when Aunt Bertha reaches the age of 80, and they later test their hypothesis ..... as a 'big brother') and finally to Ben's independent line of thought. Such a.
MICHAL TABACH, RINA HERSHKOWITZ
and BARUCH SCHWARZ
CONSTRUCTING AND CONSOLIDATING OF ALGEBRAIC KNOWLEDGE WITHIN DYADIC PROCESSES: A CASE STUDY
ABSTRACT. Studies of knowledge constructing often focus on the analysis of a single episode, without considering enough the history of the learners, or the future learners’ trajectories with regard to the concepts learned. This paper presents an example of knowledge constructing within the context of peer learning. We show how the design of the task and the tools available to the students afford the constructing of conceptual knowledge (the phenomenon of exponential growth and variation, as it is expressed in its numerical and graphical representations). We trace the constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment. We show that knowledge is constructed cumulatively, each activity allowing for the consolidating of previous constructs. This pattern indicates the nature of the processes involved: knowledge constructing and consolidating are dialectical processes, developing over time, when new constructs stem from old ones already consolidated, which gain consolidation through the new construction, creating a new abstract entity. We also discuss the potential of the tool the students used (a spreadsheet program) to such processes of learning mathematics.
I NTRODUCTION Many researchers have shown that every human activity interweaves between the subject and the environment (see Rogoff, 2003 for a review). In addition, many have recognized the role of the relationships between semantic tools and human thinking, among them language (Vygotsky, 1936/1987), and computerized tools (Pea, 1999). As a human activity par excellence, knowledge construction is now commonly viewed as a sociocultural-cognitive process. Roschelle (1992) showed how the use of educational computer software in the field of physics enabled collaborating students to construct relationships between the concepts of speed and velocity. Roth and Bowen (1995) also described the phenomenon of constructing, or what they sometimes refer to as diffusion of knowledge in classrooms, where various tools (computerized or not) were available to the class. These descriptions were based on the analysis of protocols that included not only verbal expressions, but also gestures or computerized actions. Hershkowitz, Schwarz, and Dreyfus (2001), and Dreyfus, Hershkowitz and Schwarz (2001) have described the Constructing of mathematical knowledge within a chain of speech acts for two models of social learning, guided participation and peer collaboration. They developed a method of Educational Studies in Mathematics (2006) 63: 235–258 DOI: 10.1007/s10649-005-9012-2
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analysis to examine the Constructing of knowledge. The studies mentioned generally focused on the analysis of a single episode. This paper presents an example of knowledge Constructing within the context of peer learning, offering a threefold contribution to the existing body of research. First, we show how the design of the task and the tool available to the students afford the Constructing of knowledge. Second, the choice of knowledge represents a substantial part of the mathematics curriculum: conceptual knowledge about the phenomena of growth and variation. Here we focus mainly on exponential growth. Third, we trace the Constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment.
METHODOLOGIES
FOR KNOWLEDGE CONSTRUCTING IN INTERACTION
Researchers interested in studying knowledge construction have adopted various methodologies. For example, Pontecorvo and Girardet (1993) have studied the construction of historical knowledge by observing historical epistemic actions (such as appealing to a source) during a conversation and by looking at the inferences collectively agreed among students. Resnick, Salmon, Zeitz, Wathen, and Holowchak (1993) used a double coding of argumentative moves, on the one hand, and speech acts (claim, opposition, elaboration, explanation, etc.) on the other hand, in order to observe reasoning in conversation. Hershkowitz et al. (2001) presented a theoretical and practical model for the cognitive analysis of mathematical knowledge Constructing, namely the RBC model. This model suggests that the Constructing processes for new knowledge are expressed through three observable and identifiable epistemic actions, Recognizing, Building-With, and Constructing (RBC). Constructing of new knowledge is largely based on the re-organizing of existing knowledge elements in order to create a new knowledge construct. The second action, the Recognizing of an existing mathematical construct, takes place when the learner recognizes that a specific knowledge construct is relevant to the problem he or she is dealing with. The third action, Building-With, is an action comprising the combination of recognized knowledge elements, in order to achieve a localized goal, such as the actualization of a strategy or a justification. The actions of Recognizing and Building-With are often nested within the action of Constructing. We can even find sub-Constructing nested within larger Constructing. Hershkowitz et al. (2001) also claim that revealing further stages of abstraction1 might consequently reveal stages of Consolidating. The stages of Consolidating can be revealed or can even occur during the course of later
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activities. At such a stage, the previous Construct serves as a cornerstone in the new Constructing since it fuels the new actions of Recognizing and/or Building-With. The learner’s growing acquaintance with constructs, through the actions of Recognizing and Building-With serves as evidence for Consolidation (Hershkowitz et al., 2001; Dreyfus and Tsamir, 2004). The RBC model is a methodological tool that was used by Hershkowitz et al. (2001) in an interview with a single student. The three epistemic actions, which are nested within each other, describe the cognitive processes of this student during the interview. Hershkowitz et al. also pointed out that their hypotheses about the Consolidation of constructed knowledge have not yet been empirically tested. In another study, Dreyfus et al. (2001) combine cognitive and interactive aspects in a double coding methodology to investigate the work of two student dyads, performing the same algebraic task based on the use of a spreadsheet. The double coding deals with interactional functions and epistemic actions taken in the course of the task. Consequently, this methodology led to a better understanding of the learning processes by both members of each dyad. Dreyfus et al. (2001) have found that within a complete learning episode that lasted about an hour and a half for each pair, the cognitively differentiated segments were also differentiated with regard to the character of the interactions taking place, and vice versa. In the present research, we have examined the processes of knowledge Constructing and dyadic interaction within the context of computeraided 7th grade successive algebra lessons over a period of a few months. We will first present a brief framework of computer-mediated learning in algebra.
EXCEL
AND THE LEARNING OF ALGEBRA
The availability of computers in schools and their incorporation into instruction have led to new approaches in teaching algebra. Importantly, these tools have enabled researchers to express a more holistic approach to early algebra. Especially the functional approach became popular (Heid, 1995; Hershkowitz et al., 2002; Radford, 2000; Yerushalmy and Schwartz, 1993). Hershkowitz et al. (2002) defined three criteria that computerized tools should fulfill to be appropriate to mathematics learning and teaching in the classroom. We list here these criteria to show the appropriateness of Excel tools: (i) The extent to which a tool is generalizable, and its cultural status: Excel’s potential to support the creation of numerical series out of other series, using a kind of algebraic rule, and representing numeric
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data in graphic form, enable it to be successfully applied for many content areas, both within and outside the school setting. (ii) The mathematization ability of the tool: Excel has a potential for the development and support of mathematical processes, through the possibility of using patterns or formulas followed by “dragging” (Kaput, 1992). (iii) The communicative power of the tool: Excel has power to afford communication. The symbolic language of Excel mediates between the symbolic algebraic language and informal languages. In the following we will describe some additional characteristics of Excel, which are relevant for this study. There is a difference between Excel formula and the regular algebraic syntax; still, Excel’s symbolic system may scaffold the development of the algebraic symbolic language; the variable-cell (like A1 ) is a symboliccomputer object with several meanings (Haspekian 2003, p. 6) in learning algebra. The possibility of moving between the various meanings is in itself a mediator between the formal and in-formal languages. The symboliccomputer object might be seen by the students as an entity which is between a specific mathematical object, like a number, and an abstract object like a variable (Sutherland and Balacheff, 1999). In many cases, finding an explicit relationship between variables in a certain situation and its formal expression through an algebraic formula is a difficult and complex task. Excel enables the user to discover local interrelations between variables, and through operating the “dragging” option, to generalize them so that the relevant phenomenon is expressed numerically or graphically. In this way, the range of mathematical phenomena accessible to young students might be substantially broadened. In particular it is possible to write Excel formula using a variable-cell of a recursive nature, which expresses the local recursive relations between cells in the same column. It is also possible to write an Excel formula using a variablecell of an explicit nature, which in that case is expressed by a connection between two (or more) columns. In an Excel environment, both forms of generalization are global, since by the “dragging” option they create the numerical phenomenon. In both cases, when “dragging” the formula, Excel copies the index of the cell as a relative index. That is, writing the formula = A1 ∗ 2, and dragging it down to the next cell, Excel will change the index and it will be = A2 ∗ 2. When teaching algebra in a functional approach, this might emphasize the nature of the input-output relations of formula and variables. The students may realize that each output is created from an input via the same underlying rule. Several studies have shown that the theoretical suggestions concerning the potential of Excel are justified in general (Ainley et al., 2004; Healy et al., 2001), and for the passage from arithmetic to algebra in particular.
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IN THE COMPUMATH PROJECT
This project aims at developing a computer-supported mathematics curriculum for Junior High school. Following Kieran (1992), Yerushalmy and Schwartz (1993), and Heid (1995), the development team, in teaching Algebra, adopted an approach based on the concept of function (Hershkowitz et al., 2002). With this approach students construct generalizations of co-variational phenomena in word problem situations (Friedlander and Tabach, 2001) or present patterns like visual series of dots where they have to find relationships between the variables involved. Writing such generalizations always includes algebraic symbolic generalization. The development of 7th grade algebra course included the development of the entire learning environment, from a socio-cultural-cognitive approach to learning, stressing the individual’s active role in creation of his or her knowledge through interactions of different types (peer work, whole class discussion, work with the computer). It was theoretically assumed and confirmed by research that interaction in small groups or in dyads supports mathematical argumentation (Hershkowitz et al., 2002, p. 667). Because of the limited time available in the computer laboratory, the course was designed for 1-2 weekly hours in a computer laboratory, and 3-4 hours in a pencil and paper environment. Mathematical concepts are first approached through situations in which students hypothesize numerically and graphically, describe phenomena verbally and algebraically, and only then test their own hypotheses. The crystallization of mathematical concepts, with the help of the teacher, is meant to take place at a later stage by way of refinement. At the beginning of the lesson, the teacher usually presents a problem situation in a short whole class discussion. Then in the main part of the lesson, the students investigate the problem in pairs, and the teacher acts as a moderator. In the final stage of the lesson, a whole class discussion is conducted, with the leading role of the teacher.
DESCRIPTION
OF THE RESEARCH
Research aim and method The aim of this research was to track the Constructing and Consolidating of conceptual knowledge during successive dyadic interactions, which took place mainly in the second part of some lessons. For this purpose, two 7th grade algebra students with high-level verbal skills were selected to work together. The work of this pair of students in some lessons of the
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learning trajectory was videotaped, while the rest of the dyads in the class also worked in a similar way. Excerpts of relevant episodes were selected and analyzed in order to identify cognitive and social changes. In analyzing the excerpts, a double coding model was adopted to distinguish between the chronological course of the interaction and its logical flow, and to discern the students’ hidden assumptions and motives (Dreyfus et al. 2001). The analysis includes examples of the dyad’s collaborative work, to possibly uncover the Constructing and Consolidating of a new mathematical knowledge (See for example Figure 7). Research procedure Students were presented with two activities in which they were requested to distinguish between linear and exponential phenomena (the first activity and the first part of the second one), and afterwards between different exponential phenomena (the second part of the second activity and the third activity). Five months separated activities 1 and 3. Finding evidence for Consolidating knowledge necessitates longitudinal research. Moreover, such a design should include a sequence of activities that enable students to construct knowledge in a certain activity, and to show Consolidating of this construct in a subsequent activity. Consolidation is characterized by reorganization of previous constructs, which are Recognized, with higher confidence while capitalizing on previous constructs in the course of a new similar activity, or by a further elaboration. For that reason, the structure and sequence of activities should provide opportunities for Consolidating knowledge: it should include relevant opportunities. Relevance may be attained through similarity of tasks that invites Recognizing and Building with previous constructs. Relevance may also be attained through a difference that enables the actualization of a previous Construct, and its adaptation. Constructing and Consolidating of knowledge are then interwoven in a complex dialectical process. In the next section, we will describe the dyad’s work in the three activities, in light of the goals of the activities’ designers. We will also analyze the interactions between the two students in the light of knowledge Constructing and Consolidating.
RESEARCH
FINDINGS
The first activity – “X times as much” After about a month of learning in the Excel environment, the teacher presented the following situation to the students:
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Figure 1. The “X times as much” activity.
She asked the students to hypothesize whether the amount of money in Efrat’s savings box would reach or exceed the amounts accumulated in the boxes of four other children, whose allowances are governed by linear rules. The students had dealt with these rules in a previous activity. She then asked the pairs of students to investigate the numerical data with Excel, eventually confirming or rejecting their hypotheses. After the students succeeded to get a numerical representation in the the B column in the spreadsheet, they were asked to sketch a graph describing the phenomenon (graphical hypothesis), and to test it with the computer. Task analysis. The students were asked to postulate a relative hypothesis; in fact, they were requested to compare familiar (linear) growth phenomena to the new phenomenon presented to them. Relative hypotheses (that compare phenomena) are easier to consider than absolute ones (Bryant, 1974). Efrat’s allowance savings began with a very low initial amount, subsequently leading to a surprise: Most learners estimate that an allowance starting with such a low initial amount will remain low. As was discussed previously, there are two possible ways to represent Efrat’s allowance in Excel. One is the explicit global algebraic formula 0.02∗ 2x , where x stands for the number of the week. How to enter such a formula into Excel is shown in column C of Figure 2. Another option is the use of a localized recursion rule, describing the relationship between the amounts of money paid as the allowance in two consecutive weeks 2∗ the amount of money in the savings box last week, and using the “dragging” option in Excel to apply it to other weeks (see column B in Figure 2). Creating an algebraic model of the phenomenon through finding the explicit formula is relatively difficult for students who are just beginning their 7th grade algebra studies. The aim of the designers of the activity was for the students to investigate the exponential change numerically and to realize that: (1) The exponential growth is very rapid; (2) the shape of the graph characterizing such growth is typical; (3) the phenomenon is basically
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Figure 2. Efrat’s allowance calculated through a recursion formula and an explicit formula.
related to repeated multiplication by two. The use of a spreadsheet enables students to view the numerical and graphical representations by recursive Excel formulas and “dragging”. Analysis of the students’ work. Like many other students, the dyad we examined (Avi and Ben – two male students), hypothesized that Efrat’s allowance would be the lowest. They generalized the allowance algorithm by using a “formula” which is a localized recursion rule (= B2 +B2 ), dragged this formula, and were very surprised to see the high results (see Figure 3). Then, they easily sketched the graph by hand (see Figure 4) and created a computerized graph which is quite similar. Figure 3 shows an excerpt of the dyad’s working dialogue transcripts. To the left of each utterance appears the name of the speaker and the turn number. To the right of each utterance appears a diagram, showing the logical flow of the utterance. Each utterance is marked by a circle connected by an arrow (or arrows) to a previous utterance/s that triggered it. In order to follow the interaction between the students, their utterances have been separated into two columns, one for each student in the dyad. Therefore, an arrow connecting circles located in different columns would indicate that an utterance said by one member of the dyad was “inspired” by a previous utterance made by the other member. Alternately, a vertical arrow within the same column would indicate that the speaker continued one of his previous utterances (Dreyfus et al., 2001). Figure 3 shows that Avi and Ben tried to enter the first element of the sequence, and to write a formula describing the numerical data in Excel. At first, it appears that the two students participated in co-Constructing: they speak to each other, and so each supposedly refers to what the other said. However, the pattern of arrows in the diagram challenges this impression: many of the arrows are vertical (for example: Avi, lines 31-36, and Ben, lines 37-44). This pattern of arrows suggests that Avi Constructs the
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Figure 3. Constructing algebraic generalizations in the “X times as much” activity (Excerpt 1).
phenomenon for himself, and so does Ben. Another indication is that Avi is speaking aloud to himself, following his own line of thought, which can be deduced from Avi’s lack of response to Ben’s request for clarification (35). It is worthwhile noting that Avi intuitively suspects that the proper way to describe the allowance is the exponent (40). This indicates a higher level of thinking, beyond the aims of the activity designers. Ben does not understand Avi’s idea, and therefore suggests an idea of his own, based on the power of the spreadsheet he became familiar with during previous activities: to translate the phenomenon to an additive recursion rule (39), followed by “dragging”, resulting in a numerical column expressing the phenomenon, up to the 20th week. They are both surprised to see the results of the amounts calculated in the Excel table, since they are very far from their own hypotheses (45).
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Figure 4. Graph’s sketches created by Avi and Ben.
From the point of view of interaction (as evident by the net of arrows), both partners seem to participate equally; their interaction is symmetric. Avi and Ben’s work is in accordance with the shared goal, representing Efrat’s allowance accumulation phenomenon, but they propose different ways of obtaining that goal. Each of them (at least at the beginning), had his own idea, and so for most of this segment, they are not aligned. In sum, in this segment, Avi and Ben are acquainted, for the first time (at least in mathematics lessons); with the numerical representation of the exponential phenomenon they are investigating. At the beginning there is evidence of the development of two different directions of thought; (1) Avi, striving for the explicit exponent model, which is difficult to express algebraically. (2) Excel enables a way to “circumvent” the explicit generalized rule using a localized recursion rule that becomes global with the help of the Excel “dragging” function, as suggested by Ben, and adopted by Avi as well. Ben’s success is a result of capitalizing on Excel’s potential. Avi and Ben are surprised by the numerical phenomenon they discover (up to the 20th week), which shows rapid growth compared with linear growth (45) – they underestimated the growth. The two draw sketches of graph. Ben first draws a sketch, and shows it to Avi. Avi sketches a bit more accurate graph, and explains that it starts slowly, slowly, and then there’s a huge progression (see Figure 4). Although Ben’s sketch is less accurate, they both drew their sketches quickly, and it appeared that both of them experienced no difficulty in converting the numerical representation to the graphical one. When they were presented
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with the Excel graph, Avi says (96): There, like I said. He seems to be happy that his hypothesis was verified. Knowledge constructing processes observed. The surprise of the numeric exponential growth and the sketches that Ben and Avi drew, suggest that Avi and Ben Constructed some knowledge about exponential growth as was planned by the designer, specifically that (a) the phenomenon is characterized by rapid numerical growth and (b) the graphical representation of this growth has a typical form, indicating the kind of rapid growth rate of the phenomenon.
The second activity – “Aunt Bertha” In this activity the students are confronted with the phenomenon of exponential growth for the second time, about five months later. The teacher presented the situation, which appears in Figure 5. The students were asked to hypothesize which option will give Yossi the largest amount of money when Aunt Bertha reaches the age of 80, and they later test their hypothesis with the help of Excel.
Figure 5. The “Aunt Bertha” activity.
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Figure 6. Patterns of recursion describing the four possibilities in the “Aunt Bertha” activity.
Task analysis. In the activity, four variation phenomena are presented, two of them linear (options 1 and 2), and two of them exponential (options 3 and 4). The range of years in this problem is of crucial importance: for the first five years, option 2 is the most profitable, between the 6th and 12th year the most profitable option is 3, and from the 13th year onward, option 4. The hypothesis the children are required to postulate is again a relative hypothesis, but in their activity they have to compare two types of exponential growth as well. As discussed earlier, one can use two generalization forms to verify one’s answer: An explicit generalization, describing the relationship between the years (input) and the amount of money received (output), or a generalization through a recursive rule, describing the relationship between amounts of money received in two consecutive years (see Figure 6 for the Excel notation of the expressions). The activity’s design was meant to Consolidate previously constructed knowledge (exponential growth is more rapid than linear growth; exponential growth can be described through a multiplicative recursion), and to enable the creation of additional knowledge (the role of the exponent base) by the comparison between two types of exponential growth 1.5x and 2x . Analysis of the students’ work. Avi and Ben immediately hypothesize that the optimal option is the fourth. Their thinking flows, however, are different. Ben’s justification (11) is that: Every time it will grow multiplied by two, this, after that reaches a hundred, will already be two hundred, four hundred. He may be referring to a ‘hundred’ in comparison to the third option, or because for him, from that point onward, the familiar growth is meaningful and he knows how rapid it is. Avi reacts, claiming (12): No, no way, it doesn’t have to reach a hundred; at ten it will reach a few million. For Avi, exponential growth “breaks through” every scale, reaching millions immediately. It is worthwhile noting that since the first activity, Avi and Ben encountered only
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Figure 7. Constructing “Aunt Bertha’s” four options (Excerpt 2).
one case of exponential growth, mentioned casually during in an “Open Day” for parents and students at their school. In the computer laboratory, Avi and Ben first try to check their hypothesis as to the optimal option. For that purpose, they need to create a numerical representation of the phenomena. The next segment (Figure 7) refers to the generation of numerical representations in Excel. In this excerpt, the students attempt to fill in the table, entering the first amount for each option, creating an Excel formula and “dragging” it so that the option will receive full numerical representation. The pattern of the arrows in the diagram indicates four separate segments (29–35, 36– 38, 39–41, 42–47), corresponding to the creation of four generalizations. Ben begins each of these creation segments; he seems to be the leader. However, the places where Avi contributes are the very same places where Ben encounters a cognitive difficulty (40–41). It appears that Avi lets Ben
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“run forward” and supports him when Ben hesitates. This conclusion is also supported by an analysis of the interaction of the fourth generalization. The logical chains produced in this activity are shorter than the ones characterizing such creations in their previous activity. Avi and Ben are more coordinated so that not much discussion is required. During the process of creating the fourth generalization (42–47), Avi and Ben develop two parallel lines of reasoning. Ben’s suggestion is the one finally accepted, but with Avi’s approval and confirmation. Both the third and fourth phenomena describe exponential growth. However, the students treat their generalization differently. Avi, who suggested a recursion formula for option 3 (41), did not hesitate to declare that option 4 is exponential (43). Ben has no suggestions for the third option (40), while positively identify the fourth option as multiplicative (44). This is in spite of their similar description: “1.5 times as much” in the third option and “twice as much” in the fourth. This difference may originate from the experience that they have accumulated during previous collaborative activities. In the first activity they had already encountered a 2x type growth, but had not yet encountered a 1.5x type growth. They may relate to 2x as a prototype of exponential growth. They did not Construct yet the general notion of exponential change. Although their hypothesis was correct (the forth option is the most profitable one), Avi is disappointed by the display of the representation of the four options, since the fourth option does not reach a few million (as he initially hypothesized). He claims that he assumed the time period discussed is 80 years, not 15 years. He tries to “drag” down the formulas to represent 80 years, but the numbers obtained are too long to be displayed; Avi cannot confirm his intuitions and remains frustrated. It seems that Avi had Constructed, in the first activity, that the growth (at least for 2x ) is accelerating dramatically, but he can not confirm it. Regarding interaction, both members of the dyad contributed equally. Avi and Ben agree about the common goal. When there is no cognitive difficulty, they also agree about the way to achieve it. When a cognitive difficulty arises, they are not in alignment with each other, the difference in their ways of thinking is revealed. To sum up, the relative ease with which the dyad produced generalizations of linear phenomena indicates the knowledge acquired and Consolidated in previous activities. In contrast, the exponential phenomena (options 3 and 4), necessitated Avi’s support of Ben. Avi is surprised by the low amount received. This is an expression of knowledge Consolidating regarding the rapid change rate of exponential growth and the difference between exponential and linear growth. The knowledge about the rapid rate of exponential growth is also refined by the overestimation mentioned.
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In the first activity, the growth was underestimated, but in this activity, it was overestimated. This process can be viewed as a refinement of the Constructing of knowledge about exponential phenomenon. Knowledge constructing processes observed. During this activity we identified Consolidating of knowledge already Constructed in the previous activity (exponential growth eventually overcomes linear growth, despite the fact that in the first years of growth, the linear options presented are more profitable). We also suggest that Constructing of new knowledge concerning the phenomenon of exponential growth occurs: (1) the role of the exponent base (2) the role of the power of the exponent. This is based on several facts: – Both students correctly choose the exponential growth in the fourth option, over the exponential growth in option 3. – Both students show progress in their suggested ways of generalization: Avi explicitly mentions exponents, and Ben progresses from a formula of addition (= B2 +B2 in the first activity), to a formula of multiplication (= 2∗ E2 ), in spite of his lack of idea regarding the third option. However, note that both formulae are localized generalizations (recursion rule). – Avi’s attempt to drag down the local formula for 80 lines suggests the refining of exponent-related knowledge. Multiplying the amount received each year for 80 years is what leads him to expect such an accelerated growth, expressed by a large final amount. In other words, the exponent power is ‘responsible’ for the accelerated rapid growth. The third activity - “The Crazy Snack” In this activity, which took place shortly after the second one, the students encountered exponential growth for the third time. The teacher presented the “Crazy Snack” situation to the students (see Figure 8). Task Analysis. This time the first hypothesis is absolute, since the students have to hypothesize first the price of the snack in a certain year in the future. Since the initial price Avi and Ben chose was 3.5 units, the explicit algebraic formula describing the relationship between the number of years and the price of the snack is 3.5∗ 1.1x , x representing the number of years. The relationship can also be expressed in Excel through a recursion rule and “dragging” (Figure 9). This activity required knowledge concerning the concept of percentage, which is quite difficult to some students. Many students use in Excel an additive formula of the type B2 + B2 /10 rather than a multiplicative one.
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Figure 8. The “Crazy Snack” activity.
Figure 9. Price of the snack expressed through a recursion formula.
The second part of this question requires a comparison between similar exponential growth formulas: 3.5∗ 1.1x versus 1.75∗ 1.11x . The exponent base increases by 1/100, whereas the initial price of the snack is halved. It is a very delicate difference between the two options: while x < 76, the second is cheaper (3.5∗ 1.176 = 4,896 > 1.75∗ 1.1176 = 4,870), but from x = 77 on, the first is cheaper (3.5∗ 1.177 = 5,386 < 1.75∗ 1.1177 = 5,406). Students have to estimate which of the two, the change in the exponent base or the scalar, affects the growth more. The question refers to a period of 107 years, and for this period the wiser choice would be to leave the situation unchanged. Analysis of the students’ work and their interaction. The snack chosen by Avi and Ben costs 3.5 units. Avi and Ben hypothesized that the price of the snack will be very high after 107 years. Ben estimated the price to be around 100,000, whereas Avi’s estimation is 11,000,000. They wanted to test their hypotheses using Excel and searched for a generalization that will describe the change in the price of the snack. They found it difficult to reach such a generalization, ending up with an additive recursion rule. Later on, they hypothesized that the second option (dividing the price by two, but
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Figure 10. Numerical hypothesis regarding the price of the snack (Excerpt 3).
increasing the inflation by one percent) is not worthwhile. However, they found it very difficult to obtain a formula to check their hypothesis. The formula they reached was incorrect ( = B2 +B2 /11), and only with the mediation of the teacher did they reach the correct generalization. Excerpt 3 (Figure 10) shows the discussion between the teacher and the dyad about the absolute hypothesis. Ben first starts hypothesizing, and then Avi joins him only after a direct request from the teacher. They both expect rapid growth. Avi overestimates this time as well, giving much weight to the power of the exponent as directly responsible for the rapid growth (A16-A19). Interestingly, Ben dramatically changes his estimation
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Figure 11. Constructing a recursion rule to describe the price of the snack (Excerpt 4).
in light of Avi’s overestimation (18), probably because he considers Avi an authority in mathematics. We should remember that their experience with exponential growth was limited to exponent base 2 and base 1.5, a fact that may explain the overestimations. In this excerpt, each student constructs his own line of reasoning (Avi 17, 13-19, and Ben 8-12). Their contribution to the discussion is symmetrical. They share a goal, but differ in their opinions about how to reach it. The first action Avi and Ben try to take is to check their hypothesis about the price of the snack in 107 years. To do so, they construct a numerical representation of the phenomenon, as can be seen in Excerpt 4 (Figure 11). Ben takes the lead, and Avi tries to give him comments (represented by the arrows from Avi to Ben in lines 51 and 53). Ben’s dominance consists of controlling the work on the computer and suggesting formulae. The interaction seems asymmetrical. However, Ben looks for Avi’s confirmation in challenging places (52). Avi and Ben agree about the common goal, and they are in alignment with one another. In Excerpt 5 (Figure 12), we see how Ben and Avi attempt to determine which option is cheaper. They compare the phenomena they represented using Excel. In this excerpt, Ben immediately hypothesizes that the change suggested is not worthwhile (194) because the price will be higher in the long run. Avi joins him. They test the hypothesis on the computer, Ben leading the creation of the formula, with Avi’s full cooperation. Interestingly, the recursion rule suggested by Ben, = C2 +C2 /11, is analogous to the one he creates for the 10% growth, = B2 +B2 /10. This mistake is the result of flawed knowledge concerning percentages. Avi does not correct Ben, and
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Figure 12. Postulating and testing a relative hypothesis in the 2nd question of the “Crazy Snack” (Excerpt 5).
it appears that his knowledge is also flawed. The incorrect formula has seemingly led to the conclusion that the hypothesis is false (the price after the change will be less than in the original situation), a fact that Avi accepts calmly, while Ben seemed surprised (211). A possible explanation to such reactions is that Ben phrased the hypothesis and apparently understands it, whereas Avi merely joined him. Therefore, Avi is not overly excited about the results. For Ben, however, it appears that the hypothesis was meaningful, so that he is very surprised. Another explanation can stem from Avi’s history: he has already been often disappointed by his own hypotheses, and may no longer expect to come up with correct ones. Later the above algebraic mistake was corrected, with the teacher’s help.
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The pattern of arrows indicates that Ben Constructs an idea, and Avi assists him, but does not develop a similar or different suggestion of his own on the subject. This analysis of the interaction supports the cognitive analysis, and our feeling that Avi’s knowledge about percentages does not allow him to take a stand. The interaction in this excerpt is asymmetric with regard to the dyad members’ relative contributions: Ben clearly leads the process. Avi and Ben agree on the shared goal, and they are in alignment with each other. Knowledge constructing and consolidating processes observed. In their first hypothesis, Avi and Ben agree that the price of the snack after 107 years will be very high (Avi’s overestimation). In the second hypothesis Avi and Ben provided evidence for Consolidating the role of the exponent base (regarding the change of the inflation rate and halving the price of the snack) and even refine their knowledge about it – the growth for a base close to 1 is not so quick.
CONCLUDING
REMARKS
This research and the analyses presented within it emphasize the necessity of using a longitudinal research design, in order to identify situations in which knowledge is Constructed, relying on evidence about its Consolidation in further stages during the same activity, or in later activities pertaining to the same mathematical content area. We have observed the co-Constructing of the concept of exponential growth throughout three activities, spread over half a year. In the first activity, this process began with a first encounter, numerical, with attributes of exponential growth: rapidity, non-constancy, and accelerating rate. The students were very much surprised by the initial numerical representation they produced with Excel. Possibly the big surprise when they originally discovered the rapid growth of the exponential phenomenon led to immediate Consolidation. In addition, once the phenomenon was Constructed, they had no difficulty in translating it to its graphical form. We may assume that on the one hand, this process of Constructing of an exponential graph, is based on the students’ Consolidating of previous general knowledge about translating one form of representation to the other. On the other hand, the graphical feedback, during the initial Constructing phase, possibly contributes further to the immediate Consolidation. During the second activity held 5 months later, we observed the Consolidation of the knowledge Constructed during the first activity through
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the students’ overestimation. We also observed a further vertical (Treffers and Goffree, 1985) Constructing, based on the above Consolidation and then expressed through more complex generalizations made by the students: Avi Recognizes repeating multiplication by two as an exponential phenomenon, and Ben represents this growth by multiplication rather than addition for his recursion rule. The difference between the expressions suggested for 2x versus 1.5x might serve as evidence that 2x was Constructed as a prototype for exponential growth, during the first activity. The decision that multiplication by two grows faster than multiplication by 1.5 shows the buds of Construction regarding the role of the exponent base. The deliberate attempt to drag the formula 80 rows down the column rather then 15 rows, might indicate initial Construction regarding the role of the exponent. (In the first activity, they drag the expression down for only 20 rows. Here they first drag the formula only 15 rows, and since the numbers were much smaller then expected, Avi tries to see the effect of dragging for 80 rows). The students ignored the scalar coefficient by which the power was multiplied. This could be a sign for further Construction regarding the relative effects of the scalar and the exponent base. During the third activity, we again observed evidence of knowledge Consolidation (the students justifiably hypothesized that the growth will be rapid). We also witnessed further evidence of the Consolidation regarding the role of the exponent base and the scalar, in the correct hypothesis that the growth 0.5a∗ 1.11x will outdo the growth a∗ 1.1x in the long run, despite its lower starting point. The new Construction in this activity regards the small base – which leads to not so dramatic growth. The combined results present an interesting picture of the knowledge Constructing process. The knowledge which was Constructed initially is about the ‘strength’ of exponential growth, and is slowly refined, as the students understand more clearly the role of the power, and later the role of the exponent base. Knowledge is Constructed cumulatively, every activity allowing for the realization of a previous Construction. This pattern indicates the nature of the processes involved: knowledge Constructing and Consolidating are dialectical processes (Davydov, 1990), developing over time, when new Constructs stem from old ones already Consolidated, which gain Consolidating through the new Construction, creating a new abstract entity. One of the contributions of this paper to knowledge Constructing relies on the social processes that accompanied this dialectic Constructing. We would like to point out that interaction does not have the dimension of “making progress”. The changes in the nature of interaction and in the status of both students in the interaction are very important, as is the role of
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interaction (or lack thereof) in the constructing of knowledge. The patterns of interaction throughout the three activities changed; at the beginning, Avi, whose level in mathematics was superior to Ben’s, dominated the interaction. But the tasks at hand were not designed for the elaboration of explicit algebraic modeling, or for an absolute estimation of the growth rate of exponential variations. In contrast, the tool afforded recursion rules and the testing of hypotheses. For this reason, Avi cannot fulfill his line of thought, whereas Ben could rely on his recursive methods. The patterns of interaction then progressed to Avi’s follow-up of Ben’s efforts (at the beginning as a ‘big brother’) and finally to Ben’s independent line of thought. Such a progression was definitely beneficial to Ben. Students were not supposed to use an explicit formula, as they just started their algebra learning. In this sense Avi has higher mathematical understanding and ambitions, which he can’t fulfill yet, but has less success in getting the numerical representation. Ben, who is acting according to the mathematical norms required, is becoming more successful. Of further interest is Excel’s contribution to the learning of mathematics. We have seen growth phenomena, normally out of reach for Grade 7 students, become accessible through the use of localized recursion formulas and the use of the spreadsheet’s “dragging” option in order to globalize them. This fact that Constructing and Consolidating stem from inferences that are unique to the use of Excel in algebra explains to a large extent the conferring of dominance from Avi to Ben, who is very skillful in using the local algebraic rules and the “dragging” action, which are typical for Excel. Therefore, spreadsheets not only enable reorganization of collaborative work but also redefinition of social forces predefined by the authority of knowledge. Teaching algebra using Excel in a functional approach emphasizes the nature of the input-output relations of formula and variables. The students realize that each output is created from an input via the same underling rule. In this sense, when the students write an Excel formula using variable-cell, they see through it and via the “dragging” action the general relationships between the variables (see, for example, Figure 7, utterances 33-35, and Figure 3, utterances 36-37). In the algebra course used in this research, because of technical limitations, the students had an activity based on the use of Excel only once per about 3 lessons, and they were given instructions where and how to use it, in the near future we will face have to face a situation where each student will have his / her own computerized tool all the time. An interesting research question might be the following: How will students benefit (or not) from “free” use of Excel in algebra learning, when the tool will be available all
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the time, and the students will be able to use it when they will feel the need? How will social mathematical interaction in the classroom be affected? ACKNOWLEDGMENTS The authors would like to thank Tommy Dreyfus, Gaynor Williams, the three anonymous reviewers and the editor for their wise and thoughtful comments. N OTE 1. We defined abstraction as a process in which students vertically reorganize previously constructed mathematics into a new mathematical structure (Hershkowitz et al., 2001).
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and RINA HERSHKOWITZ Weizmann Institute of Science Rehovot Israel MICHAL TABACH
BARUCH SCHWARZ
Hebrew University Jerusalem Israel
