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# Custom Learning Mathematics Essay

In order to write this paper I chose the book by Allen and Johnston-Wilder under the title “Mathematics Education: Exploring the Culture of Learning”. I liked this book very much as in my opinion it gives a complex view of mathematics and discusses a great number of different strategies and pieces of advice that are really helpful. The purpose of this book is to bring together readings which explore the culture of learning in a mathematics classroom. These readings show how knowledge of this culture assists teachers and learners to improve the teaching and learning of mathematics and to address concerns of social justice and the need for equity. The structure of the book is also very convenient and all the material is exceptionally organized. All the information is divided into several sections and it really helps to find the most important and interesting aspects and see what topics are covered in this book.

The main idea that I took from this book is that it is important to perceive and understand mathematics through your own words and try to avoid complexity. And my teachers also tried to show me such an approach and always asked to express my personal constructs about the mathematical topic being discussed. As the book states it is far less important if the learner can repeat the exact words of the textbook but, instead, how he had understood the matter. This book helped me to see how mathematics should be taught and perceived and what it is to be good mathematics teacher and a practical constructivist. Teachers can provide pupils with relevant contexts for mathematizing, act like a mathematician, introduce names and other terms, and help the pupils' learning processes, but they cannot give the pupils ready-made concepts of mathematics. There are also inefficient ways to learn mathematical information that seem to be easy, attractive, and quick just to remember the names and correct responses to formalized stimuli or to do the same operations that the teacher has shown. Very soon these mechanical accesses into mathematics usually become boring to teachers as well as pupils.

I totally agree with the widespread public image of mathematics that is mentioned in this book and states that this science “is difficult, cold, abstract, theoretical, ultra-rational, but important and largely masculine. It also has the image of being remote and inaccessible to all but a few super-intelligent beings with 'mathematical minds'. For many people the image of mathematics is associated with anxiety and failure”. However, the thing is that the way you learn mathematics is very important and it can either help you or create certain difficulties.

One more thing I liked is that the book is written in a rather simple style that is easy for understanding. It is not full of complex learning theories and it contains only the most important information. The book sheds more light onto the way mathematics should be learned and how teachers should teach it in order to succeed. Mathematics we learn in school usually gives us general ideas of modern mathematical thinking, but it is very far from the needs teachers have when confronting pupils. There are many difficult questions that mathematics asks and most teachers do not even try to answer them. So they usually open the textbook, trusting that the authors have penetrated these questions, and start working with the iconic models and written symbols of the textbook. Of course, the authors of modern textbooks are more experienced than ordinary teachers and have tried to follow the ideas offered by the theories of child development. However, the textbook is always a compromise of many different demands and traditions. Mathematics is divided into small and easy steps in order to smooth the learning process of pupils and to avoid situations that may lead to failure and frustrations. Working with textbooks is an easy way to organize teaching, and it also gives the teacher opportunities to help the less advanced pupils. The results are not necessarily bad, sooner or later most of the pupils learn to count, add, subtract, and many other skills appreciated in school. But some 15% of the students, who leave the comprehensive school after 9 years, do not know even elementary arithmetical skills, and many years earlier mathematics has become a difficult and boring subject for them.

Mathematizing concrete situations is necessary though they may sometimes also mislead pupils. In order to acquire mathematical knowledge, pupils have to meet the same pattern (same number, length, area, shape, mass, weight, volume) in different contexts, act with objects through suitable strategies, transform the constitutions, and so forth. The importance of measuring continuous quantities in natural situations as a starting point for mathematics cannot be underestimated. Mathematics has a tendency to cut the bonds with reality but this is not innate in the pupil's mind. Pupils have to construct it on their own and when they have really done this, they have reached another level of concept attainment. Knowledge springs from actions performed on the objects, not the objects themselves.

It was very interesting to find out that pupils do not make mistakes in mathematical tasks at random, but operate in terms of the meaning they hold at a given time. It is not easy for the teacher to detect the wrong rules pupils have followed or inefficient strategies used; it is difficult even for the specialized investigators. Correct answers can, of course, be attained by means of wrong rules or inefficient strategies and the logic behind incorrect answers is often difficult to unravel. If teachers want to guide the learning processes of their pupils, they have to know what the pupils think about the topic of concern. The only way this can be achieved is through communication.

Earlier conceptions and strategies contain the seed for more developed conceptual structures, but it is easy to regress too. A teacher has to know how to get his pupils to appreciate and use higher-level thinking. There are numerous examples even of adults who still have very primitive number conceptions of computing strategies. In order to develop mathematics teaching, it is necessary to emphasize the importance of knowing how a pupil has interpreted the mathematical situation or task, how the pupil has interpreted the goal of the problem; for example, the relation between the task and procedure that has been selected, how the pupil has implemented the procedural work with all its components, and how the pupil has understood the larger context of the task, which may include a cue for checking the sensibility of the answer. The learning process in school is not just a knower- known relationship, a pupil with a mathematical problem, but a mediated learning experience, a pupil studying with his or her teacher and peers.

Within the general framework of constructivism we can also understand and predict how a pupil typically orientates to the situation or task at present. The pupils' cognitive skills (the way they focus their attention, narrow the selection of problem category using the identified parts, compare the similarities and differences of the problem with their earlier knowledge, and select the appropriate solution) can give their teacher valuable information from which to guide the learning processes. These skills are never taught in school, singly or in conjunction with the content

I also liked that this book emphasizes the importance of communication for learning mathematics. Communication in the mathematics classroom is a socially constructed phenomenon. Such an approach in my opinion encourages children to state their own point of view to a teacher. Actual communication often reflects inequities and stereotypes of gender, race, and class. Boys, for example, may be asked more process questions and are more likely to have their responses followed through. Girls may receive praise for the neatness of their work and their ability to emulate established models of mathematics practice. Other inequities emerge when some children, because of their ethnic background, are seen as not competent in the language of instruction and as a result are unlikely to be challenged. Class inequities emerge when some children are seen as socially handicapped or deprived and as a result unlikely to succeed in school mathematics.

This book certainly proves that mathematics is a way of looking at and making sense of the world. It is a beautiful, creative, and useful human endeavor that is both a way of thinking and a way of knowing. The goal of mathematics instruction is to help students develop and deepen their understandings of mathematics as well as their abilities to communicate their ideas to others. The process of communication helps students construct as well as express mathematical meanings.

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