2014 Mathematics InstitutesGrade Band: High School Algebra

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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.

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Agenda

I. Defining Representations and ConnectionsII. Doing the Mathematical TaskIII. Examining Student Work IV. Planning for the Use of Representations and

ConnectionsV. Closure

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I. Representations and Connections

• What does it mean for students to use effective and appropriate representations in the classroom?

• What does it mean for students to make connections in the mathematics classroom?

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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.

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

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“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.

8Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

– February 2009

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“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. 200.

Principles and Standards for School Mathematics. Reston, VA

I. Representations and Connections

• When planning instruction, what considerations should you make with respect to representations and connections?

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II. Mathematical Task

The first 3 figures in a pattern sequence are shown below. Assuming this pattern continues, list the characteristics of the figures that change as the pattern progresses.

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II. Mathematical Task

• In your small groups, share the list of attributes you brainstormed.

• Ensure that the language used to define each attribute is concise and precise.

• As a whole group, we will create a “master list” of attributes.

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II. Mathematical Task

Group Work•Select one brainstormed attribute that your group would like to explore further.•Describe the step-by-step changes of that attribute in as many different ways as possible.•Use chart paper and markers to record and summarize your ideas.

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II. Mathematical Task

Sharing Your Work•Things to consider:

– Which mathematical ideas are engaged during work on this task?

– Are there similarities and differences in the growth of different attributes?

– Do particular types of attributes result in particular types of growth?

– Which representation did your group use initially? Why? Was it effective?

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II. Mathematical Task

Developing Students’ Representational Competence What is Representational Competence?

• Selecting appropriate representations• Translating between representations• Understanding how and when different

representations are the most useful

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Novick, L.R. (2004). Diagram Literacy in Preservice Math Teachers, Computer Science Majors, and Typical Undergraduates: The Case of Matrices, Networks, and Hierarchies. Mathematical Thinking and Learning.

(Vol. 6 No. 3). (pp. 307–342).

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"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."

II. Mathematical Task

National Council for Teachers of Mathematics. (2014). Principles to Actions. (p. 26). Reston, VA

III. Examining Student Work

As we prepare to look at high school student work samples for this task, think about the following:

• What representations do you predict students will use?

• In what ways might the students connect the representations they use?

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III. Examining Student Work

After examining the student work:• What representations did the students use?• What does the student work tell us about their

understanding?• What evidence of connections can be seen in the

student work?

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III. Examining Student Work

After examining the student work:• How could you use the student work to help all

students make connections- Among representations?- Among strategies?- Among mathematical ideas?

– How do the representations used communicate- Student understanding of the mathematics?- Student generalizations of the mathematics?- Mathematical relationships?

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IV. Planning for the Use of Representations and Connections

Let’s revisit our discussion of representations and connections from this morning.

What are the benefits and challenges of using tasks that elicit multiple representations and connections in the classroom?

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IV. Planning for the Use of Representations and Connections

Conjecturing About Functions - Video

• While viewing, record your list of student moves and teacher moves with respect to representations and connections on the provided document.

• What are the roles of teacher and student?

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IV. Planning for the Use of Representations and Connections

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

(2nd ed.). (Vol. II). Pearson. 22

IV. Planning for the Use of Representations and Connections

The 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

(2nd ed.). (Vol. II). Pearson.

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IV. Planning for the Use of Representations and Connections

Mathematics Vocabulary Word Wall Cards

•Which cards might address representations and connections?•How might these cards be used to develop students’ representational competence?

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IV. Planning for the Use of Representations and Connections

Read the Sorting Functions task and work on the activity. •As you work on the task, think about:

– What important algebraic ideas might students use to match a graph with an equation?

– What connections do you hope students are making to relate this information?

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IV. Planning for the Use of Representations and Connections

After completing the Sorting Functions task, consider and discuss the rubric.

•What would be considered correct in terms of the mathematical content?•What would be considered a correct explanation?

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IV. Planning for the Use of Representations and Connections

Considering the Sorting Functions Task• Analyze the student work and data analysis.• In general, did students have more difficulty with

equations, tables, or rules?• How do the student work samples and aggregate

data inform you about misconceptions? • How would it impact your subsequent conversations

with students and your planning?

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IV. Planning for the Use of Representations and Connections

Mathematical Instructional Connections for Physical and Visual Representations

• What representations/strategies might have been used to develop prior knowledge?

• What are various representations that might be used to develop and reinforce understanding of the content?

• Which representations/strategies will model the mathematics and deepen and extend students’ mathematical understanding?

• What are the strengths and limitations of the representation/strategy?

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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

Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM

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V. Closure

Revisiting our guiding question: •What questions should be considered regarding representations and connections when planning for instruction? •In what ways do we need to modify or amend today’s list of brainstormed questions?

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V. Closure

Revisiting our guiding question: •What questions should be considered regarding representations and connections when planning for instruction? •How does our list compare with the Planning Mathematics Instruction: Essential Questions document?

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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.