Category: Problem based learning

To Kahn or not to Kahn in PBL

Recently there have been some discussions going around the Internet concerning Kahn Academy and other Internet-based “teaching tools” and their applicability or acceptability in terms of pedagogically sound classroom use. You can check out Dan Meyer’s blog or tweets about the MTT2k project, which I find pretty amusing actually, or Kate Nowak’s blog entry where she stated “Enduring learning requires productive struggle and time to noodle out unfamiliar problems, posed by a teacher who knows what you’re ready for, and can provide expert scaffolding. Lecture-only instruction focused on mastering procedures is an impoverished substitute for doing mathematics, and it doesn’t matter if that lecture is in person or in a video.” To that, I, of course, say, “here, here.” I spent some time going over the Kahn Academy website this past spring when my son was having some trouble studying for his science final exam and he was looking for some review materials and I actually thought it was something of a helpful resource for him. However, I’m not quite sure that it would’ve been a helpful way for him to have learned about genetics the first time around.

On the website, Kahn Academy has a great mission of having open-source curriculum for everyone, everywhere, which I am wholeheartedly in favor of. I believe that education needs to be the great equalizer and one of the best ways to do that is to actually allow everyone equal access to the same quality of education. However, they also seem to take pride in the fact that there are now “5th graders relentlessly tackling college-level math to earn Khan Academy badges” perhaps at the detriment of their understanding or even at the skills that they should be learning at their grade level (and I am definitely not against kids exploring interesting advanced topics or even discussing non-Euclidean geometry before they get to college, for example). So it’s important for there to be balance, as I always say, between content and process.

So overall, I would say, I have no problem with Kahn Academy’s (or any online institution of learning’s) pronouncements that they are helping to “spread the wealth” of education, but I do wonder about the quality of the instruction. They have some very, very smart people working there with very good goals about making education accessible, with which I totally agree and for that I commend them. However, there are lots of theories of education – both online and face-to-face that need to be considered in order to claim that any actual learning (whatever definition of that you are also claiming) is actually happening.

Before educators who are within F2F classrooms move to using online tools to “flip” classrooms in order to substitute for other methods of instruction and claim to be using Project-based or Problem-Based Learning, I encourage everyone to really explore the pedagogical methods of that online tool. Is it congruent to what you would do in the classroom? Does it actually help facilitate the type of learning you would want your students to experience? Does it ask the questions or help with the explorations that you would want them to grapple with themselves? Do they get to the confident explanation and security in the knowledge that they would in a discussion? If not, look for something else. Or even better, ask the questions or pose the problems yourself or get the students to ask each other.

Parent resistance to PBL

I have been having a fabulous week here at the Phillips Exeter Academy Math, Science and Technology conference in New Hampshire. I have two sections of my course running on “Moving Forward with Problem-Based Learning” and it really seems like there’s a huge increase in the interest in the Exeter Teaching Materials here this year. Many teachers are finding ways to talk about the curriculum and pedagogy and learn as much as they can while they are here. I am impressed with the depth of the questions and the way people are sharing ideas and experiences.

In my course, some of the participants were sharing stories about how students and parents were resistant initially and continually to the idea of having PBL in their classrooms, specifically with the Exeter materials. I asked the class, “Why do you think parents are so emotionally invested in the way their children learn math?” For a few minutes many of them looked at me and really thought about it. It does seem that parents care deeply about their kids’ education or they wouldn’t even be saying something to the teacher or administrator. Or maybe it’s that they care about the grade – getting into college is a huge process this days, so every little thing helps and they may see this as an obstacle instead of something positive. Others even replied that the parents are frustrated that the learning isn’t clear and they don’t really understand what’s going on in the classroom.

I believe that is one of the closest reasons. It seems to me that parents want their children to learn the way that they learned in school. It is how they remember their experience and how they feel comfortable. It is safe – if they do poorly, they know why (or at least can claim they now why), if they do well, they can be proud and know why they did well. However, in this type of pedagogy, it is clear that parents do not have any frame of reference for how they can assess their child’s work.

I believe the basis of this is the fact that the traditional mathematics classroom in the U.S. is based on a very disconnected learning system. Students are in the classroom in order to listen to the teacher and make sure they understand. Any responsibility for the learning of their classmates is nonexistent and definitely not assumed. Any questions that the student asks when raising her hand is to clarify her own understanding generally. You are necessarily isolated from the rest of the class and by sitting passively and being a receiving learner, there is no mutual relationship with the teacher. In a PBL classroom, the whole way of learning is based on relationships – relationships between students, between the student and the material, the student and the teacher and the teacher and the material. If these relationships aren’t open and mutual, not much learning will occur. Responsibility for learning is everyone’s and the need for active engagement by all is paramount.

This type of classroom is so foreign to parents it scares them. The idea of placing so much of the success of learning on the relationships in the classroom is extremely foreign in a mathematics class. Are teachers capable of this? Are students able to make this switch? No wonder they are freaking out? Some students remember with fear the idea of speaking up in a mathematics class and being wrong and how embarrassed or humiliated it made them feel. One student that I interviewed relayed to me, in a sarcastic tone something like,”Sure we could ask questions, but they had to be her questions. The one she knew the answers to in order to move on, not the ones we really wanted to ask.” It is pretty clear that the culture clash of what society is afraid of and what must be done in mathematics classes needs to be addressed.

On Monday,Grant Wiggins spoke about what he thought needed to change in mathematics education in the U.S. He actually cited PBL as something that would push us in the right direction of having students practice and be more prepared to be better problem solvers in their life. Although I may not have agreed with all that Mr. Wiggins had to say that evening, I do believe that he was a great supporter of reform and the PBL curriculum that should prove to be one way that teachers can move students into being more empowered in their own learning in math.

I plan to spend the rest of the summer hopefully finishing up my dissertation and submitting in September. Any well wishes would be appreciated!

Worked examples in PBL?

Apologies for not writing in so long. Transitions can be hard and making my way through a new school, new place to live and way too many other changes in my life have caused me to put this blog lowest on the priority list. However, in preparing for my course at the PEA conference next week, I came across a wonderful article in the Interdisciplinary Journal for Problem-Based Learning by David Jonassen at the Univesity of Missouri. I have meant to write about this for quite a while because he write about “Supporting Problem Solving in PBL” and has created this great structure that describes the components and scaffolds for PBL environments. It kind of shows the different parts of a PBL environment and the cognitive ways in which students move through the environment focused on the problem, but how they navigate learning. One of the pieces that he talks about that was in this environment was “worked examples.” When I first read this, I was aghast and totally taken aback. How could this be? Why would someone propose that worked examples are part of PBL and isn’t this totally against everything that PBL stands for? I mean, why not just go back to direct instruction and have the teacher standing up at the board working examples for the whole class? OK, that’s a little extreme. I think I over-reacted and went to Jonassen’s part of the article where talks about this.

He states “The most common method for supporting schema construction is the worked example” which I believe is him stating that from a cognitive standpoint the “most common” way for a student to build a schema of seeing how to do something is for someone to show them how to do it. I do believe there is research that shows this. However, I do not believe that there is research that shows that this is the best way for the to construct a schema for understanding. In fact, Jonassen goes on to say that “worked examples should break down complex solutions into smaller meaningful solution elements, present multiple examples in multiple modalities for each kind of problem, emphasize the conceptual structure of the problem vary formats within problem types, and signal the deep structure of the problem…It is doubtful that worked examples are effectively applicable to very ill-structured problems. How can you model a solution that is unknown?” So if teachers did what he says here, they would work examples in many modalities – all representations so that the conceptual connections could be made and not just the procedural ones. Students would be able to see the bigger picture of mathematics more often because they would realize that in the real world they are not given problems that are repetitive and exactly like one that has already been worked out for them. They would be prepared for the fact that they could be given a problem that may have a solution that is unknown. We have a responsibility as teachers to prepare them for that possibility.

So I believe that what Jonassen is saying here is, don’t be afraid to, if necessary, stand up and work out a problem for a student if they are confused and ask you to show them something. That is part of your job to clarify a question or a process for them. However, it should not be part of the plan. Be open to different modalities of representing the work – not all students think the same and they need to see the differences in order to understand the full concept to big picture overall anyway. Limiting yourself to one way of working out an example is not helping them in anyway. So allowing yourself to be open to helping them “work it out” is really the best way to handle it for everyone.

Thanks for a great week!

Thanks to everyone who was in my PBL class this week. I had a wonderful time at the Anja S. Greer PEA Math Conference and met lots of wonderful people. For those of you in my week-long class, please feel free to fill out the course evaluation at:

Schettino Course Evaluation

So many teachers that I’ve met were extremely inspiring – As usual I learned so much from everyone about new ways to view technology, certain types of curriculum, ways to incorporate different topics in the classroom and even how to do a Rubik’s cube. I appreciate this converence so much and keep coming back every year. Thanks again to everyone. Special thank you goes out to Ron Lancaster for his special gift of the DVD movie version of The Housekeeper and the Professor which is a wonderful story of relationship through mathematics and creativity. I highly recommend it. Thanks so much Ron!

Many thanks

I have returned from my trip to Indianapolis and I would like to thank everyone that I met there for turning out to both of the presentations that I gave. The talk I gave on Saturday, which was with my colleague was more about our Problem-Based Learning curriculum at our school. The turn-out there was amazing and we were so impressed with the questions and comments from the group. Some of the feedback was great food for thought, especially some specific questions about our definition of PBL. It was also extremely useful to hear what teachers would feel are the challenges of implementing PBL in the classroom. I would direct some teachers to the blog of a teacher in Massachusetts named Mark Vasicek who has attempted to you PBL pretty consistently for a number of years West Side Geometry. It’s good reading.

I think that one of the reasons that so many people might be interested in even thinking about changing the way that they teach right not is because of recent work through the CCSS. There were so many opportunities at this conference to read about, talk about and learn about the details of the Common Core State Standards that I think by Saturday many people were almost tired of hearing about them. However, PBL definitely directly relates to at least a few of the CCSS standards of Mathematical Practice:

Making sense of problems and persevering in solving them.
Reasoning abstractly and quantitatively.
Constructing arguments and critiquing the reasoning of others.
Looking for and expressing the patterns used in reasoning.

We tried to give examples of how we see these standards coming to life as outcomes in student work on a regular basis in the PBL classroom. Having so many people come up to us afterwards for more information, or with interest in getting in touch was really exciting. Sunshine and I truly hope that you do. I really look forward to it.

Receiving Feedback

I received an email from a colleague a few weeks ago, that was amazingly touching. She had been meeting with an advisee and asked the thoughtful question, “Can you think of a course or a moment that changed your academic experience in a significant way.” One would think that most high school sophomores would either take that question lightly, or would at least need to pause and reflect on the depth of this question. My colleague said that her advisee responded without hesitation, “Geometry last year. I hated it at first because I couldn’t do what I had always done and do well. But by the beginning of the second semester, I had started to figure out what it was all about. And this year in my other classes, like English and history, I’m THINKING better, I’m analyzing differently because of that Geometry class.” This student seemed to be able to connect improvement in her critical thinking skills in other disciplines to the work she had done in her problem-based learning class. Did she have empirical evidence that this was the cause? Of course not, but something in her intuition and learning process was telling her that the struggle she had undergone to move through a course that challenged her in so many different ways, allowed her to grow intellectually like no other course had. For this student to even recognize this was very mature, and for her to attribute her success and skill in other courses to the learning that had occurred in this course was remarkable.

Throughout the year when teaching with PBL, I struggle with the comments I receive from students. I often wonder when they ask me to go up to the board and give more direct notes, “Why don’t I just go up and make them happy” instead of asking another scaffolding question? Why do I continue to push them out of their comfort zone and let them sit with the unknown for just that extra amount of time grappling with their peers, instead of relieving the tension and anxiety by giving them want they desire? And then a moment comes when they realize something on their own, and clarity, true understanding takes over. It is a moment of true joy on both our parts, because I know not only did they come to that understanding through their own agency and empowerment, but the way they came to it has allowed them to be the relievers of their own anxiety. They have transformed their vision of what is possible in coming to make meaning in mathematics. It is in that moment that we are together changing their understanding of what a mathematics classroom is, just a little more.

And then the next week, they are back to asking me for more direct instruction, while at the same time there is just a glimmer of increased curiosity and I know that by the end of the year, there will be a changed student sitting in front of me who may have a different belief system about learning mathematics.

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.