Middle school mathematics teachers learning to teach with calculators and computers–Part I: Background and classroom observations
Bright, George W
Rapid advances in and the availability of technology in schools have created a gap between the ways that middle/junior high school mathematics is being taught without technology and the ways that this content might be taught through the use of technology. In order to bridge this gap, teachers need to investigate the potential uses of the technology both in actually “doing” mathematics and in communicating mathematical ideas. This article is the first of two in which results are reported from a project that provided a group of experienced mathematics teachers in middle and junior high schools the chance to (a) immerse themselves in calculator and computer use for both doing and teaching mathematics, and (b) prepare themselves as leaders for communicating their new knowledge to colleagues. The teachers were expected to return to their schools and provide models for the use of technology in teaching mathematics and to produce teaching materials built around technology. These goals were addressed through a combination of course work, writing workshops, 7and supportive visits by project staff to principals and classrooms of the participants. This article provides the background for the project and discusses the instructional effects observed in classrooms.
The results of this project have helped establish a base of information on how teachers change (or do not change) their instructional techniques when technology is available for their students. It is certain that technology will increasingly be used to teach mathematics; indeed, some states have mandated that use of technology be integrated into instruction provided by textbooks. Information about teacher change will make the transitions to new technologies easier.
For several reasons, the mathematics content of this project was geared to middle/junior high school levels. First, in Texas, where the project was conducted, there is (or was when the project was conducted) a mandated computer literacy course requirement for all junior high school students. The essential elements for this content include introductory programming (typically in either Logo or BASIC), introduction to and use of utility programs (e.g., word processing, database, and spreadsheet), and brief surveys of the history of computing, the ethics of computer use, and careers in computing and computer-related fields. These essential elements we taught with hands-on use of computers, so in each middle/ junior high school there was a computer lab. This equipment would be adequate to physically support the use of technology in teaching mathematics. Each participant’s school was asked to commit the use of the computers in the school by the participant teacher during the 1988-89 academic year, following the first summer academic program, and in fall 1989, following the second summer academic program.
Second, the mathematics taught in middle/junior high schools is quite diverse. There are opportunities to explore problem solving in the areas of computational mathematics, geometry, probability and statistics, and beginning algebra. All of these areas are appropriate for problem solving and for the use of technology, but as overriding concerns, the content areas should be integrated as a whole and technology should be used to allow students to focus on problem solving (National Council of Teachers of Mathematics [NCTM], 1989). Unfortunately, the middle school mathematics curriculum historically does not introduce much new content (Flanders, 1987). Instructional techniques (e.g. use of technology) need to be up graded so that students do not “tune out” just because they have “seen all that before.”
Third, middle school is an especially important period for students of mathematics. Since it is the last time when all students take common mathematics courses, it is the last opportunity for teachers to encourage all students to continue mainline mathematics study. Making middle school instruction more exciting and more accessible might help encourage future course taking.
Finally, formal reasoning potential (i.e., cognitive capabilities of students) has emerged in these grades, and problem-solving experiences need to be structured to assist the development of those capabilities. Technology holds potential to impact all of these areas. Middle school is thus an ideal time for students to become comfortable with the use of technologies of all sorts. But first, their teachers need to investigate personally the potential uses of technology both in actually performing mathematics and then in communicating mathematical ideas with the help of technology. Once teachers are experienced with technology, they can begin to experiment with uses of technology in instruction.
THEORETICAL FRAMEWORK
Proper use of technology would exploit its unique capabilities to improve the learning of mathematics or to teach both old and new mathematics in new ways. The use of technology does not mean merely the delivery of the same instruction through alternate media. This distinction is similar to that raised by Salomon and Gardner (1986) who were concerned about the types of research questions that are being asked and that needed to be asked about the use of computers in instruction. Just asking if computers can do things better than traditional instruction is almost certainly not the best question to raise. Rather, asking how to use the unique capabilities of computers (and other technologies) to improve the leaning of content is more accurate. As Kaput (1992) wrote, “objectives are changing…in the direction of much more ambitious curriculum and pedagogy whose specification is much more problematic than the computationally oriented objectives and methods of the past” (p. 548).
Beginning with the Agenda for Action (NCTM, 1980) and continuing through a wide range of documents outlining mathematics education reform (e.g., NCTM, 1989, 1991, 1993), there have been repeated calls both for problem solving to be a primary focus of mathematics instruction ad for full use of technology (e.g., calculators and computers) in teaching mathematics. Fey (1984) called attention to the need to incorporate computing into the secondary school mathematics curriculum, including new ideas that can be taught and new instructional techniques that can be developed using computing technology. However, at the time this project began, there was no similar, generally accepted rethinking of middle/junior high school mathematics in the context of technology. More recent work (NCTM, 1989, 1991, 1993) certainly does present a strong case for the use of technology in teaching middle grades mathematics. One of the intended outcomes of this project was to equip teachers to deal with the range of concerns about the use of technology in teaching middle/junior high school mathematics.
As noted above, problem solving is also a critical concern for middle grades mathematics teachers. Students at this level begin to develop the potential for formal reasoning, cognitive capabilities, so it is important for these students to develop proper skills and abilities at the outset of their formal-operational-level study of mathematics problems. Technology lends itself to many types of problem-solving activities. Computers can be used to study problem solving through utility programs (e.g., database, spreadsheet), specially designed software (e.g., SemCalc, Green Globs, Geometric Supposer, GeoDraw, Geometer’s Sketchpad, CABRI Geometry), and through programming. These uses of technology need to be demonstrated for and studied by teachers before appropriate learning experiences can be prepared for students in the classroom.
There is still much to find out about how children learn when instruction is delivered via technology. Some research has been conducted on the use of programming to teach problem solving (cf., Blume, 1985; Liao & Bright, 1991) and from that base teachers can cooperatively begin to structure teaching techniques. Other research has investigated the effects of problem-solving software (e.g., Kaput, 1986) and the development of mathematics learning and problem-solving skills in technologically rich environments (e.g., Lampert, 1985). Current research (e.g., Kaput, 1992) is addressing effects of emphasizing the relationships among various representations of ideas that can be supported through capabilities of technology (e.g., graphs and tables). Other research has documented that “off-loading of routine or complex computations on machines also has an experience-enriching effect” (Kaput, 1992, p. 533). These computations can be arithmetical computations supported by calculators, graph generation supported by graphing calculators or computer graphing tools, or symbolic manipulations supported by an increasingly large variety of hand-held or desktop computing devices. Technology also has the potential to support students as they test conjectures, formulate problems, and manage thinking processes (Jensen & Williams, 1993), but there is not much research which would definitively guide teachers’ instructional decision making. Experienced teachers need to immerse themselves in that software and use their intuitions to plan for its use in teaching.
Teachers preparing efficient and effective use of technology need classroom environments that support their intensive studies. Regular public school teachers teaching a full load do not have this opportunity. Rather, a much better environment in which experimentation with technology can take place is a university setting, especially during summer school when teachers have no other responsibilities. This project provided that type of environment. The participants were experienced and recognized as good teachers, so attention was focused completely on learning how to use technology in learning mathematics and in teaching both old and new mathematics in new and, hopefully, better ways. In addition, the staff of the project was familiar with the range of technology to be addressed and had the experience, time, and resources to think deeply about the use of technology in teaching mathematics in middle/junior high schools.
Research on classrooms, teachers, and students has included observation as a methodology for over five decades (Evertson & Green, 1986; Hoge, 1982; Spaulding, 1983; Williams, Copley, Huang, & Bright, 1993). Applications of the methodology are based upon the purpose, program of research, and frame of reference of the researcher/observer (Evertson & Green, 1986). Within these parameters observation can be effectively utilized in a wide range of content areas and student populations. Among the diverse fields studied through the use of observation are those incorporating technology in the instructional process (Clements & Nastasi, 1988; Fish & Feldman, 1988; Hiltz, 1988; Russek & Weinberg, 1993) and mathematics (Hart, 1989; Huang & Waxman, 1992). Observation studies typically examine student behaviors or instructional processes in classroom settings. The current study investigated both, (a) student academic and non-academic behavior and (b) teacher instructional strategies in middle school mathematics classrooms in which technology (e.g., computer or calculator) was being used to teach mathematics.
INTERVENTION AND EVALUATION METHODOLOGY
Participants were recruited in Spring 1988 through advertising sent to mathematics coordinators of the districts in and around Houston There we 57 applicants, from which 20 participants were selected. Applicants were screened for minimal requirements (e.g., currently teaching at the middle grades level) and then were sorted primarily on the basis of recommendations from district personnel (such as the mathematics coordinator or principal). The general plan of work was (a) during summer 1988 three graduate courses that covered problem solving in mathematics instruction, elementary analysis, and computers in the classroom, (b) during the 1988-89 academic year, two graduate courses (i.e., using technology to teach mathematics and problem solving on computers), project meetings, and staff visitations in the teachers’ classrooms oriented toward support of the direct application of technology in teaching mathematics, (c) during summer 1989 two graduate courses (i.e., geometry and computer education research seminar) and an intensive writing workshop, and (d) during fall 1989 continuing writing/ revising workshops. The academic courses were designed to provide the information the teachers needed to expand their perspective of both mathematics and technology and to integrate technology into the teaching of mathematics. The work during the 1988-89 academic year was designed to monitor the extent of actual integration of technology, with encouragement given as needed. The writing workshops in summer and fall 1989 were designed to provide a synthesis for the teachers and to provide the staff an opportunity to evaluate the success of the integration activities. In addition, the writing workshops also resulted in products that could be shared with interested parties. This plan was followed with no particular difficulty. Four participants dropped out of the project during the first year–two because of medical reasons and two because of changing family responsibilities. Three of the remaining 16 participants had previously taken some of the scheduled courses, so they were excused from retaking these courses.
One of the techniques used to assess the impact of the project was formal observation of students while they used technology in learning mathematics. These data shed light on how the participants made use of their knowledge while teaching students. The observations were conducted between March and May, 1990, after the teachers had spent five semesters studying ways of using technology in teaching middle school mathematics. Only thirteen teachers could be scheduled; each was observed once while using technology to teach mathematics. Six of the teachers used four-function calculators only, one used scientific calculators only, one used four-function calculators and computers together, one used fraction calculators and computers together, and four used computers only. The observation form (Freiberg, 1989) was an adaptation of a standard classroom obseRvation instrument developed especially for this project to quantify academic and nonacademic interactions in classrooms where technology was being used to teach mathematics. Categories included on the form were teacher instructional activity, technology being used, grouping patterns, knowledge level of the content, and off-task behaviors of students (see Appendix 1). (Appendix 1 omitted)
The observer was experienced in the use of the instrument, having utilized a similar instrument in a study of teacher and student academic and nonacademic activities (Prokosch, 1990; Prokosch & Freiberg, 1991). The instrument was field tested and modified to be sure that it could be used effectively. Sweeps were made every five minutes during the observation. During each sweep, all five categories were coded using the fixed category designations. Each sweep consisted of a visual scan of the classroom with activities coded on the form based on the categories. Each sweep began by focusing on the teacher to determine the instructional activities and technology that was being used, the group size in which the teacher was interacting, and the knowledge level of the activity. After these data were recorded, a visual sweep of the students was made in a clockwise fashion around the room, beginning with the student(s) nearest the teacher. Observers recorded each instance of off-task behaviors observed using the seating chart on the form. This process records student off-task behaviors at the time and location they occur. Together, the data allow a comparison of total student off-task behaviors during each observation as well as an indication of what type and how many off-task behaviors occur during each instructional activity, technology use, grouping pattern, and knowledge level. The resulting data allowed inferences to be drawn concerning all five categories.
One-way ANOVAs we calculated for each of the categories, with type of technology used as the grouping variable. Data from the teacher who used fraction calculators and computers together was omitted from the analysis since the mixing of the two technologies prevented categorizing her activity clearly into either technology group. Data from the teacher who used scientific calculators was omitted since that technology is quite different from four-function calculators; we are unsure whether the teacher was using the scientific calculators only as four-function calculation or whether she was having students make use of the unique capabilities of that type of calculator. Scientific calculators may represent a third type of technology that needs study. Data from the teacher who used four-function calculators and computers was classified in the computer category since the calculators were used only minimally for computation and were not the technology focus of the teacher’s attention.
RESULTS AND DISCUSSION
Calculator students displayed more off-task instances than computer students (F=5.81, df=1,9,p
Of potentially more interest, however, are the patterns of interactions observed in the classes, especially as these patterns can be informally compared to observation patterns in traditional, nontechnology lessons. For the 13 classes, an average of 3% of the observed behaviors were classified as off-task, and an average of 97% of the observed behaviors were classified as on-task, independent of the type of technology being used. Instruction activities were coded as lecture/demonstration (36.5%), questioning (36.5%), seat work (8.6%), and other(18.3%). Knowledge level was coded as conceptual (89.5%) and procedural (10.5%). These data suggest a classroom environment that does not match the stereotypical mathematics lesson: Teacher arks questions on homework. Teacher briefly presents new material (which is often times a procedure). Students work on assigned exercises to practice. Our teachers seem to be asking questions to engage students and help them understand mathematics concepts. The teachers appeared to give minimal importance to students working on their own to master procedures.
CONCLUSIONS
At the outset, it is important to note that the types of technologies used by teachers in this project are very similar to technologies that are still in use today. Many teachers do not have regular access to technology, and even when they do, the technology is often in the form of “first-generation computers that a hard to use and of very limited computational and display power” (Kaput, 1992, p. 517) or traditional four-function or scientific calculators. Thus, the lessons learned from this project a still quite applicable to much of the mathematics instruction throughout the U.S. Knowing how teachers make the transition from no technology to some technology in their teaching will also provide a base of knowledge to help make the transition to newer technologies such as graphing calculators smoother.
The project provides at least partial insights to some key issues about the use of technology in mathematics instruction First, that students were observed off-task so little is encouraging. It has been recognized for some time that the introduction of computers often seems to motivate students, but there appears to be little similar evidence reported for calculators. Hirshhorn and Senk (1992) provide evidence of changed attitudes with the use of scientific calculators, but the mechanisms driving the changes are not clear. However, we also wonder to what extent the particular teachers influenced this effect, independent of the use of the technologies. That is, were our participants simply very effective teachers who happened to use technologies, or did the technologies themselves influence the amount of off-task time? Alternately, could the activities be used by other teachers with similar results, or did the use of technologies influence the results? Clearly, our data do not answer these questions.
Second, the difference in off-task time for calculators and computers suggests that different technologies at will indeed have different effects on students. We are not surprised at that suggestion, of course, but we don’t know what characteristics of technologies are important for determining those differential effects. Being able to characterize these differences would seem to be important for any development and implementation of new technologies.
Third, the introduction of technologies in classrooms seems to have altered what teachers did there, though it is impossible to separate out the effects of the inservice program itself as opposed to the technologies. Although we have no preproject observation data on teachers, initial interviews suggest that they were fairly traditional in their approaches to instruction. Post-project interviews indicated that many of them believed they were influenced by the ways that instruction was modeled in the courses during the project (i.e., exploring content and solving problems with technology). The post-project observations, then, suggest that personal exploration of mathematics with technology by teachers may be an effective way to alter those teachers’ behaviors in the classroom. Clearly, however, careful follow-up of this speculation is needed.
The project evaluation raises more questions than it answers: What differences in teacher behaviors and classroom climate before and after the project can be documented? How long will they persist after the inservice efforts cease? Do scientific and graphing calculators result in classroom observations more similar to those using four-function calculators or computers? Could a less intensive inservice be effective for achieving outcomes roughly equivalent to the outcomes of this project? If so, how much less? Questions such as these require further investigation.
This project was conducted before the widespread availability of graphing calculators, but it is rapidly becoming apparent that such technology is, or will be, available at reasonable cost for middle school mathematics instruction. New software along with less expensive computer hardware may change views of what is possible and what is appropriate for middle school students. These changes in technological resources need to be accounted for in future research on the effects of technology on instruction and learning.
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Author’s Note: Based on presentations made at the 1991 Southwest Educational Research Association annual meeting, San Antonio, TX and the 1991 American Educational Research Association annual meeting, Chicago, IL.
This work was supported in part by a grant from the National Science Foundation (Grant No. TPE 8751473). All positions and opinions expressed, however, are those of the authors and do not necessarily reflect the positions of the Foundation.
Editor’s Note: George Bright’s address is School of Education, Department of Curriculum Instruction, Curry Building UNCG, Greensboro, NC 27412-5001. Neil Prokosch’s address is National-Louis University, West Suburban Campus, 1 South 331 Grace Street, Lombard, IL 60148-4691.
Copyright School Science and Mathematics Association, Incorporated Oct 1995
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