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.
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The closer my pedagogy gets to supporting students in "doing mathematics," the more difficult it becomes to decide what assessment and grades should look like. The shift to doing mathematics means that you aren't looking for students to replicate processes and demonstrate their acquisition of new "knowledge." Instead, you are looking for their development in thinking mathematically - in reasoning and sense making. This is particularly difficult in the 
K-12 public education system where, in many ways, we are forced to do our best in supporting students in "doing mathematics" through a variety of predetermined concepts to be explored. 

It seems to make sense to me that in order to assess mathematical thinking, you would have to 1. determine the habits that comprise mathematical thinking, 2. have some sort of feedback system to assist students in developing those habits, and 3. figure out some way of measuring student progress in each. 

At the moment, I have been toying with the idea of having students contribute to a semester/year long portfolio. The portfolio would be based around the habits of a mathematician, with a divider/section for each one. As we progress through the year, students would complete reflections on works of their choice and file them in the appropriate section. At the end of the semester/year, each student would evaluate their portfolio and present or reflect on their growth. 

A few questions:
1. Am I even on the right track here?
2. How would this portfolio system correlate to a grade?
3. Does this system support students and give them enough feedback on how they can develop/improve/progress?

P.S. I just had another thought…what does a "unit test" implicitly tell students about what is valued/looked for in class?
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
Start Small
Be Systematic
Take Apart & Put Back Together
Conjecture and Test
Stay Organized
Describe and Articulate
Seek Why and Prove
Be Confident, Patient, and Persistent
Collaborate and Listen

I particularly like the last one because I get an amazing mental image of Vanilla Ice with his well-maintained flat top haircut and shiny "Hammer pants" ("STOP…collaborate and listen…Ice is back with a brand new invention…"). 
I have been toying with a feedback system for each that would look something like this:
Throughout the year we would keep track of how students progress in their development and implementation of each of the habits. Part of me hates to simplify and categorize the habits in this concrete way. The reality is that they are much more fluid and interconnected. At the same time, I think this helps students get specific feedback and helps them set specific goals for improvement. On that note, there are a few things I am wondering about and would like feedback on:

1. Is this a good list of habits? Are there ones that I am missing? Are there ones that are unnecessary? 
2. Suggestions for feedback systems? I hate to bring it up…but should this translate to a grade?
3. How can we rename the habits in a way that is more student friendly?


See my post here for an updated version of the habits.
Progress takes time….learning is about progress….and assessment should reflect learning. Grading usually steps in and messes all of this up. When you're grading a single assignment, you are looking at a snapshot of that individual's work at that specific moment in time. In no way does this reflect progress or, consequently, learning. 

All marketing gimmicks aside, when you look at this image it is plainly obvious to see the progress that has been made:
That's because you have an initial snapshot and you compare it to an ending snapshot. The shots are the same (besides any clever Photoshop work by the crafty individuals at Bowflex) so it is easy to make comparisons. We can learn a lot from this sort of dedication to documenting progress. I have two ideas for this:

1. Next year I'm going to give my students a problem to engage in on DAY 1. Then, I'm going to show them this infomercial. We'll talk about how progress takes time and that even though you can't see it day-to-day, by the end of the year you will. Then, on one of the last days of the year, I'll give them the exact same problem. I imagine the comparison between their initial attempt and last attempt will be stunningly different.

2. This doesn't have to be a year long project. We could use the same idea to measure student progress through a unit. The beauty of problem-based learning is that you start with a rich task that perplexes the students. Here is the one for my next unit:
The Koch snowflake is a fractal that iterates in this fashion infinitely. Each time adding a smaller and smaller piece. The paradox (or perplexing part of this) is that it creates a closed curve which indicates that it must have a finite area. My class will get this problem on the first day of the unit. I'll let them work on it individually and see how far they get. We'll talk about ideas, controversies, and what we need in order to figure this out. On the last day, they'll get the exact same problem. I'm pretty sure not every student will be able to solve it on their own, but when I compare their progress on the last day to how sophisticated their approach was on the first day, I'll have a crystal clear "Bowflex" comparison of what they learned.