Session 2: Building Place Value Concepts

1
Session 2 Building Place Value Concepts

• Using Models Effectively

2
Purposes

• To revisit concrete, pictorial, and abstract
• What do these terms mean?
• What do they look like in the classroom?
• To connect number concepts from Session 1 to
  place value concepts
• What are the prerequisite skills and
  understandings for place value?

3

• Learning mathematics requires that children
  create and re-create mathematical relationships
  in their own minds.Children need direct and
  concrete interaction with mathematical ideas
  ideas are not accessible solely from
  abstractions.

Burns, Marilyn. (1992). About Teaching
Mathematics A K-8 Resource, p. 24.
4
Definition of Models

• A model for a mathematical concept refers to any
  object, picture, or drawing that represents the
  concept or onto which the relationship for that
  concept can be imposed.

Van de Walle, John. (2004). Elementary and Middle
School Mathematics Teaching Developmentally, p.
28.
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Example of Models

• In this sense, any group of 100 objects can be a
  model of the concept hundred because we can
  impose the 100-to-1 relationship on the group and
  a single element of the group.

Van de Walle, John. (2004). Elementary and Middle
School Mathematics Teaching Developmentally, p.
28.
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Use of Models

• Models can be thought of as thinker toys,
  tester toys, and talker toys. It is difficult
  for students (of all ages) to talk about and test
  out abstract relationships using words alone.
  Models give learners something to think about,
  explore with, talk about, and reason with.

Van de Walle, John. (2004). Elementary and Middle
School Mathematics Teaching Developmentally, p.
30.
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Concrete Models Manipulatives

• Manipulatives should be used by children.
• Allow children to use manipulatives with their
  own informal methods.

Adapted from Clements, D. and McMillen, S. (1996,
January). Rethinking Concrete Manipulatives.
Teaching Children Mathematics.
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Concrete Models Manipulatives

• Select manipulatives that can serve many
  purposes.
• Introduce new topics with one manipulative.
• Over time, use different manipulatives for the
  same concept.

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Pictorial Models

• We would suggest that learning mathematics
  reflects a good deal about intellectual
  development. It begins with instrumental
  activity, a kind of definition of things by doing
  them. Such operations become represented and
  summarized in the form of particular images.

Bruner, Jerome. (1996). Toward a Theory of
Instruction, p. 68.
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Abstract Models

• The goal of manipulatives is to promote abstract
  understanding so that the manipulatives
  themselves will no longer be necessary.
  Eventually, children can work with mental images
  of manipulatives and do not need the
  manipulatives themselves.

Ginsburg, H. and Ertle, B. (2006).
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Abstract Models

• View the video clip.
• Rachel Carrot Video
• Describe to your partner what the student is
  doing abstractly.

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Concrete Pictorial Abstract

• What does modeling mathematics concepts
  concretely, pictorially, or abstractly look like?
  
• List some examples of each.

13

• Mathematics requires representations. In fact,
  because of the abstract nature of mathematics,
  people have access to mathematical ideas only
  through the representations of those ideas.

Kilpatrick, J., Swafford, J., and Findell, B.
(Eds.). (2001). Adding It Up Helping Children
Learn Mathematics, p. 94.
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Models and Representations Chart

• In each cell of the chart, find Teacher
  Presents/Student Responds
• List the models used at each level Concrete,
  Pictorial, and Abstract.

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Reflection Questions

• How does the list you completed earlier compare
  to this chart?
• Within each level (concrete, pictorial, and
  abstract), how are the models on the chart alike?
  How are they different?

Answer these questions alone, then share your
answers with your group.
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Video Clips

• Watch these three sections of one lesson.
• Refer to the Models and Representations Chart.

• What does the teacher present?
• How do the students respond?

Hearts Teacher 1 Hearts Students
1 Hearts Teacher 2 Hearts Students
2 Hearts Teacher 3 Hearts Students
3a Hearts Students 3b
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Models and Representations

• What different models did the teacher present in
  the lesson?
• How did the students show understanding of the
  mathematics content in the lesson?
• What did the students have to bring to the
  table to be successful in this lesson?

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Purposes Revisited 1

• What did concrete, pictorial, and abstract look
  like in this lesson?

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Models, Representations,and the TEKS

• How do the TEKS use concrete, pictorial, and
  abstract?

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Number Concepts leading to Place Value

• In grade level groups
• Examine your grade level TEKS on the chart. Look
  at both the Knowledge and Skills statements and
  the Student Expectations.
• When a student shows mastery of the student
  expectation, what model will you see?

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Sample Lessons 5-E Model

• Each row differentiates the same content.
• Each column sequences the prerequisites for place
  value.

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Lesson Series 1

• First Grade Lesson Groups and Singles 1

Before beginning with ideas of tens and ones,
children can benefit from separating a set into
groups of a specified size and then counting
these groups as single units. (Thompson, 1990)
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Engage

• Literature Connection
• Chicka Chicka 1 2 3
• by Bill Martin

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Explore

• Get 13 cubes.

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Explore

• By twos.

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Explore

• By fives.

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Explore

• By tens.

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Get 21 cubes.
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Explain

• What did you notice about counting by different
  numbers?
• Which way was easiest for you?
• Why do you suppose that is true?
• Did you always have cubes not in groups?
• Why do you suppose cubes were left out?

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Elaborate

• Make groups of 2 without any singles. What number
  did you make?
• Make groups of 5 without any singles. What number
  did you make?
• Make groups of 10 without any singles. What
  number did you make?

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Evaluate

• Get 29 cubes.
• Predict Will there be singles when counting by
  2s? by 5s? by 10s?
• Count the cubes by 2s, 5s, and 10s.
• Show how you made groups.
• Did the number of cubes stay the same? How do you
  know?

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Reviewing the Lesson with the Models and
Representations chart

• Where does this lesson fit on the chart?

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Looking at the other lessons in Series 1

• Divide your group in half.
• One half read the kindergarten lesson, Plates and
  Hands.
• One half read the second grade lesson, Groups and
  Singles 2.
• Be prepared to summarize the lesson for your
  group.

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Series 1 Reflection

• How are models used in each lesson?
• What are some of the different ways students
  might respond to the models used in this lesson?

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Series 2 Lessons

• Comparing Fives
• Comparing Tens
• Comparing Hundreds

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Explore Comparing Fives
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Explore Comparing Tens
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Explore Comparing Hundreds
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Series 2 Reflection Part 1

• Read your grade level lesson.
• How are models used in the lesson?
• Where do the three lessons fit on the Models and
  Representations chart?

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Series 2 Reflection Part 2

• How does the use of models in these three lessons
  change with the movement to the next grade level?
  
• How might lessons change within the course of a
  single year within one grade level?
• How are these changes reflected in the Models and
  Representations chart and in the Concept Vertical
  Analysis chart?

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In grade level groups

• Complete the Series 3 lesson for your grade.
• Kindergarten More Than Ten
• First Grade Up to One Hundred
• Second Grade Up to One Thousand
• Place your lesson on the Models and
  Representations chart.

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Back to your home groups

• Briefly share your lesson.

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Series 3 Reflection Part 1

• What is the primary mode of instruction for each
  of the nine lessons?
• List each of the nine lessons on the third Models
  and Representations chart.
• Highlight each kindergarten lesson in pink.
• Highlight each 1st grade lesson in blue.
• Highlight each 2nd grade lesson in green.

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Series 3 Reflection Part 2

• How do the Teacher Presents and the use of
  models change across these three grade levels?

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Purposes Revisited 2

• To revisit concrete, pictorial, and abstract
• What do these terms mean?
• What do they look like in the classroom?
• To connect number concepts from Session 1 to
  place value concepts
• What are the prerequisite skills and
  understandings for place value?

46
Purposes Revisited 2

• Effective use of manipulatives
• Teacher Presents and Student Responds
• How will you use what you learned today to
  improve instruction?