A friend/coworker, Andrew, and I hosted a workshop at Greater San Diego Math Council's Annual Conference this past weekend. We titled the workshop "Rich Mathematics Through ProjectBased Learning," but rather than attempt to give a "here is how you do PBL" workshop (which fails to acknowledge that there are many ways to attempt PBL) we decided to use our own efforts at projectbased learning as a lens through which participants, including ourselves, could examine the role of the teacher in fostering mathematical understanding. We each presented one approach to PBL, along with some student work and a dilemma that each approach seemed to pose about the role of the teacher in facilitating (or not?) the direction of student work/learning.
Our Two Approaches
Andrew and I decided to present two very different approaches to planning and implementing projects in our classes. I don't want to try and articulate Andrew's position too much on my own (maybe if he drops by he can leave some thoughts in the comments), but his basic premise was that a "contentbased" project might be structured something like this:
Andrew's dilemma here was, given that he is attempting to move students from A to B, "what is the role of the teacher in facilitating that progression?" This led participants to attempt to deconstruct the process and put in steps that might direct student learning. Contrasting that, I have been interested in experimenting around with a more "openended" approach to projects. I visualize it like this: This approach brings an entirely different dilemma, yet still one that largely revolves around the role of the teacher. Specifically, how does a teacher manage such divergent outcomes and how can the teacher facilitate mathematical connections/understandings for students?
A Closer Look at the OpenEnded Approach
We took an indepth look at a specific example that I have used recently with students. First, I posed the following task for students:
Create as many squares as possible using only 12 lines
Students played with that in groups for a day or so and we made some conclusions as a class. Then, I asked them to brainstorm as many questions as they could that they might be interested in pursuing based on the initial task. Here is what they came up with:
1. Can we create a rule/formula for the maximum number of squares based on the number of lines used? 2. How many triangles/rectangles/etc. can be created using only 12 lines? 3. Is there a difference between even and odd numbers of lines? 4. How many shapes can you create with 12 lines? 5. What would a graph of maximum number of squares vs. lines used look like? Linear? Exponential? Other? 6. What if by "lines" we meant "toothpicks" or "unit lines?"
I passed out some student work samples for participants to take a look at, with the prompt(s): What do you notice? What do you wonder about? What evidence of the Common Core Practice Standards do you see in student work?
Looking for Rule/Formula (Question #1)
Lines as "toothpicks" (Question #6)
Even vs. Odd # Lines (Question #3)
How many triangles (Question #2)
I'm still left with LOTS of questions about what all this means and what my role is in all of this work with students. What are the benefits of this divergence? What are the consequences? How do I help them make connections within and across each others' work? Do I push students reach certain conclusions with their work that I know are still out there? Do I let them just end where they end? If I did push, would they really own the outcome? Would they "know" it? Those last two questions have been nagging at me a lot recently (perhaps more on that in a later post?). I do think, however, that I am committed to continuing to try figure it out with my students. It is the closest I have come to truly freeing them to think for themselves, follow their own curiosities, make their own conclusions, and be honest with themselves about what is still left unanswered (versus trying to convince me that they know something that they think I want them to know). I mentioned in our workshop that our answers to some of the big questions (what is math? what is the purpose of math education? what is a project? what is the role of the teacher?) determine a lot of the small things we do in our class every day. I'm trying to let myself live in a state of constant reevaluation with those questions. I think we need to in order to be fully present in the craft of teaching.
I have been experimenting recently with different ways of having student curiosities drive our work together. In some ways it has been successful. In some ways it feels like I don't know how to do this well. I thought I'd blog about a few examples from the past two weeks and hopefully you all can help me sort this out a little more.
EXAMPLE #1
I put this up on the white board:
x/2 + 5
Me: "Someone give us a number." Student: "7!" Me: "Ok. I heard 7. We are going to put 7 in for x in the expression on the board. Then, whatever we get as the result, we are going to put in for x. And then again...and again....and again. But we aren't EVER going to stop. What do you think is going to happen? Tell your partner."
They had various ideas, we tested them out, and made some cool observations. After that, I encouraged them to experiment with anything they were curious about. What happens if we change the expression? What happens if we change the starting value? What happens if we use two rules instead of one and alternate? There were lots of options.
PROS Most students found something to pursue on their own
 Most gravitated toward something that was appropriately challenging
 CONS The initial task was relatively narrow and was defined by me, not them
 It was difficult to have students understand, respond to, and challenge each others' work because they were all working on something different
 EXAMPLE #2
I posted about this problem before, but this is an extension of my thinking about the launch of that problem. For my first two classes, I gave students the following problem: They seemed to rely on me to define the task for them and set parameters. So, for the third class, I put this up on the board:
I said, "For the next five minutes, everyone experiment with something you find interesting." They experimented for a bit and then I had them compare their activity with the rest of their group. There were a few different ideas. Most students hovered somewhere around the question of "which numbers can you create" but there was a lot of discrepancy about parameters.
PROS Students were the ones engaged in "finding the task" and setting the parameters
 It opened up the possibility for students to pursue questions outside of the one that might have been intended or suggested
 CONS It took a lot longer and was more difficult to facilitate
 It felt like I was tricking students into asking THE question versus actually giving them freedom to explore their own questions
 Students had a difficult time accepting the suggested parameters of a different group if they were not the parameters THEY saw as fitting (there might be some implications here about the questions we pose as teachers seeming unnatural to students?)
 EXAMPLE #3
I put the following images up on the projector:
I gave students some time to examine the images and then asked them to brainstorm a list of questions that were raised by the images. We put a list of them up on the board. It was interesting that a lot of the questions were clarifying questions rather than problems to be investigated (Is the first one just a zoomed in portion of the second? What is the dot? Does the line always have to cross diagonally through the small squares?) It seemed like they were so used to asking about parameters, rather than setting them, that it didn't occur to them to just set the parameters and ask a solvable question based on them.
Eventually, we got a few questions with potential. Will it always hit the corner? How many times will it hit the sides before it hits a corner? Does it matter if the side lengths are odd or even? Is it possible to hit every grid line on the side BEFORE it hits the corner? Is it possible to end up in the same corner that you started in?
I suggested that each group: 1) pick a question they were interested in, 2) set their own rules/parameters and 3) get to work. It was interesting. Groups worked for a couple days and then things really stalled out. Because there were only four people (or so) working on a problem, there wasn't the same opportunity for them to bounce ideas off of other groups, for us to work through difficult things together as a class, or the same chance that someone might have an insight that led to progress for the whole class. We eventually proved that 1) it would ALWAYS end in a corner and 2) that corner would NEVER be the starting corner (assuming you launch at a 45 degree angle from a corner). So, from there, I suggested we all work towards finding a way to predict exactly which corner it would land in based on the rectangle size.
PROS This was the closest I have come to having student curiosities drive the work; felt like students had a genuine opportunity to follow their own question
 Students were engaged in questioning, setting parameters, exploring, and then adjusting their question or parameters if they needed to
 Many students really enjoyed the freedom and creativity involved
 Almost all students were engaged in mathematical activity
 CONS Students seemed to, initially, search for a more shallow level of depth than we usually accomplish as a class
 Some students were very turned off by the ambiguity and openness
 I didn't know how to bring things together or take it further when students were all over the place
 As a result, I eventually defined a question for the whole class. Even though it evolved out of their work, it was still defined by me
 Help Me Out...
I'm really interested by idea of using student generated questions but I feel like I need help on how to make it work. Things that I'm thinking about:  I don't love the idea of tricking students into asking the question you want them to ask, but I also have trouble when students are all working on different things.
 I'm curious about the "initial event" that prompts student questions. Should I start small and welldefined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #3)?
 In most cases, I found it hard to facilitate student work. How can I get students to share work, challenge each other, and challenge themselves?
Mostly, I would love to hear about your experiences, questions, advice, or thoughts.
The 12th graders and I started a new unit/project/problem/whatever last week. It started with this image:
What is this?
We took some guesses before someone correctly identified the image as the readout from a stationary bike machine. If this was the readout, I nudged, then:
What did the course look like?
I let them play with it for a while. They identified some important questions they had and information they wanted:  What do the dots represent?
 How can we quantify this?
 Does speed matter?
 Did the bike have gears?
 What do we mean when we say, "what does the course look like?
Ultimately, they agreed upon some answers and, as a result, gave themselves some space to work within. This part takes time and I really think we need to give it the time it deserves. If we don't let students pose and answer those questions, the rest of the unit suffers. I let them know that we would examine some simpler cases first and return to the unit question somewhere down the road. So, the next day they came back and and I had them map the course for these two: This took another class period for them to sort out. By the end, though, they started to feel pretty good about the fact that the dots were a measure of resistance which they connected to the slope of the course at that point. Towards the end of class, someone asked, "What would it look like if you were going downhill?' They talked for a bit and devised a system to handle that, so the next day I gave them this:
This allowed them to have conversations about what it meant when the resistance was positive, what it means when resistance is zero, and what it means when resistance is negative. It's fun to watch the students construct an understanding of antiderivatives. The threeday span reminded me of a few things: the importance of listening to how students are thinking about and making sense of problems
 our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
 allowing them the time to sort things out together
I feel like there are a couple ways to go with this trajectory next: Stay with resistance, but use a continuous curve
 Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
 Stay with discrete and resistance, but use a more complicated readout
I'm leaning towards #1. What do you think would be best for the students?
I have been thinking a lot recently about the subtleties of problembased approaches to math education. The following is a gross oversimplification, but I think it will illustrate the essence of what I am interested in. Let's compare two wellknow approaches, Interactive Mathematics Program and Exeter Math: Interactive Mathematics Program (IMP) The Game of Pig  Year 1
MODEL: the unit starts with a "unit question/problem" and then smaller subquestions (sometimes out of context of the unit question) are explored to deepen understanding before returning to the unit question again at the end of the unit.
 Exeter Math Year 1
MODEL: students are given a set of problems that are, more or less, completely unrelated. Each smaller problem stands on its own; it does not tie in to a larger context.
 EXAMPLE: This unit follows the following progression:
1. Students are introduced to the unit question by playing games of Pig and thinking about strategy.
2. The first section detours from the unit question to help students "define" probability. There are investigations about gambler's fallacy, experimental versus theoretical probability, and measuring probability between 0 and 1.
3. The second section introduces "rug diagrams" as a way to represent probability. There are some investigations about this and the end of the section ties rug diagrams back to coin and dice games.
4. The third section looks at how things play out "in the long run." It involves investigations about the law of large numbers and expected value.
5. The last section looks at a simplified version of the game of Pig before returning to the original unit question.
 EXAMPLE: Here are some problems, in sequence, from the Year 1 Exeter problem set:
 Clearly, there are advantages and disadvantages to both approaches. What are they? Do you prefer one over the other? Why? In short, "What's the Difference?!?"
I'm going to try my best to summarize a lot of what I have been thinking about lately in terms of curriculum design and pedagogy that supports students in "doing mathematics." Although there is more to it than what I write here, the following three pillars seem to be a good place to start:
1. Each lesson should revolve around a single task, prompt, or question Students can't "do mathematics" if we don't offer them an opportunity. I think it is most helpful to think of this as a shift from the traditional lesson format of lecture, guided practice, individual practice to a more fitting lesson format of launch, explore, summarize. The traditional lesson format is designed precisely to have students mimic the teacher's way of knowing and doing mathematics. The second lesson format shifts the responsibility of doing, inventing, and creating to the student. By offering students this responsibility, we also give them the opportunity to develop mathematical habits of mind. 2. Each task, prompt, or question should be studentdriven Many people interpret studentdriven curriculum as the notion that students decide what to study. While I think that might be a possibility (by having students generate and pursue extensions of the task, prompt, or question), I prefer to think of this as curriculum that is designed to respond to a student's current way of knowing by always pushing on their equilibrium point. In this way, it becomes difficult for a curriculum to stand entirely on its own, independent of the student community. While curriculum can/should serve as a reference, it is ultimately up to the teacher to craft a lesson that, in a response to an assessment of a student's current mental model, puts the student in a position of cognitive conflict. This iterative process results in a connected mathematical trajectory that is studentdriven. 3. The "resolution" of each task, prompt, or question must come from the students
Mostly, I mean here that no body other than the students should provide resolution (ex. textbook, teacher, video, etc.) When students see that the validity of their thinking will be measured by something/somebody other than themselves, they will always end up judging their thinking against a "greater" authority. In these situations, the "doing" becomes solely about figuring out how to arrive at someone else's understanding. When a task is open, the "doing" becomes more about arriving at a resolution that makes sense to the individual student/group. There may be differences in opinion and reasoning among students, which is good. In order for a pair, group, or class to reach a resolution, the authority must come from the logic and mathematics of the students. I would even suggest that any task that prompts students to uncover a preexisting solution will inhibit their ability to engage in "doing mathematics." Even though much of this sounds dramatic, I truly don't think it is too far removed from what is possible in even the most standardsdriven schools. Thomas Romberg writes of a problembased approach similar to this in which he sees students "studying much of the same mathematics currently taught, but with quite a different emphasis." I think this is a fitting description of what I propose here. I would push for much of this type of design to occur within a problembased environment, but I don't think it is necessary. One could just as easily work with students to study, abstractly, a unit on exponential growth, trigonometric ratios, or any other traditional standardsbased unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.
In " Mathematical Thinking and Problem Solving," Judah Schwartz (somewhat) jokingly makes the suggestion that we should never ask students to engage in a problem/task that has only one right answer (if even a right answer at all). The idea really caught my attention. Schwartz knows that the suggestion may be unrealistic, but striving for that would have profound impacts on many aspects of math education. This idea resurfaced for me this week. I was finishing up a lesson sequence designed to get students thinking about exponential growth and percent change. We were working on this last problem (video) in that sequence: I was really dissatisfied with how it was went. I heard students saying "I don't know what to do" more often than usual, which was frustrating. I could be wrong here, but after reflecting on the lesson I couldn't help but think that the elusive "right answer" might have been the problem here. It is almost as if I was taunting students by covering up something that they were supposed to somehow (magically) discover. All of the students had the original dollar and they all had the final copy. It's not a stretch here to at least experiment with different percentages and see how close you can get…but nobody did that without me suggesting it. I'm still not sure exactly what went wrong here, but I would love thoughts and suggestions.
Helping students to trust their own thinking and feel comfortable experimenting is frickin' hard. That doesn't mean we should resort to easier, more mundane forms of math education that make us, as teachers, feel better about ourselves. UPDATE 3/22 I just came across this video (look for the one by Annie Fetter). It seems to be exactly in line with what I am thinking with this post.
Confession: I obsess over the details of curriculum design. For me, having the opportunity to design and create challenging, perplexing learning situations is one of the best parts of my job as a teacher. I also think that the details of these designs can have a lasting impact on our students. Last time I posted about this (read this first), the class left off knowing that population growth was not linear (as they had expected) but that, in fact, it was increasing by and increasing amount each period (as they put it). Here is how the sequence of curriculum design has gone since then and I'm wondering....did I get this wrong? 1. The next lesson was designed to get them thinking about multiplicative structures/exponential growth. We experimented with the problem below and then did a bunch of variations/extensions to eventually come up with this generalization (in their words): starting value (multiplier)^"iteration1" 2. All of the variations from the previous lesson were whole number "multipliers." So, I decided to follow up with this video (thanks Dan). We talked about how big the 100th dollar would be and how big the "nth" dollar would be. 3. Now that they had an idea about percent growth, I decided to push their thinking with this "sequel." I gave them the resulting dollar bill. As I have mentioned previously, I am really pushing myself with every lesson to put students into a place of cognitive dissonance or disequilibrium. With that in mind, I'm wondering:
1. Did I get this curriculum sequence wrong? 2. What would have made it better in terms of promoting disequilibrium?
"Piaget suggested that the creation of new cognitive structures springs from the child's need to reach equilibrium when confronted with internally constructed contradictions; that is, when perception and "reality" conflict."
"Nussbaum and Novick suggest a threepart instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)
We started our unit on exponential growth yesterday and I'm pushing myself really hard to take this approach to my curriculum design. Here's how it has played out so far:
1. I opened with this population counter and asked students to make their best estimate about what the counter would read at an exact date and time. They were just watching the counter so it was pretty difficult to get any data or make any calculated guesses.  
2. I asked them how they thought we should break down the problem and what information they would need. The responses were somewhat varied, but most hovered around something like the response at the right. It was obvious that they were assuming a linear/constant growth (their "perception").  
3. Based on their existing perception, I gave students (essentially) what they asked for and prompted them to make some predictions about the next few decades.  
4. Then, I revealed this slide. You can imagine their surprise when their constant growth didn't match up with "reality." I think this is what Nussbaum and Novick mean when they say "cognitive conflict" or when Piaget talks about "when perception and reality conflict."  
5. I asked them to revisit their predictions based on what they noticed in the new data. There was some variation in their response, but the common theme was something like the image on the right.  
This is where we left off for today. In my opinion, students have made the cognitive shift I was hoping they would. They came in thinking population was linear and left knowing that it isn't. They also have some intuitive ideas about how to make predictions based on how it actually is growing. It's sooooo tempting to step in and explain to them about percent increase, exponential growth, and (for some) derivatives. But, if I did that, I would completely rob them of the opportunity to construct that understanding for themselves. I do know one thing…they're about ready for this awesome video tomorrow!
We're just wrapping up our "Infinite Possibilities" unit in which students explored the area of the Koch Snowflake. It came to my attention that Khan Academy has a set of videos on the same problem we have been tackling. So, if it exists online and we did it in class…"what's the difference:" Khan Academy 1. Khan opens with this shot (a continuation of a previous video) in which he had shown how to calculate the area of an equilateral triangle in terms of its side length.  Infinite Possibilities 1. We open with an introduction to the figure. Students explore how it is changing at each iteration, attempt to solve for the area, and identify what they think we will need/what will be important. 
2. Khan begins to organize information by showing the viewer that you can set up a table that will compare the number of sides to the overall area.  2. We pursue the "meaning" of area and investigate these irregular shapes as a way of developing the concept of total area as a sum of parts (or taking things apart and putting them back together). 
3. Here Khan starts to show the viewer a pattern that is emerging. He points out that the number of sides is multiplied by four at each successive iteration and that the side length of the new triangle is 1/3 the length of the previous. 
3. We look at how the area of the "new" triangle is changing at each iteration.

4. Khan has now shown the calculations for the first four iterations. Each iteration is color coded. He has also written the terms of the sequence in a way that helps the viewer recognize patters. 
4. Next we were concerned with looking at the total area at each iteration. As a class, students brainstormed what to include in a chart and then worked in teams to create it, complete it, look for patterns, and create any generalizations. 
5. Khan has identified the portion of the series that is geometric and is demonstrating the calculation of an infinite geometric series.  5. Our next move was an introduction to sequences. Students attempted to look for patterns to help make predictions and used structure in the visuals to make connections to any generalizations. 
6. Khan asks for a virtual drumroll as he has solved the problem and boxes the answer in magenta at the bottom right. 
6. We debate about whether or not Tiger will get the "stupid" ball in the hole, which leads to an investigation of partial sums, convergence, and infinite series. 
I'll leave you with a series of questions: 1. The two approaches tackle the same question. Are students learning the same things in each? 2. What does each method imply about the definition of mathematics? 3. How would you choose to measure/evaluate student progress in each approach? What is valued?
I'm reading "Out of the Labyrinth" right now and even though I'm only about 50 pages in, there are some pretty powerful quotes. This one is my favorite: "To teach it now as if it were A Rule, or (even more intimidating), The Law, is to pretend that what took years of experimenting and ingenuity is as obvious as your nose. And then, because you never really had a chance to understand what was going on, whenever you need this rule again it will come as just that  an arbitrary fiat, enforced by Them. And so the whole integrity of mathematics is compromised. The only reasonable conclusion for a struggling student to draw from such pretense is that he is irremediably stupid, or that Mathematics works in mysterious ways, its wonders to perform."and further down the page:"...and so a teacher, who is supposed to develop our powers of inquiry, becomes instead a messenger of Received Truth."There are many other incredible little tidbits in this book and, so far, it is making a pretty good case for a spot in my preferred reading list. As I read the above passage, I was reminded of something that I have been thinking about a lot lately (and even alluded to at the end of this post). We talk a lot about the importance of context in math education. There are many benefits to situating mathematics into some nice context that students find engaging/relevant (mostly, it seems beneficial to help students see that math can help them describe and understand their world) and there are some sites/people that are doing this very well. But simply using an exciting context does nothing to remedy the fact that often the "whole integrity of mathematics" can still be compromised. I mean, you can start with a really interesting and perplexing question and still completely miss the boat when it comes to helping students do their own mathematics; it just becomes a better way of teaching "The Law." These types of questions could (and should) drive a whole unit because then we can explore different parts of the problem, honor different approaches and ways of thinking, and ultimately, help students create their own math along the way.
