Students’ argumentation could indicate their conceptual understanding or reasoning (Heng, Surif, & Seng, 2014). Similarly, Soekisno (2015) asserts that mathematics argumentation is a feasible method to examine students’ ability in problem-solving. Number pattern is one of the mathematics problems which require argumentation (Sadieda et al., 2018). Furthermore, Cross et al. (2008) argue that argumentation support students’ understanding of a concept and extend their knowledge by achieving know ideas. Argumentation involves reasoning which can be used to draw a conclusion of the given information and creative thinking to make an evidence-based statement (Pallant & Lee, 2015). The construction of argument demands cognitive involvement, such as analyzing and understanding data, providing an explanation, supporting ideas, and rebut the validity of an idea, all of which include in reasoning (Heng, Surif, & Seng, 2014). It shows the relation between argumentation and reasoning.

The reasoning is an important tool to learn mathematics (Baroody, 1993). In the Indonesian school curriculum, known as K13, the students are expected to be able to have reasoning and present the topic either in concrete or abstract ways (Depdiknas, 2013). Also, NCTM (2000) underlines that students should be encouraged to reason and think analytically, make and investigate mathematics conjectures, develop and evaluate mathematics argument and proof, and use various kinds of reasoning. Reasoning comprises basic thinking, critical thinking, and creative thinking (Subanji, 2011). Thus it is categorized as high order thinking (HOT). HOT skills are essential in problem-solving, especially problems given in the form of a project (Kusaeri et al., 2019). In conclusion, reasoning plays a paramount role for students in solving mathematics problems.

Zeytun, Cetinkaya, and Erbas (2010) found that teachers’ prediction about students’ reasoning is limited by teachers’ thought regarding the problems given to the students. A part of teacher and learning environment which discourage the emergent of students’ reasoning, the students also contribute to this condition- they are not used to reasoning properly in the classroom. Students often solve the mathematics problems as if they were following the process of reasoning, but actually, they did not. It is called as pseudo-reasoning (Subanji, 2011). This process makes students have an incorrect perception of reasoning. Thus, to develop students’ reasoning, a learning environment which provides a vast opportunity for students to get used to reasoning is required.

The topic of function in the schools is used to introducing to students using correspondence approach which base on the abstract definition, a bit narrow, and focus on the explicit rules (Confrey & Smith, 1994). This approach leads students to pay much attention to the rules or formulas and procedures to describe output values from known input values since students are provided with notations, manipulations, and the formula of function (Umah, As’ari, & Sulandra, 2016). Learning which centres on procedural rules will weaken the students’ reasoning (Kusaeri & Aditomo, 2019).

Covariation approach is an alternative approach to teach function which differs from a convergence approach (Subanji, 2011). It pinpoints to the relation of two structured quantities in the form of algebra, visual in a graph, or real-life situation (Confrey & Smith, 1995). Besides, it focuses on the ability to form the expression of two varied quantities and coordinate the changes within its relation. Covariational does not only confine procedural rules but also include the reasoning (Thompson & Carlson, 2017). In this case, this approach is more appropriate to support students’ reasoning when solving problems involving the graph of a function.

Covariational is defined as the coordination of several quantities, and the change of one quantity causes the change of another quantity (Clement, 1989; Carlson et al.,2002; Koklu, 2007). Slavitt (1997) defines covariational as analyzing, manipulating, and understanding the relationship between changing quantities. Confrey and Smith (1994, 1995) and Thompson (1996) even argue that covariational approach is a more intuitive one formally. Subanji (2011) explains that constructing graphs requires reasoning on covariational problems. It closely relates to the concept of function as one quantity could be an input (independent variable) and another quantity as the output (dependent variable) (Subanji, 2006). In the present study, covariational approach means students’ ability in establishing the image of two different quantities wherein a change in one quantity associates with the change of another one in a specific relation.

The use of the covariational approach in constructing graphs of a dynamic problem can be identified through the levels of covariational reasoning. Carlson et al. (2002) propose mental actions of covariation framework- from mental action 1 (MA1) to mental action 5 (MA5). MA1 (initial coordination) is coordinating the value of one quantity or variable with changes in the other variable. MA2 (the coordination of the direction of change) is coordinating the direction of change of one variable with changes in the other variable. MA3 (major coordination of change) is coordinating the amount of change of one variable with changes in the other variable. MA4 (coordination of average rate of change) is coordinating the average rate of change of the function with uniform increments of change in the input variable. MA5 (the instantaneous rate of change) is coordinating the instantaneous rate of change of the function with continuous changes in the independent variable for the all domain of the function.

Mental actions of covariational reasoning are developed through various arguments (Umah et al., 2016). Each established argument has structure or components which can be utilized to analyze the argument itself (Fischer et al., 2014). The are several models of argumentation employed to analyze covariational reasoning, for example, Toulmin (2003) and McNeill and Krajcik (2011). Toulmin's model comprises detail basic structure of argumentation and covers various informal argumentations. However, Umah et al., (2016) found that the model did not suit students’ ability since the students could only provide data, backing, warrant, and conclusion. Moreover, Inglis, Mejia-Ramos, and Simpson (2007) assert that rebuttal and modal qualifier differentiate between the amateur and advanced mathematicians in using non-formal arguments. The expert tends to focus on the two to make a claim. On the other hand, McNeill and Krajcik’s model of argumentation has been adjusted to students’ ability; thus, it is easier and more applicable to analyze students’ argumentation.

McNeill and Krajcik (2011) further develop Toulmin's model of argumentation to support students involved in a simpler mode of argumentation. It consists of a claim, evidence, reasoning, and rebuttal (McNeill & Krajcik, 2011). A claim is a statement or conjecture which explains a question or a phenomenon (McNeill & Krajcik, 2008; Berland & McNeill, 2010). Evidence is quantitative or qualitative data to answer the question, solve problems, or to decide (Aikenhead, 2005). It can be proof to support a claim. The reasoning is an explanation of why a proof could support a claim (McNeill et al., 2006). The rebuttal is the alternative claim which provides a counter-proof and reason why the existing claim cannot stand (McNeill & Martin, 2011). McNeill and Krajcik’s argumentation could be used to improve verbal and writing argumentation (Sadieda, 2019). It is also promising to analyze formal and informal argumentation (Umah et al., 2016). Students apply formal argumentation to represent their mathematical argumentations in the form of writing. Meanwhile, informal argumentations refer to students' verbal words. In this case, the argumentation model which accommodate both kinds of argumentations.

Prior studies (e.g., McNeill & Martin, 2011; Sadieda, 2019) have used McNeill and Krajcik’s argumentation. Sadieda (2019) analyzed the students’ argumentation in proving sub-group problems. McNeill and Martin (2011) used the model as a rubric to examine the strength and weakness of students' argumentation about the lever. However, the foregoing studies have not addressed covariational reasoning. Several authors (e.g., Umah et al., 2016; Santoso, Budiarta, & Sulaiman, 2019; Rodriguez et al., 2019) studied covariational reasoning, but they used Toulmin's argumentation. The present study identified students' argumentation (claim, evidence, and reasoning which refer to McNeill and Krajcik’s model) when constructing a graph of a function in the five levels of students' mental actions of covariational reasoning (Carlson et al., 2002).

The present study follows a descriptive qualitative approach. Data were collected through covariational reasoning test and interview. The test was adapted from prior studies (Carlson et al., 2002; Koklu, 2007; Stewart, 2008). The first item of the test refers to the bottle problem (Carlson et al., 2002). The second item was taken from one of the covariational problems in calculus book (Stewart, 2008). And the final item was developed from a covariational problem by Koklu (2007).

Problem 1

Imagine the bottle is filled up with water constantly. Construct a graph in which the height of the water as a function of the amount of water in the bottle! Provide a reason for your answer!

This part presents each subject's works on the covariational test and the excerpt of interview transcripts which indicate their claims, evidence, and reasoning in mental actions of covariational reasoning. We then interpret the data to formulate the findings of the present study, as summarized in Table 3. The findings are ultimately discussed to have better insight on students’ structure of argumentation and possible future studies.

Transcript 1 shows subject A’s responses to the interview of his work on problem 1.

This study found that only one student fulfils five mental actions of covariational reasoning. Meanwhile, the other three students have similar components of argumentation which do not meet the mental actions. In providing the claim, the students can not reach mental action 5 (coordination of the instantaneous rate of change) since they are not able to provide a proper statement about the instantaneous rate of change on the constructed graphs. Similarly, when giving reasoning, the students are not able to come at the mental action 5 since they are unable to relate the idea of instantaneous rate of change to their smooth constructed graphs. For the evidence, the students can only fulfil mental action 1 and mental action 2 because of missing to construct any points as important evidence (mental action 3) to mark the amount of change in the graphs. This study, although only identify students’ claim, evidence, and reasoning without rebuttal, provides insightful knowledge on the student’s complete and incomplete argumentation in mental actions of covariational reasoning which support or hinder their ability in understanding the graph of a function.

The authors thank two anonymous reviewers for their constructive critics for this article. We also give special thanks to the editors who provide valuable assistance in the revision. The errors or inconsistencies found in this article remain our own.