how we are so surprised at how much the students don’t know and are afraid to try new things.  My take on it is that not only might this be the first time that the students are being asked to think this way, but we are deliberately asking them to think this way.  In any other type of teaching, these issues might not arise because the students are not being asked to hunt down the prior knowledge of their past algebra skills or to be invested in the learning process, as we are asking them to do.  I feel that we are noticing these weaknesses mostly because we are forcing these issues with our students.  Instead of getting down on them and ourselves, we should feel proud that they are accessing and practicing these skills.

Does this make it any easier?  Not really.   Is this a reason to not do it?  Definitely not. In the article that I wrote called “Transition to a Problem-Solving Curriculum” I commented that teaching this way requires a great deal from the teacher including more patience, more insight, more flexibility and more mathematical knowledge.  This is a lot to ask of our already overworked and underpaid faculty.  But, as Schoenfeld says, true problem solving is “far more rewarding, when achieved, than the pale imitations of it in most of today’s curricula.”

I just “learned” last week in a graduate course that I’m taking, that some researchers believe that it is impossible to teach problem-solving. They say that it is an innate quality that students either have or don’t have. Well, what the hell…I better give up now.

I also “learned” that comprehension has to procede production..so does this mean that in order to produce a valid solution to a problem you have to understand how to do it first? Hmm, well, I have students every day coming up with ideas that I don’t necessary teach them and they have pretty good production, I think.

I just read in a chapter of a book by Collins, Brown and Newman (see below for citing) that the reason why teaching problem-solving is hard is because it “requires the externalization of processes that are usually carried out internally”. You can imagine how tough that is. It’s tough enough for a teacher to make adjustments to a skill like knowing how to write the equation of a line that they can see and observe. At the same time, students can have a hard time seeing an example of what they should be doing to problem-solve as opposed to seeing how to write the equation of a line. Collins, Brown and Newman say that “this externalization (can be) accomplished through discussion, alternation of teacher and learner roles and group problem solving” (p.458).

This makes me reflect on how I might be modeling problem-solving in class. In order for the kids to be able to follow a model, there has to be something there for them to watch and mimic, right? I think what I try to do sometimes is ask questions of the kids, and try to get them to answer me in moments when I really know what to do next, but want someone to help out. I know that I am not always right, so that at least models it. I also think that I do not hesitate to attack a difficult problem. I tell them, do like Ed Burger tells you to do. He came to our school last year and told the students “Do you know what to do when you get a math problem you can’t do? (Pause) Don’t do it!” (Cheers from the audience) “No, no, not what I meant, you do an easier one.” His message was that when presented with a problem that you can’t do, make it simpler by seeing what you CAN do. I thought this was a hugely important message for him to send to the students. It really speak to the message of empowerment and prior knowledge. This is something I’m still working on how to convey to students.

Collins, Brown & Newman, (1989). Cognitive apprenticeship: Teaching the crafts of reading,writing, and mathematics. In Resnick, L & GLaser, R (Eds). Knowing, learning, and instruction: Essays in honor of Robert Glaser. (pp/453-494), Hillsdale, NJ:Lawrence Erlbaum Associates, Inc.

It is always interesting to hear from parents of students who are in a PBL classroom. There are some schools who adopt PBL curricula for all of their mathematics courses, and when that happens, it’s easy to say “this is the way we teach here” so the parents don’t have a lot of choice (if they want their students as this school). In some ways, I think it might be able to work at public schools, in the case where there is buy-in by the faculty and also only if their standardized test scores remain, or become, high. My experience is only in private, independent schools where parents are paying a great deal of money to have their children educated. Wouldn’t that mean that we as those teachers have a responsibility to researching the best methods available for teaching our subjects?

This past weekend was our school’s “Parents’ Days” where parents come and sit in on classes with their children and then have conferences with teachers on Saturday. I have to admit I went into Parents’ Days with a sense of trepidation. I knew that some parents had already voiced their conerns to the head and had prepared my teachers in order for them to be ready. Interestingly, many parents who observed my class were pleasantly surprised. What they saw was students volunteering their work on the board, presenting solutions that weren’t always right, and cooperatively working on problems together. I was relieved when my class went smoothly and the girls were positive throughout the class.

One parent who had observed my class on Friday came for a conference on Saturday.  This mother had written to the head of school a few weeks earlier with grave concerns about how her daughter was learning math this year.  The mother had had an experience in elementary school where they were supposed to be learning from problems, but it seems it was not a good experience.  She admitted that she had felt as if she had lost a year of math.  When she came into my office, she said “by this time you probably know that I have written to the head about this course.”  I said, yes I knew about that and I was hoping that I might be able to help her better understand the PBL methods and make her feel a bit more comfortable with it.  Before I had a chance to move into my diatrabe of defense against a doubter, the mother stated, “Well, after the class yesterday, I really don’t need you to do that.”  It seems that she clearly saw the serious mathematics that was going on and felt much better about the method.  This was such an encouraging moment for me – I wanted to hug her!  There were many other interactions that were very positive for me.

One of my colleagues had a very negative conference with a parent who told her that “some of us are building up momentum to talk to the head” about their concerns, but it seems strange that they would wait until after the conferences to say something.  I doubt that we are out of the woods by any stretch of the imagination, but just getting through this weekend with such positive feedback in the majority was truly encouraging for the success of this program.

Well, it was an interesting meeting. When PBL is discussed in such a cut and dry way, it’s really hard to argue with. The presentation went smoothly with a number of questions afterwards. I felt like people listened, but I’m afraid that they brushed off a great deal of the philosophy because it is so abstract. Many advisors had good questions though about assessment, handling different types of learners and the such. This coming weekend is our parents’ weekend and I needed to give them a reason to relax and things to say to parents if they have concerns.

I think the hardest thing about talking to the whole faculty at our school about PBL, is I’m always afraid that externally they support it, but internally they really haven’t bought into it. Most tradtionalists really believe that math should be taught in a way that allows for exact, direct instruction. One parent wrote me an email and said that this type of thinking was too much to ask of teenagers. I guess my answer to that is that they have been asked to think creatively and critically in English and History courses so far. Why is it different for math? I think that people are in the habit of the way they have learned all their life. It really is a cultural problem that it is acceptable that people have problems with math. I mean, all over the country there are programs to help illiterate people – Literacy Volunteers of America, etc. When is the last time we heard about a volunteer group that makes sure that people know basic math? It is unheard of (and quite the social stigma) to not be able to read, but is strangely accepted if people did not understand math. One of my fellow colleauges even asked how it was that we could require our students to be able to retrieve prior knowledge in such a random way. I really thought that was a strange question. Yes, this person was speaking as an adult who is much farther removed from high school math, but at the same time, hopefully she has been using some of the skills she learned in math class all her life? If not, what a pity.

I’ve uploaded the powerpoint presentation that I used at the meeting this week. Hopefully it will be heplful to someone.PBL Explanation and Justification

Well, next week I am addressing our whole faculty at a meeting on the issue of why we changed our geometry curriculum to a PBL curriculum. I’m trying to decide whether or not to get technical with them or just to basically give them very basic ideas. I could easily talk for an hour just about the pedagogical reasons behind the instructional design of PBL. I could talk for another hour about Jo Boaler’s research and why PBL seems to work best for girls. I can give them tons of tables from my graduate classes that describe the difference between a recipient learner and the low level thought processes that go with that, and the self-regulated learner and the high-level processes that go with that? Do my colleagues want to hear all of this? Well, guess what, I have the floor so I can talk about what I want to!
So what I’ve done is started to compile a powerpoint presentation that does include all of the background information. I’m going to start with the ideas behind PBL, then move to the advantages it gives for girls’ learning. I’m going to talk about how we changed the problems and assessment strategies. How does this work for girls who might not conisder themselves good at math?

Finally, I think I’m going to give each faculty member who wants one, a hand-out with bullet points to discuss with visiting parents on parents’ weekend just in case there are some that are concerned about that. I’m working on another post right now that includes the ideas of parents’ concerns. Hopefully, the presentation will go well next week and the value of this type of teaching might be shared.

I continue to feel very lucky that my administration supports this change fully.

So the first graded exercise (aka test) which we are choosing to call “problem sets” this year has happened. many students received grades that were lower than they are used to getting. What are some reasons that this could happen?

first, it is the first test with a new teacher. Generally, every year it takes students a while to figure out what teachers are looking for in their work, what kinds of questions a teacher will ask, and how best to answer them.

second, it is the first test in a course that is supposed to be geometric, but we are teaching it from a very algebraic standpoint. So, although many students go into “geometry” thinking, oh it will be different from algebra – it does not give students who struggled with algebra skills like slope, equations of lines, factoring, etc. any leeway, as traditional geometry texts or courses do.

third, it’s not what they are used to as a “test”. Since we are trying to assess problem solving skills, it seems logical that our graded exercises are not simply just repeating material that they are shown in class. We are trying to assess the growth from a recipient learner to a very self-regulated one. How does one do this in a fair way? One of my colleagues really struggled with what she interpreted as sacrificing her own standards of grading. We had a long talk about what we are trying to assess with problem sets. If our overarching goal is problem solving, then we need to praise risk-taking and attempts at problems that are not totally done right. I feel that no student should earn a failing grade who makes attempts at solving problems. This is a very fine line for some teachers.

I assess weekly with “quick quizzes” which are more to manage students’ daily skill-level of the content discussed that week. Every other week they have a “problem set” that is a larger grade, and whose main goal in assessment is for the students to practice independent problem solvers. Of course, this is frustrating for the girls in the beginning. It is a very different type of “test” than what they are used to.

In general, I do see a direct correlation between participation in discussion and good grades on problem sets AND growth in their independence. I am honestly continually assessing students on a daily basis. We also require journal writing which is reinforcing topics and recording the process of their own problem solving. It is encouraging to see growth, but it is also frustrating to sense some of their continued bad attitude towards me.

This summer two colleagues and myself started a project to change our traditional geometry curriculum to a Problem-Based-Learning Curriculum. We teach at an independent school and the geometry course has mostly 9th and 10th graders in it. We had the support of our school head, and my experience with PBL from a former school that were pushing us forward. I feel that this is a pretty radical change for a lot of people, but do feel that the experience is worth writing about and recording. It is not supported by the whole department.
As the school year started, it was clear that some students were feeling the “uncomfortableness” that almost always comes with trying a PBL curriculum. For many reasons, students end up feeling lost, frustrated, disorganized and without focus. this generally takes a good 2-3 months of working with the kids to get them past the “out of my comfort zone” feeling that they have. Generally, though students are not afraid to talk about their feelings. Interestingly, in my class initially students kept their thoughts to themselves for about a week and a half. There didn’t seem to be much going on in class that they didn’t like, and they seemed to be trying the homework with minimum effort, but trying nonetheless. then it happened….

One Wednesday morning, the dam broke and all of their fears, frustrations, anger and emotions came flooding out. They yelled at me, calmly spoke to me, gave examples of what they needed and more. As I felt my mouth become bone dry, I started giving my usual speech about why I teach this way. Problem-solving is the overarching goal of mathematics education. What I want you to leave this class with is not knowing the pythagorean theorem or how to use it. I want you to leave with the confidence to know that if you don’t know how to solve a problem, you know how to use the resources at your fingertips, and know how to delve into the knowledge that you already have in order to find the best solution that you can. I spoke about the goals of using problem solving skills all throughout your life. They seemed to hear me, the mood of the class changed and things picked up in the rest of the period quickly.

My colleagues also were having similar conversations with their students. But there was a hum beginning on our campus about this change. I heard a few advisors who didn’t know what to say to their advisees who were complaining about it. I felt very frustrated that my colleagues in other departments didn’t feel comfortable coming and asking me about it. I received only one specific email from a parent that was worried about her daughter. There was also an email sent to my head about me and how she was in full support of this change. It is very helpful that she believes in it too. (that’s an understatement)
During the third week of school, I observed my other colleagues and noticed a few things. One of them was at the end of the spectrum where she might have talked a bit too much, but honestly, I thought she was doing a pretty good job. Highlighting the important topics that came up, but letting the kids struggle with the problems together. It seemed very fair and productive. The other colleague was trying to stay out of the conversation a little much. I could see the level of frustration rising in the students as she would ask question after question, but not tell them if they were right or not. I admired what she was trying to do, but in the beginning it was pretty important to earn their trust and confidence in the teacher. So I spoke with her and tried to show her that it was important to speak up a bit more. I think she will work with it a little more. I need to encourage them to come see me teach more too.

I will try to keep up with this blog so that I end up with a record of the successes and failures of this year’s transition – hopefully to say that we will do it again next year.