Research on the Effects of Reform on Students and Teachers

Reviews of studies of mathematics reform (e.g., Ross, 2000) indicate that when reform is fully implemented there are positive impacts on students.  For example, problem solving achievement improves (Brenner et al., 1997; Cardelle-Elawar, 1995; Fennema, Franke, & Carpenter., 1993; Fuchs, Fuchs, Karns, Hamlett, & Katzaroff, 1999; Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Schoen, Fey, Hirsch, & Coxford, 1999), as does mastery of traditional math objectives (Reys, Reys, & Koyama, 1996; Romberg, 1997; Villasenor & Kepner, 1993).

But math reform is difficult to implement. Even teachers chosen as exemplars of reform regress from the ideal, displaying the height of reform one day while regressing to traditional methods the next (Senger, 1998). Some elements of reform are more difficult to implement than others. The most challenging is the management of student talk about mathematical reasoning—encouraging student constructions without leaving them floundering or subverting ownership of their learning  (Ross, 1995; Ross, Haimes, & Hogaboam-Gray, 1996; Ball, 1993; Smith, 2000; Williams & Baxter, 1996).

The catalogue of barriers to reform is a lengthy one. Teachers must be agents of a change they did not experience as students (Anderson & Piazza, 1996). The pedagogy is not only different but harder to learn. For example, in traditional math there is a generic script that guides each day’s lesson through a manageable body of content. In reform math the day is governed by unpredictable student responses to real life problems. Teachers, especially elementary generalists, tend to lack the disciplinary knowledge required to make full use of rich problems (Stein, Gover, & Henningsen, 1996; Spillane, 2000) and texts cannot prescribe universally applicable courses of action (Remillard, 2000). Adoption of reform math can leave teachers feeling less efficacious because their contribution to student learning is less visible than in traditional teaching (Ross, McKeiver, & Hogaboam-Gray, 1997; Smith, 1996). Teacher beliefs about mathematics (i.e., a rigid set of algorithms, not understandable by most students, that must be approached in an inflexible sequence) conflict with reform conceptions of math as a fluid, dynamic set of conceptual tools that can be used by all (Gregg, 1995; Prawat & Jennings, 1997). Reform does not meet parental expectations about how math should be taught and tested (Graue & Smith, 1996; Lehrer & Shumow, 1997). Reform conceptions of mathematics conflict with mandated assessment programs that measure computational speed and accuracy (Firestone, Mayrowetz, & Fairman, 1998). Time to cover the curriculum is a major challenge. Keiser and Lambdin (1996) found that student constructions took longer than lecture-recitations, novel problems increased time taken for discussion of homework, and students with poor motor skills took longer to use manipulatives than anticipated.

Technology as a Strategy for Reducing Barriers to Implementation

There is ample correlational evidence that teachers who are more frequent users of computers in the classroom are more likely to adopt even the most difficult dimensions of reform such as constructivist teaching (e.g., Becker, 1998; Christmann, Badgett, & Lucking, 1997; Heid, 1997; Huetinck, Munshin, & Murray-Ward, 1995; Waxman & Huang, 1996). What is less clear is how the integration of computers with mathematics instruction contributes to reform. The relationship may be spurious: good teachers tend to adopt the innovations of the day, in this case technology integration and math reform (Becker, 1998). More likely is that technology enables teachers to implement their constructivist beliefs by relieving students of the tedium of calculation and providing them with visual representations to support dialogue about mathematical ideas. Some researchers (e.g., Sandholtz, Ringstaff, & Dwyer, 1997) have argued that technology encourages teachers to change to a constructivist orientation because teachers have to share control with students in a computer-based learning environment. The contribution of technology to math reform is not automatic. Providing computers and software to teachers without appropriate in-service has minimal effect on teacher practice (Robertson, Calder, Fung, Jones, O'Shea, & Lambrechts, 1996).

The purpose of our study was to investigate how technology literate teachers, viewed as effective teachers by their supervisors, and supportive of reform ideals, integrated computers into their math teaching. Our goal was to find out whether access to technology facilitated or impeded implementation of math reform. The main mathematics teaching software used by the teachers in the study was Math and More, a program developed to be compatible with NCTM Standards. The software designers intended that the program would engage children as active learners, provide meaningful problems, encourage collaboration with others, stimulate reflection on progress, and offer a variety of resources. There were three program levels distributed among grades 1-5. Each level contained three units focused on Geometry, Data Management & Probability, and Patterning & Algebra respectively. The activities in the units were also linked to other math strands and other areas of the curriculum. Each unit consisted of 6-8 investigations that included on-line and offline activities.

Sample

Eighty grade 1-3 teachers, 21% of the population in 32 schools in one school district, responded to an invitation to participate in the study by completing surveys measuring teacher support for math education reform (Ross, Hogaboam-Gray, & McDougall, 2000: 30 items, alpha=.83), confidence in math teaching (Huinker & Madison, 1997: 9 items, alpha=.77), and confidence in teaching with computers (Ross, Hogaboam-Gray, & Hannay, 1999: 8 items, alpha=.88). (There was a significant positive relationship between teacher commitment to math reform and teacher confidence in using computers to teach (r=.31, N=79, p=.005) but not with confidence in math teaching (r=-.05). The three teachers who were selected were the computer contacts for their schools, in the top quartile in commitment to math education reform, above average in confidence in teaching mathematics and with computers, and were recommended as capable teachers by district curriculum staff. The three teachers each had four computer stations in their rooms, connected in school networks, with printers. Each had attended in-service on the main mathematics software, called Math & More. This software contained activities for exploration and structured skill practice to support the reform curriculum.

Donna, the grade 1 teacher, had taught for over 20 years. She used her home computer to prepare classroom materials but rarely used e-mail, explored the Internet or recorded student grades on spreadsheets. Donna used the computer as a learning tool, emphasizing that it was not an entertainment device, even if students sometimes viewed it that way. “Instead of play, let’s say exploration with a purpose.”  She liked the convenience the software provided in generating lessons. “The thinking is done for you. Like you can go see, okay that’s done, I’ll just plan a nice lesson around that…Its great.”

Beth, in grade 2, had taught in the elementary panel for 23 years. In the early 1990s she participated in the district math curriculum development and had been a presenter at summer institutes and fall workshops. Beth was on the computer daily for her own purposes, e-mailing, accessing newsletters, and locating materials on the Internet. For Beth the computer was a valued but ultimately inessential teaching tool: “If I didn’t have it in my program it wouldn’t be a loss.”

Bruce, in grade 3, had taught for over ten years. He had been a presenter in the district’s mathematics software in-service. Bruce used computers regularly to accomplish his personal goals, including accessing M.Ed. courses through computer-mediated communication. Bruce saw advantages in using computers in math class. For example, the production values contributed to the self-esteem of less able students: Computers are “more successful with weaker students because the end product was really nice. They were really pleased with it and it looked the same as everyone else in the class”. He also expressed a number of concerns such as the software offered too many choices: for a “child who was struggling, it was…overwhelming.

Source of Data

Each teacher was individually interviewed for 60 minutes in January 2000. Questions concerned their implementation of each dimension of reform displayed in Table 1 and how they used computers to teach math and other subjects.

The site visits followed procedures established by Simon and Tzur (1999). We visited each classroom in February on two consecutive days during their math period (all three teachers allocated 75 minutes per day for math). We interviewed teachers before, during, and after each math lesson to elicit the teacher’s intentions and reflections on the lessons we observed. We recorded the appearance of the room (e.g., location of computers, display of math posters) and noted events that occurred when students were learning mathematics, with and without computers. We anticipated that all teachers would be in transition, manifesting some aspects of traditional math teaching as well as reform elements. (Frykholm, 1996 and Spillane & Zeuli, 1999 have demonstrated that self-reports exaggerate the degree of math reform implementation).

We compiled individual case reports describing the teacher’s implementation of the 9 dimensions of reform in Table 1. We returned to each classroom in May for a second round of two-day visits, searching for evidence confirming or disconfirming generalizations developed in the first round. We created a narrative record of the lessons and collected artifacts. We completed site observation records that recorded evidence about the 9 dimensions of Table 1.

We interviewed four student focus groups in each class about their perceptions of the lessons, focusing on implementation of reform elements. Students were randomly assigned to 3-person, same-gender, mixed-ability groups. In the first round, one interview guide focused on experiences when students were learning math with computers; the other guide elicited student responses to learning math without computers. Both guides asked students to describe what they were doing in math class that day and provide reflections on their experience. The focus groups were repeated with different students in the second round of visits with questions focused on teacher implementation of the nine dimensions from the students’ perspective.

Following the site visits and preliminary analysis of the case study data, we met with each teacher to review the findings from their case. The purpose of the meeting was to obtain feedback on the accuracy of our interpretation of the case and identify which aspects of their practices would be of greatest use to other teachers. Teachers designed a 30-minute presentation on their implementation of reform that was delivered to groups of 20 teachers on 6 occasions, each of which we observed. 

In June 2000 teachers were individually interviewed for 30-60 minutes about the relationship between their use of computers and mathematics education reform. The interview guide was individually tailored to each teacher to fill gaps in our database. Teachers also participated as a group in a final interview to identify key themes emerging from the data. All interviews were audio recorded and transcribed verbatim in whole (teachers) or part (students).

Data Analysis

All data were entered into NUD*IST, a qualitative data analysis program that lends itself to the induction of themes and summarization of evidence through constant comparison. Our coding scheme was framed by the characteristics of reform described in Table 1 and was elaborated as we processed the data. The outcome of the first round of site visits, student focus groups, and teacher interviews were individual case reports describing teachers' implementation of the 9 reform dimensions. We elaborated the initial case study reports following review by the case study teachers and additional data collection (second round of site visits, focus groups, teacher interviews, workshop planning sessions, and workshop observations). The credibility of the claims was established through triangulation (of data sources, observers and interpreters, and over time), accurate recording of information (e.g., audio recording of interviews), maintaining an audit trail (tracking themes and assertions from raw data through to conclusions), member checks, value auditing (reviewing the data in terms of researcher bias), and rich description.

Results

We compiled case studies (Ross et al., 2000) by comparing within-teachers across data sources, over time, and between analysts. We began the cross case analysis by summarizing the impact of technology on implementation of reform for each teacher on each dimension using a 5-point rating scale as shown in Table 2. By comparing data between-teachers we generated a series of assertions and assembled evidence for each.

1. Technology had its greatest impact on teachers’ implementation of reform by helping them expand the scope of their programs and by promoting positive attitudes toward mathematics. The impact on dimension (i) (scope) was most visible in the grade 2 classroom. Beth extended the number of mathematical strands (branches of mathematics) that she taught by assigning much of the burden of instruction for those emphasized in reform to the computer. For example, her teaching of geometry and spatial reasoning was primarily delivered by Math and More software. Patterning was addressed mainly through the computer as well because it freed students from the repetition of physically creating different displays, made it easier for students to make patterns (“for little kids with fine motor problems, the dexterity isn’t great.”). It took far less time to generate products and students enjoyed the activities. Beth also reported that some math strands were easier to address on-line than off-line, for example, storing data, constructing graphs, and editing graphs in data management problems. In addition she used the software to complement her teaching of time, especially the correlation of analog and digital timepieces, concepts students had difficulty learning off-line. The other two teachers also reported that the software provided an array of engaging activities to supplement their teaching of a variety of types of mathematics. The computer was especially helpful in the areas designated by reform as requiring more attention: patterning and algebra, geometry and spatial sense, and data management and probability.

The computer also helped all three teachers promote positive attitudes to mathematics (dimension ix). Almost all students progressed rapidly and easily through the activities. In the structured practice section, the computer provided immediate positive feedback in the form of colorful, sticker-like displays and congratulations. Incorrect responses were not highlighted and if students chose to do so they could access an audio clue (but not the right answer).

2. The computer promoted equity of access to all forms of mathematics available in the classroom but this did not mean that reform ideals were achieved. All teachers used a rotational system to ensure equal access to computer activities. Within each activity the software offered multiple levels of difficulty, ensuring that all ability levels could participate. For example, in the structured sections, money tasks varied in terms of the number and variety of coins to be counted. The grade 1 teacher offered the following example of an exploration activity that could be completed by all ability levels: “When you are generating your own pattern you could make it as complex or as difficult as you wanted…[high ability] students would be doing diagonal patterns whereas the little fellow would be just doing your basic ABAB pattern”.

But we observed that Beth and Bruce assigned tasks at the same level of difficulty in the structured section to students of all ability levels, thereby reducing the capacity of the software to provide differentiated experiences. In addition Beth rarely used the exploration section--we observed students immediately went to the structured section. Students rarely had access to higher-level mathematics in Beth’s classroom because of her conception of mathematics and how it should be taught. Beth believed that mathematics, with a few exceptions in geometry, was a linear subject that lent itself to a structured approach with careful teacher monitoring of progress before students could pass from one step to another. She viewed each branch of mathematics (e.g., algebra) as being relatively independent of others (e.g., geometry), in contrast with the reform view that each branch of mathematics is connected to all other branches by a common set of principles for creating knowledge. She also believed that although mathematics at the university level might change, in grade 2 knowledge was fixed and that it was the teacher’s task to ensure students mastered essential procedures. Access to higher-level understanding was limited in Beth’s classroom because she did not make student thinking a central focus of her teaching in the way that Bruce and especially Donna did.

3. Teachers adapted technology to fit their off-line activities, heightening or depleting its contribution to reform implementation. The variations in Table 2 indicate that the contribution of technology to the implementation of teacher was mediated by teacher decisions. For example, in Bruce’s grade 3 classroom computer software provided rich opportunities for discovery of mathematical ideas (dimension iii). These exploration tasks were complemented with activities not using computers. For example, the mapping activities in the software were reinforced at an off-line centre in which students were given a provincial map with the task of finding alternate routes from their own city to another. We observed Bruce modeling the discovery process by asking who, what, why questions. He encouraged students to elaborate their responses before calling on other students to provide a more complete response. Bruce focused on students’ thinking and probed their reasoning. We also observed whole class discussions in which students asked each other understanding type questions and pushed for explanations. In one class students spent over 30 minutes exploring a single estimation problem and a variety of ways of solving it were elicited. Bruce also debriefed students on what they had learned at the computer centre, clarifying misconceptions, and assessing progress. When students were asked whether the teacher told them how to solve math problems or whether they and the teacher figured it out together, most indicated that they were responsible but the teacher helped them. For example, one child said “Me and the teacher figure it out, like he gives me clues and stuff”.

Donna consistently operated as a facilitator of student construction of mathematical ideas. She encouraged grade 1 students to problem solve to develop deep understanding. She relied heavily on open-ended questions, for example, “what can you tell me about?” in response to displays of mathematical objects (such as a frequency table) at computer and off-line stations. Donna had students make a hypothesis, test it by collecting data, and describe a pattern based on the data collection. She was not upset by cognitive dissonance. In recounting student responses to tasks requiring probabilistic reasoning, Donna reported “they sort of came up baffled about probability, because it didn’t follow any pattern, which was wonderful! Because that was basically what I wanted them to establish.” Although Donna intermixed discovery with guided discussion, she tended to have students discover things for themselves first: “I try to listen and learn from their direction and go from there.” As with Bruce, the computer provided additional opportunities for discovery learning that were not qualitatively different than what Donna was already doing.

Beth’s choice of activities meant that the computer supported off-line teaching in which student activities were tightly controlled, focused on practice of specific skills, discourse was dominated by the search for correct answers, and there was little overt attention to how students thought through mathematical problems. When students were invited to perform a procedure Beth broke the sequence into pieces, cuing students on each step. She began lessons with whole class demonstrations of procedures, followed by applications at the centres. The computer did not contribute to implementation of dimension (iii) (discovery) because the computer activities emphasizing exploration were not made available to students  (“kids see them as games”). The structured computer activities students completed provided mainly correct/incorrect feedback. Some students may have benefited from clues provided by the software through earphones, although we observed students accessing the help function rarely. Other parts of the software provide a graphic display of how to solve the problem when students were unsuccessful but we did not observe this function being employed by students, because Beth discouraged its use. By having students work independently on computer tasks there were few opportunities for deeper conceptual understanding through student-student discourse. Students confirmed that curriculum choices were the exclusive prerogative of the teacher.

The differential impact of technology on teachers’ implementation of reform was also observed for three other dimensions: (iv) the creation of mathematical community, (v) the use of the computer as a problem solving tool, and (vi) encouragement of student-student interaction.

4. Although the impact of technology was largely positive, computers impeded implementation for some teachers on some dimensions. All three teachers reported instances in which their computer activities were not as rich as off-line learning opportunities. Bruce was affected more than the other two. Computers impeded Bruce’s implementation of dimension (ii), provision of rich mathematical tasks embedded in real-life contexts. Bruce was trying to make greater use of Quest 2000 texts because of the rich array of mathematical tasks this series offered. But he was not able to use the technology package for the series, which Bruce believed to be “wonderful”, because the network did not permit him to load the disk. Bruce was critical of the activities in the software (especially in the Geometry section) because they provided too many routine tasks in which a single solution was required. Bruce described a program that enabled students to see on their screens the solutions being generated by three other students as well as their own (which would contribute to understanding that math problems have multiple correct answers and solution paths) but technical glitches affecting computers in other classes ended his use of it. Some aspects of dimension (vii), authentic assessment activities, were modestly supported by computers. For example, assessment was well integrated with instruction and the computer displayed student progress: “Because they are so small, I’m able to see the screen from wherever I am in the room” But the software did not support two main aspects of Bruce’s assessment policy: performance assessment and the use of rubrics for self- and teacher-evaluation of student work.

5. Hardware problems constrained the impact of computers on instructional practice. All teachers reported that the computer was a risky vehicle. For example, we observed students struggling to make sense of a money activity in which the software had mysteriously brought up a display of US rather than Canadian coins on some computers. Crashes occurred often enough that teachers had off-line substitute activities available. None of the teachers used a computer display in whole class teaching because the screen projection function rarely worked. Bruce noted that using an unprojected screen was not effective: “While I was working on one computer I would have kids unfocused and goofing around, somebody standing on a chair, somebody tattling on them. It was just crazy.” Teachers were limited to the software on the school network because the operating software did not allow them to load CDs brought in by students or acquired from professional sources. There were also individual teacher concerns: Donna color-coded all her off-line activities to guide grade 1 students through her centers but she could not code computer activities in the same way. Bruce thought Math and More offered students too many choices. Beth felt that the computer offered insufficient practice to some students “after three attempts the computer gives them a blue star which allows them to go on to the next question" even if they had not mastered the algorithm.

6. None of the teachers realized the potential of the computer to support student-student constructions. In the three classrooms off-line activities were collaborative tasks or individual assignments undertaken in a group setting in which students were required to explain solutions to peers and seek help from other students if they needed it. All three teachers provided explicit training in how to seek and give help to peers. Opportunities for student-student interaction were greatly reduced on-line. In all three classrooms students worked on computer tasks individually. There were overt tutoring opportunities. When a new computer activity was introduced, teachers taught four students how to work through the program. When the first group finished, they taught four new students and so on. But the task was, in Donna's words, "to teach…the buttons and functions and to explain how they work", and did not address the mathematical concepts embedded in the activities. This contrasted with the focus on student thinking in Donna's off-line procedure of appointing students "teacher for the day" to model mathematical reasoning. All three teachers encouraged incidental tutoring at the computer but most students worked through the activities without seeking help. We observed students displaying their work to peers but there were no substantive conversations about mathematical ideas, in contrast with the high levels of off-line collaboration.

Discussion

The first finding of the study is that technology did not have a huge impact on the implementation of mathematics education reform. Computer-based instruction was not an innovation for these teachers. They had institutionalized the computer as a mode of instruction, in part because an infusion of technology in the district two years earlier had positive effects on the computer skills and attitudes of students (Ross, Hogaboam-Gray, & Hannay, 2001) and teachers (Ross et al., 1999) Teachers adjusted the technology to fit their conceptions of how mathematics should be taught; there was no mechanistic process of technology driving teachers to constructivism. The computer was an amplifier providing additional means for teachers to accomplish their instructional goals. It was not a reorganizer that challenged existing ways of teaching. The implication for instructional reform is that the provision of computer-delivered curriculum packages is unlikely to have a dramatic effect on practice nor, in this study, did it have the negative effect on teacher professionalism observed by Apple and Jungck (1992). As persuasively argued by Fullan (2000), change does not occur by importing successful innovations but by creating the conditions that made the innovations successful in their originating sites. This finding might be limited by the fact that the teachers espoused high commitment to the ideals of mathematics education reform and had made substantial progress toward implementation of these ideals. One might argue that the effect of technology could be considerably greater for teachers who had not progressed as far on the reform path.

The second finding is that technology facilitated implementation of math reform of all teachers with respect to dimensions that were universally supported (i.e., improving student confidence in their mathematical abilities) or could be added to their programs (i.e., increasing the range of mathematics taught) without disrupting their core teaching beliefs. Previous studies (Anderson & Piazza, 1996; Grant, Peterson, & Shojgreen-Downer, 1996; McCarthey & Peterson, 1993; Sherman, & Richardson, 1995; Widmer &. Sheffield, 1994) reported that teaching with manipulatives is frequently accomplished more readily than other dimensions of reform (the three teachers in our study used manipulatives extensively) because manipulatives can be added without changing other aspects of math teaching. The implication of this finding is that making hardware and software that is compatible with reform ideals available to teachers is likely to extend the scope of their program and promote student confidence in their mathematical abilities.

The third finding is that the contribution of technology increased with enactment of reform ideals. There were benefits for Donna on every dimension of reform because she had developed the capacity (in her beliefs about mathematics teaching and prior adjustment of her instructional practices) to take advantage of the opportunities that the computer offered. Bruce also benefited on several dimensions but he experienced negative effects as well. Technology positively impacted Beth's implementation on two dimensions (with a large impact on the scope of her program) but the gap between belief and practice was larger for her than the other teachers and on many dimensions of reform she was not able to take advantage of the opportunities afforded by the technology. The implication of this finding is that teacher in-service on the integration of computers into the mathematics classroom needs to be differentiated by teachers' levels' of reform implementation because their adaptations of the technology will vary depending upon their off-line instructional preferences. Transitions or stages of math reform implementation have been identified (Bright, Bowman, & Vace, 1998; Nolder & Johnson, 1995; Slavit, 1996; Spillane & Zeuli, 1999).

The fourth finding is that all teachers missed opportunities to promote student-student construction of mathematical ideas, a central tenet of reform. The computer can provide a microworld for representing real world problems, manipulating models, communicating symbolically, and inferring mathematical concepts and principles. These processes are much more likely to occur when a skillful teacher mediates between the child and the screen, scaffolding the student's interaction with the software (Sutherland & Balacheff, 1999). The same processes can be undertaken by peers because the screen displays student thinking, thereby heightening opportunities for students to jointly fill gaps in understanding, challenge assumptions, and experiment with alternatives (Heid, 1997). But these processes are unlikely to occur when students work alone. The implication of this finding is that teachers should assign students to computers in pairs with collaborative tasks that stimulate discussion of alternative solutions and solution strategies. But without access to better software that promotes student-student discussion of mathematical ideas, the teachers will continue to find it extremely difficult to use the computer as a constructivist tool. 

The fifth finding also has implications for in-service. In reviewing similarities between the current round of math reform and the New Math movement of the 1950/60s that ultimately failed to influence teacher practice, Bossé (1995) noted that inattention to teacher in-service was the key deficiency of both movements.  Experience since then (e.g., Knapp & Peterson, 1995; Schifter & Simon, 1992; Smith, 2000) demonstrates that ongoing professional development can increase implementation of reform, particularly  when in-service is focused on providing teachers with examples of constructivist teaching (Bitter & Hatfield, 1994) and explicitly addressing their beliefs about mathematics as a teachable subject (Borko, Davinroy, Bliem, & Cumbo, 2000; Grant, Peterson, & Shojgreen-Downer, 1996). The last finding from this study suggests that in-service should include persuasion that students can learn from each other and provide strategies for teaching students how to give effective help in math class (e.g., generic help as outlined as described by Farivar & Webb, 1991 and math-specific help as in Fuchs et al., 1997).

Finally, the teachers in the study used the particular software that was loaded on their school networks, principally Math and More. Since the study was conducted, the school district modified its hardware to enable teachers to load other math programs and the district continues to add new software to school networks. We doubt that the provision of additional software options, in itself, would dramatically increase the contribution of technology to implementation in this site. The existing software had the potential to support the ideals of mathematics education reform and was in fact designed to do so. It was the way that teachers used the software, for example, by encouraging the use of the structured rather than the exploratory activities in one case and by assigning students to activities alone (in all cases), which determined technology’s impact. However, if the characteristics of the software had pushed teachers away from reform (e.g., by emphasizing repetitive practice of algorithms) the characteristics of the software would have loomed much larger in the study.

Conclusion

This study found that technology contributed to teachers' implementation of math education reform and suggested ways in which the impact could be strengthened. The study was limited by its sample. These teachers were purposefully selected to represent those who were implementing reform (Donna), those who had moved substantially toward it (Bruce) and those who had changed on some but not all dimensions (Beth). In addition, these teachers were more committed to computer-based instruction than their peers and more confident of their abilities. The impact of technology might be weaker with teachers who preferred a traditional approach to math teaching and were less literate technologically. On the other hand teachers of higher grades might be more influenced by technology because its benefits are more obvious for problems addressed in higher grades (e.g., graphing functions in algebra). However, the decision to integrate computers into the curriculum does not rest on their instrumental value in enhancing the teaching of mathematics. Although we regard the improvement of mathematics teaching and subsequent increase in student understanding to be worthy ends, the ability to use a computer to accomplish personal and academic goals is intrinsically valuable and sufficient to warrant students spending extensive classroom time on computer tasks.

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