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Chapter 7 – Option 2 (Connection)

03/05/2019 8:07 AM | Karen Campe

We’ve worked through the 4 routines in the book (and you may have explored 2 more at www.fosteringmathpractices.com).  Which one do you feel most capable of trying? Which one seems to connect well with your goals for math content or student thinking habits?  (Note, we asked about thinking habits in Ch 2 here https://atomicmath.org/Winter2019_Prompts_Responses/699659 ).


  • 03/05/2019 3:27 PM | Alison Foley
    I am going to try the "Capturing Quantities" routine. I am an elementary math coach so I think that routine is the most fitting for younger grades. I know that one struggle so many teachers have is that students are not comprehending problems and then are too quick to jump to any operation to solve them. I think the capturing quantities routine will help students at all grade levels as diagrams are so helpful. Diagrams can help students (even primary students) to visualize the relationships between the numbers. My plan is to work with 1-2 classes at first to practice the routine myself and as I feel more comfortable with it to share with all of our teachers. I went through our math curriculum and found problems at each grade level that I think would work well with the capturing quantities routine. I look forward to trying this with students and then teachers!
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  • 03/06/2019 10:18 AM | Todd Butterworth
    I think Decide and Defend would fit best as a routine to get into the habit of using. I'm often asking my students to critique their and others thinking and giving structure to that practice would be really useful. We have a rotating schedule and I think the 4th day of each cycle would be an ideal time to implement the routine. That would give me three days to wander into new material with the students and then a day for them to use Decide and Defend to reinforce/challenge their thinking so I can see where their misunderstandings lie. As I begin thinking about next year, I think I'll try to see if I can find places to implement the routine. The key for me will be having them in place well in advance so I can think deeply about whatever question I'm asking them and make sure I provide them with appropriately challenging and interesting problems to explore. As I've said before, this is in some ways easier in an AP course where there are student work samples for the kids to analyze, which cuts down on some of the prep work for me. We do have built in collaboration time, so, depending on what classes I'm teaching next year and when we have that time, perhaps that would be a good use of our time.
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    • 03/11/2019 9:48 PM | Sarah Giaquinta
      I can totally see this being a great tool for when we are teaching proofs in geometry. There are so many crazy ways to prove some of the stuff and letting them muddle around in some of the more obscure proofs would be great for them! I'll bring this up tomorrow and we should write this down in our CPT notes for geometry so we don't forget for next year!
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  • 03/07/2019 9:36 AM | Jennifer Rianhard
    Last summer, I worked on creating elementary word problems for grades K-5. I went to another school district to see it in action since they have been using the format they have in place for years and they have seen a great correlation between the word problem work and their scores on the PT section on the state test. I began by going into each classroom three times to model the lesson for the teacher and the students. I have been working on the three reads instructional routine with my teachers and their students. We have the students read and highlight what the question is asking them to do. Then students will work for 10 minutes in a small group or individually, they are allowed to choose. Then they come back together as a whole class to discuss their findings so far. They talk about strategies they used to help them start the problem. The students have an anchor chart hanging in the room of strategies they can choose from to help begin this word problem process. The majority of the students are not close to being done with the problem, which is fine. Sharing information and their math thinking can help another student trigger something that can help them work on their problem. When the students are done completing the problem, they are asked to write about their math thinking. Sentence starters, transition words, and math vocabulary are hung in the room for students to access. The instructional routine, three reads, has been a big part of my year so far, we have more work to do with this routine, but overall, it seems to be heading in the right direction. Next, I would like to have conferences with the kids to see what they thought their strengths and weaknesses were with a specific problem and if there’s anything they need help with.
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  • 03/10/2019 5:03 PM | Katie Chuchul
    Last year when I taught fourth grade, I implemented Contemplate then Calculate consistently and tried out Connecting Reps a few times. I felt that Contemplate then Calculate worked really well for 4th graders and Connecting Reps was a little tricky to start with but students got the hang out of it, especially if the tasks were just right.

    I am thinking of continuing to try Connecting Reps with my 5th graders in my current unit (multiplying and dividing fractions). Having students match the representation to the equation is essential for them to recognize the relationship and difference between multiplying and dividing fractions.

    I also want to try Three Reads and may recommend this routine to Title 1 math tutors and special educators to use with their small groups as well.
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  • 03/11/2019 9:45 PM | Sarah Giaquinta
    I am planning to do more work with the Decide and Defend routine. Out of the all the routines I have tried, this one allowed my precalc classes to get into some more "meaty" material and pull it apart. I often find myself asking my students "but why??" when they are working through the precalc curriculum. It can be so abstract at times, that they forget why they are doing what they are doing. I don't want them to think of the course as memorizing random things, but in fact seeing the connections between all of the topics, especially as we dive into the unit circle, trig graphs, proofs and equations. With this routine, I am hoping to work more on "the mathematical structure of a problem situation." I think this is a great thinking habit to continue to work on throughout the course, as sometimes the structure is not always clear! I spoke a bit in the chapter 6 posts about my experiences with Decide and Defend and trying to get students to not rely on me, but rely on themselves and their partners more. What I also really like about this routine is that is is very applicable to many different topics I teach throughout the course. I want to start it earlier in the year next year, so that by the time we are trying to pull trig proofs apart, the routine isn't the new part and they can concentrate more on the math in front of them. I could see this working well with graphing trig equations and looking at how the transformations work within the graphs. I really like how it forces students to make their own judgments about the work in front of them and really have to back up what they are seeing. It is great to be able to produce something, but to actually be able to explain someone else's work to other students forces them into a much deeper understanding of the material.
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  • 03/12/2019 1:49 PM | Tofer Carlson
    I've played with a couple of the routines this year, but feel the most comfortable with Contemplate then Calculate.

    I use CtC at the start of the school year as a way to get students actively into math (very important for me for students to be working on math from day 1). We've got half days with 30ish minute periods for the first three days, so I did CtC in each class, each day, and then twice more over the course of the next two weeks (Algebra II). I liked the way the low stakes entry points helped make students more likely to participate, and in at least one of the three classes I did this in, that has continued into the year, making it easier to build a class culture of supportive discourse.

    There was a monumental difference in the productivity of this routine dependant on class size. In my two classes that started the year with about 22 students this worked well, and was easy to manage. In the class that started the year with 30 students, this routine didn't work well, I found it hard to impossible to listen to students talking in groups, and in a bizarre turn of events, the students in the huge class were much less willing to take the risk of offering their own thoughts publicly.

    In terms of content goals, I think Connecting Representations connects well with trying to help students form connections between numerical, graphical, and algebraic representations of functions (in Algebra II), but I haven't had time to make my own to spend more time doing the routine. Whenever we have a disruption - snow day/delay, early release, surprise guidance meetings, these extra things, the experimenting, are the things that get cut.
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  • 03/12/2019 2:53 PM | Walter Pohle
    Students struggle when faced with solving a math problem that is embedded in a word problem. All they want to do is cut to the case and solve the problem using some type of operation. All too often, they choose the wrong operation. I feel the "Three Reads" will force the students to slow down and systematically understand fully what question is being asked. I have given the same math problem to students. One as a straight up math problem such as 5 divided by 1/4 and the other (5 divided by 1/4) embedded in a word problem. It was no surprise that students who struggle with comprehension also struggled with the math calculations. Although they were able to do the "straight up" math problem just fine. Once the students learn to read like mathematicians through the use of the Three Reads routine, the math at hand will become more and more in focus.
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  • 03/14/2019 8:51 AM | T Onofreo
    Three Reads connects so well with where the teachers in my school are in teaching problem-solving behaviors. We have begun using Exemplars problems and teaching students how to fully represent their thinking but students still struggle with how to enter the problems and begin their thinking. Three Reads will fill that void in our teaching.
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  • 03/19/2019 9:01 AM | Laura Larson
    I have been able to implement several of the routines one or more times in my classroom, and I continue to find time/pacing the biggest challenge with the "meatier" ones. I started out with CthenC, which I find nice and bite-sized, appropriate for both visual tasks and numeric tasks (works well as a modified Number Talk).

    Then came Connecting Reps, which is really nice for anything bordering on algebra, but I can see where it would work nicely in other areas as well. It's also among the less "intimidating" for students because at its core it feels like matching to them -- I have to really lean on them to focus on the structure that led them to make the connections.

    Decide and Defend is loaded with potential but takes a LONG stretch of time and nearly always seems to have unexpected results (for better or worse), often bringing to light a misconception that never even occurred to me!

    My newest favorite is Capturing Quantities, which we've been using to help students do a better job of defining the variables in a relationship. I really feel like an incomplete/inaccurate understanding of variables is at the center of struggles students have with writing, using, and solving algebraic equations. Again, my biggest constraint here is always time -- as I tell my fellow 7th-grade teachers, I am SLOW at this one, so it always takes me a full class period, which I don't always feel justified in using, but as the benefits begin to be apparent in class and beyond, I might use it more and more.

    CapQ is my favorite because of how it fosters the habits of precise language, using visual models, and annotation. I am also a fan of the routines that have students creating something (a representation, a diagram, etc.) because it automatically increases engagement and is a higher order of thinking.

    Looking forward to working these into my teaching more -- obviously it's a process!
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    • 03/19/2019 1:46 PM | Amy Lucenta
      I love how you talk about unexpected results in Decide and Defend - it does take a while, but it is SO revealing and valuable.
      Interested to hear how students progress with Capturing Quantities - I think you'll see evidence of attention to quantities quickly!
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  • 03/26/2019 2:30 PM | Hala Sahlman
    I have been really loving the connecting representations routine. I love the amount of entry points for all types of learners, while still challenging students to think critically throughout the routine. I also like the part of the routine where students share their connections with the class. Having students verbalize their thought process without pointing is a great strategy to help students explain their thinking in great detail. The next step of having another student repeat the first student's connection not only helps students learn from one another, but also encourages discipline and listening skills which are vital at the middle school level. I will definitely be using this routine for years to come.
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  • 04/30/2019 12:29 PM | Kerin Derosier
    I love the Decide and Defend routine. So often in my class I try to lead the discourse towards an "argument" about which answer makes the most sense when there are a couple answers that have been revealed. I also have done an activity in the past where the students have to figure out if a made-up student has made any mistakes and if so, what they are, and then try to fix them. I like the structure to this routine and the idea that it combines these two things that I already do in my classroom.
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