This qualitative study with phenomenology design aims to investigate the use of backing and its relation to rebuttal and qualifier in prospective mathematics teachers’ (PMTs) argumentation when constructing a mathematical proof about algebraic function. The data were collected through subjects' works on the proof, recorded think-aloud data, and in-depth interviews. Data analysis was guided by Toulmin’s argumentation scheme. The results show that the PMTs used three types of backing, i.e., backing in the form of definitions or theorems (

In mathematics, proofs have a very important role, and thus they are a key area of mathematics education research

Toulmin's argumentation scheme has been used by many mathematical education researchers to analyze the process of formulating proofs (

The Toulmin scheme has not to be used in the exact way it was initially proposed: It can flexibly be re-structured or focused in various ways

A warrant can be supported or reinforced in different ways

The current research pursued the investigation of the backings as a small number of prior researches did (e.g.,

The study follows a qualitative research method: A phenomenology design is used to explain the phenomena that appeared in the argumentation structure of PMTs, i.e., specific types of backings used when constructing a proof and its relation to rebuttal and qualifier. Following the phenomenology design, data was collected through subjects’ works on the proof, recorded think-aloud, and in-depth interview. The in-depth interview aims to analyze, identify, understand, and explain the students' thinking processes underlying each of their reactions and perceptions

For analyzing the argumentation structures, an algebraic problem was given to the subjects: A wrong mathematical statement was given and the students were expected to be able to determine counter-examples (rebuttals according to the Toulmin scheme). The problem was designed to show the components of argumentation (data, warrant, backing, claim, and rebuttal) and allows various ways of completion by using various forms of warrant and backing. Prior to use in research, the problem has been validated by experts. The tasks were given as follows:

^{2} dan g:R⟶R by the formula g(x)=x.

The data collection began with providing the proving problem to forty-four (42) PMTs for individual completion. During their problem-solving process, they were asked to voice what they thought (think-aloud method). They were allowed to explore, write, and state all their thoughts and ideas without being limited by time. They should finish when they felt that they were not able to finish it or had no further ideas. When they worked on the problem, we observed and recorded all behaviors, including verbalized thoughts (according to the think-aloud method; the think-aloud-data was recorded via camera and then transcribed for further analysis). The subjects were then interviewed individually to explain their process of thinking when constructing the proof. For the interview procedure, a semi-structured clinical interview form was used

We then analyzed all data (subjects’ works on the proof, interviews, video graphed think-aloud data) and narrowed them through a sequencing process to the research-question-relevant parts. The analysis of the data was according to a multi-case-study approach by subsequently categorizing the narrowed data

In this part, we will provide; the subjects’ works on the proving problem which used backing along with some supporting excerpts of transcript, findings which drawn from the subjects’ works and a discussion on the findings. The subjects’ answers on the given problem were categorized into three as follows:

This first type of answer was done by S1 and S2 by clustering the real numbers into some groups of numbers. S1 expressed the real number as ^{a}/_{b} where ^{+} where ^{-} where

The excerpt of the interview below reveals the thinking process of S1 and his reason for using the sample of numbers.

^{2}=a^{2}/b^{2} (pause) ... for a^{2}/b^{2} be f (x)

^{2}/b^{2} ?

^{2}/b^{2}

S2 grouped real numbers into positive numbers, null, negative numbers and fractions (Figure 3). S2 used samples of the numbers to produce the claims. The interview below shows that S2’s claim was based on samples of the numbers from the grouped numbers.

^{2}≥x

^{2}≥x), what is it? Please, explain slowly

^{2} must be positive. So, I have x^{2}≥x

^{2}≥x is positive

^{2}=4, it is 4>2. So, x^{2}>x. Take x=1 where x>0. 1^{2}=1 provides x=x^{2}

S3 and S4 did not use a sample of the numbers at all, but they used the properties of numbers. Figure 4a and Figure 4b show their works.

S3 had x=1/a as counter-example. S4 had a short and correct answer but the interview shows that she used the properties of numbers before coming to ^{2}>x

In this category, there are two forms of answers (Figure 5a and 5b). S5 drew the graphs of ^{2}^{2}≥x

Referring to all subjects' works, as shown in Figure 2 to Figure 5b and the excerpts of interviews, we found that

It is also found that three types of backing were used by the subjects namely

The subjects began proof by exploring the data

The excerpt of the interview (EoI-1) shows that for ^{2}/b^{2} . The justification a/b <a^{2}/b^{2} is based on a ^{2}/b^{2} (^{2}/b^{2} . Figure 6 illustrates S1’s argument based on the Toulmin scheme.

A modal qualifier (_{2}) and the third claim (C_{3}) for the relationship between the other

S1 constructed the three initial claims of the first claim (C_{1}), second claim (C_{2}) and third claim (C_{3}) before producing a final claim (C). The three claims are based on four deductive warrants W_{1}, W_{2}, W_{3}, and W_{4}. W_{2} and W_{3} are supported by numerical backing (Bn1 and Bn2) which are selected based on grouping of _{2} (_{3} (

The Toulmin argumentation scheme for the first claim is explained in Figure 8 with two warrants and two backings. S2 claimed that

Before arriving at the final claim, S2 made several claims. ^{2}≥x

S3 used a

Based on Figure 10, at this stage, S3 had not found the counter-example (

S4 also used a ^{2} <1/a so that

^{2}

^{2}

^{2}>a and 1/a^{2}

^{2}>a then 1 divided by a large number, it will be less than 1 divided by a small number

The

^{2}≥x

^{2}

S5’s argumentation about using graphs concerns the graph as a representation of the real numbers and functions ^{2} ^{2}

The ^{2}≥x

This false claim was not realized by S6. Then he was aware of proving the claim ^{2}≥x^{2}≥x ^{2}≥x^{2}≥x

In proving, S6 did not only use a

There are rebuttal differences generated by these three types of backing.

In the above three sections, we have described three types of backings (

A

The word ‘investigating’ of the task given in this research is understood by the students as a hint to a

The current research focused on a false mathematics statement. The students only need to provide a counter-example to prove it. In this case, the proof takes place shortly. We could not investigate a complex systemization in case the students apply various types of warrants and backings. If a correct statement is used,

The research found that, in a mathematical proof, the students do not only employ definition or theorem (

The backings

A backing plays an essential role in (1) strengthening the warrant when the warrant is unable to justify the claim; (2) finding counter-examples (rebuttal); and (3) providing certainty (qualifier) for the claim.