THE INTEGRATION OF INTERNET RESOURCES INTO HIGH SCHOOL MATHEMATICS - COLLABORATION BETWEEN THE COMPUTER SCIENCE DEPARTMENT FACULTY AND HIGH SCHOOL TEACHERS

ABSTRACT

The Internet contains resources which can significantly improve the teaching of mathematics in the secondary schools. Instructional models from the National Council of Teachers of Mathematics focus on active student-centered learning involving collaborative classroom activities. Integrating the use of the Internet into these instructional models increases their efficiency. The graphical components of web pages enhances the visualization of complicated concepts. The multimedia heightens student motivation by substituting color, graphics and sound for white chalk on a blackboard. The wide range of current data sources enables teachers to quickly create real world problems in fields that interest their students. The use of existing web pages cuts material development time for any individual teacher. Finally, the Internet fosters connections with colleagues and experts beyond the classroom.

A collaborative effort is required to boost utilization of the Internet in the secondary school. At the College of Staten Island, this collaboration occurs between the faculty of the Computer Science Department and the teachers of the BASIS high school district and District 31. Pre-service and in-service teacher training programs are run by college faculty. In all the courses, the technology is integrated with relevant course content. The training courses are given in the student-centered style that is being promoted. Students are involved as aides or mentors. Technical support is provided to the high schools by the college wherever feasible.

Initial evaluation of these efforts shows that they are succeeding in increasing teacher awareness of the resources available on the Internet. The number of teachers developing lessons using web materials is increasing. As more district schools are connected to the Internet during the coming year, a cadre of teachers will be ready to implement and evaluate these Internet-based lessons.

INTRODUCTION

BACKGROUND

Faculty from a Computer Science Department and high school teachers can collaborate to improve the teaching of mathematics in the intermediate and high schools. The National Council of Teachers of Mathematics (NCTM) standards provide an excellent template for student-centered learning in a technology-rich environment. An incredible variety of curriculum resources are available on the Internet to support the teacher who wants to follow the NCTM guidelines. Integrating the use of the Internet into these instructional models increases their efficiency. The graphical components of web pages enhances the visualization of complicated concepts. The multimedia heightens student motivation by substituting color, graphics and sound for white chalk on a blackboard. The wide range of current data sources on the Internet enables teachers to quickly create real world problems in fields that are of interest to their students. The use of existing web pages cuts material development time for any individual teacher. Finally, the Internet fosters connections with colleagues and experts beyond the classroom.

Teachers and faculty, however, have many hurdles to overcome before they can incorporate the Internet and the recommendations of the NCTM standards in a curriculum that is fettered by time, budget and assessment constraints. First, Internet connections or even computers are not easily accessible to the classroom teacher. Second, teachers are not all computer experts. Third, state mandated curriculum with associated assessment measures limit the teacher's ability for innovation. Fourth, because there is no control over the quality of web sites, all resources must first be evaluated and tested by groups of teachers. The formation of professional partnerships between faculty with computer expertise and teachers can assist in the removal of these obstacles and improve mathematics instruction.

Faculty from the Computer Science department (http://indy1.cs.csi.cuny.edu) of the College of Staten Island (CSI) have been involved in collaborative projects with the local school districts for many years. The department's efforts are closely coordinated with those of the Discovery Center of the college. The Discovery Center, run by Drs. Leonard Ciaccio and James Sanders ( http://www.cs.csi.cuny.edu/discover/discover.html), seeks to renew teaching in grades K-12 by transforming the entire college into a resource for pre-service teacher education and in-service collaborative projects. Projects run by the Computer Science department and the Discovery Center all have the following characteristics. First, all participants are treated as knowledgeable professionals with expertise to contribute to the project. Second, all projects include "discovery learning" in which students conduct independent research on multi-disciplinary topics in order to construct their own understanding and knowledge of a particular discipline. Third, educational technology is a key component in facilitating both independent research and the communication of the results.

The NCTM has taken the lead in shaping a new mathematics curriculum which emphasizes mathematical reasoning, problem solving and communication. The following excerpt describes the change in focus.

• ... However, the 9-12 standards call for a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures and by proficiency with paper-and-pencil skills to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving.⁽¹⁾

NCTM also has published standards which describe a different style of instruction in which the mathematics teacher is a facilitator of learning, encouraging students to construct their own understanding of mathematical concepts.

These alternative methods of instruction will require the teacher's role to shift from dispensing information to facilitating learning, from that of director to that of catalyst and coach. The introduction of new topics and most subsumed objectives should, whenever possible, be embedded in problem situations posed in an environment that encourages students to explore, formulate and test conjectures, prove generalizations, and discuss and apply the results of their investigations. Such an instructional setting enables students to approach the learning of mathematics both creatively and independently and thereby strengthen their confidence and skill in doing mathematics.

The role of the students in the learning process in grades 9-12 should shift in preparation for their entrance into the work force or higher education. Experiences designed to foster intellectual curiosity and increasing independence should encourage students to become self-directed learners who routinely engage in constructing, symbolizing, applying, and generalizing mathematical ideas. Such experiences are essential for students to develop the capability for their own lifelong learning and to internalize the view that mathematics is a process, a body of knowledge, and a human creation.⁽²⁾

Several topics outlined in the NCTM curriculum standards are of special interest to educators working with Internet resources. Real-world problems, applications and modeling are included in the list of algebra, geometry and trigonometry topics. The standards recommend computer-based graphing, computer-based methods to solve equations and computer-based explorations of 2-D and 3-D figures. The increased emphasis on statistics and probability is also noteworthy.

The emphasis on real-world problems and mathematical modeling is a significant departure from the tradition of existing textbooks which isolate the study of mathematical tools from the problems which they were developed to solve. The ability to manipulate visual models and graphical representations has only recently become widely available to students. The requirement that students communicate their mathematical knowledge with verbal, written and graphical representations is considerably different from the expectation that the student will arrive at the correct answer showing all the work of a prescribed method.

Both the content of this curriculum and the recommended style of teaching need a technology-rich environment to ensure success. The most obvious role that the Internet fills in this new instructional model is as a source of information. As students pursue independent research projects, they will require information that is not available in the classroom or school library. This is especially true in modeling problems, where the students need access to current statistical data from a wide variety of fields. In addition, the typical mathematics teacher may not have the expertise in every topic to answer the questions generated by the students in the course of their research. The Internet affords the opportunity to inspect large databases, read scholarly research and contact experts which makes the students' research efforts more productive.

The ability to post results on web pages or send them to collaborative partners via e-mail is a powerful motivator for student research projects. Students crave recognition for their work. Several on-line collaborative projects award formal certificates for participation. Students are able to count the number of people who have visited their web site. They can also post e-mail addresses to receive individual feedback.

Using the Internet aids the teacher in customizing instruction to different skill levels. One of the toughest tasks faced by the classroom teacher today is balancing the needs of the "average student" with the needs of those with greater and lesser ability. With the introduction of the Internet, schools can meet this challenge by incorporating the open-ended lessons which are available on many mathematics sites. The web sites often pose a set of questions ranging from the routine to the complex. Multimedia extends the attention span of the less motivated student. For visually oriented students, the graphical or pictorial representations facilitate their understanding of abstract topics. In addition, almost all sites provide links for students who are eager to know more.

PROJECTS

The authors are currently involved in two separate collaborative projects with teachers from K-12. The first project is a graduate course, CSC 813, Utilizing the Internet and CD-Roms in the Teaching of Secondary Mathematics, which is funded by a NYNEX Excellence in Education Award. Fourteen mathematics teachers from grade 7-12, accompanied by sixteen of their computer-literate students attend the course. The teachers explore and evaluate Internet resources for mathematics in class. They test the activities on the students who come to class with them or in their own classrooms. Issues surrounding the Internet such as student safety and equity questions are discussed each week.

The second project is a graduate course, CSC 812, Windows, Word-processing and the Web, which was designed for the teacher on sabbatical program. Its goal is to produce computer- literate teachers who view the Internet as a resource that can be incorporated into the day-to-day life of their classrooms. Assuming only limited competency with the computer, the class gives the teachers survival skills in windows, some advanced word-processing techniques, Internet surfing competence, e-mail proficiency and Web page creation. The teachers in the class during the current semester have taught all levels of classes, special education through gifted and advanced placement ranging from grades K-12.

TEACHERS ATTITUDES TO COMPUTERS, THE INTERNET AND STUDENT CENTERED LEARNING

The teachers come to the projects with pre-existing attitudes towards the Internet and its place in the classroom and with varying degrees of competence. Collaborative projects must acknowledge these initial levels and build upon them. A survey on attitudes towards educational technology in the classroom was administered to both groups In addition, teachers were questioned about their current utilization of computers. All but one of the fourteen teachers in CSC 813 came to the class with computer skills in windows and word processing software; five were familiar with the Internet. Only five of the thirty-three teachers in CSC 812 classified themselves as experienced computer users. Of this self-selected group, only two had knowledge of the Web. On the attitude survey, twenty-three of the forty-three teachers responded that they moderately or strongly agreed with the statement "I am competent in the use of computers. Fourteen of the teachers also either strongly or moderately agreed with the statement "I can usually figure out what to do when something goes wrong with a computer." Both groups believed that computers would be an important part of the classroom of the future and that students need computer skills to succeed.

The teachers were positive toward the potential of the Internet in establishing connections outside of the school. They viewed the web as an accessible source of data for classroom activities. The mathematics teachers however were less sure of its usefulness in obtaining resources related to their discipline.

Table 1 Teachers' Attitude Toward the Internet

The mathematics teachers were administered a separate survey form to assess their current utilization of the instructional practices outlined in the NCTM standards. The data clearly show that the teachers currently teach in a traditional style, quite different from what the NCTM is proposing. To their credit, the majority of the teachers incorporate the following practices into their teaching styles. The students use calculators to solve problems (11 teachers, 92%). The teachers use real-world data when presenting statistics lessons (11 teachers, 92%). Two solutions are frequently presented for each problem (10 teachers, 83%).

In contrast to the recommendations of the NCTM, the majority of the teachers rarely, if ever, give a problem with no known solution (10 teachers, 83%). Reading or literature assignments never supplement the current topic (11 teachers, 92%). The students do not set up models with demographic or economic data (9 teachers, 75%). These results show that the teachers are not working in a multi-disciplinary project oriented style.

The Internet and indeed educational technology of any kind is not present in the classroom. The students do not have access to graphing calculators (9 teachers, 75%). The teachers do not use computer software for demonstrating concepts (12 teachers, 100%). Students do not complete assignments with the aid of computer software (12 teachers, 100%).

From the initial survey data, it is clear that major work must be done both to move the Internet into the classroom and reorient the teacher to a new role as a facilitator of student learning.

SIGNIFICANT FACTORS IN THE COLLABORATIVE PROJECTS

From experience in previous collaborative projects, the faculty believe that any instruction in educational technology must be in an environment that is discipline-based. Discussions, lessons and assignments must be set in the subject matter which is relevant to the teacher. In CSC 813, all activities revolve around the teaching of mathematics. In CSC 812 , activities are either general, such as using a spreadsheet to calculate grades or varied to suit the needs of each teachers, e.g. customized web sites for each discipline. The ability to immediately use the material presented is strong motivation for continued exploration. Examining materials within one's own discipline also gives teachers the confidence to seriously evaluate the quality of the proposed lessons.

The NCTM standards promote a new role for the teacher, one as a catalyst and coach. Students in these classrooms work on independent research projects, solving real-world problems. During the course of their explorations, they learn the requisite knowledge and skills. The traditional chalkboard lecture is being replaced with self-directed learning. In this learning process, students construct their own knowledge based on structured explorations of topics that interest them. NetLearning, the textbook of both courses, describes the change in the following manner.

This is a significant departure from the manner in which these teachers are currently teaching. It is also a radical change from the experiences that these teachers have had in their college courses Discussing or reading about student-centered learning will not change the teachers' behavior in the classroom. So, rather than lecturing on the NCTM recommended mode of instruction, the classes are conducted in that style. Student-centered learning and the use of the Internet is a natural marriage. In the Multimedia Laboratory for Undergraduate Instruction, the teachers experience first-hand the wealth of on-line information resources available and the ability to collaborate with individuals who share similar interests as well as with known experts in the field. It becomes natural for them to approach the Internet as the perfect tool for implementing the NCTM standards in their classrooms.

One of the hardest things for a seasoned teacher to relinquish is control of the classroom. The NCTM standards mandate that the student lead and the teacher support. Each lesson in the graduate courses is designed to give the teachers control over knowledge content, presentation format and analysis of the problem. However, the expectation is that the teachers will request instruction in the skills necessary to complete the task at hand. These skills are the core of the curriculum which the professor is trying to cover. Insuring that all students acquire a set of tools is one of the challenges of self-directed learning. The teachers are given independent and open-ended projects to complete in the multimedia laboratory and at home. A representative project is a newsletter designed to showcase the home curriculum. Teachers were initially exposed to the minimum word processing skills necessary to produce a newsletter. As they developed their own product, the students recognized the need to learn advanced techniques. Experiencing student-centered learning while they are participating in the graduate class is the most effective incentive for incorporating this project-based style in their own classroom teaching methodology.

The College of Staten Island has an impressive technology infrastructure which consists of over 2,500 computers which are connected by local area networks within buildings and by fiber optic links between buildings. Students and faculty have access to standard commercial software packages, multimedia equipment, e-mail, and the Internet. The Office of Information Technology maintains the teaching laboratories and the network in addition to providing technical support for faculty and departments. The K-12 teachers do not have similar support at their institutions. Computer laboratories are for the most part off limits to the classroom teachers. If the school has an Internet connection, it is generally located in the library. The teacher has limited possibilities for classroom computer connections. First, the teacher can bring a portable or a computer on a cart into the classroom. Second the teacher can assign activities which require students to use computers in the school or public library. Third, the teacher can explore web sites or make e-mail connections at home and bring a hard copy of the work into the classroom. The technical support which is generally available to college faculty is not readily available to K-12 teachers. Technical support at home and in the school is a critical part of incorporating the Internet in the curriculum. Until the K-12 schools are set up to provide that support, the college faculty and student assistants must provide what they can. In CSC 813, funding from NYNEX is paying a college assistant to travel to the teachers' home to help them connect to an Internet Service Provider. The college is also providing a portable computer for use in the teachers' classroom. A college assistant works with the teachers in their classrooms to set up the portable for experimenting with Internet-based lessons.

The teachers on sabbatical who have registered for courses have requested that special sections be created that are restricted to only teachers. Having a course solely inhabited by teachers allows collaborative partnerships to form. These students bring with them the desire learn new teaching techniques and the ability to share "tried and true" techniques with their fellow teacher. A class composed solely of teachers offers a collegial atmosphere as opposed to the more formal structure of a hierarchical classroom. This collegiality is noticeable in the work ethic of the teacher. They have much to gain from the ability to focus on shared teaching issues. With a uniform population, the professor can structure the classroom projects in such a way as to stress the importance and integration of the activity back in the home school. It is also anticipated that the fellowships that develop in this class will lead to collaboration between schools and students after the class ends. A support network, maintained through e-mail, will encourage these teachers to share their on-going computer and Internet experiences.

In CSC 813, the teachers are accompanied by computer-literate students of their choice. These students act as aids and mentors. They are beneficial in the graduate course as they work through the exercises with the teachers. When the teachers return to their classroom, the students are an easily accessible source of help in dealing with the technological challenges of using computers in the classrooms. The response of both teachers and students to this partnership has been positive. The students enjoy the use of networked computers which access the Internet through a fractional T-1 line. They appreciate the chance to work with students from other schools who share their interests in web page graphics and animation. The teachers benefit from having every how-to question answered immediately without the need to wait for the professor or college assistant to get to them. The enthusiasm and energy generated by the students as they explore the math sites has infected the teachers. The students often draw clusters of teachers around them as they see some fascinating multimedia visualization or have questions about an intriguing problem. Finally, the presence of students leads the teachers to maintain a professional atmosphere in the class which has contributed to their effectiveness in covering an extensive collection of web sites.

INITIAL RESULTS

With positive experiences from the class, the teachers are willing to try these new methodologies in their own classrooms. In the beginning, they take hard copies of web materials into class with them. By the end of the semester, they devise means to have the students access the web independently.

Three of the teachers used material from a the web site - The Most Colorful Math of All, http://www.c3.lanl.gov/mega-math/workbk/map/map.html . This site explains the four color map problem. It provides vocabulary, background information, concepts, activities, and curriculum materials. The activities include two interesting stories, a discussion of the computer proof, a set of four maps for experimentation and questions about changing the design of the maps. Approximately 90 students completed this exercise. The teachers reported that the high school students at first believed that the problem was too simple for them. As the students worked, they began to realize the complexity of the task. Those students who were able to reason their way through the problem felt a sense of accomplishment and enjoyed the whimsical element of coloring, while other students were simply frustrated by the exercise.

Four of the teachers used lessons from the site What Good is Math?, http://www.urich.edu/~ed344/math/mathhome.html. Three of the teachers used the activity, How do I get the Most for My Shopping Dollar. This site poses shopping problems for the students. For example, the problem listed under "Shoes Galore" requires that the students decide whether it is better to buy one pair of shoes and get the second pair free, or to buy two pairs of shoes with 25% off each pair. Approximately 60 students completed the lesson. The teachers reported that the students loved the practical side of the problems. At first, they were overwhelmed with finding a solution on their own, but when they approached it slowly and logically, they were able to figure out the best buy. The students felt a great deal of pride in being "educated consumers". The students appreciated the fact that the web site had a solution, so they could understand mistakes they had made in their reasoning. Two of the teachers chose the activity, So You're Planning a Trip, which has an exercise concerning gas mileage. Eighty-five students participated in this exercise. In both classes, the students did not want to leave the subject and continued the research on their own, figuring out what gas mileage their family cars obtained.

Other teachers in the class experimented with a variety of word problems, geometry problems or logic puzzles. The following is representative of the teachers' selections. Fifty students accepted the Pyramid of Cans Challenge on the Vermont Portfolio Program page, http://plainfeld.bypass.com/~union/pyramid.html. The teacher reported that the initial solution arrived at by both classes used the erroneous assumption that the cans only constituted the outside perimeter. The second day, the class realized that a pyramid built in that way would collapse upon itself. The students went back to resolve the problem using their more complete understanding of the problem.

As the teachers gain confidence with the new mode of instruction and with the quality of the materials, they want to design more challenging projects. One teacher is asking his class to do a homicide investigation. He found web pages which cover the mathematics of body cooling, http://sol.cms.uncwil.edu/~herman/deproj/death/death.htm, and the speed at which sound travels, http://www.cis.ohio-state.edu/hypertext/faq/usenet/physics-faq/acoustics/faq.html. He plans to download the sites onto a portable computer and have his students use them to determine the innocence of a fictional suspect in murder investigation.

Initially the teachers view the Internet as a source of interesting lesson plans, but slowly they begin to appreciate the communication aspects of the world wide web. One teacher has already joined a collaborative project on the web site Green Issues, http://www.rmplc.co.uk/eduweb/sites/egriffin223/index.html. Her students' graphical analysis of the shared environmental data will be featured in a school exhibition during the spring open house. Another teacher was frustrated by her inability to find algebra-related sites that would intrigue her students. She joined two listserv's for teachers using the Internet and asked for help in locating sources.

These initial anecdotal reports indicate that the courses are effective means of influencing teachers to experiment with student-centered instruction. They also succeed in bringing Internet resources into the classroom.

SUMMARY

Collaborative projects which combine the expertise of Computer Science faculty and the instructional experience of secondary school teachers show initial success in incorporating Internet materials in student-centered mathematics education. Teachers in two graduate courses, the majority of whom had never explored the Internet nor used any educational technology in the classroom, began to experiment with web materials in their classrooms. In addition, the teachers experimented with a departure from the traditional chalk-board lecture approach to the assignment of student-centered projects.

The factors which contributed to the success of these projects include discipline-based lessons, teaching the graduate course in the style recommended by NCTM, providing technical support to the teacher at home and in the K-12 school, classes which involve only to teachers, and involving K-12 students in the training process.

The faculty of the Computer Science Department of the College of Staten Island plan to continue their collaborations with secondary school teachers. As more district schools are connected to the Internet during the coming years, a cadre of teachers will be ready to implement and evaluate these Internet-based lessons.

SELECTED REFERENCES

Berenfeld, Boris. "Linking Students to the Infosphere" T.H.E. Journal, April, 1996:76-83.

Flake, Janice L. "The World Wide Web and Education" Computers in the Schools 12:1/2 1996:89-100.

Fraser, Alistair B. "The Web, a Classroom sans Walls" Syllabus 10:4, November/December 1996: 18 - 20.

Giagnocavo, Gregory, ed. Educator's Internet Companion. ( Lancaster, PA: Wentworth Worldwide Media, Inc.: 1996).

Goldberg, Bruce & Richards, John. "Leveraging Technology for Reform: Changing Schools and Communities into Learning Organizations" Educational Technology September-October 1995:5-15

NCTM Curriculum Standards for Grades 9-12 (1989) on-line
http://www.enc.org/reform/journals/ENC2280/280123.htm

Schumacker, Randall E., Young, Jon I., and Bembry, Karen L. "Math Attitudes and Achievement of Algebra I Students: A Comparative Study of Computer-Assisted and Traditional Lecture Methods of Instruction" Computers in the Schools 11:4, 1995:27-33.

Serim, Ferdi & Koch, Melissa. NetLearning, Why Teachers Use the Internet. (Sebastopol, CA: O'Reilly & Associates, Inc.: 1996)

Topp, Neal W. & Grandgenett. "Year 1 Evaluation of Nebraska's Statewide Plan: Connecting Schools to the Internet" Computers in the Schools 12:1/2 1996:115-129.

Williams, Steven R. & Juliet A. Baxter. "Dilemmas of Discourse-oriented Teaching in One Middle School Mathematics Classroom" The Elementary School Journal 97:1 1996:21-38.

Willis, Jerry "What Conditions Encourage Technology Use? It Depends on the Context" Computers in the Schools 9:4, 1993:13-27.

1. ^i.NCTM Curriculum Standards For Grades 9-12 (1989) On-line http://www.enc.org/reform/journals/ENC2280/280123.htm, p.3

2. ^ii.ibid. p. 5.

3. ^iii.Serim, Ferdi & Koch, Melissa. NetLearning p. 100

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