A friend/co-worker, 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 Project-Based 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 project-based 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 "content-based" 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 "open-ended" 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 Open-Ended Approach

We took an in-depth 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 re-evaluation 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:
  1. 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.
  2. I'm curious about the "initial event" that prompts student questions. Should I start small and well-defined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #3)?
  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 three-day 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:
  1. Stay with resistance, but use a continuous curve
  2. Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
  3. 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?
 
 
For one hour of my school day, I work with an amazing group of Seniors in a class I have called "Mathematical Thinking" (more about that in a later post, perhaps). Mostly, the class is a mixture of problem-based units and other miscellaneous open-ended puzzles, problems, and mathematical games.

Yesterday, we worked on a puzzle/game called Cartesian Chase. I played a few games against students to demonstrate the rules (we confined ourselves to a 3x7 rectangle) and then just let them play for while. Then, I had them stop and record anything they were noticing in terms of a strategy that seemed to be working. Then, they switched and played with new partners for a while longer. I stopped them again after a few games and had them record updated strategies. We ended with a class a few "undefeated" people playing each other. It quickly became apparent that there was a winning strategy at play.

In the process of all this, here is what I noticed:
- nearly ALL of the students were engaged and playing for the whole time
- students were having fun with each other
- we had a few early conjectures in place about what strategy might be best
- students uncovered structure in the problem, used it to win every time, & were able to clearly explain it
- after the game was "solved," a few students were curious: "what if we added another column?" and "what happens with other board sizes?"

I work hard to bring this same spirit of playfulness to other lessons. I work hard to make every day feel like a puzzle in our class. For some reason, I can never quite bridge that gap in the way I would like. I think I get pretty close most days, but for some reason "how many burger combinations are possible?" still feels more like a math problem and less like a puzzle to students. Maybe it has to do with our intent as teachers? Do we place too much emphasis on students "knowing" something specific by the end of the lesson? Could we set up the task better (slower?) so that it emerges as a puzzle? I have a lot of questions, but I do know that I value what students are learning about themselves as mathematicians and thinkers from a lesson just as much, if not more than, I value students knowing some piece of the thing we call "mathematics."
 
 
Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem.  I would love some blogger critique so please let me know what you think...


HORIZON: How far away is the horizon?

This is a pretty famous problem in mathematics (I think). I remember seeing it and being intrigued. I think there is an opportunity for pretty rich math here. Students would have to create an abstraction to represent the situation, develop rules about their abstraction, and use them to approximate an answer to the unit question.


PROBLEMATIC PACKAGING: How can you optimize this packaging?

This one arose out of a curiosity I had about optimizing parking lot design. I thought it might be fun as a puzzle where students get a package and different "items" (blocks of two or three sizes and shapes and worth various points) and they have to figure out how to maximize their point value. Could be extended by looking at different point values or different size boxes.


GET ME OUTTA HERE: How do you know if a game is solvable?

I love these puzzles. I'm not exactly sure how to turn this into a full-blown unit, but I'm pretty sure it can be done. This might not be the best unit question. I have also considered giving students a specific puzzle and asking "What is the fewest number of moves?" Maybe we can extend it from there and move into "solvable" setups?

7 CLICKS FROM KEVIN BACON: What is the minimum number of clicks to get to Kevin Bacon from ANY person on Facebook?

This one I'm REALLY not sure about. Obviously, it comes from the popular game about social connection (although it might be a good idea to use a celebrity my student will have actually hear of). I thought there might be some connections to graph theory here and it might be a nice extension of a combinatorics unit that I will do again next year ("How many combinations are there at Chipotle?"). Feedback please.


WIN, LOSE, or DRAW: Can you draw this without lifting your pencil or going over the same line twice?

The Bridges of Konigsberg is such a GREAT problem that I want to turn it into a whole unit next year. I'm not sure I love the context of the original problem because, to students, I think it feels like a pseudocontext. I thought about giving them a crazy network diagram and asking the unit question about that (maybe with "bridges" showing up along the way?). However, I usually prefer to start with a concrete situation and abstract from there....not sure.


...something more complicated but this was the best picture I could find for now

GOING COASTAL!: How long is the California coastline?

I did a similar project this year where students figured out the area of the Koch Snowflake. I'm thinking about trying to start concrete and move abstract next year with this one. I'm sure the Koch Snowflake will still rear it's ugly head somewhere in our investigation.
 
 
I'm sure I have explored/solved/assigned this problem before but I revisited it on a recent plane flight to Northern California. I thought I would just post the problem for now in case you want to give it a shot. I'll post my work tomorrow.

Which numbers can be written as the difference of two perfect squares?


p.s. As it turns out, doing this on a plane is also an excellent way to alienate the person sitting next to you. Apparently, Sudoku is totally acceptable but if you dare get any "math-ier" you are guaranteed to be met with shaming eyes when you look up (trust me).

UPDATE 2/27

Here is the work I did on this problem. I added some commentary in light blue typeface. I'm no number theorist and there are a few fuzzy uses of variables but I think the overall idea is pretty close...