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REASONING AND EXPLAINING: MATHEMATICAL PRACTICES OF THE COMMON CORE Statewide Instructional Technology Project

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Introduction This is the third in a series of five webinars on the Standards of Math Practice. Links to resources, including recordings of previous webinars can be found at ( http://tinyurl.com/StandardsMathPractice )

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To view the official ADE documents 2010 Arizona Mathematics Standards 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

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Mathematical Practices

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Standards of Math Practice Reflect Changes in how we Approach Teaching and Learning Teacher Centered Student Centered TeachersFountains of Knowledge Create an environment that supports learning StudentsReceptive Learners Active Learners Teacher/Student Relationship AdversarialCollaborative

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MP2. Reason abstractly and quantitatively. MP3. Construct viable arguments and critique the reasoning of others. Reasoning & Explaining

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MP2. Reason abstractly and quantitatively. Mathematically proficient students can… Make sense of quantities and their relationships in problem situations. Take math in and out of context Work with symbols as abstractions Use a well-developed number sense Apply properties of operations Understand what units represent Which of the above skills do you find to be the greatest challenge to develop in your students?

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MP2. Reason abstractly and quantitatively. Take math in and out of context: Decontextualize Translate a problem situation into a number sentence to solve it. Work with symbols as abstractions. 7 + 9 = ? --------------------------------------------------- If my house is at 0 and my friend’s house is 5 units away where could my friend’s house be located? |5|

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MP2. Reason abstractly and quantitatively. Take math in and out of context: Contextualize Recognize the connection between all the elements of the number sentence and the original problem – and then make sure that the units in problems make sense. Simon Howden / FreeDigitalPhotos.net Stuart Miles / FreeDigitalPhotos.net Matt Banks / FreeDigitalPhotos.net

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MP2. Reason abstractly and quantitatively. Number Sense. Part-whole thinking Counting Estimation skills Place value Unit Mental Calculation The lookback time tL (the difference between the age t0 of the Universe now and the age te of the Universe at the time the photons were emitted. Used to predict properties of passive stellar evolution for galaxies. T=13.7 x [1 - (1 + z)-3/2].

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MP2. Reason abstractly and quantitatively. Properties of Operations 16 + 21 = 10 + 20 + 6 + 1 = 30 + 7 = 37 16 + 21 = 16 + 1 + 21 - 1 = 17 + 20 = 37 16 + 21 = 16 + 20 + 1 = 36 + 1 = 37 Consider the properties of multiplication: Does multiplication always make the product greater than either of the factors? 2 x 8 = 16 (yes) 8 x ¼ = 2 (no) 8 x -3 = -24 (no) Can you subtract a larger number from a smaller number?

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MP2. Reason abstractly and quantitatively. Understanding Unit After calculating, we need to make sure that the units, or objects in a problem solution make sense. 4 th graders decide how many vans it would take to carry 36 students if each van could carry 8. After calculating, they found a calculated answer of 4 r 4. Successful solvers had to understand that the unit was a bus, and that the solution was 5 buses, not 5 children or just 5.

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TECH RESOURCES FOR SUPPORT MP2. Reason abstractly and quantitatively.

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Thinkfinity http://www.thinkfinity.orghttp://www.thinkfinity.org Online Resources

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Main page of free teaching materials: http://www.geogebra.org/en/wiki/index.php/English#Number_Sense Area of a circle: http://www.geogebratube.org/student/m279 Classifying Quadrilaterals: http://www.geogebra.org/en/upload/files/english/dfreeston/Quadrilaterals.html http://www.geogebra.org/en/wiki/index.php/English#Number_Sensehttp://www.geogebratube.org/student/m279 http://www.geogebra.org/en/upload/files/english/dfreeston/Quadrilaterals.html Geogebra (Mid/ HS) http://www.geogebra.org/cms/http://www.geogebra.org/cms/

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REASONING The Math Forum @ Drexel Technology Problem of the Week (free, but requires sign up) http://mathforum.org/tpow/list.htmlhttp://mathforum.org/tpow/list.html

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MP2. Reason abstractly and quantitatively. Resources for further exploration Illuminations http://illuminations.nctm.org/Activities.aspx?grade=1&grade=2&grade=3&grade=4 http://illuminations.nctm.org/Activities.aspx?grade=1&grade=2&grade=3&grade=4 Illuminations Circle Tool http://illuminations.nctm.org/ActivityDetail.aspx?ID=116 http://illuminations.nctm.org/ActivityDetail.aspx?ID=116 ThinkPort http://www.thinkport.org/Classroom/math.tphttp://www.thinkport.org/Classroom/math.tp Wolfram Demonstrations Project: Open-code resource to illuminate concepts in science, technology, math and a range of other fields. Currently almost 8,000 demonstrations. Download free format. http://demonstrations.wolfram.com/ http://demonstrations.wolfram.com/ Shodor Interactivate http://www.shodor.org/interactivate/http://www.shodor.org/interactivate/ Math Assessment Project http://map.mathshell.org/materials/tasks.php http://map.mathshell.org/materials/tasks.php

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MP3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students can… Understand and use assumptions, definitions, and prior results. Use observations about data to form conjectures and build logical progressions to support them. Listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Think about and use precision. Compare and contrast various solution strategies. Justify and communicate their conclusions Recognize and use counter-examples.

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MP3. Construct viable arguments and critique the reasoning of others.

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An argument is how you justify, or explain your thinking to others. An argument is a logical statement or discussion in which reasons are put forward in support of and against a proposal. It is the answer to the question, “How do you know that?” It is not a personal attack on your opinion.

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MP3. Construct viable arguments and critique the reasoning of others. Strategies Start with the great problems. Regularly engage students in mathematical discussions. Ask students to explain their thinking. Encourage precision. Solicit multiple answers. Have students justify why a solution is correct. Model and provide scaffolding. Explore the difference between opinion and fact. Embed logic into your discussions: Does one part depend on another part? Does changing one aspect of the problem change the result?

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MP3. Construct viable arguments and critique the reasoning of others. More Strategies: Great Questions How can you show that your computation is correct? Can you use a different tool or strategy to do that computation and see if you get the same answer? Can you compare your work with someone else? I didn’t understand… What possibilities did you consider? What criteria did you use? Why did you reject some options? What made you choose this option ? Wonderful Math Questions from PBS: http://www.pbs.org/teachers/_files/pdf/Microsoft%20Word_FINALMathTipDoc.pdf

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MP3. Construct viable arguments and critique the reasoning of others. More Strategies: Growing Mathematical Discussions A researcher analyzed videos of 4 th and 5 th grade math classes in the United States. What percentage of the classroom discourse was made up of teacher talk? A. 50 – 64% B. 65 – 74% C. 75 – 84% D. 85 – 95%

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Clip : http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-numerical- patterning/220-numerical-patterning-introduction-part-b http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-numerical- patterning/220-numerical-patterning-introduction-part-b As you view the video, please focus on what type of discourse you hear the teachers (mostly) using, and what type you hear students (mostly) using.

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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.

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1st Steps to Implementation of Mathematical Practices How to transition How to transition Begin with the Mathematical Practices of CCS Begin with the Mathematical Practices of CCS Look closely at the Critical Areas provided for each grade Look closely at the Critical Areas provided for each grade What shall I do first What shall I do first Focus on the Mathematical Practices 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 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.

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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

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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/

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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/ Wiki on Standards of Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice http://enhancingmypractice.wikispaces.com/Standards+of+Math+Practice

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To view the official ADE documents 2010 Arizona Mathematics Standards 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

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More Questions Contact ADE Mary Knuck Mary.Knuck@azed.govMary.Knuck@azed.gov Suzi Mast suzi.mast@azed.govsuzi.mast@azed.gov

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Abstract and Quantitative Thinking Abstract thinking Using concepts and to make and understand generalizations Can discern patterns beyond the obvious Use patterns or clues to solve larger problems. Quantitative reasoning Understand magnitude and proportion Visualize those abstractions Apply those abstractions to a problem

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