Network of influences in an implementation of a mathematics curriculum innovation

Julie Sarama
Douglas H. Clements
Julie Jacobs Henry

As the births of living creatures, at first, are ill-shapen: so are all Innovations...

-- Francis Bacon

There is widespread interest in reform in U.S. mathematics education, engendered by influences from comparative educational research (National Center for Education Statistics, 1996; Stigler, Lee and Stevenson, 1990) to position documents from prestigious organizations (National Council of Teachers of Mathematics, 1989; National Council of Teachers of Mathematics, 1991; National Research Council, 1989). One route to such reform is the development and implementation of innovative curricula. We studied the adoption of one such innovation, emphasizing teachers’ social construction of knowledge and beliefs.

The innovation was designed by two of us (Clements and Meredith, 1994; Clements and Sarama, 1995) as part of a research project funded by the National Science Foundation (NSF)¹. In this context, we developed a software environment with correlated curriculum materials. During the field-testing phase of development, we worked with a school that was attempting to alter its mathematics program according to recent reform recommendations. Given the apparent convergence of goals of the school staff and designers, we believed the adoption process began with a good chance of success. We planned to assess the effect of the innovation on the knowledge and beliefs of students and teachers. During the project, however, concerned and confused about what we observed, we altered the direction of the research study to examine, and attempt to positively affect, the implementation of the innovation.

Theoretical Framework

The philosophical approach underlying, albeit sometimes nebulously, the national organizations’ reform recommendations, cooperating school’s recently written mission statement, and designers’ innovatory materials, is constructivism. Taking a constructivist perspective, one may accept two major goals for mathematics education (Clements and Battistza, 1990; Cobb, 1988): (a) students should develop problem-solving abilities based on rich conceptual knowledge structures, rather than learn to seek out and apply teacher-prescribed methods to complete tasks; and (b) students should become autonomous and self-motivated. Such students would believe mathematics is a way of thinking abut problems and their responsibility in the mathematics classroom is not so much to complete assigned tasks as to explore, talk about, and learn mathematics. Socially-oriented constructivists emphasize that knowledge is constructed in collaboration with others as “a communal activity, a sharing of culture” (Bruner, 1986, p. 127). Such a framework is relevant in examining not only how students construct ideas about mathematics but also how teachers construct beliefs about mathematics teaching and learning. Few previous studies on implementations of technology-based educational innovation have used this framework.

While not dictating specific teaching strategies (Clements and Battista, 1990; Simon, 1995), individual and social constructivism can serve as general philosophical and psychological frameworks of design and evaluation of educational environments. For example, materials and teaching based on such a framework would support students’ construction by guiding students’ mathematical tasks, peers, and the teacher, the students’ intuitive mathematical thinking would gradually become more abstract and powerful. The computer-based software environment, and the activities that accompany it, are based on these frameworks.

Research Model and Background

Teachers’ knowledge and beliefs constitute important mediators in students’ experience with an innovation, especially one based on frame-works inconsistent with traditional approaches to teaching and learning (Farley, 1992; Thompson, 1992). These knowledge and belief systems influence implementation and can, in turn, be affected by experiences with the innovation (Cohen and Ball, 1990; Fullan, 1982). Thus a model for the network influences must be complex; our model (a modification of the level 4 model of Koehler and Grouws (1992)), is presented in Figure 1. We shall review research relevant to each main social group in our model.

Figure 1. Research model for the network of influences influencing technology innovations in the classroom.

The first social group is teachers. We investigated teachers’ knowledge of mathematics content, pedagogy, and student learning and their beliefs about mathematics, pedagogy and technology. Teachers come to an innovation with different ideas and experiences (Cohen and Ball, 1990) and thus varied knowledge and belief systems. Reformers often fail to acknowledge such individual backgrounds of teachers (even when they do so with students, Darling-Hammond (1990)). Changing teachers beliefs requires three general condition: (a) Teachers must be dissatisfied with their existing belief in some way; (b) they must find the alternatives presented to be both intelligent and practical; and (c) they must figure out some way to connect new beliefs with earlier conceptions (Ely, 1990; Etchberger an Shaq, 1992; Posner, Striken, Hewson and Gertzog, 1982). Reformers can attempt to influence beliefs by first acknowledging the complexity of the process. Time for teacher training must be adequate, and strategies must be used that account for teachers’ prior knowledge and experiences. In addition, teachers examine the time and energy that must be put forth in implementing an innovation and weigh this personal cost relative to rewards attained (Wright, 1987). Teachers reject innovations that appear unconnected to the curriculum, viewing it as yet another “add on” (Ferris and Roberts, 1994). Such rejection is more likely when the innovation requires the use of computer technology, especially when teachers are not comfortable with the technology. Access to up-to-date computers with appropriate software, space to operate them, and local, ongoing technical support may be difficult to arrange (Ferris and Roberts, 1994).

The second social group, administrators (Olson, 1988; Sykes, 1990), may provide such support as (a) supportive organizational arrangements (hardware, software, space, materials); (b) time for ongoing training, geared towards enabling teachers to grow, learn, adapt, integrate, reflect, and continue to develop new skills (Borchers, Shroyer and Enochs, 1992; Ely, 1990; Hord, Rutherford, Huling-Austin and Hall, 1987) and (c) initial encouragement, vocal ongoing support and commitment from authority structures and peers are highly influential in determining the extent of implementation (Bresler and Walker, 1990).

The third social group is the designers of educational innovation. They often emphasize the importance of computers as a took of intellectual enhancement rather than as a machine for focusing student attention on routine fact and basic skills learning (Becker, 1991; Clements, 1997). To achieve these goals, designers should involve teachers in each step of software development and implementation (De Diana and Collis, 1990; Ely, 1990; Ferris and Roberts, 1994). Support personnel connected to the designers, the fourth social group, may assist with implementation in the following ways: (a) encouraging participation, shared decision making, and communication; (b) one-to-one consultation and reinforcement (Bresler and Walker, 1990); and (c) monitoring, reminding teachers that their attention is required for the program, that the innovative project is a priority, that a commitment has been made to it, and that somebody cares about them (Hord et al., 1987).

The fifth social group, the students, is rarely addressed in the literature. We take the position that students are an essential component of classroom research (cf. Koehler and Grouws, 1992). Finally, the wider social context in which these social groups are embedded, the American educational system, has a persistent “grammar of schooling” (Tyack and Tobin, 1992) composed of rigid cultural beliefs about proper teaching, learning and knowledge, as well as the inviolable legitimacy of the age-graded school with self-contained classrooms, a fragmented curriculum, and rigid time schedules. An historical analysis of innovations reveals that those challenging this grammar tend to be short-lived (Tyack and Tobin, 1992). In planning the specifics of the implementation of this educational innovation, we attempt to integrate the suggestions and cautions from the literature.

The present study can be distinguished from much of the extant literature on innovations in that (a) the impetus for the implementation came from school personnel, and (b) the designers did not approach the teachers as university researchers or authorities but as curriculum designers who were seeking teachers’ input as professionals and offering assistance in using the innovation for reform. Given these conditions, we believe our findings are relevant to others working cooperatively with schools.

Method

The Curriculum Innovation: Turtle Math

The designers created a modified Logo Environment, Geo-Logo (Clements and Meredith, 1995a; Clements and Meredith, 1995b), for an NSF Curriculum development project that emphasized meaningful mathematical problems and depth rather than exposure. They based the design on curricular considerations and a number of research implications for the learning and teaching of geometric concepts with turtle graphics (Clements and Meredith, 1993; Clements, Meredith and Battista, 1992). The designers believed the power of Logo as a catalyst for mathematical thinking will only be realized when it becomes a regular tool for thinking (Clements and Meredith, 1993). Therefore, they wrote additional materials so that the software would form a complete supplemental package, Turtle Math^TM (Clements and Meredith, 1994), designed to support students’ development of problem-solving abilities based on right conceptual knowledge structures. The goal was for students using Turtle Math to view mathematics as a way of thinking about problems and view their responsibility in the mathematics classroom less to complete assigned tasks and more to explore, talk about, and learn mathematics. The designers believed they were serving a need in the teaching community, providing a resource for teachers who wanted to use Logo but did not have ideas for activities, especially in areas outside of geometry. We briefly describe the principles that guided the design of Turtle Math and some features of the software based on those principles (for a complete description, see Clements and Sarama, 1995).

Philosophical- and Research-Based Design of the Turtle Math Software

1. Encourage construction of the abstract from the visual. Logo can help children construct mathematical strategies and conceptions out of their initial intuitions and visual approaches (Clements and Battista, 1992; Clements and Meredith, 1993). For example, there is a large literature on children’s difficulty with turns and angles. Children can build more robust ideas of these concepts using Logo because they give turn commands and receive feedback. However, if children’s Logo experience is not mediated they maintain misconceptions (Hoyes and Sutherland, 1989). Turtle Math provides several measurement tools; for example, an on-screen protractor, placed at the turtle’s position and heading, can measure turns. One arrowhead shows the turtle’s heading. The other follows the cursor, which students move with the mouse. When they click this arrowhead “freezes” and a turn command is displayed. Rulers and other measurement tools are also available. Even if students do not adopt our mathematical goals, and use visual and empirical strategies, the environment should continue to support their activity. The visually-oriented measurement tools allow students to approach task in a wider variety of ways. Another main way Turtle Math supports the growth of the abstract from the visual lies in its overall structure, described in the following section.

2. Maintain close ties between presentations. The nature of programming creates the need to make relationships between symbols (code) and drawings explicit. But students may lose the psychological connection between the two when involved in long projects (Clements and Meredith, 1993; Clements and Sarama, 1995). In Turtle Math, students enters commands in “immediate mode” in a command window (Figure 2). Any change to commands in either location, once accepted, are reflected automatically in the drawing. The dynamic link between the commands in the command window, long and narrow to the side of the graphics screen, instead of the traditional short but wide placement below this screen. This permits the immediate inspection of more commands, which facilitates connecting symbols and drawings, as well as pattern searching. Further, students can easily modify the code, encouraging experimentation and supporting later work with procedures.

3. Facilitate examination and modification of code; encourage procedural thinking. The environment should support easy creation, alteration, and use of procedures and highlight procedural-conceptual connections (Clements and Meredith, 1993; Clements and Sarama, 1995). The dynamic link also means that all commands in the command window represent a proleptic procedure.

Figure 2. Turtle Math^TM screen (Clements and Meredith, 1994).

That is, they constitute exactly the code necessary to draw the geometric figure on the screen and they can be defined immediately as a procedure with a tool. A “Step” tool allows students to “walk through” commands to find errors or explore mathematical properties. Other palette tools allow easy editing and erasing.

4. Encourage procedural thinking. The benefits of using the command center and immediate mode programming have already been described. However, Turtle Math’s structure also encourages use of procedures from the beginning. First, as a proleptic procedure, the code in the command center enables students to edit a set of commands that constitutes on graphic object on the screen even before procedures are formally introduced. Second, a tool is provided by simply naming the set of commands in the command center. Third, changes made to procedures within the teach window also are immediately reflected on the graphic screen upon existing that window.

5. Provide freedom within constraints. The goal is to encourage problem solving with structured activities as well as student directed exploration, both within the constraints of the Turtle Math environment (e.g., structures that encourage increasingly mathematical thinking). The set of provided activities is not intended to be static. Teachers and students can invent and save their own. In addition, within this structure, students work at a variety of levels. Students and teachers can enable or disable the tools though an options menu. Students can use the enabled tools to analyze figures or to work in a visual, empirical manner via measurement. Thus, they are allowed to solve problems at different levels of mathematical sophistication.

6. Allow a domain in which to explore diverse areas of mathematics using a consistent metaphor. The benefit of having a consistent way to talk and think about different areas of mathematics can serve as a way for students to bring coherence to a varied mathematics curriculum.

7. Facilitate learning of mathematical ideas. In several additional ways, Turtle Math encourages students to build solid ideas about mathematics. For example, the Turn Rays option shows rays during turns. If you type rt 120, a ray is drawn to show the turtle’s initial heading. Then as the turtle turns, another ray turns with it, showing the change in heading throughout the turn. A ray also marks every 30&Mac176; of turn. Turtle Math also provides coordinate grids, scaling of distances (i.e., “fd 1” can mean forward 1 cm or forward 1 inch, allowing interesting use of fractions), geometric motions via menu, mouse control, and commands (including motions and scaling)

Sample Lesson

The Turtle Math manual gives teachers the following suggestions for using Turtle Math in the classroom:

Do all three parts of an investigation (off computer activities, a main activity, and a challenge activity). Off-computer experiences are essential to building concepts, and challenges are not “extensions” or “extras,” but are an integral component of the investigation.
Post copies of the Turtle Math Help sheets near each computer.
Have students keep a Turtle Math journal. Duplicate activity sheets and have students store their journal entries in a loose-leaf notebook, or have them enter similar categories in a blank notebook of their own.

Each activity as written in the activity book includes background information, off- and on- computer activities, and challenges. For example, the triangle activity includes the following off-computer activities:

a silent sorting activity and discussion meant to build a definition of triangles, based on properties:
an art activity in which students make pictures with as many different shapes of triangles as they can;
a discussion about equilateral triangles and what makes them special; and
a drawing activity in which students are asked to evaluate whether the triangle is equilateral or not to justify their answers.

The main computer activity in which students are asked to evaluate whether the triangle in Logo, including having the students stand up and turn their bodies to help figure out the turn measure.

On computer the students follow and activity sheet that contains a chart to organize in their turn estimates and challenges them to draw smaller triangles, to draw a triangle with a perimeter that is twice as large, and to write a procedure that will draw an equilateral triangle of any size. As a final suggestion, teachers are encouraged to have students carry out a triangles project.

School and Participants

The cooperating school was a suburban elementary school in Western New York. The school had recently crafted a mission statement, detailing the school’s commitment to change. The outcomes for mathematics include statements heavily influenced by NCTM’s Standards (especially the process standards of problem solving, communication, reasoning, and connections). For example, the outcome for mathematical communication is, “All students will construct and express the meaning of mathematical ideas in a variety of ways using the language of mathematics.”

During the year of our study, the school was in the process of redesigning their mathematics curriculum. The administrators in the district encouraged teachers to embrace a “constructivist” approach to teaching mathematics. Administrators viewed this as consistent with another district goal: Taking a whole language approach to the language arts. According to Ms. Cas (Communications arts specialist), the teachers in this school had done particularly well in adopting whole language. Several teachers at this school (including one of the participating teachers) were involved in a shared-decision making process to draft new mathematics goals for the district. Finally, the school administration embraced technology as a medium for attaining their goals. They acquired a well-equipped computer laboratory with 30 Macintosh stations and established technological literary as a criterion for hiring new teachers.

The participants belonged to the social groups previously described. The first social group included teachers. This was a voluntary project within the school; the nine participants were those who continued using the curriculum throughout the year, especially three fifth grade teachers (Ms. Jack, Ms. Moore, and Ms. Gaughan) who used Turtle Math extensively and were willing to meet with the researchers. The second social group included administrators: Ms. Cas, a literacy specialist who asked the researchers to work with the teachers; a district supervisor; and the principal. Third were the designers (DHC and JS), who wished to garner advice from teachers to aid in the development of the curriculum. Fourth was the support person working for the designers, Mr. Barrett. Fifth were students of the participating teachers. Three students were chosen from each of the focus classroom identified by their teachers as average ability and willing to talk with the researcher.

Procedure and Data Collection Method

Here we briefly describe the chronology of the implementation of the innovation and our concurrent research. Impetus for the implementation began in the year preceding the study, when Ms. Cas asked the designers to suggest software for an after-school gifted math program. The designers asked if she and her teachers would be willing to field test their own software, Turtle Math. Ms. Cas and two other teachers agreed and asked for training. The teachers did not request any further help and reported that was a success.

The following autumn, Ms. Cas asked the designers to meet with the principal to discuss the possibility of additional teachers using the innovation. The principal and Ms. Cas recruited several teachers to use Turtle Math for mathematics reform. Believing that this could be a fruitful area for the exploration of how teachers implement an innovation and how their philosophies are affected by this work, we began the present study with a written questionnaire developed by another research team (Cobb et al., 1991). This questionnaire asked teachers to respond “yes” or “no” to 30 statements which could be grouped as consistent with either constructivist (e.g., “Teachers should encourage children’s own solutions, even if they are inefficient”; “teachers are most successful when children judge for themselves whether their answer is right and wrong”) or transmissionist (e.g., “Students need much repetition and practice”; “Effective teaching requires rewarding right answers and correcting wrong answers”) views on the learning of mathematics. The questionnaire was distributed by Ms. Cas.

At the initial meeting, the designers gave teachers software and written materials describing Turtle Math objectives and activities. Throughout this stage, the designers attempted to help teachers meet the goals espoused by the district. Mr. Barrett taught demonstration lessons to the teachers’ classes, was available during the day to help the teachers, and organized optional after-school sessions during which teachers explored the materials. He kept a detailed record of these activities. In addition, he and the authors, as participant-observers, recorded field notes during teacher-led classroom lessons.

At this time, then, we believed that conditions existed that would facilitate successful implementation of the innovation. Administrators and teachers were enthused about using the computer lab and Turtle Math to reform their mathematics curriculum, and it appeared that ample resources and support were available.

Teachers returned the questionnaires over a period of several weeks. They expressed, orally and in writing, that they felt uncomfortable with the questionnaire. First, they objected to the Yes/No dichotomy that the questionnaire imposed on complex issues. Second, they did not like the sense of being classified and judged by others based on their answers to this form. (No such reactions from their participants were reported by the developers of the questionnaire; in retrospect, the objections seem reasonable and should be considered by researchers.)

During the next several months, several teachers’ use of Turtle Math decreased. The designers believed that a third party might form a different relationship with teachers and thus uncover valuable information about their perspectives. Therefore, the designed formed a research team and the additional member of that team (JJH) conducted a series of semi-structured interviews with the three teachers we selected as the focus group and follow-up interviews with all other teachers. When these interviews revealed the teachers’ belief that computer access was inadequate, the designers loaned classroom sets of computers to the focus group. Throughout, Mr. Barrett and the authors continued to observe and participate in classrooms. At the conclusion of the project, semi-structured interviews were conducted with students and the principal. All interviews were audio-taped and transcribed.

The authors interpreted these data in collaborative sessions in which themes were identified, discussed, and evaluated in the context of the five social groups and extant research literature (e. g., factors that affect implementations), using a constant comparison methodology.

Results

We present findings by describing the perspectives and beliefs of each main social group according to our network of influences model (Figure 1). The fifth group, students, are presented separately by their teacher to illustrate specific teacher-student influences and a as a group when results were consistent across teachers.

Designers

As stated, the designers were initially optimistic. They believed that Ms. Cas would be unable to continue her initial level of involvement and that the initial meeting with teachers and specialists went well.

The designers initially received advice about the activities, but only through the assistant. They assumed that the teachers would choose activities relevant to what they were teaching in their class; however, this was not the case.

Ms. Moore:	I tried to pick something that was not being explored by other teachers so that Doug did not get feedback on a real popular item and then nothing on others.
Interviewer:	So you picked them to be helpful for Doug, not necessarily to be helpful for your kids?
Ms. Moore:	It didn't really apply, because a lot of Turtle Math is geometry, and that's our next unit that we're just getting to. And geometry we get to later on in the year

At another meeting, the designers presented a chart outlining the mathematical topics involved in each activity and re-emphasized the importance of integrating the activities into the mathematics curriculum. The teachers, however, remained focused on the computer. They chose activities by reading titles on the screen. Because most activities needed documentation to be used in the way the designers had intended (or even understood at all), the teachers’ inattention to the paper materials concerned the designers.

Nearing a publication deadline, the designers requested the written comments from the teachers. The designers were surprised that several teachers seemed unwilling to write short evaluations of the activities, even with compensation for doing so. When the designers asked the six less-involved teachers why they seemed less interested in the project, two mentioned personal problems and two mentioned they did not have the time on the proper computer access to work with the activities at home.

After the Turtle Math package went into publication, the designers continued their efforts with those teachers still interested in using Turtle Math. When prodded, the teachers asked for help linking activities with their syllabus, as outlined in the teacher’s manual of their textbook. They also said that they would like to see the designers teach an activity for them. Though Mr. Barrett had modeled lessons in the lab, one designer readily agreed to model a lesson after school with a small group of students. During this lesson, the teachers talked to each other and the graduate assistant more than they observed.

In a discussion following the lesson, the teachers expressed concern about aspects of the designers’ pedagogical philosophy. The teachers relayed an incident that had taken place earlier where children were satisfied that a non-closed figure was a triangle: “They made their equilateral triangle, and when we looked there was a little bit of a space between the line segments. And there was nothing there to verify if it is correct or is not correct.” The designer explained why having a built in checking device was not consistent with the philosophy of the software. The discussion continued, but both sides remained frustrated at its conclusion. This episode confirmed the designers’ developing belief that these teachers did not share their philosophy of mathematics education. It also confirmed the teachers’ belief that the designers were out of touch with what was practical in the classroom. Even though the teachers continued to use Turtle Math, the designers essentially stopped expecting (and asking for) advice on the software, although they visited and talked with the teachers in other capacities.

School and District Administration

The principal and Ms. Cas were enthused about Turtle Math and let the teachers know that they supported the project. Thus, although participation was strictly voluntary, the administration encouraged involvement. The teachers later revealed to JJH that they felt pressured to get involved in different projects every year as the principal’s focus changed.

When Ms. Cas had worked with the teachers to reform their language arts teaching program, she spent considerable amounts of time with each teacher. She helped them teach and integrate language arts lessons with other subjects, and provided feedback when necessary. Because she was the initial liaison between the designers and the teachers, the teachers assumed she would be functioning in a similar manner with Turtle Math. Ms. Cas’s new district-wide position, however, necessitated her spending considerable less time with the teachers at this school, and the person who replaced Ms. Cas was much more tentative in her interactions with teachers. She spent more time with pull-out programs for small groups of students, rather than working with teachers in their classrooms. Therefore, the teachers did not receive the continued support that they expected from the administration.

When teachers voiced their concern about having too little computer time to accomplish all their goals, the designers communicated this to the principal. The principal stated there were free lab periods (not assigned) available. She did not acknowledge that the existing computer lab organization and scheduling system was problematic.

In the spring, the principal asked the designers to work with interested teachers during their staff development day. This, once again, communicated to the teachers that she still thought Turtle Math was important. However, the principal was not involved in the meeting during the staff development day.

Near the end of the year, the district math coordinator visited the fifth grade rooms to teach a geometry lesson. The students at this school did well, and the mathematics coordinator complimented their performance:

Ms. Moore:

Yesterday (the mathematics coordinator) came into my classroom and started speaking to the children about geometry…and she was amazed how much they knew ...not just knew, but understood more of the concepts.

For these teachers, the math coordinator’s comments validated the usefulness of the program. This validation by someone at the district level (an authority not connected this Turtle Math) had more impact than positive comments by the project staff. When this was suggested to the principal, she did not agree that such administrative validation was important. She felt that the teachers could offer each other validation.

Teachers

Teachers are the key agents in the institution of an academic innovation. Teachers must incorporate the innovation into daily classroom life, making crucial decisions about how it will function. This section will describe the three teachers central to this research and their knowledge and beliefs of software and technology, mathematics pedagogy, and mathematics. These three fifth-grade teachers possessed distinct teaching styles and personalities as noted by Ms. Cas, “If the three of them were a Venn diagram, there would be no middle.” Nevertheless, they worked more as a team than did the teachers at the other grade levels, planning together twice weekly. Two of the teachers, Ms. Jack and Ms. Gaughan, even began experimenting with team teaching during mathematics time. This provided an interesting dynamic because Ms. Jack was as confident in her mathematics teaching as Ms. Gaughan was unsure.

Ms. Jack. Beliefs and knowledge about software and technology. Ms. Jack was interested in using the computer as a “powerful learning tool.” She emphasized the technological capabilities of the software, the ability to quickly and neatly portray geometric figures or many tosses of the dice: “And this way, they could see real quickly, well what happens when you make it 20 degrees, or what happens when you make it 30 degrees wider, or when you make a 180 degree turn.” She also viewed Turtle Math as a more efficient way to get through her regular geometry curriculum: “I think that the most useful thing is that it made what they did with paper and pencil much easier, because we started out geometry this year with Turtle Math. And it was much more hands-on, more visual than how we usually start geometry … It seemed to be much easier for them to draw conclusions about angles and about what would happen, after they’d had the opportunity to manipulate them.”

The portrait that emerged of Ms. Jack was that of a teacher dedicated to the use of technology based on its attractiveness and efficiency in communicating the required curriculum. To a lesser degree, Ms. Jack acknowledged the role of the innovation in opening up a new level of understanding.

Ms. Jack emphasized the need for whole-class discussion of computer work away from computers. “[I would ask] ‘How did you go about doing it?’ We would draw on the board whatever they were talking about, … but unless they’re back in a more confined space, that conversation was very hard to have.” Even so, Ms. Jack did not conduct follow-up discussions of the first few activities. Once Ms. Jack had computers in her room, working with Turtle Math became integrated in her teaching. Although Ms. Jack valued co-operative work (“when it’s more exploratory, then I need to have them in pairs”), she mostly had students work alone. “I prefer it that way…. I think it’s because it’s my nature …. I know what it’s like to be sitting and waiting to touch.”

Ms. Jack thought that one of the most powerful aspects of the software that students viewed it as a game. “To play a game, they needed to know those things. They would have never developed as much of an interest so quickly.”

Ms. Jack thought the computer activities were mostly a motivating way to get children to practice. She presents an off-computer example: “Most of the time, 90% of what they learn they learn during the last 15 minutes when we play a game using fractions. We spend a little time, ‘Here’s the concept, this is how you reduce things.’ But then when you play a game with using the lowest terms, they learn more in those six minutes … because it’s a game … They don’t see it as work.” Time on computer was an issue for Mrs. Jack as it was limited and all the teachers felt pressure to emphasize work processing. “It also became kind of a problem too because we were in there two days a week, and both times were being dedicated to Turtle Math and mathematics, and then they weren’t getting any of the writing time.

When computers came into the classroom, this was eased for the teachers. “Yes, when we had the machines, even for the week, in the classroom, they were much more accessibles, and they didn’t require the whole class.”

Beliefs and knowledge about mathematics pedagogy. Ms. Jack is a confident, dedicated teacher. She quickly embraces reform efforts. However, her explicit beliefs as stated during interviews, and her implicit beliefs as revealed by classroom observations were not consistent. Ms. Jack expressed mathematics goals that were holistic and consistent with NCTM standards: “What I want to really accomplish in math is that they [students] are all mathematically literate. Beyond what does it mean to add, subtract, multiply or do long division, that they understand mathematics is involved in every way possible… The most devastating thing to me is to hear a kid say that ‘I’m no good in math, or I hate math.’” Such statements are related to her concurrent work on the math outcomes committee where beliefs about mathematics education were discussed extensively.

Ms. Jack’s responses to the initial questionnaire were the most popularized. She agreed with ever constructivist item and she disagreed with every transmissionist item (except the item regarding the need for practice). She expressed no ambivalence toward the questionnaire. Consistent with her responses, she made statements such as, “I think that for me, my students are more successful when they discover it for themselves … they’ll come to their own generalizations and they’ll come up with their own rules and nine times out of ten they will come up with exactly what you would have tried to teach them.”

Her implicit pedagogy as observed in her teaching and interaction with students, however, was traditional. When Ms. Jack taught regular mathematics lessons, she led students through the problems. She was eager to supplements these lessons with “hands-on” activities, but these activities did not follow her professed constructivist model. She describes one such activity: “Last spring, when we were doing surface area, I handed everyone a tape measure and sent them out to find the surface area of three things on the playground before they could turn in their clipboard. Kids who weren’t really sure what they should be doing found somebody who thought they knew what they were supposed to be doing, and worked each other through it, and they did a great job.”

The activity was fun and cooperative way to practice the surface area algorithm; however, the actual lesson diverged from Ms. Jack’s proclaimed constructivist pedagogy in two ways. First, the students did not own the mathematics; rather they worked through assigned problems to earn free play. Second, they did not develop their own strategies and generalizations, but instead practiced a previously learned algorithm. Another example occurred during the after-school demonstration. Students walked rectangular paths off computer; later, on computer, one student entered FD 180 for one side of his rectangle and FD 90 FD 90 for the parallel side. The designer asked him if the two sides were the same length, and the students responded, “No.” The designer invited the student to “play around with it.” Ms. Jack, however, would not let the issue rest. The field note stated: “she jumped up to walk the path and show the kid the right answer. She converted 90 to 9 and 180 to 18 (without explanation) and started walking the path (forward 9, forward 9). When she started running out of room, she changed the size of her step so that she was practically waling in place, then asked the student if that was the same as eighteen. He seemed utterly bewildered.”

Beliefs and knowledge about mathematics. Ms. Jack felt torn between her district’s plan to eliminate textbooks and the school’s culture of covering a mandated curriculum. Ms. Jack resolved this conflict by describing her curriculum as “in process”; she will work through the teacher’s manual until she has time to reflect on and rework the curriculum.

Ms. Jack’s prior business career convinced her she is competent in mathematics and capable of “thinking on [her] feet.” Ms. Gaughan often called upon her expertise. Ms. Jack believed students should view mathematics as an important part of everyday life. When she spoke of teaching her students mathematics, she described mathematics as an inductive science in which closely examining multiple examples can lead to the extrapolation of generalizations. When asked about concepts that should be taught, she said, “I would like to eliminate some of the things that I do classroom-wise, even though there are some kids who need a certain kind of work, and replace it with Turtle Math.”

Students. The students in Ms. Jack’s class considered her the math authority, although they still considered her manual to be a valid source of answers: “She looks out of the book or just knows it.” Ms. Jack’s competence in mathematics led students to seek help from her first. They doubted themselves when their answer differed from hers. One student indicated that her only recourse was to wait until the teacher gave the correct answers the next day and she could “correct it with (her) red pen.”

This lack of autonomy was also observed in the computer room, in which Ms. Jack’s students sought affirmation from her or other adults. This sometimes led to a frenzied computer lab session as the adults ran around the room to answer questions until time ran out and the students were rushed out of the room. Ms. Jack’s students held unique views regarding feedback. They liked that the turtle responded to their commands without judgement and that they could change their commands if they made a mistake. They spoke of feeling frustrated in non-computer lessons after giving a wrong answer and not being able to follow up with another try: “With Turtle Math, the computer doesn’t correct what you’ve done wrong… In class the teacher will say, ‘Are you suuuuure?’ and you say it again and she’ll call on a different person”; “Turtle Math was a lot easier and it gave you another chance if you got something wrong and then you could fix your mistake.”

Ms. Jack’s students’ statements reflected their teacher’s view of Turtle Math as a tool for reviewing content previously studied in class:”…if you do isosceles and then go do isosceles triangles in Turtle Math, it will help you review.” They also shared her view of Turtle Math as a motivating environment: “Well it was more fun to type in some numbers instead of writing so it was more fun using Turtle Math.”

Ms. Moore. Beliefs and knowledge about software and technology. Ms. Moore was also comfortable with computers and quickly began using Turtle Math, only asking for assistance in coordinating Turtle Math activities with the scope and sequence of her textbook. Ms. Moore valued the innovation a novel approach to learning: “The most useful part of Turtle Math, I think, was allowing the children a chance to explore geometry and apply geometry in their own way.”

Ms. Moore saw the value of the turtle serving as a nonevaluative agent (cf. Johnson-Gentile, Clements and Battista, 1994). “It’s like you get to see if this third person or this machine will do what you think. If you are making it yourself, your brain is just kind of moving your hands to create it. But when it goes onto the computer, [it] has a set way of understanding you, and you have to put it in that way, or it’s not going to get across.”

In contrast to Ms. Jack, Ms. Moore used Turtle Math for exploring mathematics: “I used it more for …exploring and drawing conclusions …. And they could share things and see if it always happened that way … Not practicing. It was more a discovery thing.”

Ms. Moore also preferred to have her students work alone on computer: “I thought it was better alone, because if someone was making errors or was more advanced than the other partner, the arguments could start.” However, her students were encouraged to help one another: “I prefer that the students try to teach each other, because then they’re learning as well.”

Time was an issue for Mrs. Moore, as she realized Turtle Math would take time to use properly: “It wouldn’t be worth it for someone who says, ‘Hey, I should be able to catch it within an hour,’ and understand all of it.” There were also perceived pressures to work though the curriculum, especially because the teachers were worried about their students moving on to the middle school. Given that the teachers began their Turtle Math experience by using Turtle Math activities that did not coincide with their off-computer lessons, the teachers felt additional time pressure when using Turtle Math. “Each year we are asked to teach something else, that doesn’t necessarily take the place of something else, so you are left with another chunk to fit into the same amount of time.”

Beliefs and knowledge about mathematics pedagogy. Like Ms. Jack, Ms. Moore showed an interest in educational reform. She sought teacher education opportunities and consequently modified her teaching. At the same time, she declared certain innovations as impractical and viewed herself as a competent professional able to decide for herself what was educationally appropriate. Ms. Moore wrote an introductory comment on the questionnaire: “These questions are not yes or no questions.” Several items were answered with either a question mark or a combined “Y/N.” Ms. Moore answered half the transmissionist items in this ambivalent manner; she answered the other half “no.” She marked all constructivist items “yes” except that promoting “children judging right and wrong.” This pattern reflects her belief that reform does not require one to abandon previous teaching strategies.

Ms. Moore began each interview expressing her resistance to being coerced by “the experts”: “I just believe that at the elementary level, you have to have a basic understanding to then formulate and grow with it…. I don’t think people understand, children may discover that 2 + 2 = 4, but they may not. Sometimes it has to be direct instruction.”

Ms. Moore expressed misgivings about administration’s plans to eliminate mathematics textbooks and endorsed the teacher’s manual for planning and instruction. While clutching her textbook she said, “When I teach my lessons, they are hands-on as possible, as real world as possible, but there are certain things, I just have to look and see.” In her classroom (and while using Turtle Math), however, students were consistently helping each other without relying on her or their textbooks as sources of knowledge. Observations indicate she used a traditional approach while demonstrating the mechanics of the program but an exploratory approach to the mathematics. She encouraged students to experiment and required that they write about and share their experiences.

Beliefs and knowledge about mathematics. Ms. Moore stated, “I think my biggest goal is to have them realize that math is real world … that it is applied in everyday life and that it is something that is useful and will be useful for the rest of their lives.” She included technology, manipulatives, art, and writing in her math class. She believed mathematics is not only working through exercises. However, she viewed the field of mathematics as a list of topic. Specifically, she saw her school math curriculum as a list of required topics imposed by outside forces. “Our district may say, ‘Less fractions and more geometry or something.’ Then we have to adjust.”

Ms. Moore was competent in mathematics and confident in her ability to teach unfamiliar mathematical topics using innovative means. She stated that she is only limited by resources.

Students. Ms. Moore’s students mentioned they liked the different atmosphere of the computer lab. They did not notice a marked difference between on- and off-computer activities, which is unsurprising given Ms. Moore’s consistent teaching approach in these two settings.

Ms. Moore successfully taught her students to first check their answers on their own, then ask a peer, and only then ask her. They trusted their answers when they differed from hers saying she could have “messed up.” Although students in other classes mentioned checking long division problems by multiplying (recently “covered” in all the classes), only Ms. Moore’s students listed other methods for checking answers, including using calculators, protractors, and textbooks.

On the surface, Ms. Moore’s class appeared structured and teacher centered (an impression expressed by Ms. Cas). However, we observed her students learning mathematics on computer autonomously and cooperatively. They asked each other for help, or observed a peer’s screen, attempting to solve their own problem. Students were eager to show Ms. Moore their work when she passed by, but few waited for her to come around before moving on.

Ms. Gaughan. Beliefs and knowledge about software and technology. Ms. Gaughan did not use Turtle Math activities as much as other teachers; she attributed this to her own shortcomings. Nevertheless, she expressed an interest in using computers in her teaching, mostly as a way to provide motivating practice. I think it might be more interesting to them, they might attend to it better if they were ding some exercises [in] Turtle Math. Ms. Gaughan opined on the best aspects of Turtle Math, “It feels to the kids almost like a game format, is very appealing…. That’s its strength and its weakness probably.”

Ms. Gaughan talked about Turtle Math as facilitating a different type of learning. She described the difference between physically tossing a die versus using a probability program: “When you are rolling dice or tossing coins, you are actually physically doing it, but when you are sitting in front of a screen and manipulating the keys or the mouse to do the same things… it’s a step removed. And so it’s more abstract.”

Ms. Gaughan had her students work alone as well, simply because “we had enough [computers].” When asked about the possibility of students working in pairs, she responded as though she had not previously considered the idea: “Well, it might be …I was thinking …it might be nice to have it as pairs because then they would have someone to give feedback and help with problem solving.” Ms. Gaughan primarily used Turtle Math in the computer lab. When asked about any discussion at the end of the activities, she responded, “Oh, you mean together? Generally I didn’t take time to do that.”

Ms. Gaughan felt that she underutilized the program due to the amount of time she felt the software would take for both teacher and students to learn: “I didn’t get in there to play with the games ahead of time as much as I should have to really be able to present it well to the children.” Ms. Gaughan also shared her colleague’s feelings about pressure to complete the mathematics curriculum: “We were in these early chapters and I went ahead and did Turtle Math, but it didn’t fit with the math program…. You’ve got to be pretty shrewd to be able to be sure that you teach Turtle Math well and make sure that you are meeting the curricular needs.”

Beliefs and knowledge about mathematics pedagogy. Ms. Gaughan, recently hired by the district, views herself as limited by her mathematics knowledge; however, she also honestly agrees with the district’s plans for curricular change. Ms. Gaughan felt conflicted between what she knew she should say and what she felt was best for her. Her frustration with this conflict was apparent from her comments on the initial survey in which Ms. Gaughan agreed with all the constructivist items (except for children being able to judge right and wrong) but said, “politically correct responses are obvious.” Ms. Gaughan also agreed with almost half the transmissionist items.

Ms. Gaughan described her mathematics pedagogy as a step-by-step adherence to the teacher’s manual. “You show them the basic good steps or good rules to follow, and it’s all paper and pencil at that point.” Ms. Gaughan stated that discovery learning has a place but struggled with implementation. “Without exploration …they may not truly understand. They can go through the functions and not really understand what they’re doing.” Ms. Gaughan relied on Ms. Jack to provide her with ideas and support for any innovations in mathematics that she attempted.

Beliefs and knowledge about mathematics. All teachers were concerned about preparing their students for middle school mathematics. They viewed mathematics as a fixed body of knowledge and their job as teachers to ensure students posses prerequisite knowledge for the next mathematics course.

Ms. Gaughan described herself as “not math gifted,” someone who has to struggle to learn the concepts she is responsible for teaching. She knew her schol was emphasizing innovation, but she did not feel up to the task. “I can remember last year getting up to teach a math lesson, and I was just shaking in my boots because I just barely understood what I wanted to teach the children, plus I had a mandate to teach it hands-on and to teach in a higher thinking level mode, to encourage communication and verbalizing, and I was just overwhelmed.”

Like Ms. Jack and Ms. Moore, Ms. Gaughan viewed mathematics as a list of topics to be covered. When Ms. Gaughan was asked to describe her goals in mathematics, she immediately referred to the textbook teacher’s manual and started to list the topics from the table of contents. Ms. Gaughan also communicated to her students that math is something to be suffered through; she gave “challenge” math sheets as punishment: “Tell the [other] children that [Ms. Gaughan] has plenty of good math sheets for children who like detention.”

Students. Ms. Gaughan’s students believed the teacher’s manual had all the right answers; like their teacher, they relied on it heavily. They viewed Ms. Gaughan as a moral authority: “Well, usually, like in math, she let's us use her math book but she has to check our papers to make sure we don’t cheat.” In absence of the teachers manual, they were comfortable with a consensus among their peers, even willing to accept inaccuracies in their own work when their answer was close to that of their peers: “…like if I’ll say 45 and the other kids say 43 or 44 I still feel I have it sort of right because I’m right near to them at least.”

A group meeting. One incident illustrates the three teacher’s views of mathematics and how their views contrasted with those of the designers. The teachers felt their students needed constant correctness feedback from the computer. Because most Turtle Math activities do not provide such feedback, the teachers felt the need to give it to students individually. Teachers agreed it is “nice” when students make judgements for themselves, but believed that fifth graders are too immature and accustomed to video games to be content without a “checker.” Ms. Moore stated, “they need that immediate feedback, and if they don’t get it, they start to ask, ‘Is it right?’” Ms. Jack added, “The children feel that [they] ‘got it’ and now [they] can move on to something else instead of standing there and jumping up and down saying, "Ms. Jack, come look, come look!" The teachers suggested the designers reprogram the software so there would be a “checker” that would tell students whether or not they had completed the required task, such as making an equilateral triangle.

The designer explained why this was neither plausible not desirable. She said, "You want them to look at a rectangle that is not square and have them tell you why not. If the computer always did that, they would not need to justify or examine it." Later, the teachers were asked if they agreed with this point of view. Ms. Moore said, “[Designers think that students] should be able to recognize their accomplishment …share their accomplishment, and that’s gratification enough. [But the students'] level of maturity is not the middle school yet. They still need the stickers on their papers.” Similarly, Ms. Gaughan stated that “There are times when the teacher needs to say, this is the truth in the sense that these are the facts. When you have an equilateral triangle, you either don’t or you do.” Ms. Jack added, “My preference would obviously be that they take down the measurement tool and check it themselves …But, that’s not going to happen for everybody.” Teachers were concerned students may do something incorrectly on computer, not be corrected, and leave with inaccurate knowledge. This illustrates the differences between the views of the designers and teachers (even those who explicitly stated views consonant with those of the designers).

Support Staff

The support person for this project, Mr. Barrett, was a graduate student who had experience helping teachers integrate technology into curricula. He viewed his role as “getting teachers to use the programs no matter what it took.” He began by modeling the teaching of Turtle Math lessons in the computer room. He encouraged the teachers to use Turtle Math on their own, while providing technical and teaching support -- “an extra pair of hands.” Because he felt the project’s mission was to provide whatever support teachers requested, he felt uncomfortable denying a teacher’s request for him to teach. He did not, however, think teachers who asked were taking proper initiative. He believed the designers should have given teachers a written timetable at the start so teachers would have better understood how their roles were to evolve throughout the project.

In the spring, the researchers played ancillary support roles, providing answers to the teachers’ concerns as revealed in interviews. As a result of these interviews, a meeting was scheduled in which the designers helped the teachers choose specific Turtle Math activities to correspond to specific lessons in their teachers’ manual, and the designers gave each teacher a set of five Macintosh computers to use in their classroom for the duration of their geometry unit.

Students

Beliefs about software and technology. Students across all classes expressed several benefits in the software not reported by the teachers. First, they described the benefit of solving problems in context: “You weren’t just multiplying numbers. You were multiplying numbers to find the area or degrees.” Second, students mentioned geometry concepts similar to those their teachers cited, they also mentioned arithmetic quite frequently: “…it would help us a lot like adding, subtracting, multiplying and dividing and learning shapes and 90°; angles and other angles.”

Third, students'; views of certain critical components of Turtle Math pedagogy differed from those of their teachers. The teachers had communicated on several occasions that the students were accustomed to games at home and needed the “bells and whistles” to let them know they were on the right track. Much of the educational software available in their school, although sometimes masked as a game, was drill and practice with reinforcement for correct answers. Turtle Math did not offer such correct-answer feedback (the introductory activities were path and angle games, but even here, the feedback was a natural part of the situation, rather than a simple reward or reporting of correctness). When the students were asked about their general impressions of the software, they described it as fun, but they did not mention any game-like features. When they were asked what they would do if they could do whatever they wanted in Turtle Math, several students mentioned the games but also talked about exploring the tools. Asked, “What if you had to do only Turtle Math during free time?”, one responded, “I would do turtle things like make shapes and stuff because that’s fun …and measure them and stuff.” We interpret these statements to mean that, contrary to their teachers’ concerns, the students did not view the software as a game in and of itself but saw the games as part of the software. They did not expect or need the software to provide “bells and whistles.”

Beliefs about mathematics education. These fifth graders were as concerned as their teachers about their preparation for middle school. Students liked working on computers because it prepared them for the future. They mentioned “exploring the computer world,” “getting used to technology,” and learning specific math content. Although students believed exploring on their own helped them learn, we rarely observed such opportunities during the school year. The students also espoused different views on grouping in the computer lab. When asked what they thought was the best learning situation, students stated that working in pairs offered benefits: “I would have them work in pairs because they could learn more together than they can alone.” In contrast, teachers usually had each student work on a separate computer.

Beliefs about mathematics: “Right” answers and autonomy. All interviewed students believed there is usually a single right answer to a math problem. The students had recently worked on word problems to which there were no right answers so they acknowledged the existence of such problems but believed these were a special case that only arose during particular lessons. They were not comfortable with such problems.

Interviewer:	Is there always a right answer in a math problem?
Students:	No. If there is a word problem and they give you a certain amount of coins, and certain coins cannot fit into that number, there is no solution to that.
Interviewer:	Do you like those kinds of problems?
Students:	No. Even though there is a basis for it, I really don't think it is helping me.

Although the interviewer identified herself as being on the staff of the Turtle Math project, the students did not indicate the type of problems they worked on in Turtle Math as problems that can have more than one right answer. They seemed to focus in on their normal math class for examples of such problems.

Discussion

As we examined the data, a theme that consistently emerged was unrealized potential and missed opportunities for facilitation due to divergent beliefs of the social groups. Most studies of adoptions of innovations have concentrated on external factors as entities that could be controlled or altered. In contrast, we emphasize that the factors influencing implementation in this study -- those aspects of mathematics, technology, and pedagogy that emerged as critical influences -- were filtered through the belief structures of the people involved. Thus, though these factors appear to be practical issues, they are viewed differently by each individual and therefore are conceptual issues as well, not as easily addressed. These factors, people, and their beliefs about the factors are summarized in Table 1. We will discuss each factor in turn. In general, even ostensibly straightforward issues can become problematic when different people appear to agree, but hold views sufficiently divergent to prevent effective communication and problem solving.

An example of this is the issue of computer access. The principal and the teachers initially thought the lab was adequate; they were proud of the new lab, which displayed the school’s progress to parents and administrators. The lab also purportedly provided equity because teachers had equal access to technology. When teachers used computers more, they believed access was inadequate; however, the principal never agreed. Without another way to increase access, teachers could not increase integration of technology. Thus, although everyone agreed teachers need adequate computer access, different views of what “adequate” entailed prevented a viable solution for the teachers. The more teachers found ways to use computers: for word processing, Turtle Math, or other software, the more apparent the inadequacies and therefore the disparity in beliefs became.

A similar disparity in perspectives vitiated support for teaching reform. The principal alone believed extant support was adequate. In addition, although Mr. Barrett and the designers believed they were modeling a philosophical stance, pedagogical approach, activities, and software tools, teachers saw a model only of the latter two. Further, out own view changed: We now conclude support should be mediatory. Teachers have benefited most from meeting a support person frequently, for a short period of time (as Ms. Cas had) to plan activities and provide time for reflection. In such meetings, special care should be taken in establishing connection between the innovation and the teacher’s regular mathematics program. Teachers need these connections as a springboard. However, if too much is made of such explicit connection, teachers may limit their vision or the innovation to a surface-level extension of traditional mathematics curricula.

Given that the innovation in this study was a software package, technical proficiency was important. Mr. Barrett was available to the teachers after school to provide technical assistance. Attendance at these after-school sessions dwindled, however; the teachers preferred to have the software to “play around with” on their own time. We concluded that a synthesis of these two approaches is necessary. Teachers should have the time and the technical means to play with the software on their own; however, without appropriate guidance (e.g., written materials, tutoring), “playing” may be unproductive.

TABLE 1

Beliefs of social groups concerning factors influencing implementation

	Beliefs
Factor	Designers	Support	Administraton	Teacher	Researchers
Computer access	Lab adequate	No opinion	Lab and flexiable scheduling adequate	Lab adequate --> lab indequate	Classroom set needed
Support for teaching reform	School personnel and Mr. Barrett adequate	School personnel and Mr. Barrett adequate --> modeling also needed	Teacher collegiality adequate	Modeling needed	Scaffolding needed
Technical support	After-school sessions and time to work with software helpful	After-school sessions and assistance in lab needed	Help from designers and technology literate staff adequate	Assistance in lab helpful	After-school sessions adequate --> scaffolding needed
Validation	Feedback from designers and students adequate	Feedback from designers and students adequate	Feedback from other teachers adequate feedback from district and school level needed	Feedback from district and administration adequate --> multiple sources needed
Collegial support	Helpful for sustaining motivation	Not needed --> helpful for sustaining innovation	Needed (and, in itself, adequate)	Needed	Needed, though relationships are complex
Extra time for implementation	Monetary compensation helpful	Assistance in lab helpful paid inservice and after-school meetings adequate	Time alone to "play" with software needed	Scheduled time for reflection, work with software, and integration needed
Multiple reform efforts	Unaware of various efforts	Unaware of various efforts	Needed to keep faculty current	Not needed; exhausting, ephemeral, and therefore not worth effort	Need coherent theory and integration
Collaboration with designers	Better way to develop activities	Benefits and motivates teachers	Benefits and should be self-sustaining	Benefits designers more than teachers	Communication and mediation needed

The designers are two of the researchers. In our interpretation, the designers became reserchers once the study proper began (i.e., with the inclusion of JJH); therefore, the change in their views can be seen by comparing these two columns (the resercher's final views, of course, reflect what they learned from their own experience and the experiences and opinions of others). The support column reflects the views of Mr. Barrett. The administration column reflects mainly the views of the principal. Change of viewpoints during the study are indicated by arrows (-->).

The various groups disagreed on the desirability and need for external validation. The designers originally thought that validation from university professionals, students (their observable growth), and colleagues would be sufficient for the teachers. Mr. Barrett, being a part of the university team, believed that the teachers, as professionals, offered each other validation and that no other feedback was necessary. The teachers, conversely, were not sure of themselves until a district supervisor observed their classes and confirmed that the innovation was worth their investment of time and energy. Our conclusion is that teachers attempting an innovation need administrative feedback to complement peer validation and the internal validation of observing students’ growth. Each plays a distinct and necessary role in sustaining the teachers’ use of the innovation. While the administrators’ model appears consistent with a constructivist philosophy, the reality of this school environment provided conflicting messages about locus of control and independent decision making for teachers.

Before this study began, we believed that collegial support was an important factor (e. g., Ely, 1990). The administration believed the collegiality of the teachers was the most important support the teachers could receive and the teachers valued their team members. It is our conclusion, however, that the relationship between these individuals is complex. Collegiality may support or hinder individuals’ use of technology.

Available time is a ubiquitous factor in educational research. During the course of this study, the teachers frequently mentioned that they wish they had more time to explore the software. Mr. Barrett believed that he could ease the time pressure by helping schedule the teachers in the computer lab, as well as generally being available when the teachers were teaching The principal believed that time for inservice training and after-school sessions would suffice. However, these conditions were inadequate. It is out conclusion that it is not the quantity of time spent with the software that is as important, but the way the teachers’ time is spent. We believe that the best use of personnel time would be to have the teachers meet regularly with an administrative assistant, such as Ms. Cas. Giving the teachers time for reflection is important, but the teachers in this study did not reflect on their experiences and needs until they were interviewed. Time spent conversing with such an assistant could facilitate the desired reflection. Time with the software also needs to be scheduled so that the teachers can not only have time to explore the software, but work with someone on integrating the software activities into their math curriculum.

Multiple simultaneous reform efforts overwhelmed the teachers. The principal believed that she was serving her faculty’s needs by offering them a variety of opportunities for reform and the teachers did not mention these multiple reforms when initially asked whey there were not using Turtle Math. It was not until the more in-depth interviews, with time to reflect, that the teachers specifically mentioned the pressures of the reform efforts. We conclude that offering multiple reform efforts without ensuring a consistent coherent theoretical base and integrating their implementation is a course doomed for failure (support for this notion also can be found in Sosniak and Stodolsky (1993)). Though the reform efforts themselves are non-reductioninst, the method of implementation -- multiple, separate one-day workshops -- was reductionist. The teachers then had difficulty seeing the forest for the trees. Although a program intended to increase higher order skills, a hand on science curriculum, and Turtle Math had consistent and overlapping goals, the teachers did not perceive this, impeding their implementation.

The designers initially believed that they would provide software that could help teachers attain their educational goals and simultaneously work collaboratively with teachers to design activities. However, teachers’ goals were not the same as designers’ (or administrators’) goals for the teachers. This accounted, in part, for the minimal initial influence the designers had on the teachers’ beliefs and implementation of the innovation (Figure 3). In the most extreme cases, the teachers viewed the philosophy that underlies the software and the recommendations for mathematics reform as a utopian, if pedantic, vision but not as a practicable pedagogy. This was due at least in part to the teachers’ belief that mathematics is different than (whole) language; for example, in mathematics, answers are “right or wrong.” Without engaging in the present study, we would not have realized the extent or source of the discrepancies.

An overarching issue is teachers’ knowledge and beliefs about mathematics and computer technology. A deficit of mathematics knowledge hindered teachers’ implementation of the innovation. The teacher who lacked both confidence and ability in mathematics was unable to mentally coordinate this innovation and her everyday ordeal of mathematics even with considerable assistance. However, even extensive knowledge of pedagogy did not ensure that teachers exhibited decisions and practices consistent with the innovation’s philosophy. Their continually held belief that computer activities were fun “add-ons” also severely hindered the successful adoption of the innovation.

Figure 3. Revised research model

For two teachers, Ms. Jack and Ms. Moore, we observed specific contradictions between their explicit and implicit beliefs. (Though in different ways. In one meeting, we referred to them as “mirror images”; whereas Ms. Jack professed beliefs congruent with constructivism, but often reflected traditional approaches to instruction, Ms. Moore expressed a traditional philosophy but evinced many constructivist-consistent teaching approaches.) Such contradictions are not surprising, but they do have additional relevance to the adoption of an innovation. Beliefs are held in clusters and more and less in isolation and protected from relationships with other sets of beliefs. This clustering prevents cross-fertilization or conflict between clusters or beliefs and makes it possible to hold contradictory sets of beliefs (Thompson, 1992). This was observed especially in Ms. Jack. In our roles in curriculum and software development and staff development, we saw contradictions that Ms. Jack did not. This situation hampered communication, support, and the implementation in general.

Students agreed with the teachers, and thus, we assume, were influenced by them, concerning matters about which teachers had communicated their beliefs, either explicitly or implicitly. Students’ image of themselves as autonomous vs. dependent learners similarly reflected their teachers’ beliefs and knowledge of mathematics. However, students did not agree with their teachers when the students had the opportunity to form their own opinions and conclusions; for example, about aspects of the software such as the desirability of games, the need for “entertaining” right-answer feedback, the benefits of mathematical explorations, and working in cooperative groups. Teachers’ beliefs about what their students wanted and needed educationally influenced their curricular choices. In some cases, these diverged from what the students said that they wanted and from what, in our opinion, they needed. Lastly, students voiced advantages of the software that their teachers did not, including meaningful context for mathematics and enriched experience with arithmetic operations.

In summary, the designers and teachers did not share a vision, or image, of mathematics education. The teachers shared an implicit belief that to achieve their explicitly expressed constructivist goals, they had to “finish” the textbook -- even given that these goals did not come from the textbook and were inconsistent with it. In this way, the textbook actually determined their goals. We believe that such a vision provides an internal gyroscope that guides the myriad activities of an educator: curriculum development, the course of a lesson, and moment-by-moment interactions with students. The teachers did not possess such an internal guidance system for innovatory mathematics education (though they undoubtedly do for other approaches to mathematics).

These results imply revisions to our network of influences model (Figure 3). Under the present conditions, far fewer influences were active that we originally presumed. The designers and support staff had limited effect on the teachers, although the teacher interviewer (JJH) played an important role as mediator -- a role we thought would be assumed by the administration. The administration had less (direct) influence than we originally postulated or than we still feel is possible; that is, the teachers needed more administrative support and validation than was given. This is not to say that results were all negative, however. Our revised intervention strategy, applied with the focus group of teachers, had successful components. Further, the focus group of teachers have continued to expand their use of Turtle Math for two years following the end of this study.

Implications For Innovators

Several conclusions in the extant research literature were born out in this study, including the assimilation of an innovation to existing pedagogical practice and the influence of practitioners’ perceptions of support and personal cost. These findings go beyond mere replication, however, in that (a) school personnel contacted the designers, stating that the teachers felt a need for the innovation and would thus readily implement it and (b) the designers considered these implications when planning the implementation. Thus, those planning innovation implementations should consider teacher and school resistance to change robust, even if the teacher and administration explicitly express both acceptance of the innovation and educational beliefs consistent with its philosophy. For example, the teachers viewed the philosophy that underlies the innovation as an acceptable pedantic vision for reform, but not as a practicable pedagogy.

Our findings have several specific implications for those supporting the adoption of educational innovations.

The possible limiting of teachers’ implicit, mixed, beliefs, and inconsistencies between explicit and implicit beliefs, may deny teachers a clear vision for reform, an internal gyroscope to guide reformatory, technology-rich mathematics education. Thus, more discussion of varying philosophical and pedagogical frameworks, and their validity for classroom practice, may be critical
Designers may have relatively little direct influence on the teachers’ beliefs and their implementation of the innovation. They may have to modify their vision of mathematics education innovation for, or in collaboration with, teachers if they wish to help those teachers construct an internal guidance system that is consistent with that larger vision. Also, simple demonstration lessons alone may be ineffective. Possible contrasts between pedantic vision and practicable pedagogies may need to be addressed explicitly.
In addition, other mediation may be necessary if innovations such as this one are to be adopted and if beliefs of teachers are to be affected in ways consistent with the designers’ intentions. Teachers should meet regularly with support personnel for reflection and planning in the early states of adoption; this may have a powerful influence in mediating and translating the philosophy that girds the innovation. A general conclusion, that social interactions are as important a structural changes, is consistent with previous studies (Evans-Andris, 1995; Louis, Marks and Kruse,1996).
In these meetings and all phases of implementation, the deleterious effect of multiple simultaneous reform efforts should be considered. Where these efforts are consistent and overlapping, integration and coordination should be planned; otherwise, clear priorities should be set.
Affirmation of the consonance of the innovation with school goals and the positive effect of the innovation on learning outcomes form school staff may be critical to maintaining teachers’ commitment and motivation beyond any such validation that comes from outsiders associated directly with the innovation. Teachers may benefit from multiple sources of such validation, from administration and peer validation to the internal validation of observing students’ growth. Each makes a distinct contribution to sustaining the teachers’ use of the innovation. Further, collegiality often supports but may actually hinder individuals’ use of technology; possible overdependence should be monitored.
Given the divergence between the opinions of the present study’s teachers and administrators subsequent to the initial meetings, a strong implication is that administrators should remain involved with the project (e.g., attend project meetings). They should also provide the time for teachers to experiment, discuss, and, in general, construct their own meaning of constructivist pedagogies.
Different approaches should be considered. For ecample, teachers could obtain personal experiences and success using computers to learn mathematics themselves. Only after such experiences, over the course of a year, for example, would they be expected to provide such experiences to their students. At this point, they might also be asked to build their own notions of how to use the innovation in their classrooms. Such an approach might be attempted in future projects.
Another possible approach presents a dilemma. Teachers may possess insufficient ownership of an innovation if they are not integrally involved in its development. Thus, one might argue that all curriculum development start in the school level. Such a view, while seductive, presents problems, including the genesis of reform in every school or the design and dissemination of a software innovation such as that employed herein if that design must be reconstructed at the school level.
Computer laboratories may hinder the full use and integration of innovations; computers in the teachers’ room place teachers in control.
Those designing innovative computer materials, and implementation programs for these materials, should remain aware of the tendency for teachers to ignore documentation. Instead, teachers often run the program, apparently expecting that the full extent and worth of the software and the activity will be immediately palpable.
Innovators should also be aware that teachers might assimilate many software environments to a “computer game” scheme that may distort and diminish the program’s educational significance.

Footnotes

¹ "An Investigation of the Development of Elementary Children's Geometric Thinking in Computer and Noncomputer Environments," a cooperative research project at the University of Buffalo and Kent State University. National Science Foundation grant number ESI-8954664.

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Acknowledgements:

The work described in this article was funded in part by the National Science Foundation, grant number ESI-8954664 (“An Investigation of the Development of Elementary Children’s Geometric Thinking in Computer and Noncomputer Environments”; funding the research) and grant number MDR-9050210 (“Investigations in Number, Data, and Space: An Elementary Mathematics Curriculum”; a cooperative curriculum development project among the State University of New York at Buffalo, Kent State University, and Technical Education Research Center). The opinions expressed are ours and not necessarily those of the Foundation. We appreciate the help of several additional members of the Investigations in Number, Data, and Space team in data collection, and the helpful comments of Mary Lindquist, Leslie Steffe, Grayson Wheatley, and three anonymous reviewers.

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