common!corestatestandards!mathematics!5:!...

Common Core State Standards Mathematics 5: Purposeful Pedagogy and Discourse

October 15, 2012 Dr. Linda K. Griffith, University of Central Arkansas

This session will discuss the Purposeful Pedagogy and Discourse Model that is the centerpiece of new Arkansas mathematics professional development and how to support teachers as they implement the Common Core Mathematics standards.

Topics covered:

• The instructional model defined by the article Purposeful Pedagogy and Discourse Model

• A vision for what mathematics instruction looks like when it supports the Common Core Mathematics standards

• How planning for mathematics instruction needs to change in light of the Common Core

• Creating a mathematics professional development plan for your school; awareness of PD resources that are available

38 PARCC Model Content Frameworks for Mathematics Version 2.0—August 31, 2012 (revised)

Key: Major Clusters; Supporting Clusters; Additional Clusters

Examples of Linking Supporting Clusters to the Major Work of the Grade Know that there are numbers that are not rational, and approximate them by rational numbers:

Work with the number system in this grade (8.NS.1–2) is intimately related to work with radicals (8.EE.2), and both of these may be connected to the Pythagorean theorem (8.G, second cluster) as well as to volume problems (8.G.9), e.g., in which a cube has known volume but unknown edge lengths.

Use functions to model relationships between quantities: The work in this cluster involves functions for modeling linear relationships and rate of change/initial value, which supports work with proportional relationships and setting up linear equations.

Investigate patterns of association in bivariate data: Looking for patterns in scatterplots and using linear models to describe data are directly connected to the work in the Expressions and Equations clusters. Together, these represent a connection to the Standard for Mathematical Practice, MP.4: Model with mathematics.

The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.. Expressions and Equations

Work with radicals and integer exponents..

Understand the connections between proportional relationships, lines and linear equations..

Analyze and solve linear equations and pairs of simultaneous linear equations..

Functions

Define, evaluate and compare functions..

Use functions to model relationships between quantities..

Geometry

Understand congruence and similarity using physical models, transparencies or geometry software..

Understand and apply the Pythagorean Theorem..

Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.. Statistics and Probability

Investigate patterns of association in bivariate data..

Written by Linda Jaslow in collaboration with Aimee L. Evans

Purposeful Pedagogy and Discourse Instructional Model: Student Thinking Matters Most In studying the Common Core State Standards for Mathematics (CCSSM), and in particular the Standards for Mathematical Practice (SMP), it becomes clear that what we do in the classroom will change both from the perspective of the teacher and the student. The teacher will need a deep and connected understanding of the mathematics content and, during instruction, will need to provide experiences that allow the students to construct meaning for themselves through carefully crafted tasks and conversations. Students will need to reason, communicate, generalize and challenge the mathematical thinking of themselves and others. Student thinking matters most. The purposeful pedagogy and discourse instructional model that we are using in the Arkansas CCSS Mathematics Professional Development Project, is based on the research of four sets of researchers:

• Jacobs, Lamb, and Philipp on professional noticing and professional responding; • Smith, Stein, Hughes, and Engle on orchestrating productive mathematical discussions; • Ball, Hill, and Thames on types of teacher mathematical knowledge; • Levi and Behrend (Teacher Development Group) on Purposeful Pedagogy Model for Cognitively

Guided Instruction. This model is intended to support teachers to deliver strong mathematical content using critical best classroom practices as well as to develop a learning environment where their students regularly use the 8 Standards for Mathematical Practice. Assessing Students, Professional Noticing, and Teacher Mathematical Knowledge At the core of our model is assessing students (TDG-‐CGI model), which refers to taking a close look at student understanding. While assessing students, we apply the concept of professional noticing (Jacobs et al.). Professional noticing is comprised of 3 teacher skills:

• Attending to children’s strategies, • Interpreting children’s understanding, and • Deciding on how to respond on the basis of children’s understanding.

In order to assess a students’ understanding, we must look at the details of their thinking (what did they do) and then mathematically interpret these details. While this may seem trivial, students’ strategies are complex and many deep mathematical operations and properties are embedded implicitly in their work. It takes time to identify the important details in students’ thinking and then mathematically interpret the relationships and properties of operations that are embedded. The ability to notice will help the teacher identify the mathematics available for exploration during the lesson(s) to follow. Since student thinking matters most, in the Arkansas professional development courses the beginning of most classes will involve just making sense of and deepening our understanding of the details of students’ strategies and the mathematical ideas embedded in their strategies. The deeper and more connected a teacher’s mathematical knowledge is, the easier it is to see and interpret the details of student thinking. Teaching mathematics requires a variety of types of knowledge as shown in Figure 1.


One type of teacher mathematical knowledge is specialized content knowledge – the mathematics behind the mathematics. For example, it is not enough to know we can divide fractions by inverting the second fraction and multiplying. A teacher must understand the mathematics that allows that strategy to work. Teachers must also understand how

children will approach various problems, how their thinking develops, and how students’ thinking is different than adults’ thinking. This knowledge is called knowledge of content and students. All of this comes together to create the critical part of professional noticing, identifying the details of children’s thinking and mathematically interpreting the details, which allows us to assess students’ thinking, which of course matters above all else. Exercising Professional Noticing A fourth grade student solved the following problem: Kathy is making ____ cupcakes. She put ____ cups of frosting on each cupcake. How many cans of frosting will she need to make her cupcakes? Two sets of numbers: (36, ¼) (36, ¾)

What did this student do? What big mathematical ideas are embedded in her strategy? Take a few minutes to follow her trail of thinking. How would you mathematically notate her reasoning? See Figure 2. What is it that teachers have to know to be able to understand the mathematics of this students thinking? It is not enough to know the properties of operations, teachers need to have a deeper understanding of

Figure 1: Domain map for mathematics knowledge (Hill & Ball)

Figure 2: Student work on 36 x 1/4


these properties and be able to interpret this important mathematics embedded in student informal strategies. To solve the problem using the first set of numbers, the student first transformed the problem with commutative property 36 x ¼ = ¼ x 36. She then solved by first finding that ½ of 36 = 18, and then finding that ½ x 18 = 9. What mathematics allows for this sequence of thinking? 36 x ¼ = ¼ x 36 Commutative property ¼ x 36 = (½ x ½) x 36 Decomposing (½ x ½) x 36 = ½ x (½ x 36) Associative Property The student then used the relationship between ¼ and ¾ to solve the problem with the other set of numbers. Professional Responding, Purposeful Pedagogy, and Orchestrating Classroom Discourse Critical instructional decisions are based on the mathematical interpretation of students understanding. With specialized content knowledge and knowledge of content and students in place, we are ready to focus on our mathematical practice. The Purposeful Pedagogy Model (TDG; Cognitively Guided Instruction) and Orchestrating Classroom Discourse (Stein et al.) come together to give us a vision of such practice centered around the all important student thinking. The Purposeful Pedagogy Model has three components: assess students, set a learning goal, and design instruction. Elements for the design of the instruction are defined by the Orchestrating Classroom Discourse research. Orchestrating Classroom Discourse outlines 5 practices for doing so:

1. Anticipating likely student responses to cognitively demanding mathematical tasks; 2. Monitoring students’ responses to the tasks during the explore phase; 3. Selecting particular students to present their mathematical responses during the discuss-‐and-‐

summarize phase; 4. Purposefully sequencing the student responses that will be displayed; 5. Helping the class make mathematical connections between different students’ responses and

between students’ responses and the key ideas. We will use the details of student understanding to set learning goals for our students, design instruction, and orchestrate classroom discourse. In doing so, we are engaging in the comprehensive practice of professional responding. This is best understood by taking a look at a classroom vignette from Kindergarten. The students in this class have been solving problems that begin with 10 and add some more. The teacher has elected to present this problem by beginning with an amount other than 10 and then adding on 10 to see how students will respond. Before reading the classroom exchange and the teacher’s


professional responding, look at the student work from a kindergarten class for the following problem and answer these questions for yourself: Zayeqwain had 6 pennies. He gets 10 more. How many pennies does he have now?

• What did the students do? • What is the mathematics embedded in their strategies? • How are the strategies alike and different? • Why do you think the teacher would have selected these two students to share? • What conversation do you think the teacher would like to have?

Pretty Moniqua Classroom Vignette The classroom teacher, Mrs. J asked the two students to share their solutions with the class and then engaged the class in a discussion around their strategies. Pretty: There are 10 [pennies], (then she counted on) 11, 12, 13, 14, 15, 16. Moniqua: There are 6, (then counted) 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. And look I came up with 2

number sentences (excitedly) 6 + 10 = 16 and 10 + 6 = 16. See I can do it two ways!

Mrs. J: Look at these two strategies. Are they alike or different? Sandia : They are alike. They both counted up. Mrs. J.: I can see that they both used a counting up strategy. What do the rest of you think? Theo: No, they are not alike. They started counting from a different number. Moniqua started

counting from 6 and Pretty started counting from 10. Mrs. J.: (pointing at Moniqua’s number sentences) So, which one of Moniqua’s number sentences

go with the problem? Class: 6 + 10 = 16. Mrs. J.: Why? Claudette: Because Zayeqwain has 6 pennies and then he gets 10 more. Mrs. J.: Do any of these number sentences represent Pretty’s strategy? Claudette: 10 + 6 = 16.


Mrs. J.: Why? Maria: Because she started with 10 first and then added her 6 seconds. Mrs. J.: Is that okay to do? Class: Yes. No. (mixed answers) Mrs. J.: Will they both get the same answer? Cecilia: I just counted it on my fingers. They are both 16. (The class is surprised.) Mrs. J.: Really? Do you think this was just an accident, or do you think this will always happen? Cecilia: It won’t always happen, just on this problem. Mrs. J decided to stop after this exchange and let her students’ ideas percolate. About a week later, when she posed a similar problem (4 + 10), five additional students switched the order of the numbers to solve the problem and counted on from 10 instead of 4, utilizing the commutative property of addition. After further discussion, many of the students were beginning to think that this might be something that would always work. How is this episode related to the purposeful pedagogy and discourse instructional model? The teacher posed a problem to her class and allowed the students to solve the problem the way that made sense to them. She identified student work that had the potential to help her students discover and make sense of an important mathematics concept. Specifically, when Pretty counted on from the larger number, the teacher understood Pretty’s strategy was based on the commutative property. The teacher also noticed Moniqua’s number sentences, 6 + 10 = 16 and 10 + 6 = 16. Based on her analysis and observation, she made an instructional decision to use this as an opportunity to have class discussion about the commutative property and how number sentences relate to the structure of the problem. As opposed to telling the students that this was a “turn around fact” or to “just count on from the larger number,” she put the students in the position to consider these complex ideas for themselves by facilitating the dialogue to help them make meaning connected to their existing thinking. While this type of exchange requires the classroom teacher to think very purposefully about instructional decisions and to think deeply about the mathematics embedded in students’ solutions, the effort is worthwhile. The evidence comes from Cognitively Guided Instruction, an instructional model that emphasizes these very practices. Visits to CGI classrooms in Arkansas will reveal that children are thinking more deeply and flexibly about mathematics. They are not simply solving problems that have no meaning to them; they are becoming young mathematicians capable of explaining their thinking, which matters most, and grappling with and making sense of the complexity of the mathematics. How do we now take the information we have about students’ thinking and professionally respond in a way that is based on students’ understanding and designed to facilitate children’s thinking along a learning trajectory? We must select or design appropriate mathematical tasks or problems. Mathematical tasks should be selected that will facilitate children’s development. Once we have identified the task, we should consider the following questions:

• What do we anticipate students will do with the task? • Will this task provide the experiences needed to further students’ development? • Which of the strategies we expect are likely to help the most in making sense of the

mathematics in the goals we have set for them?


The next stage is to pose the task or problem and allow the students to solve the problem in a way that makes the most sense to them. Our job is to monitor students to identify what students are doing, guide them as they work, and decide which students’ papers should be shared. Back to Teacher Mathematical Knowledge Once we have identified the best student strategies to meet the learning goals, we need to decide in which order to share students’ strategies and what mathematical connections should be the focus of the classroom discussion. Again, the teacher’s mathematical knowledge, specifically her knowledge of content and teaching (Hill & Ball, Figure 1), will be critical in making decisions by being able to envision how the mathematics available through the students’ strategies connect to one another and to the mathematics concepts that are desired. At this juncture, the teacher’s knowledge of the mathematics meets the need to design or plan the discourse to take students deeper into the mathematics. This involves both the sequencing of the presentation and also the selection and phrasing of the questions posed during the discourse. There are likely multiple productive paths, but there are certainly some unproductive or problematic paths as well, and the teacher will need to choose well. Student thinking matters most. Seeing It All Together The research of these four sets of researchers come together to create the instructional model that we are using in the courses for the Arkansas CCSS Mathematics Professional Development Project. While this model, being a blend of the work of so many different projects, may seem complex at first, it is perhaps more straight forward when viewed using the graphic organizer below. The key ideas that hold the model together are the importance of noticing the details of student thinking, interpreting those details, and using that information to design instruction comprised of discourse around student strategies aimed at a specific mathematical goal. In other words, what details do we see in our children’s work, how do we interpret their thinking, and where mathematically do we go from there? In maintaining this focus throughout the professional development courses, it is our hope to support teachers in their journey toward achieving mathematical proficiency for their students as described in the CCSSM. And always remember, student thinking matters most.

REFERENCES:

Hill, H., Ball, D. L., & Schilling, S. (2008). Unpacking “pedagogical content knowledge”: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39 (4), 372-400.

Jacobs, V. R., Lamb, L. L. C. & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41 (2), 169-202.

Jacobs, V. R. & Philipp, R. A. (2010). Supporting Children’s Problem Solving. Teaching Children Mathematics, 17 (2), 99-105.

Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating Discussions. Mathematics Teaching in the Middle School, 14 (9), 549-556.

Stein, M. K., Engle, R. A., Smith, M. S. & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10 (4), 313-340.

Thames, M. H. & Ball, D. L. (2010). What math knowledge does teaching require? Teaching Children Mathematics, 17 (4), 221-229.


Arkansas CCSSM Professional Development Purposeful Pedagogy and Discourse Instructional Model

Angeles uses ___ of a bag of beads to make a necklace. If she makes ___ necklaces, how many bags of beads will she need?

(1/3, 12) (1/3, 24) (1/3, 36) (2/3, 12) (2/3, 36)

Planning Sheet – Mrs. Kasnicka – Multiple Grouping (Multiplication) p.1

1. Sort student work to determine what mathematics students brought to bear on the problem and what mathematics is available for instruction through a discussion. Sort by…strategy, representation used, level (correct/complete, productive failure), missing elements.

2. Determine where most of the class appears to be in terms of the mathematics they understand and the mathematics they are ready to learn.

3. Select/create a learning goal or goals that will address where your class is. In doing so, think about how to support students that might be working below the rest of the class so the discussion helps them as well.

4. Select student papers that can be used as the basis for a discussion/lesson directed at the learning goal(s). Determine in what order to use the student work. Determine if the student will present the work (P) or if you will allow the class to interpret it (I).

5. Develop questions to pose about the work, either to the student it belongs to (O) or to the class (C).

Selected Work Learning Goal(s) Questions to Pose

Guillermo Drawing of 4 rectangles, with 3 sections in each numbered 1-‐12.

(Focus attention on students who are still struggling with making sense of the problem context) Use a direct modeling strategy to make sense of the problem and to find “12 sets of 1/3”

(I) What did Guillermo do? (C) How did he represent a necklace (1/3 bag of beads)? (C) How did he represent a bag of beads? (C) What do the numbers 1-‐12 in his picture show? (C) How can we write an equation to show his thinking?

Blake Drawing of 8 rectangles, shaded in groups of 2/3 (incomplete drawing), showing 3 sets of 2/3.

Interpret Blake’s incomplete picture to try to relate 3 groups of 2/3 = 2 in a multiplicative way to 12 groups of 2/3 = 8.

(I) What did Blake do? (C) What does the picture with the two rectangles, red shading, arrow at the top and numeral 3 represent? (C) How did he represent a bag of beads? (C) Where is a necklace in his picture? (C) How can we write an equation to show his thinking? (C) How can this help us solve the problem?

Planning Sheet – Mrs. Kasnicka – Multiple Grouping (Multiplication) p.2

Selected Work Learning Goal(s) Questions to Pose

Angeles Table labeled B (beads) and N (necklaces) with entries beginning with (2, 3) and counting by that “chunk”

Relate Angeles’ table to Blake’s picture and determine how Angeles extended her table to answer the question (multiplicative relationship). Connect table to the number sentences.

(I) Compare Angeles’ work to Blake’s (and Guillermo’s). What connections do you see? (C) What does the first row in Angeles’ table show? What is the 2? What is the 3? (C) How is “2 bags makes 3 necklaces shown in the table? In the picture? In the number sentences? (C) What do the other rows in the table represent? How did Angeles know what to put there?

Cristina Pair of number sentences 3 x 2/3 = 2 and 9 x 2/3 = 6 with a total of 8 circled

Relate Cristina’s number sentences to Angeles’ table and Blake’s drawing. Relate Cristina’ number sentences to each other.

(I) What did Cristina do? (C) How could Cristina’s equations be used to solve the problem? (C) How do Cristina’s equations relate to Angeles’ and Blake’s work? (C) How do Cristina’s number sentences relate to each other?

Student Work for the Lesson Angeles uses ___ of a bag of beads to make a necklace for her store. If she makes ___ necklaces, how many bags of beads will she need? (1/3; 12) (1/3; 24) (1/3; 36) (2/3; 12) (2/3; 36)

Guillermo Blake

Angeles Cristina

VOLUME 9, ISSUE 2 PAGE 19

How Might Future Mathematics Instructional Materials Look?

Submitted by Dr. Linda K. Griffith University of Central Arkansas

As I have led professional development sessions in recent months, one question has been asked of me repeatedly, “What instructional materials do you recommend?” The only response I’ve had has been to refer the questioner to the Arkansas Ideas Common Core website and to the statewide strategic plan, where under the curriculum section there is a document that describes a two-‐tiered approach to evaluating instructional materials in light of Common Core State Standards for Mathematics Content and the Standards for Mathematical Practice. I continue to give this response, but I have begun to wonder, how will the next generation of instructional materials look?

I think almost every set of instructional materials contains good “problems” and/or “tasks”. The issue, pervasive in all of the materials, is they take the good problem or mathematical task and provide all the information needed to solve the problem or complete the task, provide all the mathematical structure needed to solve the problem or complete the task, and finally break the problem or task down into a step by step procedure. Now the problem or task is no longer a problem or interesting task but an exercise in following the steps provided to get to an answer or outcome. I have found this to be the case in even the best resources I have used in the past.

There is a TED video by Dan Meyer entitled The Math Curriculum Needs a Makeover that does a great job of illustrating the point I am trying to make. This video can be found at the URL: http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html

I encourage all of us in the mathematics education community in Arkansas and especially those of us that are in leadership positions to consider how instructional materials will really need to look if we are to implement both the Common Core State Standards for Mathematical Content and the Standards for Mathematical Practice. I do not yet have a clear vision but I have begun to have a few interesting ideas. Student materials will look very different from teacher materials. The student materials will just be visuals and short problem statements and the visual may include graphics, pictures, videos, etc. Teacher materials will need to answer three important questions:

1. What might students do with this problem or task? This should include a variety or approaches. 2. What mathematics does this problem or task have potential to reveal? A good problem or task

will not focus on a single standard but multiple standards that are coherent and share a common focus.

3. How can a teacher orchestrate classroom discourse to reveal the important mathematical ideas from this problem or task? This might include the kind of student work that would be productive to share or the kind of common errors that need to be shared with a discussion about why that approach did not result in a successful solution to the problem or completion of the task.

(continued on next page)

Newsletter

(Leadership Corner continued from previous page)

In order for the teacher materials to include the answers to these three questions, the problem or task will need to be carefully field-‐tested and the result of the field tests used to formulate answers to these three questions.

There will still be a need for practice or exercises, but the order will just need to be reversed. Right now we see the examples of how to complete the exercises, then the practice exercises for the students and then in the problems that use the procedure that students have been practicing. I propose that the problems or task that lead to the students developing the procedures need to come first and then they can do the practice exercises to increase their effectiveness and efficiency with the procedures that have created, that is allow students to develop fluency with the procedures.

I encourage each of you to watch the video recommended in this article and find a problem or task in your instructional materials that you can pare down to its essence. Pose the problem or present the task to your class after first thinking about the three questions listed earlier in the article.

After you have completed this process, reflect on what you have learned. I hope that many of us that are instructional facilitators or professional development providers will begin to think about how to use professional learning communities and embedded professional develop to support efforts to help teachers find good problems and/or task and use these to foster the kind of mathematics education experiences that will make the implementation of the Common Core State Standards for Mathematics Content and the Standards for Mathematical Practice a reality. § AAML would like to invite all ACTM members to consider joining AAML. This organization hopes to serve all members of ACTM who are providing leadership in mathematics education in Arkansas. This includes teacher leaders as well as those in more formal leadership positions.

ARKANSAS ASSOCIATION OF MATHEMATICS LEADERS (AAML) MEMBERSHIP FORM 2010-‐2011

Name Job Title Home . address

Home phone

Email

Cell phone

School . District

Work phone

School . Name

School . Address

Check the organizations to which you currently belong:

_________ Arkansas Council of Teachers of Mathematics (ACTM)

_________ National Council of Teachers of Mathematics (NCTM)

_________ National Council of Supervisors of Mathematics (NCSM)

Please mail this form with a $15.00 check for annual dues made out to AAML to: Tim Brister, AAML President, Harding University, Box 12254, Searcy, AR 72149 [Online registration for ACTM and AAML available at http://www.actm.net/]

Newsletter

What is Our Goal? Dr. Linda K. Griffith, University of Central Arkansas Recently my friend, Tim Brister (President AAML), sent me a link to a video of Phil Daro, one of the writers of the Common Core State Standards for Mathematics, who has spent a great deal of time studying the Trends in International Mathematics and Science Studies (TIMSS) videos. The URL that will take you to the video is: http://vimeo.com/30924981. I learned a great deal from watching this video multiple times and would like to share some of what I learned with you and encourage you to watch the video for yourself.

Based on his observations, Daro proposes that the difference in Japanese and American teachers is not their pedagogical abilities nor their content knowledge, but their goals. He says that he has realized from hours of watching the TIMSS videos that when American teachers look at a problem their goal is for their students to get the correct answer to the problem. While the goal of Japanese teachers is to help their students learn some mathematics as they work on the problem.

He begins by saying he has often been asked what is one thing we could change that would make a difference in mathematics education. He goes on to say that it would not be a magic pill but if we could get American teachers to change their goal by asking, “What is the mathematics kids are to learn from working this problem?” This would replace what they currently ask, “How an I teach my kids to get the answer to today’s problem?”

He makes the point that getting the answer to a problem is one consideration but there are two others: making sense of the problem situation and making sense of the mathematics you can learn from working on the problem. Much of what he says in this video helped me to really see the reasons for and importance of the eight Standards for Mathematical Practice that are an integral part of the Common Core State Standards for Mathematics.

He makes an excellent point about the importance of sharing approaches to problems that did not work. He says it is often easier to get to the mathematics through thinking about why an approach did not work. In some of the work that has been done for the Arkansas statewide professional development plan for implementation of Common Core State Standards for Mathematics, the term “productive failure” has been used. This is another way to say that not only are the approaches that produce correct answers important but understanding why an approach fails is of at least equal value.

He gives several examples of answer getting-‐techniques– procedures that yield correct answers, but mask or hide the true mathematics. His examples include: the butterfly method for adding fractions, setting up proportions and cross-‐multiplying, FOIL, and canceling. It is well worth the 17 minutes that it takes to watch this video to hear what he has to say about these commonly used procedures and think about the ramifications.

He concludes by agreeing that the mathematics curriculum in the United States has been a mile-‐wide and an inch-‐deep, but he says we (teachers) have contributed to this problem by incorporating these answer getting techniques into the curriculum in addition to or in place of the important mathematics.

I hope you will take the time to watch this video and reflect on what Daro is proposing and how this is related to the implementation of the Common Core State Standards for Mathematics in Arkansas. §

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