Wednesday, September 26, 2012

Implementing Rich Mathematical Tasks

My school system is partially through the transition to Common Core Mathematics.  Our lower elementary grades have fully transitioned, HS Geometry has transitioned, and 8th grade has transitioned to the point where we're teaching both the old and new together this year!  As part of the transition, 8th grade students who previously passed Algebra I in 7th grade are enrolled in a course we've titled "Intermediate Algebra."  The course resembles traditional Algebra II in a lot of ways and is designed to fill the gap that would be made by students moving directly from our state's Algebra I/Data Analysis curriculum into CC Algebra II.

One of the expectations for the course this year is that I provide my students with rich mathematical tasks with an open-ended quality.  I was a bit hesitant to get started with these as the students I teach haven't had many experiences with open-ended problem solving.  I feared they would be frustrated by the lack of rigid structure and be very needy.

Here's how I organized the task process in my room.
  1. I presented the concept of a mathematical task as different from a BCR (the style of constructed response used on MD state testing which is quite formulaic) and explained to them that there are so many correct ways to answer a task that they should answer a task in the way that makes the most sense to them mathematically.  I stressed that I was not going to be answering lots of questions during this task, so they were to rely on their group and "figure it out."  (I still coached a minimal amount when groups were quite astray, but I wanted them to feel independent.)
  2. I assigned groups of four, making sure each group had at least one or two strong math students and a mix of boys and girls.  (Side note: I use pocket charts for grouping and it has been a breath of fresh air this year.  Something about this system, perhaps the groups being on the wall when kids enter the room, has kept the whining/eye rolling/ugliness entirely at bay this year.  I so much as mention the word "group" and their little heads spin to the back wall to check out who they'll be with, but they don't complain!)    
  3. I presented the task they were responsible for, indicated the variety of materials available, and set them to work.  
  4. After 30 minutes the first day, none of the groups were even close to fully answering the question.  Most had talked for 90% of their time and written very little.  All the groups stayed on topic and remained focused during that time frame.  
  5. The next day, I made some general comments about the task and gave a small insight or two into the problem to give a few hints related to common misconceptions.  The groups reconvened and were charged with finishing their answer. 
  6. We did a jigsaw grouping to share out.  Two students stayed put and presented their group's work to two students who came from a different group.  We rotated again so all students had a chance to present and a chance to review another group's work.  
  7. The original groups had some time to debrief about what they would change after getting input from other classmates.
  8. Homework was to write a reflection on the process. 
The tasks I'm using this year are from the Dana Center.  This is a great PDF resource to download if you teach Algebra I or II, or possibly even pre-Algebra with motivated students.  You can scaffold more or less depending on your group of students and the relative difficulty of the task.  This week, students worked in groups of four on the "Extracurricular Activities" task, part B.  The task gave students a scenario and asked them to write a function to model the scenario and find its domain and range.  From my description, I'm sure you're wondering why I needed to devote even 15 minutes to this problem.  We actually spent nearly 90 minutes in class spread over two days plus one night's homework on this task.  Here's why: 
  • The problem is embedded into the task in a way that students must make meaning of the situation.  Since the meaning wasn't immediately clear to the students, most every group restarted or made significant revisions to their work during the process.  (Make sense of problems and persevere in solving them.)
  • Students have various ideas about the meaning of the problem and how to go about solving it, so they have rich discussions. My students were holding each other accountable for their ideas, asking pertinent questions to better understand their classmates' points of view.  (Construct viable arguments and critique the reasoning of others.)
  • Students were able to use any of the materials in the room, so some groups gravitated toward calculators, while others preferred to sketch a graph by hand or write out a table.   (Use appropriate tools strategically.)
  • The real world context leveled the playing field in an unexpected way in at least one group.  I overheard one student who is a weak math student leading his group through this task at the beginning because he was the only one with the real-world background knowledge to understand the problem.  His more "book smart" teammates were lost without his guidance.   He was more engaged in this task than I've seen him since school started. 
I was very proud of my students this week.  I wasn't really looking for their work as the final product.  I was hoping to see them demonstrate the eight Standards for Mathematical Practice.   I highlighted a few above, but I can honestly say that I saw all 8 standards in play during this process. 

Some students were already asking to do more tasks on Tuesday before we finished our first task.  They loved the interaction and I loved how involved they felt in solving the problem.  

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