2014 Mathematics InstitutesGrade Band: K-2

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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. Looking at Student Work IV. Facilitating the Use of Effective

Representations and ConnectionsV. Planning Mathematics InstructionVI. Closing

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I. Defining Representations

Rally Robin1. Choose a partner2. Think: When you hear the term

representations in mathematics, what does it mean to you?

3. Take turns sharing your thoughts with your partner.

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

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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.

Defining Connections

Table DiscussionWhy do we want students to make

connections in mathematics?

What is an example of a connection we want young students to make?

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.

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

Planning for the Use of Representations

Table Group discussion:

What questions should be considered regarding representations and connections when planning for instruction?

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II. A Mathematical Task

Jo made two trays of cookies. One tray had 19 cookies on it. Together, both trays had 43 cookies. How many cookies were on the second tray? Use pictures, numbers and words to show how you found the answer.

• Work individually.• Solve this task in two different ways. • Be ready to share your ideas.

Mathematical Task

• Share your solutions with your table group.– What representations did you use to solve

the task and explain it to others?– What connections are evident?

• Capture at least 3 different ways that someone in your group solved the task on chart paper.

Mathematical Task

Gallery Walk• One person from group stays with group

poster to explain why representations were chosen.

• Participants ask questions and look for connections between the representations their group chose and the representations used on other charts.

Mathematical Content

Large Group Discussion

• What is the mathematical content of this task? What big ideas are students working with?

• Where do we find this content in the Whole Number/ Number Sense Learning Progression?

Early Number Sense

Read pages 107 – 108 (stop at Spatial Patterns)

What are the early number relationships that move students from counting to reasoning?

Part-Part-Whole

“To conceptualize a number as being made up of two or more parts is the

most important relationship that can be developed about numbers.”

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 K-2

(2nd ed.). (Vol. II). (108). Pearson Education Inc.

Part-Part-Whole as a Big Idea

Turn and Talk with Shoulder Partner:

Thinking about the mathematics that students learn as they go through elementary school and beyond, what connections/applications do you see to this idea of thinking about a number in terms of parts?

Jot down a few ideas.

Part-Part-Whole

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Addition Fact Strategies – Near Doubles

3 + 4

3 + 3 + 1

Place Value

2,1342 thousands (2000)

1 hundred (100)

3 tens (30)

4 ones (4)

Multi-digit Addition and Subtraction

28 + 26

How would you add these numbers mentally?

Multiplication Facts

4 x 7

2 x 714

2 x 714

Fractions

½ + ¾Inside the ¾ is ½ plus an

extra ¼

Algebra

2x + 6 xx

x

x

2(x + 3)

Part-Part-Whole

Decomposing and recomposing numbers is a huge idea and it all starts in kindergarten!

What are some additional part-part-whole connections you and your shoulder partner discussed?

Part–Part–Whole: Stages in Thinking

• Recognizing number as a quantity• Seeing and describing smaller parts inside

a number, but not remembering them• Realizing and remembering all the

different parts that make up a number• Decomposing numbers in flexible ways to

facilitate computation

Adapted from: Richardson, K. (2012). How Children Learn Number Concepts: - A Guide to the Critical Learning Phases. Bellingham, WA: Math Perspectives Teacher Development Center,.

III. Looking at Student Work

With a partner(s), choose a set of student work - K, 1, or 2. Be sure someone at each table looks at each collection.

– What representations are evident in the student work? Can you find similarities and differences?

– What does the student work tell us about their understanding? How do the representations give us clues to understanding?

Note that K and 1 use the same problem, but with different numbers.

Do we see evidence of the different types of representation in student work?

Looking at Student Work

Looking at Student Work

• Do you see evidence that students are making connections within their own work?

• How could you use the student work to help all students make connections?– Among representations– Among strategies– Among mathematical ideas

IV. Representations for Part-Part-Whole

Guiding Questions• What representations can be used to help

students develop strong part-part-whole ideas?

• What are the benefits and limitations of these representations?

Dot Representations

• Dot cards• Dot plates• Dominoes• Dice

Subitizing

• Perceptual subitizing – recognizing small quantities without counting

• Conceptual subitizing – recognizing patterns and groups to help determine a quantity

Subitizing

“Children use counting and patterning abilities to develop conceptual subitizing.

This more advanced ability to group and quantify sets quickly in turn supports their development

of number sense and arithmetic abilities.”

Clements, Douglas H. 1999. “Subitizing: What Is It? Why Teach It?” Teaching Children Mathematics 5 (7); 401

Subitizing

Questions to ask children when using Dot Representations

1. How many dots did you see?2. How did you see it?3. What did the pattern look like?4. Did you see any parts that you know?

Subitizing - Let’s Try Some Examples

Activities for Using Dot Representations

• Activity 8.8 Find the Same Amount• Activity 8.9 Learning Patterns• Activity 8.10 Dot Plate Flash

What are the benefits and limitations of using dot representations?

Five-Frames and Ten-FramesHow do these representations help anchor numbers to 5 and 10?

Ten Frames – Number Talk

Kindergarten Class

How is the ten-frame helping students?What questions is the teacher asking to help students focus on parts within a quantity?

VIDEO

Activities for Using Five- and Ten-Frames

• Activity 8.13 Five-Frame Tell-About• Activity 8.14 Number Medley• Activity 8.15 Ten-Frame Flash Cards

What are the benefits and limitations of five- and ten-frames?

Rekenreks

How is this representation similar to the other representations explored? How is it different?

Rekenreks

Let’s make a rekenrek!

You need:• A piece of cardboard• Two pipe cleaners• 10 red beads• 10 black beads

Rekenrek

VIDEO

What are the benefits and limitations of rekenreks?

How is this number talk different from the previous number talk?

How does the teacher honor and value various strategies while encouraging part-part-whole thinking?

VIDEO Click Here

RekenrekYour turn!!!!1. Choose one person to be the teacher. Everyone else turns back to screen.2. Teacher uses rekenrek to show the combinations below. After each combination, ask…

How many beads did you see?How did you know there were ___ beads?

6 on the top, 6 on the bottom6 on the top, 7 on the bottom6 on the top, 5 on the bottom

Additional Physical & Visual Representations

Activity 8.16 Build It in PartsActivity 8.17 Covered PartsActivity 8.18 Missing-Part CardsActivity 8.19 I Wish I HadActivity 8.20 Number Sandwiches

Part-Part-Whole Mat

Use the mat and some counters to model the problems.

1. Sam had 6 red balls and 5 blue balls. How many balls did Sam have?

2. Sam had 12 stickers. Four of his stickers were torn. How many were not torn?

3. Sam had 14 goldfish. He gave some to Mary. Now he has 7. How many goldfish did he give to Mary?

Part-Part- Whole Mat

How does modeling problems using a Part-Part-Whole mat help students?

What are the benefits and limitations of part-part-whole mats?

Number Lines

Number Path – discrete counting model

Number Line – length or distance model

Challenges or Limitations with Number Lines

Young children try to use a number line as a counting model –

– They count the numbers or tic marks, not the segments

– They don’t start from 0, because they typically begin counting from one

Can you identify the misconception?

Number Lines

• Number Lines are not recommended as a representation at K and 1 because of the conceptual difficulties they present

• Introduce at grade 2 with an emphasis on ‘hops’ or lengths, but be cognizant of the difficulties young children have with number lines.

Fuson, K., Clements, D., & Beckman, S. (2010). Focus in Kindergarten: Teaching with Curriculum Focal Points. Reston, VA:

National Council of Teachers of Mathematics.

Open Number Lines

Read page 135.What is an open number line?

How do you introduce open number lines?Going back to our student work, which students used open number lines while solving the problem?

What are the benefits and limitations of number lines?

Representations

“Representations do not “show” the mathematics to the students. Rather the

students need to work with representations extensively in many contexts as well as move

between representations in order to understand how they can use a representation to model

mathematical ideas and relationships.”

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 208). Reston, VA.

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

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

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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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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.

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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."

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

V. Planning for Instruction

Let’s revisit our list of questions regarding representations and connections when planning. Do you have any additions or revisions?

Planning Mathematics Instruction: Essential Questions

• Work with a partner.• Highlight questions you already think about

when planning.• Which questions are new for you to think

about?• Are the questions on our list reflected in

this document?

Planning a Lesson

• Revisit the student work for student 1- D.• Assuming that student 1-D’s work is

representative of your class, plan a short lesson that will continue to move the students forward in their thinking. Consider the representations that you will use and the connections you want to have students make.

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

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

VI. Closing

3 Name three representations you will use to develop part-part-whole thinking.

2 Describe two ways that you will help your students make connections between representations.

1 Identify one key question that you will incorporate into your planning process to focus on representations and connections.