This module was developed by Amy Hillen, Kennesaw State University; DeAnn Huinker, University of Wisconsin-Milwaukee; and Victoria Bill, University of Pittsburgh Institute for Learning. Video courtesy Patterson 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. Principles to Actions Effective Mathematics Teaching Practices The Case of Millie Brooks and the Half of a Whole Task Grade 3 Principles to Actions Effective Mathematics Teaching Practices The Case of Millie Brooks and the Half of a Whole Task Grade 3

Overview of the Session Solve and discuss the Half of a Whole Task. Watch a video clip of a third grade teacher facilitating small group work using this task. Discuss ways the teacher supports her students’ learning of mathematics. Connect specific teacher actions seen in the video to the Effective Mathematics Teaching Practices.

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)

Half of a Whole Task

The Half of a Whole Task Identify all of the figures that have one half shaded. Be prepared to explain how you know that one half of the figure is shaded. Write a written description giving your reason why a figure is showing halves. If a figure does not show one half shaded explain why the figure is not showing halves. (Adapted from Watanabe, 1996, p. 461)

The Half of a Whole Task What do students need to know and be able to do in order to engage with the task? What mathematical ideas does the task provide an opportunity for students to learn?

Ms. Brooks Learning Goals and Connections to the Common Core State Standards Ms. Brooks Learning Goals and Connections to the Common Core State Standards

Establish Mathematics Goals to Focus Learning Learning Goals should: clearly state what it is students are to learn and understand about mathematics as the result of instruction; be situated within learning progressions; and frame the decisions teachers make during a lesson. “Formulating clear, explicit learning goals sets the stage for everything else.” (Hiebert, Morris, Berk, & Janssen, 2007, p.57)

Ms. Brooks’ Mathematics Learning Goals Students will understand that: A fraction describes the division of a whole region or area into equal parts. A fraction is relative to the size of the whole unit. If the numerator is half the quantity in the denominator then the fraction is equal to a half. Fractions have an infinite number of equivalent forms, regardless of whether or not the pieces of the whole unit are adjacent.

Connections to the CCSSM Grade 3 Standards for Mathematical Content Number and Operations – Fractions (NF) Develop understanding of fractions as numbers. 3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.A.3Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. (a) Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. (b) Recognize and generate simple equivalent fractions (e.g., 1/2=2/4, 4/6=2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Common core state standards for mathematics.

Connections to the CCSSM Standards for Mathematical Practice 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. Retrieved from 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.

Ms. Brooks Third Grade Classroom Ms. Brooks Third Grade Classroom

Context of the Video Clip Teacher:Millie Brooks Grade:3 School:#26 District:Paterson Public Schools This lesson takes place in February. Students have just begun their study of fractions. After reading the instructions aloud, Ms. Brooks asked students to work on the task in small groups. The clip shows the interactions that Ms. Brooks had with one of the small groups, in which there was a disagreement about Figure (d).

Lens for Watching the Video: Viewing #1 As you watch the video clip, make note of: Ways Ms. Brooks supports her students’ learning, and The mathematical ideas students are grappling with as they discuss, explain, and try to justify their reasoning.

Effective Mathematics Teaching Practice Effective Mathematics Teaching Practice

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.

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.

Facilitate Meaningful Mathematical Discourse Mathematical discourse should: Build on and honor students’ thinking; Provide students with the opportunity to share ideas, clarify understandings, and develop convincing arguments; and Advance the math learning of the whole class. Discussions that focus on cognitively challenging mathematical tasks, namely those that promote thinking, reasoning, and problem solving, are a primary mechanism for promoting conceptual understanding of mathematics. (Smith, Hughes, Engle, & Stein, 2009, p. 549)

Pose Purposeful Questions Effective Questions should: Reveal students’ current understandings; Encourage students to explain, elaborate, or clarify their thinking; and Make the mathematics more visible and accessible for student examination and discussion. Teachers’ questions are crucial in helping students make connections and learn important mathematics concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding. (Weiss & Pasley, 2004)

Lens for Watching the Video: Viewing #2 As you re-watch the video clip, pay particular attention to: The ways in which Ms. Brooks facilitated meaningful mathematical discourse, and The questions asked by Ms. Brooks. Be prepared to give examples and to cite line numbers from the transcript to support your observations.

Applying Ideas to Your Own Classroom

As you reflect on the Effective Mathematics Teaching Practices examined in this session, identify one or two ideas or insights that you might apply to your own classroom instruction.

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: Ensuring mathematical success for all. Reston, VA: Author.

References National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Watanabe, T. (1996). Ben’s understanding of one-half. Teaching Children Mathematics, 2(8),