Conceptual Visibility and Virtual Dynamics in Technology-scaffolded Learning Environments for...

Jl. of Computers in Mathematics and Science Teaching (2012) 31(3), 283-310

Conceptual Visibility and Virtual Dynamics in Technology-scaffolded Learning Environments for

Conceptual Knowledge of Mathematics

Julie McleodGood Shepherd Episcopal School, USA

[email protected]

Sheri VaSindaOklahoma State University, USA

[email protected]

Mary Jo dondlingerRichland College of Dallas County Community College District, USA

[email protected]

This article adds to the growing body of research surround-ing the use of virtual manipulatives for mathematics learning. The study reported herein used five different virtual manipu-latives from four different websites, all within a sixth grade unit on proportional thinking. While many studies have re-ported on student achievement when using virtual manipu-latives, the researchers in this study consider achievement a baseline and evaluated the manipulatives from the perspective of the student, examining aspects of the virtual manipula-tives that led to self-regulated learning behaviors, discovery and learning of the underlying mathematical concepts under study, constructive emotional connections to learning and more. From these findings of the study, a Multidimensional Virtual Manipulative evaluation (MVMe) Tool is developed as a means to evaluate the potential effectiveness of a virtual manipulative. Students’ perceptions of welcome versus un-welcome scaffolds are also discussed within the context of virtual manipulatives.

284 McLeod, Vasinda, and Dondlinger

Introduction

Teaching mathematics to greater depth and understanding has nev-er been more important in today’s hyper-competitive global marketplace (Friedman, 2006). indeed, professions from science to engineering require strong understandings of mathematical concepts to support an increas-ing reliance on digital technologies. Without a doubt, mathematics educa-tion is critical to preparing our students for their future (griffin, case & capodilupo, 1995). in response to this demand, organizations such as the national council of Teachers of Mathematics (ncTM) have called for in-creased depth of understanding and conceptual knowledge and have empha-sized representation as an important area of mathematics education research (ncTM, 2000). Specifically, the ncTM expects instructional programs to enable students to create, use, select, and apply mathematical representa-tions to model and interpret mathematical phenomena.

historically, mathematics has been taught procedurally, as a list of steps that would most efficiently and effectively bring the solver to the correct an-swer. anderson et al. (2001) define procedural knowledge as knowing how to do something or recalling the algorithm to solve a problem. This abstract type of knowledge can typically be memorized without any understanding of the underlying concepts. conceptual knowledge is defined as knowledge of the interrelationships of basic elements that make up larger structures (anderson et al., 2001), including knowledge of the concept models. con-ceptual knowledge is more concrete than procedural knowledge and also of-fers learners more flexibility in solving problems because even if one step of a procedure is forgotten, students with conceptual knowledge can fill in the gap or use an invented method of solving rather than the algorithm. current research suggests that teaching both procedural and conceptual knowledge in mathematics is necessary (donovan & Bransford, 2005) as each supports the other in helping students develop valid and reliable mental models to facilitate strategies for solving problems. rapp (2005) defines mental mod-els as internal, dynamic organized knowledge structures changing as new information is added and hypotheses tested. When students learn only one type of knowledge, their overall mathematical thinking and reasoning is im-poverished (rittle-Johnson, Siegler, & alibali, 2001), and mental models built from incomplete, or impoverished understanding can be faulty (rapp, 2005), further hindering mathematical thinking and reasoning. although the construction of accurate mental models is complex, rapp (2005) identifies three factors that influence the development of mental models: cognitive en-gagement, interactivity, and multimedia learning; these factors also corre-

Conceptual Visibility and Virtual Dynamics 285

late with principles of learning that support the use of manipulatives, both physical and virtual. Studies of teaching mathematics with an emphasis on conceptual knowledge and development of mental models using virtual ma-nipulatives responds to the ncTM’s call for research on mathematical rep-resentations.

This study used five different virtual manipulatives from four different websites, all within a sixth grade mathematics unit on proportional think-ing. While many studies have reported on student achievement when using virtual manipulatives, the researchers in this study consider achievement a baseline and evaluated the manipulatives from the perspective of the stu-dents. This article draws upon the findings of a larger study which exam-ined whether, how, and why students demonstrated curiosity in a sixth grade mathematics classroom with technology-integrated learning (Mcleod, 2011). one research question from that qualitative case study examined stu-dents’ conceptual knowledge building when using these five virtual manipu-latives. in this article, we further analyze the data from student interviews, examining aspects of the virtual manipulatives that led to constructive emo-tional connections to learning, self-regulated learning behaviors, and ul-timately discovery and learning of the underlying mathematical concepts under study. By doing so, we synthesized and created a Multidimensional Virtual Manipulative evaluation (MVMe) Tool that can be used by educa-tors and designers interested in supporting students’ development of strong conceptual knowledge.

ThEorETICaL PErsPECTIVE

Moyer, Bolyard, and Spikell (2002) define a virtual manipulative as an “interactive, web-based visual representation of a dynamic object that pres-ents opportunities for constructing mathematical knowledge” (p. 373). Both physical manipulatives and virtual manipulatives have demonstrated bene-fits for students who are learning mathematics. indeed, studies have linked manipulatives with higher achievement and increased flexibility with alge-braic concepts (Suh & Moyer, 2007). a growing body of research evaluat-ing the benefits of virtual manipulatives is beginning to reveal their value. reimer and Moyer (2005), who used virtual manipulatives with third grad-ers learning fractions, found that over half of the students in the study im-proved their conceptual understanding. in additional studies, Moyer, niez-goda and Stanley (2005) researched second graders studying place value and kindergarteners studying patterns. They found that second graders us-


ing virtual base-10 blocks were able to use more sophisticated strategies and explanations of place value. even english language learners were able to demonstrate their thinking about place value when their words escaped them. in their kindergarten study, students used wooden pattern blocks on the first day, virtual pattern blocks on the second day, and drew patterns on construction paper on the third day. results indicate students created more sophisticated patterns using more blocks with the virtual manipulative than with either other method. Finally, Steen, Brooks and lyon (2006) studied first grade students’ use of virtual manipulatives during a geometry unit. Posttest results showed that the treatment group had significant gains on conceptual understanding of geometry. The teacher in the study also report-ed more efficient use of instructional time, more student time on task, and higher student motivation.

Pedagogically speaking, not all virtual manipulatives are created equal-ly. Because manipulatives in general are representations of a mathematical concept, they are figurative, similar to a metaphor in language. lakoff and Johnson (1980) understand that metaphors not only serve us linguistically, they also are a part of our way of thinking. lakoff and Johnson examine and categorize metaphors, assessing how they are used to create or obfus-cate meaning. They contend that metaphors highlight some aspects of the concept to which they refer and at the same time they hide other aspects of that concept. Manipulatives, like metaphors, function in the same way: highlighting some aspects of the underlying mathematical concept while hiding others. a manipulative such as virtual pattern blocks might be very successful with patterning in kindergarten because it highlights the patterns that students can easily create but less successful with fractions in older el-ementary grades because it hides some of the important aspects of break-ing up a whole into fractional pieces. This manipulative both highlights a concept, patterning, while also hiding a concept, fractional relationships. it is important pedagogically to understand which parts of the concept are the most crucial and perhaps the most difficult to understand, and then to match that with virtual manipulatives that appropriately highlight that aspect of the concept. if properly selected, conceptual knowledge building websites such as the national library of Virtual Manipulatives (Bouck & Flanagan, 2010; reimer & Moyer, 2005) can assist students in building mental models of the mathematical concepts, or completing project work designed to deepen con-ceptual knowledge while using technology as a tool.


rEsEarCh DEsIgn anD METhoDs

Because the growing evidence of the effectiveness of virtual manipu-latives and achievement has been established, this study begins to examine why the virtual manipulatives are effective from the perspective of the learn-er, drawing from weekly, semi-structured interviews in which students were asked to describe their learning with the manipulatives, including what drew them into the work and how and why they continued to explore and learn when the work was complex or challenging. We then further analyze these data to draw conclusions about what makes a virtual manipulative effective as a representation of a mathematical concept in the implications and con-clusion.

Participants

Participants were drawn from all sixth grade students at a Title i ele-mentary school in a north Texas suburban town (see Table 1). a subset of the population was purposefully selected for the interviews, including cri-teria such as access to participants, their probability of speaking honestly, and their “capacity to express [their] experience in words” (Wengraf, 2006; p. 95). in addition, this sample was selected to maintain similar demograph-ics to the sixth grade in general and include a broad range of mathematical abilities (see Table 1) as indicated by the response to intervention Tier sys-tem (Fuchs & Fuchs, 2006).

Virtual Manipulatives Used in This Study

during the first week of this study, students were beginning to learn ratios and proportions, a new concept in sixth grade that builds upon past learning of fractional and decimal equivalents. after demonstrating an un-derstanding of ratios, students moved to conceptual understanding of pro-portions as two equivalent ratios using the virtual manipulative displayed in Figure 1, from the learnalberta.ca website’s Math interactives. during their interaction with these virtual manipulatives, students connected pro-portions with photography and decided whether a targeted ratio or enlarged photograph was equivalent to the original ratio or original photograph. This manipulative includes a multiplier across the top of the workspace which connects the students to the mathematics necessary to decide whether ratios are proportional. Students also received visual mathematical feedback when they clicked a multiplier and the outline of the photograph responded. dur-ing the interviews, this work was called “resizing images.”

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Figure 1. Screenshot of proportion conceptual knowledge work.

once students conceptually understood the definition of a proportion, the next virtual manipulative assisted students in building a model of using proportions to solve for an unknown. MathPlayground.com hosts the Think-ing Blocks manipulative for this work and is shown in Figure 2 below. The manipulative first presents students with a problem and prompts them to build a model of the ratio that they then label and use to solve the problem. during the interviews, this work was called “Thinking Blocks.”

Figure 2. Screenshot of Thinking Blocks used for students’ proportional

conceptual knowledge.

during the second week of the study, students learned about percents as a ratio and also modeled percent, decimal and fractional equivalents.


To help students develop a mental model of a percent, the teacher used the virtual manipulative shown in Figure 3, from the national library of Vir-tual Manipulatives (nlVM). Students were presented with a blank ten by ten grid and the manipulative asked students to model a certain percent and provided feedback on their answers. during the interviews, this work was called “Percent grid.” This same grid with 100 squares was used throughout the week to continue to develop and reinforce students’ conceptual knowl-edge of a percent.

after students gained an understanding of a percent, they needed to in-corporate this understanding into their existing knowledge of decimals and fractions. earlier in the year, students had studied decimal and fractional equivalents and built mental models of those concepts. For this work, stu-dents used the nlVM manipulative for representing decimals shown in Fig-ure 4. This manipulative offered students random decimals that they mod-eled. Then, they sketched the model of the decimal along with the percent model using the hundreds grid. They also wrote the decimal and the percent. This work was called “decimal to Percent” during the interviews.

Figure 3. Screenshot of nlVM percent conceptual knowledge manipulative.


Figure 4. Screenshot of students decimal to percent work.

Finally, students incorporated their models of fractions into their new percent knowledge using a virtual manipulative on Shodor.org and shown in Figure 5. Students were presented with a fraction. They then used the next blank box to create their percent grid by making 10 rows and 10 columns. as they shade their percent model, the arrow on the number line moved to offer mathematical feedback as to the size of the model they created. once students had an equivalency match between the fraction and the percent models, they documented their work by sketching their results on paper. This work was called “Fractions to Percents” during the interviews.


Figure 5. Screenshot of students work to build fraction and percent equiva-lents.

Data Analysis

data analysis used a qualitative, constant-comparative approach to al-low themes to emerge (gall, gall & Borg, 2007; Strauss & corbin, 1998). analysis began by developing codes which were constantly compared to other codes and to new data and then were grouped with other common codes into categories. after approximately half of the interviews had been initially coded, the researchers began to look at the codes for emerging cat-egories. Since new codes emerged as the interviews were initially coded, researchers reviewed all the transcripts again, re-coding lines in light of fi-nal definitions of each code as well as new codes that had emerged. The researchers also re-examined the categories, merging the five initially iden-tified categories into the four categories presented in the results. Validity checks of the data included triangulation using the different data and dia-logic validity of critical colleagues.

FInDIngs

interviews from the first two weeks of the larger study addressed the mathematical conceptual knowledge work. Table 2 depicts the percent of the first two weeks’ interviews that were coded with each category. clearly,


discovery and learning drew a significant amount of the discussion dur-ing these weeks, followed by the digital tools students were using to help construct that knowledge. This coupling indicate to us that building concep-tual knowledge requires multiple and varied opportunities for learners to test hypotheses and continue to reshape their mental models. Some virtual manipulatives fostered positive emotional connections to learning, which seemed to have a positive effect on learners’ self-regulatory behaviors and ultimately their discovery and learning. in order to identify which manip-ulatives were most conducive to learning, we present results from all four categories, beginning with and providing the most detailed analysis of the codes within the digital tools category. We then limit results of the remain-ing categories to codes most relevant to these digital tools. note for read-ability, codes are italicized and categories are bold.

Table 2Percent of First Two Weeks’ Interviews Coded with each Category

Category Conceptual Knowledge

Self-regulation 14%

Discovery & Learning 31%

Digital tools 28%

Emotional Connections 27%

Digital Tools

For the digital tools category, four codes occurred most frequently in the student interviews: technology scaffold, technology impediment, it’s vir-tual and addicting. The technology scaffold code was assigned when stu-dents noted that technology was scaffolding for their learning. For example, elizabeth describes a scaffold from the Thinking Blocks manipulative that she appreciated, saying “i’d say on the Thinking Blocks… how at the very end it had the hint button so i just wanted to see what i have i been doing in all these different steps… So, yeah, i’d just say the hint button [helped me] because then i knew what i was doing.” The hint button on this virtual manipulative aided elizabeth’s understanding of the work, allowing her to make connections from the manipulative to the more abstract and symbolic mathematics.


When coding began, researchers coded all comments related to technol-ogy scaffolds as this general code. as the study progressed, the research-ers found several types of technology scaffolds emerging from the students’ interview responses: some positive, some not. consequently, we recoded those technology scaffolds that hindered students as technology impedi-ments. among the data remaining, we noted the prominence of three scaf-fold types to which we assigned additional codes: visual modeling, material intelligence, and amplification of effort. in describing learning principles that are present in strong and compelling video games, gee (2003) lists sev-eral principles that apply to other kinds of virtual learning environments. Two of those principles, material intelligence and amplification of input, were described by students as they discussed technology scaffolds.

• Visual Modeling: This code was assigned when students discussed the benefits of “seeing” the mathematical concept or model using visual vocabulary in their statements. For example, Thomas said, “i liked the resizing the images more because … you could see what you were doing… and you could see if it was right or not and it gave choices to where you can see exactly what you were doing.” Being able to “see” and thus “understand” more deeply was particularly important during conceptual knowledge building work as students were building their mental models of the concepts being studied.

• Material Intelligence: according to gee (2003), material intelligence means that “thinking, problem solving and knowledge are ‘stored’ in material objects and the environment. This frees learners to engage their minds with other things while combining the results of their own thinking with the knowledge stored in material objects and the environment to achieve yet more powerful effects” (p. 210). in the manipulative used for the fraction and percent equivalent work, the number line below the models of the fraction and percent had material intelligence. as students shaded their percent model, the arrow on the number line would automatically move slightly up the number line to indicate the value had increased Karen appreciated that scaffold, saying “because it was kind of fun how if i clicked on it and moved and then click on it and move.” This freed her mind from having to consider with each click how close she was to creating an equivalent fraction and percent and offered important mathematical feedback about the percent model she was building.


• Amplification of effort: This code is based on gee’s (2003) learning principle of amplification of input, which he describes as, “for a little input, learners get a lot of output” (p. 208). This occurs when students feel that they only have to expend a small amount of energy for a considerable gain from the manipulative. For example, olivia noted on the percent grid work that “you didn’t have to do a lot, you just had to do the percentage and color in the box thingy [sic].” Karen discussed this scaffold with the resizing images work, noting that it “was really easy because all you have to do is zoom it out with the times [multiplier radio button] and if it was on the line or not, just drag it [the equal or not equal sign].” Both students noted the ease with which tasks were accomplished, leaving their cognitive energy available to consider the connections and implications of the work, rather than working with physical sketches, manipulatives, or tedious calculations.

not all of the technologies used during the conceptual knowledge work provided scaffolds that the students appreciated. indeed, at times, the technology was seen as impediment to the learning. during the conceptu-al knowledge work, the most significant technology impediment was not-ed from the Thinking Blocks virtual manipulative. For some students, the multi-step problems created issues. Kevin was quite confused by the pro-cess, commenting that “they give you a question, you put the blocks down and then you have to answer another question after you answer that ques-tion then you go back to the main question.” he clearly could not envision how the questions were related to each other or to the “main question.” The complexity of the manipulative and the lock-step manner in which it led stu-dents through their thinking also caused issues for Marissa, who noted that when she “tried to solve the problem and drag the boxes to the little rectan-gles… it was kind of too much.” Marissa is describing the process in which the manipulative asks the students to model the proportion with blocks, then use a multiplier to solve for an unknown, which is the important learning for this work. however, for Marissa, the technology steps interfered with her success in building the model, which is a significant impediment to learning the underlying mathematical concepts. Moreover, John noted that “when-ever it said build your model on the Thinking Blocks, it asks you to put the question mark no matter what… you don’t need a question mark. you al-ready know the answer so it was really annoying.” in this situation, the tech-nology scaffold was actually in the way of John’s thinking and learning.

The third digital tools code was it’s virtual, applied when students ad-opted the stance that tasks can be completed more easily, quickly and ef-


ficiently if they use technology to complete it or described technology as compelling, authentic or realistic. The label for this code came from a stu-dent’s utterance during the interview when asked why technology was better than any other tool. Marissa said, “it’s virtual!” noting that “When things are on the computer, it makes you more interested. on paper… you just sit there and you’re just putting all the stuff on the paper. People like it more when it is on the computer because they feel like it is better and its very fun.” in this situation, Marissa is expressing a general dislike for paper work and overt appreciation for anything digital, which she views as fun. The ease with which students can complete multiple problems using tech-nology connects back to the amplification of effort code. Further, similar to the material intelligence code, when students spend less cognitive energy on working the model, they can spend more cognitive energy to explore the limits of the mathematics via manipulation of the model and evaluation of the results of that manipulation. as Karen expressed it, “i don’t really like the papers because… if you mess up people will know that you messed up but if it’s on the computer, only you know that you messed up.” Karen found that computer work allowed her to test different ideas without feeling that she “messed up,” sparing her the embarrassment of the natural mistakes of learning. While considerable effort is made in the classroom to create a safe learning environment where mistakes are viewed not as a source of embarrassment but as an avenue to deeper learning, adolescents often have a heightened sensitivity to peer criticism and will work hard to make sure they are not embarrassed.

The final important digital tools code was addicting, a characteristic of learning with technology that it made it difficult for students to stop. as Kevin describes it when asked why he completed so many of the resizing images problems, “because it was fun, addictive and it was learning.” alli-son described her work with the percent grid similarly saying, “…once you told us to stop i couldn’t stop. i kept on doing it until i had to stop. But i didn’t want to. i just wanted to continue doing it because it was like really fun.” allison notes that this is both good and bad, “i think of addicting as a good and bad thing because it’s like fun to do so you don’t want to stop but the bad thing is, you don’t want to stop.” For these students, what deter-mined whether technology addiction was good or bad is whether or not they see it as productive or beneficial. For example, elizabeth explained, “i think how some of you were talking about being addicted to technology would be that it’s like your own little world so it’s another world so if you mess up on one world you can create a whole new one and it will be perfect and stuff but it can really take up your life. you know in the classroom if you are


addicted to like a game or something, that’s good because it’s academic and you’re learning and it’s productive.” She indicates, and the others concurred, that if the addiction furthered learning, it was productive. in most contexts, students used the word addicting or addictive to describe their reasons for persisting or for doing more than they were required to do.

The remaining categories and important codes in those categories are described briefly in this article to offer a more complete context of the learn-ing. For more detailed information, please see Mcleod (2011).

Emotional Connections to Learning

The use of virtual manipulatives also afforded emotional connections to learning. The four codes most prominent in this category were feelings of competency, easy, fun and confused. The feelings of competency code is used when students’ discussed their understanding of the concept and spe-cifically when students felt that they understood the concept. For example, Becky noted “i got to level 2 ratios and i was ahead of everybody.” also, Sarah said, “i knew that if i kept doing it, i would get better at it and it was fun.” allison offers another example when she describes how her feelings of competency developed during her work with percents. She said, “before whenever it was like Target Math [a daily review] or something and we had to find a percent i would have trouble with that because i didn’t really get how to do percents. But since after we learned about how to figure out per-cents i sort of liked it because it wasn’t that complicated.” allison is offer-ing an important example of the competency students feel after they have worked with the concept using the virtual manipulative as opposed to only learning the procedural knowledge. only after she interacted with the vir-tual manipulatives and built her conceptual knowledge did she gain her feel-ings of competency.

Many of the feelings of competency were coupled with a feeling that the work was easy, which was the second most important code in the emo-tional connections category for conceptual knowledge. The easy code was used when students said the work was easy or that they did not have to put forth much effort. The third most prominent code in this category was fun, and was applied anytime the students said the word fun while describing the work or comparing work from different days or weeks. not all the work was fun or easy though. The final important code from this category was con-fused. This code was assigned when students described being confused, ei-ther using the word confused explicitly or describing uncertainty about what course of action to take.


self-regulatory behaviors

For the category of self-regulatory behaviors, two codes seemed to dominate the students’ discussions: persistence and leveraging resources. The persistence code was applied when students talked about continuing even when confronted with challenge or frustration or when the allotted time had expired. in many cases, the positive emotional connections to learning fostered by the virtual manipulative are what compelled students to persist. Madison claimed that “the thing that made me explore more was the decimals to percents because it was fun.” The second self-regulatory behavior code that occurred most frequently during the conceptual knowl-edge work was leveraging resources. This code was used when students expressed some strategy that they used in a challenging situation, demon-strating that they could self-assess and select from different tools before they asked someone else for help, although asking for help was one strategy available to them. among the strategies that students described were reading more carefully, jotting down notes on paper, or looking for clues embedded in the virtual environment.

Discovery and Learning

For the category of discovery and learning, four codes were perva-sive: invented play, constructing knowledge, trial and error and pulling together fragments into coherent knowledge structures. Invented play was used when students created an environment of play beyond that which was requested by the teacher or the virtual manipulative. Students created com-petitions that involved many students or just played mind games with them-selves. For example, during the resizing images work, students competed to see how many problems they could answer or they created pictures when they completed their percent model in the fraction to percent work (See Fig-ure 5). The next code, constructing knowledge, occurred when students felt compelled to tell the story of some evolution of their mastery of a concept or of how they approached their work. Constructing knowledge is a process, and the code was assigned even when the process wasn’t completely fin-ished but was overtly in progress in the students’ account. The third code, trial and error, occurred when students described a time when they made an important discovery, but it appeared accidental rather than by purpose-ful and thoughtful action. The fourth code in the discovery and learning category was pulling together fragments into coherent knowledge structures.


This code was used when students demonstrated some new understanding, particularly by identifying small bits of information that they now see are connected.

DIsCussIon

Based upon our analysis of learners’ reflections on their experiences with the digital tools during their conceptual knowledge work on propor-tions, we found three affordances of well-designed virtual manipulatives: 1) Technology-based scaffolds; 2) Visibility of the conceptual metaphor; and 3) dynamics of virtual manipulation. These affordances seem to have posi-tive effects on students’ emotional connections to learning and self-regu-latory behaviors which in turn have an impact on discovery and learning as students strive to construct viable mental models. The nature and type of technology-based scaffolds offered by the manipulative are critical to pro-moting the feelings of competency and fun that then encourage persistence to deeper understanding and more complete constructing of knowledge. ad-ditionally, the visibility of the mathematical concept within the virtual rep-resentation will determine the degree to which the manipulative hides or re-veals a mathematical concept. Finally, we find that it’s the dynamic nature of virtual manipulatives that fully enables the discovery and construction of conceptual knowledge.

Technology-based Scaffolds

Scaffolds are not conceived as permanent structures; rather, they are intended to provide structure and guidance to novices while performing within the zone of proximal development. This guidance is only intended to be offered as long as the novice needs assistance. The ultimate goal is that the learner can perform the task without assistance from the expert or without the use of the scaffolds (Vygotsky, 1978). nevertheless, some tech-nology based scaffolds were permanent structures within the digital learning framework. Moreover, Sharma and hannafin (2007), in their review of the literature for technology based scaffolding, noted two types of technology based scaffolds: cognitive and interface. cognitive scaffolds will scaffold a process with supports for the steps involved while interface scaffolds focus on communicating representations.

examining the interview transcripts and evaluating the manipulatives used during both conceptual knowledge building weeks, we found that the


scaffolds used for the Thinking Blocks manipulative were different from those used by any of the other manipulatives. Several scaffolds in the Think-ing Blocks manipulative were permanent structures that frustrated students when they no longer needed them. This manipulative was also the only one which provided cognitive, or procedural, scaffolds (Sharma & hannafin, 2007). These structures frustrated students and based on the students’ per-spective were coded as technology impediments. First, students noticed that the screen was filled with information that was difficult to assimilate and use. although all of the information on the screen is designed to guide stu-dents through the problem solving procedure, the sheer volume of informa-tion made the procedure feel overwhelming. Moreover, the built in scaffolds could not be eliminated. John regretted that “whenever it said ‘build your model’ on the Thinking Blocks, it asks you to put the question mark no mat-ter what… you don’t need a question mark. you already know the answer so it was really annoying. i don’t want to put a question mark.” This question mark was meant to denote the unknown value that the students were work-ing to solve, but to the students, the question mark only seemed to commu-nicate the sense that they were incapable of finding the answer without be-ing led in a lock step fashion. in another example, Marissa noted, “when we were done with the whole entire problem and we already had the answer in there, the little box would pop up and say you need to put it in there again. it was kind of like pointless to have to just do it again and … that was the part i didn’t really like about it.” in this situation, the manipulative asked students to use the model to work out the problem. once students found the answer, they were asked to input the answer again as a response to the origi-nal problem that was provided, which felt repetitive to students. all of these impediments combined to bring a level of frustration to students’ problem solving that actually interfered with the mathematics that it was designed to scaffold.

in contrast, the other manipulatives offered scaffolds that students ac-knowledged as helpful and that illuminated the underlying mathematical concepts, which Sharma and hannafin (2007) call interface scaffolds. Scaf-folds that afford the student control of pacing and meaningful interaction with the mathematical concepts represented in the manipulation lead to the construction of accurate mental models that allow students to generalize understanding in order to solve problems (rapp, 2005). Student dialogue coded as material intelligence and visual modeling both pointed to scaf-folds that allowed students to explore the limits of the mathematical concept and provided important mathematical feedback to students along the way. other scaffolds embedded into the environment did not necessarily provide


mathematical feedback, but offered students clues as to how to proceed or to solve. The up arrow button on the percent grid manipulative and the mul-tiplier button on the resizing images work, for example, represented actions that students could take but did not present the next step in a rigidly-struc-tured procedure to the student. Moreover, some scaffolds provided amplifi-cation of effort to students, particularly those provided in the resizing im-ages and the percent grid manipulatives, which allowed students to explore purposefully and to use their cognitive energy for other learning activities.

clearly, the students in this study preferred scaffolds that enabled ex-ploration and evaluation of the concept under study rather than those which led them through steps to the solution to a problem. interface scaffolds pro-moted exploration and were welcomed by students while procedural scaf-folds were viewed as technology impediments to conceptual knowledge building. ultimately, designing technology-rich learning to include as many scaffolds as necessary to aid in students’ understanding while also not inhib-iting their learning is a challenge that must be more universally met. While Sharma and hannafin (2007) found that learners often lack the ability to se-lect and use appropriate and helpful scaffolds, this study finds that students understood exactly which scaffolds helped them and which ones did not, and were frustrated most when the technology prohibited choice.

Visibility of the Conceptual Metaphor

Student perceptions coded as visual modeling have close connections to lakoff and Johnson’s (1980) work on metaphors which build human thought processes, or mental models. “i see” has long been a metaphori-cal expression for “i understand.” When students stated that a manipulative helped them “see” a mathematical concept, they found that the manipula-tive helped build their mental models and deepen their understanding of the concept. lakoff and Johnson (1999) posit that the mind is embodied and sensory-motor activity defines the ways in which we understand abstract thought. on some level, students’ use of the “seeing is understanding” meta-phor highlights the role sight plays in the way the brain processes informa-tion. indeed, evidence generally supports pictures as better for promoting memory than words (rieber, 2000).

nevertheless, not all virtual manipulatives maximize this affordance. The difference between those that do and don’t lies in the visibility of the representation of the underlying concept. all five virtual manipulatives used in this study represented various concepts related to proportions. The prob-lem with the Thinking Blocks manipulative was that it represented much more than an underlying mathematical concept; indeed, it even represented


more than a mathematical procedure. Thinking Blocks layers typical class-room procedure over the mathematical procedure which is layered over the mathematical concept. Students are asked to read the problem, review what their “[virtual] math teacher says,” work through the problem solving steps, and finally build their models. Several students commented that they “just really got confused with so much” and that when they “had to do all the extra stuff it got confusing.” Because Thinking Blocks is a representation of or metaphor for so much more than the mathematical concept, the visibility of the mathematical conceptual metaphor was dramatically reduced, hidden underneath the process of reading the problem, hearing from the teacher, following the problem solving procedure, and then modeling the concept. This is not to say that we find no value in the scope of this entire process, but merely to point out its limitations for building conceptual knowledge. Findings related to the impact of learning technologies on the acquisi-tion of procedural knowledge are reported elsewhere (Mcleod, 2011).

Dynamics of Virtual Manipulation

a virtual manipulative offers more than just static images that students can “see” and then understand. The interactive, dynamic nature of a virtual manipulative allows students to see changes in the digital display as they manipulate variables or other input. This dynamic affordance allows them to test their own theories and intuitions and get immediate mathematically based feedback. From this feedback, learners can modify their conceptions, generate new theories, and then test their understanding again. in describing the principle of material intelligence, gee (2003) draws upon the galileo’s work with pendulums to point out that “just staring at and playing with pen-dulums in the real world is not actually a good way to ‘discover’ the laws of the pendulum’s movement.” in addition to observing the motion of a swing-ing chandelier, galileo discovered these laws “by using geometry and draw-ing, on paper, arcs and circles and paths of movement along them and figur-ing out their geometrical properties.” From this example, gee argues that “geometry is a powerful tool that stores much knowledge and skill that the learner does not have to invent for him- or herself” (p. 110). in short, ge-ometry has a great deal of material intelligence, as does all of mathematics. The problem is that the novice has little visibility into this powerful capac-ity of mathematics. indeed, much formal learning is required before learners can access the geometrical properties of arcs and circles that galileo used.

however, students found that the dynamic nature of virtual manipula-tives allowed them to play with the concepts as they learned their limits.


Sarah noted that the resizing images virtual manipulative allowed her to “see it [the proportion] being equal” as she manipulated the dimensions of an image. Virtual manipulatives by definition have a visual component; however, it is the dynamic nature of the manipulative that reveals much more about the underlying concept than merely the visual component (Moy-er, Bolyard, & Spikell, 2002). as Madison expressed it, “you could really see, like once you clicked on something you could really see it.” For Madi-son and others, it wasn’t just the image displayed on the screen that aided understanding; the ability to “click on something” and observe what hap-pens allowed them to really “see”—to learn and understand. rapp (2005) identifies this type of interactivity as critical to building accurate mental models, as opposed to faulty or incomplete ones which may result when stu-dents cannot test and retest their conceptions. indeed, we find that the in-teractive nature of some virtual manipulatives helped learners see changes, and that this powerful affordance enabled students to build dynamic mental models as they interacted with virtual manipulatives.

Far beyond mere visual representation of a concept, a virtual manipula-tive can enable the discovery of laws and properties. in another example, Marissa found that the virtual environment afforded trial and error which allowed her to proceed even when she was stuck. in her work with the per-cent grid, she was presented with a problem to model that was greater than 100 percent. Because the grid displayed was a hundreds grid, modeling this problem required more than one grid. in recounting her experience, Marissa said, “i had a big number… i was kind of freaking out about it because i only had like one grid.” however, the virtual environment provided an up arrow button that Marissa decided to try even though she didn’t know what would happen. “i clicked on it [the up arrow button] on accident and i was like relieved … a lot of these grids just popped up and i was kind of sur-prised.” While this action seemed to be accidental to Marissa, the virtual en-vironment was rich with material intelligence that enabled her to try some-thing and ultimately discover what she needed to solve the problem.

indeed virtual manipulatives that offer material intelligence are a pow-erful alternative to physical manipulatives that offer more limited math-ematical feedback. Pea (2004) notes that one of the benefits of technology-enhanced scaffolds is that they focus learners’ attention away from unneces-sary or even misleading activities. For example, base 10 blocks are a com-mon physical manipulative used by mathematics teachers to build conceptu-al knowledge of place value. These blocks have some material intelligence built in to their structure as a student can line up ten of the ones cubes which match the length of the tens rod. however, when these base 10 blocks are


used, many students like to play with them by building towers, an activity that offers students spatial feedback, but no particular feedback about place value. in contrast, students’ play on the base blocks virtual manipulative offers mathematical feedback directly related to place value. For example, when students add more than 10 ones blocks, the number disappears indi-cating that the value that the student has modeled cannot be directly rep-resented in arabic numerals using base 10. it is this dynamic manipulation and mathematical feedback that offers considerably more visibility into the powerful mathematical concepts that students can then access to solve prob-lems in other contexts.

IMPLICaTIons

We have presented the perspective of the learner as s/he navigates building mathematical conceptual knowledge, a stance we share with dew-ey (1902) in connecting the curriculum and the pedagogy with the child. From the analysis of this perspective, we further contend that building dy-namic mental models of a mathematical concept with virtual manipulatives requires that educators evaluate which virtual manipulatives maximize and aide learning. To assist this evaluation, we have developed a Multidimen-sional Virtual Manipulative evaluation (MVMe) Tool (See Figure 6). This tool can be used both to help educators evaluate virtual manipulatives and also to help designers create new virtual manipulatives that are powerful op-tions for students’ learning. The nature of technology scaffolds frames both dimensions of our tool. one dimension of the model is the extent to which the technology scaffolds provide conceptual visibility, while the other di-mension involves the virtual dynamics enabled by the scaffolds.

as previously noted, lakoff and Johnson (1980) offered insights into how metaphors or representations can hide certain aspects of an underly-ing concept and highlight others. Because the students clearly appreciated certain types of scaffolds, one criterion is the extent to which a representa-tion or virtual manipulative hides or highlights the underlying mathemati-cal concepts through the interface scaffolds built into the manipulative. The best virtual manipulatives move beyond just highlighting, to something even more directly clear to the learner: revealing. Thus our scale moves from hiding to highlighting, attributed to lakoff and Johnson (1980), to reveal-ing, which we consider an even more illuminative uncovering of the math-ematical concept under study, therefore, the highest end of this spectrum. This highest level is present when the technology scaffolds allow students to


uncover important mathematical concepts through their interaction with the manipulative, thus refining their mental models of the concept. Such a vir-tual manipulative would likely embody all three of the learning characteris-tics rapp (2005) identifies as contributing to more accurate mental models: cognitive engagement, interactivity, and multimedia presentation. The hori-zontal continuum of this modeled is labeled with hiding, highlighting and revealing to indicate whether the technology scaffolds, such as those which provide visual modeling and material intelligence, are present to assist the learner in building or strengthening their mental models. When comments regarding a manipulative’s conceptual visibility trended toward the reveal-ing end of the scale, we noted an increased frequency of comments coded as constructing knowledge and pulling together fragments. The manipulative with low conceptual visibility (Thinking Blocks) led to comments coded as confusing (see Fig. 7).

Figure 6. Multidimensional Virtual Manipulative Evaluation (MVME) Tool

While the horizontal axis reflects the visibility of the conceptual meta-phor enabled by the technology scaffolds, the vertical axis gauges the vir-tual dynamics of the manipulative’s scaffolds. We developed this scale in response to students’ perception that when using digital tools, learning is easier, faster, more fun, and less tedious than working through problems


on paper. This perception, coded as it’s virtual, pertains to interactions and feedback a virtual manipulative provides. Students found that the dynam-ics in some virtual environments made learning more immediate, engaging, and effortless. in short, the virtual dynamics had an amplifying effect. The y-axis in our MVMe model represents the degree to which the technology scaffolds provided this amplification of effort. Those scaffolds that become technology impediments and thus hinder interaction with the mathematical concepts lie at the low end of the scale. We found that when the technology actually interfered with students’ interactions, students became frustrated and their desire to continue learning using the manipulative suffered. We have labeled this end of the virtual dynamics spectrum as “impeding.” Scaf-folds that were permanent structures and could not be removed when the learner no longer needed them tended to impede learners as they gain mas-tery. Moreover, feedback on learner performance, such as “awesome work!” or “Fantastic!”, lies on the lower end of this scale than mathematically based feedback, such as the number line in the fraction to percent manipula-tive and the multiplier radio button in resizing images, which students found considerably more helpful. indeed, sometimes students felt that the technol-ogy made learning so compelling that they didn’t want to stop exploring, playing, and learning with it. They described this characteristic as addicting because they found themselves unable to resist trying one more problem, confronting a new challenge, and making another discovery. We labeled this end of the spectrum as “compelling” to denote this powerful and irresistible effect. When comments regarding a manipulative’s virtual dynamics trended toward compelling, we noted an increased frequency of comments coded as feelings of competency, persistence, and invented play (see Fig 7).

additionally, we found that a manipulative with low conceptual visibil-ity and impeding virtual dynamics tended to yield feelings of incompetence. We also found that scaffolds with a relatively high amplifying effect, but somewhat low conceptual visibility required leveraging resources outside of the manipulative. Figure 7 shows where these and previously mentioned codes related to emotional connections, self-regulatory behavior, and dis-covery and learning resulting from this study aligned within this model.

Based upon the interview data from the study, we plotted the five vir-tual manipulatives used in this study within the MVMe (see Figure 8). The Thinking Blocks manipulative appears in the lower left quadrant because its procedural scaffolds limited its conceptual visibility and the permanent nature of the scaffolds impeded the dynamics between the student and the underlying concept. in contrast, the resizing images manipulative was the most revealing as it garnered the most comments coded for the powerful


ways in which it revealed the concept: visual modeling and material intel-ligence. Students also noted that this manipulative produced the greatest amplification of effort and found it the most addicting or compelling. The Percent grid and decimals to Percent manipulatives were both highly re-vealing, but somewhat less compelling. The Fractions to Percent manipula-tive was a bit less revealing and compelling, but still much more effective in terms of conceptual visibility and virtual dynamics than Thinking Blocks.

Figure 7. Multidimensional Virtual Manipulative evaluation (MVMe) Tool with codes


Figure 8. Multidimensional Virtual Manipulative evaluation Tool with vir-tual manipulatives

ConCLusIon

clearly, digital tools offer students powerful ways to interact with mathematical concepts, constructing the conceptual knowledge much need-ed in today’s world. like simulations found successful for learning complex scientific processes and concepts, virtual manipulatives can offer students ways to interact and build viable and dynamic mental models, providing a basis with which to apply concepts in novel situations. our Multidimension-al Virtual Manipulative evaluation Tool is intended as a heuristic to help ed-ucators evaluate the available virtual manipulatives when selecting concep-tual knowledge work for students. The tool could also aide designers in de-veloping new virtual manipulatives so that their products maximize student learning. Those virtual manipulatives that offer strong interface scaffolds reveal or highlight the underlying mathematical concept. Virtual manipula-tives that provide dynamic mathematical feedback appropriate to learners’ level of mastery amplify student effort and compel learners to persist to new problems and challenges.

directions for future research include examining additional affordanc-es of virtual manipulatives that allow learners to build dynamic and viable mental models of an underlying mathematical concept. although this study


found that students preferred virtual manipulatives with strong interface scaffolds, studying student perceptions of cognitive scaffolds that aren’t explicitly procedural is another area for future research. Finally, examin-ing whether virtual manipulatives that are highly compelling and revealing have a significantly different impact on student achievement than those with low conceptual visibility and virtual dynamics is another area that would be of interest to study. The more educators understand about why certain scaf-folds or affordances of virtual manipulative produce desired effects while others do not, the more consistently they will be able to design and select effective virtual manipulatives that provide meaningful learning experiences for students.

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