Karlsson, N., and Kilborn, W. (2020). “Teacher’s and student’s perception of rational numbers,” in Interim Proceedings of the 44 th Conference of the International Group for the Psychology of Mathematics Education, Interim Vol., Research Reports. Editors M. Inprasitha, N. Changsri, and N. Boonsena (Khon Kaen, Thailand: PME), 291–297. Citation: Klang N, Karlsson N, Kilborn W, Eriksson P and Karlberg M (2021) Mathematical Problem-Solving Through Cooperative Learning—The Importance of Peer Acceptance and Friendships. Front. Educ. 6:710296. doi: 10.3389/feduc.2021.710296 Although cooperative learning approach is intended to promote cohesion and peer acceptance in heterogeneous groups ( Rzoska and Ward, 1991), previous studies indicate that challenges in group dynamics may lead to unequal participation ( Mulryan, 1992; Cohen, 1994). Peer-learning behaviours may impact students’ problem-solving ( Hwang and Hu, 2013) and working in groups with peers who are seen as friends may enhance students’ motivation to learn mathematics ( Deacon and Edwards, 2012). With the importance of peer support in mind, this study set out to investigate whether the results of the intervention using the CL approach are associated with students’ peer acceptance and friendships. The Present Study

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Fujita, T., Doney, J., and Wegerif, R. (2019). Students' collaborative decision-making processes in defining and classifying quadrilaterals: a semiotic/dialogic approach. Educ. Stud. Math. 101 (3), 341–356. doi:10.1007/s10649-019-09892-9 Areas of mathematics used in the social sciences include probability/statistics and differential equations ( stochastic or deterministic). [ citation needed] These areas are used in fields such as sociology, [143] psychology, [144] economics, finance, and linguistics. [ citation needed] Supply and demand curves, like this one, are a staple of mathematical economics. Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. [38] [39] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' [40] that he used for naming one of these methods in the title of his main treatise.In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. [126] [127] A fundamental innovation was the ancient Greeks' introduction of the concept of proofs, which require that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates), or are part of the definition of the subject of study ( axioms). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements. [32] [33] A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. [124] A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. [125]

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The results of this study show that the effect of the CL intervention on students’ problem-solving was associated with students’ initial scores of social acceptance and friendships. Thus, it is possible to assume that students who were popular among their classmates and had friends at the start of the intervention also made greater gains in mathematical problem-solving as a result of the CL intervention. This finding is in line with Deacon and Edwards’ study of the importance of friendships for students’ motivation to learn mathematics in small groups ( Deacon and Edwards, 2012). However, the effect of the CL intervention was not associated with change in students’ social acceptance and friendship scores. These results indicate that students who were nominated by a greater number of students and who received a greater number of friends did not benefit to a great extent from the CL intervention. With regard to previously reported inequalities in cooperation in heterogeneous groups ( Cohen, 1994; Mulryan, 1992; Langer Osuna, 2016) and the importance of peer behaviours for problem-solving ( Hwang and Hu, 2013), teachers should consider creating inclusive norms and supportive peer relationships when using the CL approach. The demands of solving complex problems may create negative emotions and uncertainty ( Hannula, 2015; Jordan and McDaniel, 2014), and peer support may be essential in such situations. Limitations At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. [10] It results that "rigor" is no more a relevant concept in mathematics, as a p Rzoska, K. M., and Ward, C. (1991). The effects of cooperative and competitive learning methods on the mathematics achievement, attitudes toward school, self-concepts and friendship choices of Maori, Pakeha and Samoan Children. New Zealand J. Psychol. 20 (1), 17–24.Khan Academy has a full question bank to draw from, ensuring that each student works on different questions – and at their perfect skill level

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In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. [35] [10] In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space. [36] Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more. Is Khan Academy a company? The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, [71] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. [72] [73] The Babylonian mathematical tablet Plimpton 322, dated to 1800BC

Webb, N. M., and Mastergeorge, A. (2003). Promoting effective helping behavior in peer-directed groups. Int. J. Educ. Res. 39 (1), 73–97. doi:10.1016/S0883-0355(03)00074-0 The teachers in the control group received 2days of instruction in enhancing students’ problem-solving and reading comprehension. The teachers were also supported with educational materials including mathematical problems Karlsson and Kilborn (2018b) and problem-solving principles ( Pólya, 1948). However, none of the activities during training or in educational materials included the CL approach. As seen in Table 1, only 10 of 25 teachers reported devoting at least one lesson per week to mathematical problem-solving. Measures Tests of Mathematical Problem-Solving In light of the limited number of studies on the effects of CL on students’ problem-solving in whole classrooms ( Capar and Tarim, 2015), and for students with SEN in particular ( McMaster and Fuchs, 2002), this study sought to investigate whether the CL approach embedded in problem-solving activities has an effect on students’ problem-solving in heterogeneous classrooms. The need for the study was justified by the challenge of providing equitable mathematics instruction to heterogeneous student populations ( OECD, 2019). Small group instructional approaches as CL are considered as promising approaches in this regard ( Kunsch et al., 2007). The results showed a significant effect of the CL approach on students’ problem-solving in geometry and total problem-solving scores. In addition, with regard to the importance of peer support in problem-solving ( Deacon and Edwards, 2012; Hwang and Hu, 2013), the study explored whether the effect of CL on students’ problem-solving was associated with students’ social acceptance and friendships. The results showed that students’ peer acceptance and friendships at pre-test were significantly associated with the effect of the CL approach, while change in students’ peer acceptance and friendships from pre- to post-test was not. Khan Academy’s practice questions are 100% free—with no ads or subscriptions. What do Khan Academy’s interactive math worksheets cover?