Mathematics involve the materials that can develop thinking skills, especially **reasoning** ability. As mentioned by [1] mathematics has axiomatic deductive characteristics that require the **reasoning** and thinking ability in order to comprehend it. Furthermore, [2] states that there are some different characteristics of mathematics learning for secondary and high education levels. In college, especially mathematics education study program, the materials focus on its fundamental aspects and in details. Meanwhile, mathematics for the secondary schools is on the implementation of theories or **mathematical** laws that taught in college level. It requires the students to have a high concentration level during the mathematics learning process. The development of mathematics is inseparable from **reasoning** and proofing. As stated [3], [4], [5], [6] that between mathematics and **reasoning** cannot be separated one another because understanding mathematics requires **reasoning** which can be trained through **mathematical** material. It indicates that **mathematical** **reasoning** is very crucial during mathematics learning. In line with this, [7], [8], [9] explain that **reasoning** is one of the crucial competencies in mathematics as a supporting feature in mathematics learning. [1] adds that **reasoning** plays a role in solving **mathematical** problems and the implications of **reasoning** are usually found in real life. Moreover, [10] points out that **reasoning** and proofing are the most basic aspects of mathematics learning. One of the courses taught in mathematics education study programs is abstract algebra that aims at developing students' proofing ability. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. [11] mentions that one of the major problems

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Developing **Mathematical** **Reasoning** Using Word Problems The word problems at the end of each legend in the preceding section are based on detailed analyses of various problem situations involving number. These problems are organized within a framework that makes it possible to distinguish among problems in terms of **reasoning** difficulty. An important aspect of applying this knowledge when teaching is sequencing word problems from easiest to most difficult. Such sequencing allows children to develop **mathematical**

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Abstract. This study examines students' **mathematical** **reasoning** based on Discovery learning models in terms of gender. This research was conducted at the SMPN 3 Kendari with quasi-experimental methods involving two classes with different treatments. The simple random technique is used to determine the class of research. Class VII. 6 (experimental class) consisted of 15 women and 11 men, while class VII.9 (control class) consisted of 15 men and 7 women. The instrument used was a student's **mathematical** **reasoning** ability test consisting of four items in the form of essays tested. Data processing using 2-way ANOVA with further tests using Scheffe. The conclusion obtained is that students are given a learning discovery model, the **reasoning** ability of male students excels in the ability to give **mathematical** problems verbally and in writing provided in the form of logical diagrams that contain existing data, perform **mathematical** manipulation related to the problem, and ensure validity as an argument, whereas women excel in the ability to draw conclusions based on relationships between **mathematical** concepts. The discovery learning model can increase students' **mathematical** penalties and overcome gender discussions.

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With regard to **mathematical** **reasoning** abilities, the teacher has a very important role in developing **mathematical** **reasoning** abilities in students both with the learning method used, and the type of evaluation used. Improving students' **mathematical** **reasoning** skills also needs to be supported by the right learning approach so that learning objectives can be achieved. One important aspect of planning rests on the teacher's ability to anticipate needs and materials or models that can help students to achieve expected learning goals[4]. In this case, Teachers must have a method in learning as a strategy that can facilitate students to master the knowledge provided [5]. One of the predictable and possible learning approaches to improve students' **mathematical** **reasoning** abilities is learning through problem posing approach.

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The article reviews the stable logic analysis model of complex systems or steady multilateral systems with two non- compatibility relations. Energy concept in Physics is introduced to the multilateral systems and used to deal with the multilateral system diseases. By using **mathematical** **reasoning**, it is demonstrated that the treatment principle: “Virtual disease is to fill his mother but real disease is to rush down his son” and “Strong inhibition of the same time, support the weak” which due to the “Yin Yang Wu Xing” Theory in Traditional Chinese Medicine (TCM).

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mathematics items reached 40% of the allocated time and high- complexity mathematics items reached only 3% of the allocated time.Unfortunately, TIMSS’ mathematics items consist of **mathematical** problems ranging from moderate to high complexity. In summary, the time allocated for mathematics learning in Indonesian schools is still mostly spent to solve **mathematical** problems which can be categorized as relatively easy. The low achievement of Indonesian junior high school and Madrasah Tsanawiyah students in **mathematical** **reasoning** may be caused by an internal factor, which is the previous competency of **mathematical** **reasoning** when they were in elementary schools.According to the findings of a research conducted by Watts, et al. (2016: 8 & 11), there are indications that **mathematical** skills demonstrated by children from the preschool stage to the first year at elementary schoolscan predict their mathematics achievement in a positive and significant wayuntil they reach the age of fifteen or enter senior high schools.This statement highlights the likelihood that the **mathematical** knowledge and skills showed by students in elementary schools will be reflected in the development of their abilitieswhen they later enter junior high schools, senior high schools and the next higher levels.Therefore, it can be said thatthe **mathematical** abilitiesof students at the junior high school level are directly affected by their prior **mathematical** abilitiesat the elementary school level. TIMSS assessed students’ **mathematical** abilities through two dimensions, namely content domain and cognitive domain. The content domain was in line with the material in the content standard, for first-grade junior high school includes number, algebra,

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Zhang’s theory, multilateral matrix theory [1] and multilateral system theory [13,14,19], have given a new and strong **mathematical** **reasoning** method from macro (Global) analysis to micro (Local) analysis. He and his colleagues have made some **mathematical** models and methods of **reasoning** [2-17], which make the mathemat- ical **reasoning** of TCM possible [13] based on “Yin Yang Wu Xing” Theory [18]. This paper will use steady multi- lateral systems to demonstrate the treatment principle of TCM: “Real disease is to rush down his son but virtual disease is to fill his mother” and “Strong inhibition of the same time, support the weak”.

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Brousseau and Gibel (2005) defined **mathematical** **reasoning** as a relationship between two elements: a condition or observed facts and a consequence. Duval (2007) described **mathematical** **reasoning** as ‘a logical linking of propositions’ (p. 140) that may change the epistemic value of a claim. G. J. Stylianides (2008) viewed **mathematical** **reasoning** as containing a broader concept; besides providing arguments (non-proof and proofs), it also encompasses investigating patterns and making conjectures. In Shaughnessy, Barrett, Billstein, Kranendonk, and Peck (2004), the National Council of Teachers of Mathematics (NCTM) described **reasoning** in mathematics as a cycle of exploration, conjecture, and justification. This is in line with G. J. Stylianides (2008), who focused on **mathematical** **reasoning** as a process of making inquiries to formulate a generalisation or conjecture and determine its truth value by developing arguments, where argument is understood to mean a connected sequence of assertions. In a Danish context, the definition of **reasoning** competence in the primary school mathematics curriculum is often based on the competence report (Niss & Jensen, 2002), which, among other factors, is very specific about students being able to distinguish proof from other kinds of argumentation and makes a clear distinction between ‘carrying out’ argumentations and ‘following’ the argumentations developed by others (e.g., other students or textbooks). In the present paper, the definition of **reasoning** that will be tested in the developed testis a composite of definitions from the NCTM, the competence report (Niss & Jensen, 2002), and G. J. Stylianides (2008). This will be described in more detail in the Methods section.

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The learning stages of cycle I started with the planning stage, developing the lesson plan based on the PBL steps for the topic of area and perimeter; student worksheet I, and **mathematical** **reasoning** test problems for cycle I and II. At the implementation and observation stage, the teacher started by giving apperception through questions related to the elements of the circle previously studied. The teacher motivated the students by providing examples of the circular edge of the pool. The teacher posed a question, such as "how do you determine the area of the pool edge? How do you calculate

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This study aims to determine the use of **reasoning** strategy considering different variables which were consisted of students’ problem solving skills and gender, location of schools and parents’ income and ucted in this research. The sample of the study was composed of eighth grade students from different six schools in both Kırşehir city centre and districts. In sample group, 64 girls (%54,4) and 52 boys (%45,6) were chosen randomly. For the tifying **mathematical** **reasoning** level of students, twenty open-ended questions from **Mathematical** Communication and **Reasoning** ”, which was developed by Suzuki (1998) and adopted into Turkish by Taşdemir (2008), was conducted in the study. Reliability of the inventory used in the study has been retested and it has been As a conclusion, the overall average of students’ using **reasoning** strategy was tion, a significant difference was seen in favour of girls in using **reasoning** strategy considering gender. In addition, there is a significant difference in favour of central districts with respect to towns and villages according to school location. Likewise, higher income levels of families and higher education status of parents were found significantly related to the

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Exploratory analyses of the different components of the overall Progress Test in Maths score suggest the programme may have had a positive impact on some aspects of **mathematical** skills. The results are indicative of a positive effect on the **mathematical** **reasoning** subscore. Given the findings of the process evaluation regarding teachers’ views that the programme was less suitable for the highest and lowest ability learners, we also conducted exploratory analyses to investigate whether there were differential impacts by prior attainment. This suggested no significant variation in the impact of the programme by prior attainment, although the estimate for pupils in the low prior attainment group is statistically significant at conventional levels. It is perhaps also worth noting that we also found some indication that pupils of lower prior attainment were more likely to drop out from the analysis sample and thus we should be cautious in drawing definitive conclusions about impacts for this group. Furthermore, given the negative skew apparent in the distribution of the raw scores, it may be the case that the assessment was more sensitive to differences in performance at the lower end of the distribution.

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Students often make mistakes when solving math problems is wrong **mathematical** concepts, math procedures mistakes, errors in constructing concepts, etc. This results in an improper math problem solving when working on the problem. Student misconduct when solving the math problem needs to get the attention of all circles. These mistakes have an impact on the understanding of students on the next **mathematical** concept. Brodie (2010) in the results of his research explained that students' errors in building **mathematical** **reasoning** include: basic error, appropriate error, missing information, partial insight. Gal & Linchevski (2010) found that student difficulties in geometric representation included: (1) perceptual organization: Gestalt principles, (2) recognition: bottom-up and top-down processing; and (3) representation of perception-based knowledge: verbal vs. pictorial representation, mental images and hierarchical structure of images. Bingobali, dkk (2010) explores the causes of student math difficulties based on lecturers' views, which include: Epistemological causes, Psychological causes, Pedagogical cause. It was further discovered that students had difficulty in understanding concepts, abstracting concepts, and relating mathematics to everyday life. Therefore **reasoning** has a very important role in overcoming student difficulties. This study examines more in depth the mistakes made by mathematics students when solving **mathematical** problems by looking at the **mathematical** **reasoning** structure of students in solving **mathematical** problems. The study was conducted to see in more detail the types of student errors, especially from the aspects of thinking when constructing **mathematical** concepts. By knowing the types of errors made with the structure of **reasoning**, latter can be designed a learning through scaffolding or remedial scheme that will be used to restructure students' thinking.

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Table 1 shows that students’ completeness criteria is still low in which there is no class that is able to reach more than 50% of minimum completeness criteria. Based on the problem as mentioned above, it is of vital importance to improve mathematics learning in order to achieve learning purpose. Mathematics learning achievement can not be separated from learning equipment employed. The learning devices being used are Students Work Sheet (LKPD) and Lesson Plan (RPP). LKPD is one of the device to help as to facilitate teaching and learning activity in order to create an effective interaction between students and teacher to improve learning activity and learning result. Widjajanti (2008:1) stated that LKPD is one of the learning resources that can be developed by teacher as a facilitator in learning activity. The designed and developed LKPD is in line with situation and condition of learning activity that will be encountered. The benefit of using LKPD is to ease teacher in conducting learning. The benefit of LKPD for the students are to help them to be independent learner and are able to comprehend and to perform a written test. By using LKPD, teacher can direct students’ activities. These activities may help students build their knowledge as well as their mathematics ability. LKPD also help teacher in conducting learning as for students it will help to them to learn independently and to learn to comprehend a task in a written form. LKPD is used by the students as a work sheet, hence they have an adequate learning resources to be learned anywhere. The development of LKPD is supported by Lesson Plan (RPP). RPP facilitates teacher to conduct learning process in order to achieve the learning purpose. The interview result with teacher of National Vocational High School (SMK N) Kayutanam which was conducted during preliminary research on August 29 th to September 29 th, 2018 indicates that the implemented lesson plan is in line with curriculum 2013. However, the main activity in the lesson plan has yet facilitate students in constructing their own knowledge. Therefore it is important to develop lesson plan to facilitate it. Hence, it is necessary to develop learning device based on constructivism approach to encourage the students to build their own knowledge in order to improve their **mathematical** **reasoning** skill. The research about learning device based constructivism approach has been conducted by many national

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Slot Quiz merupakan latihan yang disediakan bagi setiap subtopik yang terkandung di dalam topik “**Mathematical** **Reasoning**”. Soalan yang dibina adalah berpandukan buku-buku rujukan dan buku teks tetapi jawapan yang disediakan adalah terdiri daripada pola-pola kesilapan yang sering dilakukan oleh pelajar beserta dengan maklumbalas bagi memudahkan penyemakan semula dilakukan. Soalan yang disediakan terdiri daripada soalan aneka pilihan (multiple choice). Masa yang diperuntukkan bagi soalan tersebut adalah bergantung kepada bilangan soalan yang dikemukakan. Terdapat lima soalan bagi setiap subtopik. Pembangun lebih cenderung menggunakan soalan aneka pilihan kerana ia lebih efektif dan memudahkan penyemakan jawapan dilakukan.

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Valid Assessment Forms. Well over 88 tasks were developed, trialled and validated to create the learning progressions. Tasks generally comprised more than one item and scoring rubrics for each item were provided to reflect the definition of **mathematical** **reasoning** used in the project. The tasks generally enabled all students to make a start and provided opportunities to display their **reasoning**. For example, the Hot Air Balloon task requires students to (i) construct a graph from a table of values (time vs height), (ii) determine how long the balloon stayed at or above 250 metres, and (iii) identify when the balloon was at 400 metres and explain their **reasoning**. The tasks with their component items were presented as Forms with 5 to 7 tasks per form. Mixed Forms (tasks from two areas) and Standard Forms (tasks from one area only) were trialled to explore **reasoning** both within and across strands. Feedback from project schools suggested that they would be more interested in standard forms. As a result, four Standard Forms for each strand have been developed together with the associated scoring rubrics. Maximum score totals are different for each Form to prevent the inappropriate use of raw scores. This necessitated the provision of a Raw Score Translator for each Form that can be used to locate students on the respective learning progression for **mathematical** **reasoning**. The Forms can be used as pre-tests to determine where students are in their learning with respect to the relevant learning progression and the information derived from this can be used to inform planning and teaching. A parallel Form can then be used as a post-test to determine if there has been a qualitative shift in student behaviour and to provide feedback on the effectiveness of what was planned and taught.

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Zhang’s theories, multilateral matrix theory [1] and multilateral system theory [2-19], have given a new and strong **mathematical** **reasoning** method from macro (Global) analysis to micro (Local) analysis. He and his colleagues have made some **mathematical** models and methods of **reasoning** [20-35], which make the mathe- matical **reasoning** of TCE possible based on “Yin Yang Wu Xing” Theory [36-38]. This paper will use steady multilateral systems to demonstrate the intervening prin- ciple of TCE: “Real disease for economic over-heating is to rush down his son but virtual disease for economic downturn is to fill his mother” and “Strong inhibition of the same time, support the weak”.

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Overall, the three students in this study present impor- tant cases. Their pre-intervention performance showed no multiplicative **reasoning** to solve multiplicative problems correctly. Their improved performance on the posttest sug- gests that they could learn if taught in ways that tilt the cognitive work to the students while adapting to their avail- able and evolving **reasoning**. Specifically, this student- adaptive instruction might have enabled Tiffany’s advance to a level of conceptualization supportive of successful solutions to mostly multiplication problems. It enabled Kathy’s advance to a higher-level conceptualization sup- portive of correctly solving multiplication and some divi- sion problems. Furthermore, it enabled Sally’s advance to a level of conceptualization supportive of correctly solving all problems. To this end, as the comparison problem types were not directly taught during the intervention, only one student (Sally) demonstrated successful transfer of multipli- cative **reasoning** gained from the intervention to solving a range of multiplicative word problems. As supported by existing literature (Wagner, 2006; Xin & Zhang, 2009), it seems that systematic programming for transfer is neces- sary for students with LD.

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Indeed, in his second proof Gauss used the genus theory of quadratic binary forms, which consisted of establishing a bound on the number of existing genera of the quadratic forms of a given determinant and subsequently investigating the only two cases for the primes p and q. The resulting proof is far shorter than his first proof. We will not look at his second proof here, but the original proof can be found in Disquisitiones Arithmeticae, and another version of it can be seen in The Quadratic Reciprocity Law: A Collection of Classical Proofs. Instead we will investigate Gauss’s fourth proof, which made use of quadratic Gauss sums and eventually helped advance the field of algebraic number theory. The first step of our journey showed us how observation and induction helped establish one of the fundamental theorems of number theory. Next, we will look at how further attempts to refine and strengthen this theorem resulted in the discovery of new territories and gave rise to new features of the **mathematical** landscape.

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In sum, the purpose of this study is thus important in a few ways. First, with the research advancements made in the area of **mathematical** **reasoning** and proving, aspects of students’ performance in proof and proving are no longer limited to proof productions as studied in the past (Alcock & Weber, 2005; Antonini & Mariotti, 2008; Deer, 1969; Epp, 2003; Mueller, 1975; Selden & Selden, 2003). While logic training has been increasingly advocated as central to these additional aspects (Epp, 2003; Selden & Selden, 2003), an empirical inquiry of the role of logic training is in need. Without a clear understanding of the extent of logic training with regards to these additional aspects, the instructional theory and goals of developing students’ logical **reasoning** and **mathematical** proving in classrooms remained as individual teachers’ pedagogical beliefs. Second, the role of counterexamples had only gained emerging research attention for the purpose of mathematics learning and still at the stage of theory-building via case study methods (A. Stylianides & Stylianides, 2009b; Zazkis & Chernoff, 2008). This study clarified further the role of counterexamples in developing students’ **mathematical** **reasoning** and proving in

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not about a proposition or assertion. Yet Walton too, while alert to this difference, has often tended to blur it even while stating it. Commenting on a distinction between practical and “theoretical or discursive” **reasoning**, he states: “In the action type of critical discussion, the proposition is a practical ought-proposition that contains an imperative” (1996a, 177). This phrase is puzzling. Evidently, Walton senses that the issue in practical **reasoning** is not merely an “ought-proposition,” and so he adds that it “contains an imperative”—thereby inventing a new hybrid. But is a proposition that “contains” an imperative still a proposition? Is an imperative not a wholly different kind of speech act? Apparently Walton, at this stage, cannot abandon the ingrained idea that **reasoning**, and hence also practical **reasoning**, is about propositions. But that for which we argue in practical **reasoning** is not a proposition about what we “ought” to do, nor about what is “prudent,” although both these considerations (which are far from identical) may enter into the **reasoning** as premisses. The issue in practical **reasoning** is a proposal about what to do (a point I have contended, and whose implications I have tried to explore, in Kock 2003a, 2003b).

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