2014 Mathematics SOL Institutes Grade Band: 6-8

Making Connections and Using Representations The purpose of the 2014 Mathematics SOL 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

I. Defining Representations and Connections THINK, PAIR, SQUARE What does it mean for students to represent mathematics in the classroom? Think for a moment about the question and jot a few notes to share with an elbow partner. After you and your partner have shared, share a thought with a nearby partner group. 4

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.

I. Defining Representations and Connections THINK, PAIR, SQUARE What does it mean for students to make connections in mathematics classroom? Think for a moment about the question and jot a few notes to share with an elbow partner. After you and your partner have shared, share a thought with a nearby partner group. 8

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. 9 Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

10 “Connections are useful because they help students see mathematics as a unified body of knowledge rather than a set of complex and disjoint concepts, procedures and processes. Real world contexts provide opportunities for students to connect what they are learning to their own environment. Their mathematics may also be connected to other disciplines which provides opportunities to enrich their learning.” National Council of Teachers of Mathematics. 2000, p Principles and Standards for School Mathematics. Reston, VA

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Implications for Instruction Reflect, Record, and Share How do we purposely plan for representations and connections in our instruction? Think for a moment about the question. Share your ideas with your partner. After you and your partner have shared, join with others at your table and make a list of your top three actions you will take to include representations and connections in your instruction. We will use your responses to make a group list. 12

II. Doing the Mathematical Task Jay is making flower pots. It takes ¾ of a package of clay to make 1 flower pot. How many flower pots can Jay make with 4 ½ packages of clay? 13 Solve the task in two different ways. Show how you solved it and explain your thinking. Also consider potential misconceptions student might encounter when working on the task.

Doing the Mathematical Task Share Out Share your solutions with your table group. During your discussion be sure to mention: Mathematical Content Strategies for your solutions Representations Connections 14

Modeling the 5 Practices Anticipating what students will do--what strategies they will use--in solving a problem. Monitoring their work as they approach the problem in class. Selecting students whose strategies are worth discussing in class. Sequencing those students' presentations to maximize their potential to increase students' learning. Connecting the strategies and ideas in a way that helps students understand the mathematics learned. 15 Smith, M.S.,& Stein, M.K. (2011). Five Practices for Orchestrating Productive Mathematical Discussions. Reston, VA. National Council of Teachers of Mathematics.

III. Looking at Student Work While looking at the collection of student work, consider the following questions: What representations are evident in the student work? What does the student work tell us about their understanding? How are the representations alike? How do student representations communicate understanding? 16

IV. Facilitating the Use of Representations and Connections It is important to expose our students to a variety of fraction models and representations to help increase their understanding. What are the models we want students to use and think about? How can we help them work flexibly among and with each of these representations? 17

Fraction Models Students must also work with a variety of models: area length set or quantity 18

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 19 MORNING REFLECTION – PROMPT RESPONSE On the front our your index card respond to the following: Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM

? + MORNING REFLECTION – JUST CHECKING IN Use post it notes to write comments and post them on the wall chart. POSITIVES, THINGS WE COULD CHANGE, QUESTIONS 20

A great way to teach fraction division is to use equation strings involving progressively difficult problems. It is also important to use word problems as often as possible with students and ask them to create own. Fraction Division 21

Fraction Division with Equation Strings These problems are arranged so that as students move down the column, they can extend their reasoning on one problem to solve the next, more challenging problem. 22 Empson, S.B. & Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. (p.203). Portsmouth, N.H.: Heinemann

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Counterintuitive ? ? The idea that the answer to a division problem can be greater than the number being divided (or that multiplication can result in a smaller number) is counterintuitive. Students will only come to realize this understanding after many opportunities to visualize the impact of dividing (and multiplying) by a fraction less than Petit, M.M., Laird, R.E., & Marsden, E.L. (2010). A Focus on Fractions: Bringing Research into the Classroom. (p. 164). New York, N.Y.: Routledge

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28 Please use an area model or pattern blocks to illustrate your thinking.

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Use your choice of models. 30

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With your partner, create a “real life” word problem and model it. What are the advantages of teaching fraction division this way? Disadvantages? Were there limitations of specific models? 32

Decimal Division with Equation Strings 33

Integer Patterning -3 x 3 = x 2 = x 1 = x 0 = 0 -3 x -1 = 3 -3 x -2 = 6 -3 x -3 = 9 34

35 "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

The 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. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). Pearson. 36

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. 37

38 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.

VI. Closing What questions should be considered regarding representations and connections when planning for instruction? 39

Afternoon Reflection What will be your first steps when you get back to your division? 40