I am nearing the end of my first year in grad school. Even more than I anticipated, blogging and getting feedback from readers has been exceptionally helpful in pinpointing the things in mathematics education that really matter to me the most and in finalizing my action research proposal. With that said, I wanted to post my preliminary research proposal in case anyone is interested in reading it. It is rather long and still relatively "un-polished" but I owe much of it's creation to readers who regularly push my thinking. Thanks!
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:
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.
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.
What does it mean to "educate?" Here is dictionary.com's definition of education:
The next section was alarming:
I'm just (very) concerned with the perception that education is all about the transfer of "facts." I know this isn't a new argument, but I can't help but feel this is a pretty critical time in education (particularly for math). National and state budgets are in bad shape. People perceive math as a fact-based discipline. Computers provide fact-based instruction effectively. I find all of this really alarming.
There was a great post by Grant Wiggins recently in which he suggests:
"I propose that for the sake of better results we need to turn conventional wisdom on it is head: let’s see what results if we think of action, not knowledge, as the essence of an education; let’s see what results from thinking of future ability, not knowledge of the past, as the core; let’s see what follows, therefore, from thinking of content knowledge as neither the aim of curriculum nor the key building blocks of it but as the offshoot of learning to do things now and for the future."
And another section from the same post in which he references Ralph Tyler:
"According to Tyler, the general aim is "to bring about significant changes in students’ patterns of behavior.” In other words, though we often lose sight of this basic fact, the point of learning is not just to know things but to be a different person – more mature, more wise, more self-disciplined, more effective, and more productive in the broadest sense."
To me, these quotes speak to what it means to "educate." I think we would be well suited to redefine math education as one that nurtures and cultivates habits of mind. These are the habits that influence future action, that inspire innovation, and that, in my opinion, are what it means to be "educated." More to come...
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
I experiment methodically and systematically
I make small changes to look for change and permanence
I use charts and other methods to organize information
I organize my work in a way that can be referenced by others
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
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.
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.
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?).
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 will often bring these rich tasks to our class for students and I have found that students not only enjoy them, but that these tasks have also been great for developing and reinforcing the "habits of a mathematician." I usually title these tasks "POWs" (for Problem of the Week). Yesterday, as part of the progression in figuring out the area of the "infinite star" (aka Koch Snowflake) I gave students this problem hoping that we could emphasize the importance of 'taking things apart and putting them back together':
I was pretty surprised by the difficulties students had with this problem. In fact, I had even included a more difficult one in anticipating students blowing right through this one. About 50% of the class was completely stumped by the original problem (which I appropriately named the 'Crooked House'…in case you haven't noticed, I appreciate a cheesy title). I asked them at the end of class to reflect on and discuss strategies that were helpful and even asked them why this problem was so difficult. They said things you might expect:
"we didn't have a formula"
"I needed the distance formula"
"I didn't know where to start"
A little piece of my soul died. We have spent all year de-emphasizing formulas and talking about habits that give you a place to start. Then, this happened:
student: "It was like a POW, but not."
me: "What do you mean?"
student: "Well, it seemed like a POW-type problem but it wasn't one."
me: "So, just because it didn't say 'POW' at the top of the page you didn't approach it in the same way?"
It was clear that students see math compartmentalized into separate worlds. I haven't helped them see that these habits ARE mathematics. They aren't just useful for puzzle-like problems, these habits are at the foundation of the creation and authoring of ALL mathematics. With my last two units I have started to include 1 or 2 of our habits as the central focus. Today's prompt looked like this:
It was a great reminder of the importance of redefining mathematics for students, helping them see connections, and making sure the message we send is clear and consistent in ALL that we do.
There are a few themes in math education that are particularly interesting to me…habits of mind is one of them. They are the building blocks of "doing mathematics." The more I work with students, the more convinced I have become that they are also the strongest indicators of success in school mathematics. They are the things that good mathematicians of any age do on a regular basis. I seen a few collections of mathematical habits and there are certainly some great resources already (here, here, here, and here to start). After collecting and distilling, here is what I have so far:
Look for Patterns
…helping students DO mathematics