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.
 
 
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."
 
 
I love the Khan Academy videos. I love them because they have opened up a discussion about education, learning, theory, and bringing all of it together in meaningful ways to support students in their mathematical development. Relatively new to the education scene, Khan Academy receives a lot of criticism. In the most recent critique of Khan Academy, we have been amused by the "Mystery Teacher Theater 3000" videos.

First, I think that Khan Academy gets a bit of a bad wrap. I'm pretty sure it was not Sal Khan's intent to "revolutionize" or "reinvent" education by posting his videos online. In my opinion, a public misconception about mathematics, learning, and education has allowed KA to rise to it's current position. The public views mathematics as a body of factual information, one that is passed from the "knower" to the "student." Learning, then, is perceived as memorizing/understanding factual information. This public perception applies to much of education in general, but especially to mathematics education. 

Recently, the MTT3K video exploited some mistakes in Khan's explanation of multiplying positive and negative numbers. Among other criticisms, the makers of the video point out that Khan mistakenly refers to the "transitive property" and explains "-4 x 3" as "negative fours times itself three times." In addition, the video points out some questionable pedagogical decisions by Khan which might be confusing or difficult for students first learning the topic and that fail to explain why the multiplication facts are true.

I welcome the dialogue that is happening surrounding these videos, but want to take the discussion of why Khan Academy is an ineffective learning tool one step further. In doing so, I want to reference Constance Kamii and her application of Piagetian principles to teaching:

"Piaget's theory of memory is very different from the empiricist belief that 'facts' are 'stored' and 'retrieved.' According to Piaget, a fact is 'read' differently from reality by children at different levels of development because each child interprets it by assimilating it into the knowledge he has already constructed."

In this next quote, she refers to facts about addition. I think the same could also easily be said for multiplication:

"In Piaget's theory, there is no such thing as an 'addition fact.' A fact is empirically observable. Physical and social knowledge involves facts but not logico-mathematical knowledge. The fact that a ball bounces when it is dropped is observable (physical knowledge). The fact that a ball is not appreciated in the living room is also observable (social knowledge). But logico-mathematical knowledge consists of relationships, which are not observable. Although four balls are observable, the 'four-ness' is not. When we add 4 to 2, we are putting into an additive relationship two numerical quantities that each of us constructed, by reflective abstraction; 4 +2 equals 6 is a relationship, not a fact."

At first, these quotes seem hard to swallow. However, Piaget (and others) have conducted many experiments to show that even children who "knew" facts about addition lacked the ability to perform on different tasks involving those same operations.

We construct in a way that is personal to us and relative to our current ways of understanding. My point in all of this is that, video or no video, you can't make someone learn. The 'knowing' that is so prized in education comes from every person's natural ability to think and from the human tendency to maintain internal/cognitive equilibrium. My suggestion is that we don't tell students what they should know or how they should think. Give them a task that pushes on their way of knowing, let them do mathematics, and watch and listen closely as they sort things out together.
 
 
I hear people talk a lot about the importance of context in mathematics....usually, I think, for all the wrong reasons. There is a common misconception that if we show students how they can "use math" that they will find it more enjoyable and see the value in learning it in school. I'm pretty sure the problem isn't that students really want to know how this stuff is valuable in usage. The problem, I think, is that when the subject is relegated to procedures/facts/tricks it often disrupts the way a student naturally thinks.


Applied Math

By it's very name, this approach implies that we are "applying" something. Usually, there is a mathematical topic presented and then problems are selected in which students have to apply what they have learned to solve these problems. Sometimes, the problem might come first and then topics are presented (as facts) to provide students with the "tools" necessary to solve a problem ("necessitating" content). The focus here is on math as a tool to solve a problem. The assumption is that the mathematics exists before, and inspires, the problem.


Doing Math

Doing math is an act of creation by students. Usually, there is a problem/task/situation that requires students to think in the form of reflective abstraction. Mathematical "facts" arise as generalizations students make by looking for patterns and consistencies. These facts might, then, be explored in the abstract. The focus here is on problems as an entry to creating mathematics. The assumption is that the problem exists before, and inspires, some mathematics (I say "some" because although that may be anticipated by the teacher, it is ultimately defined by the students).

As I see it, yes, context is important but not to show students how they can use math. The context is important to show students how their mathematics is a natural extension of how they think and live in the world. Constance Kamii says it best:

"Most math educators think about verbal problems (word problems) as applications of computational 'skills,' rather than as the beginning point that eventually leads to generalized computation, without content, context, or practical purpose."

Kamii, C. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York, NY: Teachers College Press.
 
 
I'm worried about my career in math education. I'm worried because I'm starting to wonder if it is a fool's errand to attempt to teach math in a way that goes against the mainstream. It's tiring...and I haven't even been doing it long. Our society has a clear, and in my opinion misguided, perception about the nature of mathematics and particularly about the way that mathematics should be taught in schools. I gave a year-end survey last week and, while there were many things to celebrate, students clearly echo the thoughts/opinions of the masses. Generally, I read a lot of things like:
       
        - We didn't cover enough math topics
        - We should have prepared for the STAR Test
        - We didn't do Algebra II (or whatever math we were supposed to be doing)
        - I didn't like when there wasn't a clear/right answer

As you might guess, I try to create a class in which students are doing math. We have worked on some interesting problems this year (How many combinations are there at Chipotle, What is the area of the Koch Snowflake, How can we predict global population, What is the best strategy for the game of Pig, etc) and students have done some really interesting mathematics as a result. Unfortunately, our society views mathematics as a thing and not an act of creating. Our national content standards, for many, serve as a definition of what mathematics is (and, by comparison, what it is not). In schools, society dictates, math should be compartmentalized and taught by transmission; if you perform well on the STAR/SAT/Whatever then you are good at math and if you don't then you're not.

I have tried really hard to open up this narrow definition of mathematics and to provide all students the opportunity to be mathematical every day, yet some of my students don't even recognize what we do in class as mathematics. It's difficult to be a part of a system while simultaneously doing things that run against it. You can't, and I don't, blame students for their outlook on things. I'm not sure my writing here captures the drama of all this, but I find it really alarming that this is what our educational system is teaching students about mathematics and about themselves. As I reminded my students today, the root of "educate" is "educe," or "to draw out." My job with them has been to teach by offering them an opportunity to express what was already a part of them, the ability to create mathematics, to create powerful ideas.
 
 
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.
simplify_the_problem.pdf
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You always hear people say, "kids don't like math!" Correction...kids don't like feeling dumb. People don't like feeling dumb. Feeling dumb comes from being told you're "wrong" over and over. Math education effectively does this better than just about any other subject in school. It's no wonder that people are sending out tweets like this:
Things get even worse when we start throwing grades in the mix. Now, all of a sudden, not only are we telling kids they're wrong...we're punishing them for it. It shouldn't be surprising that students are reluctant to take risks, persist with difficult problems, and trust their own thinking; they don't want to get it wrong. Today, I saw this tweet:
To which, I replied:
On second thought, I should have written: "It CAN'T exist independent of a specific group of kids." I mean, of course we know it can, and it does, but isn't this putting the cart before the horse?!? When your curriculum is decided in advance, you've already told students what the measure for "knowing" is and if they don't meet that then something is wrong with them. I've written about other curriculum/lesson structures that respond more directly to students but my concern goes beyond that.

I have had the pleasure of "mentoring" (which is a total misnomer...we really just learn from each other) one of our new teachers this year. She's amazing. Yesterday we were planning a probability unit together and we were trying to figure out an answer to the unit question that would drive our investigation of probability. In the process, her and I were collaborating, conjecturing and testing, investigating smaller problems, drawing diagrams, and listening closely to each other. Honestly, I don't really care if, as a class, we are successful in calculating an exact answer to this problem. I want students to come up with an answer that makes sense to them, that responds to their ways of knowing, and that is reflective of their deepening understanding of chance and probability. To quote a brilliant mentor of mine, "I guess my point is, 'solving' the unit problem will certainly be in the discussion but we'll be 'successful' moreso because students invented the solution, rather than being told." Mostly, I just want the students to have the same experience we had; the experience of playing with and doing mathematics.

I know there are "realities" in a lot of schools that make this difficult. Benchmarks, AYP, API, etc, etc. There are all sorts of measures of success and progress out there (most of which, I would argue, are false indicators of "learning"). Well, here's my vote for success/progress:
I want people to know that we are all mathematical in our thinking...maybe just not in the ways that school has defined mathematics. I want fewer people to hate math.
 
 
I have written about these "habits of a mathematician" before, but the list is changing, improving, and evolving so I wanted to share my most recent thoughts. Mostly, this list has grown out of a desire to show students that "doing mathematics" is not about trying to find an answer that exists at the back of a book (or elsewhere); doing mathematics is about creating, exploring, inventing, and authoring. I owe a lot to these people for their work before mine in helping shape this list:

Look for Patterns

I am always on the hunt for patterns and regularity
I bring a skeptical eye to pattern recognition
I look for reasons why a pattern exists

Solve a Simpler Problem

I start with small cases
I "chunk" things into small sub problems
I take things apart and put them back together

Be Systematic

I experiment methodically and systematically
I make small changes to look for change and permanence

Stay Organized

I use charts and other methods to organize information
I organize my work in a way that can be referenced by others

Visualize

I draw pictures and diagrams to represent a situation
I invent notation or representations to facilitate exploration

Experiment through Conjectures

I ask "what if..."
I invent numbers to test relationships
I use specialized cases to test my ideas
I modify conjectures to continue/deepen exploration

Be Confident, Patient, and Persistent

I am willing to take risks
I will explore even when I'm unsure about where to start
I do not give up easily

Collaborate and Listen

I listen carefully to and respect the ideas of others
I effectively use strengths of people around me
I ask people for their thoughts when they seem hesitant
I don't dominate a conversation

Seek Why and Prove

I seek to understand why things are the way they are
I look for clues that help me understand why
I create logical arguments that prove my ideas

Generalize

I look to understand ALL cases
I invent notation that helps me generalize
I algebraically represent the structure I see


I'm most interested in how a curriculum that is based on these might help students in "doing" their own mathematics. It's not enough to simply slap these up on wall and reference them often (although that might be a start). As I see it, there are some serious implications on curriculum, pedagogy, and assessment that go with these.

Curriculum

Most importantly, if we expect students to create their own mathematics we need to offer them the opportunity to do that. I think the best response to this is to center curriculum around rich, engaging, interesting problems (in short, a problem-based curriculum). This can still be done within a traditional, standards-based classroom but it definitely would suggest relying less on textbooks as an instructional tool. Here are two examples of what I see as possible problem-based units.

Assessment

At the end of each unit (or possibly periodically through the unit) students will select work(s) that demonstrate their use/growth in relation to these habits. They will fill out a reflection sheet (coming soon) and then submit that work to their portfolio. Three times a semester, students will evaluate their portfolio and will receive evaluations from a peer and myself (and possibly a parent/guardian?).

Pedagogy

In the past, I have prompted students to implement these habits (both explicitly and implicitly). The more I think about this, I'm inclined to think this is doing them a disservice for two reasons: 1) I'm telling them what to think or encouraging them to understand in my way and 2) they aren't the ones creating mathematics. Essentially, all I should do is help introduce the habits, provide students with opportunities to do mathematics, and help give them feedback as they grow and develop.

As always, a work in progress...feedback welcome!!!
 
 
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 student-driven

Many people interpret student-driven 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 student-driven.

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 pre-existing 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 standards-driven schools. Thomas Romberg writes of a problem-based 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 problem-based 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 standards-based unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.