Affinity Learning in Mathematics

I. Abstract Teaching complex subjects such as mathematical modeling is intrinsically challenging. It is more so in a typical classroom setting. In this paper, we explore the use of technology to provide an electronic tutor that interacts with both teacher and student to provide a personalized and focused learning experience. Affinity Learning is an environment that captures the skills of a master teacher in a dynamic but simple technical embodiment and presents lessons and assessments online to a student. Initial results not only indicate that learning has occurred, but also distinguish male from female performance and give interesting insight into the learning process itself. II. Introduction Mathematical modeling of a phenomenon, event or process, is challenging. The modeler, or student in this instance, must often translate observations of objects, forces, and time sequences into descriptive and predictive equations. For the learner to comprehend what they are observing mathematically, they must have an intuitive understanding of what is occurring and the intellectual ability to abstract that understanding into mathematical representation. Teaching individuals to have these skills is difficult. It is believed that accomplishing effective learning of difficult topics can be best accomplished through a Socratic or tutoring method rather than through didactic or static text instruction. Crowded classrooms and unavailability of teachers deeply skilled in specific complex topics frequently renders didactic instruction as the method of choice. Affinity Learning is a concept for using computers to leverage educators and provide instruction attuned to the skills of the student. In effect, we sought to build an electronic tutor. At the very least we hoped to prove that such a thing could help learning. Our initial effort, funded by a NSF Proof of Concept grant, has given results beyond our expectations. We find that not only can we influence learning, but that we can distinguish male and female learning characteristics and actually track the learning as it is taking place. The insight gained leads to many more questions such as "Can we predict performance by early student interaction and adjust course presentation to fit student characteristics?," "Can we distinguish poor presentation or assessments methods from poor student performance?" and "Can we adjust instruction based upon learner characteristics to increase student performance?" The Affinity tool is based on relatively simple database-driven software. We have, in effect, used this tool to capture the teaching skill of a master mathematics teacher. In an online setting, the student is guided through a set of activities and assessments in accord with their skills and rate of learning. When a student outcome on a particular activity is unanticipated in the software/database, the teacher is solicited for help. In offering that help, the teacher designs a new activity and assessment that is appropriately incorporated into the environment. The Affinity environment grows from an initial state to more and more sophisticated capabilities. II. Background and Principles for the Project Design Our primary target goal was to increase the effectiveness of developmental mathematics courses through the use of technology. A developmental mathematics sequence initially was chosen because it offered an innately assessable discipline driven by a significant need. It has been noted that 72 percent of four-year institutions of higher education in the U.S. and nearly every (99 percent) two-year college offer developmental math courses, as do 93 percent of institutions with high minority enrollments (NCES, 1996). Whether computer-based instruction is or is not superior to human instruction was in and itself an interesting question to our development team, and which generated a great deal of thought and discussion within our design team meetings. However, we felt that a purposeful blending of such instructional partners was an important key to the eventual success of our project. Human instruction is notably weak or missing in many developmental math courses, and many universities are reluctant or unable to commit scarce instructor resources to what are viewed as remedial courses. The learning environment we conceptualized within Affinity Learning provided the advantage of more interactive and personalized instruction than what is usually now available in developmental math courses. Often in such university courses, students work nearly completely on their own, left to work their way through a textbook with only a graduate student instructor available to answer questions and offer periodic assistance. We felt such students needed instructional help beyond their typical resources and setting, and we believed that well designed software, could provide such help. Mathematical modeling was targeted as the primary content area within the project, because mathematical modeling is both an important topic in today's mathematics classroom, and an unusually difficult process to teach in the traditional classroom. Mathematical modeling can be defined as a mathematical process that involves observing a phenomenon, conjecturing relationships, applying mathematical analyses (equations, symbolic structures, etc.), obtaining mathematical results, and reinterpreting the model Mathematical modeling often requires considerable student involvement, which made it a rich instructional context for targeting within our project. Part of the difficulty in the instruction of mathematical modeling, is that considerable flexibility and feedback is often needed to work with the student (Smith, 1997). As a student's understanding evolves, their conceptual model may go through many different evolutions, hopefully becoming more refined over a period of time, and with more instruction and feedback. Often, if a formula can be used to represent the model, the formula evolution itself may somewhat represent the evolution in the modeling process. This characteristic of mathematical modeling makes it particularly useful for documenting and examining student thinking within the instructional process, and could eventually become a key feature in our instructional design. Mathematical modeling is in essence a "scientific inquiry" process for mathematics, and can be thought of as being undertaken in a series of four stages, which become cyclical as the Our proof-of-concept project was accomplished in three phases. In Phase 1 a specific sample module was designed and developed in the affinity learning environment. After considerable discussion, the project targeted the concept of mathematical acceleration as a rich content area for the demonstration of our design principles. We reviewed and closely examined a variety of potential mathematical models from high school and college classes that could be used within the content for the prototype. Master teachers from the public schools were key participants within these discussions. Embedded assessments were also integrated into the environment and built upon the tracking technology implemented in CLASS™ Project courses. The team also drew upon the concept of a Knowledge Garden being developed for the CLASS™ Project by Dr. Scott Henniger, from the University of Nebraska at Lincoln. Phase 2 consisted of observing a sample set of students as they used the developing module. This provided a use sample for refining the software and interfaces, and also enabled us to develop a graphical representation of student progress through the module. Finally, during Phase 3, students were tested using the affinity learning environment and using a conventional environment. A concerted attempt was made to make both presentations as engaging and educationally rigorous as possible. At the end of this three phase effort, we now have the following outcomes: 1) a demonstration module for the affinity learning concept as it relates to mathematical modeling within the construct of acceleration, 2) a graphical profiling process to track student progress students, and 3) a variety of related research papers and presentations describing our fundamental design principles, its resultant prototypes, and strategies for dealing with student learning within this context. Our design principles: In order to establish a vision for the project that was consistent with current literature, feasible, and our own experiences, our design team carefully conceptualized seven "design principles" for the project. Our project design principles are refined to be consistent with the vision of new technology based resources as recommended by documents such as the 1996 NSF document "Shaping the Future: New Expectations for Undergraduate Education in Science, Mathematics, Engineering and Technology." The design principles undertaken within this project targeted the instructional topic of acceleration as a context for the mathematical modeling process. The construct of acceleration is a common topic covered in a variety of developmental mathematics and science courses. Acceleration is also commonly taught in many remedial classes, where students are often nearly on their own, and left to work their way through a textbook with only a graduate student instructor available to answer questions and offer assistance. Thus, our vision for the project was represented by our seven design principles, which are now described. These principles were used to continually help focus the project, and to help ensure that the project was designed to produce a solid educational environment, conducive to individual student learning. Design Principle 1) The adaptive instruction of the project seeks to use technology to help students learn through involvement with real life problems, real life data, and true examples of mathematical modeling as they apply to today's world. The use of real life problems, data, and tools within the context of technology based mathematics instruction has long been recognized as a beneficial contribution to student learning This exposure in the lower grades (i.e. algebra or geometry) would set the stage for much more meaningful problem solving and mathematical modeling when the same students reach Calculus and study optimization as a formal topic. Design Principle 2) The adaptive instruction within the project sought to actively rather than passively involve students, in deep conceptual questions and encourage them to be both dynamic and flexible in their thinking and problem solving. A fundamental instructional idea behind mathematical modeling is that students, through modeling activities, discover patterns and consistencies in data that will allow them to test, refine, and build generalizations by creating a "mathematical machine" which represents a particular situation Design Principle 3) The adaptive instruction system sought to be an additional resource to teachers and classrooms, rather than a replacement for these valuable assets to student understanding. The project we developed sought to enhance rather than replace the important synergy that often happens between a teacher and student in the learning process. Within this context, our project tried to build instructional nodes that might facilitate the shared thinking between a student and their teacher, or a student and other students. Design Principle 4) The adaptive instruction sought to enhance human interaction, by connecting students more effectively with the teacher, their peers (other students), and appropriate mentors (professionals) during the mathematical modeling process. Based upon the student level, interests, and local resources and professional availability, the system we conceptualized also would permit opportunities for students and professionals to work together electronically and collaboratively to confront modeling challenges as a "affinity learning group". Similar to electronic special interest groups or listservs, but more focused on a particular task or set of activities, these affinity learning groups might move forward together to share ideas and activities occurring on the system. In this way, students can tap the thinking of other students, as well as designated professionals. This targeted capability for the prototype was well planned within the project, but funding levels didn't permit the direct integration into the project prototype. Design Principle 5) The adaptive learning system seeks to help with the ongoing assessment of student understanding, through a systematic use of embedded assessments, as well as student self-assessment. The systematic assessment of student understanding is a very important piece of the interactive technology integrated within the project. As educational technology continues to rapidly advance, new assessment opportunities and techniques are surfacing based upon these new technologies Our design work has shown us that there is indeed a rich context of potential assessment information that exists within an on-line or technology based learning environments, and that the technology itself can indeed be a very useful tool in the organization of such information as described in some researc