yTaking Action Book Study: Chapter 6

Chapter 6

Use and Connect Mathematical Representations

Reflect:

After reading Chapter 6, please reflect on the questions below and post your response by Monday. Feel free to respond to any of the questions provided or share something else that you intentionally did differently in regards to connecting mathematical representations.

Please note: the prompts below are to help you reflect. There is not an expectation for you to respond to all {or even any} of the provided questions!

Respond:

Assessing Proficiency to Translate Among Representations

Select a mathematics topic that you are currently teaching and design an assessment to check in on your students' ability to translate between specific modes of representations. Below are some translation suggestions to keep in mind as you design your assessment:

From a tape diagram, array or open number line {visual} to telling a story {contextual}
From an array {visual} to describing it {verbal}
From an array or open number line {visual} to writing an equation {symbolic}
From an equation {symbolic} to explaining the meaning of each number {verbal}
From an equation {symbolic} to telling a story {contextual}
From an equation {symbolic} to drawing a picture or diagram {visual}
From an equation {symbolic} to acting out the operation using objects {physical}

Then give the assessment to your students, analyze it and determine implication for your teaching of that topic.

Which translations were solid for your students as a class?
Which translations were problematic for many students?
What else did you notice as strengths or limitations in your students' representations?
What are some next steps you can implement now to strengthen your students' representational competence?

Interact:

On Tuesday, read your colleagues' reflections and respond to at least one other post by sharing a comment, insight, or interesting possibility by Friday.

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shawnseeleydotcom 44p · 268 weeks ago

Recently in my math classes, I had students make connections between expressions and word problems in small, randomized groups at Wipebooks. Students were told that each word problem connected to only one expression, and that they were expected to be able to explain the connections they made. In addition, students were instructed to create a word problem for the one unmatched expression (reversing directionality like Mr. Martinez).

I designed the task to use similar numbers and included extra information in one of the word problems. I anticipated that students would make incorrect connections between some of the representations and I was not disappointed. Nearly 1/2 of my class' groups disagreed about which word problem was represented with which expression, but all agreed on which expression did not represent each word problem (see photos in link below).

After discussing as a class why we all agreed that certain expressions didn't connect to the word problem representations, I had students share their thinking with someone from a group they disagreed with and asked them to have a "math fight" where they attempted to persuade the other person (MP3). I gave students two minutes to discuss and then we came together as a class, where I asked who had changed their mind. About 1/2 of the students who were incorrect raised their hands and I asked some of them to share out their thinking regarding the representations, the connections they made before, and why they now thought differently. At the end, we took a look at each group's word problems and made connections within those representations. It was awesome!

Link with a PDF handout I gave student groups and some pictures of the activity: https://drive.google.com/open?id=1-CKjZijpicTkJdR...

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Christine Wilson · 267 weeks ago

I have not yet tried a task that includes representations but have in the past asked students to draw models or make connections with word problems by "visualizing" the situation. I make connections to visual representations often when I'm teaching. It's usually with measurement like how long things are. I do this as a way to help students evaluate their answers to make sure they "make sense." I also know that our curriculum has students write expression that represent word problems. One lesson asks them to write "situation equations" and then "solution equations." I look forward to trying to work the practice of making representations. I think it's very valuable to help students engage in them to create clearer conceptual understanding. Especially when working with decimals and fractions within multiplication and division. It gets very tricky with these types of numbers.

1 reply · active 267 weeks ago

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Caty Carino (TSD) · 267 weeks ago

Hi Christine,

I know I felt the same way when reflecting on my teaching and using models, visualizing, pictures. I feel like I heavily model drawing pictures during fractions, measurement, and geometry. But then I lack the pictures during multiplication and division because it is harder to visualize those. I do see the benefits of using drawings and having students analyze the problem visually reeks benefits because then it helps them understand what is going on.

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Caty Carino · 267 weeks ago

While reading this chapter one thing that resonated with me is the use of drawings and models in math. I have seen this in the classroom where students are hesitant to draw a picture or make a model of a problem. I know that math is taught to be quick and efficient, which I am not against in any way. But I do believe drawing models and representations can help create a deeper understanding of the problem. I especially try to do this when I am modeling a performance task or a word problem, so students can see that a model sometimes helps us understand what is going on in the problem. Especially when we dive into writing equations.

For unit 5, measurement, my team and I decided that we wanted to do a clothesline with metric units of length. Students will get different cards like 1 m, 1 km, 80cm, etc. and they will need to place the cards in order or what they think might be in order prior to learning about the metric system. Then our goal is to teach about the metric system and revisit our clothesline to see if we need to make any adjustments. I think having this visual representation will help others understand the values of metric measurement better and also offers the opportunity for them to edit and revise their work.

2 replies · active 255 weeks ago

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shawnseeleydotcom 44p · 267 weeks ago

Caty,

I agree with drawing models. I think that too often, we race to the quick and efficient methods with students (and so do parents) because we hate to see kids frustrated and struggling. Some of my students think drawing models is a waste of time, but then I see those same students making errors that show me they don't have conceptual understanding. I'm wondering how much modeling and drawing is going on in primary grades.

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Susan Heater (TSD) · 255 weeks ago

I did a clothesline for our lesson on place value. I had benchmark numbers and allowed the kids to place their numbers on the clothesline. I think this representation made it so much more visual and hands-on for the kids. They also seemed to be very engaged!

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Julie Rodriquez · 267 weeks ago

As with the previous chapters, I had MANY take-aways from Chapter 6, but I will focus on my top two. First, when I read about assessing students' knowledge of representations on pages 121 - 123, I thought to myself, " This is so simplistic but so valuable." We are halfway through Unit 5 (Division of Whole Numbers and Decimals), so I decided to do a division check-in with both whole numbers and with decimals on our first day back after break. I gathered a lot of data about my students from giving them three check-ins today. I wasn't surprised that drawing a picture/model would be the most challenging. I was pleasantly surprised that the majority of my students are able to use an algorithm to solve division of whole numbers and division of a decimal by a whole number. The majority of my students were able to write a realistic, meaningful story for the division of a whole number by a whole number by when they had to write a story involving a decimal by a whole number or a whole number by a decimal, they still need a lot of guidance. This check-in provided me feedback on my next steps and also gave me new tools to share with the class. I would like to try the strategy of giving my students some of the pictures created and lead a math discussion comparing and contrasting the representations. I think it could also work to analyze one or two pictures that are not quite finished and discuss how we might make them work for the situation.

The second take-away I had was on page 141. I love the idea of incorporating mathematical experiences into a real-world, home-town cultural task. I think the kids would get really engaged in an activity like this one, especially because I am currently having my students create a town, neighborhood, and their own dream house. They are using all kinds of math to create their work but are so enthusiastic about their work! They are having to represent polygons, scales, and more, and will eventually create the area (floorplan) and volume (house structure) for this project. I had not thought of adding the component of creating mathematical stories to go along with their project, but this would add another component of representations to the students' work.

I'm excited to continue incorporating a variety of representations into my lessons and helping students build their conceptual understanding of mathematics.

2 replies · active 265 weeks ago

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Stephanie Clement · 267 weeks ago

I like your idea of a check-in. What valuable information for what to focus on next. Making a representation is definitely the hardest, but if we focus on doing these more I'm sure we will see a lot of growth.

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Meribeth · 265 weeks ago

I totally agree with you statement about assessing student knowledge of representations. I constantly worry about whether or not my students are actually listening to what their peers are sharing during math as they explain their thinking when solving problems. I am going to prepare a set of sticky stems (Figure 5.4) for kids to place on their math books. Hopefully the stems will result in greater oral participation and a broader set representations.

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Stephanie Clement · 267 weeks ago

One thing that really stood out to me in Chapter 6 is the importance of teaching and encouraging representations. My students would benefit and deeper their understanding of content by making a representation of the skill being asked. This will not only benefit them now, but as the skills become more challenging and the scaffolding happens, they will have a better understanding of how it works.

I would like to spend more time on representations. Giving the students more practice and purposefully sharing in a sequential order their strategies/work. I love giving my students the opportunity to be exposed to a variety of strategies and learning from others.

My new focus will be on giving students more opportunities to show representations. This way I will know that they understand the concept, not just the procedure.

1 reply · active 253 weeks ago

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Julie Rodriquez · 253 weeks ago

Stephanie,
Your last statement speaks volumes - when students create representations of their learning, it shows conceptual understanding (or that they do not conceptually understanding, which is just as valuable for educators). I agree with you that students mastery and retention of new learning is deeper when they are able to represent their understanding in a variety of ways. It also deepens the understanding of their peers when they have see many ways to visually represent their work.

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Meribeth Rowe · 265 weeks ago

Chapter six opened my eyes to representations. I am seeing so many new ways to assess student understanding with a variety of representations. For my struggling math learners I can see them making the same mistakes as Dylan and Mateo on the ALT 6.1 task. My current cluster of ELL students are also MAP students and struggle with the verbal part of mathematics. I enjoyed the tape diagram concept since it so clearly links with addition and subtraction comparison bar diagrams in our current unit.

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Susan Heater (TSD) · 255 weeks ago

The part of chapter 6 that stood out to me whas 6.3 Shifting to a Focus on Representations. The example they give about Ms. Robinson's question: Clare has 12 bears and 8 chairs. If all of the bears want to sit on a chair, how many bears won't get a chair?" makes me think of our lessons using comparison bars in 3rd grade. What i would like to do is allow for more flexibility in those lessons and let them go beyond comparison bars. I would like to have them use comparison bars, but also counters, drawings, and number paths. I think that the comparison bars can be too abstract for some of the kids, and starting with a more concrete manipulative would help scaffold them in the right direction. They can still share their work with a partner that approached the problem a different way. I can share the students' work from the most concrete to the most abstract, and encourage them to use a different form of representation the next time they are solving a similar problem.

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