Answer & Explanation:First, read the article: Enhancing Mathematical Learning in a Technology-Rich EnvironmentThen, go to the National Library of Virtual Manipulatives (NLVM) website provided by Utah State University. Select a grade band and mathematics topic that most interests you and choose two different virtual manipulatives to review.

For your initial post:

State which grade band and mathematics topic you chose. For each of the two virtual manipulatives you chose: list two elements of each manipulative that made it stand out to you, and describe one way each manipulative could be used by students in the mathematics classroom.

Explain how technology plays an

important role in today’s mathematics classrooms. Refer to both the

article and the NLVM website in your response.Your initial post must be at least 150.

suh__2008__technology.pdf

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Enhancing Mathematical

Learning in a

Technology-Rich

Environment

I

n Principles and Standards for School Mathematics (NCTM 2000), the Technology Principle

asserts: “Technology is essential in teaching and

learning mathematics; it influences the mathematics that is taught and enhances students’ learning”

(p. 24). More specifically, a technology-rich environment for mathematical learning influences five

critical features of the classroom (Hiebert et al.

1997): the nature of classroom tasks, the mathematical tool as learning support, the role of the teacher,

the social culture of the classroom, and equity and

accessibility. An essential question when working

in a technology-rich mathematics environment

is how technology can be used (appropriately) to

enhance the teaching and learning of mathematics.

This article describes teachers working collaboratively in a technology-rich environment to plan

mathematics lessons that address the needs of their

diverse students, in particular, English Language

Learners (ELLs) and students with special needs.

Through classroom examples, we discuss how a

technology-rich learning environment influences a

classroom’s critical features. Moreover, we define

By Jennifer M. Suh, Christopher J. Johnston, and Joshua Douds

Jennifer M. Suh, [email protected], is an assistant professor of mathematics education at

George Mason University in Fairfax, Virginia.

Her research interests include lesson study and

the development of preservice mathematics

teachers. Christopher J. Johnston, [email protected]

gmu.edu, is a doctoral student at George Mason University interested in preservice and beginning mathematics teachers’ use of technology. Joshua Douds, [email protected], is a

fourth-grade teacher at Westlawn Elementary School in Falls Church, Virginia. His research interests include center-based mathematics and the effects of cooperative learning in mathematics.

Teaching Children Mathematics / November 2008

Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

235

Figure 1

Student work showing mathematical knowledge mapping

unique technological properties that amplify opportunities for extending mathematical thinking.

The participating school is a Title I elementary

school in a major metropolitan area with approximately 600 students: 51 percent Hispanics, 24

percent Asians, 16 percent Caucasians, 3 percent

African Americans, and 6 percent others. More

than 50 percent of the student population receive

free or reduced lunch, 44 percent receive services

for English for Speakers of Other Languages,

and 49 percent are identified as limited in English proficiency. On the basis of student need at

this school, an important school initiative sought

to incorporate nonlinguistic representations into

students’ daily activities to help build their background knowledge. Representing knowledge (nonlinguistic representation) is one of nine categories

of instructional strategies proven to advance student achievement (Marzano et al. 2001). An essential part of the initiative promotes the integration

of technology in all content areas to provide the

diverse student population with interactive, visual,

and multimedia tools. To enact the initiative,

teachers in each grade level collaborated with a

university mathematics educator to design lessons

incorporating technology tools and nonlinguistic

representations to engage, motivate, and respond to

the needs of a diverse group of learners.

To begin planning collaborative lessons, teachers identified a mathematics area at their grade level

that presented a teaching and learning challenge.

For one of the highlighted classroom examples, we

will share a third-grade money lesson that was part

of the measurement strand. The lesson objective

was to count a collection of mixed coins and then

find and record a variety of ways to show a given

amount of money. The future building-block target

was to make change for amounts up to five dollars.

236

The second featured lesson was a fourth-grade

fractions lesson with the objective of renaming fractions and finding equivalent fractions. This lesson

was a prerequisite to adding and subtracting with

unlike denominators using models. Once teachers

identified the lesson objectives, the lesson-planning

team worked together to construct a mathematics

knowledge map outlining the key components of

both interrelated prerequisite and future knowledge

mathematics concepts building blocks. Additionally, they identified effective representations or

models to teach each concept (see fig. 1).

In these two lessons, the planning team

included third- and fourth-grade classroom teachers, the special education teacher assigned to those

grade levels, the mathematics specialists, and the

university mathematics educator. Teachers ranged

from novices to experienced teachers with varied

strengths and weaknesses in the areas of technology integration, mathematical content knowledge,

and teaching practice—a range that provided

opportunities for all participants to develop deeper

pedagogical content or technology knowledge.

Creating Technology-Rich

Mathematics Learning

Environments

When creating a technology-rich mathematical

learning environment, teachers must understand

what using technology “appropriately” (Garofalo et

al. 2000, p. 67) means when integrated into teaching mathematics:

1. Introducing technology in context

2. Addressing worthwhile mathematics with appropriate pedagogy

3. Taking advantage of technology

Teaching Children Mathematics / November 2008

Figure 2

An Advancing Mathematical Thinking planning sheet

Advancing Mathematics Thinking with the Use of Technology

Web site

Addition of Fractions

Math Strand

Number and Operations

Grade Level

Fourth Grade

Description of

mathematical concept

(NCTM)

National Library of Virtual Manipulatives at Utah State University, copyright 1999–2000. All Rights Reserved.

Analysis of Mathematical Representations and Models

X

Concept tutorial/skill practice

Investigation/problem solving

Open exploration

Representation

• Create and use representations to organize, record, and communicate

Connected pictorial

mathematical ideas

and numerical

• Select, apply, and translate among mathematical representations to solve patterns

representations

• Use representations to model and interpret physical, social, and mathematical

phenomena

Communication

• Organize and consolidate their mathematical thinking through communication

• Communicate their mathematical thinking coherently and clearly to peers,

teachers, and others

• Analyze and evaluate the mathematical thinking and strategies of others

• Use the language of mathematics to express mathematical ideas precisely

Peer talk:

Discuss applet’s

function and the

mathematics

process (step by

step)

Connections

• Recognize and use connections among mathematical ideas

• Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole

• Recognize and apply mathematics in contexts outside of mathematics

Connecting

renaming before

combining; finding

common multiples

with arrows

breaking pieces

Reasoning and Proof

• Recognize reasoning and proof as fundamental aspects of mathematics

• Make and investigate mathematical conjectures

• Develop and evaluate mathematical arguments and proofs

• Select and use various types of reasoning and methods of proof

Analyzing and

making sense of the

algorithmic process

Problem Solving

• Build new mathematical knowledge through problem solving

• Solve problems that arise in mathematics and in other contexts

• Apply and adapt a variety of appropriate strategies to solve problems

• Monitor and reflect on the process of mathematical problem solving

Discovering what

happens when

fraction pieces

are renamed and

combined

Teaching Children Mathematics / November 2008

237

4. Connecting mathematics topics

5. Incorporating multiple representations

In planning the lessons, we used these guidelines

to structure the learning environments with virtual

manipulatives and applets.

In addition to knowing how to integrate technology appropriately, teachers must focus on worthwhile

mathematics and effective pedagogy when using technology. An effective way to optimize the mathematical thinking opportunities presented by technology

is to plan the mathematics task focused on the five

Process Standards (NCTM 2000): Problem Solving,

Reasoning and Proof, Communication, Connections,

and Representation. We used a template during planning to guide the activity and the classroom discourse

so that teachers were focused on advancing students’

mathematical thinking processes (see fig. 2).

The remainder of the article describes two lessons in which technology was used as an instructional strategy. More specifically, we describe the

task and the technological tool that supported the

learning, the role of the teacher in capitalizing on

learning in the technology-rich environment, and

how the technology gave more access to learning

opportunities and more equity to diverse learners.

students were able to touch the screen and drag the

coin directly onto the hundreds chart to help them

count by twenty-five, ten, five, or one. We used the

highlighting pen to shade in money amounts and

to show the value of each coin. For many children,

counting money is especially challenging because its

representation is nonproportional; that is, although

a dime has more value than a nickel, the dime is

Figure 3

Student work using tech tools

(a) The SMART Board to count change

(b) A virtual hand to drag coins and skip-count

Counting Change

Makes Sense

The third-grade lesson objectives were to teach

students to count a collection of mixed coins and

find and record a variety of ways to show a given

amount of money. Making change for amounts to

five dollars was a future knowledge building block.

To address the objectives, we designed the task on

the SMART Board with a hundreds chart and coins

that had infinite clones to count change. Using

the hundreds chart (see fig. 3a), students worked

with benchmark numbers such as five, ten, and

twenty-five, learned to skip-count when counting

change, and practiced using numbers flexibly. The

second activity, “Show Me the Money,” embedded

two tasks. First, students counted the money in the

virtual hand (see fig. 3b) by dragging the coins and

skip-counting. The following scenario offered the

other task: “I have in my hand a total of thirty-three

cents. Show me all the possible ways to make that

amount” (see fig. 3c).

We used technology to provide students with

multiple representations. The electronic hundreds

chart helped students see the relationship between

coins and their value. Using the SMART Board,

238

(c) Multiple coin combinations for thirty-three cents

Copyright © 2003 Arcytech. All Rights Reserved.

Teaching Children Mathematics / November 2008

physically smaller. The task was designed to relate

the proportional representation of the hundreds chart

with the nonproportional representations of the coins

so that students would gain better understanding of

each coin’s magnitude and worth. Demonstrating

how to count up to thirty-three cents at the SMART

Board, one student commented, “With twenty-five

cents, I need to shade in a lot more: two rows for the

two tens and five ones; and to get to thirty cents, I

need to add a nickel and then three more pennies.”

Shading the hundreds chart was an instructive visual

representation of the coin values.

The teacher’s role

The SMART Board technology facilitated the

teacher’s ability to give students opportunities to

show multiple ways to count change. Important

teacher tactics included allowing students to display different solution paths on the SMART Board

simultaneously, asking students to compare different thinking strategies for making compatible numbers, and initiating productive discussion on efficient change-counting strategies. Simultaneously

displaying multiple student solutions allowed students to compare and make some important generalizations about counting coins. For example, when

given the coins (a quarter, dime, dime, and nickel),

one student shared, “I count the quarter first and

then the nickel to get to thirty cents [and] then add

the two dimes to get to fifty cents.” Another student

said, “It is easier for me to add the quarter, then the

two dimes to go from twenty-five, thirty-five, fortyfive cents, [and] then add the nickel to get to fifty

cents.” Many students began to adapt their thinking

and model the strategies shared in class that made

it easier to skip-count money. The task also allowed

them to discover ways to compose and decompose

numbers using different coin combinations.

Equity and access

for diverse learners

Technology enhanced students’ learning by allowing

diverse learners to understand the concept through

multiple representations. Students recorded the

numeric value right next to the coins as they counted

change on the hundreds chart, thereby allowing

the visual representations to be closely tied to the

numeric representations. For some English Language

Learners, being able to write words such as quarter,

dime, nickel, and penny next to the coin gave them

better access to the lesson. The technology features

allowed for better communication, problem solving,

reasoning, and connections among concepts. In fact,

Teaching Children Mathematics / November 2008

the dual representations of the coins and the

hundreds chart allowed for some highability students to engage in more challenging tasks. By using the hundreds

chart and counting on, these students

used the tools to determine how

much change one should get back

if one pays with a dollar bill. For

example, the cost of a candy bar is

sixty-eight cents. The child counts on,

“Sixty-nine, seventy,” using pennies and

then counts on, “Eighty, ninety, one hundred,” using

three dimes; the total is thirty-two cents in change.

Having multiple tasks embedded within each task

also allowed for differentiation in instruction.

Exploring Equivalent Fractions

The lesson objectives for the fourth-grade fractions

lesson were to rename fractions and find equivalent fractions; the subsequent lesson focused on

using models to add and subtract fractions with

unlike denominators. The virtual manipulatives

called Fraction Equivalence, found on the National

Library of Virtual Manipulatives Web site, allowed

students to explore the relationship between equivalent fractions. On the Fraction Equivalence applet,

students were presented with a partially shaded

circle or square and the fraction symbol. They were

directed to “find a new name for this fraction by

using the arrow buttons to set the number of pieces.

Enter the new name and check your answer.” To do

this, students clicked on arrow buttons below the

whole unit, which changed the number of parts.

When students had an equivalent fraction, all lines

turned red. When a common denominator was identified, students typed the names of the equivalent

fractions into the appropriate boxes. They checked

their answers by clicking the “Check” button.

Each step of the way, the pictures were linked to

numeric symbols that dynamically changed with

the students’ moves (see fig. 4a). To help explore

the relationship between equivalent fractions, the

applet prompted students to find several equivalent

fractions. This applet was specifically designed

to develop the concept of renaming fractions.

Although constrained to one specific objective,

the tool allowed for more exploration than do

physical manipulatives, such as fraction circles or

bars, which are usually limited by the number of

fractional pieces. This applet allowed students to

equally divide a whole, up to ninety-nine pieces,

and generate multiple equivalent fraction names.

239

The teacher’s role

The teacher’s role in extending students’ thinking during this task was to encourage students

to record a list of equivalent fractions, look for a

pattern, and generate a rule. For instance, using

the applet on a SMART Board, a student found

1/3 = 2/6 = 3/9 = 4/12. As we recorded this on the

board, students’ eyes started to widen and hands

started waving in the air: “Oh, oh, I know the

rule!” Some students noticed the additive rule.

One student stated, “The denominators are going

by a plus-three pattern.” Another student echoed,

“It is like skip counting.” And another voiced, “It

is the multiples of three.”

To get students to further explore the relationship, the teacher asked them to examine the

multiplicative pattern for both the numerator

and the denominator in 2/3. Students listed 2/3 =

4/6 = 6/9, and again they quickly saw the additive

pattern and the multiples of two for the numerator

and three for the denominator. Then the teacher

posed the questions, “Are 2/3 and 20/30 equivalent

fractions? What about 2/3 and 10/15?” To find a

rule beyond the additive rule, students were asked

to use the applet and talk to their partners while

exploring the relationships between the equivalent

fractions and to other fractions. When students

came back together as a group, several of them

shared their discoveries: “The fractions 2/3 and

20/30 are equivalent, because you multiply both

numerator and denominator by ten. And in 2/3 =

10/15, you multiply both numerator and denominator by five.” These comments led to a lively conversation about how 10/10 and 5/5 both equal one

whole. The teacher connected this idea to the identity property of multiplication by asking, “What

happens when we multiply one by any number?”

The ensuing discussion reinforced the idea that

no matter how you rename the fractions, as long

as you multiply them by one or n/n, you will have

an equivalent fraction. To challenge the students,

the teacher posed a question: “What would the

equivalent fraction be for 1/3 if the denominator

were divided into ninety-nine parts?” This type of

questioning encouraged students to extend their

thinking by making conjectures and testing their

rule or hypothesis.

Equity and access

for diverse learners

Instead of merely teaching an algorithm, we used

the fraction applet to allow all the students to

think and reason about the relationships among

240

equivalent fractions. The teacher gave students

Figure 4

Fraction equivalence applet

(a) in English

National Library of Virtual Manipulatives at Utah State University,

copyright 1999–2000. All Rights Reserved.

(b) in Spanish

National Library of Virtual Manipulatives at Utah State University,

copyright 1999–2000. All Rights Reserved.

the opportunity to work with a partner. As the

pairs worked together with the applet, they were

able to make sense of the mathematics by talking through the processes. The teacher paired

limited English-proficient students with students

who spoke the same language and could better

explain what was happening. The ability to switch

to Spanish gave many ELLs better access to the

mathematics (see fig. 4b). And finally, while other

students explored with a partner, the special needs

learners worked together in a small group with

the mathematics educator, who scaffolded their

experience by working collaboratively in front of

the SMART Board.

Traditionally, special needs learners are often

given direct instruction on how to perform an algorithm using mnemonic devices or procedural steps

without being given opportunities to construct

Teaching Children Mathematics / November 2008

conceptual understanding of the procedure. One

of the biggest challenges of working with physical manipulatives, such as fractions circles, is that

actually manipulating multiple pieces creates so

much of a cognitive load on students’ thinking

processes th …

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