Virtual
Place
Value
Compare how third graders
think mathematically when using
virtual versus concrete base-ten blocks
to learn place-value concepts.
T
echnology permeates every
aspect of our daily lives, from
the sensors that control the
traffic signals on our morning commute to the cameras that allow
real-time video chats with family
around the world. At times, technology may make our lives easier, faster,
and more productive. However,
does technology do the same in our
schools and classrooms? Will the
benefits of technology translate to
learning reading, mathematics, science, or social studies?
As elementary school mathematics teachers become aware of
interactive math simulations, virtual manipulatives, graphics, and
228
dynamic models, we must question
ourselves and our motives:
• Is this technology important to
my classroom?
• When should my students use
technology?
• How will this technology influence student achievement?
Although these questions have been
researched ever since the first computers were placed into elementary
school classrooms, we do not have
solid answers even today.
Researchers have attempted to
demonstrate through test scores
the advantages of students using a
Boy using a laptop computer: Lightpoet/Veer, Background: Techno cubes Andreus/Veer
By Justin T. Bur ris
November 2013 • teaching children mathematics | Vol. 20, No. 4
Copyright © 2013 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.
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particular type of technology, virtual manipulatives. Studies linking
technology to student achievement
provide a mixed bag of results and
conclusions for students’ use of virtual manipulatives in mathematics
(Bolyard 2006; Drickey 2000; Kim
1993; Smith 2006; Steen, Brooks,
and Lyon 2006; Suh and Moyer 2007;
Takahashi 2002; Terry 1995). Questions that link student technology
use to student achievement are
often difficult to answer. However,
teachers at Kennedy Elementary in
Alief Independent School District
asked a different question. Students
are no longer being asked to learn to
use technology but to develop skills
and learn with technology (ISTE
2007). Following this new research
agenda, third-grade math teachers joined a school’s math coach
in attempting to discover how students were thinking and interacting
mathematically while learning with
technology. Using video, the teachers captured interactions of students
using virtual base-ten blocks while
studying place value. Rather than
trying to link technology use and
student achievement, the teachers asked, How do students think
mathematically when using virtual
base-ten blocks to learn placevalue concepts?
Examining classroom
assessments
In the study, third-grade classes
engaged in a unit to learn place
value by using base-ten blocks to
build and identify quantities and
write the corresponding numerals. Third graders are expected to
read, write, and describe numbers
through 999,999 and compare numbers through 9,999. To compare how
students interacted with base-ten
blocks during the place-value unit,
the team had students in two classes
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Vol. 20, No. 4 | teaching children mathematics • November 2013
229
Figure 2
Students and teachers used Interlox™
Base-Ten Blocks for concrete
manipulatives in this study.
Figu r e 3
Figu r e 1
The eTools Virtual Place-Value Blocks workspace allowed
students to work with ones, tens, hundreds, and
thousands blocks.
The video analysis included the
components of number.
randomly videotaped one pair of students using
concrete base-ten blocks. Video recordings
provided the opportunity for repeated viewings
of student thinking and interactions with both
types of blocks (Knoblauch 2009). Field notes
and student work were also collected to help
describe the interactions.
Virtual and concrete base-ten blocks
use concrete base-ten blocks while students in
two other classes used virtual base-ten blocks.
Teachers in all four classes followed the same
lesson plans, enVisions curriculum, and Investigations textbook series. Students worked in
pairs in all four classrooms. However, students
who worked in pairs at the computer switched
at ten-minute intervals to give each student an
equal opportunity to manipulate the mouse.
During the study, the math coach randomly
chose one pair of students to be videotaped
each day so that the team could observe
students’ interactions with the virtual blocks,
hear students’ conversations (including
counting), and describe their construction of
quantities (Burris 2010). Similarly, the coach
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November 2013 • teaching children mathematics | Vol. 20, No. 4
The virtual base-ten blocks used in the study
were designed by enVision as part of their
e-Tools software. The blocks can be described
as groupable models because the blocks can
be grouped and regrouped into units, tens,
hundreds, and thousands (Van de Walle and
Lovin 2006). Additionally, a numeral counter
was available. As students placed blocks onto
the screen, the counter recorded the results.
These dynamic virtual models allowed students to interact with and manipulate the
blocks using different tools. For instance,
students could use the hammer tool to break
quantities apart and the glue tool to regroup
quantities (see fig. 1).
Students who used the concrete models worked with Interlox™ base-ten blocks
(see fig. 2). These blocks are also considered
groupable because students can build representations of one ten, one hundred, or one
thousand. The blocks were chosen because
they can be manipulated similarly to the virtual blocks.
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Instructional timeline
Day 1
Exploration with virtual
or concrete blocks
Day 2
Building hundreds
Day 3
Making numbers with
hundreds, tens, and
units
Day 4
163 stickers:
Noncanonical numbers
Day 5
Build and write:
Expanded notation
Day 6
Thousands
Day 7
Thousands
Day 8
Greater numbers
Day 9
Comparing numbers
Day 10
Posttest: Place value
Place-value learning experiences
Working in pairs, students engaged with the
virtual base-ten blocks in a computer lab or with
the concrete blocks in the classroom each day.
To align with Dienes’s (1969) dynamic principle
of unstructured play with manipulatives, the
teachers began the unit with time for students to
explore the blocks and become familiar with the
tools (see table 1).
How did students interact?
Early elementary mathematics education
focuses on the three components of number:
the written numeral, the quantity, and the
verbal or spoken number (Wright et al. 2002).
While the teachers observed student interactions, they used a video protocol that accounted
for these components (see fig. 3). They wanted
to watch and record how students used the
concrete and virtual blocks to build a quantity,
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table 2
table 1
Teachers began the instructional
sequence of the place-value unit
by allowing time for students to
explore the blocks and become
familiar with the tools.
The UDSSI (unitary, decade, sequence, separate, integrated)
model for multidigit numbers describes the conceptual
structures that children use for place value.
Conceptual
structure
Description
Unitary
Students use a count-by-ones strategy and
can identify “a whole word to a whole
quantity.” They cannot partition 32 into
3 tens and 2 units.
Decade
Students use a count-by-ones strategy and
may start to relate the number words to
separate quantities. For example, when
counting 32, a student relates “thirty” to
thirty objects and “two” to two objects.
Sequence
Students understand tens and units; a
student may count, “Ten, twenty, thirty,
thirty-one, thirty-two.”
Separate
Students understand that digits separated
are tens and units. Students may count
groups as “1 ten, 2 tens, 3 tens, and 1,
2 units.”
Integrated
Students can move fluidly between sequence
and separate conceptualizations. Students
can identify the 3 in the 32 as both thirty
and as 3 tens.
count the quantity, and write the numeral. The
video recording sheet also included research
about place value (Fuson et al. 1997b; Fuson
et al. 1997a) and the UDSSI (unitary, decade,
sequence, separate, integrated) model (see
table 2).
The purpose of including the model was to
help clarify the sequence of place-value learning
and identify those characteristics of students.
The model provided a lens to observe the students’ conceptual structures of place value while
they interacted with the manipulatives. After
the conclusion of the unit, the teachers used
the video protocol as they watched students’
interactions.
Components of number
Using both concrete manipulatives and virtual manipulatives, students built numbers by
starting with the largest digit, whether it was
Vol. 20, No. 4 | teaching children mathematics • November 2013
231
Figu r e 4
This example of the construction of quantity with concrete
models depicts the building and similar recording of the
numeral 873.
Figure 5
The example of the construction of quantity with virtual
manipulatives shows that Bin and Andy moved from
greatest to smallest place value.
hundreds or thousands, and moving to tens
and units, or from left to right. For example,
Terris and John constructed 873 with (concrete) Interlox blocks. Terris began by pulling
out 8 hundreds flats, 7 tens rods, and 3 units.
He counted as he built: “OK, let’s just do 873.”
During the construction, John talked through
the solution, “OK, so that’s one, two, three, four,
five, six, seven, eight hundreds and one, two,
three, four, five, six, seven tens and one, two,
three ones.” The quantities were also recorded
in a similar fashion, with the largest digit (thousands) recorded first (see fig. 4).
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November 2013 • teaching children mathematics | Vol. 20, No. 4
Students used similar constructions with the
virtual models. On day 1, Bin and Andy built
the number 345. Andy counted the blocks as he
clicked the mouse, “One, two, three hundreds;
one, two, three, four tens; and one, two, three,
four, five ones.” The boys clicked on the Arrange
tool to organize the blocks for recording (see
fig. 5). Of 138 numbers constructed and captured on video for both the virtual and concrete
groups during the study, 137 were from the largest digit to the smallest.
A key to place-value instruction is to match
the symbol to the coordinated quantity (Fuson
1998). Teachers observed that more than
90 percent of students in both groups correctly wrote the expanded form of numbers
after or during the construction of quantities.
From collected student work, the team determined that the group working with concrete
manipulatives correctly wrote the expanded
form of the numerals with 95 percent accuracy, whereas the group working with virtual
manipulatives wrote the expanded form with
92 percent accuracy.
During the instruction of place value, students should also be writing and saying numbers
that match the corresponding representation
(Baroody 1990). As these third graders constructed each number, their teachers observed
different counting strategies in both groups.
Specifically, students favored one counting
strategy. Of 138 representations of number analyzed by video, 128 constructions were observed
using a count-by-tens and a count-by-groups
strategy, more specifically, the integrated conception (Fuson et al 1997b). Students in both
groups demonstrated what Fuson (1998) calls
“the place-value meaning of number words.”
The counting showed students’ ability to count
by tens or count by groups. These two strategies demonstrated a conceptualization of single
groups of ten (Fuson 1998). Different types of
number word sequences directly align to the
conceptualization of place value within the
UDSSI model (see table 2).
Why is this important? Students in both
groups used virtual and concrete base-ten
blocks in similar ways: They built quantities from left to right; they could count the
quantity and could write the corresponding numeral and expanded form similarly.
Regarding the components of number, they
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Place value (UDSSI)
table 3
The instructional team incorporated the
UDSSI model into the study to frame students’ thinking and abilities regarding place
value. The model provided the teachers with
a conceptual model for the learning and
understanding of place value (Fuson et al.
1997a; Fuson et al. 1997b). They attempted to
delineate the conceptual stages for students by
listening to the method of counting, watching
the construction of numbers, and observing
the representations of numerals.
Video analysis suggests that students in
both groups were able to correctly describe the
quantities as tens and ones and make the connection to the written symbol; this is described
as the integrated structure (Fuson et al. 1997b).
The integrated conception of the UDSSI model
describes the ability of students to move
seamlessly between the count-by-tens and
the count-by-groups structures. For example,
students may be able to link seventy-two and
seven tens and two ones with the numeral 72.
The integrated conception suggests that students can move between the two structures.
The connection is made with the written
numeral and the quantity (Fuson 1998).
While constructing 6256 with virtual manipulatives, Portia and Jacquelin talked through
their work as they clicked on the virtual blocks.
Figu r e 6
interacted with virtual base-ten blocks in the
same way as they did with the concrete ones.
Portia and Jacquelin used a box with an x to represent the
thousand cube, moving fluidly by identifying the numeral
and the component places of 6256.
“We are building 6256.” Portia stated.
As Jacquelin clicked on the quantities, beginning with the thousands and moving to the
units, she counted, “One, two, three, four, five,
six thousands; two hundreds; one, two, three,
four tens; and six ones.”
Portia saw a discrepancy: “Wait, you need
one more ten; you only have four.”
Using the arrange tool, Jacquelin organized
the blocks while Portia recorded the solution
(see fig. 6). Note that Jacquelin used a box with
an x to represent the thousand cube. In this
respect, Portia and Jacquelin moved fluidly by
identifying the numeral and the component
places, thereby demonstrating the integrated
conception of the UDSSI model (see table 3).
Regardless of which manipulative they used,
most students operated within the integrated
Using virtual and concrete manipulatives, students constructed numbers within the UDSSI structures.
Place-value structures by manipulative
Unitary
Decade
Sequence
Separate
Integrated
Students
use a countby-ones
strategy.
They cannot
partition 32
into 3 tens
and 2 units.
Students use
a count-byones strategy
and may start
to relate the
number words
to separate
quantities.
Students
understand
tens and units;
a student may
count, “ten,
twenty, thirty,
thirty-one,
thirty-two.”
Students understand that digits
separated are
tens and units. A
student may count
groups as “1 ten,
2 tens, 3 tens, and
2 ones.”
Students can move
fluidly between
sequence and separate
conceptualizations. A
student can identify
the 3 in the 32 as both
thirty and as 3 tens.
Concrete
0
0
0
9
55
Virtual
0
0
0
1
73
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Vol. 20, No. 4 | teaching children mathematics • November 2013
233
Figu r e 7
Kim and Nuri used the numeral counter to help them build
quantities of 163.
Picture
Hundreds, Tens, Ones
conceptualization of place value. Students could
count by tens and count by groups. The virtual
models offered the same support and interaction as the concrete base-ten blocks.
Renaming numbers
As a precursor to computation, building nonstandard forms, or noncanonical representations, of number can assist students in the
renaming of numbers (Van de Walle and Lovin
2006). For example, a student using the algorithm to solve 62 – 27 would rename 62 as 5 tens
and 12 ones to subtract. During the two lessons
for renaming numbers, students in both groups
generated a total of twenty-five nonstandard
numbers. Using concrete manipulatives, students constructed ten representations, and
students using virtual manipulatives created fifteen nonstandard forms. Although the students
using the virtual manipulatives generated more
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November 2013 • teaching children mathematics | Vol. 20, No. 4
nonstandard representations, the information
supplied by the videotaped sessions and student
work offered a clearer picture of the differences
in each groups’ thinking about place value.
One dissimilarit y that teachers noted
between the groups was the ease of construction of the nonstandard numbers for those
using the virtual manipulatives. A distinct
difference was that students using the virtual manipulatives “reused” the quantity on
the screen. They used the hammer and glue
tools to show various representations without
clearing the screen or starting over. However,
students using the concrete models had to
“trade” the blocks to construct the quantity.
When Kim and Nuri were finding nonstandard forms of 163, Kim built 163 by first
adding 1 hundred followed by 6 tens and 3 ones
to display the standard form. Using the hammer
tool to break apart the blocks, Kim broke the
hundreds into 10 tens.
“Is that still 163?” asked Nuri.
“Yes, look: It’s over here.” Kim pointed to the
counter in the lower portion of the screen.
Teachers noted that the pair had used the
numeral counter to help them build quantities of 163. Next, the girls created 16 tens and
3 ones. Nuri counted the tens as “ten, twenty,
thirty, forty,” through 160. The pair continued
with 1 hundred and 63 ones on the board, using
the hammer tool to experiment. They broke the
hundred flat into 10 tens. “We could do that,
10 tens and 63 ones.” Kim suggested. By using
tools available with the virtual manipulatives—
including the hammer tool and the place-value
chart—Kim and Nuri found seven nonstandard
representations for 163, including 14 tens and
23 units (see fig. 7).
Within the virtual manipulative group, the
standard form of the number was constructed
and students used available virtual tools,
including the counter, hammer, glue, and
place-value tools, to find solutions. Within the
concrete manipulative group, students used
counting strategies and benchmark numbers to
generate the nonstandard form. The counting
strategies and benchmarks may have benefited
students with regard to the construction of
some non­standard representations. However,
the strategy may have limited the number of
ways that students could rename numbers,
depending on students’ facility with the forward
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number word sequence and visualization of the
model. The virtual tools allowed students to
realize the potential for renaming numbers
without those restrictions.
What are the benefits
of virtual base-ten blocks?
This study suggests that students interacting
with virtual or concrete base-ten blocks are
capable of mathematical thinking of place
value. Because students use and interact
with both manipulatives in similar ways, evidence supports the use of virtual or concrete
manipulatives. Regarding place value, the
study suggests that students construct quantities, write numerals, and count or identify
quantities similarly with concrete or virtual
manipulatives.
Virtual manipulatives provided support
for learning nonstandard representations or
renaming numbers. These nonstandard forms
are distinctly interconnected with multidigit
addition and subtraction, specifically the multidigit algorithm (Hiebert and Wearne 1996). The
available virtual tools, including the hammer
and glue tools to break apart and reorganize
quantities, allowed students to construct multiple nonstandard forms.
Using virtual blocks, students could accurately build quantities, could write numerals,
and could count quantities related to placevalue concepts. However, students could also
build nonstandard forms or could rename
numbers with the tools provided by the virtual
models. Students who used virtual blocks could
compose and decompose numbers more readily
than those who used concrete blocks. These virtual tools are a viable manipulative for students
to use when constructing knowledge of placevalue concepts. This study suggests that by
interacting with virtual manipulatives, students
can demonstrate mathematical thinking of
place-value concepts with the added benefit of
constructing nonstandard representations that
are directly linked to the multidigit algorithm.
Conclusions
Before elementary school mathematics teachers adopt a technology—and more important,
before students interact with technology—we
must ask, What is the purpose of the technology? As students at Kennedy Elementary School
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Before elementary school
math teachers adopt a
technology and before
students interact with it,
we must ask, What is
the purpose of the technology?
learned place-value concepts with virtual models, the purpose of the virtual base-ten blocks
became clear. Students used the virtual blocks to
construct and count quantities and to write and
identify numerals just as they would with concrete models. Surprisingly, the virtual models
benefited students in renaming numbers.
After reviewing the videos, team members
were amazed that students used the concrete
and virtual blocks in a similar way. Students’
interactions with virtual base-ten blocks in this
study were similar to students’ interactions
with concrete blocks. The virtual models were
advantageous to students as they generated
nonstandard numbers more efficiently using
technology.
When thinking about using virtual manipulatives in your classroom, do not ask whether
virtual models are “concrete” but rather how
students will interact with the models and how
they will think mathematically when using
them. As Clements (1999) suggested, children’s
interactions with manipulatives should be the
emphasis, not the manipulatives themselves.
Before teachers adopt a technology in elementary school math classrooms, they must ask
themselves (1) What is the purpose of the technology or virtual manipulative, and (2) how
will students interact with and think mathematically when using the technology? Kennedy
Elementary School students interacted with
virtual manipulatives to think mathematically
about place-value concepts.
REF EREN C ES
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Should Place-Value Concepts and Skills Be
Vol. 20, No. 4 | teaching children mathematics • November 2013
235
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through the Use of Concrete and Virtual
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University of Houston.
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Knoblauch, Hubert, 2009. “Videography:
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Schnettler, Jargen Raab, and Hans-Georg
Soeffner, pp. 69–83. New York: Peter Lang.
Smith, Lorraine A. 2006. “The Impact of Virtual
and Concrete Manipulatives on Algebraic
Understanding.” PhD diss. Fairfax, VA:
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Steen, Kent, David Brooks, and Tammy Lyon.
2006. “The Impact of Virtual Manipulatives on
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Suh, Jennifer, and Patricia Moyer. 2007. “Developing Students’ Representational Fluency
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Takahashi, Akihiko. 2002. “Affordances of
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Van de Walle, John A., and LouAnn H. Lovin.
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Justin T. Burris, justin.burris@
gmail.com, is a math coach at
Kennedy Elementary School in the
Alief Independent School District
in Houston, Texas. He is a visiting
assistant professor of Mathematics Education at
the University of Houston and is interested in
students’ mathematical interactions with virtual
models and manipulatives.
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