Current Situation and Thinking on the Thinking Problems in the Elementary Mathematics Textbooks——Taking the Textbooks Published by Southwest Normal University Press as an Example
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In the elementary mathematics textbooks published by Southwest Normal University Press,the proportion of the number of thinking problem is high in the knowledge field of number and algebra,while low in the field of statistics and probability;the proportion of thinking problem in which the situation is based on mathematics is high,while the proportion based on cartoon and others is low;the number of thinking problem balances on some aspects such as knowledge characteristics,thinking level and open-ended in all grades;the proportion of thinking problem which has embedded questions is low on self-explanation;most of them provide some space to explore for students,reflecting higher order thinking.Suggestions on thinking problems writing are as followed: studying the spirit of mathematics curriculum standard on thinking problems writing deeply;making overall arrangement in the full set of textbooks;paying attention to pupils' reading level;adding some tips properly in some difficult thinking problems;adding some embedded questions properly in the end of some thinking problems.Keywords:
Higher-Order Thinking
Thinking processes
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Mathematical thinking skills are very important in mathematics, both to learn math or as learning goals. Thinking skills can be seen from the description given answers in solving mathematical problems faced. Mathematical thinking skills can be seen from the types, levels, and process. Proportionally questions given to students at universities in Indonesia (semester I, III, V, and VII). These questions are a matter of description that belong to the higher-level thinking. Students choose 5 of 8 given problem. Qualitatively, the answers were analyzed by descriptive to see the tendency to think mathematically used in completing the test. The results show that students tend to choose the issues relating to the calculation. They are more use cases, examples and not an example, to evaluate the conjecture and prove to belong to the numeric argumentation. Used mathematical thinking students are very personal (intelligence, interest, and experience), and the situation (problems encountered). Thus, the level of half of the students are not guaranteed and shows the level of mathematical thinking.
Mathematical problem
Analytical skill
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Summary The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child's work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is 'Turtle Geometry', which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a 'cybernetic turtle' controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical performance.
Constructive
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Mathematical logic
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It is well known that solving mathematics problems is one of the most difficult parts for students of any stage and how to become a successful mathematics problem solver requires complex cognitive skills. Like many other countries, in Saudi Arabia teaching solving problems is a main goal of teaching mathematics and it is a central concern of mathematics educators and mathematics curriculum developers (Ministry of Education, 2007). As textbooks are fundamental resources in Saudi classrooms, this study aims to examine to what extent Saudi textbooks reflect the mathematics curriculum goal in developing students' abilities in mathematical problem solving (Ministry of Education, 2007). Most specifically, the study investigates how the national middle school mathematics textbooks in Saudi Arabia represent problem solving heuristics. The study basically adopted Schoenfeld’s definition about heuristics, that is, a heuristic is “a general suggestion or strategy, independent of any particular topic or subject matter, that helps problem solvers approach and understand a problem and efficiently marshall their resources to solve it” (Schoenfeld, 1987). We established a framework for coding heuristics into different categories such as guess and check, look for pattern, draw a diagram, etc. The data were collected from three textbooks, which are used in Grades 7, 8 and 9, through analyzing all examples problems in the textbooks, and then coded the result following the framework. The initial findings show that all the textbooks represent a good number of problems and there are fourteen heuristics presented in the textbooks, though most of which are found in Grades 7 and 8. It is hoped that the findings of the study will shed light on the potential challenges facing textbooks developers in terms of mathematical problem solving
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In this study, about 200 middle school students solved an augmented-quotient division-with-remainders problem, and their solution processes and interpretations were examined. Based on earlier research, semantic-processing models were proposed to explain students' success or failure in solving division-with-remainder story problems on the basis of the presence or absence of an adequate interpretation provided by the solver after obtaining a numerical solution. In this study, students' solutions and their attempts and failures to “make sense” of their answers were analyzed for evidence that supported or refuted the hypothesized semantic-processing models. The results confirmed that the models provide a solid explanation of students' failure to solve division-with-remainder problems in school settings. More generally, the results indicated that student performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their mathematical thinking and reasoning.
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Introduction The Indian National Curriculum Framework (NCF) (NCF, 2005) and the National Council of Teachers of Mathematics (NCTM, 1980) both focus on development of problem solving in school mathematics. A majority of research work done on problem solving in mathematics has been conducted in the US, and yet, according Trends in International Mathematics and Science Study (TIMSS-2007), East Asian students perform better than the US students (Gonzales et al., 2008). It is true that, over the past decade, a few Indian students have won gold and silver medals in International Mathematics Olympiads; however, studies of students' mathematical learning conducted in Indian metros show that even students from top schools perform below the international average (EI Reports, 2009). The Program for International Student Assessment (PISA) (PISA, 2012) reported that the mathematical proficiency of students from Indian states such as Tamil Nadu and Himachal Pradesh was lower than the average of countries of the Organization for Economic Co-operation and Development (OECD: http://www.oecd.org/). A problem is defined as to search consciously for some action appropriate attain a clearly conceived, but not immediately attainable aim (Polya, 1981). For this study, the term mathematical problem refers a problem that is solved by using mathematical models, formulas, mathematical logic, and rules. In the historical review of problem solving, there is a dichotomy between the terms problem solving and doing exercises. The term problem solving refers the use of various heuristic strategies, pattern searching, and control functions for selecting the appropriate strategy, whereas doing exercises refers the use of known procedures and methods (Schoenfeld, 1985). For the scope of this paper, we consider mathematical problems at K-12 level. Problem solving is introduced at school level when students learn word problems including a real world scenario. At that point students experience difficulties because they need understand the real world scenario, connect it the mathematical language and convert it into the mathematical model solve. Hence, the paper focuses on teaching word problems. Despite more than seven decades of work in teaching problem solving (Polya, 1981; Polya, 1946; Schoenfeld, 1985; Silver, 1985; Marshall, 1995; Jonassen, 2011), classroom teaching of solving mathematical problems at school level has remained a great challenge. The paper is organized as follows. Initially, the theory of mathematical thinking, knowledge and behavior is discussed, drawing from the literature of mathematics education, give a theoretical framework for the argument. Secondly, the state of the art for teaching word problem solving is discussed, and the gaps are identified. The next section contains a discussion of the methodology for developing ontology and of the proposed ontology, considering one particular domain with an example. The next section compares the proposed ontology with existing teaching ontologies described in the literature. The final section includes an evaluation of MONTO, the conclusion, and possible future work. State of the art Mathematical thinking, knowledge, understanding Once students are taught mathematical problem solving, they should be able explore patterns and seek solutions the problems and not just memorize procedures, and should be able formulate conjectures and not merely do exercises. The implicit objectives of teaching mathematical problem solving at school level--such as development of mathematical thinking, logical thinking, and critical thinking--are expected be achieved after some years. While teaching mathematical problem solving, Schoenfeld (1985) conducted a study of students and came up with a framework for the analysis of mathematical thinking and behavior that contains four components (Table 1). A student who has been taught mathematical problem solving is strong in analyzing a large amount of quantitative data, uses mathematics in practical ways, and is analytical both in thinking on her own and in examining the arguments put forward by others (Schoenfeld, 1992). …
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A lot of advanced mathematics knowledge are added in the senior middle school mathematics,because of the mathematics curriculum reform of senior middle school.Some knowledge about Taylor formula and Taylor series can not adapt to the students' study in the book advanced mathematics published by China Renmin University Press.In this paper,We have given a problem put forward by students,and have given a comparatively right method to resolve this problem.Finally,we discuss the relationship between Teachers and textbooks.
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th grade primary school textbooks from Romania. These textbooks are analyzed using two classifications. The first classification is based on how much creativity and problem solving skills pupils need to solve a given task. In this classification problems are gouped in three categories: routine problems, grayarea problems and puzzle-like (non-routine) problems. The results show that most of the problems from textbooks are routine-problems. Only about 15% of the problems are more difficult, which can be solved in few steps, but even these problems are not challenging. The second classification divide problems based on how the operation chain they have to solve is given: by numbers, by text or in a word problem. The results show that there are big differences in the percentage of problems from these three categories in different textbooks. In one of the studied textbook half of the problems are word problems, in the other one only one quarter.
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In the most general sense, mathematical thinking can be defined as using mathematical techniques, concepts, and methods, directly or indirectly, in the problem-solving process.In this study, efforts were made to include the Graph Theory of mathematics, which is found abundantly in physics, chemistry, computer networks, economics, administrative science, data communication, transportation planning, engineering, and similar areas of daily life, into the mathematics instruction process.Because suitable problems are selected in graph theory, the mathematical thinking skills of the students can be developed.A fundamental, qualitative research approach was adopted in this study, in which the mathematical thinking processes of 12th-grade students were examined.Two girls and two boys with average and high mathematics success who were in the 12th grade of a public school in Balıkesir in the 2018-2019 Academic School year constituted the working group of the research.The data in the study were collected with the two worksheets the researchers prepared, the clinical interviews held during the application, and through unstructured observations.The results obtained by analyzing the data in the process of mathematics teaching, in which graph theory included are: When students solved the problems they faced and when they ascend top steps of mathematical thinking, they showed a better performance when compared to the other studies.Because, in the steps of privatization, generalization and assumption, only 1 of the total 8 responses given by the students is empty.At the stage of proof which is the final stage of mathematical thinking, the students have achieved a success rate of 75% despite they did not have a lesson about making proofs.This situation was interpreted as the visual model of problems selected from graph theory and attracting the attention of the questions as stated by the students.The results of this study show that, in the process of mathematics teaching, especially if teachers provide enough diversity students by using different fields of mathematics, they can increase the performances of students in mathematical thinking stages.
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The mathematics applied consciousness means the pupils can solve similar questions use the mathematics knowledge they have learned, especially use the knowledge according to real situations. In fact, they considered little of the realistic meaning when they solved mathematics word problem, they lacked of applying consciousness of mathematics knowledge in the real situation. Some research showed that the essential difference between the school mathematical class and the real situation was the main cause of this issue and it will develop the pupils' applying consciousness of mathematics through true mathematics learning situation.
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