Student Engagement and Mathematical Justification

Student Engagement and Mathematical Justification

The following paper begins with the description of student engagement. It moves further to identify the effect of student engagement on student performance. In addition to that, the paper also focuses on the importance of mathematical justification. The paper also highlights the importance of student engagement in mathematical justification.

Furthermore, the paper comments on the options that the teachers have for improving the student engagement in their class. It also gives recommendations related to those options which would further foster the engagement and participation of the students.

STUDENT ENGAGEMENT AND MATHEMATICAL JUSTIFICATION

Introduction

The concept of student engagement is strongly related to reduced dropout rates in schools. The students who participate in the class have strong chances of clearing and getting good grades in the exams. Despite the advantages, the task of motivating and encouraging the students to participate in the class gets challenging year after year. There has been much ambiguity in defining the term student engagement. Recently, however, Parsons and Taylor 2011, have defined student engagement in their study as the willingness of a student to participate in school activities such as, attending classes, submitting homework and answering the questions asked by the teacher in the class. This relates the concept of student engagement to compliance. (Parsons & Taylor, 2011)

In the same work, Parsons and Taylor mentioned student engagement as the intensity and quality of participation from a student. This definition takes student engagement as a comparative scale measuring the emotional attachment of the student with the school activities and the teacher. (Parsons & Taylor, 2011)

Student engagement is based on many factors. One of these factors is the teaching style. There are ways in which the work can be made more interesting and engaging. The teachers can also introduce several activities and tasks to increase the interest of students. Therefore, teachers play an important part in developing the interest of students in a certain subject and helping them in getting good grades. Effective teaching strategies can act as a lever to boost the grades and participation of students. (Parsons & Taylor, 2011)

Mathematics is a fundamental subject. It is used in almost all the sciences. It is a key subject included in the curriculum of almost all the schools. The increasing importance of mathematics has increased the demand of capable mathematicians to tackle the issues of the new technological era. Ever-increasing technology means that the amount of mathematics used in different fields is also increasing. Therefore, it is indispensable that the students perform well in mathematics.

As mentioned earlier, there is a strong relation between the good performance of students and their engagement in school activities. If the student engagement is increased in the mathematics classes, good results in the subject will be brought forward which will then help the students in tackling the issues of this new technological era. Student engagement in mathematics leads to better mathematical justification as well.

Justification in mathematics means supporting your work with evidence. The students should be able to justify their solutions in the mathematics classes. This indirectly highlights the importance of student engagement in mathematics classes. If the teachers encourage student engagement in the class, the student will be able to communicate his mathematical ideas and concepts and will be able to prove his point as well. As a result, the students will improve their mathematical skills which in turn will help them in further studies. (Parsons & Taylor, 2011)

Purpose of the Study

The purpose of this study is to investigate the concept of student engagement. It also furnishes a relationship between the performance of students in the field of mathematical justification and student engagement. The study explores the research of different authors and finds out ways and strategies to encourage the students to participate in different school activities. Basically, the paper revolves around two major questions:

Why is academic justification an important part of mathematical justification?

How does a math student benefit from a teacher who uses best practices and student engagement strategies?

Based on the aforementioned questions, the study aims to investigate the relationship between the concept of student engagement and mathematical justification. In addition to that, the study also aims to find out the strategies which can be employed by teachers and administrators to improve student engagement and thus mathematical justification.

Research Methodology

This study is based primarily on secondary data. Secondary data is the data collected by someone who is not related to the research directly. Secondary data can be available in written or electronic form. The usage of secondary data brings about advantages with it: (McCaston, 2005)

The basic advantage of secondary data over primary data is that it is relatively cheaper to access, store and use. It is even free of cost sometimes.

In addition to that, secondary data gives a chance to the researcher to explore the topic by reading the work of most renowned authors in the world.

Moreover, exploring secondary data also gives the researcher a direction in which he may take his research. In this way the researcher gets a guideline to follow.

Lastly, secondary data is relatively easy to obtain as the researcher can surf on the internet and find thousands of articles on the internet. (McCaston, 2005)

There is, however, a problem of irrelevancy in secondary data. As the work of other authors may not be on the same topic, the researcher can get astray as well. Therefore, excessive care was taken in selecting the articles for this study. Only peer reviewed work was included. In addition to that, obsolete work was also avoided. After the selection of articles to be included, the work was initiated on the study.

Literature Review

Student engagement, as defined in the work of Parsons and Taylor 2011, is the willingness of the students to engage in school activities. In a second definition, student engagement is defined as the intensity with which the student participates in these activities. The latter definition explores the concept more qualitatively than the previous one and thus makes student engagement a scale of comparison as well. (Parsons & Taylor, 2011)

There are three types of engagement as described in the work of Parsons and Taylor. The three types are: (Parsons & Taylor, 2011)

Behavioral Engagement

Academic-Cognitive Engagement

Social-Psychological engagement (Parsons & Taylor, 2011)

The first type of engagement is related to the non-academic engagement. This refers to the engagement of students in extracurricular activities and sports. The second type of engagement that refers to pure academic stuff. A student will be called academically engaged if he submits homework on time and participates in the class as well. The third type of engagement indicates the emotional attachment that a student develops with his school, teachers and friends. (Parsons & Taylor, 2011)

Measurement of Student Engagement

Student engagement can be a very difficult thing to measure. The level of interest the student is having in a certain topic is almost impossible to measure accurately. However, as mentioned in the research of Parsons and Taylor, researcher use two different techniques in order to measure the student engagement level. The techniques are: (Parsons & Taylor, 2011)

Self-report

Teacher rating (Parsons & Taylor, 2011)

The first method is based on the reports brought forward by the students themselves. If a student is interested in a subject he can report it to the researcher and answer a few questions about his interest. If a student is not interested, he can also report his scenario along with reasons. (Parsons & Taylor, 2011)

The second method takes into account the remarks and perspective of the teacher. According to the teacher, the level of academic engagement of a student is indicated by the class participation and the grades he gets. Therefore, teacher's remarks on class participation and grades can provide a lot of information about student engagement. (Parsons & Taylor, 2011)

Importance of Student Engagement

As mentioned earlier, student engagement is indispensable to the good performance of students. Lack of student engagement indicates a lack of interest which eventually hinders the academic progress of the students. It also negatively impacts the lives of students in the long run. As the work of Parson and Taylor 2011 cites the work of Willms, Friesen, Milton 2009, 'What happens today in education profoundly influences the lives of individuals and the health of whole communities for decades to come.' Therefore, if student engagement is neglected today, the students, who are the builders of the future, will be having a severe lack of interest in their future studies and eventually their jobs. This will inflict a loss on the society and these students as well. (Parsons & Taylor, 2011)

Students nowadays want to participate in the class and other activities in their own way. The students cannot be forced to do a certain thing nowadays. This makes the role of teacher even more critical as the teacher has to understand the problems of the students and then encourage them to participate further in the classroom activities. (Parsons & Taylor, 2011)

Student engagement helps the students in getting involved in their own learning. It removes the element of the teacher as a supervisor and gives the students an option to use their own way of learning. In this way, a student is able to use his own method to learn about a certain subject. The student can also share his way of learning with others by telling them. This will lead to positive criticism of the methods of that student and the idea of involving a student in his studies brings about fruitful results. (Theilheimer, 1991)

Academic engagement pushes a student to take risks. A student will take different routes to solve a problem in order to find the fastest way out. This will lead him to undesired dead ends as well but if the student is truly interested and engaged in a certain subject, then his interest will push him to take risks which will eventually get him in the right place. (Magdol, n.d.)

According to Lynn Magdol 2009, Academic engagement helps a student is avoiding many risks which can be serious. Some of these risks are related to low confidence, poor self-control, alienation and other significant behavior problems. In this way, the student can avoid several problems if he is given a proper environment to engage in activities. (Magdol, n.d.)

Mathematical Justification

Mathematical justification refers to the act of supporting mathematical supports and ideas using facts and formulae. Mathematical justification is very crucial as it forms the base of mathematical learning and working. The whole study of mathematics is based on proofs and concrete calculations. If the students are not able to justify their steps and calculations, they will never master the subject of mathematics. (Holdan & Lias, 2009)

These justifications are also important to the teacher as well. The teacher, most of the times, is not interested in the answer but in the method via which the student arrived at that answer. Therefore, the mathematical justification of steps and formulae is an important part of mathematical studies. (Werndl, 2009) (Members of The National Council of Teachers of Mathematics, 2009)

As mentioned in the work of Charlotte Werndl 2009, mathematical reasoning has the following three steps which do not follow a definite order: (Werndl, 2009)

Conjecturing

Generalizing

Justification (Werndl, 2009)

In the first step, the student uses the given facts or assumptions in order to draw preliminary conclusions about the problem. These conclusions are preliminary, as mentioned earlier, and are not certified to be true. (Werndl, 2009)

In the second step, the student finds a relationship between this problem and the other problem he has solved. In this step, the student endeavors to find a common link between the two or more problems of the same kind so that they can be solved in a similar way. (Werndl, 2009)

In the third step, the student justifies the steps he has taken. Justification completes mathematical reasoning. If the student does not know the reason behind applying a certain formula, his reasoning will be useless. He may get the correct answer, but the method will be incorrect. (Werndl, 2009)

Mathematical reasoning helps the students in supporting their proofs. If a student claims a certain formula to be applicable in a certain scenario, he will need reasoning in order to prove it to others. If mathematical reasoning is not given, the claim will just be considered as a guess. In addition to that, Martin et al., also mentions the mathematical reasoning as a basis of sense making in mathematics. If a student is able to provide correct mathematical reasoning, it means that he has understood the concept clearly. (Members of The National Council of Teachers of Mathematics, 2009)

The connection between mathematical reasoning and student engagement is quite obvious. It is a very simple notion that a student who is engaged a lot in studies and school activities will be able to make good justifications as he will take interest in understanding the concepts and formulae clearly. In addition, the student will be better at communicating the justification as he will develop a handsome level of confidence. Hence, academic engagement benefits a mathematics' student in making and communicating his justifications better. (Members of The National Council of Teachers of Mathematics, 2009)

Facilitation of Student Engagement

The role of the teacher is instrumental in facilitating the student engagement. The teacher is the one who triggers the thoughts of the students and thus he can control the thoughts better. The following factors affect the student engagement level in a class and the teacher can manage them to get better results:

Teachable Moments

Teachable moments, as defined in the research of Beverly Scholz 2010, are time periods where the process of learning is the easiest. During teachable moments, the students can learn the maximum in a small amount of time. For instance, students easily understand a chemical reaction in a chemistry lab which they were unable to understand in the classroom. (Scholz, 2010)

These teachable moments should therefore be used and managed with immense care. The teacher should try to make the students grasp the most by giving detailed answers to their questions. As the teachable moments are time periods in which the level of interest of a student is at its peak, the students rarely forget the things they learn during these moments. (Scholz, 2010)

As a result, the students have sufficient knowledge to indulge into questions and answers with the teacher and other students. This will increase the level of participation in the classroom activities and hence, the level of student engagement. Moreover, the reputation of the student will improve in the eyes of the teacher and the students and his confident level will also improve. (Scholz, 2010)

Exploration vs. Instruction

According to Adelson 2004, instruction is a direct guideline about how to do a certain thing. It is a teaching style in which the teacher gives the students methods to solve their problems and getting answers. (Adelson, 2004)

Although it seems to limit the potential of the students, Adelson quotes Klahr saying that direct instruction is more organized and disciplined and can get the students out of certain complex situation more easily and efficiently than exploration. Klahr supports his point by saying that most of the things we know about science were taught to us and we did not discover them. (Adelson, 2004)

Exploration, on the other hand, is defined as a method in which the students are given situations and tools to deal with those situations. In this method, the students do not get a way out from the teacher, they find their own way out. In this way, the potential of the students is harnessed. (Adelson, 2004)

Exploration learning is slower than instruction-based learning but this method gives full knowledge to the students. In addition to that, exploration does not limit the learning ability of the student. Moreover, exploration gives way to further studies which increases the innovative ideas as well. (Adelson, 2004).

Active and Passive Learning

Michel et al., 2009, defines active learning as a broad term encompassing several styles of teaching in which the students participate in the teaching process as well. He further states that active learning refers to all the teaching methods in which the students do a task and think about that task at the same time. Some characteristics of active learning are momentary pauses during the lectures for writing and discussing purposes. (Michel, Cater & Varela, 2009)

Michel et al., 2009, also mentions the prerequisites of active learning. The first prerequisite is the atmosphere of the classroom which should be friendly and relaxed. If the teacher creates a strict environment, the students will not actively participate in the discussions and other activities. (Michel, Cater & Varela, 2009)

In addition to the atmosphere, planning is also an important requirement of active learning. The pauses and discussions should be planned in a way that they bring about the greatest benefit to the class. Moreover, the teacher should also implement the plan made or else the main purpose of the active learning method will be lost. (Michel, Cater & Varela, 2009)

Lastly, continuous improvement is also necessary for active learning. Continuous learning means discussing the class plans and methods with the students and getting their recommendation in order to improve the system. (Michel, Cater & Varela, 2009)

Passive learning, as defined by Michel et al., 2009, is the teaching method in which the teachers enter the class, deliver a tedious lecture and leave. In this method, the students are not given time either to note the lecture or to discuss it. In addition, the syllabus and the class schedule is also decided by the teacher and handed over to the students to follow. This makes it a rather stringent method of teaching. (Michel, Cater & Varela, 2009)

The biggest disadvantage of passive learning is that the students have trouble in understanding and remembering information imparted by the teacher. Michel et al., 2009, describes the lack of pauses and lecture notes as being the reasons behind this problem. In addition to that, the student participation in the teaching method and material is minimal and therefore the students frequently get tired. (Michel, Cater & Varela, 2009)

Field Dependent and Independent Styles

Mutlu and Temiz describe independent workers as those who do not trust the surrounding environment and people. They consider the whole class as a group of components and not as a whole. This makes it difficult for them to comprehend the concepts of group work. In addition, during a discussion, independent workers rely on their own ideas and believe them to be correct. (MUTLU & TEMIZ, 2013)

The independent workers have an ability to discern between the components of something treated as a whole and thus they are good at evaluating things. Independent workers are not good at making relationships as they place minimal trust on people and their ideas. (MUTLU & TEMIZ, 2013)

Dependent workers, as defined by Mutlu and Temiz, tend to trust their surroundings and find it easier to grasp a general theme as portrayed by the components. They are more inclined to consider things as a whole. Therefore, they prefer working in groups. (MUTLU & TEMIZ, 2013)

One particular strength of dependent workers is that they can get a general idea of the situation very easily as they are not bothered about the separate components that much. In addition, they also have the ability to share their thoughts with others and trust others thought as well. (MUTLU & TEMIZ, 2013)

Mathematical Connections as Scaffoldings

Mathematical connections, as the name suggests is to make connections between different mathematical concepts and the real life. The idea behind the use of mathematical connections is to make the solution of mathematical problems easier for the students. If the students can see the connections between mathematics and real life, they will take more interest in it as they will see the things happen in front of them. (Glacy, 2011) (Siemon & Virgona, 2008)

In the work of Siemon and Virgona, scaffolding is described as 'adult interactions in children's learning, in particular, the support that an adult provides in helping a child to learn how to perform a task that cannot be mastered alone.' This describes the intervention of teacher with mathematical connections in order to help the students with complex things that they cannot understand. (Glacy, 2011) (Siemon & Virgona, 2008)

Efficient usage of scaffoldings and mathematical connections will help the students in understanding the different concepts easily and in making their own justifications and reasoning. Therefore, it is also a critical factors affecting the student engagement. (Glacy, 2011)

Discussion and Analysis

Student engagement, as mentioned earlier, is a very important determinant of student success. Student engagement is an indicator of a learning student. If a student is learning and grasping the knowledge imparted by the teachers, then he will participate in the class and other activities as well. This shows the progress of the student. (Parsons & Taylor, 2011)

In addition to that, academic engagement helps the students in participating in the studies. Normally, the teacher has the syllabus and the teaching schedule but if the teacher encourages the students to participate, they will have a say in their studies. The students will have information about what they are studying and the reason behind studying a certain subject. (Parsons & Taylor, 2011)

Student engagement is an important part of mathematical justification. As mentioned earlier, mathematical justification is a very important part of mathematics in which the student proves his formulae and problem solving. Student engagement fosters the process of mathematical justification as the student gets to know about the concepts and starts to participate with the class. Mathematics requires student-teacher interaction which is facilitated by the student engagement. Student also discusses the mathematical phenomena with other student which is again shows student engagement. Without necessary skills to participate in activities, the student will face difficulties in mathematical reasoning and justification. (Members of The National Council of Teachers of Mathematics, 2009)