The developments in math eLearning have made it possible for math learners and practitioners to use technology for different kinds of tasks from learning math concepts to collaboration on new ideas and sharing already established knowledge on the web. But there is still a lot of room for growth. The ability of technology to replicate math interactions between humans is in a rudimentary state.
To enable machines to operate and communicate as humans is not a new idea. In May 2001, Tim Berners-Lee, inventor of the World Wide Web, expressed his view on the future of web technologies in the Scientific American. His vision of an interconnected web world has since been referred to as Semantic Web. At the core of the Semantic Web vision is the ability of computer programs (agents) to understand and manipulate human knowledge, as well as to interact and share meaningful information through Semantic Web. Not all Semantic Web ideas have been realized, but the attempts to bring us closer to the vision has already been made.
Semantic Web as an innovative concept appeared independently from innovations in the eLearning field. Yet, the two are connected. Semantic technologies empower most present-day eLearning developments. Semantic Web emphasizes interoperability of systems, but systems can not communicate without understanding the meaning of content. Similarly, a major challenge of the eLearning industry today is to replicate a real-life teaching process, yet to do that machines have to understand the meaning of human knowledge. While we’ve seen some success with translating text-based knowledge into a machine-understandable format, other areas of knowledge remain untackled, math being one of them. This article is intended to explore the state of Math eLearning as it relates to semantic math technologies.
The current state of Math eLearning
To understand where Math eLearning field is going it’s necessary to keep in mind that the field is multidisciplinary. From the teacher’s point of view, any new tool should aid teachers in their mission. At the same time, increasingly, the focus is put on self-learning and reducing the teacher’s burden. In a way, these are two opposing goals. The master/student relationship is considered to be the most effective one for learning and it has been this way since Greek-inspired civilization. On the other hand, with the advent of eLearning the teacher’s role isn't of the owner of knowledge anymore but a mediator between the knowledge and the student with a perspective of eliminating the teacher’s role. This tendency creates worries in society, but it might be a necessary step in the learning process optimization. Today, most eLearning objectives are centered around learners and their learning activities 1. This transition can be seen in the types of eLearning systems available:
Math eLearning Systems. Technology has been integrated into education multiple times throughout history. From Computer Based Instruction (CBI) to Internet-based Learning (IBL)2 - the way courses have been offered changed significantly allowing for an increase of distance between teachers and students. The student doesn’t have to appear in class anymore as long as they have access to a learning management system (LMS) and the Internet. Yet, despite all these advancements, the teacher remains a major participant in the eLearning process. The majority of modern Math eLearning Systems are set to deliver information from the teacher to students. The courses are designed by the teacher. Creation and administration of the learning plans, setting up the sequence of courses, and course upload into the system are also performed by the teacher.
Math Knowledge Testing Systems - knowledge testing systems are used by organizations and educational institutions to test knowledge of students or employees. Most such systems require substantial effort from the teacher’s side. Tests are preloaded into the system. Teachers have to compose a great number of variations of tests to prevent students from cheating. The teacher also has to ensure tests include a wide range of questions so that the knowledge can be tested to the full extent. Facing these obstacles, teachers resort to limiting either the number of tests created or the frequency of tests. Math Knowledge Testing Systems tend to judge students’ answers and scores rather than evaluate the whole problem-solving process, whilst in mathematics, students are required to not only know how to solve problems but also to understand the meaning of calculations3.
Both Math eLearning Systems and Math Knowledge Testing Systems require teacher involvement for administering and knowledge delivery. None of the systems can yet imitate real, live teaching process with all the benefits it holds for students. In real life, a human teacher facilitates the math learning process by guiding the student toward the right solution. A teacher can instantly spot mistakes and answer questions in real-time. A teacher can compose personalized tests, even though their ability is limited by their professional level and time. In general, teachers still stand behind any successful eLearning system, as most content cannot be generated without a teacher's involvement.
Intelligent Math Tutoring Systems
An intelligent tutoring system (ITS) in eLearning provides immediate feedback to the learners without human teacher intervention4. The general philosophy behind intelligent tutoring is that successful teaching and efficient learning aren’t possible without individual student support. ITS are also called cognitive systems. They usually employ some artificial intelligence techniques. The three main components of Intelligent Tutoring Systems were defined back in the 90s and haven’t changed since. The knowledge base contains the knowledge and expertise and can use it as a base for instructions and interventions, the learner module represents the current state of student knowledge, the tutor module provides instructions that correlate to the student’s current state of learning, and user interface enables communication between the student and the machine5. Most current research and developments are focused on the technological advancement of Intelligent Tutoring Systems in either one or more areas presented above.
Various research on ITS emphasizes one or two modules as key modules for ITS success. More practical inventions in the mechanics of ITS user interfaces are preferred over student module, that was deemed to be problematic6. However, the following section of the article focuses on the tutor module as this is where semantic technologies can bring the most change. A tutor module enables a conversational dialog between the human and the machine and is an essential part of an ITS. The tutor module decides on the whole structure of the course, determines which problem to present next based on the previous assessment of student knowledge, answers questions, and provides feedback or suggestions.
Outside mathematics, Intelligent Tutoring Systems use natural language recognition to conduct a conversational dialog, deliver instructions and enable spoken (speech-to-speech) interactions. In mathematics, the effectiveness of a dialog is determined by the state of semantic math technologies. Until recently, semantic math technologies were practically non-existent. Mathematical representation and search were limited to non-interactive books, articles, and pdfs. Tutor modules in Math eLearning Systems were unable to provide conversational feedback or assessment of the process of thinking wherever math notations were involved. The poor state of semantic math technologies explains why Intelligent Learning Systems found a wide application in the areas such as critical thinking and computer literacy, but not in subjects related to math such as algebra, geometry, physics, and economics.
Semantic math technologies use custom strategies and individually chosen programming mechanics to present and understand the meaning of math. Majority of such technologies are TeX based. Currently, there is only one ITS math system on the market that employs semantic math recognition - Gradarius.
Gradarius is an Intelligent Math Tutoring System based on semantic math technologies and developed by researchers at the Stevens Institute of Technology. Thanks to semantic math recognition, Gradarius understands the process of problem-solving and recognizes the meaning of solutions, which enable it to provide useful real-time feedback, resembling a human teacher and student interaction. As a result, substantial time and cost savings can be achieved for teachers and educational institutions. Gradarius presents an opportunity for the education field to optimize the math learning process at schools and universities, making Intelligent Math Tutoring possible.
A semantic component in Gradarius works to recognize the meaning of entered formulas in a browser, translates formulas to a format understandable by the machine, and reads formulas out loud when speech instructions are required (for people with disabilities). The innovative character of the system, that acts like a real tutor, allows it to replace the old teacher-LMS-student workflow. Apart from that, the introduction of Gradarius into the eLearning system brings the following benefits to participants of the eLearning process:
The main implication of using semantic math technologies in eLearning is that formulas can be entered directly into a browser. First math processing tools converted formulas from a manual drawing to a machine processed image, where symbols could not be separated. With semantic technologies, symbols that comprise a math formula are communicated to the machine separately, and yet the computer understands the meaning of the entire formula as well as reads the meaning of everything that constitutes a path to the solution as different parts and hypotheses are being entered into a browser.
This advanced formula understanding function enables the system to provide real-time feedback to students just like a human teacher would. In case the student finds the problem too difficult to solve for their level, the system can help with automatic problem-solving. Additionally, the system can evaluate the student’s progress and store the necessary data points in the form of a report for subsequent analysis and reuse. All these benefits contribute to the personalized nature of the learning process in Intelligent Math Tutoring Systems.
For teachers, automation of activities that take away from the time spent on the actual delivery of knowledge and instruction is the main task that stands before Intelligent Math Tutoring Systems. With semantic math technologies, homework assignments can be checked by the system, which reduces the burden on teachers. Any assessment, in general, can be handled by the system, allowing the teachers to focus on teaching.
Similar to the student module, data reports help teachers see when the problem arises and guide the student toward the right solution. The same reports enable teachers to design courses tailored to individual student needs that they can then adjust based on statistics received from the system.
Educational institution benefits
The main benefit of using semantic technologies in Intelligent Tutoring systems is understanding the meaning of math formulas and solutions entered by the student, which in turn enables a conversational dialog between the human and the machine. This dialog helps achieve a new level of engagement and develop a personalized tutoring approach for each student. There is statistical evidence that such an approach to computer-based learning improves test results and leads to a reduction of DFW rates. Wherever Gradarius was used in the education system, the following results have been reported:
- Student evaluation results improved by 0.6 - 0.4 points
- Grade D, Withdraw, and Grade F rates decreased by 19.2 - 21.3 points.
Understandably, better results and personalized instruction bring increased student satisfaction with the course of study. Additionally, for educational institutions, significant cost savings can be achieved in terms of HR hours (i.e. graders) and classroom space.
The ability of semantic math technologies to provide a highly accessible learning environment for people with disabilities deserve special attention. Semantic Math tools convey the exact and correct meaning of math formulas. In other words, while other systems perceive the expression such as 2+3*4 as “two plus three multiplied by 4”, semantic math technologies interpret it correctly as it would have been communicated to a student by a human mathematician - “the sum of two and the product of three and four”.
These and other benefits can be witnessed after Gradarius installation in an educational environment - school or university. Apart from the field of education, semantic math technologies can be used in a scientific or corporate environment, as well as on the Web to improve math presentation and knowledge online. In the latter case, semantic math technologies can become a convenient tool for math problem-solving in a browser for practicing mathematicians. For search engines, such technologies provide search functionality that can identify and find the information presented in the form of math notations and formulas rather than PDFs.
While there are no Intelligent Math Tutoring Systems that completely replicate the mind and emotional intelligence of a human teacher, semantic technologies bring a new level of understanding between the human and the machine. In math eLearning, this means that a consistent dialog can be established between the system and the student, which provides a stream of feedback crucial for effective acquisition of knowledge and steady progress in the learning process.
Softaria is a software development team, that participated in the implementation of semantic math technologies in Gradarius. We offer services in the eLearning development using math recognition and understanding technologies. For more - see Math eLearning