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Mathematical Commognition: An Investigation Using Repetition with Complex Variations

Santos O. Ombay, Dennis B. Roble*

University of Science and Technology of Southern Philippines, Lapasan Highway, Cagayan de Oro City, Philippines

Abstract: This study focused on the investigation on how repetition with simple and complex variations may influence students’ mathematical commognition (communication and cognition). The study was conducted at Bulua National High School of the Division of Cagayan de Oro City to Grade-10 students enrolled for school year 2019-2020. This study utilized the pretest-posttest control group design. A teacher-made test that measures students’ mathematical commognition comprises of 5-item open-ended questions with a rubric scale that measures mathematical cognition and mathematical communication skills was the main instrument used in the study. The analysis of covariance (ANCOVA) was used to determine the effects of simple and complex variations on students’ mathematical commognition. Results of the analysis revealed that students exposed to repetition with complex variations approach had significantly higher performance scores in terms of mathematical commognition as compared to its counterpart. The researchers then recommends that mathematics teachers may use this method as a basis for future studies for more insights on instruction that uses repetition with complex variations.

Keywords: repetition with complex variations, repetition with simple variations, commognition

Received April 04, 2020; Revised May 06, 2020; Accepted May 13, 2020

Citation: Santos O. Ombay, Dennis B. Roble, Mathematical Commognition: An Investigation Using Repetition with Complex Variations. American Journal of Educational Research, 8 (5). 267-271. January 2020. doi:10.12691/education-8-5-6

1. Introduction

A new feature of the Department of Education (DepEd) K-12 curriculum of the Philippines is the spiral approach of teaching where the basic facts of a course are first learned, then as learning progresses, more details are introduced to develop the same concepts from one level to another in an increasing degree of complexity and sophistication. Revisiting concepts with a deepening level of difficulty helps students gain profound understanding, for a concept is best learned through repeated experience such that, learners can recall the basic information to be used for future purposes. Through, repeated experiences, students learn covertly to plan their activities thus internalization occurs through repeated encounter of mathematics problems, that is, the activity is structured in spiral, passing and at the same point at each new revolution while advancing to a higher level [1]. However, as time passes, when a student initially learned something, they tend to forget the basic and important information. Hence, in order to better retained the acquired information a teacher needs to bring out critical features of the object of learning into students’ focal awareness to recall the concepts [2] and design mathematical tasks for students to practice increases students’ confidence, speed and skills [3]. Hence, repetition provides practice to learners’ need to master new skills and it helps strengthen their learning comprehension.

On the other hand, the concept of repetition with variation across Eastern and Western countries as a way of developing student skills and attitudes towards problem solving varies differently [4]. Given a set of practice exercises that vary systematically, repetition of this practice may become an important route to understanding. In order to fully grasp the importance of repetition in the classroom setting, concepts must be properly introduced in a repetitive way with varied level of complexity. However, after the information introduced, it is not a guarantee to have better comprehension to the expected output. Many students perceived mathematics as a field of study that is difficult to understand because mathematics was presented in a way that is less appealing and difficult for students to learn which made them feel bored and not responding well to the lesson.

In order to engage students meaningfully, the mathematics teacher should present the material in a way that it involves all of them in activities which motivates them to become active participants. One way to achieve this is by incorporating questions that promotes discussion and collaboration which ensures formative opportunities in the classroom [5] and presenting the lesson with a deepening level of complexity can enhance their commognition (communication and cognition). In addition, cognitive factor is the ability to reason from premise to conclusion, or to evaluate the correctness of the conclusion [6]. Likewise, this factor is also called a logical evaluation. They pointed out that what is called for in syllogistic reasoning tasks is not deduction but the ability to evaluate the correctness of the answers presented. The factor can be confounded with verbal reasoning when the level of reading comprehension required is not minimized [7].

2. Literature Review

Conceptual understanding or cognition is the student’s comprehensions of mathematical concepts, operations, and relations. The successful student comprehend the concepts of a mathematical ideas, and has the ability to extract the things out to illustrate the new concepts and be able to transfer it into new situations and apply it to new contexts that is learn more than isolated facts and methods that is allowing the student to connect new ideas to previously learned knowledge and skills.

Sarwadi and Shahrill, conducted a study on mathematical errors and misconceptions revealed that the teacher has a vital role in developing conceptual understanding and thus, teachers should also provide exercises that will stimulate detecting, reasoning about and correcting errors or erroneous worked-out solutions in tests and examinations. Students' written work allows teachers to detect errors and misconceptions and therefore will help teachers to plan diagnostic instruction. This is related to the present study because the students are given mathematics problems with different levels of difficulty. The teacher researcher also required the students to solve problems and ask a volunteer from the class to write the answer on the board [8].

Hidayat & Iksan, stressed out that when there is a participation or active student engagement during discussion conceptual understanding is obtained. Accordingly, one way to enhance conceptual understanding is trough worksheet based. Environment presented to students was really close for them so that they not only saw the relevance of mathematical concepts with real environment but also made learning process meaningfully. Students who had a good understanding of the concepts connected to the achievement. In addition, doing mathematics is also one factor that can improve the mathematical skills and thus, conceptual understanding is developed. This is related to the present study because the learners are reinforced by holding a class wide discussion around the different ways to use a formula. Subsequently provide worksheets that require solving questions with the formula [9].

Fi and Degner, believed that problem is an effective way to teach mathematics for understanding. It provides students a way to learn mathematics with conceptual understanding because it requires deep analysis. They present teaching through problem solving as a pedagogy that engages students to think critically to facilitate students’ learning of important mathematics concepts and mathematical processes. The study is related to the present study because the learners are given an open-ended problem related to the topic and ask to show the step by step procedures supported with a mathematical explanation to justify the answer [10].

Lai, found out that doing mathematical tasks in a form of procedural variation developed conceptual understanding, hence creating a structure of students’ mathematical activity is of great help to experience the object of learning. This experience aims to provide opportunities among students to master the procedural processes of a problem that may lead to mathematical abstraction, thus solving problems in mathematics help to make connection between new knowledge and previous knowledge. Moreover, classroom activities are designed to help students stablish a connection of certain dimension of variations for meaningful learning. In other words, this form of procedural variation offers a structured and structural approach to exposing underlying mathematical forms and therefore, can enhance students conceptual understanding of a series of related concepts. This is related to the present study because the students are required to solve mathematics problems showing the step by step procedures of the solution whose level of complexity is varied [11].

Commognition is a theoretical framework developed by Sfard, that focuses on both social and individual aspects of thinking and learning; it merges and combines tenets from theories of communication and cognition. The framework uses a set of principles and notions that discuss thinking as a special type of interpersonal communication and learning as a change in discourses [12].

Sundayana, Herman, Dahlan & Prahmana states that educational researchers’ interest in human communication has been steadily growing classroom communication and discourse is amongst the main foci of educational studies. Within this structure, mathematical communication skills need to be developed among the learners such that their ability to translate mathematical statements into symbols will be further enhanced. Hence, the capacity of the teacher to let the students engage in the discussion is the significant factor that can flourish mathematical thinking. This study is related to the present study because the students are required to work in groups to answer an open-ended problem from which they are required to present their output afterwards [13].

In addition, the study of Kosko & Gao found out that that communication has a positive effect on mathematics achievement. Activities of students in small groups provide an opportunity for students to perform mathematical communication through a number of metacognitive questions that focused on the problem, build prior knowledge with new knowledge, and the use of appropriate strategies to solve problems. This is related to the present study because the students are asked to work in small groups and ask to share the answer to the class [14].

3. Methods

This study sought to determine the influence of repetition with complex variation on students’ commognition. It utilizes the pretest-posttest control group design. It was conducted at Bulua National High School of the Division of Cagayan de Oro City to Grade-10 students enrolled for school year 2019-2020. A teacher made test was used in this study that measures commognition (cognition and communication) which comprises of 5 items open ended questions. It is evaluated using the rubrics scale adapted from the study of Lomibao, Luna & Namoco. The participants answers were rated according to their ability to explain, interpret and apply, with a corresponding point in each given category. The highest score of four points was given to the response of participants whose answers were indicative for strong or superior; three points for moderately strong, satisfactory, and with minor flaws; two points for developing progressing, nearly satisfactory, with serious flaws and one for beginning unsatisfactory, very weak [15]. And there were three mathematics teachers, including the researcher, who rated the answers of the participants. The structured questionnaire used, underwent face and content validity from experts in the field of mathematics education.

The researcher handled the two classes to minimize the possible effect of the teacher factor that may affect the outcome of the study. To avoid Hawthorne effect, students were not informed that they were the subjects of the study. The teacher-researcher delivered the lesson through applying repetition with complex variation. In order to ensure that the two approaches were implemented appropriately and distinctively in the control and experimental groups and to avoid bias researcher invited two (2) grade 10 math teachers of the department to observe the two classes under study. These teachers observer have been teaching grade-10 mathematics subjects for more than a decade and earned units of master’s degree in mathematics.

The teacher-researcher delivered the lesson through applying repetition with complex variation and repetition with simple variations in the control group. It started with a lecture of the basic terms such that students would familiarize the basic concepts of the lesson. However, in the control group, the researcher illustrated the concept of repetition with simple variation where the class starts with a discussion on the basic steps on how to solve the problem by identifying what is asked in the problem, listing down the given facts, sketching the graph and indicating the part that needed to be solved. Next, they were given exercises with different unknown variables of the same concept as an example. The class were given time to do the solving on their seat while the teacher-researcher monitored the learners by asking them relevant and essential question pertaining to the topic as a guide to arrive at the correct answer. After a couple of minutes, a volunteer was asked to write the answer on the board and explain it before the class, once the lesson had been fully understood the teacher-researcher asked the class to group up into small groups for collaboration and asked to work on the given problems with different unknown variables of the same concept and have them presented the output in the class afterwards. After which, the teacher-researcher gave problems on real life situation of the topic and discussed it to the class. While in the experimental group, researcher employed the method which is the repetition with complex variation. The concept involves the repetition of the lessons integrated with varying degrees of complexity. The class tackled on the topic about permutation, combination and probability and then asked specific questions to launch exercises that built skills and understandings of concepts. The presentation of the lesson began with level zero or the low-level questions, in this level the students were given an activity that engage recall or basic information about the topic or finding the messing variable of the problem. Next is level one the average level or moderate order of thinking, in this level the students were applying what they were already know or knowledge application. In this case, students were tasked to solve problems and asked to justify their solution, then called a volunteer to show his or her solution on the board and explain the answer to the class. Level two, in this level the students were given problems that provoke higher order thinking skills (HOTS), this time the students were tasked to do the collaborative work from which the learners were grouped into a minimum of tree to do the task given to them. After the activity, a discussant from each group would present the output. Level three which is the last level, the students were tasked to pose or create problems related to the topic.

To ascertain if there is a significant difference between the experimental and control groups in terms of mathematical commognition as influence by teaching using repetition with complex or simple variations. The analysis of covariance (ANCOVA) was used to determine the effects of the treatment because the samples were intact. The performance on students commognition of both groups were described using the mean and standard deviation. In testing the hypotheses, alpha is set at 0.05 level of significance.

4. Results and Findings

The students’ commognition scores in both groups in the classroom was shown in the following table:

Table 1. Mean and Standard Deviation of the Pretest and Posttest of Students’ Commognition Scores in Mathematics

Table 1 shows the mean and standard deviation of the experimental and control group. In the pretest, the control group obtained a mean of 9.48 while the experimental group got 11.42 which indicates that the participants from both groups got a very low scores in the pretest. This means that both groups were at the beginning level that means that the participants from both groups had a poor background or a little knowledge about the lessons. The results further revealed that the student’s cognition as well as the communication skills are not yet manifested at this time due to unfamiliarity of the selected topics.

It can be observed further from the table that after the treatment, the experimental group got a mean score of 88.12 while the control group obtained 48.39, which means that there is an increase when it comes to their commognition scores for both groups. However, the experimental group has a huge increase compared that of the control group that is from the beginning level to approaching proficiency level for the experimental group while the control group was still at the beginning level despite of the increase in the mean scores. This indicates that, the experimental group has a better understanding of the lessons. This result revealed that all participants were not able to reach the proficiency level which means that they should strive more to improve their cognition and communication skills.

With regards to pretest of students’ standard deviation in the commognition test, the experimental and control groups got 7.11 and 5.99, respectively. The slight difference of the standard deviation value indicates that the scores of the experimental group was a little bit dispersed than that of the control group. Moreover, the standard deviation in the posttest of the experimental group was 36 and 28.82 for the control group. This means that, the experimental group test scores became more scattered compared to the control group. The effect of the treatment varied from learner to learner because of their heterogeneous ability that is some students got a high score while others scored very low. In other words, some students assimilated the concepts very well for they took the subject seriously and they studied more on the lessons while others scored extremely low which might due to the fact that the time frame was not enough for them to fully understood the concept of each topic. To determine if there was a significant effect of the treatment, further analysis was done using analysis of covariance.

Table 2. One-way ANCOVA Summary for students’ Commognition Scores

Table 2 shows the analysis of covariance of pre-test and post-test scores of students’ commognition scores. The analysis yielded a computed probability value of .000009 which is less than 0.05 level of significance which led the researcher to reject the null hypothesis. This implies that there is a significant difference in the students’ mathematics performance in favor of the experimental group. It indicates that the experimental group exposed to repetition with complex variations approach performed better than those exposed to repetition with simple variations. This implies further that using repetition with complex variation as an approach in teaching is an effective method because it enhances cognition and communication. It is interesting to note that in this method the teacher gave problems in mathematics of different level of complexity. As observed, there was a lively interaction among the groups because they had to find the correct answers to the problems showing a multiple clearly defined solutions to be presented in the class, hence, oral and written communication have been developed since the participants were required to write their solution in paragraph form. Furthermore, one of the most important factors that helped the students increased their math performance is emphasizing the foundation of each mathematical concept that is the students are intellectually engaged in learning by means of reasoning, and problem-solving skills that are fundamental for learning. Moreover, when the degree of complexity increases, students become acquainted and comfortable with the skill therefore making the application of the skill can improve comprehension of the mathematics problems. The results of the study confirmed the earlier findings of Lai & Murray, who found out that in creating a form of procedural variation, such as finding a different way of solving a problem, students are able to comprehend different components of a concept and hence upgrade their structure of knowledge [11]. Likewise Sundayana et al., Kosko & Gao and Lomibao, Luna & Namoco said that employing both written and oral communication in the class could improve students’ performance in mathematics [13,14,15].

5. Concluding Statements

Based on the analysis and findings of the study the researcher concludes that repetition with complex variation is an effective method in teaching student mathematics to improve commognition. Hence, the researcher recommends that teachers may use repetition with complex variation as an approach in K-12 mathematics classes to improve mathematical commognition, similarly teachers may design activities in their mathematics classes that requires students to provide with clearly defined solutions and a written explanation in paragraph form such that the cognitive and communication skills will be developed. Moreover, teachers and researchers may use this method as a basis for future studies for more insights on instruction that use repetition with complex variation. Finally, further research may be conducted to wider scope using different population in different institutions for better generalizability of the method.

Acknowledgements

The researchers would like to express their sincere gratitude to Bulua National High School especially to the school principal, Dr. Minda S. Rebollido for allowing the researcher to conduct the study in the said school. The Department of Education (DepEd) headed by the Schools Division Superintendent Dr. Cherry Mae L. Limbaco in the division of Cagayan de Oro City for approving this research study. Likewise, to the Department of Science and Technology for awarding a scholarship under project STRAND. The author also wishes to express their gratitude and appreciation to Dr. Laila S. Lomibao, the Dean of the College of Science and Technology Education of USTP and Dr. Rosie G. Tan, the chair of the Department of Mathematics Education of the same university, for their tireless support, professional guidance, countless supervision, encouraging comments and patience in reviewing the research works with special care and attention. Finally, the researchers would like to thank the Department of Science and Technology (DOST) for their professional and financial support to the main author.

References

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