Math Instruction and the Common Core--How to help your students Master the Content
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The document provides an overview of effective instructional strategies for teaching the Common Core State Standards for Mathematics, including focusing instruction where the standards focus, thinking across grades and topics to promote coherence, and pursuing conceptual understanding, skill and fluency, and application of mathematical concepts. It discusses the three shifts in instruction required by the CCSS and examples of how to implement tasks, scaffolding, modeling and other strategies to develop students' mathematical understanding and skills.
Math Instruction and the Common Core--How to help your students Master the Content
- 2. Math Instruction and the Common Core: How to Help Your Students Master the Content Clare Wurm, Consultant State Education Resource Center
- 3. Agenda: o Review Common Core State Standards for Mathematics o Learn effective instructional strategies o Practice strategies o Questions
- 4. Three Minute Video Overview of CCSS
- 5. CCSS Math: 3 Instructional Shifts • Focus strongly where the standards focus • Coherence: Think across grades and link to major topics within grades • Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application
- 6. Focusing attention within Number and Operations Operations and Algebraic Thinking Expressions and Equations Algebra Number and Operations— Base Ten The Number System Number and Operations— Fractions K 1 2 3 4 5 6 7 8 High School 6
- 7. Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction, measurement using whole number quantities 3–5 Multiplication and division of whole numbers and fractions 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra and linear functions Priorities in Mathematics 7
- 8. CCSS Math: 3 Instructional Shifts • Focus strongly where the standards focus • Coherence: Think across grades and link to major topics within grades • Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application
- 9. CCSS Math: 3 Instructional Shifts • Focus strongly where the standards focus • Coherence: Think across grades and link to major topics within grades • Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application
- 10. Required Fluencies in K-6 Grade Standard Required Fluency K K.OA.5 Add/subtract within 5 1 1.OA.6 Add/subtract within 10 2 2.OA.2 2.NBT.5 Add/subtract within 20 (know single-digit sums from memory) Add/subtract within 100 3 3.OA.7 3.NBT.2 Multiply/divide within 100 (know single-digit products from memory) Add/subtract within 1000 4 4.NBT.4 Add/subtract within 1,000,000 5 5.NBT.5 Multi-digit multiplication 6 6.NS.2,3 Multi-digit division Multi-digit decimal operations 10
- 11. CCSS Math Organization Two distinct sets of standards: Content Standards (math concepts) GRADE or COURSE SPECIFIC Practice Standards (habits of mind) APPLY to K-12
- 12. CCSS Math Practice Standards 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
- 13. How do the CCSS change instruction? • Three Aspects of Rigor • Mathematical Practices • Content Standards
- 14. How do the CCSS change instruction? • Three Aspects of Rigor • Mathematical Practices • Content Standards
- 16. How do the CCSS change instruction? • Three Aspects of Rigor • Mathematical Practices • Content Standards
- 17. CCSS Math Practice Standards 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
- 18. How do the CCSS change instruction? • Three Aspects of Rigor • Mathematical Practices • Content Standards
- 19. How do the CCSS change instruction? 5.G.4 Classify two-dimensional figures in a hierarchy based on properties. Rather than: Define the following: SQUARE RECTANGLE POLYGON RHOMBUS QUADRILATERAL
- 20. How do the CCSS change instruction? POLYGON QUADRILATERAL RECTANGLE RHOMBUS
- 21. CCSS Considerations for Students Receiving Special Education Services Standards are for ALL students. Standards are intended to make all students college and career-ready. IEP goals should align to grade- appropriate standards
- 22. Level of Support • General Education “as designed” • Supplemental Instruction •Direct instruction •Address gaps • With Accommodations (How) •Changes in instruction and materials •Create access (remove barriers) • With Modifications (What) •Change in curriculum (amount or performance)
- 24. CCSS Math Instruction 1. Ask student to explain thinking. 2. Break problem down into discrete tasks. 3. Use a variety of models and manipulatives. 4. Use scaffolded instruction.
- 25. Brainteaser! How many different rectangles are in this figure?
- 26. 1. Ask student to explain thinking. clarifies vocabulary allows speaker to hear and clarify his/her thoughts allows others to hear different solution and compare to their own informs teacher about student’s understanding
- 27. 1. Ask student to explain thinking. o Discourse facilitates process that deepens learning. (Barnes, 1993; Cazden; 2001) o Constructivists believe higher-level thinking is enhanced through discourse with peers and teacher. Vygotsky (1934, 1978); (Almasi, 2002)
- 28. 1. Ask student to explain thinking. 1. Ask student to explain thinking. Usually not enough
- 29. 1. Ask student to explain thinking. Keep asking! Students need practice articulating ideas. How did you start? What do you need to find out? Does your method always work? How is your strategy different than _______? (book’s, classmate’s, teacher’s) Convince me that you’re right.
- 30. 2. Break problem down into discrete tasks. AKA: Task Analysis Each discrete task can be taught separately.
- 31. 2. Break problem down into discrete tasks. “Task analysis can be used effectively with all children, regardless of cognitive level and/or expressive communicative abilities.” “This evidence-based practice can be used for any skill that can be broken down into smaller steps, including academics, behaviors, communication, and social skills.” Franzone, E. (2009b). Overview of task analysis.
- 32. 2. Break problem down into discrete tasks. Example: Adding fractions
- 36. 2. Break problem down into discrete tasks. Example: 7.RP.2 Recognize and represent proportional relationships between quantities. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships. Represent proportional relationships by equations.
- 37. A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how the constant of proportionality was determined and how it relates to both the table and graph.
- 38. Tasks involved: Read and interpret table. Create a graph from data in table. Determine if quantities are proportional. Find unit rate or constant of proportionality. Explain how you found it. A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how the constant of proportionality was determined and how it relates to both the table and graph.
- 39. Read and interpret table. Create a graph from data in table. Determine if quantities are proportional. Find unit rate or constant of proportionality. Explain how you found it.
- 40. 3. Use a variety of models and manipulatives. Students think and learn differently so diverse models will “click” for different students. Seeing a concept in different forms helps students make connections among math concepts.
- 41. 3. Use a variety of models and manipulatives. 4. Use a variety of models and manipulatives. Concrete Representational Abstract
- 42. 3. Use a variety of models and manipulatives. Concrete = manipulatives Representational = drawing, diagram Abstract = mathematical symbols
- 43. 3. Use a variety of models and manipulatives. 3.MD.7.a: Relate area to the operations of multiplication and division. 5 x 3
- 44. 3. Use a variety of models and manipulatives. 5 x 3 Concrete:
- 45. 3. Use a variety of models and manipulatives. 5 x 3 Concrete:
- 47. 3. Use a variety of models and manipulatives. 5 x 3 : http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?from=category_g_1_t_1.html
- 49. 3. Use a variety of models and manipulatives.
- 50. 3. Use a variety of models and manipulatives.
- 51. 3. Use a variety of models and manipulatives.
- 52. 3. Use a variety of models and manipulatives. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- 53. 3. Use a variety of models and manipulatives. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Seniors are selling candygrams for $3 each. The principal donates $5 to start off their fundraising. Write an equation to show how much money they will have(y) after selling x candygrams.
- 54. 3. Use a variety of models and manipulatives. X = candygrams Y= Money raised 0 5 1 8 5 20 10 35 25 80 50 155 0 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 60 Moneyraised Candygrams sold Seniors' Fundraiser
- 55. 3. Use a variety of models and manipulatives. Seniors are selling candygrams for $3 each. The principal donates $5 to start off their fundraising. Write an equation to show how much money they will have(y) after selling x candygrams.
- 56. 3. Use a variety of models and manipulatives. Seniors are selling candygrams for $3 each. The principal donates $5 to start off their fundraising. Write an equation to show how much money they will have(y) after selling x candygrams.
- 57. 4. Scaffold Instruction Instructional Scaffolding… ..is support that teachers provide to facilitate students’ development of math proficiency. …is essential for teaching students with or at risk for mathematics disabilities. Doabler, et al. Enhancing Core Mathematics Instruction for Students at Risk for Mathematics Disabilities
- 58. 4. Scaffold Instruction As students become more independent in their learning, the scaffolding is gradually withdrawn. Doabler, et al. Enhancing Core Mathematics Instruction for Students at Risk for Mathematics Disabilities
- 59. 4. Scaffold Instruction Gradual Release I do. We do. You do.
- 60. Gradual Release Teacher or Para Role I do it. We do it. You do it together. You do it alone. Examples • Think- Alouds •Modeling •Demon- stration •Coaching •Guided practice •Small group • Partner work • Group work • Prompting • Independent work Watch it Practice it Try it Student Does, Teacher Watches Teacher Does, Student Watches Lots of Guided Practice 60
- 61. 4. Scaffold Instruction Scaffolded instruction will incorporate some of the other instructional techniques that we have discussed.
- 62. Brainteaser! How many different rectangles are in this figure?
- 63. How many different rectangles are in this figure? • Think aloud • Create a table • List some sizes; ask student(s) to find other sizes • Make multiple copies to color/highlight possibilities
- 64. 4. Scaffold Instruction A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit are proportional for each serving size listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how the constant of proportionality was determined and how it relates to both the table and graph.
- 65. 4. Scaffold Instruction 1.OA.a.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. TASK: a. There are 8 children and 6 chairs. A child sits in each chair. How many children won’t have a chair? b. There are 8 children and some chairs. A child sits in each chair. 2 children don’t have a chair. How many chairs are there?
- 66. CCSS Math Instruction 1. Ask student to explain thinking. 2. Break problem down into discrete tasks. 3. Use a variety of models and manipulatives. 4. Use scaffolded instruction.
- 68. Practice There are math problems posted around the room. Choose one at your students’ level and try one (or more) of the instructional strategies to plan instruction. Be ready to share. Share questions that you have on QUESTIONS board. Peruse the Common Core State Standards for Mathematics.
- 70. Bibliographical Notes Doabler, et.al. Enhancing Core Mathematics Instruction for Students at Risk for Mathematical Disabilities, Teaching Exceptional Children, Mar/Apr 2012. Franzone, E. (2009b). Overview of task analysis. Madison, WI: National Professional Development Center on Autism Spectrum Disorders, Waisman Center, University of Wisconsin. Wakeman, Karvonen & Ahumada, Changing Instruction to Increase Achievement for Students With Moderate to Severe Intellectual Disabilities, Teaching Exceptional Children, Nov/Dec 2013.
- 71. Math Resources Achieve the Core -- CCSS basics, background, examples http://www.achievethecore.org/ Illustrative Mathematics -- Standards, progressions, problems and tasks http://www.illustrativemathematics.org/ National Library of Virtual Manipulatives -- Java interactive math activities http://nlvm.usu.edu/ CCSS basics High School Mathemtics Instructional Toolkit (p. 5= math discussion tips) http://celenza.wikispaces.com/file/view/Math_Instructional_Toolkit.pdf IXL Learning -- Practice problems by standard http://www.ixl.com/math/ NCTM Illuminations http://illuminations.nctm.org/ Reproducible Resources (Problem-Solving Organizer) http://www.solution-tree.com/free- resources/specialneeds
- 72. Clare Wurm, Consultant tel: 860-632-1485, x 383 email: wurm@ctserc.org SERC LIBRARY, Offers more than 10,000 resources www.ctserc.org/library – Books – Instructional materials – Tests – Journals – Online databases – DVDs, videos, CD-ROMs – Professional development materials for staff Contact Us or Visit the SERC Library