THE PROCESS OF STUDENT COGNITION IN CONSTRUCTING MATHEMATICAL CONJECTURE I

This research aims to describe the process of student cognition in constructing mathematical conjecture. Many researchers have studied this process but without giving a detailed explanation of how students understand the information to construct a mathematical conjecture. The researchers focus their analysis on how to construct and prove the conjecture. This article discusses the process of student cognition in constructing mathematical conjecture from the very beginning of the process. The process is studied through qualitative research involving six students from the Mathematics Education Department in the Ganesha University of Education. The process of student cognition in constructing mathematical conjecture is grouped into five different stages. The stages consist of understanding the problem, exploring the problem, formulating conjecture, justifying conjecture, and proving conjecture. In addition, details of the process of the students’ cognition in each stage are also discussed.

Mathematical conjecture is important in mathematics.It plays a vital role in mathematical development as formalization of conjecture is good and inevitable for mathematics, as large mathematical theories get bigger (Mazur, 1997).Constructing mathematical conjecture involves abstraction and generalization processes related to ideas that are initially hypothetical in nature (Norton, 2000;Nurhasanah, Kusumah, & Sabandar, 2017).In addition, constructing mathematical conjecture and developing proofs are two fundamental aspects of professional mathematical work (Alibert & Thomas, 2002) and is the first step in invention (National Council of Teacher of Mathematics-NCTM, 2000).
Beside Mathematics, mathematical conjecture also plays an important role in mathematics instruction.NCTM (2000) stated that a program in mathematics instruction should enable all students to recognize reasoning and proof as fundamental aspects of mathematics, make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs, and select and use various types of reasoning and methods of proof.Many researchers such as Boero, Garuti, Lemut, and Mariotti (as cited in Manizade & Lundquist, 2009) argue that the student must work through internal arguments and sort through solutions that are plausible, similar to ones that a mathematician goes through when building a proof during the process of constructing a conjecture.Boero, Garuti, Lemut, and Mariotti propose that the process of constructing or building conjecture should be emphasized more in mathematics instruction.Besides, constructing mathematical conjecture or making a prediction has three benefits in the mathematical classroom since it can reveal students' conception, plays an important role in reasoning, and fosters learning (Lim et al., 2010).Indonesian current national curriculum known as Kurikulum 2013, stipulates a scientific approach as the common learning approach for all subjects taught in all school levels where building or constructing conjecture is one of the activities in reasoning (Kemendikbud, 2013).
Different researchers give different definitions for mathematical conjecture.Ponte et al. (1998) state that a mathematical conjecture is a statement that answers a certain question and is considered to be true.Pedemonte (2001) states that a conjecture is a statement that is strictly connected to an argument and a set of conceptions where the statement is potentially true because some conceptions allow the construction of an argument that justifies it.However, Norton (2004) states that conjectures are ideas formed by a person (the learner) that satisfies the following properties: the idea is conscious (though not necessarily explicitly stated), uncertain, and the conjecture is concerned with its validity.
Conjecture in this paper is synthesized from these researchers.Mathematical conjecture is a mathematical statement that is hypothetical in nature, where the statement is potentially true and is constructed by the students using their own knowledge based on the information provided or the given problem.
The truth or falsity of conjecture is proven through a reasoning process by using logical rules or a counter example.Once a conjecture has been proved, then it becomes a valid statement (Pedemonte, 2001).Proving a conjecture for a mathematician or a novice, such as a student, is generally different (Fiallo & Gutierres, 2007).In general, students prove conjecture empirically, narratively, visually, and algebraically (Healy & Hoyles, 2000).
Most research on the process of student cognition in constructing mathematical conjecture focuses on how students construct mathematical conjecture.In addition, testing conjecture constructed by other students is also a topic of interest (Jiang, 2002).Conjecturing in mathematics teaching and the learning process has become an important topic of research in many mathematical fields.
Constructing mathematical conjecture involves a lot of processes of cognition.Some of these processes have been studied by many researchers (Ponte et al., 1998;Pedemonte, 2001;Norton, 2004;Morseli, 2006;Canadas et al., 2007), however, none of these researchers discusses in detail the process of cognition from the beginning, rather, they put more focus on the stages of constructing and proving the conjecture.In this paper, the process of student cognition in constructing mathematical conjecture is discussed in detail from understanding a problem to proving the conjecture.
Understanding the problem is a crucial stage since students start their planning to find a conjecture when they understand the problem.Based on synthesizing the process of cognition in constructing mathematical conjecture proposed by Ponte, et.al (1998) and Morseli (2006) and in combination with Polya's (1945) first step in the problem-solving process, we study the process of student cognition in constructing mathematical conjecture using five different stages.The stages consist of understanding the problem, exploring the problem, formulating the conjecture, justifying the conjecture, and proving the conjecture.Table 1 shows the relationship between the process of student cognition studied here and that studied by Ponte (1998) and Morseli (2006).

METHOD
A qualitative research method was used in this study.In the academic year 2013/2014, six students from the Mathematics Department of Ganesha University of Education were chosen as research subjects.The students were chosen based on their mathematical ability and gender.One male and one female student were chosen respectively from high, moderate, and low mathematical ability.
Data on the process of cognition was collected from each subject through a task-based interview.Each subject was given two mathematics problems from which he or she constructed a mathematical conjecture.After constructing mathematical conjectures from the problems, subjects

Problem 2
For any natural number n, the function related to n such as its value, area, sequence, series, or arch length.
The problems used in the second interview were similar to those in the first interview but with different point names or variables.The interviews were recorded, transcripted, and then analyzed using Miles' and Huberman's (1994) method for analyzing qualitative data.

RESULT AND DISCUSSION
The process of student cognition in constructing mathematical conjecture was explained in five different stages, namely understanding the problem, exploring the problem, formulating conjecture, justifying conjecture, and proving the conjecture.
The results of the data analysis for all subjects were abstracted to obtain a general idea about the students' process of cognition in constructing mathematical conjecture.Abstraction as a process of cognition for all subjects in constructing mathematical conjecture was done by determining common processes for all subjects and eliminating the condition of the problem used.The process of student cognition in constructing mathematical conjecture is specified in the five stages mentioned previously.
The detailed process of cognition in each stage is given in Table 3. problem to be formulated as conjecture. Writing conjecture by referring to the result from an exploration of a problem, mathematical language, and sentence type as a point of reference. Believing the formulation of conjectures can be understood by other people.Justifying the conjecture  Explaining the reasons for conjecture by using a picture or graph, measuring or counting, or a mathematical connection by relating relevant mathematical knowledge to its corresponding conjecture. Generalizing the conjecture is done by observing some cases to find a property or pattern and then visualize it so it is valid for all other cases. Being aware of the deficiency or mistake underlying the formulation of a conjecture.Identifying the deficiency or mistake underlying the formulation of conjecture or its reason enables students to correct the conjecture.Proving the  Being aware that the truth of conjecture must be proved and giving expression to this in steps.conjecture  Choosing kinds of proof according to the constructed conjecture. Organizing the proof.Proving the conjecture is done by showing figures or graphs, writing mathematical sentences, counting, connecting relevant mathematical knowledge, and concluding that the conjecture has been proved.Some of these processes have been discusses in Canadas, et.al (2007).
An interesting finding was obtained in stage 2. To find conjecture, students drew some figures Researcher : Can you explain them?Student S2 : These graphs are families of power function graph with natural numbers as its power.For x = 1, the graph is a straight line, for x = 2, the graph is part of a parabola, and so on.Researcher : Why did you make four graphs?Student S2 : For ease of seeing the relationship between them.By drawing some graphs, I can see its value for a given x in its support area enclosed by the arch and x-axis, and arch length when n changes from 1, 2, 3 and 4. A similar dialogue was obtained from other subjects in this stage when they were asked how they would make someone believe their conjecture.
The proof of conjecture organized by students was consistent with Healy and Hoyles' (2000) findings.Almost all students use narrative and visual proof for their conjecture in problem 1, whereas they used empirical and algebraic proof for their conjecture in problem 2.An example of narrative proof was given by Student S1 as depicted in Figure 3.He made a conjecture about the form of KLMN by stating, "The form of KLMN is always rectangle".To prove his conjecture, he drew a parallelogram, ABCD, and its bisector lines.Then he showed that both KL and NM and KN and LM were parallel and its angle is 90 respectively by using the congruence property of the triangle.

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were interviewed based on their work.The interview was carried out twice for triangulation purposes with the second interview taking place two months after the first.The following problems were used in the first interview.ProblemGiven a parallelogram ABCD with length AB as a unit and length BC as b unit, draw a bisector line from each point A, B, C, and D. The bisector line from point A intersects the bisector line from point B at point N, from point B and point C at point M, from point C and point D at point L, and from point D and point A at point K. Construct a conjecture about the quadrilateral KLMN such as its form, position, side length, area, or the like.

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Figure 3.A proof by Student S1 An example of visual proof was given by Student S4 as depicted in Figure 4.She made a conjecture about the arch length by stating, "If n increases then the arch length increases".To prove her conjecture, she drew a lot of arches and said that from the graphs, the arch length for n=2 is longer than for n=1 because the two graphs have two common terminal points but the points are connected by a straight line for n=1 and part of a parabola for n=2.A similar reason was used when comparing the length of arch for n=2 and n=3.

Table 1 .
Comparison of steps in constructing mathematical conjecture  Constructing a proof that must be acceptable to the community of mathematicians  Proving the conjecture Table2shows the focus of analyses on the process of student cognition in constructing mathematical conjecture in the five stages.

Table 2 .
Stages in constructing mathematical conjecture and its analysis focus

Table 3 .
Detailed process of student cognition in constructing mathematical conjecture Reading the problem to determine what is given and what is asked.Determining what is given in the problem by using his/her own words.Determining what is asked in the problem by using his/her own words.