# habits of mind - mathematical practices of the common core statewide instructional technology...

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HABITS OF MIND - MATHEMATICAL PRACTICES OF THE COMMON CORE Statewide Instructional Technology Project Slide 2 How many of your feel the same? http://www.tubechop.com/watch/303621 Slide 3 Introduction An introductory webinar was offered March 6 th. Please watch the recording for the explanation introducing Common Core and the changes to the AZ 2010 Math Standards. Slide 4 To view the official ADE documents 2010 Arizona Mathematics Standards Overview of the 2010 Mathematical Standards PDFPDF Standards for Mathematical Practices PDFPDF Mathematics Introduction (Coming Soon) Mathematics Glossary PDFPDF Summary of Updates to Explanations and Examples PDFPDF Slide 5 Gauge the Audience What grade level do you teach? A = K-5 B = 6-8 C = 9-12 Slide 6 http://www.tubechop.com/watch/290412 How does your classroom compare? Slide 7 Slide 8 Slide 9 Slide 10 The Mathematical Practices 10 Slide 11 The Mathematical Practices 11 Slide 12 1.Make sense of problems and persevere in solving them 6. Attend to precision The Habits of Mind of a Productive Mathematical Thinker Slide 13 MP1. Make sense of problems and persevere in solving them In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? or they may try another strategy. In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They are willing to try other approaches. Slide 14 MP1. Make sense of problems and persevere in solving them In first grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They are willing to try other approaches. In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They make conjectures about the solution and plan out a problem-solving approach. Slide 15 MP1. Make sense of problems and persevere in solving them In second grade, students realize that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They make conjectures about the solution and plan out a problem-solving approach. In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. Slide 16 MP1. Make sense of problems and persevere in solving them In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. Slide 17 MP1. Make sense of problems and persevere in solving them In 4th grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, Does this make sense? They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? Slide 18 Make sense of problems and persevere in solving them In 5th grade, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way?. Slide 19 MP1. Make sense of problems and persevere in solving them In grade 6 grade, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way?. Slide 20 MP1. Make sense of problems and persevere in solving them In grade 7, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way?. In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? Slide 21 MP1. Make sense of problems and persevere in solving them High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Slide 22 Good problems: Mathematics questions or tasks that are challenging enough so it becomes useful to take in ideas other than just ones own to engage thinking. Productive Thinking vs Reproductive Thinking Connect to student experience or interest Incorporates rich mathematics Entry points solutions pathways are not obvious Slide 23 MP1. Make sense of problems and persevere in solving them Mathematically proficient students make sense of problems: Mathematically proficient students persevere in problem solving: Make sense of problemsPlan a solution pathway Explain the meaning of the problem to themselves Consider similar cases and alternate form Look for entry pointMonitor progress and change course if necessary Analyze givens, constraints, relationships, goals Explain correspondence and search for trends Check their answers using alternate methods Continually ask themselves, Does this make sense? Listen to and work to understand the approaches of others Slide 24 MP 6. Attend to precision Mathematically proficient students: Communicate precisely to others. Use clear definitions in discussion with others and in their own reasoning. State the meaning of the symbols they choose Specify units of measure, and labeling axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Give carefully formulated explanations to each other. In high school, they have learned to examine claims and make explicit use of definitions. Slide 25 Slide 26 Technology Resources To solve this problem you might explore (Connecting Cubes) http://www.eduplace.com/kids/mw/manip/mn_2.html (Pattern Blocks) http://gingerbooth.com/flash/patblocks/patblocks.php (Geoboards) http://nlvm.usu.edu/en/nav/frames_asid_129_g_2_t_3.html?open=activities (Graphs) http://nces.ed.gov/nceskids/createagraph/default.aspx Slide 27 Technology Resources A Short Introduction to Using the Connecting Cubes eManipulatives (Click to play) Slide 28 Technology Resources A Short Introduction to Using the Virtual Pattern Blocks (Click to play) Slide 29 Technology Resources A Short Introduction to Using the Virtual Geoboard (Click to play) Slide 30 Discuss how you solved the problem After solving the problem, complete the form linked (https://docs.google.com/spreadsheet/viewform?formkey=dHZQa XZDUmxOdzFENV93UkpqOXBMNmc6MQ#gid=0) in the chat area to answer the following questions:https://docs.google.com/spreadsheet/viewform?formkey=dHZQa XZDUmxOdzFENV93UkpqOXBMNmc6MQ#gid=0 1. What answer did you get? 2. Which technology tool did you use? 3. Why did you choose this tool? 4. How did this tool help you solve the problem? Slide 31 Solutions to the Problem Click https://docs.google.com/spreadsheet/ccc?key=0Ao xuu3aQOMKNdHZQaXZDUmxOdzFENV93UkpqOX BMNmc to access the results of the form. https://docs.google.com/spreadsheet/ccc?key=0Ao xuu3aQOMKNdHZQaXZDUmxOdzFENV93UkpqOX BMNmc Slide 32 Problem Solving w/No Technology Slide 33 Problem Solving with Technology Slide 34 Slide 35 Slide 36 . Slide 37 Student Example: Problem Solving Number of TablesNumber of Seats 15 28 311 414 517 2062 I found there would be 17 seats at 5 tables. I noticed that each time I added a table, the number of seats increased by three. That is because we are adding five new places but losing two on the sides of the tables that connect. To answer question 2, I counted out 15 more trapezoids to make a total of 20 tables. Then I skip counted by threes for each new table, starting with 17, until I came to 62 seats for the 20th table. Student 1 : List created after student modeled with pattern blocks. Slide 38 Student Example: Problem Solving Student 2: (Draw a diagram, then direct calculation) I used pattern blocks to help me see the pattern. After adding several tables I discovered that each table added three seats to the total. At the ends there were always two more seats, no matter how long the row was. Here are my results: (organized list) To answer question 2, I figured out how many more tables I would need: 20 - 5 = 15 more tables Each of those 15 new tables would add three seats to the row: 15 * 3 = 45 more seats I added the seats from 5 tables and the new seats: 17 + 45 = 62 total seats at 20 tables Slide 39 Student Example: Problem Solving Student 3: (Direct Calculation based on seats lost) I multiplied the number of tables by the number of people who can sit at one table: 5 * 5 = 25 For each of the places where two tables connect, we lose 2 seats. There are 4 of those places, one less than the total number of tables. 4 * 2 = 8 I subtracted the number of seats lost from the total places: 25 8 = 17 seats at 5 tables I did the same thing for 20 tables: 20 * 5 = 100 total seats 19 * 2 = 38 seats lost at the connections 100 38 = 62 seats at 20 tables Slide 40 Use words or numbers and symbols to write a rule for calculating the number of volunteers that can sit at any given number of tables. How many tables would it take, arranged in a straight line, to seat 85 volunteers? Slide 41 Extend the Problem Slide 42 Slide 43 Discuss your attention to precision Communicate terms and thinking process Link to Spreadsheet from form. Slide 44 Mathematical Practices Expertise that we each seek to develop in our students What does it mean to do mathematics? What does it mean to understand mathematics? As teachers, our goal is to provide regular and consistent opportunities to develop and build these habits of mathematical thinking. Slide 45 1st Steps to Implementation of Mathematical Practices How to transition Begin with the Mathematical Practices of CCS Look closely at the Critical Areas provided for each grade What shall I do first Focus on the Mathematical Practices How do your students model the Mathematical Practices? How do your students model the Mathematical Practices? In what ways do your classroom strategies foster development of the Mathematical Practices? In what ways do your classroom strategies foster development of the Mathematical Practices? Implement the Critical Ideas Look at the ADE website for standards, crosswalk, and summary of changes. Look at the ADE website for standards, crosswalk, and summary of changes. Slide 46 Resources for Further Exploration The Illustrative Mathematics Project http://illustrativemathematics.org/ http://illustrativemathematics.org/ Math Common Core Coalition http://www.nctm.org/standards/mathcommoncore/ http://www.nctm.org/standards/mathcommoncore/ Achieve the Core http://www.achievethecore.org/http://www.achievethecore.org/ National Council Teacher of Mathematics http://www.nctm.org/standards/content.aspx?id=23273 Guiding Principles for Mathematics Curriculum and Assessment http://www.nctm.org/standards/content.aspx?id=23273 Mathematics Problem Solving http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html http://jwilson.coe.uga.edu/emt725/PSsyn/Pssyn.html Mathematics Through Problem Solving http://www.mathgoodies.com/articles/problem_solving.html http://www.mathgoodies.com/articles/problem_solving.html Slide 47 Resources for Further Exploration Inside Mathematics http://www.insidemathematics.org/index.php/standard-1 http://www.insidemathematics.org/index.php/standard-1 Curriculum Exemplars from EngageNY http://engageny.org/resource/curriculum-exemplars/ http://engageny.org/resource/curriculum-exemplars/ Indiana Dept of Ed Implementing the Standards for Mathematical Practice Indiana Dept of Ed - Implementing the Standards for Mathematical Practice http://media.doe.in.gov/commoncore/2011-05-10- StandforMath.htmlhttp://media.doe.in.gov/commoncore/2011-05-10- StandforMath.html Tools for the Common Core Standards http://commoncoretools.me/http://commoncoretools.me/ Slide 48 Resources for Further Exploration New Jersey Center for Teaching & Learning Progressive: Mathematics Initiative http://njctl.org/programs/ Free digital course content for over twenty courses, these initiatives span K-12 mathematics and high school science. http://njctl.org/programs/ Rich problem source (middle/HS) http://psc.stanford.edu/psc- materials/problem-bank.htmlhttp://psc.stanford.edu/psc- materials/problem-bank.html Kenken http://www.kenken.com/index.htmlhttp://www.kenken.com/index.html Math Forum http://mathforum.org/kenken/ http://mathforum.org/kenken/ Online Pattern Blocks http://gingerbooth.com/flash/patblocks/patblocks.php http://gingerbooth.com/flash/patblocks/patblocks.php Wiki on Standards of Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice Slide 49 To view the official ADE documents 2010 Arizona Mathematics Standards Overview of the 2010 Mathematical Standards PDFPDF Standards for Mathematical Practices PDFPDF Mathematics Introduction (Coming Soon) Mathematics Glossary PDFPDF Summary of Updates to Explanations and Examples PDFPDF Slide 50 More Questions Contact ADE Mary Knuck [email protected]@azed.gov Suzi Mast [email protected]@azed.gov