Challenges for Teachers Attempting to Integrate a Mathematics Innovation

Julie Jacobs Henry
Fredonia State University

Douglas H. Clements
State University of New York at Buffalo

Abstract

When teachers decide to use innovative materials in their classrooms, several questions arise. The fourth-grade teachers in this study answered these questions in different ways that greatly affected the success of the innovation. Data were collected through audiotaped semistrctured interveiws, teacher journals, and field notes. The data were analyzed using a social constructivist framework to determine how these teachers constructed a rationale for their use of the innovation. One teacher was able to foster mathematical conastrxurtions, relisquish control over the classroom and the curiculum, and feel validated in these processes by paretns, admistrators, students, and colleagues. Educators and researchs are encouraged to help teachers refine mehtods and establish links between their beliefs and effective innovation. (Keywords: Elementary, Logo, mathematics, reform, teacher beliefs.)

The Research Advisory Committee of the National Council of Teachers of Mathematics (NCTM) (1990) has called for a transformative agenda for mathematics education, research that deals with what ought to be. Researchers are also asked to consider and respond to the problems and issues that arise as practitioners work in the transformative process. There is a need for research that not only focuses on the use of technology in classrooms but also examines the ways in which teachers choose to reorganize their instruction and their curriculum using technology. The committee has suggested that one route to reform is the development and use of innovative curricula that exemplify the recommendations in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989).

However, any reform effort in education eventually gets translated through the classroom teacher (Glidden, 1991; Grant, Peterson, & Shojgreen-Downer, 1996; Kinnick, Driscoll, & Strouse, 1990; Rogers, 1995), an agent whose goals and beliefs may elude reformers (Ely, 1990). Teachers must make substantive decisions and thus answer fundamental questions concerning how any innovation will take shape in the classroom (Ben-Peretz, 1990; Dwyer, Ringstaff, & Sandholtz, 1991; Sarama, Clements & Henry, in press). The responses they generate are based, in part, on their prior beliefs (Thompson, 1992). Teachers come to the innovation with different ideas and experiences (Cohen & Ball, 1990). Reformers are quick to acknowledge this process with children but sometimes slow to do so with teachers.

Teachers' prior learning, beliefs, and attitudes are rarely considered as an essential ingredient in the process of teaching itself, much less in the process of change… Of course, interpreting the new through the lens of the familiar is, as cognitive science now tells us, how all of us construct meaning from the information we process using our existing schema. There is no reason why teachers should behave any differently (Darling-Hammond, 1990, p. 342).

Teachers find it time-consuming and demanding to examine their beliefs and adjust their teaching practice (Cohen, 1988; Grant et al., 1996) and may be inhibited by conditions within the schools. For example, teachers often feel pressure to appease administrators by implementing an innovation without examining their own beliefs and developing their own adaptation of an innovation. The purpose of this study was to examine how teachers answered fundamental questions about their adoption of a mathematics innovation, and how these answers impacted their success with the innovation.

METHOD

Participants

Six teachers at an upper-middle class suburban school participated in this study, particularly the two initial contacts, Carol and Beth. The research began, when Carol, a fourth-grade teacher who had participated in the fieldtesting for one version of the innovation, contacted the designers and asked for a year-long curriculum innovation. Beth, another Grade-4 teacher at Carol's School, had also participated successfully in the fieldtesting project, and expressed interest in the new innovation. The four other Grade-4 teachers at this school were invited to participate by these two colleagues.
One of the district's goals was to increase the use of technology. Each classroom had five Macintosh computers, and each teacher received a Macintosh PowerBook laptop computer.

The Innovation

The designers of this innovation created a modified Logo geometry environment for a National Science Foundation 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 & Sarama, 1995, 1996). The software was intended not only to play a major role in the study of two-dimensional geometry but also to support mathematical thinking in many areas of mathematics. Two versions of this environment were employed. The main one was Turtle Math (Clements & Meredith, 1994), a stand-alone product that provides off- and on-computer activities in seven areas of mathematics: geometry, similarity, number, fractions, arithmetic, statistics, and probability. Geo-Logo Sunken Ships (Clements, Battista, Akers, Rubin & Woolley, 1995) is software supporting geometry unit from the "Investigations in Number Data and Space" curriculum. These units include step-by-step instructions for teachers, sample dialogue for class discussions, and activity sheets for students.

Procedure

We studied teachers' beliefs, questions, and decisions using a social constructivist framework. This approach emphasizes the processes through which meanings are created, negotiated, sustained, and modified within a specific context (Schwandt, 1994) and how they emerge through a process of social exchange (Gergen, 1985). We periodically conducted semistructured interviews with the teachers to elicit their beliefs about mathematics and mathematics education, the innovation, and the local teaching environment (Bussis, Chittenden, & Amarel, 1976). We probed for beliefs and perceptions of what assisted and hindered them in using the innovation and the questions they asked themselves during their implementations. Interviews were audiotaped and transcribed. We observed teachers' practices to help characterize their implicit beliefs. Researchers functioned as participant observers. Videotapes of the classroom were reviewed and used to supplement the field notes.

The data analysis task was to systematically analyze these materials and synthesize across categorized data to answer research questions about the relationships among (1) teacher beliefs, (2) teacher support, and (3) teacher's decisions regarding their use of the innovation. Analysis of data was concurrent with data collection so that the data analysis could guide further data collection. The data collection and analysis had the benefit of triangulation, using multiple data sources and multiple researchers (Denzin, 1989) in formulating conclusions. The teachers were invited to participate as research partners, collecting their own field notes and helping formulate themes, while providing member checks on key assertions. Special attention was given to discussing emerging categories, themes, and conclusions and asking other members of the research team to uncover confirming or disconfirming evidence for their field notes.

RESULTS AND DISCUSSION

The teachers in this study were found to continuously ask themselves questions about the fit between the innovation and their classrooms. Five questions emerged as particularly compelling for these teachers:

Which students would benefit from using this innovation?
How can I best foster students' mathematical constructions?
Who is in charge?
What is the relationship between the innovation and the regular curriculum?
Whom am I trying to please?

The answers that teachers developed to these questions greatly influenced the effectiveness of this innovation. Each question is examined in the following subsections.

1. Which students would benefit from using this innovation?

Students in this school were grouped by ability for mathematics instruction each day, moving to a different classroom in the fourth-grade team. Carol and Beth taught the groups identified as "high ability." There were one "low-ability" and three "average-ability" groups. For all these teachers, the identified ability level of the students led to decisions about the practicality of an need for an innovation that was perceived as enrichment beyond the normal curriculum.

The teachers of the high-ability groups defined themselves as having not only time, but also a need for enrichment activities. They felt required to go "above and beyond" the other classes to indicate to themselves and to the students, other teachers, administrators, and parents that they were fulfilling their responsibilities to these students. Carol expressed that:

This is a top class, and if I just do the bare minimum with them, I'm doing them an injustice. Which is why I like this computer program. It's so much more sophisticated, and they're doing so much higher-level thinking (Interview, February).

Parents in this district were especially vocal, and teachers frequently mentioned the importance of challenging students and making sure that they were not bored, a situation that would displease the parents. A teacher of an average-ability math group indicated that ability grouping was a bit of a relief in mollifying parents. He noted, "They don't want it too easy. That's why I'm happy that we have the two upper classes so those kids can be challenged" (Interview, June). These beliefs about the ability of students and innovation as enrichment for them encouraged use for Beth and Carol, and relieved the other teachers from any need to go "above and beyond" in implementing this innovation. The other teachers agreed to use activities from the more structured version of the innovation (Sunken Ships) and found it successful but maintained that time did not allow them to use more than a limited set of these materials with their students. (We shall return to this issue.)

2. How can I best foster students' mathematical constructions?

This innovation is predicated on the belief that (1) with proper teacher guidance students can fruitfully learn important mathematical ideas and (2) that these ideas will be more meaningful that those learned from more transmissionist approaches. Carol and Beth shared this belief to varying degrees throughout the school year. Their struggle to find their role in fostering the mathematical constructions of their students is the essence of the story of the innovation of these two classrooms. We examined the teacher's answers to this question at three times of the year: early, middle, and end.

Early on, Carol expressed a belief in the value of "teaching math through projects." She entered the study with an emerging constructivist epistemology, believing that students learn best when they are given projects and guidance to help them construct mathematical concepts for themselves. She described a project as a long-term endeavor, somewhat self-designed, during which children would learn all sorts of things that they almost would not even know they were learning. During the year, she sought out opportunities to have children use Turtle Math to do projects. She explained:

Well, I find if you do something that is important to them, they're more interested in it, and they seem to delve into the problem more, and reason and think about if things are sensible and meaningful for them, as opposed to, just get this assignment done, and don't even think about if it makes sense or has any relevance to you at all (Interview, February).

Beth pursued this innovation as an opportunity to "teach more open-ended," a phrase she associated with the NCTM (1989) Standards and other reform documents and a technique that she felt she had implemented successfully during her pilot-testing of the Sunken Ships unit. (The authors agreed that her implementation of the unit was successful.) Beth had not embraced the wider constructivist framework of these reform efforts, however, and was attempting to achieve higher-level thinking in her students without sacrificing her traditional transmissionist view of teaching and learning. Beth viewed the innovation as her ticket to a more "open-ended classroom," anchored in the exciting, yet comfortable (to her) milieu of technology. For Carol, the innovation was a comfortable extension of her current beliefs about teaching and learning, but Beth was ambivalent, hoping that the use of this innovation would help her to move in the expected, albeit unsettling, direction.

Midyear marked a running point for both Beth and Carol. During the month of November, both teachers devoted class time to discussing hierarchical categorization of squares and rectangles, a concept suggested for study by the innovation, but one they had not examined in previous years. Students in Carol's room struggled to understand the relationship during a day-long discussion. At the end of class, Carol could not refrain from telling students that squares were indeed a type of rectangle. At this point in the year, she had some ambivalence about the role of the teacher and the need for the teacher to bring closure even when the students' thinking had not brought them there.

In the following days, however, Carol tried a different approach and had her students apply agreed-upon properties to classify shapes. She explained:

I think they finally got the idea that a square is kind of rectangle, but it's got special properties all its own that classify it as a square. And as we went along farther down, we learned the different quadrilaterals and the rhombus. The rhombus is a parallelogram, but because it has all sides equal, we give it a special name. And I think just the more we did these types of things, the more it is clicking in them that certain things can be in more than one category. It was a natural process (Interview, February).

Carol was developing a distinction between knowledge that can be constructed (e.g., hierarchical classification of shapes), and knowledge that she was transmitting (a name given to a category). She was becoming more comfortable with her role of providing some information, posing tasks, and ten allowing children to work through the "natural process" of making sense of the concepts.

Studying the relationship between squares and rectangles affected Beth in the opposite direction. Until this point, Beth had been ambivalent about the role of class discussion; she thought that perhaps students could "discover" things as a group by talking through their thinking. Students were able to articulate their thoughts, but Beth would become lost in the various constructions, following each comment or thread, until she and her students lost faith in her ability to discern the pertinent factors. In her frustration, Beth would eventually attempt to transmit the concepts to her students, usually at the end of the math period by using unclear metaphors or having them copy points into their journals. For example, during the square versus rectangle discussion, she used the analogy of square is to rectangle as stool is to chair. This part-whole relationship was difficult for many students and inadvertently flipped midway when Beth started talking about chairs and stools as kinds of seats, and a chair as a kind of stool with arms. After 15 minutes on this example, she seemed to forget her point, leaving the students befuddled.

In an October journal entry, Beth writes about her confusion over the role of the teacher:

I told the students that I know this is confusing to them, because I'm probably the first teacher who ever said to you that a square is a rectangle. And one of the kids said, "No, Mrs. C. last year told us that." And I just find it so interesting, because a teacher can tell them that a square is a rectangle, but they still don't want to believe it. So even through this child admitted that last year her teacher did tell her that a square is a rectangle, and this year her teacher told her that a square is a rectangle, she still doesn't want to believe it. And I'm not sure why. And I'm sure today, when we talk about this, there will still be some children who are still going to say, "No, a square isn't a rectangle." I'm going to start my lesson today with having them write the properties of a rectangle again, and we'll all once again agree that one of the properties of a rectangle is not that two sides have to be longer than the other two.

Here Beth revealed a tendency to envision mathematics teaching as an instrumental activity, where the teacher explains and defines while the students practice (e. g., writing the properties of a rectangle repeatedly). In this situation, Beth was being confronted by the fact that even when the class was told repeatedly, and even when the class agreed to the properties, this relationship was still obscure for some of the students. Rather than using this perturbation to discredit her beliefs about the value of telling and practice, however, Beth cited "the square-rectangle thing" repeatedly as a prime argument against her use of the innovation. She was not able to conceptualize or facilitate the "natural process" that Carol saw as students developed shape classification systems that made sense to them through their continuing work with different shapes and their discussions. In fact, Beth had her students write down the properties of rectangles on seven separate occasions, desperately seeking to solidify this concept before moving on.

Beth was operating under a reductionist epistemology where concepts are viewed as discrete pieces to be passed along one at a time to students. Carol's use of multiple examples and discussions in which students abstracted from these examples illustrated her constructivist perspective. Beth implicitly believed that any concept that she brought to the attention of the students needed to be consolidated for every student before she, in good conscience, could move on. This brought the curriculum in her classroom to a standstill on this and other occasions, during which Beth vacillated between open discussion and teacher transmission of information.

By spring, Beth lost faith in her ability to foster students' constructions of mathematics. During class, she allowed the students to discuss their ideas without a clear idea of how to resolve them. When the class was discussing, "Is an inch always an inch?" (a student-generated question), students argued that an inch must be an exact standard for important measurements (e. g., volcano holes), and that measurements must "match up" if you are the same height at the doctor's office and at home. Other students, however, pointed out that when they held different rulers next to each other, they could see differences. Beth allowed the discussion to meander for most of the class period, then with a desperate look at the clock, she said,

I like the discussion, but let's think logically. In California, the number 7 is still 7. We have all agreed than an inch is still an inch, but nothing is perfect, so things might be off a little bit (Class Observation, March).

The California metaphor seemed lost on these children, and the abrupt ending of the discussion combined with Beth's demeanor seemed to communicate to the children that these discussions were a waste of time. At the end of this class, she expressed frustration that she was never able to just run through something. Instead she got stuck on pointless topics. "How does this happen?" she asked the first author in disgust.

For many months, until Beth discontinued the use of the innovation, this was a consistent pattern in Beth's classroom. She experimented with the role of facilitator, lost focus, abruptly shifted back to a telling mode with an obscure analogy or set of didactic notes on the board for students to copy, and felt frustrated. This new role did not work for Beth, and she was profoundly aware of and upset by it. This conflict ultimately destroyed Beth's confidence with the innovation and convinced her to terminate her use of the curriculum in mid-April. For Beth, the innovation was meant to engender a new role for her as a teacher, one that was in conflict with her implicit beliefs about how students learn. When the role did not develop successfully, Beth, Rather than reconstructing her own epistemology, scrapped the innovation.

Beth agreed that her class discussion remained unproductive.

For some reason with Turtle Math, I have this direction to begin with, and then one kid pipes up and says something, so I start veering to the left, and then someone else says something else, and so I veer to the right, and pretty soon I have no idea where I'm going to. Because there are so many great ideas that I could be doing, but I've lost my focus. And I lose my focus so easily with Turtle Math, and I don't know why still. But it's very easy for me to get off (Interview, April).

Beth lacked a sense of the critical features of the field of mathematics that could have kept her on a coherent, consistent path. She felt unable to distinguish between "great ideas" that were pertinent to her goals as a mathematics teacher and ideas that were not relevant. She believed she was responsible for directing all learning and therefore struggled with myriad possible paths.

Carol was coming to different conclusions at the end of the year. She had discovered methods for fostering students' mathematical constructions, and this success encouraged her to pursue this innovation. Although she often admitted that there were gaps in her knowledge of mathematics and computers, she believed that she and the students would "stumble upon" these together. One day, Carol decided to use a concept attainment lesson ("What's My Rule?") suggested in the Turtle Math manual. In this lesson, students take turns moving shapes into columns at the blackboard while the class guesses which rule the student was using to sort the shapes. During this lesson, Carol planned to explore the distinction between parallel and nonparallel lines. In fact, students used this format to crystallize many of the other concepts that had become salient to them through their experiences with the innovation, shorting by right angles and no right angles, symmetric and nonsymmetric. Students described their thinking and argued productively among themselves. When it was Carol's turn, she sorted according to parallel and nonparallel, and students were able to competently discuss these properties as well. Carol had found ways to structure situations, sit back and allow the students to take the lead, and then intervene when appropriate to accomplish her goals.

3. Who is in charge?

The innovation provided teachers with an invitation and an impetus to alter their traditional classroom control patterns and to view the curriculum as more fluid. The computer activities encouraged teachers to alter their role from "sage on the stage to guide alongside" (Schofield, Eurich-Fulcer, & Britt, 1994, p. 581). These challenges to traditional patterns were viewed as exciting, demanding, or threatening by different teachers at various points in their use of the innovation. Beth's instructional style was more structured than Carol's, teaching a concept and then assigning problems from the book for practice. Beth started out using Turtle Math the same way. She explained:

What I do personally is, I'll beat something to death, because I want them to have the meaning of it, and then let them go on the computers, and even Carol has said to me, let them discover it on the computer. The computer has been more of the reinforcement for me (Interview, February 1995).

When she tried to replace this format with what she viewed as the innovative format, she was not successful. Beth had heard of a new role for teachers as "facilitators," although she seemed sketchy on the specifics. Beth hoped that having the children work on the computer would help her develop a more facilitative role. One day, frustrated by the logistics of manipulating the hardware to demonstrate a new Turtle Math activity and knowing that Carol was having success with less effort, she decided to forego her usual structure. She met with one of the researchers on the first day of her new technique:

Today I'm going to do something different. I'm going to introduce a new game and not show them how to play it, and let them figure it out. Because my problem is, I think I just have the grips on the kids too tight, and Carol will just let them go more and say, "Oh go ahead and try this." So I'm going to play Carol today and just let them go. And to make things even worse, I'm going to work with small groups, so I'm not going to be around to help the kids on the computer (Interview, October).

Her skepticism was apparent; she did not believe that this new pattern of control was going to be successful and set herself up to make things "even worse." Predictably, she was not happy with the functioning of the class, and as the class ended, she insisted that she would not do it that way again.

Carol saw herself as a teacher with a more open instructional style, one that matched the innovation:

I don't think you always have to activities where you come to a solid conclusion, where everyone comes to the same conclusion. Which is probably why I did so well with Turtle Math. I have found that some of the best things I do are things where you start the kids on a path and you give them a little direction, and you let them go (Interview, June).

In another classroom, the students in the low-ability math group worked in a structured classroom environment. During class activities, there was no talking or fooling around as was evidenced in the other classes. Students sat quietly and either paid attention or drifted. During on Sunken Ships activity, however, students were calling off coordinates as guesses for where the teachers had hidden a ship on the overhead projector grid. Students became highly animated and infused and hid the ship for a second game. After this game, the students were even more insistent: "We want to hide it. Come on, and you find it." The teachers appeared uncomfortable with this unusual outspokenness but eventually acquiesced. When the student at the overhead had difficulty reading the coordinates, other students called out suggestions and reminders, while the teachers stood to the side of the room. The innovation stood out as an occasion where these teachers relaxed their vigilance and let the children take control of their own learning. The class was deemed a success. As the first author was leaving after this class observation, the teacher ran up to her in the hall, exclaiming:

I told you! They may be slow, but they get it! They've got all the potential in the world. (Observation, March).

Unfortunately, the Sunken Ships unit was the only time that the low-ability group used the innovation. The teacher cited "the time thing" as the reason that she was unable to pursue further activities, but we believe that their unfamiliarity and discomfort with the new classroom patterns was a significant influence.

Another issue regarding who is in charge? was the need to manage multiple simultaneous activities. Only one-half of the students in a class were on the computers at any given time, and each pair was at a unique point in the activities. Teachers needed to plan off-computer activities for the remaining students that were "more than just playing around." Teachers struggled with planning each of these experiences and allocating their attention appropriately while they were taking place. There were also the technical concerns of working with computers. Beth described the balancing act:

The kids are on the computer, and I feel like I'm not giving them enough attention, and the kids off the computer I feel like I'm not giving them enough attention. And I'm just torn too much, and then there are always, with computers, there are always little fundamental things that are important, like, the child over here crashed. So I'm running around doing that, and then the class is over, and I don't really know what Jimmy got from the class (Interview, April).

Teachers solved this problem in distinct ways with varying degrees of success. Carol decided to primarily focus on the computer group:

I feel that I want to spend my time with the computer people. Because they're coming up with things, and they want me to look at them. With the off-computer group, they want me to see things, too, but it's like I walk and say, "Great, great ideas." I think they need more supervision on the computers because of the things they're coming up with or if they have problems (Interview, February).

Carol also frequently used manipulatives such as pattern blocks and geoboards as her off-computer activities. She liked those activities because they were "free-thinking," she did not have to go back and "go over it," yet they were not a "waste of time." During our observations, the students on the computer were involved and focused, and the off-computer students were relaxed and playful. When an off-computer student would approach Carol with some extraneous concern, Carol was likely to chase them off with a friendly, "Go, I don't want to hear it." Students seemed to take this in stride and either work out their problems or save them for later.

Beth was never comfortable with the off-computer activities she chose, except when she was working with the Sunken Ships unit. She found that students using the manipulatives would "think that it was fun and games," and then she had to try to "quick with them something else." She often used a worksheet as the off-computer activity, which frequently required her to read aloud problems and answers. Students in this group were then left sitting around while she answered computer questions. When she was working with the students on computer, other students would approach her with their individual concerns. Beth did not deflect them as Carol did, but instead allowed herself to be pulled in many directions. For instance, during one busy math period, when a student asked her how to spell a word to name her file, Beth spent several minutes guiding her and eventually helping her look up the word in the dictionary while other students floundered. In a similar vein, Beth felt strongly that the students needed to receive homework assignment every day except Friday. She would grab a sheet to give them that often did not correspond with the topic of the day or the way she was teaching it. This often required extra time to teach the material necessary for the homework sheet. Carol was more comfortable in letting days go by without assigning any homework assignments.

Beth felt more successful teaching the Sunken Ships unit. The unit guide gave explicit instructions for assigning off-computer activities, guiding discussions, managing off- and on-computer work (including printed directions for students), and assessing student work, as well as homework assignments segmented by days. Beth's earlier experience with that unit made her acutely aware that Turtle Math did not provide a similar structure. When asked if the unit would be a nice break for her this year, she replied:

Well, it's funny, because honestly, when I found out that that's what the other teachers were doing [the Sunken Ships unit], I found myself being a little envious of "Hey, look at all you got for this." And I was surprised at myself for feeling that way (Interview, April).

Another critical part of the answer tot eh question, "Who is in control?" revolves around the issue of who controlled the expertise related to this computer program in a classroom. Carol allowed and encouraged the students to become the experts on many features of the innovation. Instead of personally mastering every aspect of the software, she savored the complexities that the students uncovered on their own. Carol was comfortable handing over this control:

And I don't even know how to do the colors, to tell you the truth. Beth said that someone in her room today wants to know how they add color, and I said ask Mark. And that's the way it's been going. Anyone who wants to add color asks a kid (Interview, March).

Other teachers were no able to hand over that control. The innovation involved a complex computer environment, and most teachers felt that they needed to master every aspect of the environment before allowing their students access so that they would be counted on to be able to predict and troubleshoot every outcome. One teacher on the team had no computer experience and did not use this innovation despite significant pressure from administrators, colleagues, and students. When he was asked about the possibility of the students being the experts in a particular program, he replied, "I feel that the teacher should know more. But you are right that sometimes the students know more, and I'm embarrassed." For teachers lacking technological savvy, permission to share expertise could be a crucial distinction.

The innovation also required new means of assessment. Beth and Carol both expressed confusion about how to assess the learning that was taking place during Turtle Math activities. They felt that students were acquiring or contrasting concepts, but they as teachers were not able to specify which students were in need of assistance in a given area. Beth talked about the assessment she was accustomed to:

Last year, I could look at their papers, if they were all working on the same thing, I could easily go around and just pick up a lot easier on who was getting it. It was just much, much more controlled (Interview, April).

This was contrasted with the innovation where "I personally can't get to know which kids and what they're doing and what their strengths and weaknesses are."

When read these comments later, Carol agreed, punctuating each phrase with "yes, yes, and yes." She added:

When they are on the computer, it is very hard to go around and assess them, because you are so busy answering questions about the programming, and helping them design it, and I felt like I was missing finding out if they were getting the point of the activity (Interview, April).

On another occasion, however, she describes the value of the projects created by her students in helping her assess students on a higher level:

You really see the ones who have misconceptions. Whereas if they were doing a pencil-and-paper activity, they might have gotten it right. But when they really have to design a parallelogram and make the turtle draw a parallelogram, if they can't do it, you can see why they can't do it. What is there about a parallelogram that they don't understand? (Interview, February).

Teachers felt a responsibility to assess each student's level of functioning, to guide them to the next level, and to document that this had occurred. When they had time to probe a student's understanding, their assessment was enriched. Overall, though, given the constraints of their skills and environment, they believed that learning outcomes were less distinct and documentation less accurate than usual.

Carol tried to assess through journal entries in which students described their learning. She noted that journal entries were too vague most of the time, with summations such as: "I learned how to draw my design. The tricky part was getting it to do the top part" (Student Journal, March). She also tried to build Turtle Math portfolios for the students but was thwarted by a temperamental printer. She planned for the following year to circulate with sticky notes and a clipboard so that she can note the thinking and learning that she sees and record the progression for each child. She believes that there is a way to systematically assess this learning and is willing to put in the extra work to find it. Overall, these assessment concerns and techniques are similar to those that emerged during the genesis of whole-language instruction, when teachers "felt in their bones" that the students were learning but lacked specific ways to document and flesh out that feeling.

In summary, a teacher's comfort relinquishing control and his or her success in orchestrating changing classroom patterns was critical in determining their satisfaction with the innovation. This orchestration included the ability to coordinate activities, manage multiple groups of students, and take charge of mathematical learning goals, discussions, and assessment yet maintain students' freedom to explore and invent. Carol shared control deliberately and happily, while Beth was unable to negotiate a comfortable compromise. She believed that she needed to manage each student's learning at all times and could not fulfill this expectation while using Turtle Math. Sunken Ships provided more structure, which helped Beth and the other teachers adjust their teaching styles, albeit only for the duration of the unit.

4. What is the relationship between the innovation and the regular curriculum?

These teachers work in a district that had defined the fourth-grade mathematics curriculum as a set of six strands: (1) numbers and numeration, (2) operations on whole numbers, (3) geometry and measurement, (4) fractions, (5) decimals, and (6) probability/statistics. Each was assessed through a districtwide strand test. Teachers were able to choose their own textbooks and materials and when they would administer each test. Carol repeatedly referred to this freedom as a force encouraging her to experiment with innovations:

No one tells us that we have to use such and such a textbook. So that kind of forces us and encourages us to really go and examine what is out there and adapt it to our own use. And therefore you're always looking for something better. Whereas if someone says you have to use the Addison-Wesley textbook, you're like, "OK, I'll use that." Done. (Interview, October).

Further, even the stand tests were "on their way out," reported Carol to be replaced with a mathematics portfolio approach. A district Math Assessment Project Committee bulletin stated that teachers were not required to use the strand tests and were encouraged to develop their own assessments. Despite the relaxed standards, Beth, Carol, and the other teachers on this team followed the strands explicitly. They divided the curriculum up into the six strands and taught them in the traditional sequence. Beth explained:

We have the strands that we have to follow, and it's sort of like, there was a schedule even before I was born, and I've sort of gotten into that routine when I started teaching, I don't think it's so wonderful, and I don't know who came up with it (Interview, February).

Beth and Carol used the innovation not to supplant but to enrich the strands for their students, with differing results. Carol viewed her curriculum as a satisfying mix of sources and did not report much trouble integrating the innovation with her regular curriculum. When asked how she chose activities, she explained:

I'm just going a natural flow. I started out going with the Turtle Math guide, and we've basically been following that pretty well, and then I shoot off for a day or two into some activity [needed for the strand test] that I have done before (Interview, February).

She believed that she needed to use specific activities to prepare students for the routine computations on the strand tests. Even though she dedicated class time primarily to Turtle Math and other projects, Carol often gave "these crummy old worksheets" for homework to prepare students for the strand tests. This innovation fit into Carol's curriculum as another helpful resource as she worked to meet her two primary goals of getting the students through the strands and enriching their learning through higher-level projects.

Beth felt much more of a conflict between her regular curriculum and the innovation. She described her concern: "The [text] book is really nice, because it's fourth grade, all right there, I wanted Turtle Math all laid out for me the way the book is all laid out for me" (Interview, April). Beth lacked confidence in her ability to construct a learning plan without assistance and expressed great difficulty making connections between the innovation and the other elements of her instruction. She repeatedly requested assistance in linking her on-computer activities, off-computer activities, and homework assignments. She wanted a unified program and felt extremely frustrated by the lack of integration. Without the "rock" of her textbook, she found that she lacked the focus necessary to create a meaningful curriculum.

Beth and Carol expressed a shared belief that interacting with the innovation could be depriving their students of other curricular components, primarily those required for standardized tests, or for Grade-5 work. Carol voiced this conflict:

I feel that I look at the Turtle Math, and I know it's great stuff, but in the back of my mind, I'm always thinking, are they learning to multiply decimals? Are they learning the real core things that somewhere down the road are probably going to turn up on a standardized test. And they didn't learn it in Mrs. Jenkins' math class. It's always in the back of your mind (Interview, June).

The other teacher, committed to their primary goal of covering the strands, were also discouraged by what they perceived as an innovation to be used in addition to the regular curriculum.

5. Whom am I trying to please?

Teachers have many clients. In choosing when and how to use this innovation, teachers considered the input of administrators, parents, colleagues, and students. The principal at this school was perceived by the teachers as supportive of students working on computers for whatever purposes. Carol remarked, "She likes having our kids be on the cutting edge and keeping up with the times." Beth did not see her as particularly informed about the distinction among different computer applications. She explained:

That's the problem we're having with her right now. We have these IBM programs, and they're basically dittos on the computer, and she doesn't see the difference yet. She just looks at computer time and says, "Oh gosh, it's wonderful that computers are in the classroom, and the kids are using them" (Interview, October).

Parents of children at the school were perceived as extremely vocal, involved, and in favor of technology. Carol noted:

They love to see anything with computers. This is what they want to see. They want to see the computers in [the classroom]. They want to know that the children are learning the technology (Interview, October).

Here teachers portray the parents as in sync with the principal, supportive of technology only for its own sake. At least one parent, however, demanded deeper explanations of the innovation. Carol met with the disgruntled parent and described the outcome:

Once I started showing her how we were applying things, and I turned on the computer, she said, "Oh that's so much better; I'm so glad that Noelle is in your math class." She liked everything I was telling her (Interview, February).

Overall, the technological aspect of this innovation proved to be popular in helping teachers convince parents and principal that this was worthwhile. However, there was little support from these constituencies to encourage teachers to pursue this time-consuming innovation rather than simpler computer activities.

Teachers did not perceive this innovation as helpful in attaining another goal of interest to parents and administrators: raising test scores. Although district reform documents preached alternative assessment, these teachers still felt pressure to prepare their students to achieve high scores on traditional tests. California Achievement Tests were scheduled for early May, and for the weeks proceeding them, teachers did not want to even talk about innovation. Beth said, "I don't want Turtle Math until that pressure is off, and I can relax a little bit."

The strand tests also constrained instruction. For the geometry and measurement strand, there was one question on the test about symmetry and one question about coordinate geometry. Teachers were not very excited about innovative activities that would require days to explore a concept that received such short shrift on the test.

Teachers perceived the innovation as negatively impacting performance on the strand test. Beth notes, "For this last strand, if I wasn't doing Turtle Math, I think my scores would have been a little higher" (Interview, October). Carol also thought that the innovation reduced her children's test scores. When asked in June if her students got more or less out of math this year than they had in the past, she equivocated:

I cannot say whether they got more of less; I can say that they got different. When I looked at their standardized test scores, I didn't think they were really as good as they could be. I think some of the computation scores were down, and hopefully, next year, when they don't get into the top math group, parents won't come back to haunt me (Interview, June).

Teachers felt pressured by parents to rationalize the quarterly grades that were given to students, to have a row of objective test scores that they could show to parents at a conference time. As they were planning units and activities, this consistently emerged as a key concern, often overshadowing other pedagogical concerns. Beth explained, "You only have to be attacked once by a parent, and it's ruined you for years. These parents are managers at work and are used to telling people what to do" (Meeting Notes, October).

Balancing the need for high test scores was a district emphasis on embracing reform efforts. Carol describes the feeling:

Change, we're so used to it. I can't even say enough how. I mean, that's our mentality here: "Change, change, change. After15 years, aren't you supposed to feel like you know more?" … But my philosophy is, if you know it's coming, you don't resist it, you get on at the beginning and learn a little bit at a time, when you still have to be on your case (Interview, October).

This attitude encouraged Carol to put in the effort now with this innovation to stay ahead of the curve. Other teachers had learned to make judgments about which reforms worked for them. The teachers who had been teaching the longest best explained how they sift through reform efforts.

And those of use who have been around as long as I have know that so many theories in education come around, and they say, "Oh research shows that this works wonderfully," and then you try it for a few years, and then research shows that something else works wonderfully. So you don't go into it 100%, and you have to get the feel of what you feel comfortable with as a teacher (Mary, Interview, May).

Lately, change had been outpacing thoughtful planning in this district, and teachers were feeling the strain. Carol warned that "It's going to take all your best people who are willing to try things burning out" before the district would finally ease the pressure. The district people were not seen as fulfilling their responsibility to facilitate change in a thoughtful manner.

Students also pressure teachers to use the innovation. The innovation was first used in September for the high-ability math groups, and it soon became identified among the students as a special activity for the high-ability group and therefore enticing to all. Students intermingled during bus rides, lunch, classes, and specials. This gave students many opportunities to forge friendships, discuss mathematics instruction (including this innovation), and demonstrate innovative computer activities for each other. The intermingling also helped to create a cascade effect (Watson, Cox, & Johnson, 1993) where students were sharing their enthusiasm and knowledge with other students, increasing the efficacy of the innovation. Note that Watson et al. identified the cascade effect as involving teachers (and they noted that it did not occur frequently). In contrast, the cascade effect we observed was frequent and involved students. That is, knowledge and motivation spread from student to students rather than from teacher to teacher.

When Beth discussed the likelihood of other teachers implementing the innovation, she noted that "they see the enthusiasm of our math classes, and if they can replicate that, they're excited about that." Later, when Beth had discontinued her use of the innovation, the cascade turned on her. She reported:

I didn't do this last unit that Carol is doing, and the kids are bummed…. Because they keep very close tabs on both classrooms. And they see what the kids in Carol's math class are doing (Interview, May).

The students who were exposed to the innovation in their math classes were eager to show off their fellow teammates. This brought a level of expectation that made these students exceptionally receptive to the innovation and encouraged teachers to implement, and to feel regret if they did not continue, the innovation.

Throughout the year, university researchers frequently visited these classrooms, another constituency demanding attention. Gangha, a doctoral student, was conducting a case study were she observed two pairs of students in Carol's classroom almost daily throughout the year. Gangha and Carol eventually established productive team relationship. During interviews, Carol always talked about teaching decisions and activities in terms of "we," referring to Gangha and herself. She also indicated that she was more fully able to implement the innovation while she had Gangha's support.

Beth had a variety of personnel in her classroom, some studying the thinking processes of the students, others collecting field notes on her use. She often appeared disconcerted by our presence, but also noted that is was helpful in general to have university people in the room. She said:

I have learned so much form you, even though your idea of coming here wasn't to teach me how to teach math or anything. But still, it just made me look at how I'm teaching in a different way, having another professional in the room. I think sometimes you can take it for granted what you're doing, where suddenly if there's another professional there, it's like, "Whoa. Why am I doing this?" (Interview, May).

Carol and Beth also referred to the second author's impact on the way that they as teachers regarded children's thinking. During the early days of the innovation, he repeatedly mentioned that the research team wanted to focus on children's thinking and what happens with children's thinking while using the innovation. At first, teachers weren't sure what this meant. The first time that Carol was observed, she was carrying around a yellow pad and writing notes on it. When asked what she was writing, she answered anxiously, "Just the names of the groups; do you want me to write about their thinking?"

By the middle of the year, Carol observed, "That is where working with you guys is really helping. Because it's forcing me to focus more on their thinking instead of their answers. It really is." She described several episodes where the cognitions of the students were interesting and informative to her as a teacher. In June, she commented:

The advantage of this and working with you is that it has gotten me to focus on how kids think, because a teacher's goal is more, how do I present this material, and did the child get the material. So I've tried to focus, when I walk around the room, to listening more, as opposed to talking more. And listening more to what the children are saying, and how they're perceiving it, because that gives you a great insight into how they learn (Interview, June).

Discussion with university personnel underscored Carol's experiences with the innovation and enabled her to refine her teaching style and develop greater insight into student learning. For Beth, our presence eventually served mainly to reinforce her discomfort with the way the curriculum was working in her classroom.

Relationships with colleagues and family members also influence teacher's answers to the question of whom they are serving. Prior to this year, Beth and Carol did all their math planning together, and gave the strand tests on the same day. At first, Beth and Carol were planning to implement this innovation together. This year, however, Beth had a new baby at home, and she was not able to spend the time after school collaborating with Carol in their usual style. By October, it was clear that the two teachers had to split paths. Carol offered an explanation:

With team planning, you end up doing things together, and then you might step back and say, "that's not my style." Beth can be so structured. But I like to sometimes deviate. I guess I'm more of a bird walker than she is. We have taken totally different direction, and we finally agreed between ourselves that we've got to do that. It was best for us to just follow the track that each of us was on (Interview, May).

Both teachers grieved this loss. They missed the relationship that they were accustomed to sharing. At the end of our weekend research meetings, they were the last to leave, eager to spend a few minutes together chatting. In time, it became apparent that Carol's use of the innovation was much more successful that Beth's. Her students were generating exciting projects, and the research team was seeking copies for publication. This contrast was painful for Beth. Early in the year, Beth and Carol exchanged pointers about their use of the innovation. Midyear, they acknowledged their different styles and amicably split paths. By the end of the year, Beth avoided discussions about Turtle Math and stopped attending meetings.

This innovation was both a casualty of and a cause of friction between Beth and Carol. The new baby kept these two teachers from reinstating their accustomed relationship during this school year, a relationship that, under other circumstance, might have served to facilitate the innovation for both teacher through the exchange of ideas and experiences. Eventually, Carol was forced to break free of any semblance of this accustomed partnership to accomplish the things that she wanted with the innovation.

Overall, the question of whom a teacher serves is a complex one. The teachers in this study considered parental and district emphases on using technology, increasing test scores, and embracing reform. They balanced this with student affinity for the innovation and pressures from their colleagues.

CONCLUSIONS AND IMPLICATIONS

Teachers in this study struggled to answer vital questions guiding their implementation of this innovation. Educators and reformers can study the answers generated to gain insight into the reform process. When these teachers asked themselves which students should use an innovation perceived as an enrichment activity, high-ability students immediately came to mind. Although this encourages teachers of such students, we may need to reconceptualize enrichment and educational reform to include an emphasis on how innovation can be useful for students at all ability levels. In this study, students identified as low ability experienced great success with the innovation and were eager to take responsibility for their own learning, which exposed their teachers to capabilities that they did not know their students possessed. Their teachers believed, however, that there was little room for more such activities, especially considering their student's ability level.

When teachers struggle to find ways to foster student's mathematical constructions, they are often urged to facilitate open-ended discussions. As Beth's experience demonstrates, this can be difficult, especially when a teacher lacks the epistemological assumptions that underlie this pedagogy. Teachers might be better encouraged to find ways to pose tasks, observe, and redirect, the model used successfully by Carol with both computer activities and off-computer activities such as classifying shapes and playing "What's My Rules?"

When teachers ask themselves who is in charge in a classroom, they are asking about who is planning, guiding, and assessing the learning. They are also asking who should be the expert when using computers. If teachers are able to share this control over the content and flow of lessons, remarkable learning can take place. This requires innovative methods of "student watching" and assessment procedures, methods even the most proficient teacher in this study found difficult. Reformers can assist teachers in developing strategies for facilitating multiple simultaneous activities, controlling general mathematical learning goals (albeit, often indirectly), and assessing learning while reassuring teachers that not all learning must be under the teacher's control. When teachers encourage students to become the experts in a computer innovation, students are empowered to be problem solvers, and teachers are freed to use software in the classroom without excessive preparation time.

The match between an innovation and the "regular curriculum" must be explored, especially if the regular curriculum is validated through high-stakes testing. The teachers in this study perceived the innovation as a layer on top of the regular curriculum and not, as the developers intended, as a way of learning the regular curriculum in innovative ways. One implication is that teachers need more explicit freedom from existing curriculum and testing requirements if they are to be free to explore the value of an innovation.

The final question again reveals the complexity of a teacher's job. When teachers ask whom are they to please, administrator, parents, students, other teachers, and, in this care, researchers, all vie for attention. Administrators discouraged this innovation through their emphasis on high test scores and their mondiscriminating support for the use of computers. Parents encouraged this innovation for students in the high ability groups as evidence of their enriched education. Students encouraged teachers to use this innovation as they sought the high-status, engaging computer activities and showed evidence of meaningful learning. Teachers need to weave together the support of all these groups in validating their curriculum choices. Reformers should be aware of these complex factors and the resulting questions teachers ask themselves and strive to aid teachers in formulating viable answers.

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Acknowledgements:

Time to prepare this material was partially provided by National Science Foundation Research Grant NSF MDR-8954664, "An Investigation of the Development of Elemenary Children's Geometryic Thinking in Computer and Noncomputer Environments." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Author Info:

Julie Henry is an assistant professor in the School of Education at the State University of New York College at Fredonia. Her research interests center on teacher beliefs, education change, and early childhood education. Douglas H. Clements is a professor of mathematics and computer education at the State University of New York at Buffalo. His research involves the learning and teaching of geometry, computer applications in mathematics education, the early development of mathematical ideas, and the effects of social interactions on learning. (Address: Julie Henry, State University of New York College at Fredonia, Thompson Hall, Fredonia, NY 14063; Henry@ait.fredonia.edu.)

Other Articles by Julie Jacobs Henry:

Network of Influences in an Implementation of a Mathematics Curriculum Innovation

Other Articles by Douglas H. Clements:

7 Ways to Add Math to Everyday Play

Analyzing Children's Length Strategies with Two-Dimensional Tasks: What Counts For Length?

Building Blocks of Early Childhood Mathematics PDF

Building Blocks for early childhood mathematics PDF

Building Blocks for Young Children's Mathematical Development PDF

A Case for a Logo-Based Elementary School Geometry Curriculum

Computers in Early Childhood Mathematics PDF

Computers and Mathematical Assessment

Computers Support Algebraic Thinking