Technology in the mathematics classroom: conceptual orientation.

        Teaching mathematics through multiple representation frameworks
        presents an open-ended but complex learning structure. This
        article elaborates on teaching mathematics for conceptual
        understanding in the Information and Communication Technologies
        (ICT)-based environment, by exploring conceptual understanding
        through multiple representations and addressing some effects of
        the cognitive tools to teaching and learning mathematics in such
        environments.

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Jl. of Computers in Mathematics and Science Teaching (2003) 22(4), 381-399

... the defining characteristic of knowledge workers is that they are themselves changed by the information they process. (Kidd, 1994, p. 186)

Mathematics curriculum development in ICT-based environment is a complex task requiring ongoing feedback and refinement. (Dreyfus, 2002). For many teachers understandings of key mathematical concepts and phenomena are grounded in the ways they have learned them before emerging technologies have been accessible to them. And even if they had some exposure to ICT integration, they are now teaching new generations, born and educated surrounded by emerging technologies. Bridging this digital divide requires revisiting roles that representations, and translations among representations, play in mathematical learning and problem solving (Alagic, & Langrall, 2001).

This article brings together teaching mathematics for conceptual understanding in the technology-based environment, by considering mathematics teaching and ICT in general terms, exploring conceptual understanding through multiple representations and addressing some effects of the cognitive tools to teaching and learning mathematics. It is grounded in existing research and reflects the author's experiences from ongoing collaborations with teachers learning how and integrating ICT in their teaching and learning of mathematics (Alagic, 2002a, 2002b, 2002c; Alagic & Langrall, 2002).

ICT INTEGRATION: MATHEMATICS

... what theoretical reflection we need if we want to really help teachers to adequately use technological tools ... (LaGrange, 2002, p. 15)

The National Council of Teachers of Mathematics (NCTM, 2000) identified the Technology Principle as one of six principles of high quality mathematics education. It stated: "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (p. 24).

Successful technology integration in mathematics curriculum/classroom depends on many factors such as mathematics teachers, mathematical connections, and appropriate integration approach.

Mathematics Teachers

The research on use of technology reveals the complexities of the interplay of technology and teaching in the learning of mathematics. One thing remains constant. It is ultimately the mathematics teachers, not the technological tools that continue to be the key to the success of the mathematical learning environment. Their own perspective on the nature of mathematics, on the potential of the technology, and the training that they receive determine their effectiveness in integration of the technology in mathematics teaching and learning. (Garofalo, Drier, Harper, Timmerman, & Shockey, 2000; Kaput, 1992; NCTM, 2000). The external world is interpreted according to one's own experiences, beliefs, and knowledge and therefore, each person visualizes the external world at least slightly differently. As learners, teachers are able to comprehend a variety of interpretations and use them. But they cannot map their own interpretations of the world directly onto their students, because they do not share a set of common experiences and understandings. Some students have an exceptional fluency with technology tools, and some are very far from it, challenging their teachers even further. Yet another essential piece of teacher knowledge for building a technology-based learning environment is how to teach for transfer. Teaching practices congruent with a metacognitive approach to learning include those that focus on sense making, self-assessment, and reflection on what worked and what needs improving. These practices have been shown to increase the degree to which students transfer their learning to new settings and events (Schoenfeld, 1992).

Mathematical Connections

The use of ICT in mathematics teaching should support and facilitate conceptual development, exploration, reasoning, and problem-solving, as described by the NCTM (2000). Technology enables users to explore topics in more depth and in more interactive ways. By removing computational constraints it makes accessible the study of mathematics topics that were previously impractical, such as recursion and regression. Technology-augmented activities should take advantage of these capabilities of technology, and hence should extend beyond or significantly enhance what could be done without technology. These activities can facilitate mathematical connections in a variety of ways as: (a) linking multiple representations, (b) interconnecting and integrating mathematical topics and ideas, and (c) connecting mathematics to real-world phenomena. Therefore a collection of representations for a certain concept is richer, providing more access points for diverse learners at different levels of their understandings.

Studies in complex domains such as solving science problems (Heller & Reif, 1984) have suggested that conceptual understanding is associated with connections--connections between science concepts and everyday life and connections among the different concepts in a discipline. Someone who is good at solving transfer problems does not randomly connect concepts (which might occur when using memorized algorithms to solve problems) but rather integrates the concepts into a well-structured knowledge base in the context.

Integration Guidelines ICT

What further factors determine the success of the use of ICT in learning mathematics? According to Flick and Bell (2000), the following interconnected guidelines provide the essential ideas for strengthening mathematics instruction while integrating technology. ICT should (a) be introduced in the context of mathematics content, (b) address worthwhile mathematics with appropriate pedagogy, and (c) make scientific views more accessible. Furthermore, ICT instruction in mathematics should take advantage of the unique features of technology and develop students' understanding of the relationship between technology and mathematics. Preparing teachers to integrate technology appropriately, with above guidelines, requires professional development that focuses on both conceptual and pedagogical issues as well as ongoing support.

TEACHING MATHEMATICS: CONCEPTUAL ORIENTATION

Students do not necessarily interpret results in the manner that is obvious (to the mathematics teacher). (Dreyfus, 2002, p. 26)

When students attain understanding, what have they achieved? What students do, in response to the questions that put understanding into action, show their levels of understanding. Students might be able to solve an equation, but if there is no understanding of where the equation is coming from or where and how to use it, they may just be using a memorized skill that is going to be useful only for that type of equation, nothing more. Understanding a concept or a topic of study is being able to carry out a variety of actions or performances with the topic by the ways of critical thinking: explaining, applying, generalizing, representing in new ways, making analogies and metaphors, and so on. It is being able to take knowledge and use it in new ways (Perkins, 1993). Of course, teaching for understanding can be promoted with traditional materials, but new media such as a variety of existing software and the Internet enhances the collection of educational tools. For example, ICT tools such as simulation and other types of visualization can deepen understanding by making semi-abstract concepts visible and expand students' means of expression.

Conceptual Orientation

Teachers set the classroom atmosphere for the types of inquiry in which students engage, whether with the teacher or among themselves or individually. The images and beliefs that they have about the nature of mathematics influence the way they are teaching. These images reveal themselves in two main orientations: calculational and conceptual (Thompson & Thompson, 1994; Thompson, Philipp, Thompson, & Boyd, 1994). Students also have varying degrees of conceptual or calculational orientations to mathematics. Those who have adapted to calculationally-oriented instruction will expect that the classroom discussions will be about getting answers. They will not only have difficulty focusing on both their and others' reasoning, they may also consider such a focus as being irrelevant to their images of what mathematics is about. At the other end, students who have adopted a conceptual orientation will likely engage in longer, more meaningful discussions (Cobb, Wood, & Yackel, 1991). A conceptual approach aims for students to solve problems by working from their own understandings. But it is not an orientation that can be created easily, and once created, easily maintained (Thompson, et al., 1994; von Glasersfeld, 1988; Wood, Cobb, & Yackel, 1991).

Facilitating Students' Thinking

Students who memorize facts or procedures without understanding often are not sure when or how to use what they know and such learning is often quite fragile. The interplay of factual knowledge, procedural, and conceptual understanding makes more sense and is easier to remember and apply when students connect new knowledge to existing knowledge in meaningful ways. Learning with understanding also makes subsequent learning easier. For teachers, it is neither sufficient to know how to solve the problem with which the students may be grappling, nor is it sufficient to know several solution methods. To be able to facilitate students' thinking in productive ways, teachers need to have an image of students' thinking as they develop these ideas. Any teacher can begin building this image by encouraging students to reason and express him or herself accordingly, by listening to their reasoning, respecting it, and asking students to do likewise. Creating effective educational use of new technologies in this context is not a simple task. Sometimes the goals are clear but the potential contribution of the new technology is not. Other times the technology tool seems attractive, but exactly how to integrate it is uncertain (Schoenfeld, 1992; National Research Council [NRC], 2000).

CONCEPTUAL UNDERSTANDING THROUGH MULTIPLE REPRESENTATIONS

Things before words, concrete before abstract. (Johann Heinrich Pestalozzi, 1803)

Multiple Representations

How is the information processed once it has been perceived and has entered the cognitive system? The answer to this question depends on the way information is represented in the system. Understanding this constitutes an important piece of teacher's knowledge for designing an effective technology-enhanced mathematics learning environment. Some types of knowledge representation preserve much of the structure of the original perceptual experience. Those are called perception-based representations. Our minds have an ability to best remember what is most important. Meaning-based representations are abstracted from the perceptual details and incorporate the meaning of the experience (Anderson, 2000). Representing knowledge requires students to think in meaningful ways to represent what they know, actively engage in creating knowledge that reflects their understanding of mathematical ideas rather than absorbing predetermined presentations of knowledge. Students learn and retain the most from thinking in critical and creative ways and some of the best thinking results when they try to represent what they know.

The same concept of the same mathematical phenomena may be captured with different representations and different modes of the same representation. If we think about conceptual understanding of a mathematical idea through a congruence relation on the collection of representations, we can begin to grasp how conceptual understanding develops in the process of translations among various representations. Figure 1 depicts a part of that complex idea of interconnectedness among some modes of representation(s) (Alagic, 2002a, 2002b).

[FIGURE 1 OMITTED]

Well-known distinction among representation types/modes is the concrete-abstract dimension. Piaget and other developmental psychologists picture abstraction as layered on top of multiple concrete representations. In ICT-based environment, the variety of representations between the concrete and abstract (levels) is getting broader. The same holds for translations among these representations. Lesgold (1998) gave an example of a computer program as an object that has some properties of both the abstract and concrete. Learning for understanding requires not only acquiring the ability to represent the environment/concept in multiple ways but also finding ways to interconnect the multiple representations and use them in combinations to accomplish cognitive goals.

Cox and Brna (1995) have shown that when people are learning complicated new ideas it helps to interact with various representations such as diagrams, graphs, models, and animations. If the learner can integrate information from representations with different formats then they often acquire a deeper understanding of the concept. If the learner fails to make the connection between the different kinds of information, then many of the benefits that multiple representations provide will not occur and it can even inhibit learning (Ainsworth, Bibby, & Wood, 2002). Multiple representations have been linked with greater flexibility in student thinking (Ohlsson, 1987, as cited in Leinhardt, Zaslavsky, & Stein, 1990). Such flexibility, in turn, has also been associated with better transfer of learning into the ill-structured domains typical of the real world (Spiro, Vispoel, Schmitz, Samarapungavan, & Boerger, 1987). Instructional representations provide a temporary context for conceptualizing student understanding. By blending familiarity and challenge to stimulate development, they are analogous to Papert's (1980) "microworlds" and Schoenfeld's (as cited in Leinhardt, et al., 1990) "reference worlds."

External and Internal Representations

Representation standard (NCTM, 2000) recognizes representations as essential elements in (a) building students' conceptual understanding of mathematical concepts and relationships, and (b) communicating mathematical arguments and understandings, to one's self and to others. The term representation, in the same standard, refers both to the process of "capturing" representation and to the product--the form itself; a tool for thinking and a finished product. Furthermore, the term applies to processes and products that are observable externally (external representations) as well as to those that occur "internally" (internal representations, mental models). There are many ways of knowing through representations: examples, models, demonstrations, simulations, analogies, and metaphors.

The use of ICT tools enhance external representations--such as tables, graphs, and text; models, visualizations, and diagrams--and reinforces their focal role in teaching/learning. Coinciding with this, the effect of external representations is of interest both for theories of learning as well as for instructional technology design, development, and use. A broader question is how specific external representations and interactions among representations in the technology-based environment affect the way we teach mathematics (Alagic, 2002a, 2002b).

By engaging their mental processes, learners translate among representations of the same kind, whether internal or external, and also among internal and external representations, constructing their own conceptual understanding of a phenomenon. Figure 2 attempts to illustrate ongoing interactions between learners' internal and external representations, as well as translations among representations of the same kind. Well designed ICT-based instructional representations of mathematical inquiry/problem solving activities provide additional opportunities for these interactions/transfers and the teacher's facilitation of such processes.

[FIGURE 2 OMITTED]

Conceptual Understanding

Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, and/or representations. Students demonstrate conceptual understanding in mathematics when they provide evidence that they can: (a) recognize, and generate examples of concepts, (b) use and interrelate varied representations of concepts through different tools, (c) know and apply facts and definitions, (d) identify and apply principles, (e) compare, contrast, and integrate related concepts and principles, and (f) recognize, interpret, and apply symbols and terms used to represent concepts (NCTM, 2000).

Students need opportunities to construct, refine, and use their own representations as tools to support their learning and doing mathematics for conceptual understanding. Teachers can gain valuable insight into students' ways of interpreting and thinking about mathematics by observing students' representations. They can facilitate building bridges from students' personal representation to more conventional ones. Also, different representations often clarify different aspects of a complex concept or relationship. Therefore, teachers who can represent a concept in a variety of ways provide a vehicle for all students to grasp the concept and make connections to previous and future understandings. Teachers' use of representations can supply a rich repertoire of access points for accommodating the different ways students have been found to learn, provided such representations are already familiar to students (Janvier, 1987). Teachers can facilitate the understanding of mathematical concepts through, (a) the use of multiple representations and its cognitive modeling (i.e., environment plus embedded model), (b) the construction and use of mental models, (c) the mapping of mental models to representations (i.e., cognitive modeling), and the translation (i.e., rerepresentation) among representations.

Research shows that many students have difficulty connecting varied representations of the same concept. For example, connecting the verbal, graphical, numerical and algebraic representations of mathematical functions is sometimes a very complex task (e.g., Leinhardt et al., 1990). Appropriate use of technology can be effective in helping students make such connections (e.g., connecting tabulated data to graphs and curves of best fit). "We, as mathematics educators, should make the best use of multiple representations, especially those enhanced by the use of technology, encourage and help our students to apply multiple approaches to mathematical problem solving and engage them in creative thinking" (Jiang & McClintock, 2000, p. 19).

Learning through Representations

Processes of learning and transfer are central to understanding how people develop important competencies. It is especially important to understand the kinds of learning experiences that lead to transfer, defined as ability to extend what has been learned in one context to new contexts (NRC, 2000). It goes to understanding defined as flexible performance (Perkins, 1993), and more specifically to conceptual understandings. The teaching for conceptual understanding through multiple representation frameworks that include ICT-based representations, presents a simple open-ended but powerful structure for developing that flexible performance. Since every transfer can be considered as a new representation, the principle that people learn by using what they know to establish their new understandings can be reformulated into learning occurs through representations.

TEACHING MATHEMATICS FOR CONCEPTUAL UNDERSTANDING: COGNITIVE TOOLS

Students do not necessarily do what seems natural (to the instructional designer). (Dreyfus, 2002, p. 27)

Students starting to use ICT, especially if they are using calculators with problems involving more than one operation, usually need some help (Wiebe, 1989). To use a calculator, a real-world problem must be restated in a symbolic form, as a symbolic representation. The fundamental problem here, as in other examples of learning to use ICT tools, is developing an understanding and becoming proficient in abstract representational systems that convey concepts. Students have to understand both mathematical representations and how to translate information depicted in question format leading to the final solution. The step from natural language to symbolic notation may be very challenging. Another example would be using spreadsheets to graph tabular data. How may understanding of the link between these two representations be accomplished? An intermediate representation would be useful to promote understanding. It is this aspect that the teacher facilitates, which is crucial and it is therefore the principal cognitive model within the environment for developing conceptual understanding.

Representations as Tools

Consideration for representations as a tool for meaningful learning has been given by a number of researchers (e.g., Kaput, 1987; Greeno & Hall, 1997). Greeno and Hall emphasized the importance of students' exploration in selecting representations when building their conceptual understanding of mathematical ideas. In the presence of technology, "the ability of students to operate within and between different representations of the same concept or problem setting is fundamental in effectively applying technology to enhance mathematics learning" (Demana & Waits, 1990, p. 218). With multiple contexts, students are more likely to abstract the relevant features of the concepts and develop a more flexible representation of knowledge. Research has also shown that developing an assortment of representations enables learners to think flexibly about complex domains and enhance their conceptual understanding orientation. Careful consideration for selecting representations that would facilitate students learning is always necessary. For example, which representations best: (a) promote conceptual understanding, (b) generalize to higher-order thinking, (c) facilitate finding solutions, and (d) reflect the learning style of the students'? Which representation is best for a given type of technology? What representations does your student prefer? Which one do you prefer?

Appropriate use of technology can make it easier for teachers and students to bring together multiple representations through intermediate representations or explicating links among different representations of some mathematical concepts. Technology "blurs some of the artificial separations among some topics in algebra, geometry, and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics" (NCTM, 2000, p. 26). Many school mathematics topics can be used to model and resolve situations arising in the physical, biological, environmental, social, and managerial sciences. Learners, both teachers and students, can use technology to facilitate such applications by providing ready access to real data and information, by making the inclusion of mathematics topics useful for applications and more practical.

Cognitive Tools

Jonassen (1992) described them as "generalizable tools that can facilitate cognitive processing" (p. 2), devices that support, guide, and extend the thinking processes of their users. Cognitive tools can make it easier for learners to process information, but their main "goal is to make effective use of the mental efforts of the learner" (Jonassen & Reeves, 1996, p. 10). These are tools that are used to engage learners in meaningful cognitive processing of information. They are knowledge generation and facilitation tools that can be applied to a variety of subject matter domains. These cognitive tools include specially designed knowledge construction tools, such as semantic networking tools and micro worlds for mediating learning. Jonassen and Reeves (1996) asserted that well designed cognitive tools should: (a) represent knowledge, (b) be generalizable (represent knowledge in different content areas), (c) engage the learner in critical thinking about the subject, (d) assist learners to acquire skills that are generalizable and transferable to other contexts, (e) be both simple and powerful to encourage deeper thinking and processing of information, and (f) be relatively easy to learn; the mental effort needed to learn the software should not exceed the benefits.

Salomon, Perkins, and Globerson (1991) best expressed the primary distinction between traditional learning applications of technologies and their use as cognitive tools as the effects of ICT versus the effects with ICT. When students work with information and communication technology, instead of being controlled by it, they enhance the capabilities of the tool, and the ICT enhances their thinking and learning. The result of this kind of intellectual partnership with the computer is that "the appropriate role for a computer system is not that of a teacher /expert, but rather, that of a mind-extension cognitive tool" (Derry & LaJoie, 1993, p. 5).

Cognitive Tools: An Example

Spreadsheets as cognitive tools for producing representations have become popular tools for exploring mathematical phenomena and building conceptual understanding of mathematical ideas. A spreadsheet program can file information, usually numerical, into a particular location (the cell). This enables information to be accessed, retrieved efficiently and analyzed as necessary. Most importantly, spreadsheets support calculation functions. The numerical contents of any combination of cells can be mathematically related in just about any way the user wishes. Cells can be added, multiplied, and factored in any combination of ways. Most spreadsheets provide mathematical functions such as logarithms and trigonometric functions. It also includes sophisticated tools for generating tables and graphs. Calculating values in a spreadsheet requires that the user identify relationships and patterns among the data that he or she wants to represent in the spreadsheet. Next, those relationships must be modeled mathematically, using rules to describe the relationships in the model. Building spreadsheets requires abstract reasoning by the user, thereby matching one of the important goals of cognitive tools. The combined calculational and graphical capabilities of a spreadsheet provide a context to engage students in analyzing and connecting multiple representations.

Spreadsheets also support problem-solving activities. Given a problem situation with complex quantitative relationships, spreadsheets can be used to represent those relationships. The what if? thinking that is supported by spreadsheets is essential to decision analysis. Such reasoning requires learners to consider implications of conditions or options, thereby engaging higher order thinking. Identifying values and developing formulas to interrelate them in spreadsheets enhance learners' understanding of the algorithms used to compare them and also the mathematical models used to describe content domains.

What Research Supports the Use of Spreadsheets as Cognitive Tools?

In one of the rare studies investigating spreadsheets as cognitive tools, Sutherland and Rojano (1993) were interested in how prealgebra students could use spreadsheets to represent and solve algebra problems. This study was conducted simultaneously in Britain and Mexico and took place over a 5-month period. During that time, students moved from a strict cause-effect local numerical notion of algebraic relationships to general rule-governed relationships that could be symbolized both in the spreadsheet and in algebraic notation. A few other studies were investigating the effects of different computer-based tutorial treatments, and spreadsheets happened to be the content or skill being learned. Baxter and Oatley (1991), while comparing the effectiveness of two different spreadsheet packages, discovered, not surprisingly, the users' prior experience level with spreadsheets was far more important to learning than the usability of the software package. These studies provided few insights about the effectiveness of spreadsheets as cognitive tools.

From an educator's point of view, based on teaching practice, teachers understanding, and the use of spreadsheets can extend from kindergarten up, through all the grade levels, from a simple 4-pane magic square, through organizing students' records/grades, to the complex applications or data analysis supported by spreadsheets software as cognitive tool.

TEACHING MATHEMATICS FOR CONCEPTUAL UNDERSTANDING: LEARNERS

Students do not necessarily see on the screen what is "evident" (to the software designer and the teacher) (Dreyfus, 2002, p. 23)

Mathematics teachers, in an early stage of knowledge development for designing an ICT-enhanced learning environment, might entail an understanding that some technology-based cognitive tools are better than others for portraying a mathematical idea. With the increased availability of these tools providing students with easy access to a range of representations, explicit instruction in linking among these, as well as how certain representations convey mathematical content more efficiently than others is being seen as a crucial aspect of mathematics education (Kaput, 1987; Dreyfus, 2002). Another early stage of knowledge development for designing an ICT-enhanced learning environment might involve the awareness that students, faced with multiple representations for the same concept, often learn to use the syntactic/formal rules of mathematics, but without developing a deeper sense of the underlying concept being represented. Premature or inappropriate use of technology-based or any other representations can cause frustration and misconceptions in learners and place undue focus on the representation at the expense of the target concept. Effective representations must be based on students' own understandings, drawings, and codes. Yet another stage of development might include the awareness that, with today's emerging technologies, the very nature of both the problems that can be solved and methods used in the process are changing. Performing calculations; collecting, analyzing, and representing numeric information; creating and using models and simulations; representational scaffolding higher levels of abstraction, and solving problems with mathematical premises are just some of the possibilities for the hands-on, minds-on learning experiences fostered through today's interactive technology applications. They empower students with a level of mathematical power they cannot achieve without technology and, if used appropriately, have a great potential for stimulating higher order thinking when freed from the mechanics of calculating. A more advanced stage could involve the understanding of the dialectic between perception and conceptualization. For example, accessing geometrical knowledge is more often presented as resulting from the ability to rely efficiently both on spatial and geometrical competencies, as opposed to resulting from rejection of some perceptive apprehension of geometrical objects. Fostering the dialectic interplay between these differing competencies, in the dynamic geometry learning environment, leads to the development of geometrical expertise (Dufour-Janvier, Bednarz, & Belanger, 1987; Alagic & Langrall, 2001; Alagic & Langrall, 2002).

ICT should not be used to carry out procedures without appropriate understanding of both mathematical and technological concepts involved (e.g., inserting rote formulas into spreadsheets). Nor should it be used in ways that can distract from the underlying mathematics. Another way to prevent ICT use from compromising mathematics is to encourage users to connect their experiential findings to more formal aspects of mathematics. ICT should not influence students to take things at face value.

FUTURE INQUIRY

... the defining characteristic of knowledge workers is that they are themselves changed by the information they process. Kidd (1994, p. 186)

To encourage learners' inquiry, instructional design in an ICT-based environment has to take into consideration opportunities that multiple representations provide. Scaffolding through these representations may bridge the gap between concrete and abstract representations of a mathematical concept and reach a reflective abstraction in a variety of new ways. A potentially abstract problem statement or an abstract arithmetic expression may be illustrated using the visualization, a powerful cognitive tool. These new representations are essential components of a learning environment in which learners are required to think harder about the topic being studied and to generate thinking that would be impossible without these new representations. This creative thinking supports learners in creating and maintaining their conceptual understanding. In short, the real power of technologies to improve education will only be realized when students actively use them as cognitive tools for building their own representations and translations among them to support their conceptual orientation in teaching/learning.

Teachers are aware of current changes and are involved in the processes of these changes in their schools. Many teachers are disillusioned by their experience with technology integration so far. High-quality training, sufficient resources and awareness of necessary change are some of the critical factors necessary to regain the trust (Cafolla & Knee, 1995). To build confidence, teachers need successful experiences and ongoing pedagogical and technological support when integrating technology into their curriculum (Byrom, 1997). Mathematics teachers need opportunities to experience and do mathematics in environments supported by diverse technologies. Understanding, using, and appreciating mathematics are essential components of the development of mathematical power. Empowering teachers through the use of technology in mathematics exploration, open-ended problem-solving, interpreting mathematics, developing conceptual understandings and communicating about mathematics is in the heart of professional development and teacher education (Schoenfeld, 1992; NRC, 2000; Dreyfus, 2002). Teachers need to experience and learn in depth, how conceptual understanding emerges in a technologically based environment to better understand "the conditions under which their students will be able to see on the screen what is evident to the software designer." Teachers are also expected to implement activities in ways that seemed natural to the instructional designer and think in a way that is logical to the mathematician (Dreyfus, 2002, p. 30). Many other questions of interest await researchers with mathematical, ICT, and educational backgrounds. Some of those, in the center of interest for the author of this article and her research-partners, are: How are specific ICT-based representations of mathematical phenomena changing the way we teach mathematics? Because technology-based representations can make conventional representations dynamic and interactive, do they provide a more immediate way to map students' developing understandings? If so, how could such "maps" provide valuable insights into students' thinking to help new teachers develop their mathematics related pedagogical content understanding more efficiently? (Alagic & Langrall, 2002; Alagic 2002c; Alagic, Yeotis, Rimmington, & Koert, 2003).

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MARA ALAGIC

Wichita State University 

USA

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