# Why should We Change Mathematics Teaching?

This is the question that I proposed to my teacher in one of my course. In the discussion, we discussed about the percentage and its relationship with decimal, fractions, ratio and proportion. I argue that we have all been taught about percents and other topics of mathematics in the traditional way, with an emphasis on learning the correct calculation procedures, and very little or not on understanding. Still we all understand what percentages, fractions and decimals are and how they are related. So why should we change the way these topics are taught? Or in any case, why should we not teach the calculation procedures first and let understanding develop later? These questions are the paraphrased ones done by my teacher.

It is an issue that we should discuss, because we hear this argument quite often. Then, my teacher proposed his own question relating to my own question: what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?

Mathematics as a human activity

I try to give an answer to my own question, why should we change the way these topics are taught? In my perspectives, I think we have to change the way mathematics has been taught because when we teach students in the traditional way of teaching, I can be really sure that they will not make sense of what they have been learned. Moreover, as Freudenthal said in his wonderful book, Revisiting Mathematics Education, mathematics is a human activity and therefore it must be connected to reality, stay close to children and be relevant to society in order to be of human value. So, in this sense, the focal point is not mathematics as a ready-made subject full of procedures, formulas or whatsoever. Instead, the focal point is on the activity. Education should gives students a guided opportunity to reinvent the mathematics.

why should we not teach the calculation procedures first and let understanding develop later?
I think, if we do it in this way, students will not develop their understanding of mathematics. Instead, what they get from this way of teaching is that they know the procedures and formulas but they do not know how to use it in their life. It can be seen from the test such as PISA or TIMSS, that many Indonesian students fail to answer questions that dealing with complex situations and asking for advance mathematical thinking and reasoning. Even, I argue that students will not develop their understanding after they know the procedures and formulas. It is because, they will always looking for the formulas and procedures when they want to solve the problems.
what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?
As Freudenthal said that learning processes, or at least part of them, can be more essential than their products (Cited from Revisiting Mathematics Education book). The focus of the teaching of mathematics is the process of learning it, not the product. In RME, the use of context is very important. In contrast with the traditional way of teaching, RME use the context problem both as a source of learning and to apply mathematical concepts. The students, in RME class, can develop mathematical tools and understanding while working on context problems. In the first time, they develop strategies or models that are closely related to the context. Later on, certain aspects of the context situation can become more general which means that the context can get more or less the character of a model and as such can give support for solving other but related problems (http://www.fisme.science.uu.nl/en/rme/).
One of a good example is the experience of a college professor who experienced the change in her son’s understanding of mathematics after following the RME class in the Netherlands. You can read her story in this link: http://mathematicallysane.com/realistic-math-makes-sense-for-students/

# Critical Analysis of Roth and Bowen’s Paper: “When Are Graphs Worth Ten Thousand Words? An Expert-Expert Study”

Purpose and Question of the Research

The authors present the research purpose very explicitly in the paper. It can be found on the page 430, second paragraph, in which they state that: “this study was conducted to better understand graphing expertise. We were particularly interested in understanding the contributions of experience (content represented, laboratory experience, and understanding of conceptual frameworks) to the particular readings provided by scientists”. The research purpose is also written explicitly in the discussion (see p. 466, first line of the first paragraph): “… to better understand readings of familiar and unfamiliar graphs by professional scientists”.

The research question is not formulated explicitly in the paper. As the reader, I do not know exactly whether the authors forgot to mention the research question intentionally or not. But, the title of the paper is written in the form of a question, when are graphs worth ten thousand words. Later, in the conclusion (see p. 470, the last paragraph), the authors say that this title is their initial question, but not the research question. In my perspective, this initial question cannot be categorized as the research question or even as a good research question. Although it is researchable, this initial question does not provide any clear explanation about the things that the study wants to investigate. There is no any explanation in the question about the kinds of graphs that are going to be used. The word ‘when’ is also somewhat ambiguous. What does the term when means? Does it mean the time of presenting the graphs, or the kinds of graphs? But, this initial question is worthwhile to investigate. The researchers formulate the reason for it (see p. 430, paragraph one): “… there is little work on the actual use of graphs in everyday science, or on scientists’ reading of unfamiliar graphs”. Continue reading

# Critical Analysis from Methodological Perspective: A case of Lederman’s Research Paper

Regardless of methodology, almost all researchers engage in a number of similar components in conducting their research. All of these components include, purpose and question of the research, research approach and methods, population and sample of subjects, data collection (e.g. tests or other measuring instruments, a description of procedures to be followed), a description of intended data analysis and interpretation, and conclusions. Those components are also considered as the key elements in doing critical analysis of research papers based on methodological perspective. In this text, I shall present my own critical analysis from methodological perspective of Lederman’s research paper titled Teachers’ understanding of the nature of science and classroom practice: factors that facilitate or impede the relationship.

# An Analysis of Misunderstanding in Mathematics: The Case of Meanings of Equal Sign

## Introduction

Many mathematical symbols and notations are figured routinely by students in learning mathematics in the school contexts. Those symbols and notations are mostly faced by students, especially when they learn about arithmetic and algebra. In fact, many students are struggling to understand the meaning of those mathematical symbols and notations, and sometimes lead them to the misunderstandings (e.g. Kiran, 1981). Realizing this issue, many researchers and experts from many different fields and backgrounds have been trying to find out what kinds of misunderstandings that happen in the students’ thinking in learning mathematics and how to deal with them.

Furthermore, misunderstandings about mathematical symbols and notations are also happened in the case of equal sign (or more holistically, equality). Question about how students understand the equality symbol have largely been discussed by many researchers and experts from many different fields persist through elementary schools to high schools and colleges (e.g. Jones and Pratt, 2007; Hattikudur and Alibali, 2010). Based on those extensive researches focusing in this issue, many students do not interpret the equal sign, as an equivalence symbol. They misunderstand about the meaning of the equal sign. Moreover, as will be seen, understanding the equal sign as an equivalence relation does not seem to come easily to the students. The purpose of this essay is to analyze about misunderstandings of the equal sign among elementary and secondary school children.

# Constructivist Approaches are Compatible with Human Cognitive Architecture: A Response to Kirschner, Sweller, and Clark (2006)

## Introduction

Learning is much more than memorizing. Learning refers to the acquisition of knowledge through interactions with, and observation of, the physical word and the creatures that inhabit it (Ashman & Conway, 1997). In order to really understand and be able to apply knowledge, students must work to solve problems, to discover things for themselves, and to struggle with ideas. The question of how to help students learn particular knowledge, skills, and concepts that will be useful in their life is at the core of the argument presented by Kirschner, Sweller, and Clark (2006). The authors compare minimally guided instructions with instructional approaches that provide direct instructional guidance of the student learning process. They define minimally guided instruction as ”one in which learners, rather than being presented with essential information, must discover or construct essential information for themselves” and then inversely define direct instruction as “providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture” (p. 1).

In their argument, Kirschner Sweller, and Clark (2006) affirm that minimal guided instruction approaches are less effective and efficient than fully guided instruction approaches because they ignore the structures that constitute human cognitive architecture. On the contrary to this, they put a strong emphasis on direct, strong instructional guidance, as an effective and efficient way to teach students. By referring to several studies concerning the efficacy of direct instruction (e.g., Klahr & Nigam, 2004), they claim that students learn more deeply from strongly guided instruction than from constructivist or discovery approaches. Opposing this claim, Kuhn and Dean (2006) have found that direct instruction does not work so well for robust acquisition or for maintenance knowledge over time.

# Am I Teaching Well? The Searching for The Ground rules of Effective Math Teachers

Presently, many practitioners, researchers and educators have given deep attention to the development and enhancement of teaching mathematics and mathematics education world-wide. Yet, many problems still exist when we are looking to the classroom and see how math teachers do their job. Teaching mathematics is always being a challenging issue to be explored, and there is no easy-single recipe for helping all the teachers to become effective and professional. In this essay, however, I try to make a very brief explanation on how teachers should teach mathematics by considering theory and several aspects of teaching mathematics in the classroom.

In order to answer this question, I propose three premises underpin what the idea I want to explain. Firstly, I argue that teachers as professional ones have to understand well about the contents of mathematics, the development of the students and pedagogical aspects of mathematics. Teachers have to know deeply about the topics that s/he is going to teach and be able to formulate the knowledge with meaningful contexts in their teaching approach. Effective mathematics teaching requires creativity and flexibility of the teachers to propose certain questions that can help the students to think. The teachers also have to understand the strength and weakness of the learners. They have to know the development of students’ understanding of mathematics and what kind of help that s/he can offer in order to guide the learners and prevent them from misunderstandings. Moreover, the teachers also need to understand the big ideas, strategies and models involving in the certain topics of mathematics. By having these kinds of things in mind, the teachers will be able to select and use appropriate tools and materials that can engage the students to develop their mathematical understanding.

# Domain-specific and Metacognitive knowledge: The Good Approaches to Effective Teaching Thinking and Problem Solving?

## Introduction

Educators and researchers for many years have been concerned about how to teach thinking and problem solving effectively. However, a lot of researches indicated that students failed to develop their ability in thinking and problem solving when facing everyday phenomena and problems. It seems that students mainly acquired knowledge that remained inert. Their knowledge cannot be useful tools to deal with everyday problems and only ended as a lesson to be learned in the schools. Domain-specific and metacognitive knowledge, however, are pronounced as two approaches that can make teaching reasoning, thinking and problem solving effectively. The goal in this essay is to discuss about domain-specific and metacognitive knowledge in order to answer the following question: do domain-specific and metacognitive knowledge can make teaching thinking and problem solving effectively to encounter everyday situations and solving problems?

## Evidence from the literature

In this essay, I emphasize to focus on the discussion about the literature-based study done by Bransford et al. (1986) presenting about two theoretical perspectives – executive or metacognitive processes and domain-specific knowledge – that can improve the teaching thinking and problem solving. Bransford and his colleagues explain the role of specific knowledge by showing several results of well-known research done by the experts and researchers. Taking as an example the study of deGroot, (Brandsford et al., 1986), comparing chess masters and novice ones, gives a description that conditional knowledge can help individuals to solve their problems effectively. In the deGroot research, the chess masters who already have knowledge base in playing chess, performed better in remembering the position of the game when it was meaningful for them. Furthermore, from the other research, it is also noticed that specific knowledge determines the learners’ strategies and their perspectives in solving meaningful problems. For example, Chi (Brandsford, 1986) found that 10-year-old chess enthusiasts can remember the position of the chess pieces more accurately than the college students do who are not experienced in playing chess.

On the other hand, metacognition also play an important role for improving teaching thinking and problem solving. Consider for example the investigation lead by A. Brown, Campione, and Day (Brandsford, 1986) who studied about the effects of different types of teaching environments in transfer tasks. It is noted that many individuals who were failed in transfer tasks are actually do not know why the strategies that they have learned are useful and when they would be used.

Combining both metacognitive and domain-specific knowledge, indeed, can strengthen the teaching thinking and problem solving. By having the ability to use their strategies and knowing the specific knowledge relating to the problems, the learners can increase their critical thinking and become an effective problem solver.

## Conclusion

The two theoretical perspectives – metacognitive and domain-specific knowledge – provide a framework for students to think and solve the problems. Students not only have the ability about the general strategies and skills in problem solving, but also conditional knowledge that available for them to understand how concepts and procedures can be used as tools to solve the problems.

## References

Brandsford, J., Sherwood, R., Vye, N., & Rieser, J. (1986). Teaching thinking and problem solving: Research Foundations. American Psychologist, 41, 1078 – 1089.