2014 Mathematics Institutes Grade Band: 3-5 1

Making Connections and Using Representations The purpose of the 2014 Mathematics Institutes is to provide professional development focused on instruction that supports process goals for students in mathematics. Emphasis will be on fostering students’ ability to make mathematical connections and use effective and appropriate representations in mathematics. 2

Agenda I.Defining Representations and Connections II.Doing the Mathematical Task III.Looking at Student Work IV.Facilitating the Use of Effective Representations and Connections V.Planning Mathematics Instruction VI.Closing 3

Representations and Connections Turn and Talk: What does it mean for students to represent and make connections in the mathematics classroom?

Mathematical Representations Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations graphical, numerical, algebraic, verbal, and physical and recognize that representation is both a process and a product. 5 Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

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7 “Representations are useful in all areas of mathematics because they help us develop, share, and preserve our mathematical thoughts. They help to portray, clarify, or extend a mathematical idea by focusing on its essential features.” National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 206). Reston, VA.

Mathematical Connections Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other. 8 Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

Brainstorm and Chart COLOR Partner: How do you consider representations and connections when planning for instruction? What questions should be considered regarding representations and connections when planning for instruction?

Common Fraction Misconceptions in 3-5 Jot Thoughts: One idea per sticky What common misconceptions, error patterns, or confusions are evident in grades 3-5 when it comes to fractions?

Rally Share Read your sticky note, and place it in the center of the table. If someone has the same thing, they can add it to the pile. Continue until all ideas have been shared. Sort your ideas and label. Share out whole group. 11

What the Research Says… Sherman, Richardson, and Yard (2005) suggest several reasons for students’ difficulty in learning about fractional concepts and skills: They memorize procedures and rules before they have developed a conceptual understanding of the related concepts. Children over-generalize what they know about whole-number computation and apply this knowledge to fractions. Estimating rational numbers is more difficult than estimating whole numbers. Recording fractional notation is confusing if they don’t yet understand what the top and bottom numbers represent. Sherman, H.J., Richardson, L.I., & Yard, G. (2005). Teaching children who struggle with Mathematics: A systematic approach to diagnosis and instruction, Pearson Education, Inc. 12

Why? Fractions Problem SolvingConnectionsCommunicationRepresentationsReasoning

Why? Fractions Problem SolvingConnectionsCommunicationRepresentationsReasoning

Rich Mathematical Task In Group A, 4 people share 3 sandwiches. In Group B, 6 people share 4 sandwiches. Who gets more, a person in group A, or a person in group B? How do you know?

Share

Orchestrating Productive Mathematics Discussions 1.Anticipating 2.Monitoring 3.Selecting 4.Sequencing 5.Connecting

Anticipate, Monitor, Select, & Sequence 18

Pair Up Stand up and push in your chair. Walk/Dance around until the music stops. When the music stops, the person closest to you will be your partner. 19

Representations Discuss: What representations are evident in the strategies that were used? What other representations might support student thinking? What might be some representations that could limit student thinking? 20

Pair Square Join another pair and discuss: What connections can be made among strategies? 21

Identify the Mathematical Content Examine the fraction learning progression. Discuss and highlight where the standards are evident in the mathematics of the task.

Student Work Sample: Grade 3 23

Student Work Sample: Grade 4 24

Student Work Sample: Grade 4 25

Student Work Sample: Grade 5 26

Analyze Student Work What representations are evident in the student work? What does the work tell us about student understanding? Misconceptions? How do students communicate understanding?

Clarify Our List Let’s brainstorm and look back at our list - Having done the task and looked at student work, what might need to be clarified?

Complete the Following Statements On a yellow sticky note:On a blue sticky note: I like teaching fractions because… I dislike teaching fractions because…

Let’s Watch… Fraction Tracks Video

Fraction Tracks What number sense do students need in order to play this game? 31 Click here to play onlineplay online

According to Empson and Levi (2011) Children come to school with conceptually sound understandings of fractions, even before instruction Research shows that the best way to build meaning for fractions is by solving and discussing word problems – starting with equal sharing problems. 32 Empson, S. and Levi, L. (2011). Extending children’s Mathematics – fractions and decimals, Heinemann.

According to Empson and Levi (2011) Children should first use pictures and words (e.g., one-half, two- fourths) to describe or represent fractions Teachers should gradually incorporate the use of symbols and equations 33

Why Equal Sharing Problems? Division into equal groups is a process that young children understand intuitively Can be solved by children without direct instruction Children do not need to know how to use fraction symbols or terminology to solve these problems Can begin as early as kindergarten 34

What are some ways to represent student thinking in these problems?

Representation should be an important element of lesson planning. Teachers must ask themselves, “What models or materials (representations) will help convey the mathematical focus of today’s lesson?” - Skip Fennell 39 Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM

Role of the Teacher Create a learning environment that encourages and supports the use of multiple representations Model the use of a variety of representations Orchestrate discussions where students share their representations and thinking Support students in making connections among multiple representations, to other math content and to real world contexts Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). Pearson. 40

41 "Students representational competence can be developed through instruction. Marshall, Superfine, and Canty (2010, p. 40) suggest three specific strategies: 1. Encourage purposeful selection of representations. 2. Engage in dialogue about explicit connections among representations. 3. Alternate the direction of the connections made among representations." National Council for Teachers of Mathematics. (2014). Principles to Actions. (p. 26). Reston, VA

Role of the Student Create and use representations to organize, record, and communicate mathematical ideas Select, apply, and translate among mathematical representations to solve problems Use representations to model and interpret physical, social, and mathematical phenomena Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). Pearson. 42

43 Students must be actively engaged in developing, interpreting, and critiquing a variety of representations. This type of work will lead to better understanding and effective, appropriate use of representation as a mathematical tool. National Council of Teachers of Mathematics. (2000) Principles and Standards for School Mathematics. (p. 206). Reston, VA.

Ideas for Instruction… To help students develop a deeper understanding of fractional concepts you can: Explore the concept of fair-shares Provide opportunities for students to share a variety of objects Introduce vocabulary without the symbolism Introduce fractional concepts in the context of things other than a pizza Expose students to fractions other than just halves, thirds, and fourths 44

45 “Problem difficulty is determined by the relationship between the number of things to be shared and the numbers of sharers. Because students’ initial strategies for sharing involve halving, a good place to begin is with two, four, or even eight sharers.” Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). (108). Pearson Education Inc.

A Mathematics Instruction Resource 46 Note: This resource was purchased for each Virginia division in attendance during the 2014 Mathematics Institute.

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Be Purposeful in the Selection of Tasks 5 brownies shared with 2 children 2 brownies shared with 4 children 5 brownies shared with 4 children 4 brownies shared with 8 children 3 brownies shared with 4 children 48

Be Purposeful in the Selection of Tasks 5 brownies shared with 2 children 2 brownies shared with 4 children 5 brownies shared with 4 children 4 brownies shared with 8 children 3 brownies shared with 4 children 49

Carousel 50

Gallery Walk 51

Planning Mathematics Instruction How does the list we created compare to the Planning Mathematics Instruction: Essential Questions document?

Closing Reflection In our efforts to assist teachers in fostering students’ ability to make mathematical connections and the use of effective and appropriate representations in mathematics, What might you want to… Stop doing? Continue doing? Start doing? 53