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?


03/09/2012 1:03pm

I'm going to try to talk you out of it, I think. As college professor for whom teaching students to do mathematics is a big part of my job, I'm suspicious of the portfolio idea because it seems likely to give the illusion of students doing mathematics without it actually happening in any meaningful way. As a parent, it would make me tear out my hair because I would feel like it was too vague for me to understand the assignment, and my child would (typically) also not really understand the assignment, and so it would all get done under great stress rather badly at the last minute.

Knowing what it means to do mathematics is subtle. I have a clear conception for myself, and I expect you have a pretty clear conception for yourself, but it's not really easy to communicate the details to students. It's even harder to teach students how to do mathematics (OK, maybe not--possibly you can't teach what it is unless you have already taught how to do it). I don't you can teach how to do mathematics without devoting a significant amount of time in class to the doing of mathematics (at least I can't).

What can you do, while still having your class learn the content you're supposed to include? Well, you can provide opportunities for students to discover patterns about the content of the class, you can provide some time in class to work on and discuss problems that require mathematical thinking to completely solve. The best of these problems allow for partial solutions by "easy" methods, and deeper solutions with mathematical insight. This is valuable! Just because your students aren't "doing mathematics" and "being mathematicians" doesn't mean that what you've done is second best. It takes a long time and lots of experience to gain the "doing mathematics" way of thinking. Your students can't jump from being novices to being competent in a year or even two. (In college we expect our math majors to make the transition in about 2 years, but even with math majors some of them can't seem to develop the right ways of thinking). Building thinking skills contributes to their being able to "do mathematics"--they don't have to show they can do it all by the end of the year.

03/10/2012 11:42am

"some of them can't seem to develop the right ways of thinking"

My sense is this isn't a concern for someone teaching children in a non-authoritarian way.

Maybe his students can tell Bryan what it means to be Doing Mathematics through these portfolios.

03/12/2012 6:26am

@lsquared I'm not so sure I would discourage Bryan from trying the portfolio route. A lot depends on how well he carries it out. When I was teaching middle school, I always assigned a project which was my way of doing "alternative" assessment. Most of time students just treated it as another asssignment (that they didn't want to do) and as result they didn't get much out it. A few students did show me that they didn't just want to get an "A", they were truly interested in the subject. My guess (which I think is accurate) was that their parents and the type of schooling they had before had a lot to do with it. Were they doing mathematics? Of course not. But they were truly interested in learning about it and had fun with it. Isn't that the goal of precollege math education? I was a math major in college and started teaching High School math right after graduation. But it wasn't until 5 years later when I started teaching middle school that I realized what "doing math" was all about. A workshop leader (madison math, early 1970s) told me that his vision was having students invent mathematics. It didn't dawn on me what he meant until I went home and "invented" Pick's Law for myself. What fun! Never did that in college because all we did was solve hard problems and were constantly being tested. There I learned how to learn math, not invent it. As you said inventing/doing math and developing the right way of thinking is difficult and I would have appreciated a professor like you in college. But 11th grade in high school? Besides the gatekeeper purpose of taking all those advanced courses which students are forced to take, it seems to me that making math interesting to kids has a higher priority than rigorous prep for students to do original research so that eventually they can become college professors is not of the highest priority for Bryan at this point.

03/13/2012 8:11am

Just a couple quick things in response:

1. I'm not trying to get students to think in the "right way." I would say that there are certain habits (as I have outlined) that are common of mathematical thinking. How and when a student/person uses them is still largely a personal and creative endeavor but their use greatly improves your opportunity for mathematical depth in thinking and analysis.

2. The portfolio system would not be absent of feedback for parents and students. I should include more detail about this but, for now, I think I'll leave it at that….still working it out.

3. I guess I am not concerned with getting every student to be an expert mathematician by the end of the year. However, I am concerned with helping them be more aware of the habits of good mathematicians and helping them realize that 1. they are capable of using them and that 2. they are things we can get better at. Progress versus mastery.

Matt Gossmeyer
07/31/2013 4:26pm

I think this is a great idea! I am going to try it this year with a divider for each of the standards for mathematical practice associated with the common core. Then students can select any assignment or project they want and include it in their binder with their citations on the work as to how they demonstrated or need improvement demonstrating the practice. Thanks for the idea....


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