Journal articles: 'Mathematical conjectures'

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Davies, Alex, Petar Veličković, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, et al. "Advancing mathematics by guiding human intuition with AI." Nature 600, no. 7887 (December 1, 2021): 70–74. http://dx.doi.org/10.1038/s41586-021-04086-x.

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AbstractThe practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

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Zeybek Simsek, Zulfiye. "Constructing-Evaluating-Refining Mathematical Conjectures and Proofs." International Journal for Mathematics Teaching and Learning 21, no. 2 (December 12, 2020): 197–215. http://dx.doi.org/10.4256/ijmtl.v21i2.263.

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This study focused on investigating the ability of 58 pre-service mathematics teachers' (PSMTs) to construct-evaluate-refine mathematical conjectures and proofs. The PSMTs enrolled in a three-credit mathematics course that offered various opportunities for them to engage with mathematical activities including constructing-evaluating-refining proofs in various topics. The PSMTs' proof constructions were coded in three categories as: Type P1, Type P2 and Type P3 in decreasing levels of sophistication (from a mathematical stand point) and the constructions of conjectures were coded in two categories as: Type C1: correct conjectures and Type C2: incorrect conjectures. In addition to classifying the PSMTs' proof and conjecture constructions, how they reacted when they needed to refine conjectures and proofs were also classified. Samples of classroom episodes were provided to exemplify these different proof-conjecture constructions-evaluations-refining processes. The results of the study demonstrated that the combined construction-evaluation-refining activities of conjectures and proofs were not only helpful to better illuminate the PSMTs' understanding of mathematical proofs, but they were also an essential instructional tool to help PSMTs comprehend mathematical ideas and relations.

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Astawa, I. Wayan Puja. "The Differences in Students’ Cognitive Processes in Constructing Mathematical Conjecture." JPI (Jurnal Pendidikan Indonesia) 9, no. 1 (March 31, 2020): 49. http://dx.doi.org/10.23887/jpi-undiksha.v9i1.20846.

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Constructing mathematical conjectures involves individuals’ unique and complex cognitive processes in which have not yet fully understood. The cognitive processes refer to any of the mental functions assumed to be involved in the acquisition, storage, interpretation, manipulation, transformation, and the use of knowledge. Understanding of these cognitive processes may assist individuals in constructing mathematical conjectures. This study aimed to describe the differences in students’ cognitive processes in constructing mathematical conjecture which is based on their mathematical ability and gender through a qualitative exploratory research study. The research subjects consisted of six mathematics students of Universitas Pendidikan Ganesha, the representative of high, medium, and low mathematical ability and either genders, male and female, respectively. The data of cognitive processes were collected by task-based interviews and were analyzed qualitatively. The differences in students’ cognitive processes in constructing mathematical conjectures were grouped into five distinct stages, namely understanding the problem, exploring the problem, formulating the conjecture, justifying the conjecture, and proving the conjecture. The results show that there were several differences in the students’ cognitive processes in constructing mathematical conjectures in the previously mentioned stages.

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Amir, Firana, and Mohammad Faizal Amir. "Action Proof: Analyzing Elementary School Students Informal Proving Stages through Counter-examples." International Journal of Elementary Education 5, no. 2 (August 23, 2021): 401. http://dx.doi.org/10.23887/ijee.v5i3.35089.

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Both female and male elementary school students have difficulty doing action proof by using manipulative objects to provide conjectures and proof of the truth of a mathematical statement. Counter-examples can help elementary school students build informal proof stages to propose conjectures and proof of the truth of a mathematical statement more precisely. This study analyzes the action proof stages through counter-examples stimulation for male and female students in elementary schools. The action proof stage in this study focuses on three stages: proved their primitive conjecture, confronted counter-examples, and re-examined the conjecture and proof. The type of research used is qualitative with a case study approach. The research subjects were two of the 40 fifth-grade students selected purposively. The research instrument used is the task of proof and interview guidelines. Data collection techniques consist of Tasks, documentation, and interviews. The data analysis technique consists of three stages: data reduction, data presentation, and concluding. The analysis results show that at the stage of proving their primitive conjecture, the conjectures made by female and male students through action proofs using manipulative objects are still wrong. At the stage of confronted counter-examples, conjectures and proof made by female and male students showed an improvement. At the stage of re-examining the conjecture and proof, the conjectures and proof by female and male students were comprehensive. It can be concluded that the stages of proof of the actions of female and male students using manipulative objects through stimulation counter-examples indicate an improvement in conjectures and more comprehensive proof.

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BARTH, PETER. "IWASAWA THEORY FOR ONE-PARAMETER FAMILIES OF MOTIVES." International Journal of Number Theory 09, no. 02 (December 5, 2012): 257–319. http://dx.doi.org/10.1142/s1793042112501357.

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In [A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proc. St. Petersburg Mathematical Society, Vol. 12, American Mathematical Society Translations, Series 2, Vol. 219 (American Mathematical Society, Providence, RI, 2006), pp. 1–85] Fukaya and Kato presented equivariant Tamagawa number conjectures that implied a very general (non-commutative) Iwasawa main conjecture for rather general motives. In this article we apply their methods to the case of one-parameter families of motives to derive a main conjecture for such families. On our way there we get some unconditional results on the variation of the (algebraic) λ- and μ-invariant. We focus on the results dealing with Selmer complexes instead of the more classical notion of Selmer groups. However, where possible we give the connection to the classical notions.

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Mollin, R. A., and H. C. Williams. "Proof, Disproof and Advances Concerning Certain Conjectures on Real Quadratic Fields." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 1023–36. http://dx.doi.org/10.4153/cjm-1995-054-7.

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AbstractThe purpose of this paper is to address conjectures raised in [2]. We show that one of the conjectures is false and we advance the proof of another by proving it for an infinite set of cases. Furthermore, we give hard evidence as to why the conjecture is true and show what remains to be done to complete the proof. Finally, we prove a conjecture given by S. Louboutin, for Mathematical Reviews, in his discussion of the aforementioned paper.

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Barahmand, Ali. "On Mathematical Conjectures and Counterexamples." Journal of Humanistic Mathematics 9, no. 1 (January 2019): 295–303. http://dx.doi.org/10.5642/jhummath.201901.17.

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Barbosa, Lucas De Souza, Cinthya Maria Schneider Meneghetti, and Cristiana Andrade Poffal. "O uso de geometria dinâmica e da investigação matemática na validação de propriedades geométricas." Ciência e Natura 41 (July 16, 2019): 12. http://dx.doi.org/10.5902/2179460x33752.

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This paper presents an activity on Geometry of position using GeoGebra software, based on Mathematical Investigation. The Dynamic Geometry, through software, becomes a tool for the formation of a mental image of abstract objects and motivation to introduce the idea of justifying its properties through arguments external to software. Allied to Dynamic Geometry, Mathematical Investigation guides the possible ways to construct conjectures and justifications and emphasizes that conjecture and looking for properties are as important as demonstrating them, since it shows the Mathematics as a knowledge in construction and favors the cognitive evolution of the relation between perception and abstraction. The activity was applied and it was observed that, from it, the student can formulate conjectures and develop logical and formal argumentation skills that are essential to construct geometric objects from their properties.

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Rizos, Ioannis, and Nikolaos Gkrekas. "Is there room for conjectures in mathematics? The role of dynamic geometry environments." European Journal of Science and Mathematics Education 11, no. 4 (October 1, 2023): 589–98. http://dx.doi.org/10.30935/scimath/13204.

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Proof, as a central and integral part of mathematics, is an essential component of mathematical education and is considered as the basic procedure for revealing the truth of mathematical propositions and for teaching productive reasoning as part of human civilization. Is there therefore room for conjectures in mathematics? In this paper after discussing at a theoretical level the concepts of proof and conjecture, both in a paper-and-pencil environment and in a dynamic geometry environment (DGE) as well as how school practice affects them, we fully explain a task involving various mathematical disciplines, which we tackle using elementary mathematics, in a mathematics education context. On the occasion of the Greek educational system we refer to some parameters of the teaching of geometry in school and we propose an activity, within a DGE, that could enable students to be guided in the formulation and exploration of conjectures. Finally, we discuss the teaching implications of this activity and make some suggestions.

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Formanowicz, Piotr, and Krzysztof Tanaś. "The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks." International Journal of Applied Mathematics and Computer Science 22, no. 3 (October 1, 2012): 765–78. http://dx.doi.org/10.2478/v10006-012-0057-y.

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Abstract It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan–Raspaud conjecture.

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Gultom, Christine Iriane, Triyanto, and Dewi Retno Sari Saputro. "Students’ Mathematical Reasoning Skills in Solving Mathematical Problems." JPI (Jurnal Pendidikan Indonesia) 11, no. 3 (September 4, 2022): 542–51. http://dx.doi.org/10.23887/jpiundiksha.v11i3.42073.

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Mathematical material will be easily understood through reasoning and that reasoning skills can be trained through mathematics learning. However, there are still many students who do not have good reasoning skills. This study aims to analyses the mathematical reasoning skills of class XI students of Senior High School on trigonometry in solving mathematical problems. This research is a qualitative descriptive study. The participants of this study were 2 students, namely student 1 (S1) and student 2 (S2) who were selected by purposive sampling. The instruments used to explore this mathematical reasoning skill were written tests and interviews. Triangulation method was employed to validate the data. The results showed that S1 was able to meet the reasoning indicator in proposing conjectures, predicting the answers and the solution process, performing mathematical manipulation and concluding sentences at the end of a completion. S2 was able to meet the reasoning indicator in proposing conjectures and predicting the answers and the solution process. However, S2 was less capable in performing mathematical manipulation and less capable concluding sentences at the end of a completion.

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Herdiyanus Warae, Putriani Ndruru, and Restu Dohude. "STUDENTS' MATHEMATICAL REASONING ABILITY IN LEARNING." AFORE : Jurnal Pendidikan Matematika 1, no. 2 (January 21, 2023): 138–47. http://dx.doi.org/10.57094/afore.v1i2.575.

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The research used a qualitative research approach with a descriptive research type. The informants in this study were class X SMK in Lolomaya Village. Data collection carried out in this study were tests and interviews. The test is used to determine students' mathematical reasoning abilities by using 5 item description questions that contain reasoning. Interviews were conducted to find out in depth students' mathematical reasoning abilities in solving mathematical problems which were attended by 6 students out of 10 students who were selected based on student abilities, namely 2 students with high abilities, 2 students with moderate abilities and 2 students with low abilities. The results of this study are (1) high ability students fulfill the indicators by making conjectures; perform mathematical manipulations; compiling evidence; draw conclusions from a statement and check the validity of an argument. (2) students who are capable of fulfilling the indicators submit conjectures and check the validity of an argument. (3) students who have low ability to meet the indicators submit conjectures

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Semanišinová, Ingrid, and Marián Trenkler. "Discovering the Magic of Magic Squares." Mathematics Teacher 101, no. 1 (August 2007): 32–39. http://dx.doi.org/10.5951/mt.101.1.0032.

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A collection of problems that allow students to investigate magic squares and Latin squares, formulate their own conjectures about these mathematical objects, look for arguments supporting or disproving their conjectures, and finally establish and prove mathematical assertions. Each problem is completed with commentary and/or experience from the classroom.

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Masuda, Ana, Didik Sugeng Pambudi, and Randi Pratama Murtikusuma. "ANALISIS PENALARAN MATEMATIS SISWA SMA KELAS XI DALAM MENYELESAIKAN SOAL BARISAN DAN DERET ARITMETIKA DITINJAU DARI GAYA BELAJAR HONEY-MUMFORD." Jurnal Riset Pendidikan dan Inovasi Pembelajaran Matematika (JRPIPM) 5, no. 1 (January 28, 2022): 56–68. http://dx.doi.org/10.26740/jrpipm.v5n1.p56-68.

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This study aims to describe the mathematical reasoning in students of class XI MIPA 5 SMAN 4 Jember in solving arithmetic sequences and series in terms of the learning styles of activists, reflectors, theorists, and pragmatists. This type of research is descriptive qualitative research. The research subjects were eight students based on the Honey-Mumford learning style. The instruments used in this study were learning style questionnaires, mathematical reasoning tests, and interview guidelines. The results of this study are Activist Students in solving arithmetic sequences and arithmetic sequences capable of presenting mathematical statements verbally and in writing but are incomplete, constructing and establishing conjectures, carrying out mathematical manipulations, giving reasons or evidence for the correctness of solutions, and making conclusions from statements, but incomplete. Reflector students are able to present mathematical statements verbally and in writing, construct and determine conjectures, carry out mathematical manipulations but are not careful in writing the methods used, provide reasons or evidence for the correctness of the solution, and make conclusions from statement, but takes a long time. Theorist students are able to present mathematical statements verbally and in writing, construct and determine conjectures, carry out mathematical manipulations, provide reasons or evidence for the correctness of the solution, and make conclusions from statements. Pragmatic students are able to present mathematical and verbal statements, construct and determine conjectures, carry out mathematical manipulations, provide reasons or evidence for the correctness of the solution, and make conclusions from statements.

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Jiang, Zhonghong, and George E. O'Brien. "Multiple Proof Approaches & Mathematical Connections." Mathematics Teacher 105, no. 8 (April 2012): 586–93. http://dx.doi.org/10.5951/mathteacher.105.8.0586.

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Atkins, Sandra L. "Listening to Students: The Power of Mathematical Conversations." Teaching Children Mathematics 5, no. 5 (January 1999): 289–95. http://dx.doi.org/10.5951/tcm.5.5.0289.

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The power of mathematical conversations was epitomized during a series of lessons in which fourth-grade students were encouraged to make conjectures and defend their conjectures to their classmates. The opening excerpt illustrates that the students had had limited experiences with angles in various orientations. Consequently, they had internalized a very narrow definition for a right angle. However, this limited definition did not become clear until we asked them to elaborate on their thinking.

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Borwein, Peter B. "Scientific computation on mathematical problems and conjectures." Journal of Approximation Theory 66, no. 3 (September 1991): 354. http://dx.doi.org/10.1016/0021-9045(91)90040-h.

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Martin, David R. "Activities for Students: Filling a Square with a Curve." Mathematics Teacher 108, no. 3 (October 2014): 218–24. http://dx.doi.org/10.5951/mathteacher.108.3.0218.

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inding patterns and making conjectures are important thinking skills for students at all levels of mathematics education. Both the Common Core State Standards for Mathematics and the National Council of Teachers of Mathematics speak to the importance of these thought processes. NCTM suggests that students should be able to recognize reasoning and proof as fundamental aspects of mathematics, make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs, and select and use various types of reasoning and methods of proof. CCSS states that students should “make conjectures and build a logical progression of statements to explore the truth of their conjectures” (CCSSI 2010, p. 6). This activity makes such reasoning accessible to high school students with some previous study of trigonometry.

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Muthmainnah, Nina, Sri Subarinah, Amrullah Amrullah, and Arjudin Arjudin. "Analysis of student mathematical investigations ability on transformation geometry in terms of cognitive style." Jurnal Pijar Mipa 17, no. 5 (September 30, 2022): 666–73. http://dx.doi.org/10.29303/jpm.v17i5.3391.

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Mathematical investigation is an activity that can encourage an experimental activity, collect data, make observations, identify patterns, make and test conjectures and make generalizations used to improve skills and develop students' mathematical thinking processes optimally. Different cognitive styles can affect students' ability to think and reason, especially in solving mathematical investigative problems. Therefore, this paper will examine mathematical investigations' ability in reflective and impulsive cognitive styles in qualitative descriptive analysis. The subjects in eleventh-grade senior high school at SMA Negeri 2 Mataram, Indonesia. Students were selected using a purposive sampling technique, and six students were selected as subjects in the interview consisting of three reflective students and three impulsive students. The instruments used are mathematical investigation tests, Matching Familiar Figure MFFT tests, and interview guidelines. The results showed that students with a reflective cognitive style were more thorough and systematic in writing down the answers to each point and always thought first in solving problems. Most students went through 4 stages of mathematical investigations: specialization, conjecture, generalization, and justification. While students with impulsive cognitive styles mostly managed to go through 3 stages of mathematical investigations, specialization, conjecture, and generalization, due to a lack of accuracy in solving questions and providing as simple answers as possible according to the question request.

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Spencer, Thomas. "MATHEMATICAL ASPECTS OF ANDERSON LOCALIZATION." International Journal of Modern Physics B 24, no. 12n13 (May 20, 2010): 1621–39. http://dx.doi.org/10.1142/s0217979210064538.

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This article discusses mathematical results and conjectures motivated by Anderson localization and by related problems for deterministic and nonlinear systems. Finite volume criteria for Anderson localization are explained for random potentials. Recent results on a phase transition for a hyperbolic, supersymmetric sigma model on a 3D lattice are also presented. This transition is analogous to the Anderson transition.

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Franklin, James. "Logical Probability and the Strength of Mathematical Conjectures." Mathematical Intelligencer 38, no. 3 (April 12, 2016): 14–19. http://dx.doi.org/10.1007/s00283-015-9612-3.

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Algreagri, Manal H., and Ahmad M. Alghamdi. "Remarks on Conjectures in Block Theory of Finite Groups." Axioms 12, no. 12 (December 6, 2023): 1103. http://dx.doi.org/10.3390/axioms12121103.

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In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results.

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Pease, Alison, Andrew Aberdein, and Ursula Martin. "Explanation in mathematical conversations: an empirical investigation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (January 21, 2019): 20180159. http://dx.doi.org/10.1098/rsta.2018.0159.

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Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; we contend that this use of explanation may help to provide an informal conceptualization of simplicity. We extracted six conjectures from recent philosophical work on the occurrence and characteristics of explanation in mathematics. We then tested these conjectures against a corpus derived from online mathematical discussions. To this end, we employed two techniques, one based on indicator terms, the other on a random sample of comments lacking such indicators. Our findings suggest that explanation is widespread in mathematical practice and that it occurs not only in proofs but also in other mathematical contexts. Our work also provides further evidence for the utility of empirical methods in addressing philosophical problems. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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Naraine, Bishnu, and Emam Hoosain. "Activities: Investigating Polygonal Areas: Making Conjectures and Proving Theorems." Mathematics Teacher 91, no. 2 (February 1998): 135–50. http://dx.doi.org/10.5951/mt.91.2.0135.

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The ability to conjecture is one component of mathematical power. This activity, appropriate for students in grades 8 and above, ofiers students an opportunity to explore relationships between and among the areas of various polygons. In a previous activity (Naraine 1993), students were able to discover that when squares are drawn on the sides of a triangle as in figure la, the four triangles will always have the same area, irrespective of the shape of triangle ABC. Similarly, the exercises in this activity allow students to make, test, modify, and prove conjectures about the areas of the various polygons in figure lb and figure lc.

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F.X. Didik Purwosetiyono, Meisy Rahma Putri Budiyanti, Rizky Esti Utami, and Achmad Buchori. "KEMAMPUAN PENALARAN SISWA SMP DALAM MENYELESAIKAN SOAL LITERASI MATEMATIKA PADA SISWA TIPE ADVERSITY QUOTIENT (AQ)." ENGGANG: Jurnal Pendidikan, Bahasa, Sastra, Seni, dan Budaya 3, no. 1 (December 21, 2022): 216–25. http://dx.doi.org/10.37304/enggang.v3i1.8576.

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This study describes students' mathematical reasoning abilities to solve mathematical literacy questions on adversity quotient (AQ) type students qualitatively. The researcher collected data from VIII-H class students at SMP Negeri 1 Gemuh, totaling 31 students using purposive sampling with the help of the mathematics teacher's consideration of where to conduct the research. The subjects selected in this study consisted of 6 subjects, namely two subjects with AQ quitters (low), two types of AQ campers (medium), and two types of AQ climbers (high). Data collection techniques used Adversity Response Profile (ARP) questionnaire instruments, written tests, documentation. The analysis technique is done by reducing data, presenting data, and drawing conclusions. Data validity checking techniques use technical and source triangulation. The results showed that 1) students with the AQ climbers type gave rise to five indicators of mathematical reasoning, namely making conjectures, manipulating mathematics, determining patterns to make generalizations, compiling evidence or reasons for correct solutions, and drawing conclusions. 2) students with AQ type campers bring up four indicators of mathematical reasoning, namely making conjectures, doing mathematical manipulation, determining patterns to make generalizations, and drawing conclusions. 3) students with the AQ quitter type are only able to bring up two indicators of reasoning, namely making conjectures and manipulating.

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Agustin, Firnanda Shofiyah, Nur Fauziyah, and Fatimatul Khikmiyah. "Analysis of Mathematical Reasoning Ability Based on Mathematics Anxiety In Junior High School Students." Mathline : Jurnal Matematika dan Pendidikan Matematika 9, no. 2 (April 19, 2024): 347–64. http://dx.doi.org/10.31943/mathline.v9i2.609.

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Mathematical reasoning ability is one of the abilities that students must master. Influencing factors include math anxiety. This study analyzes and describes mathematical reasoning ability based on mathematics anxiety in junior high school students. The type of research conducted is descriptive qualitative research. This research was conducted at UPT SMP Negeri 17 Gresik class VII B with three students as research subjects for three categories of mathematical anxiety: high, medium, and low. Research data was obtained from mathematics anxiety questionnaires, reasoning ability tests, and interviews. Based on the results of the analysis, it was found that students with low levels of mathematical anxiety could fulfill all indicators of mathematical anxiety, and students with moderate levels of mathematical anxiety fulfilled three indicators of mathematical reasoning, namely making conjectures, manipulating mathematics, and determining patterns or properties of mathematical phenomena. Learners with high levels of mathematical anxiety only fulfill one indicator of mathematical anxiety, namely, making conjectures. From the results of the study, it can be concluded that the higher the students' mathematical anxiety, the lower their mathematical reasoning ability, so it is recommended that educators and students pay attention to mathematical anxiety in learning to improve their mathematical reasoning ability.

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Blondeau-Fournier, Olivier, Patrick Desrosiers, Luc Lapointe, and Pierre Mathieu. "Macdonald Polynomials in Superspace: Conjectural Definition and Positivity Conjectures." Letters in Mathematical Physics 101, no. 1 (January 8, 2012): 27–47. http://dx.doi.org/10.1007/s11005-011-0542-5.

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McClain, Kay, Maggie McGatha, and Lynn L. Hodge. "Improving Data Analysis through Discourse." Mathematics Teaching in the Middle School 5, no. 8 (April 2000): 548–53. http://dx.doi.org/10.5951/mtms.5.8.0548.

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The National Council of Teachers of Mathematics has been advocating the importance of effective communication in classrooms since the release of its Standards documents (NCTM 1989, 1991). This emphasis is echoed in Richards's (1991) description of an inquiry classroom (see also, e.g., Ball [1993]; Cobb, Wood, and Yackel [1991]; Lampert [1990]). In this setting, the teacher's role is to guide the negotiation of classroom norms to enable the teacher and students together to engage in meaningful mathematical discussions, which include asking questions, solving problems, posing conjectures, and formulating and critiquing mathematical arguments. An increased emphasis on communication in the mathematics classroom allows students the opportunity to discuss and validate mathematical ideas and to make and evaluate conjectures and arguments.

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Thompson, Denisse R. "Connecting Research to Teaching: Learning and Teaching Indirect Proof." Mathematics Teacher 89, no. 6 (September 1996): 474–82. http://dx.doi.org/10.5951/mt.89.6.0474.

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Proof! It is the heart of mathematics as individuals explore, make conjectures, and try to convince themselves and others about the truth or falsity of their conjecture. In fact, proving is one of the main aspects of mathematical behavior and “most clearly distinguishes mathematical behavior from scientific behavior in other disciplines” (Dreyfus et al. 1990, 126). By its nature, proof should promote understanding and thus should be an important part of the curriculum (Hanna 1995). Yet students and teachers often find the study of proof difficult, and a debate within mathematics education is currently underway about the extent to which formal proof should play a role in geometry, the content domain in which reasoning is typically studied at an intensive level (Battista and Clements 1995).

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Bradford, Alexander, J. Kain Day, Laura Hutchinson, Bryan Kaperick, Craig Larson, Matthew Mills, David Muncy, and Nico Van Cleemput. "Automated Conjecturing II: Chomp and Reasoned Game Play." Journal of Artificial Intelligence Research 68 (June 7, 2020): 447–61. http://dx.doi.org/10.1613/jair.1.12188.

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We demonstrate the use of a program that generates conjectures about positions of the combinatorial game Chomp—explanations of why certain moves are bad. These could be used in the design of a Chomp-playing program that gives reasons for its moves. We prove one of these Chomp conjectures—demonstrating that our conjecturing program can produce genuine Chomp knowledge. The conjectures are generated by a general purpose conjecturing program that was previously and successfully used to generate mathematical conjectures. Our program is initialized with Chomp invariants and example game boards—the conjectures take the form of invariant-relation statements interpreted to be true for all board positions of a certain kind. The conjectures describe a theory of Chomp positions. The program uses limited, natural input and suggests how theories generated on-the-fly might be used in a variety of situations where decisions—based on reasons—are required.

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Fox, Thomas B. "Activities for Students: Transformations on Data Sets and Their Effects on Descriptive Statistics." Mathematics Teacher 99, no. 3 (October 2005): 208–17. http://dx.doi.org/10.5951/mt.99.3.0208.

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The development of concepts in statistics and functions is an important part of the school mathematics curriculum. Also important is the formulation and verification of mathematical conjectures (National Council of Teachers of Mathematics 2000). This activity asks students to examine the effects on the descriptive statistics of a data set that has undergone either a translation or a scale change. They make conjectures relative to the effects on the statistics of a transformation on a data set. Students then defend their conjectures and deductively verify several of them.

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Sari, Feti Eka Ratna, Cholis Sa'dijah, and Tjang Daniel Chandra. "Penalaran Matematis Mahasiswa dalam Menyelesaikan Masalah Trigonometri." JIPM (Jurnal Ilmiah Pendidikan Matematika) 12, no. 1 (September 19, 2023): 33. http://dx.doi.org/10.25273/jipm.v12i1.16266.

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Important for students to have reasoning abilities in learning mathematics. This is due to the relationship between mathematics and reasoning. The purpose of this study is to analyze students' mathematical reasoning abilities in solving trigonometry problems. This type of research is descriptive qualitative. As many as 36 students in class D semester 1 of mathematics education at State University of Malang were the subjects of this study. Methods of data collection obtained from the implementation of trigonometry problem tests and interviews. Data analysis with three stages namely reduction techniques, presentation, and drawing conclusions. This study resulted in the conclusions (1) students with high abilities met all indicators, namely making conjectures, performing mathematical manipulations, compiling evidence, and drawing conclusions, (2) students with moderate abilities fulfilled two indicators, namely making conjectures and compiling evidence, while (3) students who were low ability to fulfill one indicator, namely submitting conjectures.

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Rosyidah, Ana Siti, Erry Hidayanto, and Makbul Muksar. "Kemampuan Penalaran Matematis Siswa SMP dalam Menyelesaikan Soal HOTS Geometri." JIPM (Jurnal Ilmiah Pendidikan Matematika) 10, no. 2 (October 4, 2021): 268. http://dx.doi.org/10.25273/jipm.v10i2.8819.

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Abstrak: Penelitian ini bertujuan untuk mendeskripsikan kemampuan penalaran matematis siswa dalam menyelesaikan soal HOTS geometri. Metode yang digunakan adalah kualitatif deskriptif. Penelitian dilaksanakan di SMPN 1 Madiun pada 32 siswa kelas IX H tahun ajaran 2020/2021. Selanjutnya, dipilih satu subjek pada masing-masing kategori yaitu kemampuan penalaran matematis tinggi, kemampuan penalaran matematis sedang, dan kemampuan penalaran matematis rendah untuk dilakukan analisis data dengan melihat kemampuan penalaran matematis siswa berdasarkan indikator; mengajukan dugaan, melakukan manipulasi matematika, memberikan alasan atau bukti dan menarik kesimpulan. Teknik pengumpulan data dalam penelitian ini menggunakan tes dan wawancara. Hasil penelitian ini menunjukkan bahwa siswa yang mempunyai kemampuan penalaran kategori rendahmenuliskan jawaban dengan tidak tepat dan hanya memenuhi indikator penalaran matematis pertama yakni mengajukan dugaan. Siswa yang memiliki kemampuan penalaran matematis kategori sedang menjawab soal kurang lengkap dan memenuhi dua indikator penalaran matematis; mengajukan dugaan dan menarik kesimpulan. Siswa yang memiliki kemampuan penalaran matematis kategori tinggi menuliskan jawaban dengan tepat, lengkap dan memenuhi semua indikator penalaran matematis: mengajukan dugaan, melakukan manipulasi matematika, memberikan alasan atau bukti dan menarik kesimpulan.Kata kunci: Penalaran matematis; Soal HOTS, GeometriAbstract: This study aims to describe students' mathematical reasoning abilities in solving geometric HOTS problem. The method used is descriptive qualitative. The research was conducted at SMPN 1 Madiun on 32 students of class IX H for the 2020/2021 school year. Furthermore, one subject was selected in each category, namely high mathematical reasoning ability, medium mathematical reasoning ability, and low mathematical reasoning ability for data analysis by looking at students' mathematical reasoning abilities based on indicators; making conjectures, performing mathematical manipulations, providing reasons or evidence and drawing conclusions. Data collection techniques in this study used tests and interviews. The results of this study indicate that students who have low category reasoning abilities write answers incorrectly and only satisfy the first mathematical reasoning indicator, namely making conjectures. Students who have the medium category of mathematical reasoning write answer incomplete and satisfy two indicators of mathematical reasoning; making conjectures and drawing conclusions. Students who have high category mathematical reasoning abilities write answers accurately, completely and satisfy all indicators of mathematical reasoning: making conjectures, performing mathematical manipulations, providing reasons or evidence and drawing conclusions.Key words: Mathematical reasoning; HOTS Problem, Geometry

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Gerald, James A., and David Jay Hebert. "Happy Numbers: A Platform for Conjecture and Exploration." Mathematics Teacher: Learning and Teaching PK-12 116, no. 8 (August 2023): 617–20. http://dx.doi.org/10.5951/mtlt.2022.0258.

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Thompson, Alba G. "On Patterns, Conjectures, and Proof: Developing Students' Mathematical Thinking." Arithmetic Teacher 33, no. 1 (September 1985): 20–23. http://dx.doi.org/10.5951/at.33.1.0020.

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A useful strategy in solving problems is “look for a pattern.” In using the strategy, we start with simple cases or versions of the problem and from these cases discover a pattern or rule that can be applied to find the general solution.

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Maida, Paula. "Reading and Note-Taking Prior to Instruction." Mathematics Teacher 88, no. 6 (September 1995): 470–73. http://dx.doi.org/10.5951/mt.88.6.0470.

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How can students achieve mathematics literacy? One of the goals set by the NCTM to develop such literacy in students is the implementation of mathematical communication. Mathematical communication allows students to express their thought processes and is fundamental to the comprehension of mathematics. Mathematical communkation also forces students to be active learners as opposed to passive learners who simply accept and memorize procedures. For students to learn to communicate mathematically, NCTM's Curriculum and Evaluation Standards for School Mathematics recommends that teachers foster a classroom environment that encourages reading, writing, and verbal communication of students' thoughts. “Ideas are discussed, discoveries shared, conjectures confirmed, and knowledge acquired through talking, writing, speaking, listening, and reading” (NCTM 1989, 214).

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Feby Fajriatur Rohmah and Joko Soebagyo. "Investigasi Kemampuan Penalaran Siswa dalam Menyelesaikan Soal Matematika SMP Ditinjau dari Emotional Intelligence." Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika 5, no. 2 (August 1, 2022): 149–58. http://dx.doi.org/10.30605/proximal.v5i2.1854.

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In the process of solving problems, the different categories of Emotional Intelligence can affect the final results they get. The purpose of this study was to determine the process of students' reasoning abilities in solving junior high school math problems in terms of Emotional Intelligence. The research method used in this study is a qualitative research method with a case study approach. The subjects in this study were three junior high school students in South Tangerang. There are five indicators of mathematical reasoning ability in this study, namely collecting evidence, formulating conjectures, generalizing, justifying and drawing conclusions. Furthermore, the results of this study showed that subject T and subject S could cover all indicators of mathematical reasoning ability, namely indicators of collecting evidence, formulating conjectures, generalizing, justifying, and drawing conclusions. Meanwhile, subject R was able to meet the generalization and justifying indicators, but could not cover the indicators of collecting evidence, formulating conjectures and drawing conclusions.

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Mu, Yan-Ping, and Zhi-Wei Sun. "Telescoping method and congruences for double sums." International Journal of Number Theory 14, no. 01 (November 21, 2017): 143–65. http://dx.doi.org/10.1142/s1793042118500100.

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In recent years, Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.

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Cullen, Craig J., Joshua T. Hertel, and Sheryl John. "Technology Tips: Investigating Extrema with GeoGebra." Mathematics Teacher 107, no. 1 (August 2013): 68–72. http://dx.doi.org/10.5951/mathteacher.107.1.0068.

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Technology can be used to manipulate mathematical objects dynamically while also facilitating and testing mathematical conjectures. We view these types of authentic mathematical explorations as closely aligned to the work of mathematicians and a valuable component of our students' educational experience. This viewpoint is supported by NCTM and the Common Core State Standards for Mathematics (CCSSM).

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Flores-Medrano, Eric, Danae Gómez-Arroyo, Álvaro Aguilar-González, and Laura Muñiz-Rodríguez. "What Knowledge Do Teachers Need to Predict the Mathematical Behavior of Students?" Mathematics 10, no. 16 (August 14, 2022): 2933. http://dx.doi.org/10.3390/math10162933.

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The aim of this study was to explore the specialized knowledge of five mathematics teachers who participated in a continuing training project. Teachers were asked to formulate conjectures about the type of mathematical work that students enrolled in a calculus course would develop when approaching the graphical representation of functions as an introductory activity to the calculation of the volume of solids of revolution. The data collected was analyzed using the categories of the MTSK (Mathematics Teacher’s Specialized Knowledge) model. The results report how knowledge of topics and the knowledge of features of learning mathematics, particularly in relation to the knowledge of strengths and difficulties, served as fundamental pillars for the formulation of the conjectures about students’ mathematical behavior.

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Kusumah, Rahayu Tirta, Aan Subhan Pamungkas, and Jaenudin Jaenudin. "Analysis of mathematical inductive reasoning ability reviewed from student learning styles in mathematics learning." Union: Jurnal Ilmiah Pendidikan Matematika 11, no. 1 (March 30, 2023): 111–23. http://dx.doi.org/10.30738/union.v11i1.13133.

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The purpose of this study is to describe students' mathematical inductive reasoning ability viewed from visual, auditorial, and kinesthetic learning styles in the Sequence and Series material. The type of research used is descriptive research with a qualitative approach. The instruments of this study were mathematical inductive reasoning ability tests, learning style questionnaires, and interview guidelines. The subjects of this study were six students of Class XI MIPA-2 SMA Negeri 1 Kota Serang that representing each learning style. The results showed that 1) students with a visual learning style were able to do transductive reasoning, analogies, used relationship patterns to analyze situations, compiled conjectures and could estimate answers, and they were able to explain patterns; 2) students with auditorial learning styles were able to do transductive reasoning, analogies, used relationship patterns to analyze situations, compiled conjectures and could estimate answers; 3) students with a kinesthetic learning style were able to do transductive reasoning, analogies, could use relationship patterns to analyze situations and compiled conjectures, and they were able to make generalizations.

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Walkington, Candace, Jennifer Cooper, Olubukola Leonard, Caroline Williams-Pierce, and Chuck Kalish. "Middle school students' and mathematicians' judgments of mathematical typicality." Journal of Numerical Cognition 4, no. 1 (June 7, 2018): 243–70. http://dx.doi.org/10.5964/jnc.v4i1.70.

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K-12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K-12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can limit inductive generalization. Middle school students largely did not differentiate between everyday and mathematical typicality – they did not view special mathematical properties as limiting generalization, and rated items similarly regardless of context. These results suggest directions for learning mathematical argumentation and represent an important step towards understanding the nature of typicality in math.

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Nank, Sean, Jaclyn M. Murawska, and Steven J. Edgar. "It’s Off the Screen: Unearthing Megagons Through Technology." Mathematics Teacher: Learning and Teaching PK-12 116, no. 5 (May 2023): 358–65. http://dx.doi.org/10.5951/mtlt.2023.0002.

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Mathematical action technology can foster equitable student discourse. Students engage in cycles of proof to create, test, and revise conjectures through dynamic exploration of the Pythagorean theorem.

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Contreras, José N. "Exploring Nonconvex, Crossed, and Degenerate Polygons." Mathematics Teacher 98, no. 2 (September 2004): 80–86. http://dx.doi.org/10.5951/mt.98.2.0080.

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How interactive software can be used to extend mathematical conjectures and theorems to non–convex, crossed, and degenerate polygons. The author demonstrates investigating Napoleon's Theorem with Geometer's Sketchpad.

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Hadi, Ismul, Sri Subarinah, Tabita Wahyu Triutami, and Nurul Hikmah. "Analisis Kesalahan Penalaran Matematis Dalam Menyelesaikan Masalah Pola Bilangan Ditinjau Dari Gaya Belajar." Griya Journal of Mathematics Education and Application 2, no. 3 (September 29, 2022): 612–22. http://dx.doi.org/10.29303/griya.v2i3.216.

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This study aims to describe the types of reasoning errors made in solving number pattern problems. Based on mathematical reasoning procedures, error analysis was carried out based on student learning styles. The stages of mathematical reasoning in this study include the stages proposing conjectures, finding patterns, and drawing conclusions. The research subjects for class VIII A of SMPN 8 Mataram for the academic year 2021/2022 were selected by purposive sampling. To deepen the research results, two students were selected from each type of learning style. The instruments in this study are learning style questionnaires, number pattern tests, and interview guidelines. The data analysis technique used quantitative and qualitative data analysis techniques. From the results of the study, we obtained the percentage of mathematical reasoning errors, namely: (i) students with visual learning styles at the stage of proposing a guess made 42%, finding patterns 72%, and drawing conclusions 71%; (ii) students with auditory learning styles stage proposed 53% conjectures, finding patterns 67%, and drawing conclusions 70%; (iii) students with a kinesthetic learning style stage proposed 56% conjectures, finding patterns 67%, and drawing conclusions 70%; (iv) the causes of student errors with visual, auditory, and kinesthetic learning styles are a lack of understanding of number pattern material, a lack of thoroughness in calculations, a lack of thoroughness in understanding questions, and a lack of experience with concluding final results.

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Cahyani, Herlina Dwi, and Agung Prasetyo Abadi. "Analisis Kemampuan Komunikasi Matematis Siswa SMP pada Materi Bangun Ruang Sisi Datar." Journal on Education 6, no. 1 (June 2, 2023): 1733–42. http://dx.doi.org/10.31004/joe.v6i1.3143.

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In order to better understand how junior high school pupils solve arithmetic issues involving flat sided spaces, this study will examine their mathematical communication abilities. At SMPN 3 Tirtajaya, this research was done. There are five measures of students' ability to communicate mathematically, including the first measure, which links visuals to mathematical ideas, the second measure, which makes conjectures, gathers evidence, and formulates definitions and generalizations, the third measure, which expresses ideas, situations, and mathematical relations with visuals, the fourth measure, which turns everyday events into mathematical situations, and the fifth measure, which explains and creates q. A descriptive qualitative research methodology was used to perform the study on 21 students in class IX. The test asks students to answer five questions regarding how well they can communicate mathematical ideas. The description test method is the method utilized to collect data. According to the data analysis's findings, pupils at SMPN 3 Tirtajaya are classed as having moderate problem-solving skills.

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MCCULLOUGH, DARRYL, and MARCUS WANDERLEY. "NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 481–509. http://dx.doi.org/10.1017/s0017089512000675.

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AbstractWe present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

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Fix, George J. "Scientific Computations on Mathematical Problems and Conjectures (Richard S. Varga)." SIAM Review 35, no. 2 (June 1993): 318–20. http://dx.doi.org/10.1137/1035066.

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Fitri Handayani, Ucik, and Risma Cahya Setiawati. "Profile of Mathematical Reasoning Ability of Students of SMK Ma'wataibin Pagelaran in Solving Problems of Arithmetic Rows and Rows." Noumerico: Journal of Technology in Mathematics Education 2, no. 1 (March 31, 2024): 35–44. http://dx.doi.org/10.33367/jtme.v2i1.5151.

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This study describes the profile of students' mathematical reasoning ability in solving arithmetic rows and series problems. Based on the results of observations and interviews with the mathematics teacher of class X SMK Ma'wataibin Pagelaran Malang, information was obtained that 83% of students in the class tended not to be able to conclude the answer. This descriptive research uses a qualitative approach. The sample of this research is 19 students of class X SMK Ma'wataibin Pagelaran. They then selected one each in the high, medium, and low-level categories. Then, interviews were conducted with each subject to explore deeper information. The test instrument used is 1 question that contains four indicators of mathematical reasoning ability. The test results obtained that students in the high mathematical reasoning ability category amounted to 7 students, with a percentage of 36.85%. Students in the moderate mathematical reasoning ability category amounted to 8 students, with a rate of 42.10%. Students in the low mathematical reasoning ability category amounted to 4 students, with a rate of 21.05%. Based on the results of the data analysis, it is shown that high-category students fulfill four indicators of reasoning ability according to their abilities. Medium-category students correctly fulfill three indicators of mathematical reasoning ability: writing conjectures, doing work, and providing reasons/evidence. Low-category students only fulfill one indicator of mathematical reasoning ability, namely writing conjectures.

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Moyer, Patricia S., and Wei Shen Hsia. "Activities for Students: The Archaeological Dig Site: Using Geometry to Reconstruct the Past." Mathematics Teacher 94, no. 3 (March 2001): 193–201. http://dx.doi.org/10.5951/mt.94.3.0193.

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In secondary mathematics, students often see little connection between geometry and the real-life mathematical situations around them. When asked to describe geometric figures, their descriptions are sometimes no more than an identification of sides and angles. They have not had experience in using more than one property in a mathematical situation or in describing how two geometric properties are related. The van Hiele model of how students learn geometry proposes that students' understandings of geometry move from recognition to description to analysis (Fuys, Geddes, and Tischler 1988). For students to make this transition to analytic thinking, teachers need to create problem situations that enhance development of students' intuitive understandings. These investigations allow students to explore relationships among geometric shapes and to make conjectures about properties. The conjectures can then be stated formally as theorems.

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Journal articles on the topic 'Mathematical conjectures'

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