An Exploration of Computer Based Learning Technologies for the Teaching of Mathematics: eLearning, Intelligent Tutoring Systems, Computer Algebra Systems, and Dynamic Geometry Systems

Computer Based Learning Technologies

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A paper based on my term paper for EDCI 691, Spring 2008, at Texas A&M University.

Abstract

We reviewed Computer Based Learning Technologies (CBT) for the learning and teaching of mathematics. Then, based on a review of the relevant research literature, we selected eLearning and Intelligent tutoring systems as the technologies that we considered most effective in the teaching of mathematics in general and of geometry in particular. These CBTs are described in more detail and their advantages and disadvantages are examined and compared. However, these CBTs are most powerful when associated with two types of mathematics programs: Computer Algebra Systems (CAS), and, especially for the learning and teaching of geometry, Dynamic Geometry Systems (DGS). Those also were described and compared. Furthermore, we present a case for the preference of open source computer technologies over commercial ones. Finally, we proposed some questions for further research of these CBTs, with emphasis on their integration.

Introduction

Computer Based Learning Technology (CBT) or Computer Assisted Instruction (CAI) comprises a wide array of technologies that have little in common beyond the use of computers. Many educational computer programs have been written on many subjects and for all levels of proficiency. They can be delivered (sold) by CD-ROM or DVD disk and installed on computers. Other programs can be downloaded from the Internet and then installed on the computer. Some programs, such as Java applets1, do not need to be installed. These are downloaded from a remote web server through the Internet and will temporarily run in the browser of the user’s computer.

CBTs can be classified based on the platform on which the educational program is located and on which computer it runs. Some educational programs are installed on the user computer and run there locally. Other educational programs can be accessed over the Internet and are run remotely. These remote programs run on a server2 and are displayed in and interacted with through a browser running on the client computer.

Programs that run locally have the advantage that no network connectivity is required. The disadvantages are that these programs need to be installed, upgraded and maintained on the computer of the learner. Only one person at the time can use it. In a class setting this will require a lot of work by the IT staff. The programs also need to be obtained by CD-ROMs or DVD or be downloaded before being installed.

Programs that run over the Internet have the obvious disadvantage that net connectivity will be required, usually constant and broadband. This type of connection is usually costly and is not available everywhere. This is less of an issue in the USA and in other developed countries. However, in less developed countries broadband Internet connectivity is scarce outside the major cities. These programs are collectively called eLearning, which is a form of distance learning that uses the Internet. Thus, students who are not able to attend schools due to distance, illness, or disability can still have access to education. There are also cost advantages since school infrastructures would not be necessary or could be significantly reduced. Since the eLearning programs are located centrally in facilities staffed with IT professionals, the users do not need to download or install anything besides an Internet browser. eLearning programs are computationally intensive and thus need to run on powerful servers that are costly, require professional attention for updates, maintenance and backup, and necessitate abundant electricity, cooling, large bandwidth, and almost continuous up-time. That means that a remotely located student with even an obsolete computer, but with decent Internet access, can benefit from these sophisticated eLearning applications.

Another type of CBT classification is based on the level of human management and level of “intelligence” of the programs. CBTs that require the lowest level of human interaction and have little intelligence are the autonomous procedural educational programs. These programs need to be installed on the user’s computer or on a web server and students need to be given access. Further teacher input is not necessary except for checking progress and occasional replying to questions from students. These programs are fairly limited in their scope and flexibility and operate according to a more or less complex algorithm. They provides a relatively simple set of lessons, samples, exercises, and tests for the student who provides the input. These programs can be installed on personal computers, run as applets in browsers, or as web-based programs. This simple kind of educational software has not proved to be of great use to the students except to supplement lectures and activities. They have been extensively described in the literature. As an example of geometry learning see Paas and Merriëboer (1994), and Kerrigan (2002) for a review of mathematics programs for grades K-6. However, the current interest of educators and students for them is low. They are now considered obsolete, and thus we will not further discuss them.

Intelligent Tutoring Systems

There are CBTs that require very little human intervention once they are authored, installed and configured. They are the Intelligent Tutoring Systems (ITS). These program do not simply operate according to a more or less sophisticated algorithm, but rather have an intelligent agent, “a system that perceives its environment and takes actions which maximize its chances of success” (Russell & Norvig, 2002). There are many types of AI engines, however the most popular one among ITS programs are probabilistic methods for uncertain reasoning derived from Bayesian networks (Pearl, 1985), which are later described in this paper. Psotka and Mutter (1988) use the following definition for intelligent tutoring systems: “broadly defined, it is any computer system that provides direct customized instruction or feedback to students, i.e. without the intervention of human beings.” McArthur and McArdle (1990) and Steffe and Nesher (1996, p. 181) use the term Intelligent Computer Assisted Instruction (ICAI), and Baylor (1999); Baylor (2000) calls them Intelligent Agents. The importance of ITS in mathematics education has been growing tremendously lately due to the improvement of their intelligent agents, content, and authoring tools. We will describe them in more detail in this paper.

eLearning and Course Management Systems

CBT programs that require the highest level of human intervention are the Course Management Systems (CMS), a popular eLearning instructional technology, which does not typically use AI. Since they are complex and multi-functional, but also have little intelligence, CMSs require considerable and constant teacher input. CMSs are also called Virtual Learning Environments (VLE), On-line learning, or web-based education. More correctly, instead of a single program CMSs are a structured bundle of programs that run on a web server and provide a virtual classroom in a browser. The teachers will add content (text, sound, animations, and video) to the program. Teachers need to review and grade the assignments that are uploaded by the students. They monitor the discussion boards and often contribute to them. Further activity that the teacher needs to perform is checking for late submissions, communicate by discussion board, chat, or email with the students who may ask for clarifications and directions. CMSs are widely used as an eLearning platform for computer based distance education. Due to their importance in eLearning and their general usefulness in mathematics teaching they will be further discussed.

Computer Algebra Systems

Often other types of programs are used in conjunction with mathematics CBT. The most used ones are Computer Algebra Systems (CAS), software programs that perform symbolic and computational mathematics. For example, they allow the manipulation of polynomial or trigonometric expressions. They also have calculus capabilities such as integration, derivation, and the solving of ordinary differential equations. A CAS will also allow numerical calculations and matrix algebra and many other types of mathematical operations. They also can produce impressive 2-dimensional and 3-dimensional representations of functions and other types of plots. There are many CAS programs available, either commercial or free.

Dynamic Geometry Systems

A type of computer program that has been successfully used in learning geometry are the dynamic geometry programs, also called Dynamic Geometry Software (DGS), Interactive Geometry Software (IGS), or Dynamic Geometry Environment (DGE). The most common acronym is DGS. These are computer programs that allow the user to build and modify geometry constructions, usually in Euclidean plane geometry even though solid geometry and non-Euclidean DGS do exist. The user generally starts by placing points on a plane and using those to build circles and lines to construct geometric figures. Then, one or more points can be dragged to different positions and changes to the geometry construct can be observed (Goldenbert & Cuoco, 1998; Olive, 1998). Dynamic Geometry Systems will be further described.

Due to reasons explained later in this paper, we agree with Bork and Gunnarsdottit (2001) who stated that the optimal form of computer based mathematics education for today is eLearning ITS, that is, an Intelligent Tutoring System that is provided to the learners over the Internet from a central location. In our opinion, the optimal solution would be an ITS embedded in a CMS (ITS-CMS) and integrated with a CAS and a DGS.

We describe these computer technologies in more detail giving preference to free software for the obvious reasons of their zero licensing cost, Internet accessibility, and the freedom to modify, to integrate, and to distribute them to all.

eLearning – CMS

Description, Advantages & Disadvantages

As described above, eLearning simply means that the learner will connect through the Internet with the instructional program. This program may rely on artificial or human intelligence or a combination of both. In the first case the eLearning program corresponds to an ITS that is provided through the Internet. This case is discussed below. Here we describe and compare Course Management programs, which at best have simple algorithms to assist the users and the teachers in using and managing the virtual classroom.

All CMSs share some common features. The students will use their browsers to navigate to the web site of the CMS and will log-in. Usually there will be a list of available classes that the student can click to access. That will load the virtual classroom in the student’s browser. Here the student will access the various resources uploaded by the teacher starting from the syllabus and including lecture notes, articles, presentations, videos, images, and other types of files. There will be an area where the students can upload their assignments. Usually a discussion board is included, an area where the students can check their grades. There may be a calendar and support for Internet chat, email, and word processing. The CMS could contain parts that do not require the interaction with the teacher such as quizzes that will be checked and graded by the CMS itself.

CMSs are becoming increasingly popular at universities and other institutions of higher education for several reasons. Their faculty and students are generally computer literate and many students have their own computers. Students in academia usually require less supervision and are better at planning their learning efforts. Often CMSs are used more as complementary to lectures than as their replacement. Most academic assignments and projects are in electronic form and thus easily uploaded from the student’s computer to the CMS. Finally, universities generally have computer centers with UNIX/Linux servers that, unlike Windows servers, can host several applications at the same time. Thus a university can simply add the CMS program to a UNIX/Linux server that may already be in use as an email, web, or database server, obviating the need to purchase more hardware.

The use of Virtual Learning Environments for the teaching of mathematics has been described in the scientific literature. See for example Chinnappan (2006) for WebCTTM (now Vista), McClendon and McArdle (2002) for a system called ALEKS, while Dougiamas and Taylor (2003) investigated on-line classes that were built using a CMS called Moodle. For examples of non-mathematics studies see Brown, Brown, and Griffin (2006) and ICTE (1999).

Dillenbourg, Schneider, and Paraskevi (2002) discussed whether VLEs can really improve education and reduce its cost. The authors made the case that, as with all previous technologies that were adopted by schools, it is the proper implementation that will improve instruction. They stated that “media have no intrinsic effectiveness, only affordances.” Chinnappan (2006) studied the implementation of WebCTTM for a cohort of beginning mathematics teachers. The author reported that the implementation was generally successful.

Some projects are taking the concept of virtual classroom to the next level, which is stepping up from a 2-dimensional environment to a 3-dimensional one. Among them is Sloodle (http://www.sloodle.org), an open source project that integrates Moodle with Second Life (http://secondlife.com).

There are many different virtual learning environment programs available, both open source and commercial. The most popular ones are described and compared here below. The Instructional Technology Council has recently published the results of a survey of US community colleges (Council, 2008).

Since there is a large array of CMS programs that schools and academic institutions can choose from, what should the basis of their choice be? Usually complex software systems are chosen based on flexibility, features, ease of authoring, ease of administration, and cost. Considering that all CMS, commercial or free, have comparable sets of features, the distinguishing features are reduced to flexibility, ease of authoring and administration, and cost. Considering that commercial VLEs have high costs and are not significantly easier to use or manage than free solutions, a free CMS is the best candidate. It appears that for most schools and universities the choice is Moodle. Indeed, the relevant news frequently reports about educational institutions switching from a Blackboard CMS to Moodle.

Another solution would be to build a “home grown” CMS. With free software all the necessary components do exist. If the institution has in-house skills or is willing to hire outside contractors, a CMS could be build using, for example, Linux as the operating system, Apache as the web server, PHP as the programming language, and MySQL as the database. All these are free as well as capable of running on obsolete and salvaged computers if necessary.

The most popular of these CMSs is called Moodle. It is a cross-platform eLearning program that can be downloaded freely from http://moodle.org. According to the statistics provided by its web site (http://moodle.org/stats), in April 2008 there were 42,457 registered Moodle eLearning sites, 1,873,657 courses, and 19,768,999 users speaking over 70 languages. More statistical data are available on that web page. This popularity is not surprising due its high quality, extensive feature set, zero cost, and active developer and user community. This is reflected in its positive reviews, e.g. Stanford (2008) and http://moodle.org/mod/data/view.php?id=6140. Several books have been written about Moodle. Among the most recent ones we have Cole and Foster (2007) and Rice (2006) that teach how to install, to administer, and to build virtual classes with Moodle.

The system runs on all major operating systems (UNIX/Linux, Mac OS X, and Windows) and requires a web server, a PHP interpreter, and any of the major free or commercial SQL databases. Thus it is possible to build a completely free system that has complete functionality. The program is continuously improved and new versions are released weekly. The latest numbered version, 1.9.6, was released on 21 October 2009.

Dokeos is a free and open source VLE, but it is managed by a company instead of being a community project such as Moodle. It can be downloaded from http://www.dokeos.com. The company charges for support and additional features such as video conferencing. The company reports having more than 200 customers and an active user community. The CMS is most popular in French speaking Europe.

There are several other free course management systems that are somewhat less popular than the previous ones. Notable examples are ILIAS (http://www.ilias.de) and Sakai (http://www.sakaiproject.org).

Blackboard Inc. (http://www.blackboard.com) is a software company that, among others, publishes an array of course management programs. Members of these systems are the Blackboard Academic SuiteTM and Blackboard K-12. The company claims in its literature (Blackboard, 2007, p. 20) that it has 3,400 academic clients, millions of active users, and more than 1,000 external developers who are member of the Blackboard Developer NetworkTM. The company has also reported that it is deployed in over 1,200 US schools and that 41 of the top 100 American high schools named by Newsweek rely on Blackboard software (2008, p. 3).

There are several other companies that sell CMSs capable of providing on-line mathematics courses. For example Apex Learning® (http://www.apexlearning.com) sells a VLE that provides on-line standards based mathematics courses for grades 6 through 12 as well as courses in many other subjects.

Intelligent Tutoring Systems

Description, Advantages & Disadvantages

Due to their many appealing characteristics, Intelligent Tutoring Systems have attracted great interest from many, e.g. academia, schools and school districts, for-profit companies, and even the U.S. Armed Forces (see VanLent, Core, Lane, & Willis, 2005). Intelligent Tutoring Systems provide one-to-one learning environments which diagnose the strengths and weaknesses of each student, then tailor instruction to their specific needs, and provide on-the-spot feedback through appropriate hints, examples, encouragement, and exercises. This all occurs with an inhuman degree of patience, persistence, and accessibility. This process of diagnosis, tailoring, and feedback is continuous and iterative where the AI learns from the student and the student learns from the ITS. Another significant advantage of ITS is that their latest generation can be web-based and are thus a form of eLearning.

How do ITS measure against conventional and other modes of instruction? Bloom (1984) compared the three basic forms of teaching: conventional classroom (1:30 teacher:student ratio), mastery level classroom (1:30 teacher:student ratio) where the students received additional explanations and assessment, and tutoring (1:1-1:3 tutor:student ratio). The author calculated the effect size of the cross-subject class scores to compare these three instructional methods. He subtracted from the average score of the mastery level class or tutored students the average score of the class with conventional teaching and divided this difference by the standard deviation of the scores of the conventional teaching class. This way the effect size of conventional teaching is zero and thus becomes the baseline for the effectiveness of the instructional methods. Bloom discovered that the highest effect size was associated with the tutored students, plus two standard deviations (+2 σ), while the mastery level teaching had an effect size of plus one standard deviation (+1 σ, see Bloom, fig. 1). From this study we have the expression “The 2-Sigma Challenge.” That is, the goal of pedagogical research is to find instructional methods that approach and eventually will reach the effectiveness of one-to-one tutoring, which is two standard deviations above conventional teaching. This is not small a challenge because a two-sigma effect size is considered huge in social sciences. In most psychological research a 0.5 effect size is already considered large. To illustrate the significance of this study, the author noticed that a two-sigma difference corresponds to two letter grades. That is a C-student in a conventional class will become an A-student if tutored. Conversely, a tutored C-student will fail in a conventional class.

Lane (2006) reported that the best ITS offer learning gains of about one standard deviation, while computer assisted instruction without artificial intelligence improves learning by only 0.4 standard deviations (+0.4 σ). Hence the challenge of developers of “artificial tutors” to increase their effect size to match those of human tutors. The author stated that tutoring is so effective because tutors are able to “balance the need for active participation of the student with the provision of guidance. This means the student does as much of the work as possible while the tutor provides just enough feedback to minimize frustration and confusion.” Thus, it is the interaction between the student and the tutor that is critical. It is interesting to notice how these data agree with the Constructivist Learning Theory (see Hmelo-Silver, Duncan, & Chinn, 2007). As just stated, with tutoring we can obtain the largest ratio of learning activity (reading, reflection, observation, problem solving) performed by the student. We suggest that this could be expressed symbolically as

where W stands for amount of work expressed as duration of the learning activity. WSTUD is the work personally performed by the student, such as reading, problem solving, and drawing. WTOT represents the total duration of the learning process and will also include the time that the student is passive such as when listening and waiting. This representation shows that students in a conventional classroom (CONV) are the most passive and are thus less engaged in building their personal knowledge than students in a mastery level (ML) class, while the tutored (TUT) students are the ones who are the most active in constructing their knowledge.

A recent study by Beal, Shaw, and Birch (2007) seems to indicate that the 2-Sigma Challenge can be achieved by a mathematics ITS. The researchers used an ITS developed at Southern California University called AnimalWatch to compare in middle school students the efficacy of human pre-algebra tutoring and mixed human-ITS tutoring. The results were quite encouraging since the post-test scores did not show any statistically significant difference between the two forms of tutoring. However, it should be noted that the ITS was not used by itself, but was combined with instruction by a human tutor.

Matsuda and VanLehn (2005) described the strategy of interaction of a geometry ITS called AGT with the students. The authors distinguished between proactive and reactive scaffolding. Proactive scaffolding happens before the input of the student at a certain step, while reactive occurs after the students have given their input. The authors followed a tutoring strategy established by Wood, Wood, and Middleton (1978). The rule is “If the child succeeds, when next intervening offer less help; If the child fails, when next intervening, take over more control.” The ITS will increase or decrease the competence level of the students based on their sequence of successes and failures. In increasing order of competence the ITS will decide whether to ‘show-tell’, ‘tell’, or only ‘prompt.’

The reactive scaffolding will strategy will start with minimal feedback and then vary the response based on the competence level of the student. With each failure the tutor will offer more guidance and be more specific.

It appears that tutoring is so effective because it maximizes the Zone of Proximal Development (Vygotsky, 1978). The Zone of Proximal Development (ZPD) was defined by Vygotsky as the difference between the level of difficulty of a subject that the student can understand with the help of a teacher/tutor and the level of difficulty that the student can understand without assistance. A more recent (and complete) definition was given by Bork and Gunnarsdottit (2001, p. 56) as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined by problem solving under adult guidance or in collaboration with more capable peers.”

Beal, Mitra, and Cohen (2007) reported that the theoretical framework of ITS research is indeed based on the Zone of Proximal Development. The author stated that “instruction that is individualized and responsive to the student’s ongoing performance will be most effective.”(Brown et al., 1994 as quoted by Beal, Mitra, & Cohen, 2007).

It our opinion that the effectiveness of an instructional method is directly proportional to the ZPD and inversely proportional to the time (e.g. hours of tutoring/teaching/learning) required to achieve this ZPD. Thus, we suggest the following metric to quantify the instructional effectiveness (IE) of a method as3

The concept of instructional efficiency is not new. About 70 years ago Georges (1931) defined it as the ratio between the actual increase in learning and the maximum possible learning that can be achieved. We do not consider this definition to be useful since, while we can measure (or estimate) the value of the numerator of the fraction, the value of the denominator is an abstract concept that by its own definition can not be measured.

An interesting observation is that Vygotsky was a Social Constructivist according to whom a very important component of child development is culture and interpersonal communication. However, with tutoring and especially with ITS, this aspect of the learning process would be minimal or absent4. However, as we can see later there is a trend to integrate the ITS into a virtual classroom to give the students a degree of human interaction5.

Schofield, Eurich-Fulcer, and Britt (1994) presented a paradoxical research finding in their study of ITS use with high school students. The study found that students considered a teacher more helpful than an artificially intelligent tutoring system, but nevertheless preferred the artificial tutor. The authors suggested three reasons to explain this inconsistency: (1) the ITS did not actually replace the teacher, but rather provided additional help to the students; (2) the use of the ITS allowed teachers to provide more individualized help; and (3) students using the tutors had more control over the kind and amount of help they received from the teacher, with helping interactions becoming more private and potentially less embarrassing. Interestingly, these effects were not foreseen by the developers of the ITS. This stresses the importance of being aware of unintended effects on instruction by the introduction of a technology.

Anderson, Douglass, and Qin (2005) studied the response time of an ITS for high school mathematics from a cognitive psychology point of view. The ITS under study took 20 seconds to respond to an action by the student. However, acts of cognition by the learner are thought to occur in much shorter time frames. The authors and their collaborators have created a series of computer-based intelligent tutors for high school mathematics. Anderson et al. claimed that they were effective learning tools.

The authors belonged to the Pittsburgh Advanced Cognitive Tutor Center at Carnegie Mellon University (http://pact.cs.cmu.edu). The intelligent math tutor developed by them was called the PUMP Algebra Tutor (PAT), which was used for high school mathematics and has been effective in improving learning (Anderson, Corbett, Koedinger, & Pelletier, 1995; Koedinger, Hadley, Hadley, & Mark, 1997). The theoretical basis for this project was the ACT Theory of Learning and Problem Solving, which evolved into the Automatic Components of Thought-Rational (ACT-R) Theory (Anderson et al., 1995; Anderson et al., 2005). This ITS has been basically abandoned as a free project and merged with Cognitive Tutor® which is a commercial product published by Carnegie Learning, Inc. This company has released several studies that document its positive effect on mathematics learning (McGuire & Ritter, 2006). The Cognitive Tutor has also been tested in peer-review research performed at Carnegie Mellon University (Koedinger et al., 1997). This ITS is described later in this paper.

There are benefits of the ITS that we have not encountered in the literature. It reduces, or eliminates, the time spent by the teachers in grading and proctoring. It is well known that teachers usually have an excessive workload. Among the many negative outcomes of this situation is a decrease in the motivation of the teachers. Another effect is that it reduces the time that teachers can devote to professional development, lesson preparation, and face-to-face meetings with students. We are all aware of the low motivation of US public school teachers and their high turn over rate (see Darling-Hammond, 1997). Another benefit is the immediate feedback on errors and misconceptions. The student does not have to wait until the worksheet is checked or the homework graded and returned by the teacher. Teachers do not always have time to check all the problems and to quickly return the homework to the students. The consequence is that when this feedback occurs the teacher has already moved on to the next subject. Not to mention the fact that there is no guarantee that the students will review the corrections made by the teacher and thus correct their misunderstandings and fill their gaps in knowledge. The worksheet or homework has lost its relevancy. Obviously an ITS will give feedback in real time ensuring that misunderstandings are immediately addressed so that the students will quickly and efficiently construct their mathematical knowledge.

Intelligent tutoring systems have their sceptics. One of the main criticisms is that the tutor does not really allow students to learn from their mistakes because the program vigorously guides them toward the right answer.

Jim Greer of the Department of Computer Science at the University of Saskatchewan (http://www.cs.usask.ca/people/faculty_profiles/greer.shtml) and member of the laboratory for Advanced Research in Intelligent Educational Systems (ARIES, http://ai.usask.ca) stated that “Some pedagogical researchers feel that because cognitive tutors don’t allow learners to go very far off a correct path before intervening, students won’t develop skills for coping with really tough problems,” (quoted by Hafner, 2004).

The cost of producing a valid ITS is high. It requires the building of an ontology, a semantic system that contains all the required subject knowledge as well as the pedagogical knowledge. Ontology is a philosophical term that was adopted by the study of artificial intelligence (Swartout & Tate, 1999). The authors defined an ontology as “a set of concepts or terms that can be used to describe some area of knowledge or build a representation of it. An ontology can be very high-level, consisting of concepts that organize the upper parts of a knowledge base, or it can be domain-specific, such as an ontology of vehicles.” Chandrasekaran, Josephon, and Benjamins (1999) provided more information about the importance of ontologies in AI. The term “authoring” is used in the case of intelligent tutoring systems (as well as CMS). At the present state of development of ITS authoring has to be done by skilled teachers and ITS researchers. Research is being performed currently with the aim of reducing the complexity of authoring, thus making it accessible to most mathematics teachers. For examples see Murray (1999); Manzoor, Libbrecht, Ullrich, and Melis (2006), and Goguadze and Tsigler (2007). Noy and Mcguinness (2001) gave an introductory explanation of ontologies and how to build them.

Viadero (2004) published a review study that gave a sobering view of the situation of computer based mathematics tutoring systems used in middle schools in the USA. The US Department of Education reviewed the efficacy of 44 mathematics computer programs used in grades 6-9. These were compared with normal textbook instruction. This study found that only two programs offered better results than the control. These two were computer based algebra tutors called “I Can Learn Mathematics” and Cognitive Tutor® by Carnegie Learning, Inc. This report was the first one to be published by the new What Works Clearinghouse of the Department of Education. This clearinghouse evaluates research on educational programs, policies and products and then publishes these results on a web site.

The need for human interaction is never completely avoided. Students will always need to interact with real teachers. These teachers will also need to monitor the progress of the students and make sure that they do not ‘game’ the system. However, Baker, Corbett, and Koedinger (2004) presented a computer model that was able to detect whether students were misusing the ITS. The authors discovered that those students were learning only 2/3 as much as students who were using the ITS properly. Modern ITS have reporting modules that keep the teachers informed of the progress of the student. This functionality would be very important with distance education, hence the importance of ITS-CMS integration.

Artificial Intelligence

There are several types of intelligent agent, e.g. neural networks, search and logic programming, and probabilistic methods for uncertain reasoning. In ITS the last type is the most used.

Probabilistic methods for uncertain reasoning are based on the mathematical concept of Bayesian networks. Jensen (2001) defined Bayesian networks mathematically as a set of variables (nodes) and a set of directed edges between variables that represent their probabilistic relationships. Each variable has a finite set of mutually exclusive states. The variables together with the directed edges form a directed acyclic graph. Since a Bayesian network is a complete model it can be used to ask probabilistic questions about the model. An interesting characteristic of these networks is that they update themselves when some events are observed. Thus, it can be used to make probabilistic inferences. The Bayes Theorem allows us to compute posterior probabilities based on given observations

Where P(A) is the prior probability of A; P(A|B) is the conditional probability of A given B, also called posterior probability of A given B; P(B|A) is the conditional probability of B given A; and P(B) is the prior probability of B and acts as a normalizing constant (Garcia, Amanda, Schiaffino, & Campo, 2007).

Garcia et al. (2007) described a study where Bayesian networks were used to determine the learning style students using an eLearning system. The Bayesian network modelled different aspects of the behavior of those students and the AI was thus able to make inferences of the learning styles. The authors reported that the results were promising.

A generalization of Bayesian networks is the Dempster-Shafer Theory of Evidence. An interpretation of this Theory of Evidence that is used in ITS is the “Transferable Belief Model” (Smets, 2000). According to the author, the usefulness of the Theory of Evidence is that it is a “flexible way to represent uncertainty, be it total ignorance or any form of partial or total knowledge, that is more general than what the probabilistic approach provides, and a rule to combine uncertain data, called the Dempster’s rule of combination, that seems to provide an excellent tool for data-aggregation.” Without going into the mathematical intricacies of the subject, the transferable belief model provides a model for the representation of quantified beliefs. A belief function is defined by Shafer (1990, p. 1) as “a non-Bayesian way of using mathematical probability to quantify subjective judgments. Whereas a Bayesian assesses probabilities directly for the answer to a question of interest, a belief-function user assesses probabilities for related questions and then considers the implications of these probabilities for the question of interest.”

Another AI technology used in ITS is the Hierarchical Task Network (HTN). Erol, Hendler, and Nau (1996) described HTN as a representation of the ‘world’ and its relevant ‘actions.’ Each state of the world is represented by the set of atoms that are true in that state and actions correspond to state transitions. A task network is a set of the required tasks with their order constraints and the way that the variables are instantiated and what literals must be true before or after each task is performed. See Lekavý and N�vrat (2007) for a more recent description of HTN.

In simple terms, HTN planning is a form of problem reduction where we define tasks rather than goals and the methods to decompose these tasks into sub-tasks. We also enforce constraints and the planner will backtrack if necessary. This way we do not have to define all the possible combinations of states of the world. A simple illustration is given in (Nau, 2008).

The Hidden Markov Model (HMM) is an AI technology that only recently has been used in building Intelligent Tutoring Systems (Beal, Mitra, & Cohen, 2007). The authors used HMM to model the behavior of individual students. Beal, Mitra, and Cohen (2007) to model the “engagement level” fit HMMs to the sequence of events produced by individual students who were interacting with a mathematics intelligent tutor.

From a mathematical point of view a Hidden Markov Model is a statistical model that assumes that the system to be modeled is a Markov process with unknown parameters. The model attempts to determine the hidden parameters based on the observed parameters. The state transition probabilities of the system are not visible. However, these hidden variables determine the state of the visible variables. The HMM is considered to be the simplest dynamic Bayesian network (Rabiner, 1989).

Review of major Commercial and Non-commercial ITS

Cognitive Tutor.

One of the more successful ITS in mathematics education is the Cognitive Tutor® from Carnegie Learning, Inc. (http://www.carnegielearning.com). This product offers full curricula for middle and high school mathematics. According to the description of the product given by the publisher

Using the Cognitive Tutor, students receive the benefits of individualized instruction, ample practice, immediate feedback and coaching. “Just-in-time” help, “On-demand” help, and positive reinforcement put students in control of their own learning and help to keep them on task. This supports better classroom interaction, too, because teachers can spend more time with students who need additional intervention.

The software is based on research performed at the Pittsburgh Science of Learning Center, a $25 million operation run jointly by the University of Pittsburgh and the Carnegie Mellon University. The company reports striking mathematics, including geometry, score improvements. The ITS has been extensively tested and is implemented across many middle and high schools in the USA. Courses are also available for home schooling.

The program can be run in three different configurations: (1) run remotely from Carnegie Learning servers; (2) Client/Server installation local to the school, (3) stand-alone. The first configuration is thus a ITS-eLearning solution, even though a client program has to be installed on the user computers. Unfortunately, server, client, and stand-alone configurations exist only for Windows and Mac OS X computers. This, with the cost of the ITS itself and the IT administrative costs, makes it very expensive for schools to operate this form of CBT. Since Cognitive Tutor is a closed source program, no information is publicly available on its internal structure and components.

Advanced Geometry Tutor.

As previously mentioned, a very active AI-ITS research group composed of researchers of the Carnegie Mellon University and the University of Pittsburgh created the PUMP Algebra Tutor, which evolved into the Practical Algebra Tutor (PAT, Koedinger et al., 1997) end eventually into the Cognitive Tutor. The same research group recently created a geometry intelligent tutor that helps students with theorem proving called the Advanced Geometry Tutor (AGP, Matsuda & VanLehn, 2005). The authors published the results of an experiment performed in 2004 with 52 University of Pittsburgh students. The goal of the study was to compare the efficacy in learning and using the forward (FC) or backward chaining (BC) strategy in proving. The experimental design was the usual pre-test, intervention, post-test, however the students had to study a 9-page geometry booklet before taking the pre-test. The intervention consisted of solving 11 geometry problems using AGP. The results showed that students who used FC were better in proof-writing than those who used BC. Also, both FC and BC strategies gave the same frequency of incorrect proofs.

DISCOVER.

Steele and Steele (1999) developed an intelligent tutoring system called DISCOVER. This software is composed of 11 independent computer programs, some of which are using expert systems technology. The system was studied with students who had difficulty solving mathematics word problems. The software used direct instructional approaches and taught the problem solving skills in a systematic and sequential manner. However, also this project seems to have been abandoned.

LIM-G.

Wong, Hsu, Wu, Lee, and Hsu (2007) have developed in Taiwan an artificial tutor for the learning of geometry. The authors call it a learner-initiating instruction model which is based on cognitive knowledge comprehension. This system is called a Learner-initiating Instruction Model or LIM-G. The purpose of this software system is to help elementary school students solve geometry word problems. LIM-G is an interactive tool that can deal with standard and non-standard word geometry problems. The engine of this software is InfoMap which is an ontology-based knowledge engineering tool. The system was tested in Taiwanese schools where it achieved 85% full comprehension, 15% partial comprehension, and 0% incomprehension.

This ITS has some interesting features, such as providing diagrammatic problem representations to help students focus on relevant information and logic. The system is very intelligent is responding to students input and is able to diagnose their level of understanding.

Wayang Outpost.

A popular intelligent tutoring system was developed under the leadership of Carole Beal at the University of Massachusetts6. The focus of this ITS is to provide tutoring to students who are going to take the Scholastic Aptitude Test (SAT, Arroyo, Beal, Murray, Walles, & Woolf, 2004). A distinguishing feature of this ITS is the use of multimedia animations (Flash) and a colorful, playful contexts.

Beal and Qu (2007) used Wayang Outpost to study the use of a Dynamic Bayesian Network to estimate the learning goals of about 115 high school students. The authors found that these estimates were able to predict the performance on a post-test of mathematics achievement, while the pre-test performance was not able to predict the post-test performance.

Wayang Outpost was used by Beal, Walles, Arroyo, and Woolf (2007) to study the ITS to tutor a group of students to solve SAT-Math problems involving geometry skills. The results were not overwhelmingly positive. The student who had the intervention improved their post-test scores, while the control students did not. However, the improvements in problem solving skills were limited to the set of problems given by tutor.

ActiveMath.

Researchers at the University of Saarland in Germany (http://www.uni-saarland.de) and the German Center for Research in Artificial Intelligence (DFKI, http://www.dfki.de) have developed an advanced intelligent tutor called ActiveMath specifically for mathematics education (http://www.activemath.org). The program is a complete ITS, that means it is adaptive to the students, it diagnoses student errors and misconceptions, has a course generator and a semantically encoded knowledge base. Besides being a complete ITS, ActiveMath has several other useful features. The program is a Java web application and is accessed over the Internet through a web browser7 making ActiveMath both an ITS and an eLearning program.

The program is completely open, is written in Java and XML, and uses open and unencumbered standards8. ActiveMath can be freely distributed, installed and upgraded. The project also provides authoring tools that allow teachers and researchers to create new lessons. ActiveMath integrates with several free and commercial Computer Algebra Systems, e.g. Maxima. Furthermore, it is possible to integrate ActiveMath itself into the Moodle CMS to enhance its eLearning capabilities.

Melis et al. (2006) have provided a detailed description of the internal structure of ActiveMath. The learner model is the main inference component of the ITS. It makes use of the Transferable Belief Model propagation mechanism. There is also another intelligent component, the course generation model, which uses a hierarchical task network planner (Ullrich, 2005). There is an active development community that is maintaining the software and adding new features to it, e.g. reporting modules, E-Portfolios, and learning diaries.

The ActiveMath project is actively involved with many universities in Europe and elsewhere. Funding is provided by the European Union. The ITS is used in schools as well as universities and is being extended from mathematics to other subjects such as chemistry. This ITS is actively studied and documented, see Melis and Ullrich (2003); Melis and Siekmann (2004); Manzoor et al. (2006); Melis, Moormann, Ullrich, Goguadze, and Libbrecht (2007).

The pedagogical theoretical base of ActiveMath is the moderate constructivist theory. This version of the constructivist theory tries to find a balance between the needs of the student to independently construct personal knowledge and the needs of the instructional structures. The purpose of the moderate constructivist theory is to reduce the instructional interventions only to those that are considered to be beneficial to the learning process and better than any alternative (Melis et al., 2007).

Mathematics Software useful for Learning Mathematics

The mathematics programs that are most used by students and teachers of mathematics are the Computer Algebra Systems (CAS). There are many available, both commercial and non-commercial. The most popular commercial CAS is Mathematica (http://www.wolfram.com/products/mathematica/index.html), while Maple, MuPAD, MathCad are other popular commercial CAS. One of the most popular free CAS is Maxima (http://maxima.sourceforge.net). Other categories of mathematics learning programs are the Dynamic Geometry Systems (DGS) and the Geometric Supposers (Schwartz & Yerushalmy, 1992). The last type of programs, Geometric Supposers, seem to have lost popularity and are seldom, if ever, used for learning and teaching geometry today. Indeed, their last mention in the literature is in 19929.

A brief description of the two main commercial DGS and four non-commercial DGS follows. The criteria for selection are the capability to run on more than one operating system (cross-platform), preferably all three main ones10, and being under active development. All DGS written in the Java programming language are intrinsically cross-platform. The reason is that Java programs are not run directly by the operating system, but rather by an interpreter and there are Java interpreters for most operating systems11.

A DGS is considered to be under active development if it is presently maintained and improved by a viable company or an active team. In the case of free software it is preferable that the developers be members of an academic institution. The main open source license is the GNU General Public License (GPL, http://www.gnu.org/copyleft/gpl.html). Programs that are released under this license are usually free of charge and can be freely distributed, installed and upgraded.

An interesting development of mathematics education research is the integration of DGS and ITS. A prototype of cognitive tutor agent for diagram construction was developed for the Geometers Sketchpad by (Ritter & Koedinger, 1996).

Geometer’s Sketchpad

Geometer’s Sketchpad is published by Key Curriculum Press based in California (http://www/keypress.com/x5521.xml). The software runs on Windows and Macintosh computers, but not on Linux computers. It is a commercial program which requires the purchase of a license. However, student and multi-user licensing is available.

This DGS is probably the most widely used one in schools in the USA from middle school to high school. Thus, it appears often in mathematics education research, e.g. Connor, Moss, and Grover (2007). The program has many useful features for both the student and the teacher. For example, students can perform translations, reflections, rotations and dilations on the shapes that they create. Teachers can use this DGS to make classroom life presentations, create worksheets, exams and reports. The program and its manuals are published in English and Spanish.

Cabri

Cabri is the other main commercial DGS and is popular in Europe. It is published by Cabrilog, a French software company (http://www.cabri.com). According to the company, Cabri has more than 100 million users. It is developed for middle and high school mathematics. It allows the direct manipulation of mathematical objects in algebra, geometry, analysis, and trigonometry. Cabrilog publishes several different versions of Cabri under single, classroom and school site licenses. There are 2D and 3D versions, and the program is available in many languages. Several books on Cabri have been published in diverse languages.

C.a.R.

C.a.R., which stands for “Compass and Ruler”, is a free analog of Geometer’s Sketchpad released under the GNU General Public License (GPL). Since it is written in Java it is multi-platform. It can be freely downloaded from http://mathsrv.ku-eichstaett.de/MGF/homes/grothman/java/zirkel/doc_en The program is published in many languages such as English, German, Spanish, French, and Turkish. Besides the usual features of a DGS, C.a.R. has powerful macros that also allow elliptic and hyperbolic geometries. It allows the saving of constructions in many graphic formats. C.a.R. can create web pages with embedded Java applets that can be used as demos and assignments and deployed on the Internet. A worrisome aspect of C.a.R. is that only one developer is active on the project.

GeoGebra

GeoGebra is described as a dynamic mathematics software for education in secondary schools that joins geometry, algebra, and calculus. It is free software released under the GPL. It is written in Java and thus will run on any computer that has a Java run-time environment. It is available at http://www.geogebra.org/cms, where it can be downloaded as a freestanding application or run it from a web browser using WebStart which will dynamically load the latest version.

As the name implies, GeoGebra allows the users to do geometry and algebra at the same time since it includes a simple CAS.

This DGS is available in many languages such as Arabic, Estonian, Chinese, and Turkish. Several books in English and German have been published on GeoGebra. Files can also be saved as dynamic web pages or PNG pictures or postscript for publication quality illustrations.

A disadvantage of GeoGebra is that its development team is quite small. However, since it is released under an open source license anyone is allowed to continue its development.

GeoProof

GeoProof is a free GPL dynamic geometry software package that is different from most other DGS. It is an advanced academic project that combines interactive geometry with proof related features. Some unique features of GeoProof are the ability to perform calculations using arbitrary precision. Some theorems can be checked using automated theorem proving methods.

Since it is not a java application, the software has to be build (compiled) separately for each operating system. A compiled version exists only for Windows, while for Mac OS X and Linux one needs to compile the source code. The latest stable version is are dated 2006 and since the development team is quite small there is a concrete risk of abandonment. There are only French and English versions. GeoProof has good open standard import and export capabilities. The program can be freely downloaded from http://home.gna.org/geoproof.

XCas

Giac/XCas is an advanced multi-platform computer algebra system that can perform 2D and 3D geometry. It has a compatibility mode for Maple, MuPAD and the TI89. It can be freely obtained at http://www-fourier.ujf-grenoble.fr/~parisse/giac.html.

XCas is a program of advanced capabilities due to its CAS, integer arithmetic, linear algebra, calculus, and dynamic geometry modules. This CAS/DGS is programmable and even includes the popular Logo programming language. This power comes at the cost of not being user friendly. The program is under active development and the latest version (0.7.4) was released in March 2008.

Conclusions

Due to reasons explained in this paper, we agree with Bork and Gunnarsdottit (2001) who stated that the optimal form of computer based mathematics education for today is an eLearning-ITS, that is, an Intelligent Tutoring System that is provided to the learners over the Internet from a central location. It is our option that the optimal learning technology solution is an ITS embedded in a CMS (ITS-CMS) and integrated with a CAS and a DGS.

Our intent in describing these computer technologies in more detail is to support a policy of preference for free software in mathematics instruction. The reasons for this position are many, the main ones being their zero licensing cost, Internet accessibility, and the freedom to modify, to integrate and to distribute this technology to all.

Many factors are leading to an increased use of Intelligent Tutoring Systems. Among them is the realization that in today’s schools only an ITS can provide individual attention and individualized teaching to each student at a cost that is acceptable. Furthermore, an instruction program that uses an ITS conforms with the Constructivist theory of learning. The advantages of eLearning are also significant in and out of a school context. Indeed it allows a school to transcend its inherent limitation that most of the knowledge is mediated by the teachers. Even a school that has few didactic resources will be able to benefit from knowledge that is produced all over the world. The usefulness of an ITS is directly related to its pedagogical and subject content, also called its knowledge base. An ITS with adequate knowledge base and pedagogical capabilities will adapt to a very wide range of students with different knowledge, speed of learning and goals. Some students will require less practice and support than others. Some students will require more practice on certain subjects and less on others.

Research on eLearing-ITS very relevant since it deals with important developments in mathematics education: (1) Alignment of instruction with the NCTM Principles and Standards, (2) increased use of electronic learning environments, especially web based; (3) creation of a learning environment that is flexible enough to adapt itself to the needs and interests of the student.

There are other trends in education not specific to mathematics that eLearning and ITS do align with. First of all is the increase of inclusiveness of the public school system in the USA. eLearning technology is by its nature able to offer courses for disabled students, such as students who are limited in vision, hearing, and movement, to participate in a complete learning environment. Talhi, Djoudi, Ouadfel, and Zidar (2007) describe an authoring tool to create courses that satisfy the W3C recommendation for authors and learners that present disabilities.

There is another trend in education that is supported by eLearning and that is open access to education for all. The availability of education to all who desire its regardless their location, income, and school performance. The trendsetter has been the Massachusetts Institute of Technology that has implemented the Open CourseWare (OCW) program (http://ocw.mit.edu) starting in 2001. Anyone is able to download lecture notes and other teaching materials of several MIT courses. Often also video recordings and tests and quizzes, and some textbooks are available. It should be noted that these are real MIT courses and not even a registration is required. All the material is copyrighted but can be freely downloaded and used. There are about 1,800 courses available, among them about hundred mathematics courses. Some courses are even translated in Chinese, Thai, Spanish, and Portuguese. Students and educators around the world are taking advantage of these courses to improve their learning and teaching.

In November 2007 MIT launched “Highlights for High School,” an OCW based site that has resources to enhance science, technology, engineering, and math (STEM) instruction in high schools. According to MIT

Highlights for High School features more than 2,600 video and audio clips, animations, lecture notes and assignments taken from actual MIT courses, and categorizes them to match the Advanced Placement physics, biology and calculus curricula. Demonstrations, simulations, animations and videos give educators engaging ways to present STEM concepts, while videos illustrate MIT’s hands-on approach to the teaching of these subjects (http://ocw.mit.edu/OcwWeb/web/about/media/highlights/highlights.htm).

There a strong social trend today, which is the increase in collaboration among geographically dispersed people. There is a worldwide effort to freely share courses that are created for Intelligent Tutoring Systems. This activity is thus aligned with both trends of open access to education and global collaboration. An example of this collaborative effort is Intergeo which stands for Interoperable Interactive Geometry for Europe (http://www.inter2geo.eu). Its main purpose is to make accessible electronic content for mathematics teaching. The organization will ensure open access to the teaching material by ensuring a common file format based on open standards. There is also a form of quality control where the material will be tested in the classroom.

An example of what is going to be available in this repository of free geometry lessons is found at http://home.scarlet.be/~mathweb/bishop1.htm. The web page contains a GeoGebra applet that shows, using a step by step animation, how a crop circle can be constructed using compass and ruler. Freely available content for ActiveMath can be obtained at http://www.activemath.org/Content/SchoolMathematicsFractions.

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