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.
 
 
I've written about this "Habits of a Mathematician" Portfolio system before, but I have done some work on it and wanted to post on my updated version. I really want the Habits of a Mathematician to be the centerpiece of ALL that we do in class next year. In my opinion, they really get at what it means to be "doing mathematics" and are useful in helping reinvest in students a sense of agency and authority that is sometimes lost in the mathematics classroom. Of course, some content "knowledge" (I write that with some hesitation) will be an outgrowth of our work on problem-based units, but I'm leaning (heavily) towards not testing or hoping for "mastery" of any of that (the content knowledge piece is a bigger philosophical argument, which you can read about in a previous post).

The Portfolio System

At the beginning of the year, each student will purchase a 3-ring binder with 12 dividers. Each divider will represent one of the 11 "Habits" and the last section will be for "Unit Packets" (all of the other work). Students will have requirements weekly, at the end of each unit, and at every third of the semester. Here is what I am thinking for each:

Weekly

At the end of each week, students will select one piece of work that they feel best demonstrates one of the "Habits of a Mathematician." They will fill out this reflection sheet (see below) and will submit it to me. I will provide short feedback on the sheet and hand it back to them. After reviewing the feedback, the student will submit that work to the appropriate section in their portfolio.

End of Unit

At the end of each unit, students will put together all of their work from that unit (excluding the work that has been submitted as "habit" exemplars). They will complete a unit checklist and write a cover letter for their packet that summarizes the mathematical themes for that unit.

Three Times a Semester

Each student will have a "critical friend;" someone who they work closely with in evaluating their work and their progress. At each 1/3 mark in the semester, students will have their portfolio reviewed by their critical friend, by their parent, by me, and by themselves. With all of this in mind, students evaluate where they are at with the "habits" and set specific goals about how they want to progress.

Grading

I would love for this to be a grade-less system. My students tell me "the world is not ready for that yet." I can't see how it could be done any other way. My thoughts at this point are that grades would only be given at the end of each semester. Student grades would be decided on by the individual student based on feedback from their critical friend, their parent, and me. Mostly, I imagine their grade to be a representation of their progress toward their specific goals set for themselves.


I'm beginning to like this system a lot. What we assess in our classes says a lot to students about what is valued and I think this system more clearly shows students that math is about "doing" and not about "knowing." I worry a little bit about parent concerns but I'm not sure that should stop us from pushing the boundaries and redefining grading. The system is still evolving and I would love any feedback or suggestions you have.
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This is just a quick post to reflect on something that went really well this past week. We have been working on a modification of the Pig Unit from IMP. In the process of investigating, we were trying to wrap our heads around this problem:

If you roll a die two times in a row, what are the chances that you will get at least one 1?
What if you roll 3 times? 4 times? "n" times?

We usually do these problems in small groups (3-6 students) but I decided to mix it up for this one. I split the class into two large groups (each of about 12) and we formed an "inner circle" and an "outer circle." The inner circle got 5 minutes to form some initial thoughts about the problem individually and then we opened it up for a whole group, collaborative sense-making. I told them they were "done" when every person in the group both agreed with and could explain their collective solution. The other half of the students (the outer circle) were partnered with someone on the inner circle and they were to be silent observers, recording the ways in which their partner was positively contributing to the group discussion. I also removed myself from contributing to their efforts in any way.

Afterwards, we talked about ways the group worked well together, ways they didn't work well together, and we identified the habits/strategies that were useful for each group. There were a lot of amazing things that happened and some really great discussion that followed. Mostly, I loved that the students had to (and did) push, challenge, and support each other in an attempt to justify their solution and prove to each other that it made sense.

I was left wondering:
1. Why doesn't this work as well with a WHOLE class discussion/collaboration? What would support that? I think this post and research might help....haven't read it yet.
2. Does this collaboration happen with small groups also? It's harder to tell when you are monitoring multiple groups.
 
 
Constructivism and "discovery" learning are two popular methodologies in progressive mathematics education that are easily misinterpreted and, sometimes, get confused for being the same thing. I have been thinking a lot about both lately and would like to 1. try to outline what I believe to be a definition (comparison) of both and 2. propose some possible implications and questions as they pertain to mathematics education.

Constructivism

Constructivists suggest that we cannot be certain of any absolute truth but that people construct, or create, "knowledge" based on their experiential reality (or their interactions). "Truth" or "knowledge" is a social construct attained when people agree on particular mental models that appear to be consistent with our collective experiential reality. However, this social agreement does not necessarily imply "universal truth." For instance, people once believed that the world was flat (collective social agreement) but, as they discovered, this turned out not to be a "truth."

Discovery

Discovery learning is based in a different belief about knowledge and truth. This theory posits that there is a certain body of knowledge available and that teachers can help students "come to know" (or discover) this knowledge by implementing well-designed tasks in the classroom.




As it pertains to mathematics, I believe constructivists would suggest that the collective set of rules, procedures, and beliefs we call "mathematics" are not, in fact, universal truths but rather mutually agreed upon constructions from the mathematics community. I have some questions about how the implications of this theory for the teaching of mathematics:

1. Social constructivism posits that we rely on other people to both challenge and confirm our ways of knowing. When they have been mutually agreed upon, they become our ontological reality (our truth). What happens when students agree upon an ontological reality that is different from that of the teacher or that of the mathematics community at large? Whose reality is deemed correct in this instance?

2. Constructivist learning theory suggests that "learning" happens when an existing mental model is confronted with an experience that does not conform. The learner must then revise their mental model to accommodate this cognitive conflict. This suggests that the role of the teacher is to present students with situations that create this cognitive conflict (or disequilibrium) in students. Doesn't this imply that the teacher is pressing on students to rewrite their mathematical ways of knowing in a way that is more consistent with that of the mathematical community? Isn't this, in effect, discovery learning?

3. The questions above seem to indicate that it is impossible to create a socially just form of mathematics education that involves topics about which there is already a socially agreed upon "reality" (content standards). Because there is already an agreed upon reality, the students' interaction with them will always be measured against, and subject to, this greater authority. What, then, is the purpose of mathematics education and how can it be socially just?

Please, help me "sort this out..."
 
 
I have been writing a bit about the problem with "right" answers in the mathematics classroom lately. I think it is a big concern. Upon further reflection, I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:

"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students."

There is some great reading about how this relates to "mathematics for all" and teaching for social justice. Building this community is extremely difficult. Students have developed expectations about what learning and teaching mathematics "should" look like. As a result, if we are to promote this type of student discourse it becomes necessary to renegotiate the "didactic contract" (Brousseau, 1986). This "contract" both explicitly and implicitly outlines the role of both teacher and student, the expectation for classroom discourse, and, as a result, the locus of authority.

This type of discourse simply isn't going to take place if "mathematics" is practiced as "solving" a bunch of related problems (read: 1-30 odd). Instead, find the big idea, pick one rich question to lead an investigation of that idea, and then let the students sort it out. You are likely to see students doing mathematics. To return to Schoenfeld,

"This <is> their mathematics. They <have> ownership of it, not only in the motivational sense, but in the deep epistemological sense that characterizes the true mathematical knowing and understanding possessed by mathematicians."