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 the week off for Thanksgiving, so some friends and I have spent the last few days in Napa hanging out and doing some wine tasting. We got a recommendation to visit Freemark Abbey Winery, which we did, where I came across what seems like a pretty good problem-based investigation. Here is the scenario as they describe it (based on actual events):
"In September 1976 William Jaeger, a member of the partnership that owned Freemark Abbey Winery, had to make a decision: should he harvest the Riesling grapes immediately or leave them on the vines despite the approaching storm? A storm just before the harvest is usually detrimental, often ruining the crop. A warm, light rain, however, will sometimes cause a beneficial mold, botrytis cinerea, to form on the grape skins. The result is a luscious, complex sweet wine, highly valued by connoisseurs."
Basically, they had to decide: harvest the grapes now and guarantee the production of a moderately priced wine, OR wait for the incoming storm and hope that it encourages the mold growth necessary for production of high end wine. There are some accompanying details that the winery used in the decision making process (which I have somewhat shortened here):
  • There was a 50% chance the storm would hit Napa Valley
  • There was a 40% chance that, if the storm did strike, it would lead to the development of botrytis mold
  • If Jaeger pulled grapes before the storm, he could produce wine that would sell for $2.85 per bottle
  • If he didn't pull and the storm DID strike (but didn't produce the mold), he could produce wine that would sell for $2.00 per bottle
  • If he didn't pull and the storm DID strike (and did produce the mold), he could produce wine that would sell for $8.00 per bottle (although at 30% less volume because of the process to produce this wine)
  • If he didn't pull and the storm DIDN'T strike, he could produce wine that would sell for $3.50 per bottle

I can imagine several different versions of the problem depending on the group of students you were working with (by adding/removing details and alternative scenarios). I am attaching the printouts that I photographed in case anyone wants the details.
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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.
There seems to be a lot of effort recently to teach math through "real world" contexts. You may scoff at this a bit because, as we all know, this is NOT a new argument. If you stop and think for a bit though, you might realize that this is still the "heart" of the reform movement in math. Project-based learning, interdisciplinary classes, and even "WCYDWT" (which have all helped to better education) have, at their core, the desire to show students that math really does exist outside of a classroom. To be clear, in many ways I think this is a HUGE improvement but, and maybe I'm asking too much here, I think we can do more than that (as I alluded to in a previous post).

I'm lucky to have a wonderful advisor, Stacey Callier, in my grad school program. I work in a project-based school and, as she knows well, I often push back against the hidden assumption in PBL that math is a "tool" that helps us solve real-world problems (I think the terminology I used today was that math seems to always be the "servant of science"). We started talking a lot about the constructivist philosophy that is central to my work and we (mostly she) came up with this matrix:
Is it better if we situate mathematics in a "real" context that students find engaging? Of course! I just think we can do that AND STILL honor a student's way of understanding and knowing, give them opportunities to author and create their own mathematics, and help them construct their own meaning for ideas that help them solve a problem. Call me an optimist (read: delusional), but I think its possible. 


I have been thinking about this a bit since I posted it yesterday and I'm not too sure how I feel about it still. There are a few things that are bothering me:

1. The "HOLY GRAIL" label implies that this is where math education "should" be…which I'm not sure everyone would agree with (in fact, I'm not sure I can say that this is where I think it should be.

2. The top (applied vs. non) seems to get at "why teach math" while the right (constructive vs. non) seems to get at "how teach math." Is it ok to compare these two things in a matrix? Not sure.

Mostly, I labeled the top left "HOLY GRAIL" because I strongly believe that math should be taught in a constructivist fashion. If we can do this AND situate the math in a "relevant/applied" context shouldn't we do that?!?
I'm pretty sure this wasn't Sam's intention when he asked, "how do you plan" but I wanna try this anyways. I have been doing a collaboration with students in the credential program at Cal State San Marcos where I gave them an idea for a problem-based unit and then they worked together to plan it. Working with them has made me think about: 1. effective structures for designing problem-based units and 2. how to help teachers plan units for themselves. I thought this might be interesting…

Here is the problem for the unit I started today:
My request is this:
1. Do the problem yourself
2. In the comments, describe how you would design a unit around this (big chunks, progression, etc)?

p.s. YES…I renamed it the "infinite star"….I realize the decision has pros and cons.
p.p.s. This problem launch is a classic example of how paper limits the mathematics. With my limited video skills, I couldn't do better.
This one has been bugging me and I really wish I had more readers because I think there is a valuable discussion here:

What is the perimeter?

How long is the coastline?

The perplexing question here is essentially the same. This guy talks a lot about the quickness with which we move to abstraction. I'm wondering if maybe that applies here. I like the abstract because, in terms of creating a unit with a nice resolution, it easily comes full circle. I'm not sure you can say the same about the coastline. So…"what's the difference?!?"
I was writing an email to Dr. Brian Lawler, my Mathematics Methods instructor from Cal State University San Marcos when, in an unexpected moment of clarity, my thoughts and frustrations with project-based learning became incredibly clear. To be transparent, I actually work IN a project-based school. To be even more transparent, I'm not always sure I believe in it…

"I understand that an exemplar project might not be best utilized by being "replicated" because of its uniqueness to the learning community within which it was situated. However, I haven't seen anybody do a project that achieves depth in mathematical thinking. I want to see one so that I know what it COULD look like…not what it SHOULD look like in my class. My initial thoughts are that an emphasis on applied mathematics has skewed project-based learning in a direction that veers away from mathematical thought and more towards an application of formulas. I also think that an emphasis on PRODUCTS (or flashy, sexy art) has overshadowed less exciting products that COULD demonstrate thought transformation. Students have come to expect art and "relevance" (which, to them, often means how it can be used to solve a "real world" problem). Anyways….it needs work. I love working here for the freedom we have but I have actually become an opponent of PBL in the ways I have seen it done.

I have been thinking a lot about unit structure since we talked recently. I love the Interactive Mathematics Program model of perplex --> attempt --> identify what we need --> play out mathematical trajectory --> revisit initial problem. By starting and ending a unit with THE SAME PROBLEM, you can easily see where students "began" and where they "ended" which (aside from grading) helps us give meaningful FEEDBACK about their understandings/misunderstandings and development."

Without a doubt, the journey is maddeningly interesting...