# Teachers explore linear and exponential growth: spreadsheets as cognitive tools (best paper award from site 2004).

This article addresses some of the issues relevant to the cognitive
goals of information and communication technology (ICT) integration in
the mathematics classroom. It focuses on the development of conceptual
understanding through multiple representations. Specifically, it informs
about a group of middle school mathematics teachers' learning and
teaching about linear and exponential growth in a technology-oriented
environment. A particular focus of the professional development was
two-dimensional: (a) deepening teachers' understanding of linear
and exponential growth through technology-based representations, and (b)
providing effective context for students' learning from the same
technology-based representations, considering the fact that they do not
have teachers' standard representations in their toolbox. We
describe exploration of exponential and linear growth through
spreadsheets and graphing calculators, grounded on a rich, open-ended,
real-life problem. In addition, we report on lessons learned in attempts
to explicate some bridges between classical and technology-based
representations.

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In the growing intricacy of learning and teaching in the technology-oriented environment, the teachers, not the technological tools are the key to success of mathematical classrooms. Their own perspective on the nature of mathematics, their technological stance, and the training they receive determine their effectiveness in the infusion of technology in mathematics teaching/learning (Kaput, 1992; National Council of Teachers of Mathematics (NCTM), 2000).

Teachers who teach for understanding situate the classroom atmosphere for the kinds of inquiry in which students engage. Learning for understanding recognizes understanding through a "flexible performance criterion" (Perkins, 1993). Perkins identified six priorities for these teachers: (a) make learning a long-term, thinking-centered process, (b) provide for rich, ongoing assessment, (c) support learning with powerful representations, (d) pay heed to developmental factors, (e) induct students into the discipline and (f) teach for transfer. In a technology-oriented classroom opportunities are ample. The challenge remains for many teachers to "keep-up" with technology. Therefore it is important to facilitate teachers' learning with technology-based tools that are readily available and such that teachers can build new understandings based on standard representations that they already possess.

Findings reported here are part of research carried out during implementation of the professional development grant for middle school mathematics teachers, BRIDGES: Connecting Mathematics Teaching, Learning and Applications, supported by No Child Left Behind Funds (Neal, Alagic, & Krehbiel, 2002; Alagic, Krehbiel, & Palenz, 2003). The twofold goal of this study was to explore how available cognitive tools can (a) deepen teachers' understandings of linear and exponential growth through ICT-based representations, and (b) provide for students' learning from the same technology-based representations.

The first section of this article focuses on teaching mathematics for conceptual understanding via multiple representations in the technology-based environment. The cognitive tools are introduced in the context of designing multiple representations. The second section explores middle school mathematics teachers' challenges in enriching their conceptual understandings of linear and exponential growth using different ICT-based cognitive tools (e.g., spreadsheets). We describe exploration of exponential and linear growth through spreadsheets and graphing calculators, providing necessary details about the activities. In the final section, we report on lessons learned in attempts to explicate some bridges between classical and technology-based representations.

TEACHING FOR CONCEPTUAL UNDERSTANDING IN TECHNOLOGY-ORIENTED ENVIRONMENT

Teaching for Understanding: Mathematics

Perkins' six priorities (Perkins, 1993) for teachers who are teaching for understanding in technology-oriented classrooms provide an organizational framework. Within this framework an ongoing thinking-centered process must be arranged in order for the learners to think for a while with and about the ideas they are learning. Otherwise they are not likely to build up a flexible inventory of representations to support their understanding. Ongoing assessment is an integral part of learning for understanding. Students need clear goals, feedback, prompts if necessary, and opportunities for reflection. To support students' performances, the teacher has a complex task of finding appropriate representations or facilitating students' formation of those. To understand facts, concepts, and procedures from mathematics, students need experiences that clarify the way of mathematical thinking; how the discipline works; how one justifies, explains, solves problems, and manages mathematical inquiry. To apply in diverse settings an understanding that they formed, students often need guidance to make the connections and to build their skills for transfer.

Conceptual Understanding: Multiple Representations

Two main orientations to mathematics, calculational and conceptual, emerge from the images and beliefs that teachers have about the nature of mathematics and consequently about their teaching strategies. Similarly, students also have varying levels of these orientations. Conceptual approaches encourage learners to solve problems by working from their own understanding (Thompson, Philipp, Thompson, & Boyd, 1994). Learning for understanding requires acquiring the ability to represent the concept in multiple ways, in addition to interrelating among the multiple representations to accomplish learning goals (Lesgold, 1998). The teaching-for-conceptual-understanding-through-multiple-representations framework presents an open-ended structure for supporting that process. In technology-oriented environments, the variety of representations available between the concrete and the abstract is getting richer (Alagic, 2003). This article provides a couple of examples that support this claim.

Well-selected representations can provide an access point for students' deeper understanding of mathematical phenomena. The importance of students' exploration, in both selecting representations and making connections among them when building conceptual understanding of mathematical ideas, surfaces in studies about multiple representations as a tool for meaningful learning (Greeno & Hall, 1997). To encourage learners' inquiry, a teacher has to take into consideration opportunities that representations provide. Scaffolding through multiple representations may bridge the gap between concrete and abstract representations of a mathematical concept and reach a reflective abstraction in a variety of ways. For example, a potentially abstract problem statement may be illustrated using some kind of visualization. Students building their own ICT representations, and translations among them support their conceptual orientation in learning. New representations are essential components of a learning environment in which learners are required to think harder about the topic being studied and to generate ideas that would be impossible without these new representations. This creative thinking supports learners in creating and sustaining their conceptual orientation (Alagic, 2003).

Cognitive Tools: Spreadsheets

Derry (1990) defined cognitive tools as both mental and computational devices that support, guide, and extend the thinking processes of their users. Jonassen (1992) described them as "generalisable tools that can facilitate cognitive processing" (p. 2). Cognitive tools can make it easier for learners to process information, but their main "goal is to make effective use of the mental efforts of the learner" (Jonassen, 1996, p. 10). They are based on the principle that learners need to create their own understanding of new concepts. These tools give learners a way (often visual) of representing their understanding of a new concept/phenomenon and how it relates to their existing understanding of the same idea. Graphing calculators and spreadsheets as cognitive tools for creating representations are often used for exploring mathematical phenomena and building conceptual understanding of mathematical ideas.

Vockell and van Deusen (1989) described spreadsheets as tools using sets of rules that involve users in designing the rules. Spreadsheets as a tool for developing conceptual understanding of mathematical phenomena support storing data, calculating, and representing information. Data, most often numerical, is stored in a matrix-like collection of cells. The numerical contents can be manipulated in a number of different ways, starting from simple calculations and function applications to generating tables and a variety of graphical representations for stored data.

Encouraging students to manipulate spreadsheets improves their understanding of the concepts, relationships, and procedures. Beare (1992) describes spreadsheets as "flexible mindtools" for representing and manipulating data, reflecting on and speculating with quantitative information, and making the underlying mathematical reasoning explicit. The calculational and graphical capacities of spreadsheets provide a context to engage students in analyzing and connecting multiple representations. They support students' engaged, problem-based learning and open-ended investigations. They are valuable for answering "what if" questions. Considering implications of alternatives is a valuable approach to engaging higher-order thinking (Sounderpandian, 1989).

In one of the studies investigating spreadsheets as cognitive tools, Sutherland and Rojano (1993) investigated how prealgebra students could use spreadsheets to represent and solve algebra problems. The study was conducted simultaneously in Britain and Mexico and took place over a five-month period. During that time, students moved from a cause-effect local numerical notion of algebraic relationships to general rule-guided relationships that could be symbolized both in the spreadsheet and in algebraic notation. Baxter and Oatley (1991), while comparing the effectiveness of two different spreadsheet packages, found out that, not surprisingly, the users' prior experience level with spreadsheets was far more important to learning than the usability of the software package. These studies provide a few insights about the effectiveness of spreadsheets as cognitive tools. From an educator's point of view, based on teaching practice, teachers' understanding and use of spreadsheets can extend from kindergarten up, through all the grade levels, from a simple four-pane magic square, through organizing students' records/grades, to the complex applications or data analysis supported by spreadsheet software as a cognitive tool (Alagic, 2003).

IDENTIFYING THE PROBLEM: GRAPHING REAL-LIFE SITUATIONS

Context

The philosophical foundation of BRIDGES (Neal et al., 2002; Alagic et al., 2003) is based on a view of mathematics as a humanistic, socially constructed discipline and way of thinking that implies the learning and teaching of mathematics through inquiry (Heaton, 2000). Based on current research findings about successful professional development models and best teaching practices, the data available, and in collaboration with the school district personnel, the BRIDGES evidence-based professional development model is a carefully constructed interplay of two models reflecting (a) practice-based professional development connecting professional development activities of teachers to the actual classroom work of teachers (Smith, 2001), and (b) immersion in inquiry focusing on teachers' exploration of real-life problems that challenge their (mathematical) reasoning (Loucks-Horsley, Hews-on, Love, & Stiles, 1998; Loucks-Horsley & Mastsumoto, 1999). By situating teacher learning "in practice," teachers engage in activities that are at the heart of their daily work, and through guided inquiry into the standards-based school mathematics curriculum, their mathematical knowledge and understanding is deepened and widened.

BRIDGES is based on the belief that teachers must become and continue to be mathematics learners, if they are going to teach for understanding. This happens when they are challenged at their own level of mathematical competence and when their learning experiences are based on the same pedagogical principles that they are expected to implement with pupils. As Schifter and Fosnot (1993, p. 26) pointed out: "perhaps more important for [the teachers] than their investigation of any specific content area is the process of active self-reflection. By analyzing together their experience of the just-completed mathematics activity, teachers begin to construct an understanding of how knowledge develops and the circumstances that stimulate or inhibit it." BRIDGES project delivery is grounded on the principles of effective professional development (Loucks-Horsley et al., 1998; Loucks-Horsley & Mastsumoto, 1999; Smith, 2001).

The BRIDGES web site (http://education.wichita.edu/alagic/bridges/BRIDGES.htm) serves as an up-to-date report of project activities and accomplishments. It is regularly updated and maintained by the project team. Pages include information about ongoing activities, materials presented at the BRIDGES workshops and institutes, materials generated by BRIDGES teachers for use in middle school classrooms, and other related resources. In the context of this report the web pages should be regarded as supplemental evidence.

The overarching aim for the BRIDGES participants is to develop a deeper understanding of how to integrate standards-based school mathematics into the larger world of mathematics, including its applications in real-world settings. Teachers should know school mathematics at a deep level and should have an understanding of its place in the real world (Heaton, 2000; NCTM, 2000; NRC, 2000). For the proposed three-year BRIDGES grant cycle the following goals have been established: Student's Performance Goal, Teacher's Quality Goals, and Mathematics Leadership and Learning Community Goals. Teacher's Quality Goals focus on two tasks: (a) Improve the mathematics knowledge of all participants. (b) Equip teachers and paraprofessionals to be able to deliver effective mathematics instructions to ALL students using a variety of instructional strategies and approaches (Alagic et al., 2003).

Teachers had opportunities to get graduate credit hours for the BRIDGES workshops and summer institute, Mathematics Inquiry I, II, and III. Pre, Mid and Post tests of mathematical and pedagogical content knowledge, focusing on the mathematical concepts studied in BRIDGES, have been administered to participants at the beginning and end of the Summer Institute and after the Fall Workshop.

Problem Triggers

There were two explicit situations that initiated the sequence of activities, reflections, and analysis of subsequent investigations that are presented in this article. The first one had to do with an assignment of making up real-life situations to match graphical representations presented to teachers. The second one comes from a middle school classroom, as reported by the teacher, and had to do with graphing collected data.

An activity was planned to assess teachers' understanding of graphs, after a sequence of graphing activities using graphing calculators and spreadsheets. Preceding this activity, some real-life problem situations were modeled by graphing, and motion-ranger and calculator-based labs as hands-on experience were provided for a couple of activities. There was no previous discussion or preparation for this specific activity. Teachers got some sample graphs on paper and worked in small groups to create stories that would correspond to graphical representations: increasing and decreasing linear functions and exponential growth and decay. Each group was presented with a different graph and asked to make up a story to go with their graph. The discussion appeared lively and it did not look as if anyone needed guidance or support. But, when teachers shared their stories, it was obvious that stories were the simplest possible cases of time-distance graphs. While the stories always accounted for the increasing or decreasing aspects of the graph, they did not take into account other mathematically observable features, such as the rapid acceleration of an exponential curve. Without sufficient mathematical experience, all graphs and data sets were assumed to be linear. Yes, we could say that the teachers had not been challenged to do anything more complex, but the fact remains that none of them tried to reach for a challenge.

The closing assignment of BRIDGES Year One for participating middle school teachers was to implement in the classroom some of the real-world problem-solving activities which had been explored in the BRIDGES Summer Institute. Our real-world problems have been taken from Mathematics Modeling Our World, a high school National Science Foundation (NSF) supported curriculum (Garfunkel, Godbold, & Pollak, 1998). At our next session (beginning BRIDGES Year Two), in connection with an environmental study, one participant reported on student projects which involved analyzing and graphing exponential growth. All students had correct data, indicating exponential growth, but all of the graphs they developed pictured linear growth. While the students could identify their data as increasing, they were not familiar with, and therefore unable to distinguish, the shape of an exponential growth curve. The participating teachers did not readily identify the mistake, indicating that they, too, were not experienced with making the necessary distinctions between different sorts of increasing graphs. Someone could argue that this is not surprising, since exponential growth and related topics such as geometric sequences are not included in NCTM standards until grades 9-12, but distinguishing linear from nonlinear graphs should come with the introduction of the concept of "linear" in middle school grades. Also, middle school students learn to use exponents to represent repeated multiplication, so data representing exponential growth is easily introduced.

POSSIBLE SOLUTIONS

The goal of the study was to explore how available technology-based tools can (a) deepen teachers' understandings of linear and exponential growth through technology-based representations, and (b) provide for students' learning from the same ICT-based representations, considering the fact that students do not have the teachers' (standard) representations in their toolboxes. By introducing teachers and students to multiple examples and using different available technologies to contrast linear results with exponential results, conditions were provided for them to see additional aspects of their own and others' graphs. Specifically, this approach was used to review these concepts with middle school teachers participating in the BRIDGES grant for Year Two. Then they applied similar techniques to help their students learn about these new relationships. Students' projects were shared and compared to study/test their ability to distinguish different types of growth.

The remainder of the article is organized in two parts. The first part describes activities facilitated in an attempt to clarify possible misconceptions about exponential and linear growth and corresponding graphical representations. The second analyzes teachers' reflections and lessons learned. The BRIDGES participants had studied the following real-world environmental problem in the summer and then it was used with middle school students in the fall, resulting in data showing exponential growth, but graphs that looked very linear.

Moose Return (Garfunkel et al., 1998)

In 1861, the last moose in New York's Adirondack State Park wilderness area had been shot. Years later, in 1980, some moose began to reappear in the six-million-acre park. By 1988, experts estimated that 15 to 20 moose were in the park. By 1993, this number had increased to between 25 and 30 moose. A survey conducted by the New York State Environmental Conservation Department (ECD) found that the public favored an increase in the moose population. The moose gradually migrated from Canada and New England States. The numbers of moose also increased through natural reproduction. The ECD determined that a plan to move 100 moose into the park over a three-year period would cost $1.3 million.

The first activity encourages students to engage with this moose population question: Pretend you were the commissioner of the ECD, and you were responsible for making a recommendation to the governor about your findings.

1. "Pose a specific question concerning these data to which you would like an answer."

2. "What additional information do you need in order to be able to answer your questions?"

Further consideration of the given data leads to creation of mathematical models for growth of the moose population. The first model proposed is linear, assuming that the population changes only by migration of new male moose into the park at a steady rate. Analysis of this model generates questions as to its validity for the known and predicted behavior of moose. Further information leads to an exponential growth model.

After identifying the graphing challenges, we used the book, The King's Chessboard, as a springboard with the teacher-participants, to review exponential growth and study the related graphs. The story is summarized as follows.

The King's Chessboard (Birch, 1988)

Once long ago in India a wise man was summoned by the King of Duncan to appear before him. The King told the wise man that he had served him well and that he would like the wise man to choose a reward. The wise man did not want a reward, but the King insisted and the wise man finally made a decision. He asked that he be given rice each day for the 64 squares on the King's chessboard. For the first day, he wanted 1 grain of rice, for the second day he wanted 2 grains of rice, and for the third day 4 grains of rice. Each day the rice would double for each of the squares on the chessboard. The King granted the wise man's request after ignoring (out of pride) the impulse to question the implications of what seemed to be such a simple process.

Days later the Grand Superintendent saw workers carrying large sacks of rice. One grain became two and then four; grains became ounces; ounces became pounds; a bag had become two bags; and today it was four sacks each weighing one hundred twenty-eight pounds. The Grand Superintendent informed the King and the King summoned his royal mathematicians. He soon realized that there was no mistake and that he had promised the wise man two hundred seventy-four billion, eight hundred seventy-seven million, nine hundred six thousand, nine hundred forty-four tons (274,877,906,944) of rice. The King knew there was not enough rice in his kingdom to fulfill his promise and the next morning the King summoned the wise man to the palace.

The King asked the wise man how he could be satisfied. The wise man replied that he already was satisfied and originally did not want a reward from the King. The wise man returned to his simple home and quiet life. He served the King many times afterward, and the question of a reward never again arose. The King ruled wisely and justly for many years, and to the end of his days he never forgot the wise man's lesson-how easy it is for pride to make a fool of anyone, even a King.

After some hands-on preliminary exploration of the number of grains of rice in a given volume, we used Excel on laptop computers to consider the quantity of rice. Teachers were easily able to represent the amount of rice the king had promised to give the wise man each day. They made a table listing days and number of grains of rice. Then they made a graph of the data (Figure 1). Using the technology made it possible to explore the data quickly: What does the graph look like if we zoom in on it? What if the king tripled the number of grains of rice each day? What does the vertical scale have to do with the appearance of the graph? (Alagic & Palenz, 2004a; 2004b).

"Zooming" is a popular technological tool to explore graphs. These graphs gave us an opportunity to consider the advantages and disadvantages of zooming. Teachers were able to see that as we zoom in and out on an exponential graph, its shape remains essentially the same--with a relatively flat part curving into a steep rise. Also, as they zoomed out, the details and sharp rise of a section of the graph were lost--variations disappearing in the loss of detail which happens with zooming out. A follow-up activity involved beginning with a similar data table and graph, either in Excel or on a graphing calculator, and then trying to make the graph appear to have a steeper rise or a flatter section by changing the window (by zooming in or out of the graph). The lesson here is two-fold: zooming can reveal hidden detail in a graph or zooming can disguise the shape of a graph. Teachers who had not distinguished between linear and exponential growth in graphing in our previous sessions recognized differences and expected students could also comprehend them. They seemed eager to involve their students in a similar experience (Alagic & Palenz, 2004a, 2004b).

The teachers' stories and the thinking behind the creation of these stories. Teachers generated their own stories and activities to make exponential growth ideas meaningful for their students. Several of the stories introduced linear comparisons to give more impact to the examples of exponential growth. These were generally described in terms of choosing between a linear allowance or salary and one which pays 1 cent the first day and doubles each day for a period of time, usually a month or less. Other examples included spreading a rumor if each one who has heard the rumor tells one other person each day, and counting layers of paper as a large sheet is folded in half repeatedly.

Another teacher-participant reported considering exponential decay in the 6th grade classroom by cutting a sheet of paper in half repeatedly. Another 7th grade teacher used this exercise to help students think about radio-active decay and toxic waste. When the paper gets too small to cut, students can begin to understand that the pollution (or whatever) won't be entirely gone, although it may get too small to be measured.

Teachers commented on these stories as engaging for students. They reported on students surprise and puzzlement when realizing the meaning of exponential growth. Some teachers connected it to their own surprise, "Personally without knowing anything about these problems I would also go with the $5 a day instead of pennies-option." It was noticeable that teachers were trying to bring a note of humor in their classrooms by emphasizing the unexpected and impossible. One teacher reflected, "I will increasingly try to inject humor into the problems to make the experience more enjoyable." The teacher who earlier in the year had accepted linear graphs from students studying the exponential growth data of the moose return problem, later had his students consider the possibility of an exponential allowance. He reported that the graphs the students drew to illustrate their hoped-for allowance correctly showed exponential curves. Although there was no public recognition of his own (and others) misconceptions, the overall BRIDGES environment appeared more open for experimenting and risk-taking while exploring mathematical concepts.

As a follow-up to the King's Chessboard problem and to make large numbers meaningful, we asked teacher-participants to respond to the claim of the wise man at the end of the story that there is not enough rice in the world to fulfill the promise. We asked if that was accurate: how much rice is it in terms we can comprehend? The teachers came up with a variety of measurement comparisons. The volume of rice was compared to the volume of a school bus, a railroad boxcar, the Grand Canyon, or the entire earth. One group estimated the length of train needed to transport the total amount of rice by comparing the length to how many times around the Earth the train would wrap. The grains laid end-to-end were compared to the distance from the earth to the sun. The variety of comparisons and visual representations indicates the broad possibilities of integration of this estimation activity into a classroom context, which was explicated by participating teachers. After our workshop estimation efforts, we noticed that teachers began making more use and reference to estimations in other activities with their students.

DISCUSSION AND IMPLICATIONS: TECHNOLOGY-BASED MULTIPLE REPRESENTATIONS

Teachers need successful experiences and ongoing pedagogical and technological support when integrating technology into their curriculum. Also, they need opportunities to experience and do mathematics in environments supported by diverse technologies (Dreyfus & Eisenberg, 1996). Empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing conceptual understandings and communicating about mathematics is at the heart of BRIDGES professional development. Throughout the BRIDGES activities, concrete experiences have been provided to explore technology-based representations of data, graphs, and functions.

Cognitive tools allowed teachers to recognize the differences among linear and nonlinear graphs and with some guidance, their vocabulary expanded to allow for descriptions and discussions about these differences and features.

The ICT tools provided for (a) visual and graphical multiple representations interconnected with appropriate simulations, (b) meaningful explorations of a variety of cases in a smaller amount of time than if standard representations had been used, and (c) a nurturing learning environment supporting Perkins priorities, such as making learning a thinking-centered process, providing for rich, ongoing assessment, supporting learning with powerful representations, and teaching for transfer.

Many teachers will not share cognitive tools with students until they are confident that they can deal with "how-to" questions and guide students successfully. Teachers taking their new knowledge to the classroom need opportunities to practice with it and to reflect on both their lessons and students' experiences with new tools. BRIDGES provided a supportive learning environment and forum for sharing and reflecting about successes and challenges.

Being proficient with cognitive tools takes time. This investment makes sense only if the learning goals accomplished are proportional to the effort. Therefore in the technology-oriented context, selection of a problem becomes even more relevant. We spent a significant amount of time exploring exponential growth (introduced during one-day workshop, reported on in another and explored over a couple of days of a summer academy). Recognizing the need and keeping with our beliefs that deep understanding of one concept brings more value than "inch deep, a mile wide" curriculum, we revised our original curricular plans to accommodate for this situation. The rich, open-ended, real-life problem, Moose Return, was a worthwhile task for BRIDGES participants. It provided a starting point for a sequence of explorations, both in terms of related real-life problems and mathematical abstraction. The King's Chessboard allowed for exploring new representations of the concepts studied. Teachers' stories provided multiple ways of collecting, representing, and interpreting data using spreadsheets. All these activities supplied opportunities for deepening participants' conceptual understanding of exponential growth and their ability to distinguish between linear and nonlinear growth/graphs and to carry these ideas much further. It was rewarding to recognize how opportunities for modeling, simulation, mathematical abstractions and transfer (e.g., a packaging problem was brought into the discussion) unfolded. As we already mentioned, teachers began making more use and reference to estimations in other activities with their students. Another interesting example of transfer, while using dynamic geometry software for some geometry activities, some teachers attempted to model with dynamic geometry tools "halving a sheet of paper" to represent exponential decay.

What is the final conclusion? This is our story about a good investment. We diverted from our original plans to invest time in dealing with teachers' misconceptions about exponential versus linear growth. We used ICT tools that we originally planned to use, but invested more time and resources to facilitate the learning process in this unanticipated direction. Because BRIDGES provided a risk-free environment and encouraged peer support, teachers' creativity was allowed to emerge in using available tools and developing curriculum appropriate for their classrooms.

References

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MARA ALAGIC AND DIANA PALENZ

Wichita State University

Wichita, KS USA

mara.alagic@wichita.edu

palenz@math.wichita.edu

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In the growing intricacy of learning and teaching in the technology-oriented environment, the teachers, not the technological tools are the key to success of mathematical classrooms. Their own perspective on the nature of mathematics, their technological stance, and the training they receive determine their effectiveness in the infusion of technology in mathematics teaching/learning (Kaput, 1992; National Council of Teachers of Mathematics (NCTM), 2000).

Teachers who teach for understanding situate the classroom atmosphere for the kinds of inquiry in which students engage. Learning for understanding recognizes understanding through a "flexible performance criterion" (Perkins, 1993). Perkins identified six priorities for these teachers: (a) make learning a long-term, thinking-centered process, (b) provide for rich, ongoing assessment, (c) support learning with powerful representations, (d) pay heed to developmental factors, (e) induct students into the discipline and (f) teach for transfer. In a technology-oriented classroom opportunities are ample. The challenge remains for many teachers to "keep-up" with technology. Therefore it is important to facilitate teachers' learning with technology-based tools that are readily available and such that teachers can build new understandings based on standard representations that they already possess.

Findings reported here are part of research carried out during implementation of the professional development grant for middle school mathematics teachers, BRIDGES: Connecting Mathematics Teaching, Learning and Applications, supported by No Child Left Behind Funds (Neal, Alagic, & Krehbiel, 2002; Alagic, Krehbiel, & Palenz, 2003). The twofold goal of this study was to explore how available cognitive tools can (a) deepen teachers' understandings of linear and exponential growth through ICT-based representations, and (b) provide for students' learning from the same technology-based representations.

The first section of this article focuses on teaching mathematics for conceptual understanding via multiple representations in the technology-based environment. The cognitive tools are introduced in the context of designing multiple representations. The second section explores middle school mathematics teachers' challenges in enriching their conceptual understandings of linear and exponential growth using different ICT-based cognitive tools (e.g., spreadsheets). We describe exploration of exponential and linear growth through spreadsheets and graphing calculators, providing necessary details about the activities. In the final section, we report on lessons learned in attempts to explicate some bridges between classical and technology-based representations.

TEACHING FOR CONCEPTUAL UNDERSTANDING IN TECHNOLOGY-ORIENTED ENVIRONMENT

The word does not enter the utterance from a dictionary, but from life, from utterance to utterance. Bakhtin & Medvedev (1991, p. 122)

Teaching for Understanding: Mathematics

Perkins' six priorities (Perkins, 1993) for teachers who are teaching for understanding in technology-oriented classrooms provide an organizational framework. Within this framework an ongoing thinking-centered process must be arranged in order for the learners to think for a while with and about the ideas they are learning. Otherwise they are not likely to build up a flexible inventory of representations to support their understanding. Ongoing assessment is an integral part of learning for understanding. Students need clear goals, feedback, prompts if necessary, and opportunities for reflection. To support students' performances, the teacher has a complex task of finding appropriate representations or facilitating students' formation of those. To understand facts, concepts, and procedures from mathematics, students need experiences that clarify the way of mathematical thinking; how the discipline works; how one justifies, explains, solves problems, and manages mathematical inquiry. To apply in diverse settings an understanding that they formed, students often need guidance to make the connections and to build their skills for transfer.

Conceptual Understanding: Multiple Representations

Two main orientations to mathematics, calculational and conceptual, emerge from the images and beliefs that teachers have about the nature of mathematics and consequently about their teaching strategies. Similarly, students also have varying levels of these orientations. Conceptual approaches encourage learners to solve problems by working from their own understanding (Thompson, Philipp, Thompson, & Boyd, 1994). Learning for understanding requires acquiring the ability to represent the concept in multiple ways, in addition to interrelating among the multiple representations to accomplish learning goals (Lesgold, 1998). The teaching-for-conceptual-understanding-through-multiple-representations framework presents an open-ended structure for supporting that process. In technology-oriented environments, the variety of representations available between the concrete and the abstract is getting richer (Alagic, 2003). This article provides a couple of examples that support this claim.

Well-selected representations can provide an access point for students' deeper understanding of mathematical phenomena. The importance of students' exploration, in both selecting representations and making connections among them when building conceptual understanding of mathematical ideas, surfaces in studies about multiple representations as a tool for meaningful learning (Greeno & Hall, 1997). To encourage learners' inquiry, a teacher has to take into consideration opportunities that representations provide. Scaffolding through multiple representations may bridge the gap between concrete and abstract representations of a mathematical concept and reach a reflective abstraction in a variety of ways. For example, a potentially abstract problem statement may be illustrated using some kind of visualization. Students building their own ICT representations, and translations among them support their conceptual orientation in learning. New representations are essential components of a learning environment in which learners are required to think harder about the topic being studied and to generate ideas that would be impossible without these new representations. This creative thinking supports learners in creating and sustaining their conceptual orientation (Alagic, 2003).

Cognitive Tools: Spreadsheets

Derry (1990) defined cognitive tools as both mental and computational devices that support, guide, and extend the thinking processes of their users. Jonassen (1992) described them as "generalisable tools that can facilitate cognitive processing" (p. 2). Cognitive tools can make it easier for learners to process information, but their main "goal is to make effective use of the mental efforts of the learner" (Jonassen, 1996, p. 10). They are based on the principle that learners need to create their own understanding of new concepts. These tools give learners a way (often visual) of representing their understanding of a new concept/phenomenon and how it relates to their existing understanding of the same idea. Graphing calculators and spreadsheets as cognitive tools for creating representations are often used for exploring mathematical phenomena and building conceptual understanding of mathematical ideas.

Vockell and van Deusen (1989) described spreadsheets as tools using sets of rules that involve users in designing the rules. Spreadsheets as a tool for developing conceptual understanding of mathematical phenomena support storing data, calculating, and representing information. Data, most often numerical, is stored in a matrix-like collection of cells. The numerical contents can be manipulated in a number of different ways, starting from simple calculations and function applications to generating tables and a variety of graphical representations for stored data.

Encouraging students to manipulate spreadsheets improves their understanding of the concepts, relationships, and procedures. Beare (1992) describes spreadsheets as "flexible mindtools" for representing and manipulating data, reflecting on and speculating with quantitative information, and making the underlying mathematical reasoning explicit. The calculational and graphical capacities of spreadsheets provide a context to engage students in analyzing and connecting multiple representations. They support students' engaged, problem-based learning and open-ended investigations. They are valuable for answering "what if" questions. Considering implications of alternatives is a valuable approach to engaging higher-order thinking (Sounderpandian, 1989).

In one of the studies investigating spreadsheets as cognitive tools, Sutherland and Rojano (1993) investigated how prealgebra students could use spreadsheets to represent and solve algebra problems. The study was conducted simultaneously in Britain and Mexico and took place over a five-month period. During that time, students moved from a cause-effect local numerical notion of algebraic relationships to general rule-guided relationships that could be symbolized both in the spreadsheet and in algebraic notation. Baxter and Oatley (1991), while comparing the effectiveness of two different spreadsheet packages, found out that, not surprisingly, the users' prior experience level with spreadsheets was far more important to learning than the usability of the software package. These studies provide a few insights about the effectiveness of spreadsheets as cognitive tools. From an educator's point of view, based on teaching practice, teachers' understanding and use of spreadsheets can extend from kindergarten up, through all the grade levels, from a simple four-pane magic square, through organizing students' records/grades, to the complex applications or data analysis supported by spreadsheet software as a cognitive tool (Alagic, 2003).

IDENTIFYING THE PROBLEM: GRAPHING REAL-LIFE SITUATIONS

Context

The philosophical foundation of BRIDGES (Neal et al., 2002; Alagic et al., 2003) is based on a view of mathematics as a humanistic, socially constructed discipline and way of thinking that implies the learning and teaching of mathematics through inquiry (Heaton, 2000). Based on current research findings about successful professional development models and best teaching practices, the data available, and in collaboration with the school district personnel, the BRIDGES evidence-based professional development model is a carefully constructed interplay of two models reflecting (a) practice-based professional development connecting professional development activities of teachers to the actual classroom work of teachers (Smith, 2001), and (b) immersion in inquiry focusing on teachers' exploration of real-life problems that challenge their (mathematical) reasoning (Loucks-Horsley, Hews-on, Love, & Stiles, 1998; Loucks-Horsley & Mastsumoto, 1999). By situating teacher learning "in practice," teachers engage in activities that are at the heart of their daily work, and through guided inquiry into the standards-based school mathematics curriculum, their mathematical knowledge and understanding is deepened and widened.

BRIDGES is based on the belief that teachers must become and continue to be mathematics learners, if they are going to teach for understanding. This happens when they are challenged at their own level of mathematical competence and when their learning experiences are based on the same pedagogical principles that they are expected to implement with pupils. As Schifter and Fosnot (1993, p. 26) pointed out: "perhaps more important for [the teachers] than their investigation of any specific content area is the process of active self-reflection. By analyzing together their experience of the just-completed mathematics activity, teachers begin to construct an understanding of how knowledge develops and the circumstances that stimulate or inhibit it." BRIDGES project delivery is grounded on the principles of effective professional development (Loucks-Horsley et al., 1998; Loucks-Horsley & Mastsumoto, 1999; Smith, 2001).

The BRIDGES web site (http://education.wichita.edu/alagic/bridges/BRIDGES.htm) serves as an up-to-date report of project activities and accomplishments. It is regularly updated and maintained by the project team. Pages include information about ongoing activities, materials presented at the BRIDGES workshops and institutes, materials generated by BRIDGES teachers for use in middle school classrooms, and other related resources. In the context of this report the web pages should be regarded as supplemental evidence.

The overarching aim for the BRIDGES participants is to develop a deeper understanding of how to integrate standards-based school mathematics into the larger world of mathematics, including its applications in real-world settings. Teachers should know school mathematics at a deep level and should have an understanding of its place in the real world (Heaton, 2000; NCTM, 2000; NRC, 2000). For the proposed three-year BRIDGES grant cycle the following goals have been established: Student's Performance Goal, Teacher's Quality Goals, and Mathematics Leadership and Learning Community Goals. Teacher's Quality Goals focus on two tasks: (a) Improve the mathematics knowledge of all participants. (b) Equip teachers and paraprofessionals to be able to deliver effective mathematics instructions to ALL students using a variety of instructional strategies and approaches (Alagic et al., 2003).

Teachers had opportunities to get graduate credit hours for the BRIDGES workshops and summer institute, Mathematics Inquiry I, II, and III. Pre, Mid and Post tests of mathematical and pedagogical content knowledge, focusing on the mathematical concepts studied in BRIDGES, have been administered to participants at the beginning and end of the Summer Institute and after the Fall Workshop.

Problem Triggers

There were two explicit situations that initiated the sequence of activities, reflections, and analysis of subsequent investigations that are presented in this article. The first one had to do with an assignment of making up real-life situations to match graphical representations presented to teachers. The second one comes from a middle school classroom, as reported by the teacher, and had to do with graphing collected data.

An activity was planned to assess teachers' understanding of graphs, after a sequence of graphing activities using graphing calculators and spreadsheets. Preceding this activity, some real-life problem situations were modeled by graphing, and motion-ranger and calculator-based labs as hands-on experience were provided for a couple of activities. There was no previous discussion or preparation for this specific activity. Teachers got some sample graphs on paper and worked in small groups to create stories that would correspond to graphical representations: increasing and decreasing linear functions and exponential growth and decay. Each group was presented with a different graph and asked to make up a story to go with their graph. The discussion appeared lively and it did not look as if anyone needed guidance or support. But, when teachers shared their stories, it was obvious that stories were the simplest possible cases of time-distance graphs. While the stories always accounted for the increasing or decreasing aspects of the graph, they did not take into account other mathematically observable features, such as the rapid acceleration of an exponential curve. Without sufficient mathematical experience, all graphs and data sets were assumed to be linear. Yes, we could say that the teachers had not been challenged to do anything more complex, but the fact remains that none of them tried to reach for a challenge.

The closing assignment of BRIDGES Year One for participating middle school teachers was to implement in the classroom some of the real-world problem-solving activities which had been explored in the BRIDGES Summer Institute. Our real-world problems have been taken from Mathematics Modeling Our World, a high school National Science Foundation (NSF) supported curriculum (Garfunkel, Godbold, & Pollak, 1998). At our next session (beginning BRIDGES Year Two), in connection with an environmental study, one participant reported on student projects which involved analyzing and graphing exponential growth. All students had correct data, indicating exponential growth, but all of the graphs they developed pictured linear growth. While the students could identify their data as increasing, they were not familiar with, and therefore unable to distinguish, the shape of an exponential growth curve. The participating teachers did not readily identify the mistake, indicating that they, too, were not experienced with making the necessary distinctions between different sorts of increasing graphs. Someone could argue that this is not surprising, since exponential growth and related topics such as geometric sequences are not included in NCTM standards until grades 9-12, but distinguishing linear from nonlinear graphs should come with the introduction of the concept of "linear" in middle school grades. Also, middle school students learn to use exponents to represent repeated multiplication, so data representing exponential growth is easily introduced.

POSSIBLE SOLUTIONS

The goal of the study was to explore how available technology-based tools can (a) deepen teachers' understandings of linear and exponential growth through technology-based representations, and (b) provide for students' learning from the same ICT-based representations, considering the fact that students do not have the teachers' (standard) representations in their toolboxes. By introducing teachers and students to multiple examples and using different available technologies to contrast linear results with exponential results, conditions were provided for them to see additional aspects of their own and others' graphs. Specifically, this approach was used to review these concepts with middle school teachers participating in the BRIDGES grant for Year Two. Then they applied similar techniques to help their students learn about these new relationships. Students' projects were shared and compared to study/test their ability to distinguish different types of growth.

The remainder of the article is organized in two parts. The first part describes activities facilitated in an attempt to clarify possible misconceptions about exponential and linear growth and corresponding graphical representations. The second analyzes teachers' reflections and lessons learned. The BRIDGES participants had studied the following real-world environmental problem in the summer and then it was used with middle school students in the fall, resulting in data showing exponential growth, but graphs that looked very linear.

Moose Return (Garfunkel et al., 1998)

In 1861, the last moose in New York's Adirondack State Park wilderness area had been shot. Years later, in 1980, some moose began to reappear in the six-million-acre park. By 1988, experts estimated that 15 to 20 moose were in the park. By 1993, this number had increased to between 25 and 30 moose. A survey conducted by the New York State Environmental Conservation Department (ECD) found that the public favored an increase in the moose population. The moose gradually migrated from Canada and New England States. The numbers of moose also increased through natural reproduction. The ECD determined that a plan to move 100 moose into the park over a three-year period would cost $1.3 million.

The first activity encourages students to engage with this moose population question: Pretend you were the commissioner of the ECD, and you were responsible for making a recommendation to the governor about your findings.

1. "Pose a specific question concerning these data to which you would like an answer."

2. "What additional information do you need in order to be able to answer your questions?"

Further consideration of the given data leads to creation of mathematical models for growth of the moose population. The first model proposed is linear, assuming that the population changes only by migration of new male moose into the park at a steady rate. Analysis of this model generates questions as to its validity for the known and predicted behavior of moose. Further information leads to an exponential growth model.

After identifying the graphing challenges, we used the book, The King's Chessboard, as a springboard with the teacher-participants, to review exponential growth and study the related graphs. The story is summarized as follows.

The King's Chessboard (Birch, 1988)

Once long ago in India a wise man was summoned by the King of Duncan to appear before him. The King told the wise man that he had served him well and that he would like the wise man to choose a reward. The wise man did not want a reward, but the King insisted and the wise man finally made a decision. He asked that he be given rice each day for the 64 squares on the King's chessboard. For the first day, he wanted 1 grain of rice, for the second day he wanted 2 grains of rice, and for the third day 4 grains of rice. Each day the rice would double for each of the squares on the chessboard. The King granted the wise man's request after ignoring (out of pride) the impulse to question the implications of what seemed to be such a simple process.

Days later the Grand Superintendent saw workers carrying large sacks of rice. One grain became two and then four; grains became ounces; ounces became pounds; a bag had become two bags; and today it was four sacks each weighing one hundred twenty-eight pounds. The Grand Superintendent informed the King and the King summoned his royal mathematicians. He soon realized that there was no mistake and that he had promised the wise man two hundred seventy-four billion, eight hundred seventy-seven million, nine hundred six thousand, nine hundred forty-four tons (274,877,906,944) of rice. The King knew there was not enough rice in his kingdom to fulfill his promise and the next morning the King summoned the wise man to the palace.

The King asked the wise man how he could be satisfied. The wise man replied that he already was satisfied and originally did not want a reward from the King. The wise man returned to his simple home and quiet life. He served the King many times afterward, and the question of a reward never again arose. The King ruled wisely and justly for many years, and to the end of his days he never forgot the wise man's lesson-how easy it is for pride to make a fool of anyone, even a King.

After some hands-on preliminary exploration of the number of grains of rice in a given volume, we used Excel on laptop computers to consider the quantity of rice. Teachers were easily able to represent the amount of rice the king had promised to give the wise man each day. They made a table listing days and number of grains of rice. Then they made a graph of the data (Figure 1). Using the technology made it possible to explore the data quickly: What does the graph look like if we zoom in on it? What if the king tripled the number of grains of rice each day? What does the vertical scale have to do with the appearance of the graph? (Alagic & Palenz, 2004a; 2004b).

"Zooming" is a popular technological tool to explore graphs. These graphs gave us an opportunity to consider the advantages and disadvantages of zooming. Teachers were able to see that as we zoom in and out on an exponential graph, its shape remains essentially the same--with a relatively flat part curving into a steep rise. Also, as they zoomed out, the details and sharp rise of a section of the graph were lost--variations disappearing in the loss of detail which happens with zooming out. A follow-up activity involved beginning with a similar data table and graph, either in Excel or on a graphing calculator, and then trying to make the graph appear to have a steeper rise or a flatter section by changing the window (by zooming in or out of the graph). The lesson here is two-fold: zooming can reveal hidden detail in a graph or zooming can disguise the shape of a graph. Teachers who had not distinguished between linear and exponential growth in graphing in our previous sessions recognized differences and expected students could also comprehend them. They seemed eager to involve their students in a similar experience (Alagic & Palenz, 2004a, 2004b).

The teachers' stories and the thinking behind the creation of these stories. Teachers generated their own stories and activities to make exponential growth ideas meaningful for their students. Several of the stories introduced linear comparisons to give more impact to the examples of exponential growth. These were generally described in terms of choosing between a linear allowance or salary and one which pays 1 cent the first day and doubles each day for a period of time, usually a month or less. Other examples included spreading a rumor if each one who has heard the rumor tells one other person each day, and counting layers of paper as a large sheet is folded in half repeatedly.

Another teacher-participant reported considering exponential decay in the 6th grade classroom by cutting a sheet of paper in half repeatedly. Another 7th grade teacher used this exercise to help students think about radio-active decay and toxic waste. When the paper gets too small to cut, students can begin to understand that the pollution (or whatever) won't be entirely gone, although it may get too small to be measured.

Teachers commented on these stories as engaging for students. They reported on students surprise and puzzlement when realizing the meaning of exponential growth. Some teachers connected it to their own surprise, "Personally without knowing anything about these problems I would also go with the $5 a day instead of pennies-option." It was noticeable that teachers were trying to bring a note of humor in their classrooms by emphasizing the unexpected and impossible. One teacher reflected, "I will increasingly try to inject humor into the problems to make the experience more enjoyable." The teacher who earlier in the year had accepted linear graphs from students studying the exponential growth data of the moose return problem, later had his students consider the possibility of an exponential allowance. He reported that the graphs the students drew to illustrate their hoped-for allowance correctly showed exponential curves. Although there was no public recognition of his own (and others) misconceptions, the overall BRIDGES environment appeared more open for experimenting and risk-taking while exploring mathematical concepts.

As a follow-up to the King's Chessboard problem and to make large numbers meaningful, we asked teacher-participants to respond to the claim of the wise man at the end of the story that there is not enough rice in the world to fulfill the promise. We asked if that was accurate: how much rice is it in terms we can comprehend? The teachers came up with a variety of measurement comparisons. The volume of rice was compared to the volume of a school bus, a railroad boxcar, the Grand Canyon, or the entire earth. One group estimated the length of train needed to transport the total amount of rice by comparing the length to how many times around the Earth the train would wrap. The grains laid end-to-end were compared to the distance from the earth to the sun. The variety of comparisons and visual representations indicates the broad possibilities of integration of this estimation activity into a classroom context, which was explicated by participating teachers. After our workshop estimation efforts, we noticed that teachers began making more use and reference to estimations in other activities with their students.

DISCUSSION AND IMPLICATIONS: TECHNOLOGY-BASED MULTIPLE REPRESENTATIONS

Teachers need successful experiences and ongoing pedagogical and technological support when integrating technology into their curriculum. Also, they need opportunities to experience and do mathematics in environments supported by diverse technologies (Dreyfus & Eisenberg, 1996). Empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing conceptual understandings and communicating about mathematics is at the heart of BRIDGES professional development. Throughout the BRIDGES activities, concrete experiences have been provided to explore technology-based representations of data, graphs, and functions.

Cognitive tools allowed teachers to recognize the differences among linear and nonlinear graphs and with some guidance, their vocabulary expanded to allow for descriptions and discussions about these differences and features.

The ICT tools provided for (a) visual and graphical multiple representations interconnected with appropriate simulations, (b) meaningful explorations of a variety of cases in a smaller amount of time than if standard representations had been used, and (c) a nurturing learning environment supporting Perkins priorities, such as making learning a thinking-centered process, providing for rich, ongoing assessment, supporting learning with powerful representations, and teaching for transfer.

Many teachers will not share cognitive tools with students until they are confident that they can deal with "how-to" questions and guide students successfully. Teachers taking their new knowledge to the classroom need opportunities to practice with it and to reflect on both their lessons and students' experiences with new tools. BRIDGES provided a supportive learning environment and forum for sharing and reflecting about successes and challenges.

Being proficient with cognitive tools takes time. This investment makes sense only if the learning goals accomplished are proportional to the effort. Therefore in the technology-oriented context, selection of a problem becomes even more relevant. We spent a significant amount of time exploring exponential growth (introduced during one-day workshop, reported on in another and explored over a couple of days of a summer academy). Recognizing the need and keeping with our beliefs that deep understanding of one concept brings more value than "inch deep, a mile wide" curriculum, we revised our original curricular plans to accommodate for this situation. The rich, open-ended, real-life problem, Moose Return, was a worthwhile task for BRIDGES participants. It provided a starting point for a sequence of explorations, both in terms of related real-life problems and mathematical abstraction. The King's Chessboard allowed for exploring new representations of the concepts studied. Teachers' stories provided multiple ways of collecting, representing, and interpreting data using spreadsheets. All these activities supplied opportunities for deepening participants' conceptual understanding of exponential growth and their ability to distinguish between linear and nonlinear growth/graphs and to carry these ideas much further. It was rewarding to recognize how opportunities for modeling, simulation, mathematical abstractions and transfer (e.g., a packaging problem was brought into the discussion) unfolded. As we already mentioned, teachers began making more use and reference to estimations in other activities with their students. Another interesting example of transfer, while using dynamic geometry software for some geometry activities, some teachers attempted to model with dynamic geometry tools "halving a sheet of paper" to represent exponential decay.

What is the final conclusion? This is our story about a good investment. We diverted from our original plans to invest time in dealing with teachers' misconceptions about exponential versus linear growth. We used ICT tools that we originally planned to use, but invested more time and resources to facilitate the learning process in this unanticipated direction. Because BRIDGES provided a risk-free environment and encouraged peer support, teachers' creativity was allowed to emerge in using available tools and developing curriculum appropriate for their classrooms.

References

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MARA ALAGIC AND DIANA PALENZ

Wichita State University

Wichita, KS USA

mara.alagic@wichita.edu

palenz@math.wichita.edu

day doubling rice tripling rice 1 1 1 2 2 3 3 4 9 4 8 27 5 16 81 6 32 243 7 64 729 8 128 2187 9 256 6561 10 512 19683 11 1024 59049 12 2048 177147 13 4096 531441 14 8192 1594323 15 16384 4782969 16 32768 14348907 17 65536 43046721 18 131072 129140163 19 262144 387420489 ... ... ... 62 2.30584E+18 1.27173E+29 63 4.61169E+18 3.8152E+29 64 9.22337E+18 1.14456E+30 Figure 1. Sample representations related to The King's Chessboard

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Author: | Palenz, Diana |
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Publication: | Journal of Technology and Teacher Education |

Geographic Code: | 1USA |

Date: | Sep 22, 2006 |

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