^{12}]. Thus, these systems possess the ability to make intelligent decisions about the interactions that take place during learning and aim to support the learners without being directive. Furthermore, AEHS increase the functionality of conventional hypermedia combining free browsing with personalization and can support all the continuum of the learning model, from a pure system-controlled to a fully learner-controlled [

^{10}].

*adaptive and intelligent techniques*have been applied to introduce adaptation such as:

1. *Curriculum sequencing*: Helps the learner to follow an optimal path through the learning material.

2. *Adaptive presentation*: Adapt the content presented in each hypermedia node according to specific characteristics of learner.

3. *Adaptive navigation support*: Assist the learner in hyperspace orientation and navigation by changing the appearance of visible links.

4. *Interactive problem solving support*: Provides the learner with intelligent help on every step of problem solving from giving a hint to executing the next step for the learner.

5. *Intelligent analysis of student solutions*: Uses intelligent analyzers that not only tell the learner whether the solution is correct, but also tell him/her what exactly is wrong or incomplete.

6. E *xample-based problem solving support*: Helps the learners solve new problems not by articulating their errors, but by suggesting them relevant successful problem solving cases, chosen from their earlier experience.

7. *Adaptive collaboration support—adaptive group formation and peer help*: These techniques support the collaboration process just like the interactive problem solving support systems assist an individual learner in solving a problem or use knowledge about possible collaborating peers to form a matching group for different kinds of collaborative tasks.

Table 1. Adaptive Educational Hypermedia Systems/Intelligent Tutoring Systems and Their Implemented Techniques

^{48}]. Students are diverse in terms of their experience, expectation, abilities, and interests; and they vary in the learning styles as well. The learning style describes individual differences in learning. Students with different learning styles respond differently to various didactic approaches and the didactic strategies should match the learning styles of students [

^{44}].

^{23}].

^{14}]. The criteria to select the learning style model is that apart from the theoretical and empirical justification, it should possess assessment instruments, describe the didactic strategies associated with each category of learning styles, and be appropriate for the learning context as well as its cost [

^{66}]. Moreover, Merrill [

^{55}] suggests that in case the modality of adaptation is adopted to learning styles for didactic systems (live- or technology-based), it is necessary to select the appropriate didactic strategies with the learning goal of teaching and, secondary, on the basis of these strategies, we can choose the most appropriate strategy for every one of the learning styles. According to [

^{27}], the learning styles will be more effective for the learners when the technology fits didactic design principles as well as the application of pedagogical criteria is also necessary for choosing the most appropriate learning style model for the AEHS or ITS that will be developed. In Table 2, we can see AEHS/ITS using learning style for adaptation (16 out of 35). From these systems, only one uses the learning style both for adaptive presentation and adaptive collaboration (TANGOW/WOTAN). Also, the MOT only uses Kolb's learning style model (Diverger and Converger). We consider that the learning style is a factor of decisive importance when designing modern educational systems as well as the chosen dimensions of students' learning style for group formation should fit with the domain of application. For example, if the domain of application consists of abstract concepts and content (like physics), then it would be best to use the abstract and concrete dimensions of students' learning style for forming groups [

^{58}].

Table 2. AEHS/ITS and Learning Style Models, Learning Theories, or Didactic Strategies, and Domain of Applications that They Use

^{34}]:

1. *The learner modeling*: Content, structure of the learner model, and learner diagnosis.

2. *Didactic design*: Domain knowledge (content, structure, and representation), assessment process, feedback, and collaboration.

3. *Adaptive engine*: Selection of appropriate adaptive and intelligent techniques depending on the learner and the context, and learner control issues.

4. *Authoring process*: Facilitation of the use of AEHS in real conditions and exploitation of standards (IMS, SCORM, and LOM).

5. Evaluation of the efficiency and effectiveness of the adaptation.

**1.4.1 Didactic Design**diSessa [

^{21}] suggests that physics is best taught through experiments, labs, demonstrations, and visualizations which help the students to understand physical phenomena conceptually. Based on diSessa's suggestion, we design the MATHEMA by choosing didactic approaches in the frame of constructivism in order to help students learn physics conceptually. Students have particular difficulty in comprehending physics concepts which have very few real-life referents and which incorporate invisible factors, forces operating at a distance, and complex abstractions [

^{15}]. Even advanced students have difficulty grasping nonintuitive, abstract concepts such as those found in electromagnetism [

^{28}]. It is possible, therefore, for the students to have misconceptions and learning difficulties when studying electromagnetism. Indicatively, we present two

*common misconceptions*in electromagnetism that have been documented by [

^{49}] and [

^{2}]:

• The students consider that the magnetic poles exert forces on electric charges in the plane of the charge and magnet, regardless of whether the charge was moving or not.

• A constant magnetic field changes the speed (magnitude of velocity) of a charged particle which moves in it.

Also, Bagno and Eylon [ ^{2}] refer that the students have difficulty in determining the direction of the *Lorentz* force.

While teaching in the frame of constructivist environment, it is necessary to take students' misconceptions and learning difficulties into consideration by using different teaching strategies and activities in order to support them to reconstruct their own cognitive models, and design a learning environment where they can construct their ideas by themselves. How to engage younger students in complex physics thinking is a challenge, but simulations provide one intriguing way to engage students in the study of abstract, complex physical phenomena [ ^{21}]. Computer simulations have been shown to be effective in fostering conceptual change [ ^{53}]. The cognitive conflicts arising from the simulations lead the learners to discover possible misconceptions and reconstruct their own cognitive models [ ^{33}].

Much of the current work in cognitive psychology has shown that students learn better when engaged in solving problems [ ^{51}]. According to Concari et al. [ ^{16}], physics being an experimental science, observation, measuring, and theoretical speculations are processes that cannot be separated from the physical knowledge construction even in the classroom. According to [ ^{67}], educators should consider to stimulate the basic purposes of schooling curiosity, exploration, problem solving, and communication. The most effective learners should use multiple strategies to ensure that they monitor their comprehension. Thus, we need adequate didactic strategies in order to promote meaningful learning.

Taking all the above into consideration, we adopt the following didactic approaches in the MATHEMA: *questions, demonstrations, presentation of theory and examples, exercise solving, and problem solving through experimentations with simulations, explorations, guided discovery, and collaboration.* The learning goal of explorations and guided discovery didactic approaches is to motivate the students to self-direct their learning process to learn how to apply knowledge and generally develop higher order thinking. An exploration is a structured lab where the student makes predictions about a body's (e.g., particle) motion, then runs the simulation to compare the actual result with the predicted result. Guided questions help the students refine their mental models of physics [ ^{32}]. Exploration activities can be supported through a hypermedia-form presentation of the educational material, simulations linked with specific activities, and collaboration in team projects [ ^{33}]. The main purpose of the guided discovery methodology is to lead learners to discover domain concepts with various learning facilities such as simulation, demonstration environments, and others.

In order to support multiple didactic strategies in a constructivist environment, we choose Kolb's Experiential Learning Theory (ELT) [ ^{44}]. Kolb's ELT is a holistic theory of learning whereby social knowledge is created and recreated in the personal knowledge of the learner through the grasping and transforming experience. Kolb has suggested the *Learning Cycle* that includes four stages. Each stage is approached with different didactic approaches. According to Kolb, the students having better learning outcomes should go through all the stages many times.

Kolb's ELT is also a theory of cognitive learning styles that proposes four learning styles: *Diverger, Assimilator, Converger, and Accommodator.* Divergers have the ability to view concrete experiences from a number of perspectives. Assimilators have the abilities to formulate theories and prefer abstract concepts. Convergers have strength on the practical applications of ideas. Accommodators have strength in doing things. Cognitive psychologists such as Piaget, Bruner, Harvey, Hunt, and Schroeder have identified the concrete-abstract continuum as the main dimension along which human cognitive growth occurs [ ^{81}]. Kolb considers that the Divergers and Accommodators have *concrete* "learning style" and the Convergers and Assimilators have *abstract* "learning style." These two dimensions represent the major directions of cognitive development identified by Piaget. Some students may grasp abstract concepts readily while others need concrete imagery to learn [ ^{81}]. Concrete dimension enables the learners to register information directly through their five senses: sight, smell, touch, taste, and hearing. When they are using their concrete ability, they are dealing with the obvious, the "here and now." They are not looking for hidden meanings, or making relationships between ideas or concepts, and they may also communicate in a direct, literal, no-nonsense manner. Abstract dimension allows learners to visualize, conceive ideas, and understand or believe in what they cannot actually see. When learners are using their abstract ability, they are using their intuition, their imagination, and they are looking beyond at something which is of more subtle implication.

Moreover, the ELT provides a framework for understanding and managing the way teams learn from their experience. Since research into learning styles suggests that individuals learn differently, it is logical that some learners would prefer to learn individually, while others would prefer to learn from interaction in groups. People with different learning styles generate different perspectives in effective strategies for dynamic group interactivity [ ^{43}], [ ^{45}].

Kolb proposes a learning style model that we adopt because it meets all the criteria of the most appropriate learn-ing style model proposed by Sampson and Karagiannidis:

*Empirical justification*: Kolb's learning style model is supported by Kolb's ELT and some empirical studies performed by Svinicki and Dixon [ ^{73}] and Harb et al. [ ^{35}].

*Assessment instrument:* Kolb's learning style model is supported by Kolb's Learning Style Inventory (LSI) questionnaire [ ^{45}], which consists of 12 multiple-choice questions, so it makes its use easy for senior high school students or beginners in higher education. In the MATHEMA, Kolb's LSI questionnaire apart from the identification of the four learning styles that we mentioned above, is used to distinguish between abstract and concrete learners with the aim of adaptive group formation for collaborative tasks.

*Description of didactic strategies:* Svinicki and Dixon, and Harb et al. have described the most appropriate didactic approaches for each stage of Kolb's learning cycle in their papers [ ^{73}], [ ^{35}], accordingly.

*Appropriation of the context*: We conducted a research [ ^{58}] on the subject of electromagnetism, based on Kolb's ELT and the researches of Svinicki and Dixon [ ^{73}], Harb et al. [ ^{35}]. We designed the educational material according to didactic approaches of Table 3. Didactic strategies and educational material were adapted by the MATHEMA to the students according to their learning style. The results of this research showed that the participants improved their performances a lot [ ^{58}].

Table 3. Didactic Approaches Implemented by the MATHEMA

Moreover, in the development of the MATHEMA, we take care of the quality and the content of the courses, to improve the learners' achievement.

**1.4.2 Adaptive and Intelligent Techniques**In order to support all the didactic approaches that we mentioned above, taking into consideration the learner attributes and the content as well as to enrich the adaptive functionality of the MATHEMA, we implemented the following techniques:

*curriculum sequencing, adaptive presentation, adaptive navigation support, adaptive group formation and peer help, and interactive problem solving support.*In general, the MATHEMA is a learning system that dynamically generates courses of electromagnetism. The aspects of students that the MATHEMA uses for adaptation are:

*learning goal, knowledge level, total performance, prior knowledge, learning style, abstract or concrete dimension of learning style, preference for visual and/or verbal feedback*[

^{62}]

*, and preference for the kind of navigation.*

**Curriculum sequencing.** The concepts of learning goal in the MATHEMA are progressively presented following the internal structure of the concepts. The concepts of the learning goal are organized in a *layered structure* following a simple-to-complex sequence [ ^{64}], according to which at the first layer, the simplest and more fundamental concepts are included, providing an overview of the learning goal, and then, subsequent layers of concepts add complexity or detail to a part or aspect of the learning goal [ ^{59}].

**Adaptive presentation.** Taking into consideration all those that we refer to in the first three paragraphs of Section 1.4.1., the researches of Svinicki and Dixon [ ^{73}], Harb et al. [ ^{35}], as well as our own research that we mentioned above, we consider that the most appropriate didactic approaches that match with each of the student's learning style are those presented in Table 3. The adaptive presentation of the educational material according to the students' learning style, when they follow the four stages of Kolb's learning cycle, is done with the following didactic strategies:

**Diverger**.

1. *Questions, Demonstrations;*

2. *Presentation of Theory and Examples;*

3. *Exercise Solving;*

4. *Activity.*

**Assimilator**.

1. *Presentation of Theory and Examples;*

2. *Exercise Solving;*

3. *Activity;*

4. *Questions, Demonstrations.*

**Converger**.

1. *Exercise Solving;*

2. *Activity;*

3. *Questions, Demonstrations;*

4. *Presentation of Theory and Examples.*

**Accommodator**.

1. *Activity;*

2. *Questions, Demonstrations;*

3. *Presentation of Theory and Examples;*

4. *Exercise Solving.*

Fig. 1 shows how our system applies the didactic strategy for a "Converger" student. It presents an exercise to be solved by the student, and at the bottom of the educational material page, it also presents the appropriate three linked icons to educational material related to other didactic approaches. However, the student is free to choose the next educational material to study.

**Adaptive navigation support.** In adaptive navigation support, the MATHEMA helps students avoid the "lost in hypermedia" syndrome by offering them the following techniques: direct guidance, link annotation, link hiding, and link sorting.

**Adaptive group formation and peer help.** In exploratory environments, in which the students participate in experiments, we consider that it is very important for them to collaborate, so that they will share their experience, opinions, and findings. Consequently, the group formation is necessary. Thus, the MATHEMA enforces the learner's learning by involving an adaptive group formation and peer help technique. For this purpose, our system creates a priority list of possible candidate mates for a certain student, taking into account his/her learning style, his/her candidate mates' learning style, and total performance as well.

In this paper, we mainly focus upon the interactive problem solving support implemented by our system. The rest of the paper is organized as follows: In Section 2, we describe the interactive problem solving support by presenting the framework of the activity and an application of this framework in supporting students solving a problem in electromagnetism. In Section 3, we present the experimental study with senior high school students. In Section 4, we summarize the most significant points of our work and we refer to our future plans.

^{65}]. Problem solving varies in at least three different dimensions: 1) problem type, 2) problem representation, and 3) individual differences. The problem solving has four characteristics as described by [

^{51}]:

1. Problem solving is *cognitive*. It occurs internally, and thus, can only be inferred indirectly by the person's actions.

2. Problem solving is a *process*. It involves representing and manipulating knowledge in the problem solver's cognitive system.

3. Problem solving is *directed*, that is, the problem solver's processing is guided by his/her goals.

4. Problem solving is *personal*. The solver's individual knowledge and skills help determine the difficulty or ease with which obstacles to solutions can be overcome.

^{41}] began to describe the range of problem solving learning outcomes by distinguishing between well-structured and ill-structured problems in terms of their didactic design requirements. Didactic designs for well-structured problems are rooted in information processing theory, but the didactic designs for ill-structured problems necessarily share assumptions with constructivism and situated cognition. Ill-structured problems are complex problems that cannot be solved by a simple algorithm. Such problems do not necessarily have a single correct answer, but require learners to consider alternatives and to provide a reasoned argument to support the solution that they generate. Information processing theories conceive of learning outcomes as generalized skills that can be applied across content domains, while constructivism and situated cognition argue for the domain specificity of any performance, and therefore, recommend embedding instruction in some authentic context.

^{50}].

^{12}]. Systems implementing this technique may monitor the actions of the students, understand them, and use this understanding to provide help and to update the student model [

^{12}]. The AEHS supporting this methodology are the ActiveMath [

^{54}] and ELM-ART II [

^{78}].

*example-based problem solving support*. ELM-ART II contains "live examples" and short programming problems. In ELM-ART II, if the learners fail to solve a LISP programming problem, they can ask the system to diagnose the code of their solution and give detailed explanation of error. It also helps the learners to find the relevant examples from their previous experience by presenting an ordered list of examples based on their relevancy.

^{12}]. Some systems as PAT-Online cannot actively watch the student's actions and can only provide help by request. The MATHEMA is supported by a client-server architecture (JSP, Servlets, JavaBeans, and JavaScripts) offering an appropriate environment for supporting problem solving activities. The MATHEMA has the capability to store the students' actions into the student model and to act accordingly, as well as it keeps statistical data about the students' actions.

*simulations, explorations*[

^{32}],

*guided discovery*[

^{52}], and

*collaboration*didactic approaches. The specific goal of the activity is to support the students to learn electromagnetism conceptually and overcome their possible misconceptions and learning difficulties [

^{58}].

**2.2.1 The Framework of the Activity in the MATHEMA**At the beginning, the students are given by the system the learning goals, learning outcomes of the activity, and links to appropriate prerequisite knowledge. Also, links to other educational material (Questions, Theory, Examples, etc.) are given to the students according to didactic strategies that we mentioned above.

The general framework of the activity includes six steps as follows:

**Step 1:** Activation of prior knowledge.

*The students are given the formulas they have already known from previous chapters of physics, and perhaps, they know how to use them (prior knowledge). In case that the students lack sufficient prior knowledge, the system offers them additional relevant knowledge through links to prerequisite knowledge. Also, students are given the main formulas of the lesson they are studying. Then, the students are asked to synthesize all the given formulas in order to extract formulas with the aim of calculating the values of certain physical quantities (or dimensions).*

**Step 2:** Recognizing the restrictions on the parameters' values of extracting formulas in Step 1.

*The students, through a guided dialog with the system, explore if they should set any restrictions on the values of parameters of the extracting formulas in Step 1.*

**Step 3:** Application of extracting formulas in Step 1 and prediction of the kind of motion.

*The students are asked to apply the extracting formulas in Step 1 in order to calculate the values of the corresponding physical quantities (or dimensions) in various values of parameters and to write them down. Moreover, they are asked to predict about the kind of motion.*

**Step 4:** Working with the simulation.

*The students are asked to set various values in parameters of physical quantities (or dimensions) related to a certain extracting formula in Step 1 to a given simulation, then to run it. They should write down the simulation results. Then, the students are asked to compare the simulation results with the calculated or predicted results in Step 3 in order to decide which values seem to be the correct ones.*

**Step 5:** Collaboration in pairs of students.

*The students collaborate in pairs, with the aim of sharing their experience, opinions, and findings and to write down the final results. A student who does not wish to collaborate with his/her peer can get round this step.*

**Step 6:** Checking the results through a *guided dialog*.

*The students through a guided dialog with the system check their final results that they believe as correct.* The aim of the guided dialog is to detect the students' misconceptions and learning difficulties in order to help them to reflect and to reconstruct their own cognitive model. This step could be done either individually or collaboratively.

In Step 2, the *guided dialog* is carried out in three phases as follows:

**Step 2/Phase 1:** Exploration— *The students are given the values of parameters and the formulas to calculate the values of certain physical quantities (or dimensions), and he/she is asked to explore if these values of parameters should be restrictions. Then, he/she is asked if he/she agrees or disagrees that the given values of parameters are restrictions by choosing "Yes" or "No." If he/she chooses "Yes," then the system proceeds to the Step 3 of the activity, if "No," then the system proceeds to the Phase 2 of the dialog.*

**Step 2/Phase 2:** Presentation— *The system presents the values of the physical quantities (or dimensions) calculated by the formulas for the given values of parameters and the student is asked if he/she agrees or disagrees with these calculated values by choosing "Yes" or "No." If he/she chooses "Yes," then the system proceeds to Step 3 of the activity, if "No," then the system proceeds to Phase 3 of the dialog.*

**Step 2/Phase 3:** Explanation— *The system explains to the student why the given values of the parameters are restrictions.*

In Step 6, the *guided dialog* is carried out in four phases as follows:

**Step 6/Phase 1:***The student is asked to write down on the check form of the system the final value of the physical quantity (or dimension) that he/she believes as correct value or the predicted kind of the motion (we later call it final result) and the corresponding value or the kind of motion that he/she received from the simulation (we later call it simulation result).*

The cases that the system examines are four as follows:

1. *If the final result is equal to (or the same as) the simulation result, then the system informs the student that the final result is correct and also explains to him/her why it is correct.*

2. *If the final result is correct and the simulation result is not correct, then the system informs the student that his/her result is correct and it also explains to him/her which possible reasons made the simulation result incorrect. So, the system induces the student to repeat Step 4 of the activity.*

3. *If both results are not correct, then the system informs the student that both results are not correct, so it induces the student to repeat all the steps of the activity.*

4. *If the final result is not correct and the simulation result is correct, then the system proceeds to Phase 2 of the dialog.*

**Step 6/Phase 2:***In case the student's wrong result is either due to a wrong mathematical formula or due to the wrong use of the correct mathematical formula that calculates the physical quantity (or dimension), the system presents the correct mathematical formula to the student and additional help about its application. The student is asked to calculate the physical quantity (or dimension) again and to choose the correct one from among the given answers. The other answers are possible common misconceptions or learning difficulties. If the student again chooses a wrong answer, then the system proceeds to Phase 3 of the dialog. If the student chooses the correct answer, then the system returns to Phase 1 of the dialog so that the student will check any other result.*

**Step 6/Phase 3:***The system gives the student an explanation, by using mathematical arguments, why the answer is not correct. Then, the system asks the student if he/she insists on his/her point of view, and if the student chooses "Yes," then the system proceeds to Phase 4 of the dialog; if "No," then the system returns to Phase 1 of the dialog so that the student can check any other result.*

**Step 6/Phase 4:***The system gives the student a different explanation why the answer is not correct by using arguments based on the experience that the student obtained through the simulation. Then, the system asks the student whether he/she wishes to calculate the physical quantity (or dimension) again or to return to the Phase 1 of the dialog and to check any other result.*

It is important to point out that all the steps of the activity are not obligatory and the student has the freedom to go back anywhere in the system or to skip the activity any time he/she likes. The system keeps in the student model all the pages that the student had already visited, and it reminds him/her about his/her visits any time it is required.

**2.2.2 Examples from the**In our research, the activity that the students carried out through the MATHEMA belongs to the section of electromagnetism entitled:

*Guided Dialog**Motion of a charged particle perpendicular to the direction of a uniform magnetic field.*The problem that the students are asked to solve is:

(STEP 1) *Synthesize the mathematical formulas listed below in order to extract the formulas of radius, R, and period, T, of the particle circular motion.*

(1)

(STEP 2) *Apply the following pairs of values, q and v, in order to calculate the radius R and period T. If certain values of the radius R and/or period T are zero, infinite, or indeterminable, then the given values of q and v are restrictions.*

(STEP 3) *Apply the values of the parameters, q and v, of Table 4 on the formulas of the radius R and period T, and calculate the corresponding values of radius R and period T of the particle circular motion; predict the motions of the particle for the same parameters (no motion, clockwise circular motion, anticlockwise circular motion, and rectilinear motion). The number of questions in Table 4 corresponds to each pair of particle velocity v and particle charge q that are intended to identify various misconceptions and learning difficulties of the students. Given values:**and**mg.*

Table 4. Calculations of the Values of Radius and Period

(STEP 4) *Set the values of q and v of Table 4 to the simulation and run it. Then, compare the calculated or predicted results with simulation results and explain the differences.*

(STEP 5) *Collaborate with your mate.*

(STEP 6) *Check your final results.*

As examples, we present the *guided dialogs* done between the system and the student named Yiannis in Steps 2 and 6. The *guided dialog* in Step 2 was done as follows:

**Step 2/Phase 1**: System: *Apply the following pairs of values,**and* , *in order to calculate the radius**and period* . *If certain of the values of the radius**and period**are zero, infinite, or indeterminable, then the corresponding values of**and**are restrictions.*

*Are the above pairs of values or some of them restrictions?*

• *Yes*

• *No*

Yiannis' answer: *No*.

**Step 2/Phase 2**: System: see Fig. 2.

Yiannis' answer: *No*.

**Step 2/Phase 3**: System: No Lorentz force is exerted on the particle because in the formula , either , or , or both of them are zero giving the Lorentz force total result as zero. So, since in these certain cases, no Lorentz force is exerted on the particle; hence, we have no circular motion. Consequently, the given pairs of values must be restrictions.

In order to explain how the dialog for Step 6 is done, we present the number 6 of questions of Table 4: *Which is the value of the radius R of a charged particle with mass**mg and charge,**that it remains motionless (**) in a uniform magnetic field with a magnetic induction,**?*

This question is intended to identify a common misconception that the students usually have about this section of electromagnetism which is: *the magnetic poles exert forces on electric charges in the plane of the charge and magnet, regardless of whether the charge was moving or not* [ ^{49}], so as to help them overcome this.

The *guided dialog* in Step 6 of the activity between the system and Yiannis for checking the result was done as follows:

**Step 6/Phase 1:** In Fig. 3, we can see the results that Yiannis has written down on the check forms. The final result is equal to indeterminable (it is symbolized with a # symbol) and the simulation result is equal to zero. Also, Yiannis received feedback when he clicked on the "ANSWER" button.

**Step 6/Phase 2:** The system suggests to Yiannis to calculate again the value of the radius R, and to select an answer ( Fig. 4).

Yiannis' answer: *Infinite*.

**Step 6/Phase 3:** System: *Generally, the value of a fraction is infinite if the numerator is not zero or infinite and the denominator is zero. So, the value of the radius**is infinite if the numerator is not zero or infinite, and the denominator is zero. However, since the value of linear velocity**is zero, the numerator is zero. Also,**and**are not zero, so the denominator is not zero. Consequently, the value of radius**is not infinite but is zero.*

*Do you insist that the value of the radius R is infinite?*

• *Yes*

• *No*

Yiannis' answer: *Yes*.

**Step 6/Phase 4:***System: During your study of simulation you might have concluded that the radius R of the circular motion is infinite if the particle moves rectilinearly. In this case, the particle remains motionless. This means that the radius R of the circular motion is zero.*

*Are the learning performances of students improved when they carry out the activity? Do the students perceive their wrongs and revise their points of view? Do the students justify their choices and accept that they must set restrictions on the values of parameters of formulas?*The experiment was conducted in a Technical High School in Philadelphia, Athens, Greece, in January 2007. The entire processes of the research lasted 24 days. In the first week, we gave the pretest to the students. In the second week, we gave a demonstration about Interactive physics by presenting the main operations of the simulations that we have designed for the experimental study in order to facilitate the students in their exploration. Also, we described the environment of the MATHEMA. In the third week, the students carried out the activity through the MATHEMA. The experiment with the activity in the classroom lasted two didactic hours. In the fourth week, the students were given the posttest and the questionnaire to express their opinions about our system.

**3.1.1 Participants**Twelve 18-year-old students participated in the experiment. The reason why we chose 12 students is the limitation of the school laboratory (the laboratories in Greek public high schools are usually equipped with 12-14 computers).

**3.1.2 Material and Instruments**

1. *AEHS MATHEMA and Interactive Physics*. The students studied the activity through the MATHEMA as well as by using the Interactive Physics software to run the simulations we have designed for this experiment. Also, the pre-/posttest was incorporated in the MATHEMA. The system contains eight pages for the activity. Also, the system contains three pages for the prerequisite knowledge (e.g., circular motion, uniform magnetic field, etc.). For the activity, the system supports 24 guided dialogs concerning the questions of Table 4 (eight for checking the results of radius, eight for checking the results of period, and eight for checking the results of motions), and one for the set of restrictions. In addition, the system contains a dictionary of physics terms and help for more information about the system.

2. *Questionnaires*. We gave the students a questionnaire to fill in after the experiment, in order to express their opinion about our system.

3. *Pre-/Posttests*. The assessment of the students' learning performance is performed through assessment tests before and after the experiment. The pretest and the posttest for the activity included five questions of identical form. These questions are intended to detect misconceptions and learning difficulties of students before and after the experiment. For example, a question to detect a misconception is the following: *If a particle which has no charge ( ) is moving with a velocity , perpendicular to the direction of a uniform magnetic field, then its value of radius R of the circular trajectory is:*

1. *zero,*

2. *infinitive,*

3. *determinable (a certain value), and*

4. *indeterminable.*

The pretest was given to the students before the experiment. The posttest was given to the students five days after the experiment.

**3.1.3 Experimental Procedures**In our experimental study, the activity that the students carried out through the MATHEMA belongs to the section of electromagnetism entitled:

*Motion of a charged particle perpendicular to the direction of a uniform magnetic field.*The learning outcomes that were presented by the MATHEMA to the students are that the students will be able to:

1. *Synthesize* given mathematical formulas of the electromagnetism in order to *extract* the formulas of radius R and period T of the particle circular motion.

2. *Set restrictions* on the values of parameters of the extracting formulas through the *guided dialog* with the system.

3. *Apply* the extracting formulas for the values of the parameters q and v listed in Table 4, in order to *calculate* the corresponding values of radius R and period T of the particle circular motion.

4. *Predict* the motions of the particle. For the prediction of the kind of particle motion, the students made use of the *right-hand rule*.

5. *Set* the values of the parameters of the extracting formulas, q and v, listed in Table 4, to a given simulation and to *run* it.

6. *Compare* the calculated (or predicted) results with the simulation results and *explain* the differences.

7. *Collaborate* with his/her peer for the correction of results.

8. *Revise* their possible mistaken beliefs or miscalculations through the *guided dialog* with the system.

**3.1.4 Data Collection**In order to investigate the research questions, quantitative data were collected by the embedded in the MATHEMA evaluation test used as pretest and posttest. Also, data were collected from the responses of the students that our system keeps in the database. The questionnaire for the expression of students' opinions about our system was completed by hand with pencil and paper.

**3.1.5 Analysis Method**For the analysis of results, because of the small number of the participants (less than 30), we make use of the independent two-sample t-test. The one sample is the pretest scores and the other sample is the posttest scores of the participants. A significant level of was adopted for the study.

*guided dialog*in Step 6 with the system, all the students gave correct answers.

Table 5. Mean Scores and Standard Deviations of Means

Table 6. Students Who Gave the Correct Answers to the Questions of Table 4 for Each Step of the Activity

*there was a significant improvement of learning performance of the students when they carried out the activity.*

*1) perceive their wrongs and revise their points of view, 2) justify their choices, and 3) accept that they must set restrictions on the values of parameters of formulas.*

• *A. Papadimitriou and M. Grigoriadou are with the Department of Informatics and Telecommunications, University of Athens, Panepistimiopolis, Ilissia, Athens, GR 15784, Greece. E-mail: {alexandr, gregor}@di.uoa.gr.*

• *G. Gyftodimos is with the Department of Philosophy of Science, University of Athens, Panepistimiopolis, Ilissia, Athens, GR 15771, Greece.*

*E-mail: geogyf31@di.uoa.gr.*

*Manuscript received 29 Dec. 2008; revised 10 Mar. 2009; accepted 2 Apr. 2009; published online 20 Apr. 2009.*

*For information on obtaining reprints of this article, please send e-mail to: lt@computer.org, and reference IEEECS Log Number TLTSI-2008-12-0142.*

*Digital Object Identifier no. 10.1109/TLT.2009.19.*

#### References

**Alexandros Papadimitriou**received the diploma from the Department of Electrical and Computer Engineering, National Technical University of Athens (NTUA), in 1992, and the MSc degree from the Department of Mechanical Engineering, NTUA, in 2004. He is currently working toward the PhD degree in computer science in the Department of Informatics and Telecommunications, University of Athens. His current research interests include the areas of adaptive educational hypermedia systems, adaptive group formation and peer help, interactive problem solving support, and meta-adaptation techniques. He was the recipient of the Outstanding Paper Award of the ED-MEDIA 2008 Conference on Educational Multimedia, Hypermedia, and Telecommunications.

**Maria Grigoriadou**received the BA degree in physics from the University of Athens in 1968 and the DEA and doctorate degrees from the University of Paris VII in 1972 and 1975, respectively. She is now an associate professor in education and language technology and head of the Education and Language Technology Group, Department of Informatics and Telecommunications, University of Athens. Her current research interests include the areas of adaptive learning environments, Web-based education, ITS, educational software, natural language processing tools, and computer science education. She was the recipient of seven awards, has participated in 15 projects, and has four invited talks to her credit. She has 27 publications in international journals, six in international book chapters, 115 in proceedings of international conferences, 11 posters, and more than 300 citations to her research work. She is a member of the IEEE, AACE, IADIS, EDEN, Kaleidoscope, and LeMoRe.

**Georgios Gyftodimos**received a degree in mathematics and a PhD degree in informatics. He is currently an assistant professor in the Department of Philosophy of Science, University of Athens, where he participates in the Interdisciplinary Postgraduate Program on Cognitive Science and teaches courses in AI, evolutionary programming, and simulation. His research interests lie in the domains of knowledge representation and modeling for cognitive purposes. He is an IEEE fellow.

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