Hey, instructional routines, where are you?

This is my midyear wake up call.

At the beginning of the school year I developed some goals. They were ambitious, to say the least. It was around the start of December that I realized how unreasonable my expectations were for 2016-17. I’ve been in the game for a while, I should’ve known better. Shame on me.

Despite my lack of judgement and setting myself up for failure, three of my goals came with a higher priority for me. This post serves as a self-critique on my progress towards one of those three: my use of instructional routines.

I worked on these routines a lot last summer through New Visions (here and here) and at TMC16. I had a crazy vision that they would transform my teaching this year. They were going to help my students leverage mathematical structure like never before. Being routines, I was going to get better at using them as the year progressed. I was going to learn to lean on them.

Well, next week the first semester is coming to a close and my use of them has been pitiful. Sure, my first unit in algebra 2 held much promise. I used the routines five times over the course of a few weeks, which was a huge win in my book. I started off strong. Slowly, though, I got bogged down with the curriculum. I got consumed with more immediate concerns and stressors, like being at a new school, running around to three different classrooms during the day. In the meantime, I forgot all about the instructional routines that I so zealously committed myself to back in September. I’ve used Connecting Representations one time since that first unit. I haven’t used Contemplate then Calculate at all.

Accepting this isn’t easy because of how much I really wanted to use these routines. That said, I know that it’s normal to unintentionally forget about goals. But I also know that if I recognize the struggle, write about it, and let it breath, I can begin realigning myself to the vision I had back in September.

What’s great is that I’ll soon begin digging into Routines for Reasoning and have scheduled a workshop with New Visions, both of which should help me find the routines that I so desperately want to implement.



End behavior of functions via Connecting Representations

As the second and final assignment for the Structured Inquiry course I’m taking with New Visions, I was asked to create a task using the Connection Representations instructional routine (#ConnectingReps). More on Instrutional Routines here.

The instructor, Kaitlin Ruggiero, mentioned that multiple choice questions are good starting points for developing these tasks. Adopting her suggestion, I used #4 from the June 2016 Algebra 2 Regents Exam. The question focuses on roots and end behavior of a function. (F.IF.8). I chose to narrow my focus to strictly end behavior.

Here are the first set of representations, graphs of several polynomial functions:

Here are the second set of representations, statements about the end behavior of each graph:

During rehearsal, I showed graphs A, B, and D and their corresponding end behavior statements. We followed the routine to match the representations. I then revealed graph C and had them come up with the statement, which is 4. Lastly, the class reflected on what they learned using meta-reflection prompts.

I don’t have a formal write-up of the activity, but here are the above images.


  • I designed this for my algebra 2 class. My gut is telling me that it may fit in well at the beginning of my rational and polynomial functions unit. I may also consider using it if/when we review domain and range.
  • Initial noticings about the graphs had more to do with the “inner behavior” rather than the end behavior. In other words, the class was drawn to the minima, maxima, and roots.
  • There was some blank stares when I revealed the statements. This will most likely happen with students, too. The mapping symbol (i.e. function arrow) can be confusing if you’ve never seen it before. But that was the point.
  • Most of the class chunked all of the “x approaches…” statements and realized that they were the same in each representation. Since two of the given graphs (A and B) had both ends going to either positive or negative infinity and the other graph (D) didn’t, this led them to conclude that graph D had to match with statement 2. From there they reasoned that since graph A is going up on both ends, it should match with statement 1. Similar reasoning was used to match graph C with statement 4.
  • By giving the class three graphs and three statements, the third match (C and 4) was kind of boring. I still made them justify why C and 4 matched, but it didn’t feel as meaningful.
  • In retrospect, I wouldn’t change any of the representations, but I would revise what I give the class and what I have them construct on their own in order to help them move between representations more fluidly:
    • Give only A and B and their matching statements (1 and 3). Students reason through the matches.
    • Then give graph C and have them construct the corresponding statement (which is statement 4).
    • As an extension, I would give statement 2 and have them sketch a graph that goes with it. All student graphs will be similar to graph D, but the “inner behavior” will be all over the place. Because of the infinite number of possible correct responses, we could show several graphs under the Elmo to guide this part of the routine. I could save these student-generated graphs for later analysis on other properties, including even/odd, roots, maxima/minima, etc.
  • I didn’t use the words chunk, change, or connect at any point during the rehearsal. This is somewhat disappointing since I want my students to use these terms to describe their reasoning during this routine. Mental note taken.
  • Instead of me selecting the next presenter, sometimes I should allow the student that just presented to choose. This is student-centric and I like it (when appropriate).
  • Compared to Contemplate then Calculate, I feel that Connecting Representations is a slightly more complex in nature. With that said, Connecting Representations uses matching, which is really user-friendly. Both routines emphasize mathematical structure, but it seems to me like Connecting Representations emphasizes structure between different representations while Contemplate then Calculate focuses on structure within a representation. Dylan Kane and Nicole Hansen hinted at this during TMC16.
  • Though I didn’t use Connecting Representations, this past spring I foreshadowed this work with my Sigma notation lesson. Given two representations (sigma notation and its expanded sum), students used reasoning to connect the two.
  • This in-depth experience with both of these routines will allow my students to surface and leverage mathematical structure through inquiry like never before. So exciting!





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