Category: In the classroom

Doing What You Can

I just got back from a great visit to Toronto (which was also my first visit to that wonderful city.)  I spoke at a conference and also did some work at beautiful girls’ school there that was interested in PBL.  It was the first time where two of my research interests intersected (Gender and PBL) and it was fascinating for all that were involved – quite wonderful and so much fun.

Many teachers talked to me there (and it comes up everywhere I go) about the fact that they are the only teacher, or one of the few, at their school that is interested in  trying out this different method of teaching, but need to keep up with the syllabus that their colleagues are using in order for students to be prepared for the common exam either at the end of the term, the end of the semester or even on a monthly basis.

This can be problematic when there are school districts that dictate down to the homework assignment or classroom activity that you need to be doing on a daily basis.  The free that classroom practitioners need to make decisions about what is best for the learning of their students is quite important.  However, it is still possible to integrate problem solving or methods of PBL into your classroom when you can get.

I talk about the Continuum of PBL when I give workshops to let people know that you don’t have to dive in head first if you want to try your hand at classroom discourse a little at a time.  Learning to facilitate discussion as a long-time direct instruction teacher is actually quite difficult to allowing students to have more authority can be tricky.  Here is the visual aid that I use when I discuss it:

“A Continuum of PBL” cschettino 2013

The arrow tells you the level of decompartmentalization of topics – in other words how the topics are blended together or not.  In a traditional mathematics curriculum, a textbook artificially separates mathematics into what I like to think of a “compartments” that in a very linear order and most students learn to believe that mathematics must be taught in that order.  Yes, some operations and skills must come before others, but conceptually a great deal of mathematics can be learned or thought about in no particular order.  It is all extremely and equally fascinating.  In a whole problem-based curriculum such as what the faculty at Phillips Exeter Academy has written there are no chapters that mark the ending of the content and the beginning of another since there truly no time when that content is no longer applicable to the new material that is being learned (yes, of course that is true in a text book as well, but the chapter alone have come to imply that to students).

Decompartmentalization can come at different levels.  At the lowest level,many  teachers use “Problems of the Day” that challenge students at the beginning of class with logic puzzles, topics they are not seeing regularly or interesting tidbits like soduku or other fun activities to get students’ minds working.  These create discussion and allow them to see problem solving in action.  However, there is little connection to the mathematics that is being learned in the class proper.

I won’t discuss every type of PBL on the continuum or this will turn into one of my hour long talks, but I will say that if you are interested in attempting to keep up with your colleagues who are following a traditional syllabus but you might want to use less direct instruction I have a link to my “Motivational Problems” page in order for you to have them start the conversation based on their prior knowledge of material.  The problems are listed by topic and you can have them move forward in class by presenting problems and then have them practice with problems in the textbook.  Anyone who tries this – I’d love to hear from you!

This was how I first started at my last school and it worked well for me.  You can read my article from 2003 in the Mathematics Teacher to learn what it was like.  But it definitely moved me in the right direction.  Keep pushing on!

An infinite amount of thanks…

Everyone has those mentors in their life who have impacted their work or career in ways that have truly changed who they are.  In my instance, the person I am going to write about not only has impacted my life and career, but because he taught me so much about great teaching, in particular PBL, he has impacted all of the students and teachers I have worked over my twenty year career so far.  So I feel justified in taking a short break from writing strictly about professional educational work musings and just finding a moment to say thanks for the life and work of Rick Parris.

Even if you never met Rick in his time teaching at Phillips Exeter Academy, or used his wonderful opensource Peanut software for windows machines, or downloaded the faculty-authored materials that he was integral in writing by the mathematics department at PEA – if you have worked with me at all, you have been affected by Rick’s work.  Rick Parris had to be one of the most brilliant, efficient, insightful  mathematicians I’ve ever been lucky enough to work with.  He saw things in a problem that I definitely never would be able to see in a million years.  I was so extremely intimidated by him when I first started working in the same department that I would go for days confused about a problem instead of go up and ask him.  But what I soon found was that not only was he one of the most brilliant mathematicians, I’ve ever met, but he was one of the best teachers too.  Now, there is a rare combination – finding someone who has the insightful intelligence to be able to have a Ph.D. in mathematics but to also be so sensitive to others’ understanding of the subject and the patience and passion to want them to love it as much as he did.

I remember finally having the courage to go and ask him a question about a problem in the 41C materials on fall afternoon (mostly because I knew I had to understand it) and he looked at me, with what I thought was a look of disdain or horror that one of his colleagues wouldn’t understand a problem that he wrote.  And just as I was going to run in shame, he said something like, “that is such an interesting way to look at that” and I was amazed at how good that felt.  He entertained my ideas and although I felt like he was initially just appeasing me, I soon realized that he was truly and sincerely intrigued.  Our relationship as colleagues and interested problem solvers grew, even after I left PEA.  He allowed me to keep in touch constantly asking him questions and posing them over email.  He taught me so much about writing great problems, encouraging students to ask great questions and making sure that they always felt like they were they most interesting questions ever.

This past summer, the last time I saw Rick, we were talking about the game of Set (you know that really fun card game with the colors, shapes and numbers).  We were just posing really fun questions like “What’s the maximum number of sets you can get in a 12 card deal?”  We found these types of questions intriguing and even after we parted company we continued emailing with email subject lines like “a baker’s dozen of sets”, “set lore” and “the game of set redux.”  He always treated me like a real mathematician even though he was the one who I saw as my inspiration and motivation in that area.

Rick taught me about how to scaffold problems (not too much) so that students would see their way through a topic and find out exciting ideas of mathematics on their own.  I loved to watch him teach, probably observing his classes three or four times a year in order to gain insight into his questioning methods.  He made a point of trying to hear from every student in the class at least once a class.  I don’t know if he ever knew how much of an impact he had on my teaching and philosophy of learning.  I am so grateful.

So how do you say thank you to someone who pushed you in a direction that changed your life?  I guess I have just to recommit myself to learning about and researching the best practices of inquiry and problem-based learning in secondary mathematics education.  I do believe that the world needs to know about the contributions of this man and the department at PEA because without them and the model that they have created, I’m not sure that many of the schools today that utilize their curriculum would be where they are.  I give thanks to Rick and consider myself extremely lucky to have worked with him and shared his enthusiasm for problems.

A New Year: Setting Up the Dialogue

As the new school year approaches, I’m re-editing, once again, my PBL text that has been a “work in progress” for about seven years now. Every year my colleagues and I at my old school would take the input from our department and the students in the course and improve upon the work. This is what the teachers at Phillips Exeter do every year to their original materials as well. I think the idea of the problem sets being organic and dynamic is really the only way to think about problem-based learning – to believe that you can learn as much from the students and how they view the problem as they can learn from the problems themselves. In fact, while cleaning out some old folders this summer I ran across this quote, which I believe, is from Freire:

“The problem-posing method does not dichotomize the activity of the teacher-student: she is not ‘cognitive’ at one point and ‘narrative’ at another. She is always ‘cognitive,’ whether preparing a project or engaging in dialogue with the students. He does not regard cognizable objects as his private property, but as the object of reflection by himself and the students. The students – no longer docile listeners – are now critical co-investigators in dialogue with the teacher. The teacher presents the material to the students for their consideration, and re-considers his earlier considerations as the students express their own. “

Pretty amazing the way he’s got it right there, I think. That once you put that problem out there, it is no longer yours, but everyone’s to work with and the students need to be part of the responsibility for the learning. It is presented to them “for their consideration” you must reconsider your earlier consideration once they express theirs. That’s the deal you make when you use PBL – that you will do those reconsiderations. It’s part of the pact.

However, the kids need to be part of the pact too. Wait, let me back up. So, I’m sitting at my computer typing and my son, who is going into ninth grade geometry this fall, asks me what I’m doing. I tell him and editing my geometry textbook for my class for the fall. He asks me if he’ll be using a book like that – that’s not a “normal” textbook. I tell him, I don’t think so – I think his school uses a traditional geometry textbook that pretty much will give him direct instruction in the classroom much to my dismay. And he says to me with a sigh of relief, “Phew…well, that’s just fine with me.”

Of course, I’m thinking…whoa, hold the phone. Have I failed as a parent? Have I not instilled any intellectual curiosity in my son at all that he wouldn’t want to have some type of investigation going on in his mathematics classroom? I also had a very interesting experience starting at a new school this past year that traditionally had mathematics classrooms that were taught with direct instruction. It definitely took some time for students to get used to the idea of a teacher that did things a little differently. Student expectations for being “given” knowledge were extremely high and my expectations for them to construct knowledge were extremely high. It was an interesting situation.

Anderson (2005) found that many teachers who taught with PBL-type pedagogies found reluctance and resistance in students for lots of reasons. Even though they enjoyed the classroom more and even learned better in the long run, there a few downsides. Because of the habits of mind that students have formed in traditional classrooms they do not feel they are being “successful” unless the known authority figure (a.k.a. the teacher) is telling them they are right or wrong. The typical “received knower” that many students are in American classrooms today have “grown accustomed to learning in a classroom that required little from [them] in terms of engagement with mathematics” and they find it difficult for themselves to take responsibility and control for their learning in the way PBL asks them to. What can even happen sometimes is that these kids who are resistant can glom onto a student that seems to take on the attributes of the teacher or authority figure (in their perception) and a small group can become a microcosm of the traditional classroom if a teacher is not careful.

However, from working with teachers for many years and my own personal experience, students are actually very adaptable. Spending time in the classroom with this type of learning, students learn to adjust their own expectations and realize how much of a give and take there is – how much support to expect and learn that they are pleasantly surprised by what they can accomplish on their own.

I’d like to think that even my own son would be proud of himself if give the opportunity and I might give him some problems outside of class this year just to see what he does with them. However, it is just that pact that I was referring to before that the students have to buy into. If they don’t do their share and express their considerations on the problems, there is no dialogue to reflect on, there is no sharing that has gone on. And there enlies the rub – you are back to square one with direct instruction. So I’ve told myself that at the beginning of the year this year, I’m making that part perfectly clear that they have just as much say in the dialogue about what we’re learning and I hope they get the point – at least faster than they did last year!

What’s the “P” in PBL?

One of the issues I talk about a lot with people who are interested in Problem-Based Learning is the “continuum” of integration that I use to tell people how they can implement it in their classroom. How do you want to incorporate the teaching with problems in your classroom? Magdalene Lampert wrote a wonderful book called Teaching Problems and the Problems with Teaching in which she chronicled her journey of teaching a fifth grade classroom for a year with problems (it’s an awesome book, BTW). The way in which you use the problems, the pedagogy you use, and the classroom community you set (the lack of hierarchy, the authority you allow the kids to have, the safety of the risk-taking, etc.) is all hugely important parts of the PBL environment. I found another “continuum” created by someone name Peter Skillen and a colleague named Brenda Sherry. Mr. Skillen, a lifelong educator from Toronto, Canada, doesn’t claim to be an expert in PBL, but has extensive experience in the world of education and has great ideas. Check out his blog if you have time.

He created a wonderful Continua to Consider for Effective PBL which I believe is definitely worth sharing. Although his “P” in PBL is Project, I believe his Continua (since there is more than one scale) is just as applicable to Problem-Based Learning. It reminds me a great deal of what I use in terms of implementation. He also has stated that anyone who would like to add categories should feel free, and I might actually work with that. His categories to consider are

Trust
Questioning
Collaboration
Content
Knowledge
Purpose

These are amazing to start off with and I would probably add a few more to those including authority (although, I think this is what he’s getting at with trust and locus of control) and perhaps also change the “collaboration” one a bit. It is pretty tricky – this idea of interdependent, independent or dependent learning – dependent on what? The teacher, other students, a textbook? Very complex ideas at stake here. Different types of PBL are being considered and in different frameworks. But what he and his colleague have put together is amazing start to an important discussion.

In fact, it’s really important to decide what you mean my “PBL” is? Even on the public shared website for the American Education Research Association Special Interest Group for PBL there is a “Statement on Nomenclature” about what PBL might be interpreted as meaning. There is an acceptance that there is more than one, and in fact many, meanings for the acronym “PBL” and what one person thinks it is may not be the same as another. I am very open to the understanding that when some contacts me about their own school’s interest in using PBL, I have many questions for them before we start talking about implementation.

Not to belabor the great article by David Jonassen that was published in the Interdiscipliary Journal of Problem-Based Learning, (see my other blogpost Worked Examples in PBL) but I really like the distinction he makes between Project-Based and Problem-Based Learning. What it comes down to for him really is the authenticity of the problem. It’s not really about how many, what kind, or how big the problems are that you have the students do. What it is about how did you plan (or not plan) the problems. He is calling it the difference between “emergent authenticity” and “preauthentication.” (definitions by Jonassen (2011). Supporting Problem Solving in PBL. Interdisciplinary Journal of Problem Based Learning 5 (2)).

Emergent Authenticity is when “problems occur during practice within a disciplinary field by engaging in activities germane to the field.” In other words, this is more like when you pose a problem to the students that is something that a mathematician might encounter in real life and an answer is truly unknown (like in real life!!) and they are engaging in that activity of not really knowing that answer and grappling with finding the tools and resources that they need to move forward to find a solution. That is when the authenticity of the problem (or project) is actually emerging as authentic.

Preauthentication is “analyzing activity systems and attempting to simulate an authentic problem in a learning environment.” In high school mathematics classes, this when the teacher knows they want their students to learn something specific from engaging in a specific type of problem or series of problems (mostly like what I do in my curriculum, honestly) and they “set up” a problem-solving situation, but make the kids think that it’s novel. The learning experience has already been analyzed by the teacher and the teacher is giving the students the authority to do the problem at their own pace and draw conclusions, struggle on their own. However, there is some control because it is really only a “simulation” and the teacher actually has more information that can be helpful in terms of learning outcomes, etc. The authenticity has already been “preauthenticated” so that it simulates the experiences of a mathematician as much as possible, but still has the learning outcomes, goals or desired content objectives that might need to be fulfilled.

Which is better? I don’t believe there is a “better”. I believe there is what works for your school, pedagogical beliefs, student audience, teaching style, etc. All of these wonderful categories are what must be considered when you and your department start on the journey towards incorporating PBL into your curriculum. There are many great choices to be made, but it is a long journey and cooperation with lots of reflection are definitely needed. So much to consider.

 

Being sterotyped?

My daughter is starting middle school this fall and is extremely anxious about so many things that new sixth graders worry about over the summer. She’s on of those kids, though, that go to extremes – you know the type – the ones who love to shop of school supplies the first week of August, have their planner written out the second week of August, etc. These attributes only endear her to me even more (in fact, she reminds me so much of myself), so to ease her worries we went out to our nearest bookstore and she bought a book entitled “A Smart Girl’s Guide to Starting Middle School”, which I thought would give her something to read and something to help ease her worries. This it definitely did – kudos to the authors and publishers for helping high-anxiety tweens across America ease their worries with some pretty good advice actually on handling the peer pressure, rough schedule and other worries that come with the onset of middle school.

However, as I was reading through this book, I came across a section called “Teacher Types” which I, of course, read with interest. The authors of this guidebook broke teachers down into four different types, but the one I was most struck by was the one entitled “The Freestyle.” The teacher-type was the quintessential “free-love” type teacher stereotyped to let the kids do whatever they want in the classroom. The style is described as this: “Loves to brainstorm. Teaches by asking questions and fielding answers from the class. Explores different ways of doing things.” I stopped reading as a muscle spasm hit my stomach. “Oh my god, that’s me,” I thought. “But I’m not freestyle?” I got very defensive as I read on…”Pros: Lets students work on their own. Gives lots of feedback and support.” Oh, OK, yeah I do that. (feeling a little better) “Cons: “Letting everyone have her say can eat up class time, leaving some material uncovered.”

Once again, my heart started pounding…this is me. My students have never described me this way, (at least to my face) as a “free for all” teacher that let’s the conversation go wherever it may, but I can remember times when students complained that we had to go back to cover something that we did not have time for. I became very defensive of this small book that I felt was stereotyping me in a way that I was not appreciating. However, thinking more about this, I realized that over time (and from observing many teachers that have chosen to either use discovery-based, discussion-based or problem-based learning in their classrooms) it is all too easy to become too “freestyle” and allow the students to have too much independence. From talking to one of the teachers that I have trained over the past 6 years, she said to me, in all honesty, I think it’s harder to teach with PBL than in a lecture classroom because you have to be prepared, but prepared in a very different way. You have to be prepared for the unexpected.

Being “freestyle” does not mean letting the students’ ideas take over. The teacher as the scaffolder of questions must know in their mind the direction in which to move the discussion and when and how to do this without pushing too hard as too ruin the delicate house of cards that is being construction team of students. It may appear “freestyle” because it is not what students are used to, however, the teacher is doing a great deal of work. The other way in which the teacher must do a great deal of work is in what this book does make note of and the teacher must:

1. give lots of feedback on student comments and ideas
2. encourage and value ideas and insights
3. make note of time use and if time is short, change assignments regularly (the internet is great for this)
4. create a safe classroom community for students to share (must be a priority)
5. make summaries of constructed knowledge or have students summarize

If all of this happens, it is difficult for students to take from the classroom that it is “free for all” and they can leave their class for the day feelings content for the moment – and if they don’t for one day, that’s OK too. What I’ve learned from all this is is that the advice in that book pointed out the truth about being “freestyle” in some ways, but that the preconceived notions about that stereotype (or any teacher stereotype) really needs to be viewed from different perspectives. I’d hope that a student in my PBL classroom would at least see it that way.

Wrong is not always bad

I recently read an article in Education Week that was proclaiming the benefits of discussing student mistakes in class. The author, Alina Tugend who has recently published a book entitled, Better by Mistake: The Unexpected Benefits of Being Wrong, cited that in some asian cultures students can be asked to work out “math problems in front of the whole class for a healthy period of time…even if [they] are doing it wrong.” She goes on to discuss that the teacher might ask the student to discuss her thought process and why they chose to do the problem that way and the decisions they made at certain points in order for the class to see the choices that were made at certain crossroads in the problem solving process. Some researchers believe that this type of discussion allows students to help create a sort of “index” of what still needs to be learned or what has already been learned.

In other words, the class is actually viewing the errors and misunderstandings as a helpful thing. They’re using the opportunity of the mistake, of being wrong, as kind of a check point to see what else they need to know. Perhaps someone else in the room might have did something differently that might have led them in a direction that was more fruitful and everyone can learn from that as well. So there is much more to be learned from this type of environment. On the one hand, the students learn the material, but on the other hand, they are learning that they can learn from each other and they learn that being wrong in the first place was actually helpful.

This also goes to the idea of how problem-based learning is ideal for this type of learning. Posing the problem in the context of a prior knowledge base, allows students to think that they have a background that is sufficient for them to do the problem, they just need to recall what that was with a little push. It also fosters was researcher Carol Dweck calls the “Growth Mindset” allowing students to believe that their intelligence and ability to succeed to flexible and not pre-determined.

I am getting excited for my course next week at the Math and Technology Conference in Exeter, NH which I believe is full. I love this conference because I always meet lots of people who are so eager to engage in mathematics and learning. It should be a great time of dialogue and I look forward to a great time!

How do you measure success?

Last week I was being observed by a colleague and my class was doing an exercise in GeoGebra about circles, arcs and inscribed angles. I don’t think I can do the experience I had justice as I try to describe to you what happened in this class, but strangely, I just can’t believe that someone else was there to witness it. Have you ever attempted to scaffold learning in a way such that the questions you asked would move the students forward so that they came to the conclusions themselves? Well, this is what I do everyday in the PBL classroom and sometimes it’s a success, sometimes it’s in between and I do more “telling” than I’d like, but on this day I couldn’t believe what happened.

I had the students construct a circle with a central angle and measure the arc and the angle and see that they had the same angular size. This was no surprise to them. My plan was then for them to extend one side of a radius of the central angle and make an inscribed angle that intercepted the same arc so that they would measure that one and see that it was half the central angle and the arc it intercepted. Often when I do this students don’t understand that the angles intercept the same arc, or something else goes wrong. However, one this day, I wished I had been recording the flow of the conversation that went flawlessly after my simple question, “What do you observe about the two angles?” From one student to the next around the table it went:

“Well, it’s definitely smaller…”
“Mine’s almost ninety and the other one looks like it’s almost 45.”
“Maybe it’s supposed to be half?”
“Yeah because the side is a diameter and the other one’s sides a radius – like it’s in a proportion?”
“No, that can’t happen, you know the angles of a triangle don’t work like that..”
“move it around and see if you can get it to be exactly a half…”

After a while, they all agree that it seems like the inscribed angle is half the intercepted arc. So I prompt them again,”So why do you think it might be exactly half? Is there any relationship between these two angles that might make it that way?”

“Is it like because of the midsegment theorem? The radius is half the diameter so it makes them parallel?”
“Well, it doesn’t seem like the other one is parallel though…”
“It seems like the other central angle next to that one adds up to 180 with the one that intercepts the arc.”
“Hey wasn’t there a theorem about that?”
“Oh my gosh, yes”
“It was like about outside angles or something like that being …like if you add the two inside you get the one outside”
“Oh I see, the triangle on the other side is an Isosceles triangle because it’s a circle..”

At this point, I almost freaked out because I hadn’t said anything in almost 15 minutes or so, they weren’t doing it all themselves and almost every student (except maybe 2 or 3, who were still engaged at least) had contributed something to the conversation. I mean, I was in teacher ecstasy, and to top it all off, I had somewhere there to see it all. I couldn’t believe it. Besides that I had been having a really bad day, and this just turned it all around. I don’t know if this was all a function of the practice of PBL, or a function of the kids in the class, but it was truly amazing. I heard at least three of students leaving class that day say “that was a great class!”, what more could I ask for?

Modeling Proper Mistake-Making

Whether it be a small arithmetic error, or correcting a student when they were actually doing something right, we always make mistakes in class. The other day, I wrote the parametrization of the unit circle as x=cos(t) and y=sin(t) and took the derivatives as dx/dt=sin(t) and dy/dt=cos(t). It wasn’t until a student humbly interrupted me saying, “Um, Ms. Schettino, don’t you mean -sin(t)?” and I looked at it for a little while, trying to figure out what she was saying, and then I realized what I had just done. I knew I was calculating the arclength, using the arclength formula, and that I was going to square it anyway, so I just left off the sign, but they didn’t know that – skipping steps in my head is a really bad habit. So I said, “Yes, oh yes, sorry – thanks so much for fixing that for me.”

I do things like this all the time, and hopefully, I am a big enough person to admit my mistakes and give students credit for finding mine, especially if it might affect some students’ understanding. I was talking to a colleague about this with respect to PBL the other day. I asked her why it’s so important to her to admit to students when she makes mistakes and to fix them in front of the students. She told me that she likes to “model proper mistake-making” for her classes so that when they do it, they can see what she does and use the same humor, self-confidence, risk-taking and humility to fix their own mistakes, learn something and move on. I actually see this in her classes and have heard her students say that they do this too. I believe that without this attribute students do not fully take advantage of a problem-based curriculum because they cannot find the way to learn from their mistakes. I even heard a student once say that “There was one time during class that I put a problem up at the board and got the entire thing correct. I was actually, in a way, disappointed because I feel like I learn better from my mistakes.” I was amazed that a student could see that in her own learning, that the growth happened for her when she was wrong, as opposed to when she was right.

Clearly, being wrong in front of students can be somewhat embarrassing, but for me, it allows me to have bit of solidarity with my students, if even for a moment. It allows me to feel, what I ask them to do every day, to move out of their comfort zone and attempt a problem that they cannot do, and perhaps not live up to their expectations of themselves. It reveals my human side, which I do end up feeling the relational part of my teaching is all about.

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.