The AMATYC Standards for Intellectual Development, Pedagogy, and Content
The standards articulated by AMATYC (1995) provide direction for developmental mathematics programs and a “yardstick” by which programs may be evaluated, as follows:
1. The standards for intellectual development address desired modes of student thinking and represent goals for student outcomes. Students are expected to engage in substantial mathematical problem solving; participate in modeling using real-world data; expand their mathematical reasoning skills; develop the view that mathematics is a growing discipline interrelated with human culture; acquire the ability to read, write, listen to, and speak mathematics; use technology appropriately to enhance their mathematical thinking; and develop mathematical power (AMATYC, 1995, pp. 9-12).
2. The standards for pedagogy recommend the use of instructional strategies that provide for student activity and interaction and for student-constructed knowledge. Mathematics faculty are expected to model the appropriate use of technology; foster interactive and collaborative learning through student writing, reading, speaking, and collaborative activities; actively involve students in meaningful mathematics problems that build upon their experiences; use multiple approaches — numerical, graphical, symbolic, and verbal; and provide learning activities and projects that promote independent thinking and required sustained effort (AMATYC, 1995, pp. 15-17).
3. The standards for content provide guidelines for the selection of content that will be taught throughout introductory college mathematics. Students will develop number sense, translate problem situations into symbolic representations, develop spatial and measurement sense, demonstrate an understanding of function, use discrete mathematical algorithms, and analyze data and use probability (AMATYC, 1995, pp. 12-14).
The AMATYC Standards (1995) describe desired outcomes for developmental mathematics students and programs but do not provide details on how programs should achieve these standards. The document Crossroads in Mathematics: Programs Reflecting the Standards (AMATYC, 1999) provides an overview of an array of programs that attempt to incorporate the AMATYC standards but again, not specific blueprints for implementing the standards. It is left to individual developmental mathematics programs and faculty to develop an approach to implementing the standards that best serves their students. An instructional model that an increasing number of programs are incorporating, for various reasons, is mediated learning.
Mediated learning is defined as a learner-centered model of technology-mediated instruction (Gifford, 1996). In this model the individual learner is at the center of the teaching and learning enterprise and is given access to and considerable flexibility in the use of a variety of instructional support resources including interactive multimedia instruction and assessment, the instructor, and text.
This allows learners to: (a) exercise more effective and efficient control over their own learning; (b) secure real-time assessment and feedback; (c) secure more information on their own learning through individual and achievement and progress reports; and (d) receive more individualized learning assistance from instructional staff (pp. 18-19). It is technology-mediated instruction because interactive multimedia software is the primary vehicle to deliver the instruction, feedback to student interactions with the technology, and assessment. The instructional staff provides individualized assistance when requested by students.
Mediated learning environments can be structured to support important goals of developmental education, yet allow instructors great flexibility in structuring their courses. Frequent assessment and feedback, for example, can be provided by both the software and the instructor. As students navigate through the software they enter or select responses and receive immediate feedback through the software. They may also discuss with the instructor their reasoning for selecting a particular response or seek clarification of the feedback provided by the software. Feedback is also given to students as they work on, or when they complete, the “checkpoint” question given daily. Students are encouraged to work on these together, which allows them to receive feedback and assistance from classmates.
Another goal of developmental education is to enhance the retention of students. An important step in retaining students is early intervention by the student’s instructor and advisor when needed. The computer database provides the instructor with detailed information about each student’s success and time on task for each lesson, thus allowing the instructor to quickly assess the progress of each student so that intervention can take place early if the student is not progressing sufficiently. The software allows the instructor to set up courses in a way that lets each student learn in a flexible way (e.g., choice of navigation paths, pace, access to instructor as needed for individual questions). It also allows the instructor to build in a high level of organization and structure (e.g., written objectives for each topic, schedule of homework assignments and exams for the semester, daily checkpoint questions, dedicated times and location for software use and class meetings) that promotes keeping students on track to meet course objectives. This is another important characteristic of developmental education.
Mediated learning environments necessitate that students and instructors take on different roles than in traditional lecture courses. In the mediated learning model students navigate through interactive multimedia lessons that present the mathematical concepts and skills and provide immediate feedback. Students are able to navigate along a path and at a pace (provided they stay on schedule from day-to-day) that fits their individual preference. The instructor, who is freed up from having to present a lecture, provides support for students individually or in small groups by clarifying explanations provided by the software, assisting students in solving problems using paper and pencil, or engages in tasks that support successful student outcomes such as monitoring student progress, providing feedback and helping students develop good study habits. The text is one form of media, and thus is part of a multimedia learning environment. The text lists the objectives; provides explanations of concepts, procedures, definitions, and other information; and contains the homework problems. The text also serves the important role of making the course material accessible to students when they do not have access to the multimedia software.
It is also worth contrasting mediated learning with the “bolt-on” model of technology-mediated instruction. When contrasted with mediated learning there are two important distinctions worth noting. First, in the bolt-on model the technology is bolted on to the existing components of the traditional learning environment, the instructor, textbook, and the student. Technology of this type is usually designed to support student learning of particular concepts or skills but not to be the primary vehicle to deliver instruction and feedback for the entire course. Second, because of the inadequacies of the bolt-on technology to be the primary vehicle to deliver the course content and provide feedback, the learning environment remains primarily teacher-centered rather than student-centered.
Until recently the technology available for developmental mathematics and other disciplines generally fit the bolt-on description, was used for drill-and-practice, did not incorporate rich multimedia presentations of the content, and provided limited feedback. One consequence of the fact that the widespread use of high quality interactive multimedia software is a relatively recent phenomenon is that much of the existing research is related to first generation technology-mediated instruction, rather than interactive multimedia software used in mediated learning environments.
Gifford (1996) claims mediated learning enables students to:
1. Exercise more effective and efficient control over their own learning. This is achieved by enabling the student to navigate through topics and lessons over a number of distinct instructional pathways, at his or her own pace, while spending as much time as required working any given topic, exercise, or problem, until the appropriate level of mastery has been achieved.
2. Secure real-time assessment and feedback. This is achieved by enabling the student to receive performance feedback when it is most useful, new instruction when it is required, and extra assistance when it is needed and practical.
3. Secure more information on their own learning. This is achieved by enabling the student to receive individual achievement and progress reports on a timely basis, sufficiently detailed and directive that the individual student becomes more adept at monitoring and regulating his or her own learning progress.
4. Obtain situationally appropriate learning assistance. This is achieved by enabling the student to receive support from teachers or teaching assistants that is informed by detailed assessments of the individual student’s strengths and weaknesses, as analyzed and reported by a specially designed instructional support system.
5. Obtain more individualized learning assistance. This is achieved by enabling the student to receive more one-on-one and small group tutoring from instructors and teaching assistants than is feasible in the learning environment dominated by the lecture-presentational approach to instruction. (pp. 18-19)
There is evidence to support Gifford’s (1996) claims that mediated learning can be an effective instructional model. The ability to control both the pace of the learning and the navigation path provides students with an opportunity to learn mathematics in a manner that is usually not possible in a traditional setting. Students who “exercise more effective and efficient control over their own learning (Gifford, p.18)” are able to do so because of the interactivity of the software. Najjar (1996) examined the research related to interactivity and stated:
Interactivity appears to have a strong positive effect on learning (Bosco, 1986; Fletcher, 1989, 1990; Verano, 1987). One researcher (Stafford, 1990) examined 96 learning studies and, using a statistical technique called effect size (difference between means of the control and experimental group divided by standard deviation of the control group), concluded that interactivity was associated with learning achievement and retention of knowledge over time. Similar examinations of 75 learning studies (Bosco, 1986; Fletcher, 1989, 1990) found that people learn the material faster and have better attitudes toward learning the material when they learn in an interactive instructional environment. (p. 131)
Feedback is another key component of the mediated learning model. There is research that shows feedback is important to student self-regulation and self-efficacy (Hattie, Biggs, & Purdie, 1996; Kluger & DeNisi, 1996). Kluger and DeNisi found that feedback should be specific to the task, corrective, and done in a familiar context that shapes learning. In the mediated learning model feedback is available to students from both the software and the instructor. As students progress through the software they are frequently presented tasks that require interaction on their part. Immediate feedback is provided for every student response. If a student answers a question incorrectly on the first attempt, hints or suggestions are provided to point the student in the right direction. Students may then attempt the question again. Following the second attempt, a detailed step-by-step solution and explanation is provided. Students are also able to receive detailed feedback from the instructor during class as they engage in the multimedia lessons, attempt questions using paper-and-pencil, or other areas related to student performance such as course progress and study skills. In the mediated learning model the instructor has the time to provide this type of feedback and support throughout the entire class meeting because he or she does not present a lecture.
Reviews of research on the impact of technology-mediated instruction on student learning have consistently found that technology-mediated instruction can have positive effects on student learning (Becker, 1992; Khalili & Shashaani, 1994; Kulik & Kulik, 1991; Niemiec, Samson, Weinstein, & Walberg, 1987). The review by Kulik and Kulik examined 248 controlled studies covering technology-mediated learning in a wide range of courses and learners. In 81% of the studies considered students in technology-mediated settings obtained higher mean examination scores while in the remaining 19% of the studies students in the traditional settings had higher scores. In 100 of the 248 studies there was a significant difference in exam performance, with 94 of the studies favoring the technology-mediated environments.
Interactive Multimedia Software For Mediated Learning.
In developmental mathematics the technology currently being widely used is interactive multimedia software capable of presenting the course content, practice of new skills, and immediate feedback. Multimedia is the use of text, graphics, animation, pictures, video, and sound to present information (Najjar, 1996). Kaput and Thompson (1994) point out three aspects of electronic technology such as interactive multimedia software that “enable deep change in the experience of doing and learning mathematics” (p. 678). First, the ability to interact with the technology, referred to as interactivity, means that a student’s actions yield a reaction on the part of the machine, which in turn sets the stage for interpretation, reflection, and further action on the part of the student. The second aspect is the control designers have in creating the learning environments. Kaput and Thompson state:
One can engineer constraints and supports, create agents to perform actions for the learner, make powerful resources immediately available to aid thinking or problem solving, provide intelligent feedback or context-sensitive advice, actively link representational systems, control physical processes from the computer, and generally influence students’ mathematical experiences more deeply than ever before. (p. 679)
This second aspect of control in creating the multimedia environment provides the opportunity to create an environment that need not be followed in a sequential manner. The third aspect Kaput and Thompson refer to is connectivity. This is technology that links teachers to teachers, students to students, students to teachers, and the world of education to the wider world. Academic Systems Corporation (2000) currently offers the option of browser-based interactive multimedia software for developmental mathematics that includes the ability for instructors to post online notes and a feature that allows students and instructors to exchange electronic messages. Features such as these, coupled with the ability of the software to deliver the course content and provide immediate feedback, is resulting in the Academic Systems software increasingly being used in location-independent instructional formats.
Although considerable research remains to be conducted related to the effective implementation of interactive multimedia packages in developmental mathematics, there is evidence that some programs have been able to improve completion rates and grades using mediated learning. In 1998 Academic Systems Corporation reported on their website (http://www.academic.com) that data on pass rates of 23,000 students in entry level mathematics classes from campuses around the country showed that 52% of students in traditional sections passed compared to 63% of students who passed using software from Academic Systems. In a study at California State University-San Luis Obispo, students who studied introductory algebra, intermediate algebra, or both using software from Academic Systems earned a significantly higher proportion of final grades of C or better in conventional precalculus courses when compared to students who studied the same courses in conventional classrooms (Baker, Hale, & Gifford, 1997). It is worth noting the outcomes of students using mediated learning from Academic Systems Corporation because Academic Systems claims that more students purchase their Interactive Mathematics for three courses, Prealgebra, Elementary Algebra, and Intermediate Algebra, than any single textbook title (Academic Systems, 1999).
Developmental mathematics programs have been working to implement the AMATYC standards since they were published in 1995. Mediated learning appears to have merit as one means of enhancing student outcomes, at least for some developmental mathematics students. Features such as rich multimedia presentations of concepts, immediate feedback, and interactivity allow students to learn mathematics in ways not possible in a traditional lecture course and give students greater control over their own learning. Students also benefit from greater opportunities to discuss mathematics individually with their instructor and to receive feedback about their work. For students who need greater flexibility in terms of time and location for learning, the mediated learning software allows students access from any location with a personal computer (PC) and Internet access. The features of mediated learning, along with the flexibility that it affords instructors in setting up courses and students in learning, results in instructors being able to incorporate activities into their program that support the AMATYC standards. For example, our daily checkpoint questions promote mathematical communication and reasoning, the use of built-in technology tools and lessons support the use of multiple representations, and our students are actively engaged in the learning process as they read mathematics and interact with the software. In implementation models where students have access to the software outside of class, such as a lab on campus with a tutor available, valuable class time can be freed up to have students work cooperatively on problem solving activities or projects, which further supports implementing the AMATYC standards.
At the University of Minnesota General College we are in the process of developing and validating an inventory to inform students in which course format, mediated learning or traditional lecture and discussion, they will be most successful and satisfied. Students are also assisted in selecting their choice of instructional format through orientation sessions, meetings with advisors, and discussions with mathematics instructors. Through these efforts we attempt to place students in the learning environment that best matches their learning style. There is growing evidence that instruction that allows students to learn using their preferred learning style can lead to improved student outcomes (Higbee, Ginter, & Taylor, 1991; Lemire, 1998).
There has been little discussion in the developmental mathematics community about how mediated learning can support the AMATYC standards. This may be due to several reasons. First, the very process of initially offering instruction involving interactive multimedia software requires significant time and effort to review software options, ensure that the necessary hardware and technical support is available, develop a curriculum plan and an implementation plan, and communicate important information about changes in the mathematics program with others such as administrators and advisors. Second, because mediated learning represents a dramatic shift in how developmental mathematics is taught from the traditional lecture format, many instructors are still feeling their way through the basics of this type of instruction. In the early stages of mediated learning there tends to be a focus on issues such as handling technical problems, learning how to effectively support student learning as they use the software during class, and attempting to develop a course structure that incorporates the benefits of multimedia instruction while at the same time provides an environment that keeps students on task and leads to successful outcomes. However, with experience and thoughtfulness about how to best serve their students, developmental mathematics programs may find that mediated learning can be an asset when striving to incorporate the AMATYC standards into their program.
The standards for intellectual development advocate that students acquire the ability to read, write, listen to, and speak mathematics, engage in substantial problem solving, expand their mathematical reasoning skills, and use technology in ways that enhance their mathematical thinking. The standards for pedagogy state that faculty should foster interactive learning through collaborative activities, model the appropriate use of technology, and model the use of multiple approaches – numerical, graphical, symbolic, and verbal. Unlike many students in traditional lecture courses who sit passively, or at most studiously take notes of what the instructor writes on the board, students in mediated learning environments are actively engaged in the reading, listening, and the working of mathematics. The interactive nature of the software necessitates that students read and attempt to make sense of what they have read in order to enter or select appropriate responses. To facilitate students’ abilities to communicate mathematically, and to strengthen their mathematical reasoning and problem solving abilities, our students are given daily paper-and-pencil “checkpoint questions.” Students are encouraged to work together on these by sharing their strategies, explaining their mathematical reasoning, and justifying their answers. Instructional staff provide guidance and feedback when necessary as students work on the checkpoint questions, but also view this as an opportunity to communicate mathematically with students. The instructor does not lecture, making it is possible to have extended conversations with students about their mathematical thinking and reasoning. Even though a mediated learning environment makes significant use of multimedia software, it is appropriate to set aside times when students can work together collaboratively in small groups or through cooperative learning. This supports the standard of interactive and collaborative learning and is supported by research showing that it often contributes to increased academic success (Davidson & Kroll, 1991; Johnson & Johnson, 1989; Thomas & Higbee, 1996).
The standards also encourage the use of multiple representations-numerical, graphical, symbolic, and verbal, along with the appropriate use of technology. Interactive multimedia software incorporates frequent use of multiple representations such as symbolic, tabular, graphical, and written words. This frequent use of multiple representations strongly supports the development of mathematical understanding as defined by Hiebert and Carpenter (1992) in the Handbook of Research on Mathematics Teaching and Learning, who state:
A mathematical idea or procedure or fact is understood if it is part of an internal network. More specifically, the mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of the connections. A mathematical idea, procedure, or fact is understood thoroughly if it is linked to existing networks with stronger and more numerous connections. (p. 67)
The Lesh Translation Model (Lesh, Landau, & Hamilton, 1983; Post, Behr, & Lesh, 1986) describes how translations that form connections between modes of representations can be performed either between modes of representations or within modes of representation. A translation between modes would include translating from an algebraic equation to a graphical representation. A translation within the same mode of representation would include translating from an initial graph to a graph where the scales on the axes have been changed. Interactive multimedia software, with its ability to quickly and easily generate various representations, interactive nature, and built-in tools such as graphers, may help students develop the ability to translate between and within modes of representation, and thus increase the development of mathematical understanding.