Dilemma in Teaching Mathematics
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The challenge in mathematics education is finding the best way to teach mathematics. When students learn the reasoning and proving in mathematics, they will be proficient in mathematics. Students must know mathematics before they can apply it. Symbolism and logic is the key to both the learning of mathematics and its effective application to problem situations. Above all, the use of appropriate language is the key to making mathematics intelligible. In a very real sense, mathematics is a language. Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof. Mathematics is essentially a structured hierarchy of proposition forged by logic on a postulation base. However, nowadays, the teaching of mathematics is more on focusing the mathematical procedures. Students learn variety of problem-solving in mathematics and then on its application. In addition, the teaching of mathematics is moving towards the use of mathematics in the real world. This method is said to motivate students to learn mathematics and enable them to transfer this knowledge in their daily live and future career. There are beliefs that due to the current over-emphasis on problem-solving and applications, students who enjoy working the problems in high school math, and hence decide that they like mathematics and want to major in it, find that the nature of the subject changes abruptly when they encounter the proof courses. They are bewildered and dismayed. Their previous problem-solving courses did not prepare them for this abrupt changed. Is argument and proof in the applications-problem centered domain of school mathematics being postponed, suppressed and downgraded? Are we sacrificing the essence of mathematics in order to motivate students to learn and appreciate mathematics? In this paper, the authors will discuss on this issue.Keywords:
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Understanding well the concepts in mathematics is a precondition of learning mathematics well. So, the teaching of mathematics concepts seems very important. But students feel that is very abstract and difficult to be understood while many teachers feel that is very difficult to be taught and the effort is bad. I have accumulated some experience in this field for more than ten years.Now I write this article and earnestly ask each counterpart to give advice.
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Matheteizingis to a newly arising concept in the west academic field. It refers to a mathematics instruction model in which the teachers cooperate and interact with the students. As a result it helps the students correctly understand principles and rules that all kinds of mathematics demonstrate and all kinds of computing need. And therefore it will help the students form their own mathematics models about objects and situations. This paper has been based on a great number of observations on mathematics classed, interpreted the inplication and process of mathematicsand analyzed the social and psychological factors that are needed in the process. As a conclusion it provides some constructive suggestions for mathematics instruction. These will have significant implications to the primary and secondary mathematics instruction on the ground that it helps foster children's mathematical ability and develop their mathematical minds at the same time.
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Background: Elementary school must be a strong foundation to reach a higher level of education. This is inseparable from the beginning of the level of education starting from early childhood and kindergarten because it is a level of education to start the formation of children's character. One of the subjects given at every level of education is Mathematics. Mathematics is one of the sciences that has an important role in shaping the mindset of students, because in mathematics students are required to have the ability to think logically, systematically, analytically as well as problem solving abilities. Various attempts have been made by the teacher in increasing the ability of students to understand mathematics subjects. However, the expected results are still not optimal, even the learning and teaching process becomes unpleasant and tends to be boring for students. One of the math problems that is often found is the difficulty of students in solving story problems. Even though the story problem is a form of evaluation of the ability of students in understanding the basic concepts of mathematics that have been learned, in the form of questions about applying formulas. Purpose: This study aims to examine the extent to which students' ability to work on math story problems. It could help teachers to find out the problems of students in working on math story problems and make students able to solve problems in the form of stories with the right method. Design and methods: The research method used is qualitative methods. Qualitative research is research in accordance with the data, utilizing existing theories as explanatory material. Results: The results of this study show that there are many problems in students working on math story problems, and the teacher's methods during learning must be improved so that students can understand
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Mathematical problem
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Mistakes Made by Freshman Students of Science Teaching and Their Reasons during the Proving Process.
IntroductionThinking is a process required for understanding new situations. In other words, it is the conceptualisation, implementation, analysis and criticism of the knowledge obtained by observation, experience, sense, and many other ways (Ozden, 1997: pp. 79). One of the most important tools that improve thinking is mathematics (Tural, 2005). Mathematics can be defined as a process that explains the relationship that constitutes the essence of surrounding situations and it is beneficial in making decisions on both the current situation and the future. Mathematics is a process that starts with looking for patterns, discovering relationships and ends with a formal process such as 'proof (Dreyfus, 1991).The proof process holds a key role in the field of mathematics (Lakatos, 1976) and it is necessary for doing and understanding mathematics. It does not consist only of finding the answers of theorems, it also provides students with logical reasoning, which is required for mathematical understanding and reasoning (Polya, 1981). Mathematical proof may be used at all levels, including elementary, secondary etc.One of the basic levels that requires using proof is the high school level. The high school years are those during which the process of abstract thinking develops and during these years the methods of deduction and induction methods are formed. Conversely, use of geometric proofs commonly takes part in the geometry curriculum related to high school (Altiparmak & Ozis, 2005). Geometric proofs are the opportunity to educate students on the foundations of mathematical principles. Therefore, teachers tell students in the lower levels of mathematics that they will prove why the sum of the interior angles of a triangle is 180 degrees (Jones, 1994). In this framework, students should be able to understand mathematical proof within both mathematics and geometry courses at high school. However; many students have difficulty in understanding the process of proof. Consequently; these students lack proving skills when they come to university, whereas proofs are also central to university level mathematics courses such as 'General Mathematics' and 'Algebra'. In this context, students in mathematics programmes should be able to understand and construct mathematical proofs. The process of proving is also seen in mathematics courses related to science programmes. Students in science programmes can also use proof to verify or explain a statement with various methods in 'General Mathematics I or II'. While some students use the method of induction proof, some use the method of deductive proof. Conversely, some students give special examples for justifying an argument in the process of proof.Virtually, Methods of Proof' is an important topic that explains the process of reasoning, finding the true value and using it. Therefore, it has an undeniable function in the practice of university mathematics (Hemmi, 2010). However; many university students may not see the functions (meaning, purpose and usefulness) of proof (de Vilhers, 1999) and may not completely use the methods (induction, deduction, etc.) of proof. One of the most important factors regarding using these methods of proof correcdy is the foreknowledge of students. Both students in mathematics and science programmes may not make the use of proof, since they do not exactly learn the methods of proof in high school.There are many studies on the students and prospective teachers at the Mathematics Departments in Turkey. However; with the exception of Aydin, (2011), Gokkurt and Soylu (2012) and Gokkurt, §ahin, and Soylu (2014), litde research has been done on students in Science Departments. These studies have also studied science students' abilities of making proof or their views to prove it. However; in Turkey, there is not any studies on what students' mistakes are in the process of making mathematical proof and on the reasons of these errors. But the mistakes while science students are making mathematical proof and to investigate the reasons of these mistakes are important. …
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This paper describes an innovative method for teaching of mathematics which was utilized to teach abstract algebra to a class of mathematics education majors at a small liberal arts college. A variation of R.L. Moore's Discovery Method was employed in conjunction with substantial use of mathematical software. Although student reactions were initially negative, students grew to accept and even embrace the instructional methods. Anonymous teaching evaluations administered at the end of the semester were favorable. As measured by examination performance, student achievement in this experimental class section was high. This teaching experiment was extremely labor-intensive from the instructor's perspective. Otherwise this teaching experiment was a positive experience for both the instructor and the students. ********** This paper describes my efforts to create an innovative abstract algebra class at Springfield College, a small college in western Massachusetts. Students in this class typically are junior mathematics or mathematics education majors. All have completed Linear Algebra and Differential Equations, but most are not yet truly comfortable with proofs or serious mathematical abstraction. Classes for mathematics majors are small, typically five to ten students. By the time a student takes Abstract Algebra, I usually know him or her very well. Traditionally, an abstract algebra course (like most courses in pure mathematics) centers around the instructor's presentation of definitions and theorems. At best, students participate in the logical exposition of the subject by proving a few of the key theorems as homework exercises; in most courses, however, students prove only minor results in their homework exercises, and all theorems which will be cited in later proofs are proven by the textbook (and possibly by the instructor in lecture). This traditional approach is effective for some students, but surely not the best approach for all. Today's college students, according to anecdotal evidence at least, have a disturbing proclivity towards problem solving syndrome (TPSS) which was not so evident in previous generations of students. By TPSS, I mean the tendency to see mathematics as a set of problem-solving methods to be memorized and the reluctance or unwillingness to discover new methods of one's own; a victim of TPSS believes that the only way to solve a problem is to follow a template which has been provided. A TPSS-oriented student in a lecture-oriented course might not acquire critical thinking skills of the sort that faculty feel ought to be acquired at that level. Benson (2002) states a classic dilemma: It finally occurred to me that if one learns mathematics by doing mathematics (as I claimed), then perhaps the only person learning when I am up at the board is me. Many readers are no doubt familiar with the University of Texas mathematician R.L. Moore's discovery method of teaching, which is described in Traylor (1972). In planning an abstract algebra class, my first resolution was to use R.L. Moore's discovery method in a modified form. Moore's method, in its pure form, requires a class of mathematics students to prove all theorems entirely by themselves; naturally, their learning is profound and knowledge is retained. Such an uncompromising position, I thought, might leave our class stuck at square one; still, my goal was to have the students discover as much as they possibly could by themselves. Secondly, I resolved to integrate technology into the class as a primary tool for facilitating discovery. There are a number of software packages specifically designed for abstract algebra (or group theory in particular): GAP and MAGMA at the advanced level, and ESG and FGB for students. I chose to adopt the free program FGB (Finite Group Behavior) and supplement it with the more flexible Derive and Maple general purpose mathematical software packages. Having chosen these idealistic goals, I set out to teach the course to actual students for the first time in the Fall 2000 semester. …
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Mathematics teaching should not only contact with real life but also represent mathematical essence and value.It would not be successful that contacting with real life or teaching pure mathematics was too unilateral in teaching.Mathematics teaching should give attention to both abstract mathematics knowledge and colorful real life.In order to seeking the balance point of mathematics and real life,mathematics teachers should consider following three aspects: proportion of the pure mathematics knowledge,the time for effective mathematics teaching and the effect of students’ mathematics learning.
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In a first proof-oriented mathematics course, students will often ask questions – for example, What is this problem asking me to do? or What would a proof of this even look like? – that have more to do with logic than mathematics. The logical structure of a proof is a dance involving those basic logical forms – such as p or q, if p then q, for every x in A we have p(x), there exists x in A such that p(x) – that appear in the theorem being proved, and at various stages of this dance these basic logical forms are either being proved or used. How does one prove a statement of the form There exists x in A such that p(x)? How does one use a statement of the form For every x in A we have p(x)? We introduce particular “proving and using words” that encourage the fledgling mathematics major to pay attention to these sorts of questions and to the logical structure of proofs in general.
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Computer-assisted proof
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What is Mathematics? The discipline of mathematics has evolved over time and across different civilizations to become the abstract, professionalized body of knowledge within an international community of practice that we know today, and it will continue to evolve as new knowledge is developed and existing knowledge is superseded. The discourse of mathematics in all its various specializations involves certain ways of thinking and acting. From a given, axiomatic, starting point, logical deductions are made. In mathematical modeling, for example, problems are formulated in mathematical terms. However, in order to be useful, the mathematical solution needs to take into account the industrial, social, environmental, etc. contexts. Yet, the optimal mathematical solution may not be most useful in a sense and compromises must be made. There is not one single absolute and infallible mathematic, but rather, a plurality of mathematics which operates on a pragmatic basis, linked to time and place. Academic mathematics as we know it evolved through the confluence of certain socio-cultural conditions, such as the rise of commerce, the need for timesaving devices such as algorithms, as well as the spread of printed material (Restivo, 1992). Unfortunately, the public image of this mathematics is generally a cold, dehumanized process, which as Davis and Hersh (1986, 1988) point out, is actually intrinsic to the fundamental intellectual processes that are inherent to the discipline. However, a paradox exists in the seeming 'demathematization' of society. As technology becomes more sophisticated, there is an apparent reduction in the amount of explicit mathematical knowledge required for its operation, while the amount of implicit mathematics increases. Although the explicit uses in business and industry are generally valorized, they are mostly concealed from view and, with the exception of arithmetic, not visible to the general public except through their experiences of school mathematics. (For further discussion on mathematics, see FitzSimons, 2002.) What is Functional Mathematics? In the United Kingdom, the Qualifications and Curriculum Authority (QCA) website defines functional skills as practical skills in English, Information and Communication Technology (ICT), and Mathematics, that allow individuals to work confidently, effectively and independently in life. Assessment of these applied skills will include electronic and on-screen approaches, and will be based primarily on task-based scenario questions with a limited duration, delivered in a controlled environment. Assessments will use and reinforce skills-based, problem-solving learning techniques. (See Functional Skills website in reference list). The QCA website on mathematics (see Solving Problems in reference list) defines functional mathematics in terms of logical creativity: Mathematics is a creative discipline. The language of mathematics is international. The subject transcends cultural boundaries and its importance is universally recognized. Mathematics has developed over time as a means of solving problems and also for its own sake. Mathematics can stimulate moments of pleasure and wonder when pupils solve a problem for the first time, discover a more elegant solution, or notice hidden connections. Pupils develop their knowledge and understanding of mathematics through activities, exploration and discussion, learning to talk about their methods and explain their reasoning. A workshop on functional mathematics for 14- to 19-year-olds proposed several key themes: (a) relevance of content materials, (b) development of thinking skills, (c) conceptual understanding of mathematics, (d) integrated use of information technology, and (e) comprehensive assessment, including a sustained activity for learners to demonstrate their relatively straightforward mathematical abilities in complex contexts. …
Numeracy
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Tasks to Advance the Learning of Mathematics (TALMs) were developed to stimulate grades 5 – 8 students’ curiosity about complex mathematical relationships, inspire them to reason abstractly and quantitatively, encourage them to consider and create alternative solution approaches, develop their skills to persuade others about the viability of one solution approach over others, and enhance their perseverance toward problem solutions. Tasks are of nine types: Connect Calculation to Context, Rank Order Solutions, Identify What’s Wrong If Anything, Defend an Opinion, Work Backwards, Predict and Explain, Think and Choose, Place Them Right, and Make Sense of a Situation. All tasks require application of concepts and skills from one or more domains of mathematics. As students solve these problems, they quickly identify what they know and what they are not sure about; that is, they assess their own degrees of understanding and learn at point of need. The article concludes with recommendations for implementing TALMs and an invitation for students and teachers to create their own.
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Although standards of rigor in mathematics are subject to debate among philosophers, mathematicians and educators, proof remains fundamental to mathematics and distinguishes mathematics from other sciences. There is no doubt that the ability to appreciate, understand and construct proofs is necessary for students at all levels, in particular for students in advanced undergraduate and graduate mathematics courses. However, studies show that learning and teaching proof may be problematic and students experience difficulties in mathematical reasoning and proving.
This thesis is influenced by Lakatos’ (1976) view of mathematics as a ‘quasi-empirical’ science and the role of experimentation in mathematicians’ practice. The purpose of this thesis was to gain insight into undergraduate students’ ways of validating the results of their mathematical thinking. How do they know that they are right? While working on my research, I also faced methodological difficulties. In the thesis, I included my earliest experiences as a novice researcher in mathematics education and described the process of choosing, testing and adapting a theoretical framework for analyzing a set of MAST 217 (Introduction to Mathematical Thinking) students’ solutions of a problem involving investigation. The adjusted CPiMI (Cognitive Processes in Mathematical Investigation, Yeo, 2017) model allowed me to analyze students’ solutions and draw conclusions about the ways they solve the problem and justify their results. Also I placed the result of this study in the context of previous research.
Mathematical practice
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