21

Using Metacognitive Learning for Enhancing

the Mathematical Logical Thinking Ability of Students

Mimih Aminah*

Yaya Sukjaya Kusumah**

Didi Suryadi***

Utari Sumarmo****

Abstract

This research intends to study the application of metacognitive learning and its roles in students' mathematical logical thinking ability consisting of formal modes of reasoning (proportional reasoning, combinatorial reasoning, correlational reasoning, probabilistic reasoning), generalization, analogy, and mathematical proof. Two important trends that have emerged from several previous researches are that many adolescents and adults are limited in their ability to use formal modes of reasoning and that formal reasoning ability is an important mediator of cognitive achievement. An experiment with pretest-posttest control group design was conducted on the 70 students of 10th grade of a senior high school at West Java, Indonesia, during the first semester. The students were classified into three categories based on mathematical prior knowledge (MPK), namely high, middle, and low. The data were collected through the ability test of mathematical logical thinking. To the collected data the analysis of inferential and descriptive statistics was done. The statistic analysis used in this research is one way Anova, t-test, Mann-Whitney test, dan Kruskal-Wallis test. This study has found that the metacognitive learning approach is more effective to reach better result for middle achievers than conventional one. There is no interaction between learning approach and MPK towards the students’ achievement and gain of mathematical logical thinking ability.

Keywords: Metacognition, Metacognitive Learning, Logical Thinking, Mathematical Logical Thinking Ability, Mathematical Prior Knowledge

1. Introduction

1.1 Background of the research

In the daily life everyone always faces problems and challenging, either the simple or the complex one. To overcome various problems people need to think One of thinking abilities needed to solve them is logical thinking. It is defined that logical thinking is a thinking based on reasoning, not on feeling. In the logical thinking there are abilities of analyzing facts and ideas, combining the both to synthesize, generalize, explain, give reason, validate

* Student of Mathematics Education Doctoral Program, Indonesia University of Education. Postal address: STKIP Sebelas April Sumedang, Jalan Angkrek Situ No. 19 Sumedang, West Java, Indonesia. Postal Code: 45323. Email: mimih.aminah@yahoo.co.id

** Professor of Indonesia University of Education, Department of Mathematics Education. Postal Address: Jalan

Dr. Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154. Email: yayaskusumah@yahoo.com

*** Professor of Indonesia University of Education, Department of Mathematics Education. Postal Address: Jalan Dr. Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154. Email: ddsuryadi1@gmail.com

**** Emerita Professor of Indonesia University of Education, Department of Mathematics Education. Jl. Dr. Setiabudhi No. 229 Bandung, West Java, Indonesia. Postal Code: 40154. Email: utari.sumarmo@gmail.com

argumentation, hypothesize, and interprete and conclude. Such a thinking is including high-order thinking. Manipulating information and ideas through such a process enables people to solve the problems.

Piaget proposed an idea on logical thinking as an ability observed at concrete and formal operation stage. An individual at this concrete operation can use logical thinking ability to solve the problems. At the formal operation stage an individual uses operation or logical principles in the context of abstract situation. This ability refers as to individual ability to solve the problems using mental operation or ability to achieve principles and regulation by making generalization and abstract (BOEREE, 2006).

Some reserchers analyzing the result of Piaget’s experiment, such as Tobin and Capie ( 1981), and so Lawson et al. (SUMARMO, 1987; VALANIDES, 1997; KAMARUDDIN et al., 2004; FAH, 2009), identified five types of formal logical thinking, namely: (1) controlling variables, (2) proportional reasoning, (3) probabilistic reasoning, (4) correlational reasoning, and (5) combinatorial reasoning. Meanwhile, Roadrangka (KAMARUDDIN et al., 2004) identified six models of Piaget’s logical thinking as the same as the five above and added by conservational reasoning. Kinds of this reasoning consist of ability to process specific information following logical reasoning and in free-content.

Although Piaget stated that children begin developing formal reasoning at the age of eleven years, in fact, the researchers have already proven that such a reasoning for the most children cannot develop and even up to the old age (SUMARMO, 1987; LAWSON in McLAUGHLIN, 2003; LEONGSON & LIMJAP, 2003; FAH, 2009). In addition, someone cannot always give a true response. In many cases, an adult one cannot logically give an exact response to the reasoning cases (WASON in MARKOVITS & BARROUILLET, 2004). Though some researchers state that the youngest kids are able to give logical reasoning through abstract or fake premises, this reasoning is hard for some adults and even the educated (GEORGE, in MARKOVITS & BARROUILLET, 2004).

The ability of logical thinking should absolutely be trained to the students as to avoid unexact reasoning and wrong decision-making and as to analyze critical statements. One of subjects supporting the mastery of this skill is mathematics. One of five graduating competence standards for mathematics in Indonesia is skill of mathematical reasoning consisting of using reasoning to patterns and characteristics, doing mathematical manipulation in generalizing, arranging evidence, or explaining notions and mathematical statements. Those has already been regulated by the government in Minister’s Rule Number 23 of 2006 (PERMENDIKNAS, 2006).

Branford et al. (WAHYUDIN, 2012) contends that teachers have an important role in the help of empowering development of thinking habit by proposing questions, such as “Before going ahead, have we surely understood this?”, “Do we have a plan?”, and “what choices are ours?”. Those questions help students be accustomed to checking understanding when they are doing, monitoring their progress or adapting their strategies when facing and solving the priblems. This reflective skill is called ‘metacognition’. As the educators, it is urgent for us to help support of developing of students’ metacognitive skill. This is a skill that can help students learn about learning methods. The metacognitive learning that asks students to reflect what they know, pay attention, and do will help them to develop self-awareness.

According to Wahyudin (2012), children’s first experince with math is important for teachers to help them understand that confirmation must always have a reason. They need to learn and agree with what can be accepted as a sufficient argument. The metacognitive questions help them learn about how to construct and justify assumption and also to argue assumption. Therefore, the metacognitive learning has potential to develop skill of students’ mathematical logical thinking.

2. Theoretical Framework

2.1 Logical thinking

The scientific definition of thinking can refer to some of the following opinion. Plato & Aristotle (POESPORODJO & GILARSO, 1999) states that thinking is talking to oneself in the mind; to consider, to ponder, to analyze, to prove something, to show reasons, to draw conclusions, to examine a way of thinking, to look for things that relate to one another, why or for what something happens, and to discuss a reality. Meanwhile Beyer (PRESSEISEN, 2001) defines thinking as mental manipulation of the sensory input to formulate ideas, to give reason, or to judge. Lipman (2003) wrote that thinking is a process to find or make connections.

As for the definition of logic can be seen from some of the following opinion.  Kusumah (1986) explains that etymologically the word logic derived from the term 'logos' (Greek) which means word, speech, intact mind, or it could also contain the meaning of science. In a broad sense, logic is a method and principles that can separate unequivocally between the correct reasoning and the wrong reasoning. He also said that the pattern of thinking in logic is a precise, accurate, rational and objective pattern. In line with the above opinion, Geisler and Brooks (1990) describe the definition of logic as the study of correct reasoning or valid conclusions and errors that arise, formal and informal. Study on correct reasoning is about how to think properly, or how to find the truth. Here logic is interpreted as a way so that we can arrive at a correct conclusion.

Furthermore, in explaining the notion of logical thinking properly, several experts try to give the following definitions. Albrecht (1984) states that logical thinking is the process when one uses reasoning consistently to reach a conclusion. Problems or situations that involve logical thinking require structure, relationships between facts, and a series of reasoning that makes sense. Characteristics of logical thinking are also presented by Suriasumantri (1996). According to him logical thinking is interpreted as the activity of thinking in accordance with a specific pattern, or in other words in accordance with logic. Logical thinking is thinking by reasoning, not by feelings.

Furthermore Macdonald (MUBARK, 2005, p. 8) describes the logical thinking in mathematics as ”the idea that there are certain basic rules of grammar with which we can organise our discussion in mathematics is what make it possible to establish that certain things are ‘true’ in mathematics”. The same definition is formulated in Saskatchewan Learning (2007), namely that the logical thinking ability in mathematics is the ability to apply the reasoning process, skills, and mathematical strategies, on situations and new problems.

Referring to the above opinions, logical thinking is thinking following the principles of reason, free from the aspect of feelings and emotions. Logical thinking is the process when someone makes a judgment, reasons, and other forms of dynamic thinking to arrive at a correct conclusion. When the object is in the form of a mathematical problem or idea it is called mathematical logical thinking. Mathematical logical thinking ability is the ability to analyze a situation or a mathematical problem, to make judgments or estimates, and to provide an explanation on the basis of certain reasons and with certain steps to arrive at a conclusion.

On some explanations above there are other terms associated with the notion of logical thinking, namely reasoning. According to Kusumah (1986), reasoning, which is often defined as way of thinking, is the explanation in an effort to show the relationship between two or more things based on certain properties or laws that have been recognized as true with certain measures which ends with a conclusion. Reasoning is a thinking process in drawing a conclusion and has certain characteristics. There are two types of methods to draw a conclusion, i.e. induction and deduction (KUSUMAH, 1986; SURIASUMANTRI, 1996; SOEKADIJO; 2003). Induction is drawing general conclusion based on various individual cases, while deduction is drawing specific conclusion based on general statements. Furthermore, inductive reasoning consists of generalizations and analogies (SOEKADIJO, 2003; POESPORODJO & GILARSO, 1999).

Reasoning with the object is in the form of a mathematical problem or idea is called mathematical reasoning. Sumarmo (2011) details that the indicators of mathematical reasoning as the ability to: 1) draw analogy conclusions, generalization, and arrange conjecture, 2) draw logical conclusions based on the rules of inference, check the validity of the arguments, and compose a valid argument, 3) develop direct proof, indirect proof and by mathematical induction.

2.2 Piaget’s theory of logical thinking

The concept on logical thinking in detail was profound by Swiss psychologist, Jean Piaget (1896-1980) whose the study is underlying most. His theory is influential and dominating cognitive psychology of children for a half of century. Piaget’s experiment depicts cognitive behaviour in relation of logical structure degree. His theory was arranged based on clinical study to the children at various ages of middle class in Swiss in 1920s. Its conclusion is that thinking pattern of children is different from the adults’. The stage of cognitive development or the stage of individual’s thinking ability correlates with the age. The more adult he or she is, the higher his or her thinking ability is. (BOEREE, 2006; KUSWANA, 2011; WAHYUDIN, 2012).

Piaget viewed that when children interacted with social and physical environment, their mental development was lasting placing four different qualitatively satges: (1) sensori-motoric (0-2 years), (2) preoperational stage (2-7 years), (3) concrete operation stage (7-12 years), and formal operation stage (12 years and so forth). During sensori-motoric stage, children undergo physical world and get basic comprehension on symbols. At pre-operational satge, the symbols are used but thought is still “preoperational”. It means that they cannot understand that logical or mathematical operation can be reversed. Concrete operation stage is allowed children to think logically, understanding cause-and-effect for example.

At development of concrete operation children begin using logical thinking. They start developing intellectual response around conrete things. They show logical thinking in relation of physical things. The children can deal with realistic problems in realm. They can fully understand what emerges up arround them and probabilities based on reality. However, their understanding of abstract conception is still limited. They can memorize description on physically unexisting things but cannot handle ideas. At this stage the children also begin developing mathematical thinking ability, can reserve their thinking process. It means that they can think back about the past experience and correlate it with present abstract situation.

The last cognitive development of Piaget is formal operation. The most contrastive difference between formal and concrete operation is reasoning skill in abstract ways. Formal thinking is also abstract thinking, namely operational thinking that needs not to deal with concepts and concrete phenomena. Individual ability at the formal operation to think logically is signed by the ability to make hypotheses, to analyze situation to consider all factors in such a situation, to conclude, and to test its truth or validity.

At the formal operation stage, someone can use types of thinking skill operated at the three previous stages, in addition they can think in abstract ways, formulate hypotheses, determin combination of complext abstract, and consider various different proposition to accomplish problems using systematic reasoning. Deductive reasoning is major element of formal operatiuon thinking. This is ability to make a valid decision of the argument, whether the existing argument is valid or not. At the formal operation, the more modern way of thinking about mathematics covers proportional reasoning, propotional reasoning, and correlational reasoning, beginning to develop for adolescent years and up to adults (KENNEDY et al., 2011).

Based on the explanation above, in perspective of Piaget’s theory of cognitive development, the logical thinking is the observed ability at the concrete and abstract operation stage. Students of concrete operation stage can use logical thinking ability to solve concrete problems. At the abstract operation stage, students get reach of adult level in terms of the logical thinking. Through logical thinking ability, the students solve problems in various mental practice or understand principles or regulation by doing abstract and generalization. Piaget views that comprehension on formal rule in decision-making is foundation for kind of hypothetic-deductive thinking model rising up at the adolescent period.

The major characteristics of children’s thinking at this formal stage become a reference for some researchers arranging test to measure logical thinking ability. Capie and Tobin in 1980 constructed The Test of Logical Thinking (TOLT) consisting of five components: (1) controlling variables; (2) proportional reasoning; (3) probabilistic reasoning; (4) correlational reasoning; dan (5) combinatorial reasoning. TOLT covers ten double choice items with reasoning. The Lawson Classroom Test of Formal Reasoning was constructed by Lawson in 1978 (KAMARUDDIN et al., 2004). This test consists of components: (1) proportional reasoning; (2) probabilistic reasoning; dan (3) correlational reasoning. Another test is The Group Assessment of Logical Thinking (GALT) written by Roadrangka in 1983 (ROADRANGKA, 2010; KAMARUDDIN et al., 2004). The test uses format of double choice and reasoning for the answer. The questions cover six models of Piaget’s thinking: (1) combinatorial reasoning; (2) correlational reasoning; (3) proportional reasoning; (4) probability reasoning; (5) eternity (conservation) reasoning; dan (6) controlling of variables. Another researcher, Sheehan (SUMARMO, 1987; KAMARUDDIN et al., 2004) measured concrete and formal operation stage of Piaget through modifying test of Longeot consisting of 26 test items covering components: (1) formal logic of propositions; (2) formal combinations; dan (3) formal proportion.

TOLT, GALT, Lawson’s test, and Longeot’s test are not learning outcome test, not deal with specific study fields, so that can be tested to whoever and whenever and intend to distinguish somebody at both concrete and formal operation stage. The tests above has already been used in various areas nd subjects by firsly translated in pursuance of its culture. The result shows that although according to Piaget the cognitive development at the formal operation stage begins at the age of 11 years, not everyone can get reach of the formal operation stage, the abstract reasoning is not unversal too (TRIFONE, 1987; SUMARMO, 1987; MINDEROVIC, 2001; YENILMEZ et al., 2005; FAH, 2009).

Next, some researchers developed the test specifically designed to measure logical thinking ability in the field of study or particular topics constructed by Piaget’s logical operation. Norman (SEZEN & BULBUL, 2011), for example, in 1997, measured the logical thinking ability for the chemistry topic, whose compenents consist of proportional reasoning, variant controll, probability reasoning, relational, dan association. The Test of Logical Operations (TLO) in Mathematics was made by Leongson and Limjap (2003) as to assess students’ achievement on mathematics in higher education. The reasoning pattern measured covers (1) classification; (2) seriation; (3) logical multiplication; (4) compensation; (5) proportionality; (6) probability; dan (7) correlation. The items are limited to the discussed topic for the most subjects of high schools’ mathematics, such as geometry, arithmetics, statistics and algebra. Sumarmo et al., (2013b) researched the mathematical logical thinking ability of high school’ students in side of proportional reasoning, probability reasoning, correlational reasoning, combinatorial reasoning, added by analogy, mathematical proof, and analysis and synthesis of some cases.

2.3 Mathematical logical thinking

In this research, the measurement of students’ mathematical logical thinking ability has been conducted, consisting of proportional reasoning, probability reasoning, correlational reasoning, combinatorial reasoning, analogy, mathematical proof, and generalization. Understanding of such reasoning modles is explained as follows.

1) Proportional reasoning.

Proporsionality refers to relative magnitude of the increase and decrease of ratios. Proportional thinking is the establishment of relations of one part to another or of a whole with respect to magnitude, quantity or degree. This may refer to the understanding of such numerical relationships as 5 : 6 or of algebraic relationships of two variables such as y = 2x. (LEONGSON & LIMJAP, 2003). “Proportional reasoning is the ability to compare ratios or the ability to make statements of equality between ratios” (McLAUGHLIN, 2003, p. 1). A good understanding of the various functional relationships in math and science requires proportional reasoning.

2) Probability reasoning

Probability refers to reasons in time of the likelihood of possible outcomes. Probability is the chance of an event. Probability reasoning is ability to count the number of all objects (N) and the number of a certain object (n) among them, and determine the chance of selection as a fraction (n/N) (BILLSTEIN et al., 1993; LEONGSON & LIMJAP, 2003).

3) Correlational reasoning

Leongson & Limjap (2003, p. 7) defined “Correlational thinking is the establishment of correlation or causal relationship. It may also refer to the presentation or setting forth so as to show relationships.” Also, described correlational reasoning as “Can reason with relationships of variables or symbols” (p. 8). Correlational reasoning is the ability to “correlate two separate relationships between different situations and understand that if something changes in one relationship, it will also change in other” (DUGAN, 2003, p.13).

4) Combinatorial reasoning

Bernoulli (BATANERO, et al., 1997) described combinatorics as “the art of enumerating all the possible ways in which a given number of objects may be mixed and combined so as to be sure of not missing any possible result.” It can be said that combinatorial reasoning is the ability to combine different variables of a set containing those variables to make all possible combinations.

5) Generalization

Polya (1973, p. 108) defined “Generalization is passing from the consideration of one object to the consideration of a set containing that object; or passing from the consideration of a restricted set to that of a more comprehensive set containing the restricted one.” Also, Mason et al. (MUBARK, 2005, p. 7) defined the process of generalization as “moving from a few instances to making guesses about a wide class of cases.” Stacey (MUBARK, 2005, p. 7) described generalization as the process whereby “general rules are discovered by articulating the patterns observed in many particular cases.” In addition, Dreyfus (1991) defined “To generalize is to derive or induce from particulars, to identify commonalities, to expand domains of validity.” Meanwhile, according to Tall (1991, p. 11) the term “generalization” is used in mathematics to denote process in which concepts are seen in broader context and also the product of that process. Furthermore, according to Sumarmo (2013a, p. 402) mathematical generalization is “to draw a general conclusion based on limited observed data or processes.” Thus, generalization can be viewed as the formation of ideas or general conclusions based on observation of certain and specific cases.

6) Analogy

The definition of analogy can be understood from the definition of mathematical analogical thinking expressed by Kinard & Kozulin (2008, p. 88) as follows

Analyzing the structure of both a well-understood and a new mathematical operation, principle, or problem, forming relational aspects of the components of each structure separately, mapping the set of relationships from the well-understood structure to the set of relationships for the new structure, and using one’s knowledge about the well-understood situation along with the mapping to construct understanding and insight about the new situation.

Meanwhile Sumarmo (2013a, p. 402) describes the mathematical analogy as “to draw a conclusion based on similar observed data or processes.” From the above opinions it can be said that in analogy occurs an activity to find a similarity or suitability in some characteristics between the two conditions or terms that are used for comparison.

7) Mathematical proof

Milton & Reeves (MUBARK, 2005, p. 9) described mathematical proof as that which includes “the formation of a chain of ‘valid’ reasoning that leads to a conclusion. It is a process of ‘authentication’ or a process wherein the truth or fallacy of a claim is established.”

These reasoning skills are considered to be part of the formal operational thinking, and are very important for understanding in math and science.

2.3 Metacognition

Over the past 40 years, metacognition has become one of the main areas of cognitive development research. Metacognition activity research was spearheaded by John Flavell, who is regarded as the "father" in this field. Flavell was the first to propose the term metacognition in 1976. The definition of metacognition from Flavell (SCHOENFELD, 1992, p. 38) is

Metacognition refers to one's knowledge concerning one's own cognitive processes or anything related to them, e.g. the learning-relevant properties of information or data. For example, I am engaging in metacognition... if I notice that I am having more trouble learning A than B; if it strikes me that I should double-check C before accepting it as a fact; if it occurs to me that I should scrutinize each and every alternative in a multiple-choice task before deciding which is the best one.... Metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of those processes in relation to the cognitive objects or data on which they bear, usually in the service of some concrete [problem solving] goal or objective.

The above definition shows metacognition has two functions, namely monitoring and regulation of thinking process. Metacognition is also described by Flavell (SCHWARTZ & PERFECT, 2004) as “the experiences and knowledge we have about our own cognitive processes”, “cognition of cognition”, “a critical analysis of thought”, “knowledge and cognition about cognitive phenomena”, or “thinking about thinking”.

Even though Flavell is recognized as the first to introduce the term metacognition, he was not the first to study this phenomenon which was later called metacognitive. Since the beginning of the twentieth century, researchers in reading documented the importance of monitoring and regulation of one’s understanding process, and is now widely recognized that the origins of metacognition partly lies in social interaction, as proposed by the Soviet psychologist Lev Vygotsky (1896-1934) and the Swiss psychologist Jean Piaget ( 1896-1980). Vygotsky and Piaget discussed the process that is seen as metacognitive in their theories about children's thinking (BAKER, 2008).

Vygotsky (1978) states that children (novice) first learn how to engage in cognitive tasks through social interaction with other people who know better or experts, usually a parent or teacher. He said that every function in the child's cultural development appears twice: first, on a social level, and then, at the individual level; the first, between people (interpsychological) and then inside the child (intrapsychological). Transition from regulation by others (other-regulation) to regulation by self (self-regulation) is considered the hallmark of metacognitive development. In other part, Vygotsky argued that development processes throughout life depends on social interaction and that social learning actually directs cognitive development. This phenomenon is called the Zone of Proximal Development (ZPD), which illustrates that skills which can be developed with adult guidance or peer collaboration exceeds what can be achieved when working alone.

While Vygotsky emphasized the interaction of 'expert-novice', Piaget emphasized the importance of peer interaction. Piaget (BAKER, 2008) argues that when children compete thoughts with each other, their cognitive development is advancing. Encouraging children to reflect on their own thinking is actually encouraging metacognition. Discussion and collaboration helps students to monitor their own understanding and build new strategic capabilities. Instructional intervention complements peer support for monitoring, revision, and reflection.

2.4 Metacognitive learning

By reviewing the techniques proposed by several experts who assess metacognition, herewith the learning techniques applied in this study are registered.

1. Modeling

Teachers act as "expert models" who loudly voice all thoughts and feelings that arise during doing a task (think-aloud); for example solving problem, experimenting, reading notes from textbooks, and so forth; so that students can hear and follow thinking process demonstrated. Thus, students know how to effectively utilize knowledge and metacognitive skills. According to Meichenbaum & Biemiller (GAMA, 2004) modeling the process of thinking can be expressed in the form of questions, (such as, "Am I checking my work carefully enough?") or in the form of direct speech, (e.g. "This does not suit with what I expected. I had to discover where my mistake lies"). Here, the teacher alone or together with the students answers the questions posed earlier using loud voice as well.

2. Metacognitive scaffolding

Scaffolding is an effective strategy to enter the ZPD on Vygotsky's theory, which bridges the gap between what students can do on their own and what they can do with guidance from others. In scaffolding, teachers provide opportunities to students to expand their skills and knowledge they had at that time. Teachers engage students’ interests, simplify tasks as to make these tasks manageable, and motivate students to pursue learning objectives, see the mismatch between students’ effort and their resolution, manage frustration and risks, and exhibit appropriate action (KINARD & KOZULIN, 2008). So, in scaffolding teachers transform complex and difficult tasks to become easier to handle and manage.

3. In-pairs discussion, group discussion, and entire class discussion

According to Vygotsky’s theory of social constructivism (JBEILI, 2012; HUDA, 2013), learning with comprehension occurs in a social context. When students interact with each other, they usually will try to provide information, encouragement, or suggestion to a friend of the group who requires, receive feedback, show thinking and solving skills to each other, and therefore students construct understanding, knowledge and new skills. Learning in small groups is also based on Piaget's theory of cognitive conflict (HUDA, 2013). When interacting with their peer conflicts often arise, i.e. something that contradicts their belief or understanding at that time. This conflict motivates students to rethink their understanding of the problem and trying to construct a new understanding which is more appropriate to the feedback they receive.

4. Metacognitive Journal Writing

Journal is a periodical record in which the authors recorded their experiences and their series of thoughts, feelings, and opinions about their activities which are written on a daily, weekly, or monthly basis (Mower in WEST VISAYAS STATE UNIVERSITY, 2012). Students use journals to write about topics that drew their interest, to record their observations, to imagine, to wonder and to link new information with things that they already know. Students who use journals are actively involved in their own learning and have the opportunity to clarify and reflect on their thoughts.

2.5 Research Hypotheses

Based on the research questions presented above, the hypotheses stated in this research are:

1. Students who get metacognitive learning show better mathematical logical thinking ability than the students who get conventional learning.

2. Students who get metacognitive learning show better mathematical logical thinking ability than the students who get conventional teaching, seen from the mathematical prior knowledge (MPK) (high, middle, and low).

3. There is an interaction between the teaching approaches and the mathematical prior knowledge to the students’ mathematical logical thinking ability.

3. Experimental Design

3.1 Experimental Design

The experiment design used in this research is Nonequivalent [Pretest and Postest] Control-Group Design (CRESWELL, 2009), as follows:

Group A OX O

Group B O O

Explanation:

X = Metacognitive learning

O = Measuring test

On such a design, the group A is an experimental class and the group B is a controlling class. A set of given instrument at the first and last step is similar—the test on questions of mathematical logical thinking.

3.2 Participants

Subject of the research was the students of the 10th grade of a state senior high school at Province West Java, Indonesia. The experimental class consists of 36 students: 15 male-students and 21 female-students and the control class consists of 34 students: 12 male-students and 22 female-students. The implementation and the data collection at school were held during one full-semester. The materials discussed during conducting the research are (1) forms of exponents, roots, and logarithms, (2) quadratic functions and parabola, (3) quadratic equations and quadratic inequalities, and (4) system of linear equations.

3.3 Measurement instrument

1. Test of Mathematical Prior Knowledge (MPK)

Test of MPK is required to measure students’ mathematical prior knowledge about the materials of mathematics which were studied before, when they were at Junior High Schools. The materials support in learning the core of discussion which was discussed during this research. Type of MPK test items was short complete, all of it was 20 items. The right answer was given score 1, and the wrong answers was given score 0. The ideal maximum score was 20. The category of MPK was as follows.

Table 1 Category of MPK

Group

Mastery Level

Scores

High

75% 100%

15 20

Middle

55% 74%

11 14

Low

< 55%

0 10

2. Test of mathematical logical thinking ability

Mathematical logical thinking ability was measured by the test for mathematical logical thinking. The test consists of eight items, which belong to seven main aspects as described above. The scoring system of mathematical logical thinking was suited from complexity and accomplishment of every question. The range of three questions is score 1 4, the range of one question score 16, and the range of four questions score 08. So, sum of ideal maximum score is 50.

The instruments of MPK and mathematical logical thinking ability carry through the consultation step and try out to fulfill the requirement of qualified validity, reliability, difficulty index, and discriminatory power.

3.4 Techniques of data analysis

The data which were processed was the scores of pre-test, post-test, and n-gain. In each aspect, if the beginning ability is the same, the difference of average scores of post-test and the improvement will be tested. On the other hand, if the beginning ability is not the same, the difference of scores of the average improvement will only be tested. The data processing used the help of Microsoft Excel 2007 and SPSS 20 for Windows with significance level 0.05.

4. Research Findings

4.1 The analysis result of mathematical prior knowledge (MPK)

The sum of students at each of MPK level for the two sample classes, based on the result of MPK test, was got as the following Table 2:

Table 2 Data on mathematical prior kowledge (MPK)

MPK

Metacognitive Learning

Conventional Learning

n

s

n

s

High

8

16.00

1.07

10

16.80

1.03

Middle

14

12.71

1.14

11

12.36

1.21

Low

14

6.86

1.75

13

6.54

1.90

Total

36

11.17

3.95

34

11.44

4.53

To know whether or not there is equality of MPK at the same level for the two classes, the statistic test was done through the data normality test at the beginning using Shapiro-Wilk test, varian homogenity test using Levene test if the both data has normal distribution, and the last the equality test of two average points using the t-test if the data has normal distribution and Mann-Whitney test if at least one data group has not normal distribution. The significance level used is = 0.05. The sumary of statistic test result can be seen in Table 3.

Table 3 The Result of MPK Equality Test

MPK

Test of Equality

Sig.

Interpretation

High

0.127

No different

Middle

0.439

No different

Low

0.654

No different

Total

0.782

No different

Based on the result of statistic analysis, it is known that knowledge and mastery of mathematics that have already been possessed by both students of metacognitive class and those of conventional class at every level do not differ significantly.

4.2 The analysis result of mathematical logical thinking ability

Statistic scores of the result of mathematical logical thinking ability test cover the score averages, percentage averages to ideal maximum score, and standard deviation as the pretest and posttest result and N-gain presented in Table 4. N-gain formula from Hake (1999) is used to measure the increase capability.

N-gain = (posttest pretest)/(ideal maximum score pretest)

To see comparation between the inter-classes and inter-categories of KAM, score percentage to ideal score is illustrated at the Figure 1, whereas the average N-gain is ilustrated at the Figure 2.

Table 4 The Result of Mathematical Logical Thinking Ability Test

MPK

Stat

Metacognitive Class

Conventional Class

Pre-test

%

Post-test

%

N-gain

N

Pre-

test

%

Post-

test

%

N-gain

n

High

6.88

13.76

41.13

82.26

0.79

8

8.10

16.20

37.50

75.00

0.70

10

s

4.02

2.75

0.06

2.23

 

2.23

0.09

Middle

2.07

4.14

32.21

64.42

0.63

14

3.36

6.72

27.27

54.54

0.51

11

s

2.13

6.29

0.12

2.91

5.33

0.12

Low

1.36

2.72

21.36

42.72

0.41

14

1.69

3.38

20.62

41.24

0.39

13

s

2.47

4.38

0.08

1.55

 

4.33

0.08

Total

2.86

5.72

29.97

59.94

0.58

36

4.12

8.24

27.74

55.48

0.51

34

s

3.47

9.14

0.18

3.48

 

8.31

0.16

Table 4

Hasil Tes kemampuan Berpikir Logis Matem`atis

Metacognitive Conventional

Metacognitive Conventional

Figure 1

Persentage of Average Score

Figure 2

Average of N-gain

The difference of the final ability of mathematical logical thinking amongst high, middle, and low group is continuously tested by the t-test and Kruskal-Wallis test. The signifince level used in this research is = 0.05. The summary of the statistical test results is presented in Table 5.

Table 5 The Result of Difference Tests of Final and Increase of

Mathematical Logical Thinking Ability of inter MPK level

Class

MPK

Test of Difference Mean

Post test

N-gain

Sig.

Interpretation

Sig.

Interpretation

Metacognitive

High-Middle

0,002

Different

0,001

Different

High-Low

0,000

Different

0,000

Different

Middle-Low

0,000

Different

0,000

Different

Conventional

High-Middle

0,001

Different

0,000

Different

High-Low

0,000

Different

0,000

Different

Middle-Low

0,195

No different

0,011

Different

From Table 5, it is shown that in both the experimental class and the control class there are differences in the final ability and increase of mathematical logical thinking ability at different levels according to the MPK.The average difference test of each pair was done to know equality or difference of the achievement and improving ability for each of MPK level pairs. The result can be seen at Table 6.

Table 6 The Result of the Difference Test of Inter Groups to the Same MPK Level

MPK

Initial

Final

Increase

Sig.

Interpretation

Sig.

Interpretation

Sig.

Interpretation

High-High

0.473

Not different

0.183

Not different

0.109

Not different

Middle-Middle

0.261

Not different

0.043

Different

0.016

Different

Low-Low

0.154

Not different

0.661

Not Different

0.391

Not different

Total-Total

0.074

Not different

0.364

Not different

0.190

Not different

By considering Table 6, it is shown that on the students at middle level, there are differences in the final and increase of mathematical logical thinking ability; students on the experimental class have a higher final and higher increase. Students at high and low levels did not differ significantly in the experimental class and control class in the final and increase of this ability.

4.3 The Interaction between learning approaches and MPK to the mathematical logical thinking ability

The result of Tests of Between-Subjects Effects showed that there is no interaction between learning approaches and MPK to the increase of mathematical logical thinking ability (0.163 > 0.05). It is shown at picture bellow:

Figure 2 Interaction between Learning Approaches and MPK

to the Increasing of Mathematical Logical Thinking Ability

The following are the average of students’ scores on each indicators and the percentage of the ideal maximum score in both classes.

Table 7 Rank of Mathematical Logical Thinking Ability on Each Indicators

in Class with Metacognitive Learning

Indicator

Ideal Score

Pretest

Posttest

%

Rank

%

Rank

1. Proportional reasoning

4

0.81

20.14

1

3.33

83.33

1

2. Probabilitas reasoning

8

0.28

3.47

4

4.11

51.39

5

3. Correlational reasoning

4

0.17

4.17

3

2.42

60.42

4

4. Combinatorial reasoning

8

1.14

14.24

2

5.39

67.36

3

5. Generalization

8

0.11

1.39

6

3.61

45.14

6

6. Analogy

12

0.36

3.01

5

9.03

75.23

2

7. Mathematical proof

6

0.00

0.00

7

2.08

34.72

7

Mean

5.72

59.94

Table 8 Rank of Mathematical Logical Thinking Ability on Each Indicators

in Class with Conventional Learning

Indicator

Ideal

Score

Pretest

Posttest

%

Rank

%

Rank

1. Proporsional reasoning

4

1.68

41.91

1

3.32

83.09

1

2. Probabilitas reasoning

8

0.41

5.15

3

3.65

45.59

5

3. Correlational reasoning

4

0.12

2.94

5

2.26

56.62

4

4. Combinatorial reasoning

8

1.32

16.54

2

5.24

65.44

3

5. Generalization

8

0.15

1.84

6

3.21

40.07

6

6. Analogy

12

0.44

3.68

4

8.32

69.36

2

7. Mathematical proof

6

0.00

0.00

7

1.74

28.92

7

Mean

8.24

55.48

Both tables above show that before learning, the sequence of average score looks the same in both classes, except on the indicator 2 and 3. On average the highest score is the indicator1, which is proportional reasoning; while the lowest is indicator 7, which is mathematical proof. After learning, the order of the average score change with the same order in both classes, meaning that the level of difficulty is similar for students in both classes. The average of the highest and lowest scores are in a row from indicator 1 to indicator 7, the same as before the learning. For all of the given problems, metacognitive class students reached 59.94% of the ideal maximum score, slightly higher than the conventional class students who reached 55.48%. The achievement in these two classes are considered medium category.

To give a clearer illustration, the percentage of the scores above are shown on the following figure.

Figure 3 Achievement on Each Indicators of

Mathematical Logical Thinking Ability

As shown by Figure 3, the achievement percentage of mathematical logical thinking ability of students who received metacognitive learning are a little higher than students who received conventional learning, except on combinatorial reasoning. Problems containing mathematical proof are the most difficult for students. Some students did not do it altogether. This can be because students have difficulty using and exploring information, dig up hidden facts, see the connection with other concepts, make allegations, make assumptions and investigate its consequences, or to justify the results

5. Conclusion

Based on the results of the data process and analysis, the conclusions have been

obtained as follows.

1. As a whole, there has not been difference in mathematical logical thinking ability between

the students who had got metacognitive learning and those who had got conventional learning.

2. Seen from mathematical prior knowledge, the students of middle group who had got metacognitive learning had shown better mathematical logical thinking ability than those who had got conventional learning.

3. There is no interaction between learning approach and mathematical prior knowledge on the students’ mathematical logical thinking ability.

It is known that metacognitive learning has shown more about its influence on developing mathematical logical thinking ability to the students of middle group. Eventhough, it does not mean that this approach not needed to be applied to the students of high and low groups. So that, further research is still needed or required to integrate metacognitive approach which is suitable for the two groups of the students.

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Metacog.HighMidLowTotHighMidLowTotMPKMPKPretestPost test13.764.13999999999999972.725.7282.2664.4242.7259.94Conv.HighMidLowTotHighMidLowTotMPKMPKPretestPost test16.26.723.388.247554.5441.2455.48

Score (%)

Metacog.

HighMiddleLowTotalMPK0.790.630.410.57999999999999996Conv.

HighMiddleLowTotalMPK0.70.510.390.52

Average of N-gain

Metacognitive12345671234567IndicatorIndicatorPretestPost test20.143.46999999999999984.1714.241.39000000000000013.01083.3351.3960.4267.3645.1475.2334.720000000000013Conventional12345671234567IndicatorIndicatorPretestPost test41.915.14999999999999952.9416.541.843.68083.0945.5956.62000000000001265.44000000000002640.0769.3628.919999999999995

Score (%)