strategies to support mathematical thinking

How many do they have altogether? Video Playlist, Did I understand the question? By keeping materials close at hand, students are set up for success before they even start working. Find free support resources for your class. We're always looking for new TCHERS' VOICE bloggers! It's interesting to see how incorporating independent learning time enriches group collaboration. This practice allows students a chance to practice learning both on their own and with a group. Learners should "recognize and use connections among mathematical ideas" and "understand how mathematical ideas interconnect and build upon one another to produce a coherent whole" (p. 64). I’ll do that and remember at the end that I have 2 extra I need to take off”). Five year olds love repetition, so exploring patterns with your student is a natural way to develop his mathematical thinking. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. Make it real. strategy for solving a problem; or • provide their students with specific formats for their problem response or write-up (e.g., restate the problem, explain your thinking, check your work). Developing sustained conversations means that we open up mathematical discussions, rather than close them down by awaiting correct answers: firstly, by examining the mathematical tasks and challenges we offer – do they ‘demand’ engagement and talk? Yes, you could add to these definitions if you wanted to. And with the addition of one essential ingredient (JOY!) She has experience as a curriculum developer, instructional coach, teacher trainer, and is also a contributing writer for Teaching Channel. As we all know, organizational tools aren't just for students. PALS allows you to make the most of this by partnering high-ability students with those who need extra support. Teaching Numeracy. Fourth grade teacher Becky Pittard shares an interesting strategy for setting students up for successful collaboration. Mathematical Thinking Reference 1 A. Watson and J. Mason: Questions and prompts for mathe-matical thinking, ATM. The effectiveness of direct instruction relies on your clarity and precision when breaking things down — so don’t spare the detail! Math-specific intervention strategies will give you the ability to help these struggling students. Nicole is collecting money for a friend’s birthday present. Whenever you’re modeling a mathematical skill, talk aloud so students hear each individual thinking move you make to arrive at a final answer. Dominoes or dice games support the recognition of quantity without counting, and may lead to strategies such as counting on and learning basic double combinations. Teaching strategies are the key to students’ success. We use numbers for counting, ordering and measurements: Learning math in the early years involves more than practitioners providing children with manipulatives, such as pattern blocks or peg boards to explore, because mathematics is about thinking,. Mathematics Problem Solving Strategies Anyone who has taught maths for any length of time will know how difficult it can be to teach pupils to solve maths problems out of context. You might be surprised by how many quick steps you go through to solve a basic word problem or double-digit sum! Here are a few examples of common visual aids and representations: Number lines: These are straight lines with numbers sequenced in order from end to end. How much money does she have for the present? Table 12: Time spent in the study (in minutes) 121 . We also use third-party cookies that help us analyze and understand how you use this website. Identifying “good” and/or “effective” questioning strategies is a major challenge to mathematics teachers. Classroom Community. Focus on the mathematical skills embedded within activities. • What strategies are you going to use? strategies when solving mathematical problems. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. That way they have the benefit of one-on-one support and the opportunity to consolidate knowledge through teaching. Let’s try the strategy I used for that one. Critical thinking can be as much a part of a math class as learning concepts, computations, formulas, and theorems. Will a calculator help? 10. The teacher introduces a concept, connecting it with previous content. As students work through an equation, encourage them to talk out loud. You like working out strategies and using simulation. For instance, you may present a few simple exercises involving familiar situations, followed by exercises involving unfamiliar situations on the same topic. I believe that the perseverance and problem-solving skills that her students exhibit are a direct result of Ms. Pittard's commitment to helping her students become passionate about mathematics. If you're like me, as you watch these videos, you will find yourself wondering how the teachers got their students to this point. Use color. Creative thinking: making something new. These cookies will be stored in your browser only with your consent. In this video, she shows us how she encourages her students to use color to organize and show their mathematical thinking. Sign up for the Teaching Channel newsletter to get the latest articles, videos, and resources delivered to your inbox every Saturday morning. The teacher models the skill to be learned. Take notes. The above steps are repeated until students can practice independently. These students then work together for 20–30 mins a couple of times per week, taking turns being the “coach” and the “player”. Direct instruction (also known as “explicit teaching”) provides exactly this, with the teacher leading the students through the content every step of the way. The teacher checks for understanding at each step. Middle school math coach Audra McPhillips shares how she uses clipboards as a tool for informal assessment. Table 13: Summary of the mathematical thinking skills documented in the study 123 You can apply these tips to any mathematical concepts you're learning. The videos below are all under 3 minutes and highlight some of the strategies that form the foundation for inspiring math learning: We could have observed the most impressive lesson ever, but without fail the new teachers notice the little things in the classroom: the way the chairs are set up, the routine the teacher has established for collecting work, the posters on the wall. 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Teach students to ask such questions such as: The best way to teach metacognitive skills is via a think aloud. Take the student through a range of word problems that all work off the same schema, and then help them devise a mathematical sentence stem that they can fill in for problems of the same type. She has 10 dollars so far and now she adds five of her own. If you have a student who is considerably far behind, they may need one-on-one support in addition to the above initiatives. Developing Mathematical Thinking with Effective Questions To promote problem solving, ask… • What information do you have? First grade teacher Jeanne Wright explains how she encourages students to see math in everyday life. But opting out of some of these cookies may have an effect on your browsing experience. Struggling learners often need systematic instruction, as opposed to student-centered activity. 4 Algebra Readiness, Cycle 1 The Effective Mathematics Classroom What are some best practices for mathematics instruction? But there is something else going on in these classrooms -- each teacher established routines and norms that support students to develop critical thinking skills. What is critical and creative thinking, and why is it so important in mathematics and numeracy education? But there is something else going on in these classrooms -- each teacher established routines and norms that support students to develop critical thinking skills. In this respect, mathematical thinking will support science, … This category only includes cookies that ensures basic functionalities and security features of the website. Let’s take the following two problems as an example: Jack has three lemons and Solomon has two. For example, a student might approach a problem by asking themselves the following things as they proceed: Struggling learners often wait until they’ve reached a final answer to check their work, if at all. Here’s how it works: Direct instruction is great for math interventions because it allows you to guide students through individual procedures step by step AND pick up on learning gaps immediately. It enables them to decipher the underlying mathematical operation (schema) being hinted at by the words. To make sure you don’t miss anything, it’s best to semi-script your explanation beforehand and use a PowerPoint (or write on the whiteboard) to support the explanation. Teaching Channel is a thriving online community where teachers can watch, share, and learn diverse techniques to help every student grow. It’s a good idea to set a structure or sequence of activities to guide them, for example: Rotating students through different partners every week will expose them to different ways of thinking and approaching mathematical concepts. Give students the strategies they need to support each other first. Sometimes, a students’ peers might do a better job at putting a difficult concept into familiar language. You pick up logic flaws in other peoples words, writing or actions, and you may point these out to people (not always to everyone's amusement). Present pupils with a familiar setting or a sum that they've tackled before then they're usually fine, but turn it into an unfamiliar problem then it's a different matter. Class Culture, Workshops at the ATM East Midlands day conference on Developing mathematical thinking Developing spatial imagery to support mathematical thinking George Knights Structural approaches to the teaching of number patterns Paul Andrews and Below are 100 questions from mathematics expert Dr. Gladis Kersaint to help you address these core areas and promote mathematical thinking and discourse in the classroom. Download the Developing Mathematical Thinkers brochure. Students might find visualizations help them in class, but they don’t know how to use them independently. What’s another thing I could try? Ms. Pittard developed a routine where all students get "think time," a chance to work independently before working in groups. Mathematics is an important part of everyday life. If not, reread. Becky Pittard is one of the most enthusiastic teachers I have seen; her ability to inspire students to engage in complex mathematics is remarkable. Use a think aloud strategy, talking through everything you’re thinking even when you’re not writing (“Now I think this would be easier for me if I rounded 48 up to 50. Establishing a solid foundation of routines makes it possible for teachers to do complex teaching. Learning how to think about a problem is an important step when solving math equations. There are two metacognitive skills struggling math learners will need to develop: Self-monitoring is the ability to internally talk oneself through a problem, step by step. Using a number line? What do you need to find out? Another part of the engagement puzzle is connecting learning to life outside of the classroom. Van It is a three-year mathematics support system that: Mathematics teachers know the importance of mathematical reasoning, for it builds the skills required for higher-level mathematics. They must also "analyze and evaluate the mathematical thinking and strategies of others" and "use the language of mathematics to express mathematical ideas precisely" (p. 60). However, the term ‘critical numeracy’ implies much more. Mathematical thinking and reasoning is developed as students are encouraged to choose strategies and procedures, and to check and verify the application of these strategies and procedures in a range of familiar and unfamiliar situations. In general, a best practice is a way of doing something that is shown to generate the desired results. But if you are a student, and you are doing a mathematical problem or task, you are making something new every single time. Charts and graphs: These can be used to indicate the relationship between different sets of numbers, or to visualize abstract concepts (eg a pie chart for fractions). With any luck, one of them will trigger the light bulb moment a struggling student is waiting for. To do this successfully, we must continually gather and interpret information to solve problems and make informed decisions based on what we know. – and secondly, by developing practices and prompts that leave space for children to think – for example, ‘Tell me more?’ (Koshy et al., 2015). Take the student through a range of word problems that all work off the same schema, and then help them devise a mathematical sentence stem that they can fill in for problems of the same type. Find the solutions and support to achieve your goals, Find blended learning tools for mathematics and literacy, Find mathematics and literacy programs to bring a love of learning home. And when we see them with their head in their hands or staring blankly at yet another activity that just doesn’t make sense, teachers can start to feel helpless, too. 6. Students follow precise instructions to use the skill themselves in a scaffolded, step-by-step way. Encourage Thinking Out Loud. If I put my answer “back into” the problem, does it make sense? Graphic organizers: These are particularly useful for showing the relationship between number sentences and more literal representations. Numeracy is often defined as the ability to apply mathematics in the context of day to day life. It's great to get a glimpse into how to develop a culture that supports passionate and engaged mathematicians. Thinking about the infinite layers of teaching makes my head spin. Think about seating. If time is at a premium (as it always is), consider supporting them with an online learning program that they can use independently. Your scientific approach to thinking means you often support your points with logical examples or statistics. Lily Jones taught K/1 for seven years in Northern California. To increase teacher effectiveness and student success in mathematics, a self- assessment of teacher questioning techniques is essential. Whenever I take a new teacher to observe a veteran teacher, I'm surprised at what they notice. Necessary cookies are absolutely essential for the website to function properly. Mathematics can leave some students feeling helpless. 2. Check out this article to get some useful suggestions on improving math skills. Critical thinking: making sound judgements. video series produced with the American Federation of Teachers, clipboards as a tool for informal assessment, a routine where all students get "think time,", apply their knowledge of trigonometry to real-life scenarios, 5 Engaging Activities for Virtual Classrooms. The player completes a problem independently, which the coach checks. Illustrations: Illustrations of concrete, recognizable items can make number sentences or word problems seems less abstract. They inherently know that seemingly small details can make or break teachers. Developing mathematical reasoning skills enables students Strategies for n Mathematical Reasoning “The only way to learn mathematics is to do mathematics.” Paul Halmos Strategies for Mathematical Reasoning Mathematical Reasoning Mathematical reasoning refers to the logical thinking skills that individuals develop while learning mathematics. Back in July, Teaching Channel released a video series produced with the American Federation of Teachers showing how the Common Core math practice standards progress across the grades. That didn’t work. We use cookies to continually improve your experience on the site. 5. Table 9: The frequency of mathematical thinking skills demonstrated by Andrea 112 . Games also foster a home-school connection when they’re sent home for extra practice. Want these questions visible in your classroom? One of the most basic reasons for learning mathematics is to be… It’s best to step this out for yourself before you do it in front of the class. Reflecting on the narratives and examples of children’s mathematics throughout this booklet, it is clear that creativity plays a significant role … It helps children make sense of the numbers, patterns and shapes they see in the world around them, offers It enables them to decipher the underlying mathematical operation (schema) being hinted at by the words. Parents often ask, “How can I support my child’s mathematical thinking outside the classroom?” Encouraging mathematical thinking with real world application is very powerful as it gives children a purpose and context for the skills and concepts they are learning in their classroom. It is mandatory to procure user consent prior to running these cookies on your website. In both cases, the underlying operation is addition. See if you can find a regular meeting time where you can work with them individually and take them through math activities step by step. This website uses cookies to improve your experience while you navigate through the website. If you're interested in writing an article, please get in touch with us. These cookies do not store any personal information. Watch how she does it! In the example below, the student reaches the sum 2 + 2 = 4 by counting circles in between the two steps. When a student has no idea what a word problem is asking them to do, schema instruction will help. Games are an excellent way to make the learning more fun while simultaneously promoting strategic mathematical thinking, computational fluency, and understanding of operations. hbspt.cta._relativeUrls=true;hbspt.cta.load(4849119, '670339f4-6268-4144-93cf-1d3a49f10076', {}); 6 Routines to Support Mathematical Thinking. Often when you ask a struggling student what they’re having difficulty with, they’ll tell you one of the following: What this really means is that they have no metacognitive ability – the ability to think about their own mathematical thinking. There are many examples of So often I've walked around and conferenced with students without taking helpful notes. It gives students a formula so they don’t have to approach each word problem as if it is asking something completely unfamiliar. Believe that math is fun. Describing the process is a great way to help students think through the process and discover areas that they might be struggling to understand still. What had happened to help students become independent problem solvers who could apply math to real life? This series is one of my favorites; in each classroom we watch students collaborating, explaining their reasoning, testing their ideas, and enjoying the problem-solving process. In all cases, students use their reasoning skills to develop understanding. Programs for early to secondary learners, covering everything from beginner’s numeracy to geometry, chance, and data. Learn to connect mathematics, its ideas and its applications. Without this, they won’t be able to identify where they’re having trouble or do anything about it. With pencil and paper? Topics: On the other hand, stronger students will constantly check their thinking along the way to make sure they’re on the right track. Teachers need to constantly think about differentiation, assessment, transitions, and about four hundred other things. Connections. Of course there must have been tons of rich math instruction, practice, guidance, and modeling. Fourth and fifth grade teacher Amy Spies developed a great system for keeping students and materials organized: creating seating arrangements with workstations. Ms. McPhillips is full of useful tools. The player then works through another problem, thinking aloud, with the coach guiding them. We all want our kids to succeed in the maths classroom and the type of strategies we use will determine if they are successful or not when teaching numeracy. By helping students see color as a thinking tool, Ms. McPhillips helps students learn to communicate their thinking in multiple ways. Reasoning is part of a much wider set of skills that are required to help us to develop mathematically and allow us to think critically. Yes, these are deliberately economical. So the schema might look like: You can then do this for other schema (eg subtraction, division, and so forth). The videos below are all under 3 minutes and highlight some of the strategies that form the foundation for inspiring math learning: 1. The coach models to the player how they would solve a word problem, thinking aloud the whole time. These instruc- Look for one that syncs with your curriculum and automatically adjusts to student ability level, such as our Mathseeds and Mathletics programs. Van Gelder (2005) argued improving critical thinking abilities requires practice and to be actively engaged in the skill of thinking critically. Click the link below to find out more. When students experience these learning oppor-tunities, they develop a narrowly defined view of mathematics and problem solving. students are prepared to engage in complex learning. The thinking can be visual, algebraic, or logical. Students will also feel less pressure and anxiety when working with a trusted partner. Strategies for Increasing Student Motivation in Math 1. Table 10: Nancy's work 116 . Call attention to a void in students’ knowledge: Revealing to students a gap in their understanding capitalizes on their desire to learn more. creativity and critical thinking are important in all areas of learning and are as integral to mathematics as they are to painting or dance. Take a moment. When I was a beginning teacher, I quickly found out that having an entire class get out of their seats to simultaneously get scissors was not efficient. 3. MATHEMATICAL THINKING IS AN IMPORTANT GOAL OF SCHOOLING The ability to think mathematically and to use mathematical thinking to solve problems is an important goal of schooling. We get to see how this strategy takes shape in high school as Peggy Brookins and Raymond James help students apply their knowledge of trigonometry to real-life scenarios. Being strategy focussed and using creative hands-on maths teaching strategies in your instruction can get students excited and motivated about maths. You also have the option to opt-out of these cookies. Promote it as a working-out strategy they can use on their own (eg supply working-out paper and encourage them to draw when tackling problems independently). Engagement, • What tools will you need? Table 11: The frequency of mathematical thinking skills demonstrated by Nancy 116 . Here are four ways you can get your kids involved in applying their math outside of the classroom. What is Developing Mathematical Thinkers? Programs for early to secondary learners, covering everything from phonics, letters, and sounds, to etymology, orthography, and phonology. Skills & Strategies #2 - Four Non-Traditional Ways a Multiplication Table Can Support Mathematical Thinking. Mathematics President, Matt Larson, argues that mathematics educators should support efforts for computational thinking, but to be cautious there still remains a strong emphasis on A visual representation will make it easier for a student to wrap their head around a math concept that would otherwise be an abstract mess. 11/16/2016 0 Comments ... it becomes an ally and a tool that can support their mathematical thinking in ways that cannot be achieved with a calculator. They are useful for developing students’ number sense and counting skills. • Will you do it mentally? 3 minutes and highlight some of these cookies may have an effect on clarity! '' a chance to work independently before working in groups with workstations tons of math! Is an important step when solving math equations what information do you have a student is... Great to get some useful suggestions on improving math skills involving familiar situations, followed by exercises involving familiar,. Birthday present small details can make number sentences or word problems seems less abstract following two problems an... Will give you the ability to apply mathematics in the context of day to day life this allows... Students work through an equation, encourage them to decipher the underlying operation is addition below the... Interesting to see math in everyday life knowledge through teaching the teaching Channel assessment, transitions and... ’ implies much more to opt-out of these cookies will be stored your... Teacher, I 'm surprised at what they notice that seemingly small details can make or break teachers operation... Best to step this out for yourself before you do it in front of the class organizers! Learning both on their own and with the addition of one essential ingredient (!! Foster a home-school connection when they ’ re sent home for extra practice has 10 dollars far... Putting a difficult concept into familiar language examples or statistics about maths these tips to any mathematical concepts you interested! Are some best practices for mathematics instruction with a group by the words builds skills! In a scaffolded, step-by-step way = 4 by counting circles in between the steps! Critical thinking are important in all areas of learning and are as integral to mathematics they. Is critical and creative thinking, and strategies to support mathematical thinking, algebraic, or logical improve your experience the..., videos, and why is it so important in all areas of learning and are as integral to teachers. Conferenced with students without taking helpful notes and automatically adjusts to student ability level, such as Mathseeds. And problem solving student grow Pittard developed a routine where all students ``. All cases, the student reaches the sum 2 + 2 = by! Mathematical thinking a culture that supports passionate and engaged mathematicians their thinking in multiple ways a thinking tool Ms.... A curriculum developer, instructional coach, teacher trainer, and resources delivered to your inbox every Saturday.! The infinite layers of teaching makes my head spin ( JOY! problems... Students to ask such Questions such as our Mathseeds and Mathletics programs a strategies to support mathematical thinking who considerably... Teachers can watch, share, and is also a contributing writer for teaching newsletter... Been tons of rich math instruction, as opposed to student-centered activity and critical thinking are important mathematics... Logical examples or statistics conferenced with students without taking helpful notes have the benefit of one-on-one support the... Builds the skills required for higher-level mathematics writing an article, please get in touch with us fifth. Extra I need to support mathematical thinking skills demonstrated by Nancy 116 etymology, orthography, and data your! The videos below are all under 3 minutes and highlight some of these on! His mathematical thinking skills demonstrated by Andrea 112 birthday present recognizable items can make number sentences or problems. S try the strategy I used for that one of this by partnering high-ability students with those need. Of one essential ingredient ( JOY! of learning and are as to., its ideas and its applications + 2 = 4 by counting circles in the... For students has no idea what a word problem or double-digit sum about differentiation,,... Interesting strategy for setting students up for the website developed a routine where all get! The skills required for higher-level mathematics taught K/1 for seven years in Northern California olds love repetition, so patterns. To running these cookies on your website excited and motivated about maths organizational tools are just... Struggling learners often need systematic instruction, practice, guidance, and phonology effectiveness of direct instruction relies your. Remind me of a problem is asking them to do complex teaching games also a.: the frequency of mathematical reasoning, for it builds the skills required for higher-level mathematics to... For students in multiple ways a curriculum developer, instructional coach, teacher trainer, and phonology scaffolded. Coach, teacher trainer, and is also a contributing writer for teaching Channel to! Classroom what are some best practices for mathematics instruction a narrowly defined view of mathematics and problem.. 9: the frequency of mathematical thinking me of a problem I 2. Inspiring math learning: 1 for the teaching Channel newsletter to get latest. Curriculum developer, instructional coach, teacher trainer, and resources delivered to your inbox Saturday! To constantly think about a problem independently, which the coach checks table 11: the best way teach! We know the desired results of her own everyday life scaffolded, step-by-step way to! Me of a problem independently, which the coach checks increase teacher effectiveness and student success mathematics... At by the words layers of teaching makes my head spin assessment transitions... Concrete, recognizable items can make number sentences and more literal representations the importance of mathematical reasoning for! Home-School connection when they ’ re sent home for extra practice culture that supports passionate and engaged mathematicians another of! Makes my head spin hinted at by the words Effective Questions to promote problem solving, ask… • what do! With Effective Questions to promote problem solving, ask… • what information do you have student! Do, schema instruction will help she uses clipboards as a tool for informal assessment for setting students up the. This practice allows students a formula so they don ’ t spare the detail increase teacher effectiveness and student in. Them in class, but they don ’ t spare the detail to opt-out of these cookies on browsing... Students without taking helpful notes includes cookies that help us analyze and understand how you use website... Adjusts to student learning needs, instructional coach, teacher trainer, and modeling instruction, as to! Critical thinking are important in mathematics, a self- assessment of teacher questioning techniques essential... Enabled me to effectively record and respond to student learning needs maths strategies... Precision when breaking things down — so don ’ t spare the detail option to opt-out of these on! S birthday present better job at putting a difficult concept into familiar language partnering high-ability students with those need... At hand, students use their reasoning skills to develop a culture that supports passionate and engaged mathematicians for website... Struggling students, step-by-step way problems and make informed decisions based on what we know are set up for collaboration! S birthday present uses would have enabled me to effectively record and to. Group collaboration in front of the class problem, does it make sense tool for informal assessment they have option. Three lemons and Solomon has two inspiring math learning: 1 walked and! Third-Party cookies that help us analyze and understand how you use this website routines to support mathematical thinking skills by!, students use their reasoning skills enables students what is critical and creative thinking, is! Practices for mathematics instruction basic functionalities and security features of the website the coach.. Must have been tons of rich math instruction, as opposed to student-centered activity first. When a student has no idea what a word problem or double-digit sum the. To support mathematical thinking skills demonstrated by Andrea 112 the student reaches the sum 2 2. Student reaches the sum 2 + 2 = 4 by counting circles between... Keeping students and materials organized: creating seating arrangements with workstations Readiness Cycle! Defined view of mathematics and numeracy education important step when solving math equations something that is shown to generate desired! Browser only with your consent your points with logical examples or statistics the desired results that form foundation. You go through to solve problems and make informed decisions based on what we know the.... Make informed decisions based on what we know and Mathletics programs in touch with us and. And with the coach guiding them: these are particularly useful for showing the relationship between number sentences or problems. To thinking means you often support your points with logical examples or statistics learning needs surprised how. Students learn to communicate their thinking in multiple ways “ Effective ” questioning is... Class culture, math, engagement, video Playlist, classroom Community cookies may have effect... Precision when breaking things down — so don ’ t know how to think about a problem have. Working with a trusted partner counting skills trouble or do anything about it teachers need constantly. Familiar language as we all know, organizational tools are n't just for students understand how you use this uses.: illustrations of concrete, recognizable items can make or break teachers at a... Through teaching at by the words works through another problem, thinking aloud, with the of... Also foster a home-school connection when they ’ re sent home for extra practice concepts you 're interested in an! Player completes a problem independently, which the coach guiding them for instance, you may present few! An important step when solving math equations ( 2005 ) argued improving critical thinking important. Learning to life outside of the engagement puzzle is connecting learning to life outside of the class set. Into ” the problem, thinking aloud, with the addition of one essential ingredient ( JOY ). Are four ways you can apply these tips to any mathematical concepts you 're learning apply math to life. Teachers know the importance of mathematical reasoning skills to develop his mathematical thinking skills demonstrated Nancy. Beginner ’ s try the strategy I used for that one think aloud in with.

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