# Research in Mathematics Education

## Rational Number, Ratio, and Proportion

The authors of this paper acknowledged the fact that during the mathematical development of children, the greatest obstacle is experienced in learning the rational numbers (Behr, Thomas, Harel, & Lesh, 1992). They also point out that despite various researches having been done to study the children’s development of knowledge in rational numbers, there is no clear agreement about the ways of facilitating this learning process.

However, they suggested that the best way to comprehend rational numbers fully is to get exposure to various constructs. Rational number constructs are varied and may be fractions of either whole numbers or decimals. The authors suggest that this could be done through imparting this knowledge to children by making them participate in activities that involve comparing quantities of items both by feel and by sight.

The knowledge on rational numbers also shows the importance of understanding fractions and their equivalences. Knowledge in fractions gives an understanding of the size of the number. Research also suggests that two fractions can be compared to investigate whether they are equal or not, basing on their transformability.

The results show that the early attempts to impart mathematical knowledge to children involve the stressing of a higher order of thinking and processing of information. This would help the child make the necessary translations. The authors stressed on the importance of encouraging the children to go beyond the single incident and to reflect more about the general meaning. They are to be encouraged to think about their own thinking. This would improve their reasoning capabilities.

The teachers, on the other hand, should also be able to identify their thoughts and make a description about their teaching methods that should involve various levels. Teachers should not only teach content but also transmit attitudes, culture and an understanding of mathematics.

## The place of Meaning in the Teaching of Arithmetic

The author of this paper defined meaningful arithmetic as that which is aimed at teaching arithmetic meaningfully and ensures that children learn to make sense from the mathematical relationships. The authors complain that not everything meaningful is taught and that not every concept in mathematics is taught in the same degree of competency. On the other hand, meaningless arithmetic does not necessarily mean that it is useless but just that the knowledge acquired by the learner is wholly due to his or her own efforts (Brownell, 1947).

The author categorized the meaning of arithmetic into four categories. The first one was on the basic concepts such as learning whole numbers and fractions. The second involved understanding the fundamental operations. This involves the children’s knowledge of when to apply addition, subtraction, multiplication or division. The third is on the meaningful principles, generalizations and relationships in arithmetic. The fourth is on having the understanding of the decimal number system. The child should know how to use it in the computation processes and in algorisms.

From the teacher’s point of view, teaching meaningful arithmetic is interesting as opposed to just reading out facts, word for word. The teachers see the need to develop understanding to the children and they find this more stimulating than merely listening to memorized information. This, to them, resembles the mechanical drills the soldiers receive – mere repetitions and cramming.

To a student, true meaning is achieved if he or she is able to face other new qualitative problems with confidence and becoming independent. The student needs to find solutions through problem solving rather than from memorization. Meaningful arithmetic should also reduce the incidences of repetition in practice before completing the learning process.

Results further show that many teachers have proved that meaningful arithmetic yield results. In order to improve the instruction there are two alternatives. The first is to double the efforts inputted earlier in the teaching process or to change the method completely to include meaningful arithmetic. Since the first alternative does not provide good results, the second one should be adopted.

## What research tells us about teaching mathematics through problem solving

This paper addresses on the methods of teaching mathematics through skills in problem solving. For many years, there have been changes and advances in the understanding of the complex nature of the processes that one goes through during problem solving (Lester, 1994). Several discussions have been made regarding to the methods of teaching. The focus has been on problem solving. However, there are challenges to be met since this is a relatively new venture and has not been a subject of much research.

This new approach of teaching mathematics through problem solving involves the changing of certain aspects related to teaching and learning. One of them is the changing of the roles of the teachers. Another one is the selection and designing of problems for instructions. The type of learning would need to be of collaborative nature, which has not been the case in previous teaching and learning processes.

The issues of concern related to teaching through problem solving include the question of whether the children would be able to explore problems on their own and get sensible answers. The other is the issue of how the teachers would teach through problem solving. The third is on the beliefs the students have about this mode of teaching. Lastly, there is the issue of sacrifice of the basic skills that the students might be forced to do away with in the event that mathematics is taught through problem solving.

Results suggest that the teacher’s beliefs about mathematics have an effect on their teaching. Teachers with different beliefs about it will teach differently. Research also suggests that the students’ beliefs also affect their learning process, either positively or negatively. Those who believe that mathematic problems can only be solved in a particular manner only believe in memorizing rather than understanding.

On the question of whether students who learn through the problem-based curriculum would sacrifice of the basic skills, research suggest that the loss would not be significant since studies on this show that students who used problem-solving skills outperformed those who were using a traditional curriculum in all tasks.

## Cognitively guided instruction: A knowledge base for reform in primary mathematics instruction

This article provides a study that suggests that the understanding of the way of thinking of the students is important to the teachers. This gives the teachers the understanding of the subject matter, curriculum and the process of teaching. Teachers can use this information to interpret and change their knowledge about mathematical thinking in students.

The authors believe that students come to the school with a lot of information that is usually informal and may be used to develop it into formal mathematics in the primary school level. This information is usually informal and intuitive. The authors argue that what is of more concern is to get the understanding of the students thinking in order to reconceptualize the teachers’ knowledge (Carpenter, Fennema, & Franke, 1996).

The authors also acknowledge teaching to be an activity that must be done through problem solving and that it cannot be understood simply by looking at the activities engaged by the teachers during the teaching process. Therefore, the understanding of the teacher’s knowledge is a basis for understanding the teaching process, evaluating the level of competence and in shaping the mode of teaching.

In order for the teachers to understand the student’s thinking, they should consider the five facets proposed by the authors. The first is to understand of the typical understanding of the students. The other is to understand the student’s learning process. The other is to know what the students consider hard or easy. Teachers should also know the most common errors made by the students. Lastly, is to know about the understanding of the particular student.

Research has identified the two most important features of the knowledge of the learning process of students. One of them is that the students must understand the concepts in the learning process rather that learn through routine. The other is to connect mathematical concepts and theories with the experience obtained from the use of concrete objects.

## Purposes and methods of research in mathematics education

The author of this article tries to lay out some of the important perspectives in mathematics education and looks at the background information regarding its nature of inquiry. The author tries to explore some of the issues regarding mathematics education and starts by looking at the purpose of research in this education. Research would prove that there is a great difference between education and mathematics in many respects (Schoenfeld, 2000).

The author points out two main purposes of conducting research in mathematics education. The first one states that research is useful in the sense that it fosters understanding to the nature of mathematics teaching, learning and thinking. The second purpose of research is to use it to improve instructions in mathematics. Both are related and equally important.

The importance of research in mathematics education is diverse. One of the benefits of research is in the provision of theoretical perspectives for understanding the learning and teaching processes. It is also important in that it gives an understanding of such aspects of cognitions as the students’ abilities to understand or not understand concepts of limit and function.

The author argues that the results are rarely definitive but suggestive. They are not meant to provide proof but provide cumulative evidence that could be considered beyond reasonable doubt. A scientific approach could be applied but one should be careful not to overuse it. Experimental methods are not of great importance but what counts is the use of reasoning and standard of evidence.

Some theories have proposed that if students were to be given an introduction to information that they are to read, and information that gives them an idea of what is to follow, their comprehension of the subject would improve significantly.

## A history of research in mathematics education

The article records about mathematics education, which has been developed for over two hundred years as educators and mathematicians try to develop ways of teaching and the ways to facilitate the learning process. Research in mathematics has also tried to formulate theories to address the same issues that mathematics education has tried to solve.

Research in mathematics education has been affected by two disciplines. One of the disciplines is mathematics itself. Mathematicians have had an interest in studying the learning process and the methods of teaching mathematics for a long time (Kilpatrick, 1992). This process has led to disputes arising between the applied and pure mathematics and have threatened to change the curriculum in schools and colleges.

The article also addresses the fact that mathematicians have always had an interest in studying mathematical thinking. They have done this by trying to explain the mystery that underlies the processes of the creation of mathematics. They have managed to capture some processes by using some terms. These terms include intuition and insight.

The involvement of both the mathematics educators and psychologists has led the mathematics education to be organized into several categories. The first is association learning. Another is activity learning. Creative learning and problem solving is another. Others include attitudes, evaluation of achievement and learning characteristics linked to achievement.

Various researches have been done in mathematics education and generally, the quality of the published materials has increased. Most of the research that has been done is in the curriculum reforms. Apart from basic research, applied research has also been used and has seen the employment of new materials and methods.

## Motivation for achievement in mathematics: Findings, generalizations and criticisms of the research

In this paper, the authors examine recent research in the field of motivation in mathematics education. They also discuss the findings from research in this area. They noted some consistencies across research perspectives that concluded that there were some generalizations. Generalization was made in the conclusions made about the factors, processes and benefits of aspects that influence the motivational attitudes of both students and teachers.

The authors criticize the lack of the theoretical guidance that aid in the conducting and interpretation of most of the studies in this field. Only few researchers have tried to use current theories of motivation in order to have consistency in with current research on the learning process in the classroom situation (Middleton, & Spngias, 1999).

Those researchers who are interested in examining motivation in the field of school mathematics are required to assess the link that exists between the desire of the students to achieve and the field of mathematics.

Results from research conducted in the national level indicate that the American students tend to find pleasure and enjoy in doing mathematics when in the primary level. However, this interest reduces dramatically as the students’ progress with education at the high school level. Despite students understanding the importance of mathematics, many of those who had an interest in exploiting mathematics more are loosing that interest.

The author also laments about the fact that many of the children do not have the desired mathematical knowledge that is required of them in order to be functional in the current society that is increasingly incorporating technology.

## Mathematical caring relations as a framework for supporting research and learning

This article addresses on the mathematical caring relations as a method of improving the relationships between the students and the teachers. This method of study was advantageous in that it allowed the extension of previous research on the way students constructed incorrect fractions and the learning process of both teachers and students (Lloyd, Wilson, Wilkins, & Behm, 2005).

For mathematical caring relations to be said to have occurred, there must be the learning of mathematics together with responses from both the teachers and students. Assessing mathematical caring relations (MCRs) in research has been concluded to aid the interactions that aim at improving the learning process.

Social interaction between teachers and students can lead either to depletion or to stimulation of the two. The degrees of both may be negligible and tend not to show. However, at certain situations, one could outweigh the other. In the learning process, the feeling of depletion may dominate due to various reasons.

The researcher experienced learning even when the student did not learn how to make improper fractions in the course of the experiment. This was because of the depletion she showed during the experiment.

With this understanding, teachers should understand their students and develop ways to harmonize better with such students. They should incorporate the students’ current schemes in order to avoid the cases of sustained depletion from the students that are likely to affect the learning process.

## In pursuit of practical wisdom in mathematics education research

The author of this paper believes that teacher and education practitioners do not acknowledge research enough. He believes that the Journal for Research in Mathematics Education (JRME) is important for the teachers. It provides them with the administrator and curriculum consultant during planning (Birgisson, & Kehle, 1999).

He also suspects that among the reasons given about the failure of research to concur with the teachers, the one that has not been addressed adequately is that the teachers and researchers have agreed upon varied methods of framing the direction in their knowledge and beliefs about mathematics learning and teaching.

Schwandt defined the scientific rationalism as a mode of inquiry that was based on six principles. The first is that actual knowledge starts at a point where there is doubt and distrust. The process of doubt is normally a solitary one and one engages in monological activities. The third principle is that proper knowledge is acquired when rules and procedures are followed to the latter.

The fourth principle is that proper knowledge is always based upon justification and proof. The fifth is that knowledge may be taken as a possession and that individual who has knowledge has ownership to that knowledge. The last principle is in the justification of claims to knowledge. There may be no appeal but only reasoning.

Communicative rationalism argues against scientific rationalism in three ways. Firstly, instead of taking the social world as being a place waiting to be discovered, the rationalists argue that one can only study the world if he is involved directly within it. Secondly, knowledge about the world does not rely upon evidence or proof for its practicality. Lastly, we own this knowledge and incorporate it into our character.

## The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice

There is a relationship between practice and conceptions of mathematics and the teaching of mathematics. The beliefs of the teachers and their preferences and views about the subject of mathematics play a significant role in structuring their instructional behavior. The conceptual and practical differences among teachers can be explained through a discussion of properties of the different systems of concepts.

The teachers’ instructional practice is shaped by their patterns of behavior. At times, these patterns might be the manifestations of the beliefs the teachers hold or what they prefer. Such beliefs and notions may act as drivers of the behavior displayed by the teacher. They may occur unconsciously since they are behaviors that have developed out of the experiences of the teacher.

The authors were seeking to get answers for the question of whether the beliefs and views held by the teachers about the subject of mathematics and its teaching affected their instructional practices. The other question was of the behavior of the teachers and they were seeking to establish whether their conceptions influenced the behavior. The differences among the teachers were also studied and this was in their teaching of mathematics and their concepts in mathematics (Gonzalez, 1984).

The study that was conducted was one among the many that should be conducted in order to identify the major factors that may facilitate the teaching process of the teachers. The conceptions of the teacher are related to their decisions and behavior in giving instructions but in a complex way. Other things that affect such decisions include beliefs that are held by the teacher even if they are not related to mathematics.

The teachers generally have conceptions about teaching that are not necessarily related to mathematics. They also understand the emotional and social make-up of the students in their classrooms. This is important in that they play a role in shaping the instructional behavior and the decisions they make. These views and beliefs may take precedence over the teaching of the subject of mathematics for some teachers.

## Learning as constructive activity

The authors of the article acknowledge the fact that the world of education is experiencing change. In mathematics, specifically various symptoms suggest that there is willingness for change. There have been rapid changes in the methods of educating on mathematics. They have changed severally. First, it moved from the use of the simple methods to the use of complex ones but finally back to the basics.

The author acknowledges that the educators spend a lot of time and effort on the curriculum in that they do a lot in trying to find the things to teach and what methods to employ in teaching. However, the drawback is that the process of communication is not taken seriously. What they do not understand is that communication is a process on which their teaching heavily relies upon.

The results indicate that despite the impartation of knowledge to the students being the major goal of every educator, it is obvious that knowledge is not as easy to transfer. It is not a commodity for it to be easily transferable and it requires skill and effort from the side of the teachers and attention from the students (Bergeron, & Herscovics, 1983).

In this case, knowledge and competence is acquired depending on the conceptual organization of the person’s experiences. The roles of the teacher would therefore be to help the student rather than to transfer facts. For a teacher to be a good facilitator in the teaching process, one has to have enough information about the current competencies of the student and the student’s goals.

Good teachers only guide and help in the learning process. They also find ways of doing it since they understand that theirs is to make way using symbols and numeric but the rest is left for the student where he is to conceptualize and operate.

## References

1. Behr, M., Thomas, P., Harel, G., & Lesh, R. (1992). Rational number, ratio, and proportion. New York, NY: Macmillan publishing.
2. Brownell, W. (1947). The place of meaning in the teaching of arithmetic. The Elementary School Journal, 47(5), 256-265.
3. Lester, F. (1994). What research tells us about teaching mathematics through problem solving. Reston, VA: National Council of Teachers of Mathematics.
4. Carpenter, T., Fennema, E., & Franke, L. (1996). Cognition guided instruction: A knowledge base for reform in primary mathematics instruction. The Elementary School Journal, 97(1), 3-20.
5. Schoenfeld, A. (2000). Purposes and methods of research in mathematics education. Notices of The AMS, 47(6), 641-649.
6. Kilpatrick, J. (1992). A history of research in mathematics education. Georgia: University of Georgia Press.
7. Middleton, J., & Spngias, P. (1999). Motivation for achievement in mathematics: Findings, generalizations, and criticisms of the research. Journal for Research in Mathematics Education, 30(1), 65-88.
8. Birgisson, G., & Kehle, P. (1999). In pursuit of practical wisdom in mathematics education research. Nordic Studies in Mathematics Education, 10(4), 2-13.
9. Lloyd, G., Wilson, M., Wilkins, J., & Behm, S. (2005). Mathematical caring relations as a framework for supporting research and learning. The proceedings of the 27th annual meeting of the North American chapter of the international group for the psychology of mathematics education, 1(1), 1-8.
10. Gonzalez, A. (1984). The relationship of teachers’ concepts of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105-127.
11. Bergeron, J., & Herscovics, N. (1983). Learning as constructive activity. Proceedings of the fifth Annual Meeting of the North American Group of Psychology in Mathematics Education, 1(1), 41-101.