Introduction

Thinking is one of the characteristics of humans (Homo Sapiens). Since we may perceive, humans began thinking, which continues until the end of their lives. The superiority of humans compared to other creatures was indicated by the power of their mind consistently expressed in the act after going through the process of appreciation (Logan and Tandoc 2018; Sherwood et al. 2008). The ability of humans to adapt based on the power of their minds spawned technological and socio-cultural forms of life (Boyd et al. 2011; Gacel-Ávila 2005; Rustaman 1990).

The process of thinking activities includes three parts, namely problem solving, logical reasoning, and decision-making (Galotti and Mark 1994). The process describes that the thinking activities require an understanding of the problems associated with the material being contemplated, our ability to reason, intellectual ability, imagination, and flexibility of the mind that stretched into the results of thought (Cresswell and Speelman 2020; Saiz and Rivas 2011).

There is a relationship between the thinking process and mathematics (Ahdhianto et al. 2020). Someone good in mathematics will be good at thinking, and someone trained in learning mathematics will become a good thinker (Cresswell and Speelman 2020; Li and Schoenfeld 2019; Schoenfeld 2018). In terms of the process of arising mathematical ideas or concepts, Ruseffendi (1991) stated that mathematics arises because of thoughts, which are related to ideas, processes, and reasoning (Jonsson et al. 2014). Meanwhile, if viewed from the activities of mathematics done by students in learning mathematics, Runisah et al. (2017) provided an argument that mathematics activities can potentially enhance responsibility and freedom in thinking, mathematics is an arena for young students to be able to solve a problem and gain the confidence that the correct solution is not because of the words of the teacher, but because of their clear logic of reasoning (Kwang 2000). Therefore, we can see that there is a firm interconnection between math skills and someone thinking skills.

In mathematics, one of the thinking skills that belong to high-level thinking skills is the MCTS (Ahdhianto et al. 2020; Behar-Horenstein and Niu 2011; Ennis 1985; Kayaalp et al. 2020; Spector and Ma 2019). Critical thinking is a logical and thoughtful thought process that decides to believe or do. It also defines a thought process that considers every possible option to help make a logical decision and make it a challenging society recognized as a necessary condition for a responsible member (Uddin et al. 2020; Wijaya et al. 2020). There are four reasons put forward regarding the need to be accustomed to developing MCTS, namely: (1) the demands of the times which require citizens to seek, choose and use the information for social and state life, (2) every citizen always air-to-face with many problems and choices that are supposedly able to think critically and creatively, (3) the ability to look at things in a different way to solve problems, and (4) critical thinking is an aspect in solving problems creatively so that learners can compete fairly and able to cooperate with other nations (Boldureanu et al. 2018; Dewey 1916; Rapanta et al. 2020).

MCTS can be developed by studying mathematics in schools or colleges, which insist on the system, structures, concepts, principles, and strong interconnection between an element and another element (Bransford et al. 1999; Geary 1995; Marzano 1988; Oates 2011). In nature, Mathematics as a structured and systematic science, as a human activity through a process that is active, dynamic, and generative, as well as a science to developing an attitude of critical thinking, objective, and open, it becomes very important to be mastered by learners in the face of rate changes in science and technology are so fast (Hiebert 2013; Lave 1988; Stiff and Harvey 1988; Suter 2011).

Currently, most students assume that mathematics, a field of study which is difficult and disliked (Begle 1969, 1979; Mahmud 2017; Meyers 1986; Ruseffendi 2006). Only a few can explore and understand mathematics as a science that can train MCTS (Munawarah et al. 2020).

Ironically, the MCTS of students, on the one hand, is vital to have and be developed, but on the other hand, it turns out that the MCTS of these students is still lacking (Ahdhianto et al. 2020; Feriyanto and Putri 2020; Syaiful et al. 2020). It is in line with the results of a preliminary study performed for a few semesters on undergraduate students of mathematics education who have a very diverse previous educational background (Syaiful 2013). These students come from Senior High School, Vocational High School, Islamic Senior High School. In a conducted preliminary study, it was given tests of critical thinking with critical thinking indicators (Shovkova 2019; Umam et al. 2020) as follows: (1) make generalizations and consider the result of the generalizations (G), (2) identify the relevance (IR), (3) formulate the problem into a mathematical model (MM), (4) make deductions using the principle (D), (5) provide examples of inference (C), and (6) reconstruct arguments (RA). The results obtained from these tests, both for students with natural and non-science backgrounds, were unsatisfactory. It showed their scores with an average of less than 50% of the maximum score for the two groups (Bhattacharyya and Pradhan 2015; Ruseffendi 1988; Syaiful 2013).

A more in-depth review of the preliminary study illustrates that most students still did not have metacognition. The dysfunctional metacognitive beliefs in the context of learning happened (Lenzo et al. 2020). It caused difficulty in understanding mathematical concepts as well as in procedural understanding. Another indication is that students also tend to be afraid to give ideas and comments and lack confidence in mathematical communication (Moodley et al. 2015; Murphy et al. 2016; Salguero et al. 2020; Seow and Gillan 2020; Syaiful 2019).

A learning strategy and approach are vital to developing students' thinking skills, so it is necessary to have mathematics learning which actively involves students in the learning process. Thus, an alternative form of learning designed in such a way as to reflect active student involvement that instills metacognition awareness is required(Abramovich et al. 2019; Kallio et al. 2020; Laurens et al. 2017; Li and Schoenfeld 2019).

Besides, learning mathematics using the metacognitive approach is a form of constructivist learning (Erdoğan and Şengül 2017; Tachie 2019; Verschaffel et al. 2019). It views that the learning process begins with cognitive conflict and is resolved by themselves through self-regulation, which ultimately in the learning process, students build their knowledge through experience from the results of interactions with their environment (Monteiro et al. 2020).

Based on another view, metacognitive includes developing a systematic method for solving the problem, visualizing, and evaluating the productivity of the thinking process (Chew et al. 2019; García-García and Dolores-Flores 2021). Another statement that supports this is as stated in the lecture of the teaching and learning process (MKPBM) Team (2001), which views metacognitive as a form of the ability to see oneself so that what it does can be controlled optimally (Corno 1986; Susantini et al. 2021; Winne 1996).

The author considers that the MLA has many advantages when used as alternative learning of mathematics to develop the MCTS of students. With MLA, a lecturer develops students' metacognitive awareness (Al-Gaseem et al. 2020; Akben 2020). The lecturer continuously trains students to design the best strategy in selecting, remembering, re-recognizing, organizing the information they face, and solving problems (Mathabathe 2019).

By developing awareness in metacognition, students are expected to be consistently used to monitor, control, and evaluate what they have done (Faivre et al. 2020; Miegel et al. 2020; Thorslund et al. 2020; Yusnaeni et al. 2020). Often students ask themselves the question, "What will be done? What is known? What will be sought? Which is the best strategy to solve the problem? Which operation should come first? Are the steps that have been taken correctly? In which part? Is there an error? How are the efforts to correct the error?" (Nakagawa and Saijo 2020). So with critical questions that can develop metacognitive awareness like that, later will develop the MCTS of these students in learning mathematics. (Saritepeci 2020; Wilson and Conyers 2016).

The background above encourages the author to research alternative mathematics learning with an MLA to improve the MCTS of undergraduate mathematics education students. Therefore, starting from the thoughts, the problems of this study were formulated as follows:

  1. 1.

    Is the Mathematical Critical Thinking Skill (MCTS) of mathematics education students of a university, Faculty of Teaching, who receive mathematics learning using an MLA better than students who receive conventional learning?

  2. 2.

    Is there a difference in the increase in MCTS between the low, middle, and high subgroups in the students who received mathematics learning using the MLA?

Literature review

Metacognition

Metacognition refers to thought at a higher level (i.e., mental action or method of acquisition) via thinking, experience, senses, awareness, and understanding of targeting cognitions at the object level. Metacognition has gained much scholarly interest, partially because it is essential for successful learning for school children and adults. In metacognition, Nakagawa and Saijo (2020) argued that the most frequent distinction is between metacognitive information and metacognitive abilities. The former refers to declarative knowledge of a person about the interactions between the characteristics of a person, job, and strategy, whereas the latter refers to the procedural knowledge of a person for controlling problem-solving and learning activities. Procedural metacognition, the latter kind of metacognition, requires metacognitive processes, monitoring (i.e., subjective measures of existing cognitive activity), and metacognitive control (i.e., regulation of current cognitive activity).

Surveillance (Nakagawa and Saijo 2020) requires issues such as "How much effort is being made." Are they expected to bring this material into learning? "Have I understood this material enough to recall the specifics later on? and "How sure am I that this is the correct answer? Control requires behavior such as collecting examination material during the study, the differential allocation of study time to learning material, and the removal of responses or the termination of a memory quest (Nakagawa and Saijo 2020).

Metacognition is a highly disciplined thought that plays a vital role in learning and teaching practices in education organizations. In particular, Flow well was one of the first theorists to put forward a definition of metacognition. He categorized metacognition as metacognitive knowledge, metacognitive experience, and metacognitive monitoring and control (Al-Gaseem et al. 2020). In addition, metacognition involves different activities: (1) understanding one's moods, (2) understanding the moods of others, and (3) how individuals use these representations to deal with suffering and psychology and to resolve conflicts with each other (Maillard et al. 2020). Metacognition refers to both people's awareness and control of their cognitive processes and emotions, and motivations (Lysaker et al. 2020).

Five pedagogical strands in mathematics contribute to students' success: comprehending, computing, applying, reasoning, and engaging—the initial pedagogical action linked to comprehension aides. Students must be able to comprehend mathematical concepts, procedures, and relationships. The second pedagogical action assists students in performing routine computational operations functions that perform everyday operations. The third pedagogical action uses mathematical tools and procedures to resolve issues by applying conceptual and procedural techniques in mathematics. The fourth pedagogical action is related to students' mathematical reasoning, which enables them to explain and justify their actions, justifications for mathematical answers, proofs, and applications. The fifth pedagogical action in the mathematics classroom is about including students in making sense of mathematics by involving them in textbook problems and extending problems to more fields of application. These theoretical foundations for learning methods' psychological, empirical, and pedagogical components influenced our study's methodological approach and output during the analysis and interpretation processes. These theatrical controls may not be adequate. They were not explicitly stated throughout the study procedure, but they served as implicit components at each investigation stage (Khanal et al. 2021).

Metacognitive learning approach (MLA)

The metacognitive learning model is one learning model that teaches students to think creatively to solve a problem (Hargrove 2013; Hargrove and Nietfeld 2015; Listiani et al. 2014). In this case, students can plan, organize, and evaluate the activities they do. The learning focuses on the students or student-centered learning (Listiani et al. 2014). Then MLA is believed to make learning more meaningful, students' understanding becomes more profound than before, and its application is more comprehensive than others. Each learning model will have a syntax. The syntax of the MLA is as follows (Listiani et al. 2014). (1) Opening, students explore prior knowledge related to the material to be discussed at this stage. (2) Development of cognitive abilities: Students can solve cognitive-type problems at this stage. (3) Development of metacognitive abilities. Before developing metacognitive type abilities, students are first given a metacognitive type of math problem, then proceed with the following phases. (a) Planning, the teacher guides students in planning and re-implementing the completion procedure, the cognitive strategies used, and relevant prior knowledge to solve the given problem. (b) The teacher guides students in monitoring completion procedures, relevant prior knowledge, and the cognitive strategies used. (c) Reflection, the teacher guides students to understand the concepts that the students have carried out in solving metacognitive type math problems. The teacher reflects it by comparing the results that students have obtained with the statements given, so there will be a control and reflection on the previous cognitive activities. (4) Closing, students are guided in making conclusions from the previous learning at this stage. In this case, students are thinking, and the lecturer invites them to learn how to solve a problem, from planning, implementing to reflecting on the activities carried out. With this metacognitive knowledge and skills, students are aware of their strengths and limitations in learning. If the student feels he is wrong, he will immediately realize it and look for ways to correct it (Hargrove 2013; Listiani et al. 2014; Syaiful 2011; Tohir 2019). An example of the application of the MLA is listed in Table 1.

Table 1 The example of MLA application
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Method

This study was conducted at a faculty of teaching, a university in Sumatra, Indonesia. It was carried out in one semester. 83 Mathematics education students were involved in the subjects of this study. They consisted of 45 students of EG and 38 students of CG. These two groups were selected randomly from the available three classes. Each class was given a pretest consisting of 12 items of the MCTS test before the intervention was made. The pretest and posttest given were the same tests. The result of the pretest was tabulated to determine the initial ability of the students. The pretest scores are ordered from the highest to the lowest. 27% of the highest scores were grouped as students with high initial ability. 27% of the lowest score students were grouped as students with low initial ability. The remaining 46% were grouped as students with medium initial ability. Then, the students were classified as high, medium, and low levels. Finally, after the intervention was made, the posttest was given to the students to measure how the MLA enhanced the students’ MCTS.

MCTS Test

The items were developed by referring to the six indicators: generalize and consider the results of generalization (G), identify the relevance (IR), formulate the problem to the model of mathematics (MM), make deduction using the principle of (D), provide examples of inference (C), and recommend argument (RA). There are items in the MCTS test, which are then broken down into numbered questions: 1, 2, 3a, 3b, 3c, 3d, 4a, 4b, 4c, 4d, 5, and 6. The MCTS test used was a test in the form of a description, with the aim that the thinking process, accuracy, and systematic preparation can be seen in this meta-steps of solving test questions. In addition, errors and difficulties experienced by students can be identified and studied to enable improvements implementation. The MCTS test was a development of questions to measure critical thinking skills. Mayadiana designed the test under the syllabus and lecture program unit (SAP) of the Mathematics Education, an undergraduate program in a university (Mayadiana 2005). Then, the authors adapted it following the probability material. Questions 1 and 5 measure the ability to generalize and consider generalization results, questions 2 and 6 measure the ability to reconstruct arguments, and questions 3a and 4a are used to measure the ability to identify relevance. Problem numbers 3b and 4b measure the ability to formulate problems into mathematical models. Problem numbers 3c and 4c are used to measure the ability to make deductions using principles. Problem numbers 3d and 4d measure the ability to give examples of inference (please find the MCTS test items in the Appendix 1).

In order to meet the criteria as a good test instrument, before being used in research, this test instrument was first tested to determine its validity, reliability, and level of difficulty (please find attached tables in the Appendixes 2 and 3). Then, a trial was carried out on mathematics education students who had obtained probability material. Before the trial was made, the test was validated by the subject matter expert (SME), a senior Statistics lecture in a mathematics education program. After the required data were collected, validity and reliability tests items analyses were conducted. It was found that the counted r of the test items were: number 1 = 0.555, 2 = 0.599, 3a = 0.623, 3b = 0.697, 3c = 0.651, 3d = 0.514, 4a = 0.706, 4b = 0.533, 4c = 0.608, 4d = 0.452, 5 = 0.560, and 6 = 0.649. Meanwhile, the Table r of the test items was 0.403. Since the value of counted r of the test items was bigger than table r, the test items were valid and could be used to measure the MCTS of the students. Moreover, for the reliability of the test items, it was required that Alpha = 0.7674. With the alpha 0.7674, it meant that the items were reliable and could be used to measure the MCTS of the students.

Conventional Learning

The lecturer was the center of this learning because it was teacher-centered learning. The lecturer first explains the material, demonstrates, gives some examples, while students listen to the lecturer's explanation carefully and write down some things they consider important. If there are students who feel unclear, then the student asks questions. Lecturers usually answer questions from these students, or occasionally the questions are thrown to other students before being answered by the lecturer. After the material has been delivered, students are given practice questions that are done individually. These questions can come from teaching materials in commonly used math diktats or from other sources. Throughout this activity, the lecturer went around the class to assist students who had difficulties. After that, some students were asked to work on the practice questions on the blackboard. Student activities in this class tend to be passive compared to classes where mathematics learning uses a metacognitive approach. It mainly happened because students did not carry out discussion activities. If possible, students will discuss with a friend next to their seat while working on practice questions.

Metacognitive Learning

The undergraduate students of mathematics education were the center of this learning because student-centered learning was the characteristic of this learning. At the beginning of the lesson, the lecturer conveyed the learning objectives to be achieved at the meeting, then motivated students by providing metaphors. The lecturer allows students to ask questions about the previous material or ask questions and provoke students to ask questions. Thus, at the beginning of learning, there is a process of developing metacognitive awareness and developing students' abilities to ask critically. After the teaching materials in Student Worksheets (SW) are distributed, the lecturer guides students to study the materials and answers metacognitive questions that encourage them to translate concepts in their own words. Then students pour their ideas into the available teaching materials and answer all the questions on the teaching materials in their language in order. At the end of the description of the material, if there are still things that students do not understand, the students write down questions on the teaching materials to then discuss them with fellow students or with the lecturer concerned, then write down the results of the discussion in the space provided. During the small discussion, the lecturer went around the class and gave individual feedback. The effect of metacognitive feedback leads students to focus on the mistakes made and provides instructions to correct their own mistakes. Often this tiny discussion activity continues into a class discussion under the guidance of the lecturer, with one or several students presenting the results of their group discussions while other groups respond or debate. After that, the recapitulation of what students do in class is concluded by themselves, while the lecturer summarizes the essence of the student's conclusions through metacognitive questions.

Findings

Mathematical critical thinking skills (MCTS)

Results of the pretest of students' MCTS between EG and CG were not significantly different. From the maximum score of 100, the EG obtained a mean of 24.36, while the CG with 38 students had a mean score of 21.66. The obtained score pretest that less than 30% of this shows that, in general, the MCTS of the EG and CG are very low, as shown in Table 2.

Table 2 Scores of pretest of MCTS viewed from the group of students for each indicator of critical thinking
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The results of the pretest data of MCTS of students in Table 2 displayed that the student's prior knowledge of the mean scores was obtained; between the EG and CG seemed not much different. On a scale of 0–100, the EG with 45 students had a mean score of 24.35 and a standard deviation of 9.439. Meanwhile, the CG with 38 students had a mean score of 21.66 with a standard deviation of 8.609.

Based on the research data analysis results, the prior knowledge of students EG and CG was the same. Further to the two groups are enforced different treatments. The lecturer gave An EG a treatment in learning mathematics using an MLA, while the CG did not. The CG received learning mathematics using a conventional approach.

The same as the pretest, the aspects or indicators to critical thinking result from posttest measured are: generalize and consider the results of generalization (G), identify the relevance (IR), to formulate the problem to the model of mathematics (MM), make deduction using the principle of (D), provide examples of inference (C), and recommend argument (RA) as that appears in Table 3.

Table 3 Posttest scores of CTS viewed from the group of students for each critical thinking indicator
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The MCTS of students in terms of posttest scores acquisition were as follows. The lowest score of the EG was 57, while the lowest score of the CG was 34. The highest scores of the experimental and CG were respectively 92 and 85. Meanwhile, in terms of the mean scores, from a maximum of 100, the EG got a mean score of 70.02, and the CG got a mean score of 51.68. The mean score difference between the two is around 26% towards the mean score of the EG.

If referring to the benchmark categories which is usually used on campus where the research was conducted, the number of students in the EG who had MTCS in the category of sufficient was 22 people (49%), who had categories of good and very good as many as 23 people (51%). Meanwhile, the CG still had 15 people (39%) with a bad category, and 20 people (53%) were in a good category. In the CG, only three people (8%) had a good and very good category. In general, the EG belongs to have good MCTS (their mean ranged between 70 and 85%), while the CG belongs to have enough MCTS (their mean ranged between 50 and 70%).

The posttest and means-test results displayed that students who receive learning mathematics using the MLA have demonstrated an increase in the MCTS. It is much better than the students who got the conventional learning. The fact is possible because in the learning of Mathematics with the MLA, the paradigm of learning centered on teachers (lecturers) has shifted to learning that emphasizes the activity of the students to construct and reconstruct knowledge itself. In line with what is proposed by Radjibu et al. (2020), a more conducive learning atmosphere could be created by changing the outlook of the class as a collection of individuals to the direction of the class as a learning community, and teachers become a motivator, a facilitator, and a manager of learning.

Based on the calculation of the “Gain Temormalisation" results, the whole EG showed an increase in the MCTS amounted to 60.37%, while the CG only reached 38.33%. It means that an increase in MCTS that experienced by the EG was better than CG.

Before discussing more deeply about an increase in the MCTS for each subgroup of the EG, some things need to be observed regarding the learning of Mathematics with the conventional approach on the CG. The posttest results showed a significant increase in the MCTS, in which the students of CG reached the highest score, 85, with the average grade, which is quite good, and in the overall improvement to inability that is classified as medium. It indicates if conventional mathematics learning was conducted truly, it still would give positive results to improve the students' MCTS.

The increase in the MCTS in more detail to each experimental subgroup has obtained an average of improvement in the high-subgroup 65, 41%, the middle-subgroup was 59.82%, and the low-subgroup experienced an increase of 56.05%. In other words, for an EG, both overall and each subgroup, the increase in MCTS that occurred are classified into the category of the medium.

Based on testing with one-way Anova on the level of significance 0.01, obtained results showed no differences in the increase of the MCTS between low-subgroup, medium-subgroup, and high-subgroup in the group of students who received the learning of mathematics using the MLA. In other words, the use of the MLA in learning mathematics has an effect similar to increasing the MCTS of each subgroup so that every subgroup will provide the same results.

Ability to Generalize and Consider the Results of Generalizations

An ability to generalize and consider the result of generalization is measured with a test of MCTS number 1 and number 5, with a score maximum of the two questions, is 25, the prior knowledge of EG for the indicator to think critically is relatively low, which is only 9.84 (39.36%). In contrast, the final ability is classified as high, namely 20.55 (82.20%).

The description above also showed an increase in the MCTS of the EG in aspects of generalizing and considering the results of generalization of 70.65% relative to the capabilities originally (calculation-based formula gain Temormalisation). Thus, an increase of the ability to generalize and consider generalizing results to the EG, including high.

Based on Table 4, known values of JKt (total) = 2.258; JKa (Between Groups) = 0.127; JKi (Within Groups) = 2.131; RJKa (Mean Square betwen) = 0.06351; and RJKi (Mean Squarewithin) = 0.05073; thus resulting in a value of Fcount = 1.252. While at the significance level of 0.01 and degrees of freedom 2 and 42, the value of F table = 5.15 is obtained.

Table 4 Gain generalization ability and considering the results of generalizations
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Based on testing with One-way Anova on the significance level of 0.01, the obtained result showed that Fcount = 1.252 was smaller than Ftable = 5.15. It meant no further increase in the ability to generalize and consider the generalization results between low, medium, and higher subgroups in EG. In other words, using the MLA in learning mathematics has an effect similar to increasing the ability to generalize and consider the results of generalization in each subgroup so that each subgroup will provide accurate and not different results.

The ability to identify relevance

The MCTS in identifying relevance was measured by the MTCS test number 3a and number 4a. With the maximum score of the two questions being 10, the initial ability of the EG for this critical thinking indicator is very low, which is only 2.85 (28.50%), while the final ability is classified as high, namely 8.29 (82.90%).

The description of those that increase the MCTS of EG in identifying the relevance was 76.08% relative to the capabilities initially (calculation-based formula gain normalized). Thus, increasing the ability to identify the relevance of the EG included high.

Table 5 displays values of JKt = 2.105; JKa = 0.219; JKi = 1.886; RJKa = 0.11; and RJKi = 0.04491; thus resulting in a value of Fcount = 2.438. While at the level of significance of 0.01 and degrees of freedom 2 and 42, the value of F table = 5.15 is obtained.

Table 5 Identifying relevance ability
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The testing with One-way Anova at the level of significance 0.01 showed that Fcount = 2.438 was smaller than Ftable = 5.15. It meant that there were no differences in the increase of the ability to identify relevance between low, medium, and higher subgroups in EG. In other words, the use of the MLA in learning mathematics has an effect similar to enhancing the ability to identify relevance in each subgroup so that every subgroup will provide accurate and not different results.

The ability to formulate the problem to the model of mathematics

The MCTS in aspects of formulating the problem to the mathematics model was measured with a test of the MCTS numbers 3b and 4b. With a score of maximum, two questions are 10, prior knowledge of the EG for the indicator to think critically is only 1.66 (16.60%) and classified as very low, while the ability of the end is 7.06 (70.60%) and classified high.

Based on the description above, an increase in the MCTS of EG in formulating the problem to the mathematics model is by 64.75% relative to the capabilities initially (calculation-based formula gain normalized). Thus, increase the ability to formulate the problem to the model of mathematics in an EG categorized as medium.

Table 6 shows the value of JKt = 2.541; JKa = 0.02486; JKi = 2.516; RJKa = 0.01243 and RJKi = 0.05991; thus resulting in a value of Fcount = 0.207. While at the significance level of 0.01 and degrees of freedom 2 and 42, the value of F table = 5.15 is obtained.

Table 6 The ability to formulate the problem to the model of mathematics
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Based on testing with One-way Anova on the level of significance 0.01, we obtained results that Ftable = 5.15 was bigger than Fcount = 0.207. It meant that there were no differences in the increase in the ability to formulate the problem to the model of mathematics between low, medium, and high subgroups in EG. In other words, the use of the MLA in learning mathematics has the same effectiveness to improve the ability to formulate problems into the model of mathematics of every subgroup so that every subgroup would provide results that were indeed not different.

The ability to make deduction using principle

The MCTS items number 3c and numbers 4c measured the MCTS of students in making deductions using principle. With a score of maximum for the two questions is 10, the prior knowledge of EG for the indicator of critical thinking is relatively very low, which is only 0.77 (7.70%), while the final ability was 5.73 (57.30%). It was classified as sufficient.

The description also showed that an increase in the MCTS of the EG in making deductions using the principle amounted to 53.74% relative toward ability initially (calculation is based on a formula gain Ternormalisation). Thus, it could be concluded that an increase in the MCTS in making deductions using the principles of the EG was classified into medium criteria.

Table 7 displays values of JKt = 5.022; JKa = 0.09336; JKi = 4.929; RJKa = 0.04668; RJKi = 0.117; thus resulting in a value of Fcount = 0.398. While at the level of significance of 0.01 and degrees of freedom 2 and 42, the value of Ftable = 5.15 is obtained.

Table 7 The ability to make deductions using the principle
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Testing with One-way Anova on the level of significance 0.01 obtained that Fcount = 0.398 was smaller than Ftable = 5.15. It meant that there were no differences in the increase in the ability to make deductions premises using the principle between low, medium, and high subgroups in EG. In other words, the use of the MLA in learning mathematics has an effect similar to increasing the MCTS on the aspects of making deductions using the principles in each subgroup so that each subgroup would give the same result.

The ability to provide example of inference

The MCTS of students in giving an example of inference was measured with a test of MCTS numbers 3d and 4d. With a score of maximum for the two questions is 10, the prior knowledge of EG for the indicator to think critically is only 0.98 (9.80%), while the ability to reach 4.94 (49.40%) finally.

Based on the description of these, it appears that the increase in the MCTS of EG in giving an example of inference is by 43.90% relative to the capabilities initially (calculation based on formula gain Ternormalisation). It meant that an increase in the MCTS in giving an example of inference in an EG was relatively medium.

Table 8 shows that the values of JKt = 9.255; JKa = 0.174; JKi = 9.081; RJKa = 0.087; and RJKi = 0.216; thus the value of Fcount = 0.403. Meanwhile, at the level of significance of 0.01 and degrees of freedom 2 and 42, the value of Ftable = 5.15 is obtained.

Table 8 The ability of providing example of inference
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Based on testing with One-way Anova on the level of significance 0.01, obtained results Fcount = 0.403 was smaller than Ftable = 5.15. It meant that there were no differences in the increase of capabilities provided an example of inference between low, medium, and high subgroups in EG. In other words, the use of the MLA in learning mathematics has an effect similar to increasing the ability to give an example of inference to each subgroup so that each subgroup would provide the same results.

The ability to reconstruct arguments

The MCTS of students in reconstructing arguments was measured with a test of MCTS for items number 2 and number 6. The score maximum second question is 35, prior knowledge of the EG of indicators of critical thinking was only 8.25 (23, 57%), while the final ability was 23.45 (67.00%).

From the description, an increase in the MCTS of EG in reconstructing the argument is by 56.98% relative to the capabilities originally (calculations based on formulas gain Ternormalisation). It meant that the increase in the ability to reconstruct an argument on the EG was classified as a medium.

Table 9 displays the value of JKt = 1.723; JKa = 0.08718; JKi = 1.636; RJKa = 0.04359; and RJKi = 0.03895. Thus, the value of Fcount = 1.119. While at the level of significance of 0.01 and degrees of freedoom 2 and 42, the value of Ftable = 5.15 is obtained.

Table 9 The ability to reconstruct arguments
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Based on testing with One-way Anova on the level of significance 0.01, obtained results that Fcount = 1.119 was smaller than Ftable = 5.15. It meant that there were no differences in the increase of the ability to reconstruct the argument between low, medium, and higher subgroups in EG. In other words, the use of the MLA in learning mathematics has an effect similar to enhancing the ability to reconstruct arguments in each subgroup so that every subgroup will provide accurate and not different results.

Furthermore, the pretest and posttest of score acquisition percentage as a whole regarding the MCTS include aspects: generalize and consider basil generalization (G), identify the relevance (IR), to formulate the problem to the model of mathematics (MM), make the deduction using the principle of (D), giving examples of inference (C), and reconstruct arguments (RA), are presented in Fig. 1. While Fig. 2 presents the percentage of gain acquisition in the MCTS, which includes all six aspects.

Fig. 1
figure 1

Percentage of score acquisition for each mathematical critical thinking skill aspect

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Fig. 2
figure 2

Percentage of gain acquisition for each critical thinking aspect of mathematics

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Based on Fig. 1, from the six aspects of MCTS measured in the study, the highest increase in the MCTS of students was on generalizing and considering basil generalization and identifying the relevance, while the lowest increase was on the aspects of providing examples of inference. It was enabled because, based on the trial results, questions to measure aspects of generalizing and considering basil generalization and identifying the relevance were easier than the others, while questions to measure aspects that provide examples of inference were the most difficult ones compared with the others.

As displayed in Fig. 2, from the six aspects of MCTS measured, the highest gain identified the relevance, while the lowest increase provided examples of inference. For the same reason as before, it was because questions to measure aspects of identifying the relevance were easier than the others, while the questions to measure the aspects of providing examples of inference were the most difficult ones compared with the others.

Discussion

The MCTS of students who follow mathematics teaching using the MLA was much better than the MCTS of students who learned in conventional (Craig et al. 2020; Teng 2020). The MCTS of students who learned using an MLA is good, while the students who learned using the conventional can critically think that was classified into the medium. Nevertheless, the results obtained by students who learned in a conventional, showed a significant improvement in the MCTS, with the highest score are in the category of very good and the average grade is quite good. It indicates that although the learning of mathematics was done conventionally, it still would give positive results to improve the ability of the students if it was truly carried out.

In learning mathematics with an MLA, the teacher-centered learning paradigm has shifted to learning that emphasizes student activities to construct and reconstruct their knowledge (Hidayat et al. 2018; Sumarmo 2002; Tachie 2019). Also, it makes students more active (Ruseffendi 1991; Schermerhorn Jr and Bachrach 2020; van Rhijn et al. 2016; Wulf and Lewthwaite 2016). Besides, the MLA trains metacognition provides opportunities for students to learn on their own, where each student fills out the SW given according to the instructions (Atmatzidou et al. 2018; Erdoğan and Şengül 2017; Ross 1995; Xu and Ko 2019). Also, a more conducive learning atmosphere can be created by changing the view of the class as a collection of individuals towards the class as a learning community, and the lecturer as a trainer being a motivator, facilitator, and learning manager (Hidayat et al. 2018; Sumarmo 2002; Tachie 2019).

In addition, learning Mathematics with the MLA improved the MCTS significantly of each subgroup of students compared to the Conventional (Amin 2020; Erdogan 2019; Shanta 2020; Suryadi 2005). The MLA involved students’ metacognition (Amin et al. 2020; Mohseni et al. 2020). Metacognition is an essential component in learning Science like Mathematics (Kristensen et al. 2020). It made students control themselves and use their knowledge to solve problems (Teng 2020). It also challenged the students to think critically (Yildirim and Ersözlü 2013). By having good metacognition, according to Akturk and Sahin, Bonner, and Van Zile-Tamsen (as cited in Craig et al. 2020, p. 156), students were demanded to own awareness of learning and comprehension, to understand their own and other thoughts, to monitor and increase the efficiency of the cognitive procedure.

Moreover, with metacognition, students have to assess the needs of a project and eventually pick out an appropriate method for project completion, to display their development in the direction of an aim and to regulate method usage, to mirror their selection making process, and to determine the intellectual states of others (Craig et al. 2020). Shortly, with metacognition, students achieved their academic target (Craig et al. 2020; Mathabathe 2019).

Based on the statistical calculation results, it was concluded that there was no difference increase in MCTS between the low-level subgroups, medium-level subgroups and high-level subgroups of students who received the learning of mathematics using the MLA. In other words, the MLA has the same effect in increasing the MCTS of any subgroups of students (Craig et al. 2020; Teng 2020).

Specifically, an increase in generalizing and considering the results of generalization is included in the high-level categories. Also, in identifying the relevance, the increase in the MCTS of students is included in the high-level categories. Meanwhile, the increase in the MCTS in aspects of formulating the problem to the mathematical model, making deductions with principle, providing an example of inference, reconstructing arguments, got an increase that belongs to the medium-level categories. The metacognition students influenced those who had (Lenzo et al. 2020; Salguero et al. 2020; Seow and Gillan 2020).

In general, the learning of Mathematics using an MLA makes students more active during the activities of the learning takes place, the students get more opportunities in exploring the material together with the lecturer and friends through the activities of discussion. Factors that significantly supports the implementation of learning mathematics using the MLA namely: (1) work together and assistance of lecturers who acted as observer and discussion partners in completing each obstacle that faced in the process of learning (Backer et al. 2020); (2) active students' engagement to participate in learning well (Azizi and Herman 2020; Hartini et al. 2020; Suryani et al. 2020; Umam et al. 2020; Uddin et al. 2020). In short, learning with an MLA presents teaching materials that train metacognition, lecturer intervention, and class interaction (English 2016; Su et al. 2016; Suryadi 2005; Verschaffel et al. 2020; Wang et al. 2021).

In addition, some of the obstacles are encountered in the learning of mathematics using the MLA. They were: (1) the time that is available relatively little to do developments in learning; (2) difficulty in making questions exercises on sheet working students which can improve the MCTS of students; (3) the difficulty in making a group discussion with varying levels of ability in mathematics of the group members, so it is expected in each group going on the activities of productive discussion groups (Munawarah et al. 2020).

Conclusion and recommendation for future research

Learning to use the MLA can be used as an approach in teaching Mathematics to enhance the MCTS of students, especially in the aspects of making generalizations and considering the results of generalization, identifying relevance, formulating the problem to the model of mathematics, making the deduction using the principle, provides an example of inference, and reconstruct the argument. Learning Mathematics with the MLA emphasizes the activity of students in the process of learning to optimize the involvement of students and turned out to give a result that is quite effective to create an atmosphere of learning as it requires the skills of a lecturer or teacher in terms of material in Mathematics and methodology of learning. Because of it, the lecturers or teachers are always expected to keep trying to improve the ability to teach and the ability of Maths through various sources. For example, the results of research or journal. Further research will observe and report the activity of the students in the class when the MLA is applied.