Salma It is the same⦠we can call them legs and hypotenuse. 5. Inter. So, how can you express the relations? 6. C1.3 Salma [writing the answer] The sum of the ...
CONSTRUCTING AND CONSOLIDATING MATHEMATICAL KNOWLEDGE IN THE GEOGEBRA ENVIRONMENT BY A PAIR OF STUDENTS Anabousy Ahlam1,2, Tabach Michal1 Tel-Aviv University1 Alqasimi Academic College of Education2 Understanding how students construct and consolidate abstract mathematical knowledge is a central aim of research in mathematics education. Abstraction in Context (AiC) is a theoretical-methodological framework for studying the processes involved in constructing abstract mathematical knowledge as they unfold in different contexts. This research uses the AiC framework to examine the processes used by two seventh-grade students working on a sequence of three tasks to construct and consolidate the Pythagorean Theorem and its expansions in the GeoGebra environment. The findings indicate that the pair constructed the majority of the expected knowledge elements. The two students also consolidated some of the components that were built. INTRODUCTION Understanding how students construct and consolidate abstract mathematical knowledge is a central aim of research in mathematics education (Dreyfus, 2012). Some researchers have tried to understand how students construct knowledge, especially deep, abstract mathematical knowledge such as concepts and strategies in learning situations. These researchers aimed at describing and understanding processes of knowledge construction and the conditions under which these processes (fail to) happen. Abstraction occurs in different contexts, among them mathematical, social, curricular and learning-environmental. Abstraction in Context (AiC) is a theoretical framework for describing processes of abstraction in different contexts (Dreyfus, Hershkowitz, & Schwarz, 2001). These researchers defined abstraction as a process of vertically reorganizing previous mathematical constructs into a new structure. The AiC framework postulates that the genesis of abstraction passes through a three-stage process: the need for a new construct; the emergence of the new construct; and the consolidation of that construct. The emergence of a new construct is described and analyzed by the RBC model: recognizing (R), building-with (B) and constructing (C). Recognizing refers to the learner's realization that a previous knowledge construct is relevant for the situation at hand. Building-with involves the combination of recognized constructs in order to achieve a localized goal, such as the actualization of a strategy, a justification or a
Anabousy & Tabach
solution to a problem. Constructing consists of assembling and integrating previous constructs by vertical mathematisation to produce a new construct. Consolidation is a never-ending process through which students become aware of a construct, with the use of the construct becoming immediate and self-evident. Students’ confidence in using the construct increases, and students demonstrate more flexibility in using the construct (Dreyfus & Tsamir, 2004). Consolidation of a construct is likely to occur whenever a construct that emerged in one activity is built-with in further activities. Hence, consolidation connects successive constructing processes and is closely related to the design of sequences of activities that enable it. Various studies have used AiC to investigate learning processes in different contexts. Kidron and Dreyfus (2010) studied L’s justification of bifurcations in a dynamic system, and specifically how instrumentation led to constructing actions and how the roles of the learner and a computer algebra system (CAS) intertwine during the process of constructing a justification. They showed that certain patterns of epistemic actions were facilitated by the CAS context. Dreyfus et al. (2001) used the AiC framework in the context of collaborative peer interaction and identified types of social interaction that support processes of abstraction. Dreyfus and Tsamir (2004) conceptualized and studied the consolidation of students’ constructs within the AiC framework. They developed an empirically based theoretical analysis of consolidation that emerges from a sequence of interviews with a talented student on the topic of infinite sets. They showed that consolidation can be identified by psychological and cognitive characteristics of self-evidence, confidence, immediacy, flexibility and awareness. They also found three modes of thinking conducive to consolidation: problem solving, reflective activity and an intermediate mode. The current study aims at tracing processes of constructing and consolidating abstract mathematical knowledge in two seventh-grade students who solved a sequence of three tasks about the Pythagorean Theorem and its expansions in the GeoGebra environment. METHOD Salma and Sahar, two seventh-grade students from the same class, participated in the study. Their teacher attested to their high mathematical achievements. The participants and their parents gave their consent to participate in the study. Three exploratory tasks were designed for the study. The first task dealt with the Pythagorean Theorem and its proof. The other tasks were based on the "What if not?" strategy (Brown & Walter, 1993). The second task dealt with the relations between areas of squares built on the sides of an obtuse/acute triangle, while the third task dealt with the relations between areas of regular polygons built on the sides of any triangle.
Anabousy & Tabach
GeoGebra was selected as the technological tool due to its dynamic nature and ease in use. An appropriate GeoGebra applet was built for each task. In each of the three tasks the students were asked to propose a hypothesis regarding the mathematical situation and then to experiment with GeoGebra to verify or refute their hypothesis. Finally, they explained / proved the constructed mathematical concept / relation. Each task lasted about 45-55 min. and was recorded and transcribed verbatim. In the a priori analysis for the expected construction and consolidation processes, the main knowledge elements and their sub-elements were assumed for all tasks. Figure 1 presents the a priori analysis of the connections between the knowledge elements subsequently described (E1 & E2 in the first task, E3-E6 in the second, and E7-E11 in the third task). An operational definition was developed for each element to guide the analysis of the students' abstraction activity. Due to space constraints we now offer only a few definitions.
Figure 1: The connections between assumed knowledge elements E1: Articulating the Pythagorean Theorem, with the following sub elements:
E1.1: A right angle triangle whose sides are 3, 4, and 5 satisfies the Pythagorean Theorem. E1.2: Other right angle triangles satisfy the Pythagorean Theorem. E1.3: Generalization: All right angle triangles satisfy the Pythagorean Theorem. E2: Proof of Pythagorean Theorem, with the following sub-elements:
E2.1: Recognizing two squares as congruent. E2.2: The area is an additive magnitude. E2.3: The relationship between the areas of two congruent squares can be expressed as two equivalent algebraic expressions. E3: The relations between areas of squares built on the edges of an obtuse triangle. E4: The justification of E3. E7: The relations between areas of equilateral triangles built on the sides of any triangle. FINDINGS The pair successfully constructed all the knowledge elements related to the Pythagorean Theorem and its expansions, with the exception of C1.1 and C11, which
Anabousy & Tabach
were partially built. Moreover, the pair consolidated some of the constructs that were built in the sequence of three tasks. This consolidation occurred in different situations: (a) when the pair tried to explain a certain constructed element (e.g., when explaining C3); (b) when the pair used a constructed element in further R and B actions to construct another element (e.g., using C1.1 to construct C1.2); and (c) when the pair thought reflexively about a constructed element (e.g., when the pair expressed their confidence in the correctness of C1.3). The processes of constructing and consolidating knowledge in this study occurred in the context of the given sequence of tasks and the social and technological contexts. We demonstrate the analysis of three episodes: (i) the final constructing of C1.3 and the pair's confidence in its correctness; (ii) the beginning of constructing C3 and consolidating C1; and (iii) the construction of sub-element of C7 and consolidation of C3. Episode 1: The final construction of C1.3 (first task) 1.
Inter.
So, what do you conclude?
2. B1.2
Salma
The sum of the areas of the medium and small squares is equal to the area of the large square.
3.
Inter.
How can we refer to these squares other than as small, medium and large?
Salma
It is the same… we can call them legs and hypotenuse.
5.
Inter.
So, how can you express the relations?
6. C1.3
Salma
[writing the answer] The sum of the areas of the small square built on the leg BC and the medium square built on the leg DC is equal to the area of the large square built on the hypotenuse CD.
7.
Sahar
Excellent, let's answer the next question.
8.
Salma
OK
9.
Sahar
[Reading the question] Are you sure that the relations you found in question 4 are satisfied in any right angle triangle? Explain!
10.
Salma
Sure.
11.
Sahar
Why? Explain!
12. R1
Salma
Because the triangle will not be a right angle triangle if these are not equal.
13.
Sahar
Yes.
14.
Salma
Write it down. You are cleverer than me, right?
15. R1.2
Sahar
[Writing] Yes, because we have observed many cases, right?
4. R
legs,
hypo.
Anabousy & Tabach
16.
Salma
No, wrong, give me the pencil…
17.
Inter.
You are saying that…
18. R1.2
Sahar
Because we have observed many cases and the triangle was a right angle triangle.
19. R1
Salma
We saw that we could not have any two numbers, three numbers, where the sum of these two areas, the small and the medium, is equal to the large one. That is, if the area of the two squares is not equal to the area of third square, the triangle will be acute or obtuse… that’s why it's sure, sure.
20.
Sahar
OK, OK
At the beginning of constructing C1.3, the pair thought they could build a right angle triangle from any three sides. However, after working with the applet, they realized they could not do so. They also realized that to be a right angle triangle, the triangle has to satisfy the Pythagorean Theorem [turn 1]. In this construction process (generalization of the Pythagorean Theorem), Salma led the construction actions (R, B and C). At the beginning, she expressed C1.3 inaccurately [2]. By recognizing the legs and the hypotenuse [4], Salma improved the construction of C1.3 [6]. Sahar agreed with Salma in all her actions. The pair of students was confident about the correctness of C1.3 [10, 13]. While discussing the correctness of C1.3, they thought reflexively about C1.3 and consolidated it. Sahar was confident because she tried many cases [15], and Salma was confident because she began to construct the inverse theorem [12]. Statements 12 and 19 show that Salma inadvertently constructed the inverse theorem. Thus, the knowledge elements from the second task began to build the first one. Episode 2: Beginning of construction of C3 and consolidation of C1 (second task) The pair collected data (areas of squares built on edges of obtuse triangle) in three cases: 1.
Sahar
2.
Salma
3.
Sahar
4.
R1 B1
5.
We have to find the relations.
16, 9.994, 48.72 [areas of squares built on sides of triangle, fig.2a] Salma… find the relationship, I do not know how!
Salma 9 plus 4, 9 plus 4 [trying Pythagorean Theorem in the obtuse triangle, fig. 2b] Sahar
13
Salma
16 plus 7 [trying Pythagorean Theorem in obtuse triangle, fig. 2c]
7.
Sahar
What?!
8.
Salma 16 plus 7, 33, 23 increase it little
6.
B1
Anabousy & Tabach
9.
B1
10.
Sahar
Plus 11, it is not as the previous relationship, I am sure.
Salma You're right, it is not.
In Episode 2, the students tried to see whether "the equivalent relationship" found previously (C1: Pythagorean Theorem) holds even for an obtuse triangle. Thus, by building-with C1 (towards C3) they saw that the relations in the given situation are different from C1 [8, 9]. In the construction actions, Salma led for the most part. She recognized C1 [4] and built-with it the new relations [4, 6, 7, 9]. At the end of this episode, they both agreed that C1 would not hold in the case of obtuse triangles. It is important to note that in further discussions, the pair found it difficult to recognize "less than\more than" relations and that recognition of these relations was suggested by the researcher.
Figure 2a
Figure 2b
Figure 2c
Figure 2: The three cases the students worked on in Episode 2 Episode 3: Constructing C7.2 and consolidating C3 Salma This triangle is obtuse, write!
1.
R
2.
R3
3.
B3 Salma Yes, yes [checking]. It's not the same…Hhhhhh…no, sorry, it's the same, the same… we said that it is less than. Let's try another obtuse triangle 6, 6… it’s also the same… right? [Fig. 3a].
Sahar
Wait! Before we found that in obtuse triangles the sum of the areas of the medium and small squares is less than the area of the large square.
4.
Sahar
Because you did not change the length of the sides, you changed the angles.
5.
Inter.
You can change the length of the angles from here [the applet]
6.
B3 Sahar
9+10= 19, if we increase it by 2, the sum will be 21. . . so it’s right [Fig. 3b].
7.
Salma Sure it’s right.
8.
Sahar
We can say 20.
Anabousy & Tabach
9.
Salma Write: "less than"… Do you want to try more cases? Sure it's "less than".
10.
Sahar
11.
Salma "Less than" relationship.
12.
Sahar
13.
Salma "Less than" relationship! Write it down. After that we'll explain.
14. C7.1 Sahar
I will write it down. In the case of an obtuse triangle… mmm.. What did we say before? The sum…, but we have to explain our "aim". The relationship is…
15.
Salma The relationship is "less than"… the sum of the areas of the two squares
16.
Sahar
built on the edges comprising the obtuse angle …
Figure 3a
…built on the edges comprising the obtuse angle is less than the area of the square built on the edge opposite to the obtuse angle. Figure 3b
Figure 3a & 3b: The two cases explored by the pair in Episode 3 Episode 3 shows the process of constructing the sub-element C7 (C7.1, the case of an obtuse triangle). The other two elements (the case of a right/acute triangle) were constructed similarly. The episode also demonstrates the process of consolidating C3. The two students constructed collaboratively: Salma recognized the obtuse triangle [1] and Sahar recognized C3 [2]. Then, the pair built-with these constructs [3, 6] to generate the new construct and expressed it collaboratively [15, 16]. The consolidation of C3 occurred when the pair recognized and built-with this construct to generate the new one (C7). They accomplished this with relative ease and immediacy. This indicates that they consolidated the previous construct. DISCUSSION The study traced the processes of construction and consolidation of the Pythagorean Theorem and its expansions by a pair of students who worked on a sequence of three tasks in the GeoGebra environment. The findings indicate that the pair constructed the majority of the knowledge elements and consolidated some of the constructed elements. The construction and consolidation processes occurred in a technological, social and task context.
Anabousy & Tabach
The pair's working processes included exploration of different cases, generalization and explanation/proof. GeoGebra supported the learning during the exploration phase (for similar findings, see Becta, 2003). For example, in the process of constructing C1, the pair explored different cases by manipulating the triangle in the GeoGebra applet. The pair selected "representative" cases such as polygons with different types of side lengths: large/small numbers, fractions and integers. Exploring these different cases in GeoGebra enabled the students to generalize the relations (Dikovic, 2009). Furthermore, our findings show that the pair consolidated some of the constructed elements. The consolidation occurred during a task and in the following tasks. Consolidating during a task occurred in three cases: (1) when the pair thought reflexively about a constructed element while explaining (e.g., while trying to justify C3), similar to Dreyfus and Tsamir (2004); (2) while they expressed their confidence in the correctness of a construct (e.g., expressing their confidence in the correctness of C1.3); and (3) while they used a constructed element in R and B actions in order to build a new element (e.g., when the pair used C1.1 in order to build C1.2), as in Tabach et al. (2006). References Becta, (2003). What the Research Says about Using ICT in Maths. UK: Becta ICT Research. Brown, S. I., & Walter, M. I. (1993). Problem posing in mathematics education. In: S. I. Brown, & M. I. Walter (Eds.), Problem posing: reflection and applications (pp. 16–27). Hillsdale, NJ: Lawrence Erlbaum Associates. Dikovich, L. (2009). Applications GeoGebra into teaching some topics of mathematics at the college level. Computer Science and Information Systems, 6(2), 191–203. Dreyfus, T. (2012). Constructing abstract mathematical knowledge in context. Unpublished technical report, Tel Aviv University. Dreyfus, T. & Tsamir, P. (2004). Ben's consolidation of knowledge structures about infinite sets. Journal of Mathematical Behaviour, 23 (3), 271-300. Dreyfus, T., Hershkowitz, R. & Schwarz, B. (2001). Abstraction in Context II: The case of peer interaction. Cognitive Science Quarterly 1(3/4), 307-368. Kidron, I., & Dreyfus, T. (2010). Interacting parallel constructions of knowledge in a CAS context. International Journal of Computers for Mathematical Learning, 15, 129-149. Tabach, M., Hershkowitz, R., & Schwarz, B. (2006). Constructing and consolidating of algebraic knowledge within dyadic processes: A case study. Educational studies in mathematics, 63(3), 235-258.
