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:

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.