Author: Carmel

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.

Asking “True Questions”

The school year is upon us and it is with great excitement that I look toward this new school year. I just sent off the first draft of my dissertation proposal, we just got our edited version of geometry text back from the printers, and our newly edited trigonometry text is hot off the presses as well. It always seems like there is a fresh energy surrounding the beginning of the school year that emanates from the students and is just contagious – even after twenty years of teaching.

I recently read a great essay entitled “Questioning Authority” by Charles Bingham in his book about relational authority (SUNY Press). In it, he discusses Gadamer’s view on questioning as an act that “breaks open the being of the object” and “revealing the questionability of what is questioned.” Basically, Gadamer believes that a true question is one that is asked only when the answer is not already determined – it may “sow the seeds of its own answer” but the questioner is really asking about something she truly doesn’t understand. So the philiosophical question that Bingham poses is just this – can a teacher ask a “true” question? It seems that for the most part, in a mathematics class teachers, including myself, generally ask questions that they previously know the answer to and have an expectation of a response. This is really how we teach and make sure that we are fulfilling our curriculum and students are “learning” what we need them to learn.

However, Bingham posits that we can ask “true” questions if the experience of the students is that they are “true.” In other words, one must look at the intention of the question posed by the teacher – if the teacher hopes to “break open” the subject matter as can happen with a true question, and if it experienced that way by the students in the classroom, then in the students’ expeirence it is a true question. Math teachers can ask questions in a true way, questions like why did you choose that method to use? what other problems does this remind you of? Do you prefer this method or that solution? Describe your reasoning process for the rest of us here. Why does that prove your statement? What’s the contradiction in what you just said? and many others.

Think how differently the teacher-student dialogue would be if math teachers posed true questions throughout the class period. Today in a faculty meeting, an English teacher colleague of mine had us all read an Emily Dickinson poem (much to my dismay) and then everyone was able to raise their hand and say one part of the poem that was confusing to them. This was totally acceptable that without any explanation of poetry or review of Dickinson’s works, we were able to creatively and openly express our confusion and process of “not-knowing.” I wondered how acceptable this would be to my colleagues in a mathematics lesson. How many of them would’ve expected a review of some algebra and then an explanation of the problem before questions were allowed. I then wondered how different our education system would be if the questioning process in mathematics classes was more like that in the English class.

Great International Community

I just returned from a wonderful conference at Ithaca College that was the First North American Conference on GeoGebra, a wonderful Open Source Dynamic Geometry Software package that you can find online at www.geogebra.org. It has been around for a while, but it is only just getting well-known because it is free! It is also amazing because it is written in Java, so that you can easily export the ggb files and put them on web pages, so all you need to run them is a web browser. I am trying to start putting some on my web page, so if you have time, look under the technology page and check in there. I made one that does a wonderful square to rectangle puzzle with a friend named Mark Sawula and an expert named Mike May. I also did another that shows that the sum of the residuals of a data set to the best fit line is zero – you can move the points around and everything.

Anyway, the best part about this conference was how wonderful and open all of the attendees were. It was probably the first conference where the researchers, teachers, professors, graduate students and developers of the software were all working together as equals and I loved the atmosphere. I thought it was amazing, and highly recommend the conference next June 2011 to everyone. Keep you eye out for announcements.

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.

Thank you! Thank you!

What an amazing week I’ve spent in Exeter at the conference. I have met some truly wonderful people, who I feel lucky to call my colleagues in mathematics education. I’ll share some great highlights here.

Monday night we heard Steven Strogatz speak – author of the Calculus of Friendship and honored professor of Applied mathematics at Cornell University. What a touching story (and speaker!) about mentoring, teaching and the reciprocity of learning – I highly recommend the book to any math teacher as it truly captures why we do what we do. There was not a dry eye in the place. At one point I was able to get him to sign my book and he even inquired about what PBL was all about.

Yesterday afternoon I held a Special Interest Group for teachers and others who wanted to discuss issues related to gender equity for girls in STEM fields. I had been surprised to see that the conference director had organized it to be a double session (two hours long), but said, what the heck, we can always end early if people start leaving. Boy, was I surprised when about 20 people came at the beginning and stayed for the whole two hours. We had an amazing and passionate conversation that ranged from questions about best practices for teaching girls in the classroom to sharing personal stories and experiences from science and math education from the participants. I was most surprised to hear from the youngest women educators that biased practices still go on in academia to dissuade women from moving on in advanced degrees.

Everyone’s contributions were so important and I want to thank everyone that came to that session. I learned so much from that discussion, it might have been the highlight of the week for me. (Especially seeing the male teachers that came!) I would reiterate my statement of how important it is to share with our colleagues the message of encouraging girls and merely stating that the problem still exists out there. Creating mentoring programs in our schools is another great idea, as well as showing by role modeling that mathematical women can be strong and feminine.

This may have been the best PEA ASG conference I’ve attended and want to thank all of my own mentors that were here – so many people to be grateful for. Probably the most important is the woman after whom the conference was named – Anjs S. Greer who hired me at Exeter in 1995 – she changed my life and helped make me the educator I am today. I am forever in her debt and continue her work.

Enjoy your summer and thanks again for a great week!

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.

Problem-Based Assessment?

This time of year always has me and my colleagues thinking about the way we cumulatively assess our students in the PBL courses. Since cooperation and problem solving are so important to us we are committed to having some type of collaborative experience as part of the final “exam” but we also have a committment to assessing the content knowledge as well. It’s a difficult thing to balance though, as many of you know. However, since we have come to the decision that this course is really about depth and not breadth, we do not believe that the main focus should be on testing memorization of fact or to see if they can do “one of every problem” that we covered this year.

We have split our two hour final “exam” into a group problem solving experience from which the students receive part of their exam grade (and they turn in one paper as a group). The rest of the exam grade comes from an individual 40-minute performance they do on their own. In general, the grades come out very much as we expect for individual students. During the group problem solving part (approximately one hour and 20 minutes), the teachers walk around and observe the interactions in the groups – who’s leading, who’s following, who’s just sitting back and others do the work. This factors into their grade. We also allow students to give feedback into how the group worked together and for alternative answers if they require it.

If anyone has questions about the way we write questions or would like to talk about assesment in PBL, please post some questions in the Forum about it. Looking forward to doing some more work about assessment in my course at the PEA conference this summer.

The Consequence of Larger Classes

It’s very interesting to see how larger classes affects the effectiveness or productivity of a class that is based in a PBL curriculum. This year my classes are a tiny bit larger than they have been in the past. Generally, the class size has been between 12-15, and this year one of my section is as large as 17. I do think that this affects a students’ ability to get their questions answered, focus on listening to a student description of their process, see which process is correct, take good notes, and even to show respect when others are talking. It is honestly difficult enough to create a sense of community of learners with a small class – but what can you do with a larger class?

In my workshops, I have always promoted group work and discussion about the problems, but I am finding that difficult to manage as a facilitator, and when left to their devices, oftentimes students cannot facilitate a discussion themselves (even the most focused students). What I have found effective is putting them in deliberate pairs at the board to discuss their solutions. Why at the board, you may ask – why not just have them sit at their desks and show their work they did in their notebooks? Their is definitely an advantage to the physical act of getting up out of their chairs and writing (showing) the work again on the board that gives a sense of ownership for the work. It also gives a sense of scholarship when two are working together in discussion at a board (picture Charlie and Amita on Numb3rs working on a problem just writing in down on paper – not as dramatic). Students can sense this themselves. Students being at the board also gives the teachers a broad view of who’s working and who’s not – the work is all public and with a quick scan, you can see what is being done (and more importantly, what is not). In my experience, students have shown a lot of excitement when a solution has come to frution at the board – a lot more than when sitting at desks or the table.

It is the comraderie in a community of learners that grows when working together for a common purpose and there is nothing better than sharing ideas and problem solving that does that in PBL. Of course, you surely need one thing – enough board space!