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Principles to Actions Effective Mathematics Teaching Practices The Case of Jennifer DiBrienza and the Equation Strings Task Grade 1 This module was developed by DeAnn Huinker, University of Wisconsin-Milwaukee; Victoria Bill, University of Pittsburgh Institute for Learning; and Amy Hillen, Kennesaw State University. Video courtesy of New York City Public Schools and the University of Pittsburgh Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele. INSTITUTE for LEARNING Overview of the Session • Watch a video clip of a first grade class engaged in a whole class discussion of an equation string. • Discuss what the teacher does to support the students’ learning of mathematics. • Relate teacher actions in the video to the effective mathematics teaching practices. INSTITUTE for LEARNING Teaching and Learning Principle An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 7) INSTITUTE for LEARNING Ms. DiBrienza’s First Grade Classroom “Equation Strings Task” INSTITUTE for LEARNING Ms. DiBrienza’s Mathematics Learning Goals Students will understand that • Fluent addition strategies use number relationships and the structure of the number system; • Numbers can be decomposed and added on in parts, not just by ones; and • Noticing regularity in repeated calculations leads to shortcuts and general methods for adding numbers. INSTITUTE for LEARNING The Equation Strings Task Solve the set of addition problems. Each time you solve a problem, try to use the previous problem to solve the next problem. 7 + 3 = ___ 17 + 3 = ___ 27 + 3 = ___ 37 + 3 = ___ 37 + 5 = ___ After you have solved all of the problems, describe some patterns that you notice in the sequence of equations. Show how you might represent student reasoning with a drawing or on a number line so that students could visually see the relationships among the quantities. INSTITUTE for LEARNING Connections to the CCSSM Grade 1 Standards for Mathematical Content Number and Operations in Base Ten (NBT) Use place value understanding and properties of operations to add and subtract. 1.NBT.C.4 Add within 100, including adding a two-digit number and a onedigit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 1.NBT.C.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics. INSTITUTE for LEARNING Retrieved from http://www.corestandards.org/Math/Content/1/NBT Connections to the CCSSM Standards for Mathematical Practice 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. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics. INSTITUTE for LEARNING Retrieved from http://www.corestandards.org/Math/Practice Standards for Mathematical Practice (SMP) SMP 7. Look for and make use of structure. Mathematically proficient students at the elementary grades use structures such as place value, the properties of operations, other generalizations about the behavior of the operations (for example, the less you subtract, the greater the difference), and attributes of shapes to solve problems. In many cases, they have identified and described these structures through repeated reasoning (SMP 8). SMP 8. Look for and express regularity in repeated reasoning. Mathematically proficient students at the elementary grades look for regularities as they solve multiple related problems, then identify and describe these regularities.... proficient students formulate conjectures about what they notice. Illustrative Mathematics. (2014, February 12). Standards for Mathematical Practice: Commentary and Elaborations for K–5. Tucson, AZ.. Retrieved from http://commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf (pp.18-19) INSTITUTE for LEARNING Classroom Context for the Video Segment Teacher: Jennifer DiBrienza Grade: 1 School: PS 116 District: New York Community School District 2 The students are seated in a large circle on the classroom rug. The teacher poses a sequence of equations for the students to solve. The students share their strategies and discuss “what they notice” about the equations and the answers. INSTITUTE for LEARNING Lens for Watching the Video: Viewing #1 As you watch the video • Make note of what the teacher does to support student learning of mathematics, and • Identify the mathematical insights that surfaced for students. INSTITUTE for LEARNING Student Learning of Mathematics In small groups, discuss the following: 1. Identify mathematical insights that surfaced for students. 2. Summarize ways the teacher engaged students in the mathematical practices: SMP 7 and SMP 8. 3. Summarize ways the teacher advanced student understanding toward the content standards: 1.NBT.C.4 and 1.NBT.C.5. INSTITUTE for LEARNING Teacher Actions As you reflect on the teacher actions that supported student learning in this lesson, which Effective Mathematics Teaching Practices did you notice the teacher using? INSTITUTE for LEARNING Effective Mathematics Teaching Practices 1. Establish mathematics goals to focus learning. 2. Implement tasks that promote reasoning and problem solving. 3. Use and connect mathematical representations. 4. Facilitate meaningful mathematical discourse. 5. Pose purposeful questions. 6. Build procedural fluency from conceptual understanding. 7. Support productive struggle in learning mathematics. 8. Elicit and use evidence of student thinking. National Council of Teachers of Mathematics. (2014). Principles to actions: INSTITUTE for LEARNING Ensuring mathematical success for all. Reston, VA: Author. Effective Mathematics Teaching Practice Build Procedural Fluency from Conceptual Understanding INSTITUTE for LEARNING Teaching Practice Focus: Build Procedural Fluency from Conceptual Understanding Effective teaching of mathematics • builds on a foundation of conceptual understanding; • results in generalized methods for solving problems; and • enables students to flexibly choose among methods to solve contextual and mathematical problems. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42) INSTITUTE for LEARNING Teaching Practice Focus: Build Procedural Fluency from Conceptual Understanding Jot down your responses. • What does it mean to be fluent with computational procedures? • Why is it important to build procedures from conceptual understanding? INSTITUTE for LEARNING Fluency Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently. Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems. This sequence is beneficial whether students are building toward fluency with single- and multidigit computation with whole numbers or fluency with, for example, fraction operations, proportional relationships, measurement formulas, or algebraic procedures. INSTITUTE for LEARNING National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42) Conceptual Understanding and Procedural Fluency When procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations (Fuson, Kalchman, and Bransford 2005). Martin (2009, p. 165) describes some of the reasons that fluency depends on and extends from conceptual understanding: To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42) INSTITUTE for LEARNING Lens for Watching the Video: Viewing #2 As you watch the video again, attend to the teacher actions that build procedural fluency from conceptual understanding. Be prepared to give examples and to cite line numbers from the transcript to support your observations. INSTITUTE for LEARNING Video Observations In what ways did the teacher actions support students in building procedural fluency from conceptual understanding? Use the transcript to cite line numbers to support your observations. INSTITUTE for LEARNING Effective Mathematics Teaching Practice Implement Tasks that Promote Reasoning and Problem Solving INSTITUTE for LEARNING Teaching Practice Focus: Implement Tasks that Promote Reasoning and Problem Solving Effective teaching of mathematics • provides opportunities for students to engage in solving and discussing tasks; • uses tasks that promote inquiry and exploration and are meaningfully connected to concepts; • uses tasks that allow for multiple entry points; and • encourages use of varied solution strategies. National Council of Teachers of Mathematics. (2014). Principles to actions: INSTITUTE for LEARNING Ensuring mathematical success for all. Reston, VA: Author. (p. 17) Teaching Practice Focus: Implement Tasks that Promote Reasoning and Problem Solving Mathematical tasks can range from a set of routine exercises to a complex and challenging problem that focuses students’ attention on a particular mathematical ideas (p. 17). It is important to note that not all tasks that promote reasoning and problem solving have to be set in a context or need to consume an entire class period or multiple days. What is critical is that a task provide students with the opportunity to engage actively in reasoning, sense making, and problem solving... (p. 20). National Council of Teachers of Mathematics. (2014). Principles to actions: INSTITUTE for LEARNING Ensuring mathematical success for all. Reston, VA: Author. How did the teacher use the equations string task to promote reasoning and problem solving in ways that advanced student understanding toward the mathematics learning goals of the lesson? 7+3= 17 + 3 = 27 + 3 = 37 + 3 = 37 + 5 = INSTITUTE for LEARNING As you reflect on the effective mathematics teaching practices examined in this session, summarize one or two ideas or insights you might apply to your own classroom instruction. INSTITUTE for LEARNING INSTITUTE for LEARNING Extension INSTITUTE for LEARNING Equation Strings Tasks Work with a small group to create a string of equations appropriate for your students. • Select an operation and the type of numbers (e.g., whole numbers, fractions, percents). • Identify the mathematics learning goals which will be supported by your equation string. • Develop a sequence of equations that will build toward student fluency connected to conceptual understanding and reasoning. INSTITUTE for LEARNING