Looking at student work is a big and important part of my job. On the one hand, student work should be a really good indicator of what the student knows and what the student can do. Only, it isn't always. Without a doubt, the best way to understand what a student knows and is able to do is to observe a student at work and/or talk to that student. I have a class of 22 students which is fairly small. Still, in the course of one activity or learning experience it is difficult to spend adequate time with each of my students. Without this time, I'm left unable to tell what each kid really needs to take the learning to the next level. So, I depend, at times, on student work samples.
Looking at student work samples isn't always the best way to assess learning though. Few kids are able to really capture their thinking on paper. As the years pass, I feel like students are coming to me less able to complete high quality written work. There are a number of factors at play. Students' fine-motor skills seem to be decreasing. As a result, many students do not have the physical ability to record everything that is necessary to capture their thinking. Some students do not have the work ethic, stamina, or self-discipline to do the work necessary to capture their thinking. Others, do not really understand how to break down their thinking and translate it into math models and notation on the page. Some students have not been held accountable for producing high quality written work. These students do enough to be "done".
I recently gave my students a few array problems to solve. In each problem, some of the array was visible and some was either missing or obscured. The students had to figure out how many in all given what they could see. The students were instructed to explain how they found their answer. After looking through my students' work I have drawn the following conclusions:
Here is what I wonder: is it really important for students to be able to break down their thinking and show their working, step by step, on paper or is getting the right answer enough?
My gut tells me that the answer if probably some where in between. I know that I want students to be able to break down their steps and make sense of each problem. I know that the simple work we're doing around partitioning and notation is laying the groundwork for the algebraic thinking that will be demanded of my students in the future.
So, I have some work to do. I realize that I have to model more so that my students can better understand my expectations. I see a lot of effective things going on in their work but they're not pulling all the pieces together. My plan for Monday is to take the seven by seven array and ask the kids to talk me through how they solved it. Many just knew that there were 7 rows and 7 columns. Because they know that 7 x 7 = 49, there weren't too many steps to share. My plan is to use this simpler problem to model alternative solutions and appropriate notations. Then, I'd like them to retry the first two problems. I'd like to see my students partitioning and labeling their array. I'd like to see some algebraic notation that matches their partitioning. I'd also like to see evidence of effective and efficient computation. I'm doing this for a few reasons. I'd like to be able to rely on my students work as a window into their thinking. I'd like my students to be aware of their own thinking and to be able to represent it in concrete and visual ways. I want my students to know that I expect them to produce high quality work and that, when they fall short of expectations, they'll be expected to retry until they meet success. Below are a few samples of the student work I received after their first attempt. I'll try to post corresponding student work after Monday.