|AUTHOR:||LINDA M. SIMONSEN; THOMAS P. DICK|
|TITLE:||Teachers' Perceptions of the Impact of Graphing Calculators in the Mathematics Classroom|
|SOURCE:||The Journal of Computers in Mathematics and Scienc v16 no2-3 p239-68 '97|
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The National Council of Teachers of Mathematics (NCTM) Standards states that at the high school level, graphing calculators must be available to all students at all times. The purpose of this study was to compile teachers' impressions of the barriers and/or incentives associated with the use of graphing calculators on classroom dynamics, curriculum and evaluation, training, support, and overall attitude. Hewlett-Packard (HP) gave 30 HP-48S graphing calculators and an overhead projection device to 36 high schools across the United States. Each school was also given a mandatory 1-day inservice, as well as an optional 1-week summer workshop, on how to use the calculator and integrate it into the classroom. A year later, each of the schools was contacted and asked if a teacher would participate in a telephone interview which contained primarily open-ended questions. Twenty-seven teachers, representing a wide range of high school mathematics classes, participated in the audiotaped interviews. Recurring themes that emerged from the interviews were organized into these categories: (a) advantages of calculator use, (b) disadvantages of calculator use, (c) classroom dynamics, (d) curriculum and evaluation, and (e) professional support and development. The results demonstrated that the teachers' perceptions of the advantages appeared to be instructionally related, whereas the perceptions of the disadvantages appeared to be primarily logistical in nature. The dynamics of a classroom tended to shift to more discussion, inquiry, and cooperative learning. There was considerable reluctance to deviate from stringent curriculum requirements that are reinforced by standardized tests. This study gave teachers a chance to share their knowledge, experiences, and insights, which in turn will help guide the design of curriculum materials, inservice programs, and support systems to best equip teachers to handle the demands and opportunities that technology presents.
Improving mathematics curricula and instructional methods is the most prominent challenge facing mathematics educators today (National Council of Teachers of Mathematics [NCTM] Commission on Standards for School Mathematics, 1989; National Research Council, 1989; U.S. Department of Education, 1983). Part of this call for change is the proposal that technologies be used extensively throughout all school mathematics instruction (Mathematical Sciences Education Board of the National Research Council, 1990; NCTM Commission on Standards for School Mathematics, 1989; NCTM Commission on Teaching Standards for School Mathematics, 1989; National Research Council, 1989). In the NCTM Standards we find an explicit call for when, where, and how technology should be used in secondary school mathematics. Specifically, at the high school level, graphing calculators are to be available to all students at all times. More generally, traditional teacher-centered modes of instruction must give way to a more active role for students in constructing their own knowledge. It is conceivable that technology will not only alter what is important in mathematics, but may actually force entirely new classroom dynamics.
Research and discussion concerning the impact of technology in the mathematics classroom has focused primarily on the issues of curricula and classroom dynamics. Several authors have proposed that the content goals of mathematics should shift from a focus on computational skills to an emphasis on conceptual understanding (Demana & Waits, 1990b; Fey, 1984; NCTM Commission on Standards for School Mathematics, 1989; Palmiter, 1991; Rich, 1991; Steen, 1988; Schofield & Verban, 1988), enhancing student problem solving (Demana & Waits, 1990b; Dick, 1990; Wiske et al., 1988), and the ability to visualize using multiple representations (Demana & Waits, 1990a, 1990b; Dick, 1990; Fey, 1989). A study of Heid (1988) suggests that it is possible to minimize time spent on symbolic manipulation skills and increase time spent on problem solving and applications. Although such manipulation skills do not form the heart of the "ideal" curriculum, they do account for the preponderance of the "achieved" curriculum that is actually mastered (Steen, 1988). This may cause us to reflect on the necessity for change in curriculum materials if technology is to be used.
The use of technology can shift the role of the teacher to a facilitator of students' learning from a presenter of ready-made knowledge (Dick, 1990; Heid, Sheets, & Matras, 1990; Steen, 1988; Swadener & Blubaugh, 1990; Wiske et al., 1988). Students may take more responsibility for their own learning, and the dynamics of classroom activity can shift to more discussion, inquiry, and cooperative learning with continued use of graphing calculators (Dick & Shaughnessy, 1988; Farrell, 1990), as well as with the use of a variety of other technological tools (Berenson & Stiff, 1989, 1990).
For teachers to take on the difficult challenge of integrating technology into their teaching, there is a need to respect their professional judgment, listen to their concerns, and provide them with the time and opportunities to learn, experiment, and share (Lindquist, Harvey, & Hirsch, 1991; Lovitt, Stephens, Clarke, & Romberg, 1990; Wiske et al., 1988). The use of technology in mathematics classrooms raises several areas of concern for teachers: curriculum issues, classroom dynamics, training and support, and technological accessibility.
There is a general concern that there is not enough time to cover the technology and the required curriculum. New activities just added on to an existing curriculum have been shown to have only a marginal effect (Wiske et al., 1988).
The alteration in dynamics of technology-based classrooms, as discussed previously, may shift the role of the teacher to a facilitator and guide; a role that may be uncomfortable or even undesirable for teachers who enjoy directing teacher-centered classrooms. The increase in questioning by students often poses a threat to some teachers' autonomy and to predictability in teaching (Dick, 1990; Schofield & Verban, 1988).
The orchestration of integrating technology into the traditional curriculum is not a simple task. Studies have revealed that lack of appropriate training is a major concern (Schultz, 1989; Wiske et al., 1988), as well as a lack of incentives such as time, money, and moral support (Wiske et al., 1988). It is suggested that a structured, supportive, professional-development environment must be provided in order for teachers to reshape their teaching practices (Lovitt et al., 1990).
Although the availability of technology does not ensure use, the predominant obstacle cited for impeding the employment of technology is the lack of access (Dick, 1990; Schultz, 1989; Wiske et al., 1988). It is possible that this barrier will no longer be a major concern with the advent of inexpensive hand-held computers or graphing calculators.
PURPOSE OF THE STUDY
It has been suggested that there is a need for an extended discussion of the effective use of technologies in present secondary school mathematics courses and to identify the barriers that still exist (Lindquist et al., 1991). The purpose of this study was to examine teachers' perceptions of the impact of graphing calculators in the mathematics classroom. (These calculators have computer algebra system capabilities in addition to graphing features.) In particular, this study involved the examination of the following questions:
What are teachers' perceptions regarding the advantages and disadvantages associated with the use of graphing calculators in the mathematics classroom? In particular, what are the perceived barriers and incentives to the use of graphing calculators with respect to classroom dynamics, curriculum and evaluation, and professional support and development?
Since the focus of the study was on teachers who had new experience in using graphing calculators, an underlying objective for the study was to gather a more comprehensive set of ideas on how best to equip teachers to handle the new demands, and to take advantage of the new opportunities, which technology presents. Knowledge of the teachers' perceptions of the impact of graphing calculators in the mathematics classroom is critical to the development of any curricula, inservice programs, and support systems designed to improve technological implementation.
In January of 1991, a classroom set of HP-48S graphing calculators and an overhead projection device for instructional purposes were donated to each of 36 high schools across the United States. The schools were selected and sponsored by company sites in their local areas. (There were no specific criteria for eligibility for a donation other than the interest and willingness of the local Hewlett-Packard site to make the award to a school.)
A graphing calculator has the usual features associated with a programmable scientific calculator, but in addition is capable of displaying graphs of functions and parametric equations and often includes a numeric root-finder and does vector and matrix calculations. This particular model is also capable of manipulating symbols algebraically, including differentiation and integration. Indeed, the calculator could be characterized as a hand-held computer algebra system.
DESCRIPTION OF INSERVICE WORKSHOPS
Along with the donation of a classroom set of machines, each school was provided a 1-day inservice on the operation of the calculators and the overhead device. Instructors for the 1-day workshops were college instructor s with direct experience in using these calculators in precalculus and calculus. During this short workshop, the instructors used examples from their own teaching of precalculus and calculus to illustrate the various features of the calculators.
Optional 1-week summer workshops were made available for any of the teachers who were interested. (The individual or school only had to pay for the travel required to get to one of the workshop sites, and each school was within 200 miles of one of the designated workshop sites.) In these workshops, participants were given more extended opportunities to examine and work with precalculus and calculus curriculum materials taking advantage of graphing calculators. In particular, extensive examples were drawn from the C[sup2]PC (Calculators and Computers in PreCalculus) materials developed by Waits and Demana (1992) and the NSF calculus reform project materials developed by Dick and Patton (1991).
Each day of the workshop included time for direct instruction regarding some particular features of the calculator followed by an opportunity for extended practice through exercises chosen from the sample curriculum materials (two such sessions per day). Teachers were actively encouraged to make greater use of graphing strategies afforded by the technology where only paper-and-pencil techniques formerly applied. In addition, at least 1 hour each day was devoted to group discussion of issues ranging from logistical concerns (such as mechanisms for checking out calculators, classroom security, etc.) to assessment (what adjustment in testing and evaluation of homework should be made in light of the availability of the calculators). Each participant in the 1-week workshop developed a sample module of a favorite topic or concept, including appropriate exercises and test questions, that made use of the graphing calculator. (No other support, such as workshops or materials, was given to these teachers throughout the year.)
The teachers from each of the 36 schools were contacted by telephone and asked if they would participate in a study investigating the impact of the graphing calculators on teaching mathematics. It was explained that participation would involve a 30-minute interview by telephone, and although the findings would be shared, respondents would not be identified by name.
A total of 27 teachers participated in the interviews. All but 1 had attended the 1-day workshop, and 15 had chosen to attend the 1-week optional workshop (including the 1 person unable to attend a 1-day workshop). This sample consisted of 8 females and 19 males. Of the teachers in the sample, 19 had a master's degree, and 1 a doctorate degree. Most of the respondents also had considerable teaching experience (average approximately 15 years). A wide range of mathematics classes was being taught by the teachers in this sample including Algebra I, Algebra II, geometry, trigonometry, precalculus, calculus (including advanced placement), and a few others. Six of the teachers who elected not to participate in the 1-week inservice already had some experience with other graphing calculators (four for more than 1 year) and continued to use those rather than the donated calculators.
A telephone-interview format was chosen for two reasons: It allowed for a regionally diverse collection of data within a short period of time, and it allowed for a one-on-one interaction thus enabling the interviewer to clarify individual responses. With this format established, a preliminary version of the interview protocol was developed based on the review of the literature. This interview protocol was then piloted by three individuals who were experienced in teaching mathematics with graphing calculators, and appropriate changes were made. The final format of the interview protocol was derived after additional input from three authorities in mathematics and science education research. The interview protocol contained primarily open-ended questions grouped into four areas concerning teachers' perceptions of (a) advantages of calculator use, (b) disadvantages of calculator use, (c) classroom dynamics, (d) curriculum and evaluation, and (e) professional support and development (including inservice education, institutional support, and professional relationships). In addition, background information regarding extent and logistics of calculator use in the classroom was gathered. (A copy of the interview instrument is included in the appendix.)
Interviews were conducted during a 4-week period in March and April 1992. With the respondent's permission each interview was audiotaped to ensure that the responses were recorded accurately. During, and at the conclusion of each interview, the interviewer wrote extensive notes on each of the questions being investigated. In addition most of the audiotapes were transcribed completely, whereas some of them were only transcribed in the areas designated by the interview notes.
The goal of the data analysis was to identify recurring themes in teachers' perceptions regarding graphing calculator use. A qualitative approach was determined to be most appropriate due to the open-ended nature of the interview. Formal analysis of the interview data was undertaken upon completion of the data collection. As discussed in Bogdan and Biklen (1982), this analysis began with developing preliminary coding categories for each of the sections of the interview protocol. These coding categories were formed by looking for patterns both between and within the individual interview transcripts. Between 20 to 30 codes were generated for each of the sections of the interview protocol in the preliminary analysis. Each unit of data (sentence or paragraph) was then marked with the appropriate coding categories. The marked data was then sorted using a word processor. A major trend was determined if the coding category represented the perceptions of more than 50% of the teachers. A minor trend was determined if the coding category represented the perceptions of 25% to 50% of the teachers. None of the trends was found to represent the perceptions of all of the teachers.
Following the initial data analysis involving the entire group of teachers, the data was analyzed in a similar fashion based on two different groupings that developed during the interviews. The first grouping was based on the participation of the teachers in the optional 1-week inservice (15 teachers participated; 12 did not). Of the 12 teachers who did not participate in the more extended inservice workshop, 6 were using a different type of graphing calculator in their classroom. (All 15 of the participating teachers used the donated set of HP-48S's.) These six schools had previously obtained one or more classroom sets of either Casio or Texas Instruments graphing calculators prior to the HP-48S donation. Although the HP-48S has some symbolic algebra capabilities, the teachers seldom referred to these capabilities of the calculator and restricted their remarks to the calculators' graphing and numerical capabilities. (The four teachers who used the TI-81 exclusively each stated that they believed the TI-81 was easier for their students to use.)
The second grouping developed from the responses to the question of frequency of calculator usage. The amount of time the teachers reported they used the calculator in the classroom varied considerably. Although two teachers reported they used it everyday, the teachers as a whole were grouped into three almost equally represented categories: (a) those who used it more than once a week, (b) those who used it less than once a week but more than once a month, and (c) those who used it only sporadically (less than once a month) for specific short topics. The intent of the follow-up analysis by these groupings was to determine if the inservice experience, type of calculator used, or frequency of calculator use had been related to the different response patterns of the teachers.
RESULTS AND DISCUSSION
The results regarding teachers' perceptions are discussed under the following headings: (a) advantages of calculator use, (b) disadvantages of calculator use, (c) classroom dynamics, (d) curriculum and evaluation, and (e) professional support and development. Table 1 summarizes the major and minor trends elicited in each of these categories. (A major trend here is defined as a perception expressed by more than 50% of the teachers interviewed, whereas a minor trend is a perception expressed by between 25% and 50% of the teachers.) Numbers of responses are also reported according to teachers' self-reported frequency of calculator use.
Under each of these categories we include a description of the interview questions asked, representative quotes from teachers indicating each trend, and a discussion of the findings with special attention to a closer comparison of response patterns of different groups of teachers (determined by inservice participation, type, and frequency of calculator use).
ADVANTAGES OF CALCULATOR USE
What are the advantages of using graphing calculators for teaching mathematics? This open-ended question elicited the following trends in teachers' perceptions (with the percentage of teachers making a response classified under this trend followed by a quote representative of these responses).
* MAJOR TRENDS
LESS DISTRACTION WITH COMPUTATIONAL DETAIL (67%)05
It enables the students to look at a lot of different functions in a short period of time thus taking the drudgery out of looking at graphs. Graphing is fairly tedious, and heck, with the calculator you can just change the function and they can look at a lot of things in an hour.
AVAILABILITY OF IMMEDIATE FEEDBACK (63%)05
It allows them to do a lot of functions quick that would take a long time with paper and pencil. Once they get the idea they can see what's happening real quick.
ENHANCEMENT OF VISUALIZATION (56%)
Essentially, it gives the youngsters a chance to become more active in terms of the developmental stage and see things and visualize things prior to going into the theoretical development.
* MINOR TREND
DEVELOPMENT OF CONNECTIONS (30%)05
They are discovering some things that before I was giving to them and they weren't having a chance to see the patterns themselves or get the connections. For me to go through a whole series of hand drawing them at the board and then trying to point out the differences, wasn't as accurate as the calculator will do it. I don't think the connections were there, and to that extent I think it has improved the presentation of some of the topics I am teaching.
Table 2 displays the response patterns of the 27 teachers relative to these perceived advantages, with teachers grouped by participation in the extended 1-week inservice. (I01-I15 participated in the inservice, while N01-N12 did not.) Subgroups of teachers are defined by their self-reported frequency of calculator use, and those teachers using a different calculator than those donated to the schools are indicated with a superscript - t for TI, c for Casio.
The teachers' perceptions of the advantages of calculator use appeared to be instructionally related, demonstrating concern for the learning environment of the students and teacher. Many of the perceived advantages are similar to those cited by earlier research. Enhancement of visualization skills (Demana & Waits, 1990a, 1990b; Dick, 1990; Fey, 1989), less attention given to computational detail (Fey, 1984), and the improved development of problem-solving skills (Demana & Waits, 1990b; Dick, 1990; Wiske et al., 1988) are positive attributes that have been found with the use of technology in the classroom.
No particularly striking differences in the patterns of responses were detected between teachers who had participated in the inservice and those who had not, nor were there any striking differences among the subgroups defined by frequency of use. Those teachers who had previous experience with other graphing calculators (and were continuing to use them more than once a month) all mentioned at least three of the four trends identified.
DISADVANTAGES OF CALCULATOR USE
What are the disadvantages of using graphing calculators for teaching mathematics? This open-ended question elicited the following trends in teachers' perceptions.
* MAJOR TRENDS
LOGISTICAL DIFFICULTIES AND LACK OF ACCESS (59%)05
It was clearly not working, by the time we had 28 calculators and 44 kids and those things became hard to see...you look at the screen and if it's tilted into the light one way one person can see and the others can't. So it was a horrible battle, but now we have more of them and it's really nice because they each have their own. Teaching is more difficult because of the fact that it takes a certain amount of time to get them given out [sic] to the students and collect them back and get them on board. It would be easier if every student had their own and was familiar with how to use it.
PROBLEMS WITH SECURITY (52%)05
Security is the biggest disadvantage. We don't have a secure system where the kids can just come in and use them anytime they want. It would be ideal if you could have a situation like a library where you have one door with an alarm, when they walked out the door the alarm went off, then I would use them a lot more.
* MINOR TRENDS
TIME SPENT LEARNING CALCULATOR (41%)05
...it's just the learning part of it...it's so difficult to remember everything, it takes so long to get used to it.
FEAR OF CALCULATOR-DEPENDENCY (37%)05
The only disadvantage that I see is that students want to use them to get a result without understanding the result.... Basically kids get into the habit of wanting to use the calculator to multiply 2 × 3 or what's the sine of zero, and they have to have a calculator to do that and I think that that is ridiculous. There are some things they should know and be able to do quickly that are basic, like a quick drawing, they shouldn't have to have a calculator and they become calculator-dependent and the thought process disappears.
Table 3 shows the response patterns of the 27 teachers relative to these perceived disadvantages, again grouped by inservice participation and frequency of calculator use. The perceptions of the disadvantages of calculator use appeared to be primarily logistical in nature. Although each school was given a classroom set of calculators, one of the major factors impeding full implementation continues to be a lack of access. Many teachers suggested that for successful implementation each student must have his/her own calculator for use on homework as well as in the classroom. In most cases the classroom set of calculators was shared among several teachers. It is possible that this barrier would no longer be a major concern if students were required to purchase their own calculator.
Some teachers expressed a concern that students may become "calculator-dependent." This fear of calculator-dependency may have been correlated with a hesitancy to integrate the calculator into the mathematics classroom, for it was noted least often by those teachers making regular use (more than once a week) of the calculators in the classroom. Similarly, a concern over the amount of time spent learning the calculator was expressed least by those making regular use of the calculators and most by those making only sporadic use. Indeed, one could imagine that only an occasional use of the calculators could reinforce this perception, since it likely would create the need for the teachers to "refresh" their students on the operation of the calculators each time they were used.
The trends that developed in this section came from the teachers' responses to the following questions: Has the presence of graphing calculators in your classroom changed your teaching in any way? Specifically, what difference has it made in your presentation of the material, in your students' participation, and in your role as a teacher? The answers to these questions referred only to the times when the calculator was actually being used in the classroom. Some teachers reiterated enhancement of visualization, availability of immediate feedback, and less distraction with computational detail from the advantages they had mentioned previously often enough to be categorized as minor trends in this section. The following are additional major and minor trends that were found:
* MAJOR TRENDS
LESS TEACHER-CENTERED (63%)05
What you find math teachers doing more and more is that you may have a couple of interactive lectures but most of the time you're walking around facilitating the kids working, and they're working, they're not just sitting there like bumps listening.
MORE OPEN-ENDED QUESTIONS (56%)05
You can start asking the "What if" questions and have the students really check the problem out in depth where before you were hung up by manipulations.... What I like about it is usually I change the problem completely and ask "What if this happens...," "What if that happened...," "What happens to the graph if....," and then they start comparing back and forth.
FOSTERS DISCOVERY APPROACH (52%)05
Sometimes with problem-solving you ask questions a little differently, you pose the question so they are using their calculator to explore situations and discover things that you might have presented to them in a straight lecture approach before.
MORE COOPERATIVE LEARNING (67%)
It's [interaction between students] increased because they have a common ground to discuss and we generally work in cooperative learning groups of either pairs or fours.
* MINOR TRENDS
INCREASED STUDENT DISCUSSION OF MATHEMATICAL IDEAS (48%)05
They try to help each other when someone gets hung up. I think they tend to be more willing to say, well let me show you what's wrong or let me see if I can fix it for you. So I think they tend to be a little more helpful to each other.
INCREASED STUDENT INVOLVEMENT (48%)05
Kids are much more focused when the calculator is in front of them. They are active and they concentrate better.
INCREASED STUDENT ENTHUSIASM (48%)05
The kids really enjoy it. I mean it's a turn on when we pass them [calculators] out and they feel really good about the technology.
Table 4 shows the response patterns of the 27 teachers relative to these perceived impacts on classroom dynamics, grouped by inservice participation and frequency of calculator use. All of the trends indicated that the classrooms were less teacher-centered and students were taking more responsibility for their own learning as well as working together with their peers and helping each other learn. All of the teachers interviewed mentioned this pattern to some extent. These results are consistent with those reported in previous research (Dick, 1990; Heid, Sheets, & Matras, 1990; Steen, 1988; Swadener & Blubaugh, 1990; Wiske et al., 1988). Associated with a less teacher-centered classroom comes an increase in open-ended questions. Although research indicates that the increase in questioning by students may pose a threat to teachers' predictability in teaching (Dick, 1990; Farrell, 1990; Rich, 1991; Schofield & Verban, 1988), the tone of the responses from teachers in this study was that of excitement and enthusiasm for a less teacher-centered environment. These trends support the findings of Berenson and Stiff (1990) and Farrell (1990), and suggest that the dynamics of classroom tend to shift to more discussion, inquiry, and cooperative learning.
The frequency of calculator use did appear to have an impact on the responses of the teachers in the classroom dynamics section. Understandably, the less the teachers used the calculators, the less often they mentioned effects on classroom dynamics. The following quotes are examples of three of these teachers' comments with respect to the impact the calculators had on classroom dynamics:
It [the use of calculators] has not really changed my teaching or saved me any time. I think it was just something different. I teach in the same manner as before. If it changed anything, the kids just have more options now, later in college they won't be afraid to go out and get a graphing calculator.
It [classroom dynamics] hasn't changed very much at this point. I don't think we have really utilized their [the calculators] full potential at this point.... I don't think it has changed my role as a teacher much. No, not yet anyway.
Well, [classroom dynamics has changed] from the standpoint that we are taking the time to actually use the calculator and see some of the new technology that is out there for the kids. But, as far as the presentation of the material, it has made no difference really.... The only difference it has made is that I have to make sure that I show the technology that is out there.
CURRICULUM AND EVALUATION
The trends that developed in this category came from the teachers' responses to the following questions: What effects have graphing calculators had on the goals and content of the mathematics courses you teach? What effects have graphing calculators had on your evaluation of your students?
In response to the question concerning the effect of graphing calculators on the goals and content of the mathematics courses taught, more than 50% of the teachers reiterated availability of immediate feedback and less distraction with computational detail, mentioned earlier as advantages of calculator use. The following additional trends in perceived impact on curriculum and evaluation were found:
* MAJOR TRENDS
INCREASED PREPARATION TIME (78%)05
The times when it is more difficult is not the teaching of it but it's the preparation, my learning the graphing calculator and how to use it to its best advantage for the kids.
INCREASED MATHEMATICAL DEPTH (59%)05
I haven't changed a whole lot of what I ask them to do, just more of it, much faster and I would have to say that it is deeper, we can go into more depth now because we spend less time fiddling around with trying to figure out what the sine curve looks like or whatever.
* MINOR TRENDS
NECESSITY OF TECHNOLOGICAL EXPOSURE (26%)05
I think the only difference it has made is that I have to make sure that I show them the technology that is out there. I didn't use to do that before, so I would say it has forced me to at least make sure that they are aware of what's out there to help them when the time comes.
INFLUENCE OF THE ADVANCED PLACEMENT (AP) EXAMINATION (22%)05
It hasn't had that much effect [on goals and content] because colleges haven't changed their goals and content. Until they change it enough so that the AP test changes we're going to continue to make sure that the kids can do all those factoring and rational expression problems, and all the roots by hand.
Table 5 shows the response patterns of the 27 teachers relative to these perceived impacts on curriculum and evaluation, grouped by inservice participation and frequency of calculator use. The majority of the teachers stated that graphing calculators had little effect on their evaluation procedures, and a majority also expressed a concern regarding the use of calculators during exams. However, more than half of the teachers who mentioned increased mathematical depth also related changes in their test questions.
Some of our test questions change towards more open-ended questions and more mathematically thought provoking. Like, what kind of range would be most appropriate if you are graphing something.
Significantly, all of the teachers who claimed to use the calculators more than once a week also reported that their students were allowed to use the calculators on exams. Seventy percent of the remaining teachers, however, commented that their students were not allowed to use the calculators on exams.
All of the teachers of AP calculus in the sample mentioned concern about the influence of the advanced placement (AP) examination (thus, we have included it here even though it technically falls just short of our stated criterion of 25% response for a minor trend). Research (Wiske et al., 1988) has shown that there is considerable reluctance to deviate from the stringent curriculum requirements that are reinforced by standardized tests. Because of their responsibility to prepare students to pass these tests, these teachers thought that if graphing calculators are not allowed during the AP exam they should be used only minimally in the classroom. Some of these teachers indicated that they used graphing calculators extensively only during the last 4 weeks of school after the students had already taken the AP exam. Despite the fact that the AP program has actually encouraged the use of graphing calculators as an instructional tool in calculus, its own examination has proven to discourage many teachers from that use. (Evidently, the AP program has recognized this barrier, for starting in 1995, graphing calculators are indeed required on parts of the AP exam.)
Some teachers expressed a responsibility to use graphing calculators to prepare students to live in this technological world, but it appeared that they were more concerned with demonstrating and teaching about the graphing calculator as advanced technology instead of using it as a mathematical learning tool. This coincides with Wiske et al. (1988) who found a mindset in school districts where more attention is paid to the acquisition of computers than is given to the development, support, and assessment of educational applications of these tools.
Although some teachers claimed that graphing calculators had no effect on the goals and content of the mathematics courses they taught, they often indicated that they felt too inexperienced with them as a teaching tool. All of these same teachers suggested that if contacted in another year they might have more to share, particularly with regards to evaluation procedures.
PROFESSIONAL SUPPORT AND DEVELOPMENT
The trends discerned in this category came from the teachers' responses to the following question: What institutional or professional support would you like with regards to using graphing calculators in the classroom?
* MAJOR TRENDS
NEED FOR TECHNOLOGY-SENSITIVE MATERIALS AND HANDOUTS (74%)05
I would sure love to have more materials. I want to be able to draw on what other people have done, because I know they've done what I am doing. I feel like I'm reinventing the wheel when it has already been invented in Chicago or Portland or wherever.
05NEED FOR ADDITIONAL INSERVICE (71%)
I would like more training or workshops during the year as well as in the summer. I'm sure that it's going to be an on-going process of getting the more specifics.
NEED FOR TEACHER NETWORKING (52%)05
An idea would be a follow-up letter. Like a newsletter on the use of the graphics and ideas about some problems. For example, you had a hotline where you could just call in and say "I don't know how to get this function...," then once they answer the question for the individual, put that in the newsletter and send it out to all the people that have calculators. Maybe monthly or four times a year, whichever makes sense.
* MINOR TRENDS
NEED FOR ADDITIONAL CALCULATORS (37%)05
I would certainly like to have enough calculators that each student would be able to have one.
NEED FOR CONTINUED TEACHER COMMUNICATION (26%)05
We've opened up a lot of lines of communication. We talk a lot about it, we share things...somebody who has discovered a way to do something that we haven't thought of or developed a lesson.
Table 6 shows the response patterns of the 27 teachers relative to these perceived needs for professional support and development, grouped by inservice participation and frequency of calculator use. Many of the teachers commented that there was a need for additional inservice, and indicated that the short 1-day inservice workshop was overwhelming. The following quote demonstrates the teachers' perception regarding the 1-day inservice:
That one day thing was just too much. I found the functions of the calculator fairly complicated to get from one thing to another. Maybe it's just the HP's, but I didn't have enough time to absorb any particular thing and practice it before we sort of dashed on to another one. I don't think we even got a handout. Maybe that would have helped.
Overall, teachers revealed that there was considerable need and desire for follow-up support to help "troubleshoot" at all stages of the implementation process. These findings are consistent with the results obtained by Lindquist et al. (1991), Lovitt et al. (1990), and Wiske et al. (1988).
Although the qualitative approach used in this investigation allowed for a more detailed analysis of teachers' perceptions of the impact of graphic calculators, the generalizability of these findings is limited by the methodology used. First, the use of volunteers directly limits generalizability of results and conclusions. Secondly, though the results appeared to reveal internal consistency of teachers' responses, the use of interviews without direct observations of their classrooms hinders the credibility of the conclusions. The reader should keep in mind that the trends reported in this study represent only the self-reported perceptions of the teachers interviewed.
The intent of the interview process was to use open-ended questions in such a way that all responses of the teachers were not based on an a priori set of categories, as well as not due to the biases of the interviewer. To insure consistency among the teachers, the interviewer directly followed the interview protocol and did not delve more deeply into the teachers' responses. (It is possible that no matter how carefully the interview items were asked that potential biases may have been detected by the tone of the interviewer's voice, thus, producing a response effect.) Analogous to this, many of the teachers were interviewed in an area such that other people overheard their discussion and in some cases even made comments in the background. It is certainly possible that the 9 teachers that were not interviewed (out of the original 36) may have provided some additional trends in their responses.
The focus of this study was directed primarily toward the teachers' perceptions about the impact of using graphing calculators in the mathematics classroom, and the barriers and incentives for implementing technology in mathematics instruction. Most of the teachers indicated that although integration of the calculator into the curriculum is the main objective, it is a very complex undertaking for which much training and support is necessary. Specifically, for successful implementation the results indicated the following are needed: (a) an adequate number of calculators, (b) relevant curriculum materials, (c) technical assistance, and (d) a supportive environment for teachers' professional growth and development.
Past literature has indicated that inservice programs have often focused on altering teacher behaviors by familiarizing teachers with new technology and new instructional methods, but that insufficient attention has been given to teachers' concerns on the actual use in their environment (Schultz, 1989; Wiske et al., 1988). Providing a chance for teachers to share their knowledge, experiences, and insights can help guide the design of curriculum materials, inservice programs, and support systems that will best equip teachers to handle the demands and opportunities that technology presents. Finally, in future investigations it is essential that we listen to teachers at all stages of preparation and integration of new technological tools, thus, enabling continuous fostering of the transition that is necessary for successful implementation.
In conclusion, we note that this study resulted in significant changes in Hewlett-Packard's calculator donation program. The program has been greatly extended in scope over the last 3 years, with inservice workshops expanded to 2 weeks (and made a condition of award acceptance). The inservice workshops are now led by teams of secondary teachers with extensive experience in the use of graphing calculators and include particular emphasis on integrating the use of the graphing calculator in curriculum and evaluation. Different strategies for dealing with logistical difficulties are shared, and the needs for technology-sensitive materials, additional inservice, teacher networking, and continued teacher communication are being met through regular phone contact by the instructional teams, electronic mail, a newsletter, and follow-up meetings through the school year.
LINDA M. SIMONSEN
Department of Mathematical Sciences
Montana State University
Bozeman, MT 59717, USA
THOMAS P. DICK
Department of Mathematics
Oregon State University
Corvallis, OR 97331-4605, USA
Table 1 Number of Teachers Reporting Trend by Frequency of Calculator Use
More Less than Sporadic than once a use for once a week but specific Total week more than short topics Trend once a month n=8 n=10 n=9 n=27 Advantages of calculator use Less distraction with computational detail 6 6 6 18 Availability of immediate feedback 6 5 6 17 Enhancement of visualization 5 4 6 15 Development of connections 4 2 2 8 Disadvantages of calculator use Logistical difficulties and lack of access 4 6 6 16 Problems with security 6 3 5 14 Time spent learning calculator 1 6 4 11 Fear of calculator dependency 2 4 4 10 Classroom dynamics Less teacher-centered 6 4 7 17 More open-ended questions 5 4 6 15 Fosters discovery learning 5 3 6 14 More cooperative learning 7 6 5 18 Increased student discussion of math 5 3 5 13 Increased student involvement 4 5 4 13 Increased student enthusiasm 4 5 4 13 Curriculum and evaluation Increased preparation time 6 7 8 21 Increased mathematical depth 5 6 5 16 Necessity of technological exposure 1 2 4 7 Influence of the advanced placement 1 2 3 6 exam Professional support and development Need for technology-sensitive materials 6 6 8 20 Need for additional inservice 5 7 7 19 Need for teacher networking 4 5 5 14 Need for additional calculators 5 2 3 10 Need for continued teacher 3 3 1 7 communication
Table 2 Response Trends of Teachers: Advantages of Graphing Calculators
Teachers participating in Less Availability Enhancement Development inservice workshop Distraction of Immediate of of with Feedback Visualization Connections * response in category indicated Computation using more I01 * than once a I02 week I03 * * * I04 * * * using less I05 * than once a I06 * * * week, but I07 more than I08 * once a month I09 * * I10 * using I11 * * * sporadically I12 * for I13 * * specific I14 topics I15 * * Teachers not participating Less Availability Enhancement Development in inservice workshop Distraction of Immediate of of with Feedback Visualization Connections * response in category indicated Computation N01(FNt) * * * using more N02(FNt) * * * than once a N03(FNc) * * week N04 * * * N05 * * using less than once a N06(FNt) * * * week, but N07(FNc) * * more than N08 * * once a month using N09(FNt) * sporadically for N10 * * specific N11 * * topics N12 * *
t used TI-81
c used Casio 7700
All other teachers used the HP-48S.
Table 3 Response Trends of Teachers: Disadvantages of Graphing Calculators
Teachers participating in Logistical Problems Time spent Fear of inservice workshop difficulties with learning calculator and lack of security calculator dependency * response in category indicated access using more I01 * than once a I02 * * week I03 * * I04 * * * using less I05 * * than once a I06 week, but I07 more than I08 * * once a month I09 I10 * * * using I11 * * * sporadically I12 * for I13 * * * specific I14 * * topics I15 * Teachers not participating Logistical Problems Time spent Fear of in inservice workshop difficulties with learning calculator and lack of security calculator dependency * response in category indicated access N01(FNt) * using more N02(FNt) than once a N03(FNc) * * week N04 * * N05 * using less than once a N06(FNt) * * week, but N07(FNc) * * * more than once a month N08 using sporadically N09(FNt) for N10 * specific N11 * * topics N12 * *
t used TI-81
c used Casio 7700
All other teachers used the HP-48S.
Table 4 Response Trends of Teachers: Effects of Graphing Calculators on Classroom Dynamics
Teachers participating in less more fosters more increased increased increased inservice workshop teacher- open- discovery coopera- student student student * response in category indicated centered ended learning tive discus- involve- enthu- questions learning sion ment siasm using more I01 * * * than once a I02 * week I03 * I04 * * using less I05 than offce a I06 * * * week, but I07 more than I08 * once a month I09 * * * I10 * * * using I11 * sporadically I12 * for I13 * specific I14 * * * topics I15 * Teachers not participating less more fosters more increased increased increased in inservice workshop teacher- open- discovery coopera- student student student * response in category indicated centered ended learning tive discus- involve- enthu- questions learning sion ment siasm using more N01(FNt) * than once a N02(FNt) * * * week N03(FNc) * * N04 * * * N05 * * using less than once a N06(FNt) * * week, but N07(FNe) * * more than once a month N08 * * * using sporadically N09(FNt) * * * for N10 specific N11 * topics N12
t used TI-81
c used Casio 7700
All other teachers used the HP-48S.
Table 5 Response Trends of Teachers: Curriculum and Evaluation
Teachers participating in Increased Increased Necessity Influence inservice workshop mathematical preparation for student of the * response in category indicated depth time exposure to AP exam technology using more 101 * * than once a 102 * week 103 * * 104 * * 105 * using less 106 * * * week, but 107 * more than 108 * once a month 109 * * 110 * 111 * * using 112 * sporadically for 113 * specific 114 * * topics 115 Teachers not participating Increased Increased Necessity Influence in inservice workshop mathematical preparation for student of the depth time exposure to AP exam * response in category indicated technology N01(FNt) * * using more N02(FNt) * * than once a week N03(FNc) N04 * * N05 using less than once a N06(FNt) * * week, but N07(FNc) * * more than once a month N08 * * using N09(FNt) * * sporadically for N10 * * specific N11 * * topics N12 * * *
t used TI-81
c used Casio 7700
All other teachers used the HP-48S.
Table 6 Response Trends of Teachers: Professional Support and Development
Teachers participating in Need for Need for Need for Need for Need for inservice workshop technology- additional teacher additional continued sensitive inservice networking calculators teacher com- * response in category indicated materials munication using more I01 * than once a I02 * * week I03 * * * I04 * * * I05 * * using less I06 * * than once a week, but I07 * more than I08 * * once a month I09 I10 * * * I11 * * using I12 sporadically for I13 * * specific I14 * topics I15 * * * Teachers not participating Need for Need for Need for Need for Need for in inservice workshop technology- additional teacher additional continued sensitive inservice networking calculators teacher com- * response in category indicated materials munication N01(FNt) * * using more N02(FNt) * * * than once a N03(FNc) * week N04 * N05 * * * using less than once a N06(FNt) * week, but N07(FNc) * * * more than once a month N08 * * * using N09(FNt) * * sporadically for N10 * * specific N11 * * * topics N12 * *
t used TI-81
c used Casio 7700
All other teachers used the HP-48S.
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3. Highest degree
4. Number of years teaching
5. Number of years using symbolic/graphic calculators in the classroom
6. Math courses currently teaching
CALCULATOR USE IN THE CLASSROOM
1. How frequently do you use symbolic/graphic calculators?
2. How much access do students have to symbolic/graphic calculators?
3. Are your students allowed to use symbolic/graphic calculatorsduring exams?
1. What was your experience with symbolic/graphic calculators before the HP donation?
2. What are the advantages of using symbolic/graphic calculators for teaching mathematics?
3. What are the disadvantages?
4. Prior to using symbolic/graphic calculators, did you foresee these advantages or did you have different expectations? Did you foresee these disadvantages?
1. Has the presence of symbolic/graphic calculators in your classroom changed your teaching in any way? Specifically, what difference has it made in your presentation of the material? In your students participation? In your role as a teacher?
2. Discuss the impact that symbolic/graphic calculators have had on the interaction between students in the classroom and between you and your students.
3. Discuss the impact that symbolic/graphic calculators have had on student motivation.
4. In your opinion, discuss whether teaching is easier or more difficult with symbolic/graphic calculators.
CURRICULUM AND EVALUATION ISSUES
1. What effects have symbolic/graphic calculators had on the goals and content of the mathematics course you teach? On your evaluation of students?
2. What has changed in terms of your preparation time?
3. Discuss the amount of time taken to instruct the students on how to use the HP-48SX.
4. What curriculum materials do you use? How do these coordinate with the use of symbolic/graphic calculators?
5. List 3 examples of topics or concepts where your approach has changed the most. How has it changed?
PROFESSIONAL SUPPORT AND DEVELOPMENT
1. What training have you had on the use of symbolic/graphic calculators?
2. Have you used the mentor support program provided by HP?
3. What changes would you like to see in the training? What should be added? Deleted?
4. Discuss any institutional support for symbolic/graphic calculators use in the classroom (e.g., buying additional equipment, funding to attend workshops, release time).
5. What support would you like that is not available to you?
6. What effects have symbolic/graphic calculators had on your relationship with other teachers who are using technology in the classroom or who are not using technology in the classroom?
7. Discuss any new roles that the use of symbolic/graphic calculators has led to outside the classroom (consultant, trainer, etc.)? Has this been positive? Negative?