Introduction
There are disparities between different people on there relational thinking which has led to different understanding of algebraic relations. Different people will understand and explain relational questions in different students way frameworks according to how they have understood the situation. This has led to generalization of ideas. For example an equation 6 + 5 = □ – 4. There is a relational thinking in that the first function of the equation is equal to the second function of the equation. Through generalization, different people may come out with a different analysis and conclusion. People will prefer generalization of algebraic equations. There is a need for students to be taught on observation and interpretation skills in algebraic equations because they constitute an important part of critical thinking. Most students understand the = sign as a result rather than a mathematical equivalent. An application of relational thinking involves viewing the equal sign as a sign of the preceding computations and the real answer as the number after the equal sign. Like in the case 10 + 20 = □+15. Many children will think of the equal sign rather than the real outcome of the application. Relational thinking will also involve the use of number relations to simplify calculations. Students viewing an equal sign relationally can solve the equation 10 + 13 = □ + 8. They will calculate the sum of 10 + 13 to get 23 and subtract eight to get 15. Conceptualization involves making explicit general relations based on the fundamental properties of numbers and their operations. It involves a verbal or direct answer like true or false for example 4 + 8 =12 a conceptualizing question will ask whether that is true or false and doesn’t require a numerical answer.
The statement of the problem
Algebra is the basis of all mathematical calculations and a good basic understanding of algebraic relations is vital for elementary school students to get important ideas that are in the context of their study of arithmetic and calculations. The pre-service teacher’s knowledge of a subject matter and the thinking of a student are vital factors that will impact the reform of algebra in elementary schools. There is a need to address the gap in which elementary school or pre-service teachers have knowledge in the domains of the whole and a rational number but with little or no focus on early algebra.
Objectives of the study
The main objective of the study was to establish whether pre-service teachers had knowledge of the tasks that had the potential to engage students in relational thinking and about the equal sign. The study also examined the importance of the teacher’s role in the development of elementary school students’ understanding of the equal sign and relational thinking. There was a need to establish whether early thinking in terms of generalization of the algorithmic process that smoothens transition to formal algebra. Also, there was an examination of whether students had a relational view of the equal sign in algebra and the pre-service teacher’s awareness.
The hypothesis of the study
Pre-service and elementary school teachers know the tasks that have the potential to engage students in relational thinking about the equal sign. The pre-service and elementary school teachers have an important role in the development of elementary school student’s understanding of the equal sign on relational thinking. Early thinking in terms of the generalization of algorithms process cannot smooth the transition to formal algebra. Elementary school students do not have a relational view of the equal sign in algebra and their pre-service teachers are not aware.
The research will enhance the future teachers to have knowledge of tasks that will have a real potential to engage students in relational thinking and thinking about the meaning of the equal sign. It will also assist teachers to know the students understanding and misconceptions about the issues. This will enable the elementary students to move from the computation of algebra and focus on the equation structure. Pre-service school teachers will thus be able to know the real structure of the task and the main features which they will see as the real mathematical goals. The study is therefore bound to discuss detailed issues and pedagogy in relation to artifacts of students thinking and the frame works that will be developed. Pre-service teachers will be able to help students to analyze equation functions ideas such as analyzing multiple representations, of relationships and interpreting functional relationships written in words such as true or false and formulas.
Theoretical framework
Stacey (1989) investigates a student’s ability to recognize a linear pattern and express it as a rule. An alternative theory suggested that students construct based on their natural language is a cognitive statement that stimulates the semantic situation but is based on comparison rather than equality. They claimed that the model for a situation like “5 is 8 more than t” has the “8 more associated with the 5(larger and variable) and not with the “t (smaller value). MacGregor and Stacey (1997) studied a prevailing theoretical explanation for student’s inability to interpret algebraic letters as generalized numbers or even as specific unknowns. They pointed out that the explanation of the cognitive level was insufficient and suggested more specific sources such as instructive assumptions, interference from new learning, and the effects of misleading teaching.
Teachers usually see the ambiguity in solving open-ended mathematical operations such as 3 + 47, students will not. This is because they tend to solve expressions based on how the items are listed, as a left–to–right pattern. Kieran (1979) suggests that if an equation such as 36 = 18 is replaced by 3 4+2 = 18, students would realize that the usage of brackets is essential to keep the equation balanced. Elementary students believe the number immediately to the right of an equation sign is the answer to the calculation on the left hand. For example; students filled in the number sentence 10 + 5 = + 5 with 15 or 20.
Novice algebra students do not understand the meaning of letters hence they interpret letters as standing for objects or words (Macgregor and Stacey, 1997). Even if students accept that these letters are standing for numbers, they will tend to associate those letters with their position in the alphabet (Watson, 1990) and will not understand that multiple occurrences of the same letter represent the same numbers. This misconception when addressed will enable students to view letters as representing specific unknown values, as in x+y = y+ x. Kuchmann (1981) found that only a small percentage of students between 13- 15 years were able to consider a letter as a generalized number.
Tiroshi (2000) emphasizes on pre-service teachers to possess knowledge on both the existence of misconceptions by students concerning the meaning of the equal sign and the knowledge for the reason of the existence of the misconceptions. This study examines the teacher’s knowledge of why these misconceptions exist and their existence in the students.
Research methodology
The research design used was a descriptive survey. The descriptive survey involves measurement, classification, analysis, comparison, and interpretation of data. A descriptive survey also involves collecting information by interviewing or administering a questionnaire to a sample of individuals. It is used when collecting information about people’s opinions, attitudes, habits, or any of the variety of education or social issues. In the search, twenty students selected from four colleges in Ireland were selected to participate. They were majoring in mathematics as their teaching subject for pre-service students. That is why they were believed to be helpful and reliable for the research.
Research instruments
The research instruments that were used included a well-structured Questionnaire that had adequate questions for the respondents to answer. Then there was a well-structured interview in which the participants were supposed to sit for a hand-written exam. Another instrument that was used is a focus group discussion in which the participants were involved in a discussion to discuss some of the issues of the study. Also, the use of personal observation was included in the research.
Data collection
Primary data was collected from the respondents through questionnaires, well-structured interviews, focused group discussions, and personal observation. Secondary data was collected from journals, books, magazines, the internet, and former research reports.
Data organization, analysis, and presentation
The collected data was pre-processed to eliminate the unusable data, to interpret the ambiguous answers, and to remove contradictory data from related questions. It was then coded and stored on paper storage and electronic storage. Then a word processor and a spreadsheet were chosen as the statistical soft wares to be used. Data were analyzed by use of a quick impressionist summary and casual-comparative research used to analyze quantitative data. Data was presented using statistical methods in which a frequency distribution table was used to enter the variables.
Data collection
- Case 1 examined whether teachers understood the student’s conception of the equal sign. It also investigated whether the students would differentiate the numbers to the left of the equal sign and the right of the equal sign.
- Case 2 Investigated to establish whether teachers understand the student’s generalization thinking. It sought to examine the thinking capacity of the students in reasoning.
- Case 3 was used to test how the equal sign relates to two numbers on both the left and the right-hand sides.
- Case 4 was structured to examine the teacher’s perception and understanding of algebraic equations and how they can relate this to their student.
The whole results of the participant’s responses to the questions were presented in a table indicating the number of participants out of the 20 who recognized the potential tasks and the number of the participants out of 20 who had proposed relational thinking strategies. The participants were supposed to respond to the conceptions of the students and why these tasks were in place.
Table (1): Represents participants in both tasks of engaging students in relational thinking and those proposing relational thinking
The teachers goals implied the degree in which he or she can identify opportunities that can change students in relational to thinking. Some pre-service teachers may identify opportunities to engage students in case 2 which will involve conversations about the meaning of the equal sign and the significance of the equivalence sign. Other pre-service teachers will engage the students to solve the case through basic calculation procedures. The teachers should understand well which strategies they would use to approach those tasks. Strategies used will depend on the age of the students. The rational thinking and misconception in the reasoning where tested using case 3. In this case □ was equal to 31 and also the difference between 30 on the left and 31 is 1. It was noticed that most of the students assumed that the equal sign meant a grant total rather than a relational sign.
The participants cited that through generalization a student would use the first equation in case 2 to assume that the answer to the second question is 6 and the answer is false because the answer to the first one is 6. The answer could not be true because, through relational computing, the result would be 3 and is true. The fourth case involved the participants choosing the easiest equation and arranging the others according to the degree of complexity; this tested the generalization by students instead of relational thinking.
Data analysis
The participants agreed that a teacher may bring about complications to a student by developing tasks to help students to develop relational thinking. For instance, when the tasks introduce students to algebraic manipulations, they make students to practice a double-digit addition or new algebraic valuables.
Findings
Elementary teacher’s conception on tasks
Four cases were developed that were based on the tasks aimed at investigating or promoting the students development in the understanding of equivalent and relational thinking. The pre-service teachers were supposed to understand the mathematical structure before imposing them to the students. The pre-service school teachers were expected to understand that imposing tasks to students will enhance their relational thinking or to investigate and promote an understanding of the equivalence of mathematics. Only 13 out of 20 participants supported the conception of tasks. In case 1. There was an implication that most of the pre-service teachers were thinking of the symbol and were not thinking of its mathematical implications to students. This group of teachers will develop students who will be thinking of the sign but not algebraic impression of the sign, pre-service teachers also viewed relational thinking as a solution to algebra rather than significantly different thinking.
In case 2, 16 participants recognized the potential of the task. That means the majority of the interviewees could not determine the relational value of the question. They would be generalizing the meaning of the symbol instead of giving its real relational value and meaning in algebra. A pre-service teacher can perform the equation on both sides and make the equation to remain true is a notion that was supported by 11 teachers while the other 5 teachers stated that it could help to develop student’s understandings of equivalent in generalization. In case 3, 18 participants identified the use of a symbol manipulation or computational procedures as mathematical goals of the task. 9 participants identified the promotion of the understanding of the equal sign while the other 9 identified the process of helping students to solve mathematical relations in a problem.
The awareness of pre-service teacher knowledge on student’s conception
The teachers gave their opinions on which answers they expected their students to give. A half (10) of them proposed that the students will give relational definitions in response to case 1. According to them the equal sign means the variables to the left equals to the variables to the right. The other 10 gave the operational response that was implied by the separation of the problem by the answer.
In case 2, 12 participants proposed that students might simply state that the second equation is true. This is because they would see 3 subtracted from both sides. It is clear that the students may substitute 10- 4 in the left side to remain in the second statement with 6-3 = 6-3 assuming that the statement is true. This relational strategy employs the information given in the original strategy because it employs the information given in the original statement with out calculation. In case 2, a student may compute only one equation forgetting the first equation to arrive into a conclusion. The relational thinking strategy was agreed in 1 case 3 by 19 participants.
This indicates that the pre-service teachers are likely to propose strategies involving arithmetic computation or algebraic calculation. The participants were capable of recognizing opportunities the students would have to solve the tasks in relational ways. The participants had awareness of relational thinking opportunities in tasks, thus proposing students to employ such strategies.
Teacher’s misconception of student’s misconception
The teacher knowledge on students believes of equal sign was examined by having participants to examine responses to these tasks in which particular misconception was demonstrated. (Table1). Here the participants proposed students misconception of algebraic errors. In case 2, 12 participants proposed students might use relational thinking strategies, while in case 1, 15 participants out of 20 supported the use of relational thinking. While in case 4, 18 participants supported the use of relational thinking. This shows that there is a distinction between teacher knowledge why students hold particular misconceptions and knowledge why they hold these mis- conceptions (Tirosh, 2000).
Discussion
This research focused on the student’s relational view of the equal sign in algebra and the pre-service teacher’s awareness and the early thinking in terns of generalization of algorithmic process that can be a transition to formal algebra. It also focused on pre-service teacher’s misconception of students misconceptions. The research also examined the elementary pre-service teacher’s misconceptions on tasks. Another issue investigated in the report was the awareness of relational thinking and the awareness of student misconceptions about the meaning of the equal sign.
The research realized that teachers should increase their relationship with students to enhance their interest in mathematical thinking. When the pre-service teachers pay attention to students, they will improve their relational skill in mathematics. This can help the current efforts better to understand the knowledge needed for teaching (Ball and Cohen 1999). In this case they will be able to access the student’s performance and help them to increase their skill in mathematics. Children who are paid attention to by their teachers will develop an interest in relational thinking and generalization.
There is a need to establish a cordial relationship between pre-service teachers and the students. This will enable the pre-service teacher to have his strategies accepted by the students and also will find strategies emerging from the students themselves and use them to create their interest in relational thinking and generalization. The strategies should be designed in a way that will be easily adopted by the students and a simple form in which a teacher can achieve his goals of developing the strategy.
The pre-service teachers should be flexible in the use of strategies or administering of tasks they should use different strategies in different subjects. This will make the students to accustom themselves with the strategies for mathematics and algebra that the pre-service teachers will easily differentiate them from the others. The pre-school teachers should engage students in group discussions to enhance their active participation. For the elementary education is the basic education foundation for all children, the children’s mode of instructions should be based on generalization because it is the method which they will understand easily. Conceptualizing teaching methods through generalization will enable the students to keep what they have learned. The complexity of the strategies should increase as the level of study of the student’s increases. This will ensure that students develop their intellectual ideas of reasoning and grow up according to their age and a level of study. For these pre-service teachers who are unable to have a good ground of knowledge in relational thinking it would be uneasy for them to help the students to conceptualize through their teaching methods. The pre-service school teachers should have perceived the questions the way they would pass it to their students.
The participants have knowledge and are aware of relational opportunities and tasks. But the level of understanding will influence their student’s level of understanding. Every participant liked relational thinking basing on the way they responded to the written mathematics during the interview.
Relational thinking
All the participants agreed that relational thinking should be used in response to tasks, student’s tasks and application of strategies in teaching algebra. (Shifter, 1999) proposes a focus on relational thinking in the lower classes and encourages the teachers of this level to engage students in aspects of early algebra reasoning. The participants in this study were selected from a large population to represent the entire population of pre-service school teachers. The research has implications to preserve elementary school teachers on importance of relational thinking. Although, most of the participating teachers supported the relational thinking in the cases which represented tasks, they had differences across different tasks. The tasks underlying the structure and its simplicity determined the ease at which it could be noticed or chosen. Case 2 had 12 participants while case 1 had 15 participants. In case 3, 18 participants identified and approved the understanding of engagement of students in relational thinking. Case 4 had 16. Therefore, it is easy to access relational thinking in algebraic or mathematical contexts.
pre-service teachers do not value the opportunities or ways of thinking in mathematical calculations as they will identify relational thinking opportunities. Participants noted both engaging their students in relational tasks as well as algebraic tasks as shown in case 4. The participants knew about relational thinking in mathematics but were unlikely to practice this in the future, the participants also viewed relational thinking as a method rather than an idea that must be encouraged for reasons beyond solving the algebraic problems so the research suggests that teachers should be helped to reflect on the importance of rational thinking strategies. This is vital because it will prepare their students for future studies in mathematics. Pre-school teachers should emphasize on helping students to develop relational thinking rather than generalization of answers based on observations or assumptions
Relational thinking involves calculating mathematics using the fundamental use of numbers which gives the values of the variables through a basic calculation. It works as an algebraic function rather than an arithmetic function as it tries to show why the answer to a solution was realized rather than giving the answer itself. Relational thinking depends on critical conditions to arrive at solutions to mathematical problems. Relational thinking seems to create awareness among numbers and the properties of numbers for easy calculation of variables or algebraic equations to get answers and solutions to problems as it is always based on algebraic reasoning. It is a function of evaluating and accessing expressions and equations in whole and determining relationships of numbers subject to the equality symbol.
Relational thinkers seem to assume equivalent to in the two sides of the equation, they keep them as uncalculated pair. A condition referred to as lack of closure. Therefore relational thinkers can work with uncalculated pairs in equivalent expressions. They don’t make quick decisions and will look at the size and direction of the variation to find a missing number. They are able to ignore these quick decisions but see the possibilities of variations between the numbers in an equivalence relation where the direction and the size of the variation depend on the numbers and operations involved. They easily recognize variations and numbers which are closer to each other like 77 and 70. Rational thinking is marked by the capacity to see variations in numbers in a sequence. There is importance in helping people to differentiate dimensions of variations when elements in a mathematical sentence change while the elements remain constant. Given 18(A) =20(B) to intensify a good structural explanation to this relationship requires students to move beyond general calculations and use relational ideas. A relational answer will be that the above sentence is true as long as the number in (A) is two more than the number in (B). To derive a correct mathematical generalization from numerical examples is a key element of algebraic reasoning (carpenter and Franke, 2001).Critical numbers and the relational elements embodied in these expressions require that students move beyond computation and keep focusing on the underlying mathematical structure. Equivalence and compensation provides a foundation for algebraic thinking.
Compensation and equivalence have a basis in that some students using them to solve number sentences may provide a foundation for algebra thinking. This is usually referred to as relational thinking and when it is applied to a number sentence it can be properly described as algebra. More so one fundamental goal of intergrading rational thinking into the curricuculum is to facilitate student’s transition to the formal study of algebra in the later grades so that no distinct boundary exists between arithmetic and algebra. Students who are quite capable of using ideas of equivalence and compensation to solve sentences involving literal symbols are clearly different from those students who could use only computational methods.
Identifying the critical numbers and the relational elements embodied in these expressions require that students move beyond computation and focus especially on expressing the underlying mathematical structure. Equivalence and compensation provide a foundation for algebraic thinking Like in the equation if (c+2= d + 10) what is the value of (c) and (d.). Relational thinking students may solve algebraic equations using rational reasoning or both rational reasoning and computational techniques. Relational thinking students have the ability to explain relationships between literal symbols (Carpenter and Franke, 2001).Relational thinking depends on students being able to see and use the possibilities of variations between numbers in a sentence.
Equal sign misconceptions
Students might hold misconceptions about the equal sign and thus end up generalizing answers for algebraic equations. This situation will influence their ability to solve the interview tasks. Although students taught well may not have misconceptions, but the participants faced difficulties when explaining the thinking when confronted with the equal sign. Lack of awareness further explains lack of concentration the pre-service teachers made on equivalence or relational thinking. Pre-service teachers should be made aware that misconception of the equal sign exists so that they can help students at elementary level to overcome it. Tirosh (2000) proposes that pre-service elementary teachers were initially unaware of student misunderstanding and division of fraction a knowledge which could be developed in mathematics method course through discussions of hypothetical student work in which this misconceptions were illustrated. Teachers with inabilities to recognize and reflect on equal sign misconception are likely to transfer this to their future students. Misconception of the equal sign will lead students to generalize their assumptions for algebraic calculations which give false answers.
Generalizing fundamental properties of numbers
This is a principal of algebra which involves making general relations based on the fundamental properties of numbers. It is algebra viewed as generalized arithmetic. Relational thinking involves making conclusions and general relations that are arrived at as a result of the fundamental properties of numbers and operations. pre-service school teachers have an exclusive knowledge about the number relations, however majority of them have not generally examined the relations. Through questioning, a pre-service teacher can assist students to explore the generalities that often exist naturally in discussions of arithmetic, and researchers have provided evidence that even elementary school children are capable of making generalizations. It involves a simple answer of yes or no to approve or disapprove a mathematical or an algebraic equation.
According to this research, generalization was applied in case 3 where the pre-service teachers were required to analyze an algebraic expression and give a true or false answer. It involves giving answers through a word of mouth after observing written numbers. It is mostly used when comparing variables.
Conclusion
This research addressed the issue of developing a relational approach to algebra among pre-service teachers to encourage early mastering of the tasks in the teaching of mathematics. It emphasized on the concept of equivalence and rational thinking which are important in teaching mathematics. It will influence the attitude of pre-service teacher’s aspects to algebra such as reasoning. It also made a concern with relational thinking opportunity tasks and the equal sign misconception. As the result of the study showed that the awareness of relational thinking is necessary and if pre-service teachers would engage their students in this type of thinking it would be important in mathematical development. Pre-service teachers should be taught and exposed to tasks that are well-structured to challenge a student’s misconception on the meaning of the equal sign and engage them in relational thinking. The teacher should be able to transfer these tasks to the students so that they can have a good foundation of mathematical knowledge. The teachers should further understand the misconception of their students in the equal sign and help them to develop relational thinking. Developing a culture of misconceptions will encourage the development of students who will be incompetent in algebra in their future life when they will grow up. This may lead to poor performance in their advanced stages due to a poor foundation.
The study also focused on algebraic generalization which involves giving solutions to written numbers by observation and comparison to arrive to an answer. It involves conversation and giving ones opinion or a solution to a mathematical problem according to how he has perceived or understood the algebraic function. Generalization only requires a true or a false answer as it is a suggestion or expression of someone’s opinion. The study found out that students solve problems through relational thinking, generalization, while others do not identify the meaning of the equal sign; it established that pre-service school teachers have a misconception of the students misconception, they have different conceptions on different tasks and sometimes are not aware of the students conception. If the pre-service school teachers improve on these issues, they will help in developing the approach to algebra and mathematics that will be adopted by the elementary school going children. It also established that those pre-service school teachers who had a poor misconception or understanding of algebra and mathematical understanding could transfer the weakness to the students. Hence there is a need for the pre-service teachers to train well in mathematics before they start teaching algebra to students of elementary schools. This will help to lay a good foundation on the students and they are likely to grow up with relational thinking.
Recommendations
According to the findings of this research, several strategies are supposed to be adopted to improve the study of algebra or mathematics in schools. It therefore recommends several methods that should be applied in schools to improve relational understanding and generalization in mathematics. The report recommends professional development in schools to ensure that there is development of s understanding of algebra or mathematics. This should be facilitated by teachers in schools. This may take form of a work group meeting of teachers to discuss the development of mathematics in schools.
The study further recommends that teachers should develop strategies to enable their students to succeed in arithmetic calculations. This may include noticing the opportunities of students in arithmetic. The teachers should develop a continuous research in mathematics noticing any anticipated opportunities to extend their on going algebraic reasoning. Teachers should value any strategy that is suggested by students so that they cooperate in developing the mathematical knowledge of children. The research further suggests the development of on-site support structures where teachers and development facilitators will wok together at school sites after work together by sharing ideas in the development of relational reasoning and generalization. The study also suggests that teachers should take measures to ensure that there is a development of a mathematical reading culture in schools.
The research further recommends the development and establishment of algebra or arithmetic clubs in schools to develop children’s love for mathematics in schools. Also the study suggests that the teachers should create a condusive environment for students to enable them to have interest in algebra or arithmetic. This can be done by convening algebraic or arithmetic clubs in schools, mathematical contexts and some occasional relational thinking interviews for the students to access their development in mathematical knowledge.
The study recommends career development for teachers so that they can acquire modern skills to teach mathematics lesson. It also recommends that local authorities and the government should work together in ensuring that children are taught arithmetic as it is required. They can do this by ensuring that there are adequate and qualified teachers in schools. This research used a sample of a small population to represent the interest of a large population. In addition to the findings of this research which are vital for the development of relational and generalization of arithmetic, it further suggests that continuous research be carried in the future to further qualify the findings of the research.
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