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Abstract
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Introduction
Proof is a center of mathematical science and a key area of mathematics education research (Bar-Tikva & Judith, 2009). Proof and proving are the basis for doing, understanding and communicating mathematical knowledge (Stylianides, 2007), which is why the proof has gained increasing attention in recent years (Stylianides, 2007; Hanna & Villiers, 2012). Various researches on the process of constructing proof have been done (Selden, Selden & Benkhalti, 2017; Inglis & Mejia-ramos, 2009; Stylianides & Stylianides, 2009). The results of the studies indicate that in constructing mathematical proof one does not always follow deductive rules, but there are often many inductive aspects such as empirical proof, special cases or graphs (Tymoczko, 1986; Arzarello, 2007; Feferman, 2000; Mejía-Ramos & Inglish, 2008; Burton, 2004) Azarello (2001) used to understand propositions to be proved and to find relationships between propositions (Arzarello, Paola & Sabena, 2009). Arguments will occur during the process of constructing a proof to organize some of the justifications (arguments) that have been produced (Boero, Garuti, & Mariotti, 1996) and produce a valid conclusion.
Toulmin's argumentation has been used by many mathematical education researchers to analyze the process of constructing proofs (Knipping, 2008; Pedemonte, 2007; Mejía-Ramos & Inglish, 2008), constructing definitions (Ubuz, Dincer, & Bülbül, 2013) and solving mathematical problems (Ubuz, Dincer, & Bülbül, 2012; Solar & Deulofeu, 2016) because it provides space for the use of informal argumentation. Toulmin's argumentation is a scheme consisting of three main components namely data, claims and warrant and three complementery components namely backing, rebuttal and modal qualifier. [2]
According to Toulmin (2003), claim (C) is a statements or conclusions made based on data. Data (D) is a ‘foundation’ of argument based, facts relevant to the claim. Warrant (W) is like a ‘bridge’ that links data and claims and becomes tha basis of the thought or reason used to generate conclusions. A warrant may take the form of formulas, definition, axioms or theorems or create analogies, drawings or diagrams and graphs. A warrant is reinforced by backing (B) which is further evidence or addittional reasons needed. Rebuttal (R) is a statements that denies the resulting conclusion if the condition is not fullfilled. [3]
The idea of Toulmin is not final because it can be re-established in various ways (Hitchcock & Verheij, 2006)(). Many mathematical education researchers used Toulmin model by reducing it to their needs. Krummheuer (1995) and Pedemonte (2003, 2007) reduced backing, rebuttal, and modal qualifier. Knipping (2013), Ubuz et al., (2012, 2013), Pedemonte, (2007) and Chen and Wang (2016) reduced rebuttal and modal qualifier. These researchers think that backing is always theorem or a definition. This study will focus on backing component and relation to warrant (including the interpretation of formal reasoning and formal proof) and other argumentation components.
According to Pedemonte (2003), backing is required if truth warrant (authority of the warrant) is not acceptable straight away. Toulmin in Chen & Wang (2016) explains that the validity of the argument depends greatly on the backing (whether an argument was valid or not, greatly depended on its Theory backing). This shows that backing as one of the components in the Toulmin scheme plays an important role in determining the truth of claims but it is under-researched. Ubuz et al., (2012) analyzes the backing sourced from teachers in a collective argument while backing sourced from students was always viewed as a definition or theorem (Inglis, Mejia-Ramos, & Simpson, 2007; Chen & Wang, 2016). But if argumentation occurs individually, the backing that comes from students becomes very important. Thus, this study sought to answer the questions: Is the backing sourced from the student always in the form of a theorem or a definition? Do students use only one type of backing to reinforce warrants in generating claims? What is the relationship of backing to rebuttal and qualifier in the proving?
A warrant can be supported or reinforced in different ways (Toulmin, 2003). For example, 'A whale will be (i.e. is classifiable as) a mammal', 'A Bermudan will be (in the eyes of the law) a Briton', 'A Saudi Arabian will be (found to be) a Muslim'. The words in brackets indicate the difference. The first warrant is supported by a natural classification that has been accepted, a second warrant supported by law or rule and a third warrant supported by statistical information or calculations. Backing for warrants can be expressed in the form of definite statements as direct support of conclusions. The types of a backing of warrant support depend on the field of argument. Referring to the backing of warrants can explicitly occur based on statistical reports, experimental results or references.
Methods
This research was a qualitative research with phenomenology design that aims to explain the phenomena that appear in the argumentation with the reasons used when constructing the evidence. In phenomenology design, data is collected through in-depth interviews to analyze, identify, understand and explain the students' thinking processes underlying each of their reactions and perceptions (Fraenkel, Wallen, & Hyun, 2012) on the given problem. This design is considered appropriate to characterize the types of backing that are generally used by students and their relationship with a warrant.
The problem of algebra was given to the subject. It is a mathematical statement that is wrong (disproved) so that students are expected to be able to determine counter example which in Toulmin model is called rebuttal. This problem is designed to show the components of argumentation (data, warrant, backing, claim and rebuttal) and allows various ways of completion and use of various forms of warrant and backing. Prior to use in research, this assignment sheet has been validated by experts. The tasks are given as follows:
Suppose the function
by the formula f(x)=x^2 and g:R⟶R by the formula g(x)=x . Investigate whether f(x)≥g(x) for all x real numbers?
The data collection began with providing a proving problem to 42 students for individual completion. During work, students are asked to voice what is thought (think aloud). Students were given the opportunity to explore, write and state all their thoughts and ideas without being limited by time. They would finish when they feel they are not able to finish it or have no idea anymore. During the resolution of the problem, we observed and recorded all behaviors including thinking aloud students. The students were then interviewed individually to explain the process of thinking when constructing proof. The interview procedure used a semi-structured clinical interview form (Ginsburg, 1981), i.e., the researcher asks participants to share what's on their mind and provide explanations for unclear answers or writings, and asks to clarify them. During the interview, students were given the opportunity to improve their answers, without the intervention of the researcher. Stu dents who answered correctly and used backing were selected as research subjects. The selection of subject was done until the data collected has been saturated. Of the 42 participants, 23 students answered the problem correctly and 19 students answered incorrectly.
The researcher then analyzed all data (observation data, interviews, think a loud and field notes) and reduced things that were considered unimportant. The analysis of results of interview employed a multi-case study approach developed by (Bromley, 1986). The researcher classified the answers of students in three categories and discusses them based on the framework of definitions that have been made and made interpretations and conclusions. All research data were interpreted on the basis of indicators of Toulmin's component argument, then describes a complete argumentation structure to explain students' thinking processes. The conclusions were focused to answer question, Is the backing sourced from the student always in the form of a theorem or a definition? and their relation to qualifier and rebuttal.
Findings and Discussion
Based on students' work, thinking, field notes and interviews, we found that backing from students is not always in the form of definitions or theorems. Students also use other backings such as examples of numbers and calculations and graphs to help them produce claims. Both types of backing are used because students cannot make decisions based on a warrant, both deductive and inductive. The three types of backing used by students are named numerical backing, graphical backing and reference backing. The terms numerical backing and graphical backing refers to the terms used by Arzarello et al., (2009) on numerical registers and graphical registers, while reference backing refer to Toulmin's (2003) references to indicate that the backing used refers to laws or certain rules. The following will explain each backing based on the sample used by the students.
Numerical backing
Before arriving at the final claim, S2 make several claims, called as conjectures (Mason, Burton, & Stacey, 2010). The numerical backing is used as a ‘crucial experiment’ (Balacheff, 1988) to convince oneself (Laamena, Nusantara, Irawan, & Muksar, 2018b) of the possibility of the truthfulness of statements. Students who use numerical backing, verified several cases by conducting an important experiment to show the truth of the conjecture that has been produced by generating generic examples. Based on some empirical evaluations, their doubts about the conjectures are reduced to subsequent proving processes (Inglis, Mejia-ramos, & Simpson, 2007). From some of the claims they produce, they then systematize as one of the main objectives of proof and argument that is to compose their work in a deductive system (de Villiers, 1990).
No | Kategori | Nilai |
---|---|---|
1 | X | 65 |
2 | Y | 70 |
3 | Z | 90 |
Before arriving at the final claim, S2 make several claims, called as conjectures (Mason, Burton, & Stacey, 2010). The numerical backing is used as a ‘crucial experiment’ (Balacheff, 1988) to convince oneself (Laamena, Nusantara, Irawan, & Muksar, 2018b) of the possibility of the truthfulness of statements. Students who use numerical backing, verified several cases by conducting an important experiment to show the truth of the conjecture that has been produced by generating generic examples. Based on some empirical evaluations, their doubts about the conjectures are reduced to subsequent proving processes (Inglis, Mejia-ramos, & Simpson, 2007). From some of the claims they produce, they then systematize as one of the main objectives of proof and argument that is to compose their work in a deductive system (de Villiers, 1990).
References
- Arzarello F. The proof in the 20th century. Theorems in School. 2001; 2(1):43-63.
- Edwards L.D.. Odd and even: Mathematical reasoning processes and informal proofs among high school students. Journal of Mathematical Behavior. 1999; 17:498-504.
- Burton L. Mathematicians as Enquirers. Learning about Learning Mathematics. Kluwer: Dordecht; 2004.
- Gall, M, D., Gall, J. P., & Borg, W. R. (1983). Educational research an introduction third edition. USA: Pearson Education.