Virtual Place Value - Eugene School District - 541-790 … etools Virtual place-Value Blocks workspace allowed ... count the quantity, ... est digit to the smallest. A key to place-value

228 November2013•teaching children mathematics | Vol. 20, No. 4 www.nctm.org

T echnology permeates every aspect of our daily lives, from the sensors that control the

traffic signals on our morning com-mute to the cameras that allow real-time video chats with family around the world. At times, technol-ogy 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, sci-ence, or social studies?

As elementary school mathe-matics teachers become aware of interactive math simulations, vir-tual manipulatives, graphics, and

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 influ-ence student achievement?

Although these questions have been researched ever since the first com-puters 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

VirtualValue

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think mathematically when using virtual versus concrete base-ten blocks

to learn place-value concepts.

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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.

www.nctm.org Vol. 20, No. 4 | teaching children mathematics • November 2013 229

particular type of technology, vir-tual manipulatives. Studies linking technology to student achievement provide a mixed bag of results and conclusions for students’ use of vir-tual manipulatives in mathematics (Bolyard 2006; Drickey 2000; Kim 1993; Smith 2006; Steen, Brooks, and Lyon 2006; Suh and Moyer 2007; Takahashi 2002; Terry 1995). Ques-tions 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 teach-ers joined a school’s math coach in attempting to discover how stu-dents were thinking and interacting mathematically while learning with technology. Using video, the teach-ers captured interactions of students using virtual base-ten blocks while studying place value. Rather than trying to link technology use and student achievement, the teach-ers asked, How do students think mathematically when using virtual base-ten blocks to learn place- value 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 numer-als. Third graders are expected to read, write, and describe numbers through 999,999 and compare num-bers through 9,999. To compare how students interacted with base-ten blocks during the place-value unit, the team had students in two classes


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 Inves-tigations 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

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 blocksThe 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 stu-dents 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 mod-els worked with Interlox™ base-ten blocks (see fig. 2). These blocks are also considered groupable because students can build rep-resentations of one ten, one hundred, or one thousand. The blocks were chosen because they can be manipulated similarly to the vir-tual blocks.

the etools Virtual place-Value Blocks workspace allowed students to work with ones, tens, hundreds, and thousands blocks.

Fig

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E 1

students and teachers used interlox™ Base-ten Blocks for concrete manipulatives in this study.

the video analysis included the components of number.

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Fig

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Place-value learning experiencesWorking 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 interac-tions, 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,

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 stu-dents’ 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 numberUsing both concrete manipulatives and vir-tual manipulatives, students built numbers by starting with the largest digit, whether it was

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.

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

Ta

Bl

E 1 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.

Ta

Bl

E 2


hundreds or thousands, and moving to tens and units, or from left to right. For example, Terris and John constructed 873 with (con-crete) 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 (thou-sands) recorded first (see fig. 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 cap-tured on video for both the virtual and concrete groups during the study, 137 were from the larg-est 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 cor-rectly wrote the expanded form of numbers after or during the construction of quantities. From collected student work, the team deter-mined that the group working with concrete manipulatives correctly wrote the expanded form of the numerals with 95 percent accu-racy, whereas the group working with virtual manipulatives wrote the expanded form with 92 percent accuracy.

During the instruction of place value, stu-dents should also be writing and saying numbers that match the corresponding representation (Baroody 1990). As these third graders con-structed each number, their teachers observed different counting strategies in both groups. Specifically, students favored one counting strategy. Of 138 representations of number ana-lyzed by video, 128 constructions were observed using a count-by-tens and a count-by-groups strategy, more specifically, the integrated con-ception (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 strate-gies 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 quanti-ties from left to right; they could count the quantity and could write the correspond-ing numeral and expanded form similarly. Regarding the components of number, they

this example of the construction of quantity with concrete models depicts the building and similar recording of the numeral 873.

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the example of the construction of quantity with virtual manipulatives shows that Bin and andy moved from greatest to smallest place value.

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interacted with virtual base-ten blocks in the same way as they did with the concrete ones.

Place value (UDSSI)The instructional team incorporated the UDSSI model into the study to frame stu-dents’ 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 con-nection 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 stu-dents can move between the two structures. The connection is made with the written numeral and the quantity (Fuson 1998).

While constructing 6256 with virtual manip-ulatives, Portia and Jacquelin talked through their work as they clicked on the virtual blocks.

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.

Fig

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“We are building 6256.” Portia stated. As Jacquelin clicked on the quantities, begin-

ning 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 count-by-ones strategy. they cannot partition 32 into 3 tens and 2 units.

students use a count-by-ones 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 under-stand 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

Ta

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conceptualization of place value. Students could count by tens and count by groups. The virtual models offered the same support and inter- action as the concrete base-ten blocks.

Renaming numbers As a precursor to computation, building non-standard forms, or noncanonical representa-tions, of number can assist students in the renaming of numbers (Van de Walle and Lovin 2006). For example, a student using the algo-rithm 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, stu-dents constructed ten representations, and students using virtual manipulatives created fif-teen nonstandard forms. Although the students using the virtual manipulatives generated more

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 dissimilarity that teachers noted between the groups was the ease of construc-tion of the nonstandard numbers for those using the virtual manipulatives. A distinct difference was that students using the vir-tual 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 quanti-ties 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 non standard representations. However, the strategy may have limited the number of ways that students could rename numbers, depending on students’ facility with the forward

kim and Nuri used the numeral counter to help them build quantities of 163.

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Picture Hundreds, Tens, Ones


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, evi-dence supports the use of virtual or concrete manipulatives. Regarding place value, the study suggests that students construct quan-tities, 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 multi-digit 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 mul-tiple nonstandard forms.

Using virtual blocks, students could accu-rately build quantities, could write numerals, and could count quantities related to place-value 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 vir-tual tools are a viable manipulative for students to use when constructing knowledge of place-value 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.

ConclusionsBefore elementary school mathematics teach-ers adopt a technology—and more important, before students interact with technology—we must ask, What is the purpose of the technol-ogy? As students at Kennedy Elementary School

learned place-value concepts with virtual mod-els, 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 con-crete 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 manipu-latives 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 elemen-tary school math classrooms, they must ask themselves (1) What is the purpose of the tech-nology or virtual manipulative, and (2) how will students interact with and think mathe-matically when using the technology? Kennedy Elementary School students interacted with virtual manipulatives to think mathematically about place-value concepts.

REFERENCESBaroody, arthur J. 1990. “how and When

should place-Value concepts and skills Be

Before elementary school math teachers adopt a technology and before students interact with it, we must ask, What is the purpose of the technology?


taught?” Journal for Research in Mathematics Education 21 (4): 281–86.

Bolyard, Johnna J. 2006. “a comparison of the impact of two Virtual manipulatives on student achievement and conceptual understanding of integer addition and subtraction.” phd diss. Fairfax, Va: george mason university.

Burris, Justin t. 2010. “third graders’ mathematical thinking of place Value through the use of concrete and Virtual manipulatives.” edd diss. houston, tx: university of houston.

clements, douglas h. 1999. “‘concrete’ manipu-latives, concrete ideas.” Contemporary Issues in Early Childhood 1 (1): 45–60.

dienes, Zoltan. 1969. Building Up Mathematics. london, england: hutchinson educational.

drickey, Nancy. 2000. “a comparison of Virtual and physical manipulatives in teaching Visualization and spatial reasoning to middle school mathematics students.” phd diss. logan, ut: utah state university.

Fuson, karen. 1998. “pedagogical, mathematical, and real-World conceptual-support Nets: a model for Building children’s multidigit domain knowledge.” Mathematical Cognition 4 (2): 147–86.

Fuson, karen c., steven t. smith, and ana maria lo cicero. 1997a. “supporting latino First graders’ ten-structured thinking in urban classrooms.” Journal for Research in Mathematics Education 28 (6): 738–66.

Fuson, karen c., diana Wearne, James c. hiebert, hanlie c. murray, pieter g. human, alwyn i. olivier, thomas p. carpenter, and elizabeth Fennema. 1997b. “children’s conceptual structures for multidigit Numbers and methods of multidigit addition and subtraction.” Journal for Research in Mathematics Education 28 (2): 130–62.

hiebert, James, and diana Wearne. 1996. “instruction, understanding, and skill in multidigit addition and subtraction.” Cognition and Instruction 14 (3): 251–83.

international society for technology in educa-tion (iste). 2007. “National educational technology standards for students.” http://www.iste.org/nets

kim, sun-young. 1993. “the relative effective-ness of hands-on and computer-simulated manipulatives in teaching seriation,

classification, geometric, and arithmetic concepts to kindergarten children.” phd diss. eugene, or: university of oregon.

knoblauch, hubert, 2009. “Videography: Focused ethnography and Video analysis.” in Video Analysis: Methodology and Methods: Qualitative Audiovisual Data Analysis and Sociology, edited by hubert knoblauch, Bernt 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: george mason university.

steen, kent, david Brooks, and tammy lyon. 2006. “the impact of Virtual manipulatives on First-grade geometry instruction and learn-ing.” Journal of Computers in Mathematics and Science Teaching 25 (4): 373–91.

suh, Jennifer, and patricia moyer. 2007. “devel-oping students’ representational Fluency using Virtual and physical algebra Balances.” Journal of Computers in Mathematics and Science Teaching 26 (2): 155–73.

takahashi, akihiko. 2002. “affordances of computer-Based and physical geoboards in problem-solving activities in the middle grades.” phd diss. champaign, il: university of illinois at urbana–champaign.

terry, marilyn k. 1995. “an investigation of differences in cognition when utilizing math manipulatives and math manipulative software.” phd diss. university of missouri–st. louis, 1995. Dissertation Abstracts International 56 (7): 2650a.

Van de Walle, John a., and louann h. lovin. 2006. Teaching Student-Centered Mathematics Grades K–3. New york: pearson.

Wright, robert J., James martland, ann k. stafford, and gary stanger. 2002. Teaching Number: Advancing Children’s Skills and Strategies. thousand oaks, ca: sage publications.

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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Virtual Place Value - Eugene School District - 541-790 … etools Virtual place-Value Blocks workspace allowed ... count the quantity, ... est digit to the smallest. A key to place-value