About: Philosophy

When members of our Math Department began deliberating how to improve methodologies for teaching and learning Calculus at Stevens Institute of Technology in 2012, we had no idea that we would eventually create the world’s first Stepwise Calculus Learning Platform. Witnessing the widespread online literacy of students on an almost daily basis with surfing the Web, using mobile apps, and playing video games on desktop PCs, laptops, tablets, and smartphones, we suspected that information technology would play a major role.

Passing Calculus courses has long been a gateway to STEM (Science, Technology, Engineering, Math) degrees and careers. With accelerating advances in most STEM fields, we knew from personal experience that just passing Calculus was not enough. It was increasingly critical for students to understand Calculus in order to apply its topics and concepts in advanced courses. Additionally, faculty members from fields not traditionally regarded as STEM, such as Business and Economics, reinforced the growing importance of students understanding and applying Calculus concepts.

We undertook an aggressive search for digital learning systems that would help students learn and understand Calculus in a more profound manner. It was interesting to find a large collection of Websites, brochures, and articles promoting scores of systems for Math education. However, we quickly learned that most of these systems went only as far as Algebra, Geometry or Trigonometry. Many were straightforward assessment tools to help determine which initial college-level course a student should take. Some were designed to mainly support remediation. The few systems we found that covered Pre-calculus and Calculus were mostly focused on students providing a final answer with multiple choice or fill-in-the-blanks formats. In reviewing sales presentations and demonstrations of these systems, we discovered a few systems that allowed students to solve problems based on predefined steps. When a student made a mistake, the system pointed out that an error occurred., but gave the student little information on what the error was or how to correct it.

For several semesters, we tried a variety of Math learning systems at Stevens Institute, and thought we might not be using them to their full effect. In reaching out to associates at other universities and colleges to compare usage and techniques, we realized we were indeed using the systems in diligent and robust ways. We found that many of our contacts shared our skepticism about the usefulness of the systems for teaching and learning Calculus. Some believed that Calculus was force-fit into systems that were originally designed for lower-level Math subjects. Quite a few faculty members shared that students liked the systems because they were game-like and easy-to-use, but their students were not learning Calculus.

These concerns brought to mind an ancient story that is well-known to most Mathematicians. When Ptolemy I, ruler of Egypt from 323 to 283 BCE, grew frustrated at the volume of effort required to master Geometry using Euclid’s Elements, he summoned the great Mathematician and asked whether there was some shorter, easier path. Euclid is said to have replied, “There is no Royal Road to Geometry.” The Royal Road was an ancient highway built by the Persian Empire in the 5th century BCE to allow rapid communication throughout the empire. Mounted couriers could travel approximately 1,675 miles from Susa to Sardis in seven days, compared to roughly 90 days on foot via other roads. Compared to Geometry, Calculus is a much more rigorous pathway.

There is a strong belief among Calculus instructors that to effectively learn and understand Calculus concepts, students need to solve Calculus problems in a detailed step-by-step manner and be intellectually challenged during the process. They should not be content with simply knowing about a topic or focused on just supplying or identifying a final answer. Through our research, we have come to believe that classes, tools, and techniques that oversimplify Math instruction are denying students the very education that they are striving to achieve.

Frustrated by our failed attempt to find a suitable digital learning system that could motivate students to learn and understand Calculus in a more profound manner, we decided to address the matter directly by creating one ourselves. We were determined to build an online Calculus Learning Platform that encouraged students to “show their work,” in fact, all of their work, in a free-form manner, similar to solving Calculus problems on paper. We set out to develop advanced architectures and coding frameworks leveraging adaptive learning techniques to facilitate symbolic computation designed to provide students with relevant feedback and helpful instruction after entering each step of the solution.

Based on our observations of students using disparate Math learning systems, we believe it is extremely important not to simply provide solutions to students, but rather, it is imperative to provide multiple types and levels of feedback to encourage a student to independently proceed with solving a problem successfully. Additionally, a student should be able to enter as many steps as desired when solving a problem to nurture creativity and critical thinking skills. Our platform mimics the behaviors of a seasoned tutor by providing positive reinforcement and suggestions each step of the way, without solving the problem for the student. In that spirit, we decided to name the platform, Gradarius, which is the Latin word meaning “progressing step-by-step.”

Since launching Gradarius, we have received an increasing flow of tremendously positive feedback and reviews from students and instructors at a growing number of educational institutions. Students like that they can experiment with various problem-solving procedures and techniques, and then receive instant guidance so that they can learn topics efficiently and correctly the first time. Instructors are thrilled that the platform automatically assesses and grades every problem for each student while providing data-grounded evidence and reports of academic progress. They also value the robust repository containing all the detailed solutions from their students, annotated by the feedback Gradarius provided. We sincerely appreciate the growing collaboration from the user community of students and educators who have ideas on how to make Gradarius even better.

In the field of Mathematics great distinction is granted to those who solve an unsolved problem that has perplexed scholars for decades. In that spirit, we enthusiastically present the Gradarius Stepwise Calculus Learning Platform. Far surpassing the limited functionality of paper, Gradarius delivers state-of-the-art capabilities that have eluded math professors, teachers, and tutors for centuries: the ability to immediately offer every student meaningful feedback and guidance during each step of the problem-solving process for each and every problem, any time of the day or night. We hope you enjoy using Gradarius, and appreciate our philosophy of teaching and learning Calculus.

See Gradarius in action