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Integrating internet-based mathematical manipulatives within a learning environment.

The use of manipulatives within a mathematical classroom environment has traditionally been offered through the use of manufactured or teacher-created concrete objects. Manufactured manipulatives are ones such as Cuisenaire rods, color tiles, Unifix cubes, pattern blocks, colored craft sticks, or other related, mass-produced objects. Teacher-created concrete objects consist of cardstock, foam, or other paper templates similar to manufactured manipulatives. These are used to provide tactile-kinesthetic learning activities to enhance mathematics conceptualization. The Information Age offers mathematics educators the opportunity to integrate the use of digital manipulatives similar to manufactured and teacher-created ones using the World Wide Web (WWW or Web) as an innovative medium to expand the learner's conceptual framework of understanding. The availability of such web-based mathematical manipulatives is an ever-expanding possibility that can be integrated into student-centered learning environments. However, it is critical for classroom teachers to develop sound rationale when choosing a technology-based resource over other resources, (i.e., manufactured or teacher-created manipulatives). This article explores necessary rationale elements to create a successful "student-centered" learning environment involving digital manipulatives.


The integration of manipulatives within the mathematical learning environment is not an original concept. Manipulatives have been integrated into the learning environment for numerous years. Preinformation age counting activities involved the use of fingers, which could be looked upon as one of the earliest manipulatives. The primary purpose of the manipulative was to offer a concrete visualization of mathematical concepts that lead towards an understanding of the mathematical concepts as defined by learning objectives.
 All students should be able to reason and communicate
 proficiently in mathematics. They should have knowledge
 of and skill in the use of the vocabulary, forms of
 representation, materials, tools, techniques, and
 intellectual methods of the discipline of mathematics, including
 the ability to define and solve problems with reason, insight,
 inventiveness, and technical proficiency (Connected Mathematics
 Project, 2001).

This statement is directly tied to the technological needs within the dawning of the Information Age. "To justify the expensive and time-consuming task of integrating technology into education, teachers must identify specific contributions that technology can and should make to an improved education system" (Roblyer & Edwards, 2000, p. 12). Within the literature there are conflicting studies regarding the effectiveness of computer-based methods over traditional approaches. However, to impact student learning it is necessary for teachers to consider essential rationale elements to create a successful "student-centered" learning environment involving technology integration, especially digital manipulatives.

Within this framework of proficiency, manipulatives create familiar, concrete, and understandable representations of information that are often "unfamiliar, abstract, and confusing to students" (Burns, 2001a). The appropriate integration of manipulatives within the learning environment ensures the learner's conceptualization of mathematical theories and tasks at a level appropriate to the learner by providing stimulating visual and concrete representations.


Manipulatives have been integrated into learning opportunities for years. "One of the first advocates of 'hands-on learning' was the Swiss educator Johann Heinrich Pestalozzi (1746-1827). Pestalozzi asserted that students need to learn through their senses and through physical activity" (Resnick et al., 1998) and rebelliously struggling for "things before words, concrete before abstract" (Pestalozzi, 1803). However, the introduction of manipulatives within a mathematical environment has exponentially expanded the conceptualization of the theories related to the task by the learner. Numerous positive attributes can be associated with manipulatives, as manipulatives can offer the following aids within the learning environment:

1. Manipulatives help make abstract ideas concrete.

2. Manipulatives lift math off textbook pages.

3. Manipulatives build students' confidence by giving them a way to test and confirm their reasoning.

4. Manipulatives are useful tools for solving problems.

5. Manipulatives make learning math interesting and enjoyable. (Burns, 2001a)

However, the use of manipulatives w4thin the learning environment comes with specific procedural tasks. Bums (200 lb) described several specific "musts" that need to occur to facilitate positive correlations when manipulatives are used. Concurrently, our classroom experiences with manipulatives reinforce these "musts," as follows:

1. The instructor conducts ongoing dialogue with students about why manipulatives help them learn math.

2. Ground rules are set and consistently communicated as students work with manipulatives.

3. Students are encouraged to develop a system for using and storing materials in the classroom under the teacher's direction. Materials managers are appointed to help with these tasks.

4. Time given to students for free exploration provides for more "ontime" behavioral applications as necessary.

5. Manipulatives are a natural for writing assignments, giving students' a writing focus. That is, to describe, illustrate, outline what happened, and so forth.

6. Parents are given opportunities to gain hands-on experiences using the manipulatives.

The aspects discussed refer to the concrete manipulatives that are available within learning environments. Mathematical manipulatives have been integrated into the learning environments to expand the learner's conceptual framework of understanding and to develop a link between theory and concrete explanations of mathematical concepts. But we are at the beginning of a new age, the Information Age that begins the shift from mere concrete manipulatives that can only aid the learner in conceptualizing the more simplistic mathematical theories towards digital manipulatives that offer the learner a conceptualization of more advanced, difficult mathematical theories in a digital arena. Digital manipulatives can be appropriately and successfully integrated into a mathematical learning environment through the use of web-based materials. The use of digital manipulatives provides an interactive environment with immediate feedback to explore indepth mathematical theories that would be difficult to simulate with concrete models. Additionally, younger students are able to "see" (conceptualize) concepts that would normally be regulated to indepth abstract mathematical principles.


A chart outlining "Elements of a Rationale for Using Technology in Education" delineated by Roblyer and Edwards (2000, p. 13) provided an action research framework to gain insights during a recent mathematics professional development session with current in-service teachers. Teachers were from a variety of grade levels, K-5, and 9th, all involved with integrating algebraic concepts in their respective mathematics instruction. Additionally, teachers were culturally diverse as were their schools and students. All participants were female with a large range of teaching service, one year to more than 25 years. Since demographic information was not formally apart of the survey, it is given here only for descriptive purposes. First, a computer-generated survey was created using the free Internet site and e-mailed to participants near the end of the professional development session. Second, participants were provided ample opportunity and resources to visit the web-based National Library of virtual manipulatives (2002) to independently select, view, and critique virtual manipulatives based on possible future classroom implementation using the frameworks espoused in the Roblyer and Edwards chart. Third, after visiting several sites during a four hour timeframe, participants were to respond appropriately to the computer-generated survey. Results were statistically computed and reported as a featured component of the survey.


1. Motivation

* Gaining learner attention

* Engaging the learner through production work

* Increasing perceptions of control

2. Unique instructional capabilities

* Linking learners to information sources

* Helping learners visualize problems and solutions

* Tracking learner progress

* Linking learners to learning tools

3. Support for new instructional approaches

* Cooperative learning

* Shared intelligence

* Problem solving and higher-level skills

4. Increased teacher productivity

* Freeing time to work with students by helping with production and record-keeping tasks

* Providing more accurate information more quickly

* Allowing teachers to produce better-looking more "student friendly" materials more quickly

5. Required skills for an information age

* Technology literacy

* Information literacy

* Visual literacy

(Roblyer & Edwards, 2000, p. 13)


Using the concept chart (Roblyer & Edwards, 2000, page 12) 11 teachers viewed virtual manipulatives web sites and completed a similar assessment. The results are presented in Figure 1.

The distinctive skew in the number of web sites viewed was not directly addressed as a component of this survey. However, one could speculate given the participant diversity characteristics--individual interest levels, grade, and mathematical content instructional level versus related overall numbers of available web-based manipulatives, independent time on task behavior characteristics or lack of technological skills--the results indicate this is a definite area whereby additional study and/or follow-up is needed.

The survey results are designated by distinct areas of interest, as presented by the tabular format of the results. As such, the following areas of interest are addressed:

* Motivation

* Unique Instructional Capabilities

* Support for New Instructional Approaches

* Increased Teacher Productivity

* Required Skills for Information Age

Within each of the tables, the breakdown of the data is offered in percentage format in Table 1, as well as the actual number of people who responded to each level. Therefore, the percentage indicates total respondent ratio and parenthesis indicate actual number.

The classroom teachers note high levels of satisfaction, as related to the motivational factors of digital manipulatives on the Web. It is interesting to note that gaining the learner attention is considered very satisfactory, while the exact opposite response rate is noted when engaging the learner through production work is addressed. Although 91% of the respondents stated that the motivation level for the learner is either satisfactory or very satisfactory, the breakdown between the satisfactory levels and very satisfactory levels as related to gaining attention and engaging the learner is presented as opposite responses. Further, 90% of the responses noted that motivation is pronounced when related to the increased perception of learner control. Therefore, the classroom teacher's perception of learner motivation is significant, as relates to digital manipulatives on the Web (Table 2).

The classroom educators relate unique instructional capabilities to digital manipulatives on the Web within distinct parameters. Linking learners to information sources, helping learners to visualize problems and solutions, and linking learners to learner tools are overwhelmingly supported by the classroom educators' responses. However, of interest is the range of responses as related to the ability to track the learner progress using digital manipulatives on the Web. Although 54% of classroom teachers were either satisfied or very satisfied with tracking learner progress, 45% of classroom teachers were either neutral or found this to be unsatisfactory in nature. Therefore, the unique instructional capabilities are overwhelmingly apparent to classroom teachers but informative elements to support analyzing learner knowledge and understanding, such as tracking learner progress, is not viewed as inherent in all digital manipulatives on the Web (Table 3).

Classroom teachers view digital manipulatives on the Web as supportive of new instructional approaches. Cooperative learning and shared intelligences are noted by 73% and 90%, respectively, as satisfactory or very satisfactory; yet 27% and 10%, respectively, view digital manipulatives as being either neutral or unsatisfactory as related to support for new instructional approaches. Of interest is the strong response of the classroom teachers as related to problem solving and higher level skills, as 100% of respondents note that digital manipulatives on the Web are satisfactory or very satisfactory as related to problem solving and higher level skills development support structures for new instructional approaches. Therefore, the classroom educators may view digital manipulatives on the Web as bridging the gap between basic knowledge-level processes and enhancing the learner's abilities to grasp the conceptual understanding that leads towards higher order thinking skills (Table 4).

The classroom teachers were not as supportive of digital manipulatives on the Web as related to their ability to increase teacher productivity. Elements of time, accuracy of information, and the production of student-friendly materials, although noted as at least satisfactory or higher by the majority of classroom teachers responding to this survey, the questions related to increased teacher productivity received responses that were at lower satisfaction levels than the other areas of interest within the survey. This skew may be directly related to the classroom teachers level of comfort as related to the integration of technology into the instructional design of the course, the classroom teachers level of comfort as related to the integration of technology into the classroom learning environment, or to the classroom teachers personal belief in his or her own level of technological expertise (Table 5).

Overwhelmingly, the classroom teachers responded that they agreed me required skills for the Information Age were related to technological literacy, information literacy and visual literacy. The respondents may very well view the classroom learners must be literate within the areas of technology, as well as the analysis of visual, auditory, and written information if they are to be successful through out the reign of the Information Age. Therefore, it is imperative that learners are able to extend and augment their higher order thinking skills to meet their future needs.

The classroom teachers also offered several areas of concern, related to unsatisfactory ratings presented in the previous tabular format.

* inability to track learner progress and level of understanding, for either classroom teacher or learner;

* web sites were either under construction or not user friendly;

* unclear directions for use, or unclear instructions;

* inability of learner to obtain direct feedback as related to use; and

* teacher can not analyze student's strengths or areas of concern, for purposes of strengthening skills or individualizing lesson selections.

A major component of the action research led to vigorous discussion concerning "Why choose one manipulative over the other?" It is reasonable to present that the overarching goal of classroom technology implementation primary's purpose should be to create and enhance student-centered learning environments. Then, the question "Will virtual manipulatives do this better than concrete manipulatives?" Therefore, it is critical that educators are aware of "the good and the bad" regarding manipulative innovations and all elements must be considered when choosing appropriate manipulative integration. Participants in this study indicated their willingness to incorporate digital manipulatives, however additional professional development is necessary to fully incorporate "the good" from web-based manipulatives and to decrease effective classroom implementation from being hampered by "the bad."


Web-based manipulatives offer a creative, useful variety to the learning environment. These interactive materials enhance the knowledge and understanding of learners, while creating a conceptual understanding of mathematical theories beyond the mere formulaic models of traditional mathematical coursework. "These new manipulatives--with computational power embedded inside--are designed to expand the range of concepts that children can explore through direct manipulation, enabling children to learn concepts that were previously considered 'too advanced' for children" (Resnick et al., 1998).

As examples of digital mathematical manipulatives available on the Web, GeoComputer (Riverdeep Interactive Learning Limited, 2001) offers the ability to create, flip, and rotate shapes to make colorful designs while simulating the concepts of reflections, translations, and rotations. Students are also able to have interactive experiences with basic principles of geometry and measurements. Number sense concepts including fractional computations are enhanced by using pattern blocks and other virtual manipulatives embedded in web-based applets developed specifically for use within a mathematical learning environment. The use of such manipulatives within a mathematically appropriate learning environment enhances the learner's conceptual understanding of material that would previously be considered too advanced or inappropriate.

Through the appropriate and successful integration of the mathematical manipulatives within a web-enhanced learning environment, "children, by playing and building with these new manipulatives, can gain a deeper understanding of how dynamic systems behave" (Resnick et al., 1998). Further,
 Such explorations would not be possible with traditional
 (non-computational) manipulative materials. Computation and
 communication capabilities play a critical role: they enable
 physical objects to move, sense and interact with one
 another--and, as a result, make systems-related concepts
 more salient to (and manipulable by) children.
 (Resnick et al., 1998)

 Numerous web-based mathematical manipulatives are available
 for appropriate and successful integration within a mathematical
 learning environment. However, the engagement of the learner
 in innovative ways of thinking and learning about mathematical
 concepts is the focus of the exercise. The National Library for
 Virtual Manipulatives for Interactive Mathematics houses a
 vast collection of interactive web-based manipulatives and
 games hosted by the Math Forum organization at


Web-based mathematical manipulatives are available for integration into the learning environment. However, thoughtful consideration must be given to the instructional design of the course and the specific learning objectives for each module of instruction. The focus of web-based manipulatives is to enhance the learner's understanding of advanced theories and levels of understanding; "Our primary goal is not to help users accomplish some task faster or more effectively, but rather to engage them in new ways of thinking. In short, we are interested in Things That Think only if they also serve as Things To Think With" (Resnick et al., 1998). The web-based manipulatives offer the computational abilities that aid in the communication of advanced concepts and theories to the learner. The focus is on the learner and the conceptual framework of understanding that is created due to the appropriate use of digital, web-based mathematical manipulatives.
Table 1


 1 2
 Very Unsatisfactory

1. Gaining learner attention 0% (0) 9% (1)
2. Engaging the learner
 through production work 0% (0) 9% (1)
3. Increasing perceptions
 of control 0% (0) 9% (1)

 3 4
 No Opinion/ Satisfactory

1. Gaining learner attention 0% (0) 36% (4)
2. Engaging the learner
 through production work 0% (0) 55% (6)
3. Increasing perceptions
 of control 0% (0) 45% (5)


1. Gaining learner attention 55% (6)
2. Engaging the learner
 through production work 36% (4)
3. Increasing perceptions
 of control 45% (5)

Table 2

Unique Instructional Capabilities

 1 2 3
 Very Unsatisfactory No
 unsatisfactory Opinion/

1. Linking learners
 to information
 sources 0%(0) 9%(1) 0%(0)
2. Helping learners
 visualize problems
 and solutions 0%(0) 0%(0) 9%(1)
3. Tracking learner
 progress 0%(0) 27%(3) 18%(2)
4. Linking learners
 to learner tools 0%(0) 9%(1) 0%(0)

 4 5
 Satisfactory Very

1. Linking learners
 to information
 sources 73%(8) 18%(2)
2. Helping learners
 visualize problems 27%(3) 64%(7)
 and solutions
3. Tracking learner 36%(4) 18%(2)
4. Linking learners
 to learner tools 82%(9) 9%(1)

Table 3

Support for New Instructional Approaches

 1 2 3
 Very Unsatisfactory No
 unsatisfactory Opinion/

1. Cooperative learning 0%(0) 9%(1) 18%(2)
2. Shared intelligence 0%(0) 10%(1) 0%(0)
3. Problem solving and
 higher level skills 0%(0) 0%(0) 0%(0)

 4 5
 Satisfactory Very

1. Cooperative learning 55%(6) 18%(2)
2. Shared intelligence 80%(8) 10%(1)
3. Problem solving and
 higher level skills 36%(4) 64%(7)

Table 4

Increased Teacher Productivity

 1 2 3
 Very Unsatisfactory No
 unsatisfactory Opinion/

1. Freeing time to work
 with students by
 helping with
 production and
 record keeping tasks 0%(0) 9%(1) 18%(2)
2. Providing more
 accurate information
 more quickly 0%(0) 9%(1) 18%(2)
3. Allowing teachers to
 produce better-
 looking more
 materials more
 quickly 0%(0) 9%(1) 27%(3)

 4 5
 Satisfactory Very

1. Freeing time to work
 with students by
 helping with
 production and
 record keeping tasks 45%(5) 27%(3)
2. Providing more
 accurate information
 more quickly 45%(5) 27%(3)
3. Allowing teachers to
 produce better-
 looking more
 materials more
 quickly 45%(5) 18%(2)

Table 5

Required Skills for Information Age

 1 2 3
 Very Unsatisfactory No
 unsatisfactory Opinion/

1. Technology literacy 0%(0) 0%(0) 18%(2)
2. Information literacy 0%(0) 0%(0) 10%(1)
3. Visual literacy 0%(0) 0%(0) 0%(0)

 4 5
 Satisfactory Very

1. Technology literacy 64%(7) 18%(2)
2. Information literacy 60%(6) 30%(3)
3. Visual literacy 27%(3) 73%(8)

 How many virtual
 web sites
 did you view during Number Responses
 this session? of Response Ration

5 or less 2 18%
6-10 5 45%
11 or more 4 36%
 Total 11 100%


Burns, M. (2001a). A letter to parents. Retrieved on February 9, 2001 from: /letter.htm

Burns, M. (2001b). 7 musts for using manipulatives. Retrieved on February 9, 2001 from:

Connected Mathematics Project (2001). Overarching goals of CMP. Retrieved on February 9, 2001 from:

National Library of Virtual Manipulatives (2002). The math forum: Internet mathematics library. Retrieved on December 19, 2002 from: http://

Pestalozzi, H. (1803). ABC der anschauung, oder anschauungs-lehre der massverhaltnisse. Tubingen, Germany: J.G. Cotta.

Resnick, M., Martin, F., Berg, R., Borovoy, R., Colella, V., Kramer, K. and Silverman, B. (1998). Digital manipulatives: New toys to think with. Retrieved on February 9, 2001 from:

Riverdeep Interactive Learning Limited (2001). GeoComputer [Computer software]. Retrieved on March 25, 2001 from:

Roblyer, M. D., & Edwards, J. (2000). Integration educational technology into teaching. Upper Saddle River, NJ: Prentice Hall.


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Author:Brown, Evelyn
Publication:Journal of Computers in Mathematics and Science Teaching
Date:Jun 22, 2003
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