welcome to the effective use of manipulatives session complete the following task independently or...

Welcome to the Effective Use of

Manipulatives Session

Complete the following task independently or with your table partners. Save your response to share later in the

session.

Dana walks 1 mile to school every day. She passes the park, and the neighborhood store before she arrives to school. Jessie

recently moved to the neighborhood and needs help finding her way to the same school. Dana draws Jessie a map, using her own house, the park, and store as landmarks. Each location from the

house to the school is an equal distance apart. Using your strip of paper, model Dana’s map. How far apart are the locations from

each other? How do you know?

The Effective Use of Manipulatives

Bridging the Gaps from Concrete to Representational to Abstract

Presenters

Karen Bouldin: Cecil County Public Schools

Karen Jeffers: Prince George’s County Public Schools

Maryland State College and Career Readiness 2015 Summer Conference Presents

By the end of this session, participants

will…

❖ Understand the importance of using manipulatives in the classroom.

❖ Understand the three levels of abstraction as it relates to instructing students with math manipulatives.

❖ Understand how to use manipulatives to teach concepts relating to multiplication, division, and fractions.

❖ Explore and discuss different ways to manage the use of manipulatives in the classroom.

❖ Understand and explain how a specific model helped us solve a problem.

❏ Respect all thoughts and questions.

❏ Silence cell phones.

❏ Respond to emergencies outside of the classroom.

❏ We encourage you to ask questions and actively participate.

How Do You Feel About Using Manipulatives?

❖ Review the manipulative expressions poster.

❖ Place a post-it note below the expression that best describes your feeling about using manipulatives in the classroom.

I feel great! I’m interested in new ideas and new

strategies.

I feel okay. I would like to use manipulatives more.

I feel uncertain. I’m not always sure how to use

them.

I have a different feeling.

What are Math Manipulatives? &

Why are they Important?

❖ View the various quotes and article regarding the importance of manipulatives.

➢ http://www.hand2mind.com/resources/whyteachmathwithmanipulatives

http://www.hand2mind.com/resources/whyteachmathwithmanipulatives




A Silent Conversation

❖ Each table has a chart paper and markers.❖ At the center of the chart paper write the

questions, “What are manipulatives? Why are manipulatives important?”

❖ Each participant will choose a different color marker and silently respond to the question by writing the answer on the chart paper. Use evidence from the article to support your answer.

❖ If you agree with a statement, you may place a star next to it. You may also draw lines to connect one thought to another.

Why are Math Manipulatives Important?

“The more senses involved during learning, the more likely the brain will receive and process information. By using multiple

senses to learn, children find it easier to match new information to their existing

knowledge (Schiller 1999; Willis in press).”

National Association for the Education of Young Children (2008)

Why are Math Manipulatives Important? Hands-on manipulation increases the

chance by 75 percent that new information will be stored in long-term memory

(Hannaford 1995; Sousa 2006). Hands-on investigation increases sensory input, which

helps learners focus. It allows for experimentation by letting children use trial and error, which increases the chance that learners will make sense of and establish

relevancy for what they are learning (Sousa 2006).

National Association for the Education of Young Children.

(Schiller and Willis 2008)

Supported by the Universal Design

for LearningThe term UNIVERSAL DESIGN FOR LEARNING means a scientifically valid framework for guiding educational practice that:

(A) provides flexibility in the ways information is presented, in the ways students respond or demonstrate knowledge and skills, and in the ways students are engaged; and

(B) reduces barriers in instruction, provides appropriate accommodations, supports, and challenges, and maintains high achievement expectations for all students, including students with disabilities and students who are limited English proficient.

From the Higher Education Opportunity Act of 2008

Our English Language Learners Benefit

The use of manipulatives creates an even playing field for all learners.

“Manipulatives and models afford English language learners greater access to language

and mathematical terminology. A physical representation of a mathematical idea or

solution might provide an English language learner with greater confidence in his or her solution. Terms in a new language are easier to learn when used in the context of a model.

Just as student dictionaries provide illustrations of nouns, manipulative

representations of concepts and solutions provide illustrations of mathematical concepts

and ideas.”Dr. Dean M Shaw, 2002

Our Special Needs Children Benefit

“Students with special needs often have difficulty developing abstract level

understandings. Several barriers can make this situation occur. Sometimes students have never

developed conceptual understanding of the target mathematics concept/skill. Typically this occurs when students have not been allowed to develop that understanding at the concrete and representational levels of understanding. Two

ways to manage this situation are to re-teach the mathematics concept/skill using appropriate

concrete materials and then explicitly show the relationship between the concrete materials and

the abstract representation of the materials.”University of Kansas

http://www.specialconnections.ku.edu/?q=instruction/mathematics/teacher_tools/concrete_to_representational_to_abstract_instruction

How does the use of Manipulatives Connect to Common Core?

Common Core emphasizes conceptual understanding in mathematics as students utilize the mathematical practices to

solve problems. Students can develop conceptual understanding by adhering to the eight mathematical

practices.According to Van de Walle et al (2014), “Children who learn to

use these eight practices as they engage in mathematical concepts and skills have a greater chance of developing

conceptual understanding.” Using manipulatives in the classroom encompasses the

majority of these practices.

Not According to Plan“There can be some pitfalls to manipulatives,

especially for struggling students. Manipulatives are potentially confusing if their presentation is haphazard, disorganized, or lacking appropriate guidance and instruction from the teacher. They

can result in considerable time spent off-task or on activities that are not directly relevant to the needs

of certain children.”

Louise Spear Swerling, 2006

http://www.ldonline.org/spearswerling/The_Use_of_Manipulatives_in_Mathematics_Instruction

Managing Manipulatives

➢ Set up and communicate the expectations for distributing, using and collecting manipulatives in the classroom. Be consistent no matter the manipulative.

➢ Schedule “play” time when introducing a new manipulative. Students need to explore the material before they can be purposeful with it.

➢ Have students work in groups with manipulatives. This encourages communication and expands students thinking about how a manipulative can help them solve a problem.

Managing Manipulatives

➢ Train students how to work in groups with manipulatives. Use a comfortable low stress activity for the training and go slowly at first.

➢ Ensure that each group member has a role - Materials Manager; Group Leader to guide the investigation and who is the only one to ask the teacher questions; Clean-up Custodian to be in charge of making sure cleanup happens, and Recorder. Keep the groups static, but rotate the roles over a set of activities.

➢ Model mathematical thinking using manipulatives by talking out loud as you work through a problem.

In Grade 5Students are expected to perform operations with multi-digit whole numbers and with decimals to hundredths.

More specifically, they are to do the following within the 5.NBT.B.7 standard.

➢ Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

http://www.corestandards.org/Math/Content/5/NBT/B/7/

Levels of Abstraction- Concrete“Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix

cubes, base ten blocks, bean sticks, pattern blocks).

Students are provided many opportunities to practice and demonstrate mastery using

concrete materials.”http://www.specialconnections.ku.edu/?q=instruction/

mathematics/teacher_tools/concrete_to_representational_to_abstract_instruction

Levels of Abstraction- Representational (Bridging Stage)

“The math concept/skill is next modeled at representation (semi-concrete) level, which involves drawing pictures that represent the

concrete objects previously used. Students are provided many opportunities to practice and demonstrate mastery by drawing solutions.”

During this stage students use concrete or semi-concrete representations as they connect their representations to concepts. It is a purposeful

connection between the models and the numbers and operations.


Levels of Abstraction - Abstract

“The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols). Students are provided many opportunities to practice and demonstrate mastery at the abstract

level before moving to a new math concept/skill.”


Thirsty Anyone?

You are shopping at the store to buy items for your sports team’s party. You need to get Gatorade for this weekend. You buy 3 packs of 8 fruit punch and 4 packs of 6 Frost. How many bottles of

Gatorade will you purchase? Use manipulatives to solve the problem.

Tape DiagramsStudents may show their

representation of the problem using tape

diagrams.A tape diagram is a

visual thatuses rectangular models

to represent parts.

Thirsty Anyone?

You are shopping at the store to buy items for your sports team’s party. You need to get Gatorade for this weekend. You buy 3 packs of 8 fruit punch and 4 packs of 6 Frost. How many bottles of

Gatorade will you purchase? Use manipulatives to solve the problem.

It’s your Turn!Each group has a problem type,

markers, and chart paper.

1. Choosing your own numbers, draw a picture of two different manipulatives and corresponding equations to solve the problem.

2. Determine which manipulative worked best for developing understanding of the problem type.

3. Explain why.

The Connection“Students extend their whole number work with adding and subtracting and multiplying and dividing situations to decimal numbers and fractions. Each of these extensions can begin with problems that include all of the subtypes of the situations in Tables 1 and 2 in the progressions document. The operations of addition, subtraction, multiplication, and division continue to be used in the same way in these problem situations when they are extended to fractions and decimals (although making these extensions is not automatic or easy for all students).”

Progressions for the Common Core State Standards (Operations and Algebraic Thinking)

A Deeper Understanding of Fractions

First and Second grade students learned the following.➢ Partition circles and rectangles into two equal shares and

four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. (1.G.3)

➢ ...describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (2.G.3)

A Deeper Understanding of Fractions

❖ Take a look at the grade three standards for fractions.

http://www.corestandards.org/Math/Content/3/NF/

❖ How do the first and second grade geometry standards prepare students for what they are to learn in grade 3?



A Look at the ExplorationDana walks 1 mile to school every day.

She passes the park, and the neighborhood store before she arrives to

school. Jessie recently moved to the neighborhood and needs help finding

her way to the same school. Dana draws Jessie a map, using her own house, the

park, and store as landmarks. Each location from the house to the school is

an equal distance apart. Using your strip of paper, draw and label Dana’s map. How far apart are the locations from

each other? How do you know?

Another Look at the Exploration

Students may show their representation of the problem using tape diagrams.Tape diagrams may be used for fractions as well.

Extending the Problem Transitioning to Abstract

One day, Dana and Jessie decide to visit their friend Jennifer after school. Jennifer lives ⅓ mile beyond the school in the opposite direction of the park. How far is Dana’s house from Jennifer’s

house?

❖ Use your assigned manipulative to model the answer. ❖ Then model the answer with an equation.❖ Lastly, display the answer on a number line.

Why Number Lines?“Students sometimes have difficulty perceiving the unit on a number line diagram. When locating a fraction on a number line diagram, they might use as the unit the entire portion of the number line that is shown on the diagram, for example indicating the number 3 when asked to show ¾ on a number line diagram marked from 0 to 4. Although number line diagrams are important representations for students as they develop an understanding of a fraction as a number, in the early stages of the NF progression they use other representations such as area models, tape diagrams, and strips of paper. These like number line diagrams, can be subdivided, representing an important aspect of fractions.”

Progressions for Common Core State Standards (NF)

Choice Matters

3.NF.A.2.A

Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

3.NF.A.2.B

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

http://www.corestandards.org/Math/Content/3/NF/A/2/a/

http://www.corestandards.org/Math/Content/3/NF/A/2/b/

The Abstract

The swim meet coordinators are marking the lanes for 8 swimmers.

Using string, show the length of the pool and

how long each lane will be. Create a number line. Show where swimmer 6

will begin his meet.

What about equivalent fractions?Grade 3 students do some preliminary reasoning about equivalent fractions, in preparation for work in Grade 4. As students experiment on number line diagrams they

discover that many fractions label the same point on the number line, and are therefore equal; that is, they are

equivalent fractions.

Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and

denominator of a fraction by the same non-zero whole number results in a fraction that represents the same

number as the original fraction. Progressions for Common Core State Standards (NF)

Choice Matters - Equivalent Fractions

3.NF.A.3.B Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by

using a visual fraction model.

4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the

number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and

generate equivalent fractions.

Pizza Anyone?

Tim ate ½ of the cheese pizza. He ate ¼ of the sausage pizza.

Jayden ate 3/6 of the cheese pizza and 2/8 of the sausage pizza. Both pizzas are the same size. Did they eat the same amount? Solve the

problem using manipulatives.

Reflection TimeTake a few minutes to reflect on the

activities completed. How does the use of manipulatives

contribute to a rigorous environment?

What are the implications for your instruction?

What are your next steps as it pertains to working with

manipulatives?

Questions/Concerns

What questions or concerns do you still have regarding the use of manipulatives

during your instruction?

Summary/ClosingThe purpose of teaching through a concrete-to-representational-to-abstract

sequence of instruction is to ensure students develop a tangible understanding of the math concepts/skills they learn. When students are

supported to first develop a concrete level of understanding for any mathematics concept/skill, they can use this foundation to later link their

conceptual understanding to abstract mathematics learning activities. Having students represent their concrete understandings

(representational) by drawing simple pictures that replicate or mimic their use of concrete materials provide students a supported process for

transferring their concrete understandings to the abstract level. Moreover, teaching students how to draw solutions to problem solving situations

provides an excellent strategy for problem solving in the future.


Resourceshttp://www.hand2mind.com/resources/whyteachmathwithmanipulatives


http://www.ldonline.org/spearswerling/

The_Use_of_Manipulatives_in_Mathematics_Instruction





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