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.

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Enhancing Mathematical
Learning in a
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], 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. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
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.
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
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
National Library of Virtual Manipulatives at Utah State University, copyright 1999–2000. All Rights Reserved.
Analysis of Mathematical Representations and Models
Concept tutorial/skill practice
Investigation/problem solving
Open exploration
• 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
• Use representations to model and interpret physical, social, and mathematical
• 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
process (step by
• 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
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
Teaching Children Mathematics / November 2008
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,
(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.
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
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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