Technology in the Mathematics Classroom is a course that pre-service and practicing teachers take to advance their knowledge of technology integration (http://education.twsu.edu/alagic/summer2002/752r/752r.htm) within mathematics classroom. The evolving design of a learning environment (model) during facilitation of that course is reported in Alagic & Langrall (2002) and Alagic (2002). Learners’ interactions (instructor-teachers, teacher-teacher) played a significant role in developing this course. The underlying themes include inquiry-oriented approaches for technology integration and doing mathematics as problem solving, reasoning, connecting and communicating through a variety of representations in the technology-oriented learning environment.

The focus on the role and effects of classroom interaction to the learning process is increasing in both emerging research and teaching practice. The research inspired by Vygotsky's view of cognitive socialization and Piagetian ideas of cognitive conflict has focused on collaborative cognitive activity emphasizing interaction between learners as a source of development. In an ICT-based mathematics learning environment, the theoretical concept of interactions has novel characteristics. This view characterizes learning as participatory activities, and includes both peer and instructor-teacher interactions. The thinking processes behind learning are extending beyond individual cognition to include features of both groups as well as the technological tools employed. Group interactions and communications converge to the joint task to be solved. A simulation of complex mathematical phenomena or reflections on the learner’s thinking processes assists in reaching a joint goal: Improving mathematics teaching skills in ICT-based environment (Alagic, 2002; Alagic & Langrall, 2002).

The cognitive apprenticeship (Collins, 1991), an instructional model within the situated learning paradigm, is adopted for this course. Learners participate in sequenced guided activity and interact, in ways similar to that of the craft apprenticeship, but with more emphasis on the development of cognitive skills. Participatory Cognitive Apprenticeship in The Technology-based Learning Environment (PCATLE) is a variation of the cognitive apprenticeship learning model developed as a result of three years of teaching technology in the mathematics classrooms, the author’s work with prospective and practicing teachers as well as teaching experiences in mathematics and information technology. It has been evolving in association with the teachers/networking partners involved. Main characteristics of the development of this model and some of the qualitative features of class interactions during that process are reported in Alagic (2002).

The view of learning for understanding, through a "flexible performance criterion" (Perkins, 1993), is a backbone for the PCATLE model. Learning for understanding requires thinking in number of ways with what we "know", practicing and negotiating our thinking until we can make the right connections flexibly. That also means that the pillar of learning for understanding must be actual engagement in those performances. Daily assignments included metacognitive reflections via e-mail with the facilitator and either a mathematical task or reporting on teaching strategies in classrooms that integrate technology. In due process, implicit need for individualized learning is made explicit through (modeling of) differentiated instruction (Alagic & Langrall, 2002).

Scientific inquiry refers to the diverse ways in which scientists investigate the natural world. To develop competence in an area of inquiry, learners must have: (a) a deep foundation of factual knowledge, (b) understanding of facts and ideas in the context of a conceptual framework, and (c) ability to organize knowledge in ways that facilitate retrieval and application (NRC, 2000). Through the processes of scaffolding, inquiry is combined with guided reflection. Within a cognitive apprenticeship environment, the necessary conditions for metacognitive scaffolding are present and therefore self-regulated learning is more likely to occur than in traditional environments.

A quest, on-line or off-line, or a combination of the two, is introduced as an inquiry-oriented activity in which some or all of the information that learners interact with comes from a range of resources (e.g. teachers, peers, parents, library, Internet, multimedia). The instructional goal of a short term quest is usually knowledge acquisition and integration. The instructional goal of a longer term quest is extending and refining knowledge. The process scaffold can take the form of guiding questions, or directions for completing graphic organizers such as timelines, concept maps, or cause-and-effect diagrams as described by Marzano (Marzano,1992; Marzano, Brandt, Hughes, Jones, Presseisen, Rankin, & Suhor, 1988) and Clarke (1990). A conclusion that brings closure to the quest, reminds the learners about what they've learned, and perhaps encourages them to extend the inquiry into other domains. After completing a longer term quest, a learner would have analyzed a body of knowledge in an inquiry-oriented way, and demonstrated an understanding of the material by creating a challenge for others. It is important to mention that every time this class was offered, some teachers negotiated this inquiry-oriented quest as an alternative to the Web Quests. Some did it because of restricted access to both computers and the Internet. Others believe that combination of teacher-centered and technology-centered activities is the best way to go (Alagic, 2002).

Nurturing, self-reliant, self-regulating and self-evaluating environments require ongoing class negotiations where instruction-authority is based both on pedagogical content knowledge related to mathematics and technology, and expert-utilization of what the class participants bring in terms of both knowledge/understanding and classroom experiences. Learning and instruction in three cohorts, so far, was based on reciprocal understanding between the participants. That was accomplished by establishing shared goals based on common interests in learning about technology-based mathematics phenomena. The learning environment rules were simple: Try to find answers on your own before asking for help; ask your partners first, specific area-experts next, your instructor last (Alagic, 2002).

The first task for learners (learner’s level)is to get oriented to the context and determine a reasonable but challenging goal for their studies based on available opportunities that include not only existing mathematical software but also levels of available expertise. The learners can study different things depending on their individual objectives related to the field, mathematical interest, and grade level they are teaching. The final product is realized through the portfolio that includes lessons/activities of technological representations of mathematical phenomena, a (group) project/presentation on the mathematics phenomena using appropriate technology for developing the concepts, grade-level appropriate quest and reflective online journaling. The lecture materials are available for students on the Web and a possibility for an on-line discussion on these materials is available and often utilized (network level). The students' group projects are also available on the network level. Reflective discussions usually initiated by instructor’s reflective question, often take life of their own. On the local level, the studies consist of lectures and activities on integrating technology in mathematics teaching usually done by instructor or experts in the field.

Participatory Cognitive Apprenticeship in The Technology-based Learning Environment (PCATLE) model has been experienced by the participating teachers as a well-designed and a fruitful way of organizing the course, as it makes possible, for example, to share expertise and also to develop the model and the curriculum together. The combination of different kinds of activities emphasizing collaborative activities in the lab and on the Web, including lectures, and individual projects are for many teachers a new way of learning to teach. This kind of learning model is a challenge not only for everyone involved, but also for the organizing network, because it does not necessarily fit very well in the existing teaching/learning structures (Alagic, 2002).

For the researcher/learner, conceptualization of participatory cognitive apprenticeship in technology-based environment, was discovery at the time. The PCATLE as an orienting framework for the teaching practice is an effort to support teachers as they were learning to interpret and interact in the new mathematical culture they were immersed in. It also brought in the first ideas about novel world view: portals as places of interactions and resources; quests as guided (more or less) inquiry-oriented exploration and guilds as new and different communities, combining both vertical and horizontal structures of learners. For example, the instructor/researcher was a facilitator, but there was an expert in use of graphing calculators that wanted to learn dynamic geometry. She was coaching a group that was exploring functions using graphing calculator as the instructor was facilitating a designing a quest that included student’s inquiry into design of some artifacts using dynamic geometry tools. This small informal guild extended far beyond the course time and researchers’ expectations. Every year some new people join it and some part their own way. It is a support group for sharing resources and ideas, utilizing the course web-pages. The following described activities always included at least a couple of teachers that started their own quest for learning while enrolled in the Technology in the Mathematics Classroom class.

Teacher-Made (designed) Mathematics (TM MATH) is a research-based curriculum development project for engaging K-8 teachers in their own thinking and learning of mathematics via designing mathematics modules. Designing Teacher Made (TM) modules served as vehicles to bridge the existing gap between activities and conceptual understanding of relevant standards-based underlying mathematical ideas. The focus was not only on improving mathematics knowledge and skills of teachers, but also their competency in selecting and adapting standards-based curricula for their classroom environments. TM MATH is based on the belief that teachers must become mathematics learners if they are going to teach for understanding. This happens when they are challenged at their own level of mathematics competence, and when their learning experiences are based on the same pedagogical principles that they are expected to implement with their students. But, as Schifter and Fosnot (1993, p. 26) point out:

Based on key mathematical concepts, selected by participants, the set of ideas/questions were generated for guiding the development of TM modules. Modules were introduced by presenting situations that might arise in science, engineering, business, everyday life, or mathematics itself. An important part of this activity involved the identification of concepts and relations essential to understanding of the situation being studied. Typically this included an idealization and approximation resulting in a simplification of reality, but which retains the essential features and relationships of theoriginal. At this point the needed mathematics was identified and applied. In some cases mathematics was used in a descriptive way to provide insight into the physical situation, in others in a predictive way. TM modules were designed based on the learning cycle model that involves learners in active learning. Underlying was the idea that students are involved in doing mathematics and the teacher is the facilitator of instruction. There was as little lecture as possible: the teacher asked thoughtful, open-ended questions, and used students' responses and background knowledge to drive the lesson. The TM modules were aligned with the Mathematics Standards (NCTM, 2000), problem-initiated and inquiry-based. Every TM module encompassed a tiered/scaffold collection of activities for K-8 learners. The most useful modules were open-ended since they could provide a rich source of interesting mathematics.

The most important aspect of this process was that the connections between the activity and mathematics be made explicit. Without this aspect the activities might be a "dead end" and the goal of bringing meaning to underlying concepts forgotten. The leadership team of TM MATH considers this aspect to be one that is often missing in both teacher preparation and professional development programs. In order for teachers to use activities to teach mathematics they must better understand these connections and how to guide students from activity to underlying mathematical ideas (Alagic, Hutchinson, & Krehbiel, 2002).*

Project M³, Preparing Tomorrow's Teachers to use Technology (PT³) Implementation Grant from the U.S. Department of Education, was awarded to develop technologically astute and competent teacher education graduates who are capable of integrating technology into their classroom to improve the teaching and learning of their students. These funds are used to support Project M³ activities so that pre-service teachers see models of meaningful ways that technology can be used to help students reach standards of achievement. Project M³ have resulted in a higher level of ICT knowledge integration into teacher education program (http://education.wichita.edu/m3/).

The NSF-awarded planning grant Integrated and Seamless Engineering Education (I-SEE) Paradigm for Multi-Disciplinary Learning and Collaboration was developed in collaboration among faculty from engineering and education. This paradigm uses a single unifying artifact, a component or system sufficiently complex to require the application of engineering skills from each of the departments in the College of Engineering: Mechanical, Electrical and Computer, Aeronautical, and Manufacturing and Industrial Engineering. This artifact, which is problem-based and motivational, will be an integrated/overarching component for multiple courses in engineering disciplines. The artifact is being designed and analyzed from various course perspectives within the College of Engineering.

The implementation project facilitates transformation of the curriculum from a set of separate unconnected programs and courses to one in which they are connected via a common unifying theme on artifact. This transformation involves more than including the same artifact in the courses; it considers the integration of multiple perspectives into the learning process. The planning team is devising methods for assisting faculty and students with this transformation (Siginer, Alagic, Chaudhuri, Krishnan, & Rimmington, 2002).

A goal of this proposal was to explore ways to integrate engineering thinking, the problem solving skills used in engineering together with the modern, best practice and research-based teaching methods of active learning to enhance both K-16 science, technology, engineering and mathematics (STEM) and undergraduate engineering education. The plan was to establish firmer collaboration between highly qualified education and engineering faculty, preservice teachers, engineering students, and K-12 STEM teachers in an open learning environment (Hannafin, Land, & Oliver, 1999) that nurtures inquiry-oriented engineering thinking. This structure would support the development of teacher education modules and embed within them school module generation. Although the proposal was not funded, it provided a relevant step in the development of CALEM theoretical framework.

A progression of small robotics-related projects at Wichita State University started with an initial seed grant from the Boeing and continued with support from project M³ (Ellsworth, R., Gladhart, M., Carroll, J., Fillion, B., & Gibson, K., 2000). The initial effort, designed to provide teachers with training and equipment to integrate technology into their classrooms, has grown into a program that provides teachers’ technology training and equipment, an annual competition and showcase for students (through a collaboration between the Colleges of Education and Engineering), and an annual summer camp that serves both as a curriculum development and practicum experience for teachers, and a hands-on robotics invention opportunity for students. This project has served over 150 teachers and their students and involved education and engineering faculty, the Engineering Council – a student engineering group at WSU, distance experts from NASA and MIT, and industry professionals from Boeing, Raytheon, Cessna, among other notable corporations. This broad and diverse group has been successful in creating an environment of experimentation that allows for collaboration and development to integrate ICT capabilities (e.g. algorithmic thinking via robotics activities) into STEM curricula (http://education.wichita.edu/mindstorms). Robotics in the classroom is one successful model of technology integration into STEM (science, technology, engineering and mathematics) curriculum resulting from this project.

Information technology can play a special role in inquiry-based learning and transforming classroom practice, whether as the subject of instruction or as a tool for instruction. In learning situations focusing on inquiry-oriented activities, the technology is most often used (a) as a tool; (b) in the context of solving a problem; (c) to augment communication by expanding audiences; or (d) to broaden collection of representations. Cognitive tools are defined as knowledge generation and facilitation tools that can be applied to a variety of subject matter domains; both mental and computational devices that support, guide, and extend the thinking processes of their users. These are tools that are used to engage learners in meaningful cognitive processing of information (Jonassen & Reeves, 1996).

Fluency with information and communication technology is a life-long learning skill that requires an understanding of ICT to the point of being able to apply it productively, to recognize when it would assist or impede the achievement of a goal, and to continually adapt to the changes and advancement in ICT. It requires a deeper understanding and mastery of ICT for information processing, communication, and problem solving, than does computer literacy as traditionally defined. ICT builds on fundamental proficiencies, such as textual and quantitative literacy, logical reasoning, creative graphical representation, critical thinking, and knowledge of both learning context and the broader community in which it is applied (NRC & NAS, 1999).

At the center of what it means to function in today’s complex world are ICT competencies: defining the problem, retrieving and organizing, analyzing and sharing information. These competencies emerge from ICT intellectual capabilities, concepts and skills. Intellectual capabilities can be regarded as life skills that are formulated in the context of ICT. These skills are closely related to the (mathematics) process standards and skills defined by national associations (e.g., NCTM, 2000; NRC, 1996). They involve engagement in sustained reasoning, management of complexity, testing potential solutions, management of problems arising from faulty solutions, evaluation of information, collaboration, expecting the unexpected, and thinking about ICT abstractly. Central concepts for ICT fluency include such things as key aspects of computer programs, information systems, local and wide area networks, digital representation of information, modeling and abstraction, and algorithmic thinking and programming. Finally, ICT skills include such items as using a spreadsheet to model simple processes, using a graphics package to create image-based expressions of ideas, using a database system to set up and access information, and using ICT tools to learn how to use new applications/features (NCES, 2000; NRC & NAS, 1999).

Figure 2. CALEM Open Learning Environment

The components of the CALEM team-driven Open Learning Environment are highlighted in Fig. 2 (Alagic, Yeotis, Rimmington & Koert, 2003). The term open in this model represents both the way the learning goals for integrating ICT are established and the means by which they are achieved (designing a cognitive apprenticeship). Components of the environment (enabling context, dynamic resources, generating tools and metacognitive scaffolding) furnish learners/teachers with many different strategies to further their problem solving by providing for an inquiry-oriented, critical thinking and creative teaching and learning environment.

The CALEM enabling contextis such that (a) the CALEM team/facilitators provide guidance and (b) learners/lead teachers must learn enough from the CALEM experience to be able to select novel examples on an ad hoc basis. The process must involve interactive dynamic resourcesevolving as the team proceeds to develop this learning environment. How ICT is infused may change with time as the team discovers new ways of using ICT to improve learning outcomes. Generating tools to represent and manipulate concepts under study range from physical (e.g. robots) to remote (remote-sensing instrumentation) to virtual (e.g. Virtual Lab/Factory). Continuous, formative evaluation and reflective feedback provide guidance for scaffolding on what to consider and how to think about the problem under the study, from strategic to conceptual, procedural, and metacognitive scaffolding. (Alagic, 2002; Alagic et. al., 2003; Hannafin, Land & Oliver, 1999; Reigeluth, 1999).

CALEM Modules (Fig. 3; Alagic et. al., 2003) are defined as context-sensitive instructional units integrating ICT into standards-based curriculum. Metaphorically, a CALEM module has in its core K-12 module sometimes referred to as a STEM school module, encompassed with a supporting "doughnut"-like layer consisting of all the teacher’s skills necessary for successful implementation of the school module. The "doughnut" encompasses facilitation of procedures known to improve/change teacher’s practice. The modules include pedagogical strategies based on a variation of Kolb’s learning cycle (explore, conceptualize, apply) in a cognitive apprenticeship environment (Karplus & Their, 1967) as reported in Marek & Cavallo, 1997; Kolb, 1984; and NRC, 2000. The CALEM modules also include (a) ICT dynamic content (e.g. simulation models, robots) with appropriate concepts and skills, (b) a guide for teachers, (c) model applications of the module in different contexts, and (d) a link to the module development team for ongoing support.

Figure 3. The structure of CALEM modules

The conceptual framework dimensions of the modules include (a) real-world problems introduced through experimental questions, (b) appropriate developmental level activities, (c) inquiry oriented exploration, (d) open-endedness in terms of problem solutions and strategies, (e) experiential scaffolding, and (f) dynamic content environments supporting appropriate ICT concepts and skills. Thus, modules are both a content resource and tool for further development of teachers’ mathematics and/or science related pedagogical content knowledge (Schulman, 1986). Participating teachers in this open learning environment have the unique experience of designing and implementing specific thematic modules. Through this experience as cognitive apprentices they gain the knowledge and skills to continue this process by developing new CALEM modules and teaching others to do the same. Thus, these teachers become change agents for infusing ICT inquiry oriented learning strategies (http://education.wichita.edu/CALEM). The goal of "the doughnut" is facilitation of the development of content related pedagogical knowledge (PCK) in an ICT environment. Compared to Shulman’s (1986) definition, this one incorporates capability to provide guidance for using ICT cognitive tools as they relate to STEM content which in itself can be considered content.

In the process of intertwining research and best teaching practice, the CALEM environment mediates learner-centered activity and is both knowledge-centered and assessment-centered in establishing needs, determining goals, defining meaning and facilitating inquiry for and by participants in a cognitive apprenticeship (Hannafin et al., 1999; NRC, 2000) in learning for understanding (Perkins, 1993). It employs ICT interactive tools, provides metacognitive scaffolding, and promotes multiple perspectives and divergent thinking. The model shares psychological and pedagogical values of situated thinking, prior knowledge and experience, metacognitive monitoring, progressive evaluation, feedback and refinement (NCTM, 2000; NRC, 1996; NRC, 2000; Reigeluth, 1999). Intellectual capabilities and skills relate to very practical matters, getting at the heart of what it means to function in a complex technology-oriented world. Clearly, support structure is essential. This includes close collaboration within the CALEM guild (STEM teachers, graduate students/teachers, and subject discipline and education faculty) provision for the state of the art portal and generation of meaningful quests for learning.

Over a lifetime, an individual will acquire more skills and develop additional proficiency with those skills, understand information technology concepts in a richer and more textured manner, and enhance his or her intellectual capabilities through engagement in multiple domains (NRC & NAS, 1999). More details about the model and "potential impact of its dimensions on integration of inquiry and information technology into core curricula" can be found in Alagic et. al. (2003).

A range of module possibilities is being examined to determine their suitability to produce stimulating learning situations that can be adapted to each level of the cognitive development sequence. Such modules must cause students to apply existing, and develop new STEM knowledge to complete tasks and solve problems. Reflective journaling and focus group-discussions are part of the so-called Metacognitive Lab incorporated within each module. Some examples are Systems Thinking, Remote Sensing, Virtual Factory, and Reaching for the Stars.

An Example: The Robotic Systems Design Studio. The Studio allows students to solve a series of problems using a systems approach that reflects real world problem solving, reasoning, and multiple representations. Students begin constructing simple robots and progress to integrating more complex components, sensors, and advanced programming. Each project has a focus on inquiry, process skills/mathematics process standards, and algorithmic thinking necessary to design, build, program, and test robots to complete a specific job. For example, to study the terrain of Mars, students design and build a robot to traverse a simulated Mars landscape while carrying a camera designed to send images back to Earth, and an arm and sensors to collect samples for analyses. The STEM curriculum content focuses on Earth space science, spatial relationships using coordinate geometry to construct three dimensional models, and gathering and analyzing data from remote sensors to help describe object properties (e.g. temperature, color, density). ICT concepts integrated into the model include modeling and abstraction, algorithmic thinking and programming, digital representation of information, and information organization, (Alagic, et. al., 2003; Witherspoon & Reynolds, personal communication, January 28, 2002).

Technological advances, from global communication opportunities to a variety of technology-based cognitive tools support the emergence of a variety of learning and teaching frameworks. That implies the need to explicate emerging conceptual frameworks within the new paradigm of instructional theory (Hannafin et. al., 1999; Reigeluth, 1999). Our work is guided by two underlying ideas:

A number of change theorists insist that reform efforts start the process with a clear, specific vision of what they want to accomplish. Approaches based on postmodern epistemologies as well as chaos theory suggest that the reverse is preferable (Fullan, 2001). Our research outcomes (Alagic, 2002; Alagic & Langrall, 2002; Alagic, Gibson, & Haack, 2002) suggest that, for example, the PCATLE is an inviting direction/framework for supporting teachers as learners. It supports the idea of context-sensitive frameworks; adjusting as we go. The term PCASTLE emerged from a seemingly chaotic situation while teaching a group of k-12 teachers to integrate technology into their teaching of mathematics. In an attempt to facilitate cycling through the helical structure grounded on the learning cycle, puzzle pieces snapped into their places, chaotic computer lab activities became orderly, only the "term introduction" was lacking. But the learner knew that the "name" would come naturally.

The following stage of the learning cycle, expansion, awaits new challenges. In believing that a computer-supported learning environment creates an optimal environment for participatory cognitive apprenticeship, revising technology-based learning environment focusing on analyzing learning (mathematics) for understanding and social interaction in that context continues. How to create optimal conditions for PCATLE and similar environments challenges future studies. From traditional theories on mutual understanding focus needs to be broadened to the other possible aspects of social interaction. Relevant perspectives will be students' interactions in an ICT environment and embedded opportunities for differentiating instruction. Reaching every student at an appropriate, yet challenging cognitive level will increase learning for understanding, recognizing understanding through a flexible performance criterion (Perkins, 1993) and providing better conditions for differentiated instruction which ultimately provides better individualized learning for all learners (Postlethwaite, 1993; Alagic & Langrall, 2002).

Emerging learning environment, CALEM appear to be an ideal framework for necessary educational interventions (http://education.wichita.edu/calem/).

CALEM is a theoretical model, grounded in existing experiences, of the way in which ICT and inquiry can be infused in science and mathematics education. This model is meant to provide metacognitive scaffolding, promote multiple perspectives and divergent thinking while infusing essential ICT. Particular strengths include the multidisciplinary teams composed of university faculty and teachers, the action research component, and teacher student teams. Context sensitive instructional modules produced that can be used both for pre-service and in-service education.

Cavallo (2000) presents the view that the blueprints for educational reforms "have failed simply because they are blueprints" (p. 768). For a learning environment, he proposes the following set of principles: constructionism, technological fluency, immersive environments, long-term projects, applied epistemological anthropology, critical inquiry, and Emergent Design. Emergent Design facilitates the overall process – active interactions among all players during the processes of developing projects as well as the emerging projects. The emergence of the CALEM parallels in some way Emergent Design. CALEM’s theoretical framework includes in one way or another all the principles proposed by Cavallo. Team-driven, open learning environment, generation of context-sensitive entities/modules and their implementation both in schools and teacher preparation program provide for all the principles. Cavallo is definitely correct in both emphasizing that in no way we can know "for every site what will resonate and what local concerns and local knowledge exist" and "what one can assume is that there is always something" (p. 780). In face of CALEM experience, that something is definitely active presence,participation, of all players and context-sensitive process/product.An appropriate term might be context-sensitive blueprints for educational interventions.

* TM MATH modules are tested in classrooms and further refined. Modules are accessible at

http://education.wichita.edu/alagic/TM_MATH/tm_math_modules.htm.