Friday, August 31, 2018

Functions in Multiple Representations Task


I am a big fan of complex instruction (CI) tasks. And I've written about my appreciation of CI tasks and how I'm forever grateful that I went to a training by Laura Evans to learn about CI.

But today I asked myself why. CI takes a lot of work. First, I had to modify the (pretty good) worksheet a coworker made into the CI task. Then, I arrived at work super early so I could cut up the task cards. I have two classes each with six groups of four. (I recognize my class size is a huge luxury.) But it was still a lot of cutting! Finally, for me, CI is a very active way to teach. I am constantly rotating around the room or calling a particular group role for a huddle or asking checkpoint questions. It's controlled chaos. But it's still chaos. And I'm exhausted at the end of it--in a good way.

School is status. And status is school. And I recognize the power of CI tasks in minimizing the effect of status. In school students are assigned status because of race, socioeconomic status, gender, etc. There is also academic status. Especially in mathematics, being perceived as smart has a high-status connotation. And this complicates the learning. If a student feels low status, either because they self-assigned as low status or because peers/teachers assigned that student as low status, then that decreases the student's willingness to take academic risks. And if the student isn't taking academic risks, then that has a negative impact on learning.

Today was the first time I implemented a CI task with my 9th grade algebra 1 students. It is the second week of school and we had been working on learning goals related to functions for the previous four class periods. We have a block schedule and I see them every other day. And today was the day I remembered why I love CI tasks.



I watched my students persevere on this task for 45 minutes. And no one was off task. Students were thanking each other for asking good questions. Students were critiquing each other's thinking. It was great!

The Task

The goal of this task is that students represent functions in multiple representations. There are four levels and each level has a checkpoint question. Most groups get to level 2. And I'm totally fine with that because they are having amazing conversations and challenging each other to understand--not just move on. 

In each level, students are given four representations of the same function. Some of the representations are filled out and others are left blank and students have to fill in the missing representations. 


Students need to know how to represent functions using function notation. It also helps if they have some math 8/linearity exposure, but I think they can get by without it.

What Makes it CI?

I'm not an expert in complex instruction. But I've been experimenting with it for a few years now. I usually do a CI task every three or four weeks. I know other schools where all teachers use a CI model. I think that's awesome, but not possible in my context. CI can still have a powerful effect on minimizing the effect of status in my classroom even if I use it occasionally. 

The intent of CI is to reduce the effect status has on student learning. It's all about status! For me, there are four things that 'make' a CI task.

A group worthy task. You can't give a team a worksheet and expect CI magic to happen. The task needs to be group worthy. Group-worthy tasks are math tasks that can not be done by an individual or a pair. Group-worthy, CI tasks require students to use higher-order thinking skills and engage students in non-routine mathematics that creates a need for students to negotiate meaning with each other to complete the task. However, not all the tasks I give are group-worthy and not all the mathematics I teach lends itself to CI tasks. Hence, I do CI when I can!  

Meaningful, useful group roles.  I'm still experimenting with names for the roles but, in general, the roles are:
    • "Energizer"--this is super important role because this person does the team shoutouts (more on that below)
    • "Facilitator"--the purpose of this role is to have someone coaching the team on the checkpoint question (or whatever check for understanding I"m using)
    • "Liason"--the purpose of this role is to 'do stuff'. Ask other teams for help, get the next task cart, ask the teacher team questions, etc. 
    • "Unity"--the purpose of this role is to keep the team together. 

I think I can improve a lot in terms of facilitating and implementing group roles. These are pretty basic. I do have a few sentence frames for each role which helps students understand what success sounds like. I also do team huddles and that helps reinforce the expectation of group roles. 

Assigning Competence. This is the part of CI that really helps address status issues. There are a few CI tricks I use to assign competence. 
  • Shoutouts are HUGE and happen on two levels. Typically, I start students on the task. As they work, I'll circulate to look for students that are taking academic risks by asking questions or asking for another explanation. About 5 minutes into the task (read the room), I stop everyone. [I try to be dramatic when I do this] All students need to put their pencils down. All eyes on me. Then I'll thank 2 or 3 students publically for taking risks and I will be specific in my acknowledgment. "I want to thank Remy for asking his group to explain function notation a second time. I appreciate that he took a risk by asking for another explanation." When I do this, I'm assigning competence to the student (as I define competence) and I'm disrupting the status dynamic. I tend to shoutout students that are low-status OR I shoutout high-status students for doing things that are not usually seen as high-status in a math class (going slow, focusing on process not product, asking questions or sharing confusions). 
  • The other form of shoutouts is team shoutouts. After students have been working for 10 minutes or so, I'll call the Energizer person from each group. We will do a huddle and I will explain that they are going to do the next round of shoutouts. I'm careful to explain that they still need to be doing math with the group, but now they have a new job as well: to look for students taking risks or communicating productively and be ready to share that with the class. I send them back to their groups and give them a few minutes to think about it. Then, I have the entire class stops again. Each Energizer publically recognizes someone on their team. It is heartwarming to hear 9th graders say things like "I want to shoutout Marlon for asking a question when he was confused. He took a risk." Seriously, I get teary eyed when they do this. It's wonderful. 
  • I've mentioned team huddle already. This is a way to enforce group roles and assign competence. I try to do two huddles per task. I will usually call the facilitators up and talk about an important part of the task. In the Functions in Multiple Representations task, I call them up to talk about how to find the input for a given function when they are given the output (solving). Most students haven't seen this before, but understand it once they see the connection to solving. I'll call facilitators up, I'll show them an example on the board, I'll leave the example up, then have them go back to their group and explain it. If I notice groups getting stuck on a certain part of the task, I'll have a team huddle with facilitators. 
Reinforcing Positive Behaviors. I want to amplify the good work students are doing, not call attention to the negative. The actionable norm stamp card is something I stole from Laura Evans. And it is an amazing way to reinforce positive behaviors. 


Before we start the task, I'll explain to students that there are a few behaviors that I think will help them be successful in this task. And, as they are working, when I see them demonstrating these positive behaviors during the task, I'll give them a stamp. Yes, I work with high school students. And, yes, they will do anything for a stamp. The stamp DOES NOT equal points. It is just a token of acknowledgment for demonstrating positive behavior. 

Again, I am specific in my acknowledgment. I don't just stamp. I say something like, "thanks, Julie and Tim, I love how you pushed through your point of confusion" or "wow, I see everyone in this group leaning in and persistently working on this problem. Thank you!". Then I give a stamp to the individual or individuals that were demonstrating the positive behavior. 

Implementing this Task

Shoutouts, team huddles, stamping, checkpoint questions: does all this take a lot of class time. Yes! Is it worth it? YES! Because even though I'm taking class time for non-math work, the time they have to do math work is so much more productive FOR ALL STUDENTS!! 

For this task, I cut everything up.


I cut each multi rep page up so that each group member can work on the task. 

Mistake on the graph! We had a good conversation. 

At the end of each level, groups call me over. I randomly select one student from the group to answer the checkpoint question. If they get it right, the group can move to the next level. If they get it wrong, I walk away and they need to practice more. 

Do I grade any of this? No. Is there points involved at any point? No. Do students ask if I'm going to grade this? No. Do students ask if they get more points for more stamps? No. 

The task is linked below. If you try it, let me know how it goes! And if you modify it, please share your new version with me! I'd love to see how you improved the task. 

Here are the Functions in Multiple Representations Task as a word and PDF file:


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