Category: Problem based learning

A Moment from Class

The other day in my Algebraic Geometry class, we were doing this problem:

An airplane is flying 36,000 feet directly above Lincoln, Nebraska. A little later a plane is flying at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands.

This is one of the original problems from the PEA materials that we use in our PBL curriculum and I love using it for many reasons. This problem is on a page in the book where we are discussing slope and points that are collinear. So many students’ first idea is to think of the rate of change of the plane as it descends – at least that how I expect them to think about it. However, the student in my class who presented this problem, I’ll call her Robin, had a similar algebraic perspective. Robin realized that since the plane dropped 8,000 miles of altitude for every 160 miles across, she could just see how many times she needed to subtract 8,000 from 36,000 in order to get to the ground, then multiply that by 160. This was crystal clear to Robin, but other students were a bit confused.

So Sandy chimed in. Sandy drew a picture where the airplane was at a height of 36,000 feet and proceeded to subtract 8,000 a number of times drawing triangles as she did this. She did this until she got down to 4,000 (which was 4 times of course), and then realized she only needed another half of 8,000, so realized it was a total of 4.5 triangles that would go 8,000 down and 160 across to get down to the ground. So she multiplied 4.5 x 160 which of course was the total distance across the ground or 720 miles. However, this was not the answer that other students got.

So then Noa, who really likes algebra, says, “Isn’t 8,000/160 just the slope of the line?” Many of the other students agree with her and nod their heads. “So I just wrote the equation of the line as y=36,000-50x and graphed it on GeoGebra. Then I just found the x-intercept. But I knew that we were only looking for the distance from Des Moines to the landing point, so I subtracted 160 from 720, so the answer is 560.” This then inspired Sandy and Robin to check if their answers agreed with Noa and it did.

Just then, Anna said, “Can’t you just plug in zero for y in Noa’s equation? Why do you have to find the x-intercept on the graph? I just plugged in zero and solved for x.” Noa replies,” That’s the same thing…” which created a debate about finding x-intercepts of lines. Which then inspired another student to say that she saw it a completely different way and compared to triangles that had the same slope and set up a proportion giving her an equation that said 28,000/x=8,000/160, which of course set off a bunch of students writing other proportions that were also true.

After this discussion died down, and it seemed we had exhausted that problem, Sandy looked thoughtfully at the board and all of the different methods. She said, “That’s really cool. I can’t believe we all looked at it in so many different ways and we were all right.” And just having a student say that in a spontaneous way made the whole discussion worthwhile for me. It was such an amazing moment, that I sat and paused and let them all accept the pride in their own creativity and ability to use their own knowledge to solve the problem the way they saw fit. I was so proud of them.

PBL facilitation from a Yogi’s Perspective

This fall I was asked to do a small workshop for my department about PBL since almost everyone will be teaching a course that has some component of problem-based learning involved in it. I think for some department members it was somewhat daunting, but I had so much respect for those who were trying something new. It takes a lot of courage to step out of your comfort zone – especially in your own classroom.

I don’t think my professor, Carol Rodgers, would mind me borrowing her yoga metaphor and adapting it to PBL. I use it often when talking to teachers who are nervous about falling short of their ideal classroom situation or teaching behaviors. I think this can happen often, especially when learning best practices for a new technique like facilitating PBL. There are so many things to remember to try to practice at your best. Be cognizant of how much time you are talking, try to scaffold instead of tell, encourage student to student interaction, turn the questions back onto the students, etc. It really can be a bit overwhelming to expect yourself to live up to the ideal PBL facilitator.

However, it is at these times that I turn to Carol’s yoga metaphor. She says that in the practice of yoga there are all of these ideal poses that you are supposed to be able to attain. You strive to get your arms, legs and back in just the right position, just the right breathing rhythm, just the right posture. But in reality, that’s what you’re really doing – just trying. The ideal is this goal that you’re aiming for. Just like our ideal classroom. I go in everyday with the picture in my head of what I would want to happen – have the students construct the knowledge as a social community without hierarchy in the authority where everyone’s voice is heard. Does that happen for me every day? Heck no. I move the conversation in that direction, I do everything in my power for that to happen, but sometimes those poses just don’t come. Maybe I just wasn’t flexible enough that day, or maybe the students weren’t flexible enough, maybe we didn’t warm up enough, or the breathing wasn’t right. It just wan’t meant to be. I have exercises to help me attain the goal and I get closer with experience. That’s all I can hope for.

So I tell my colleagues who are just starting out – give yourself a break, be happy for the days you do a nearly perfect downward facing dog, but be kind to yourself on the days when you just fall on your butt from tree pose. We are all just trying to reach that ideal, and we keep it in mind all the time.

Thinking about the Oil Spill in the Gulf and PBL

I spent a wonderful week on Long Beach Island with my family relaxing and enjoying the waves. It made me think deeply about the effects of the BP oil spill in the Gulf of Mexico and the news reports of the large balls of tar that were showing up on some beaches in Texas. So sad – all of the time that was going by while BP employees tried to trouble shoot and problem solve. Now, I don’t claim to fully understand what went wrong and what they were trying to do to solve it, but it definitely wasn’t something that was an easy fix – that’s for sure. Talking to an administrator at my school this morning about the course evaluations for our geometry class last year helped me make the connection between the BP problem and PBL.

This made me think to myself – I sure hope they had some creative problem solvers working for BP, potentially some people who had had an education that had taught them somewhere along the lines how to think creatively, analytically and had given them some problems that they hadn’t seen before. Clearly, just practicing the same problems over and over again wasn’t going to help these engineers and team leaders come up with a solution to making that leak stop. Giving them homework where they had to just repeat what the teacher did in class each night wasn’t going to allow them to think about what they were able to do in a situation where no one else could think of an answer. Somewhere in their learning they needed to be able to practice the art of looking at a problem that they couldn’t solve and asking themselves what do I know from the past, how can I apply it to this current problem, what resources do I have to solve it now, how can I get others to communicate with me to work together and what can I learn from this situation?

Interestingly, many of the responses on our course evaluations stated that students, even though they enjoyed the course, still craved direct instruction. I am not surprised because of the habits of mind that have been embedded in our students in the U.S. and their beliefs about success in academia. It is important however, to continue to make students feel comfortable enough by summarizing topics, questioing students who make unclear statement and making sure there is clarity at the end of a discussion. Creating environments where students feel comfortable to see uncertainty as permissible in learning mathematics and problem solving is extremely important. The more we accept this in the classroom on a regular basis, the better our students will become at seeing problems openly and patiently and before we know it, they will become better at open-ended problems and problems they have not seen before. Perhaps in turn this will improve the professionals out there who are solving todays problems, and hopefully the next time a crisis happens it won’t take 52 days to come up with something to make it stop.

Patient Problem-Solving

I recently saw a wonderful TED talk online by a teacher named Dan Meyer. Here is the link to the video http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html
I highly recommend watching it because Dan talks about something he calls “patient problem solving” which unfortunately, he claims that today’s textbooks and curricula do not teach towards. After you watch this, I’d love to hear from people to start a discussion of what they think of his thoughts. In many ways what he’s talking about in rewriting the problems from textbooks (or writing your own) is basically what my colleagues and I have done, but not as technologically savvy as he has become. The goals, however, are the same. To not be as “cookie-cutter” with how we teach problem solving in mathematics. I’d actually love to see him in action sometime.

As the year winds down, I am getting geared up for the PEA conference and looking at my class evaluations from my geometry course. Summarizing student thoughts are always the way that I look at ways to improve this course for the fall. I would be happy to share all of that with you this summer. Talk to you soon! Enjoy the video.

Reality Check

On Friday, I spent the day at Harvard Graduate School of Education, at a conference where many graduate students were presenting on recent research that they have been doing – either for their dissertaion work or for courses they are taking, etc. I observed many students from other schools presenting their work which ranged from highly interesting to highly esoteric. I was excited to hear positive feedback from my research on dialogue in the PBL classroom. Many people felt that it was important work and well done. Very nice to feel validated in that way. I also heard great reseach being done on discipline in urban schools, learning differences in math education and other areas.

I’ve also had this website go public, so hopefully if you are reading this you have either received my invitation email, or have heard from a friend about my website. My hope is to really create an online community where those educators who have been in touch with me with interest in these areas can have a space for the types of discussions that we are all interested in. I think this might be a good start and hopefully it will take off soon.

Winter into spring

Hello reader – hopefully you have found this site easily. I am writing here at the beginning of March, as I sit down to fulfill about 6 goals written down on purple post-its stuck to my computer. I have many issues that I would like to address here in my first blog entry, but not knowing the issues that are on the readers minds is of concern to me. I would love to hear from people that I haven’t heard from in a long time – those from my summer courses, or those I have seen in workshops, etc. If you have thoughts, please let me know.

I will be creating forums based on topics that people inquire about or have interest in. My goal is to have this site become a resource for teachers who are interested in PBL or transitioning their school to PBL curriculum.

So for now, I am looking forward to seeing how my school will do this spring with the advent of our new Algebra II/Trig PBL curriculum. This will be the first semester that we utilize the trigonometry portion that we wrote last summer and it will be interesting to see how it goes. The students with which we will be using this curriculum did go through our geometry curriculum, so they do know what they are getting into. I wonder how they will react.

Hopefully, with spring will come new life, goals fulfilled (including my paper getting out to a journal), and new challenges faced as well. If you have a new challenge that you are facing in the utilization of PBL curriculum or movement towards it, please let me know!