Solving Mathematical Problems: A Personal Perspective

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The assumptions for the multilevel regression were assured during the analyses ( Snijders and Bosker, 2012). The assumption of normality of residuals were met, as controlled by visual inspection of quantile-quantile plots. For subgroups, however, the plotted residuals deviated somewhat from the straight line. The number of outliers, which had a studentized residual value greater than ±3, varied from 0 to 5, but none of the outliers had a Cook’s distance value larger than 1. The assumption of multicollinearity was met, as the variance inflation factors (VIF) did not exceed a value of 10. Before the analyses, the cases with missing data were deleted listwise. Results What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics? Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics An explanation of the sigma (Σ) summation notation Deacon, D., and Edwards, J. (2012). Influences of friendship groupings on motivation for mathematics learning in secondary classrooms. Proc. Br. Soc. Res. into Learn. Math. 32 (2), 22–27. Lesh, R., Post, T., and Behr, M. (1988). “Proportional reasoning,” in Number Concepts and Operations in the Middle Grades. Editors J. Hiebert, and M. Behr (Hillsdale, N.J.: Lawrence Erlbaum Associates), 93–118. 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.

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Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. [90] In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. [60] Montague, M., Krawec, J., Enders, C., and Dietz, S. (2014). The effects of cognitive strategy instruction on math problem solving of middle-school students of varying ability. J. Educ. Psychol. 106 (2), 469–481. doi:10.1037/a0035176The 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 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 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 The intervention using the CL approach lasted for 15weeks and the teachers worked with the CL approach three to four lessons per week. First, the teachers participated in two-days training on the CL approach, using an especially elaborated CL manual ( Klang et al., 2018). The training focused on the five principles of the CL approach (positive interdependence, individual accountability, explicit instruction in social skills, promotive interaction, and group processing). Following the training, the teachers introduced the CL approach in their classes and focused on group-building activities for 7weeks. Then, 2days of training were provided to teachers, in which the CL approach was embedded in activities in mathematical problem-solving and reading comprehension. Educational materials containing mathematical problems in the areas of multiplication and division, geometry, and proportionality were distributed to the teachers ( Karlsson and Kilborn, 2018a). In addition to the specific problems, adapted for the CL approach, the educational materials contained guidance for the teachers, in which problem-solving principles ( Pólya, 1948) were presented as steps in problem-solving. Following the training, the teachers applied the CL approach in mathematical problem-solving lessons for 8weeks. In this study, in accordance with previous research on mathematical problem-solving ( Lesh and Zawojewski, 2007; Degrande et al., 2016; Stohlmann and Albarracín, 2016), the CL approach was combined with training in problem-solving principles Pólya (1948) and educational materials, providing support in instruction in underlying mathematical models. The intention of the study was to provide evidence for the effectiveness of the CL approach above instruction in problem-solving, as problem-solving materials were accessible to teachers of both the intervention and control groups. However, due to implementation challenges, not all teachers in the intervention and control groups reported using educational materials and training as expected. Thus, it is not possible to draw conclusions of the effectiveness of the CL approach alone. However, in everyday classroom instruction it may be difficult to separate the content of instruction from the activities that are used to mediate this content ( Doerr and Tripp, 1999; Gravemeijer, 1999).

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Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. [48] Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. [c] Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics. [49] Lesh, R., and Zawojewski, (2007). “Problem solving and modeling,” in Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. Editor L. F. K. Lester (Charlotte, NC: Information Age Pub), vol. 2. SFS 2009:400. Offentlighets- och sekretesslag. [Law on Publicity and confidentiality]. Retrieved from https://www.riksdagen.se/sv/dokument-lagar/dokument/svensk-forfattningssamling/offentlighets--och-sekretesslag-2009400_sfs-2009-400 on the 14th of October. Frequently Asked Questions about Khan Academy and Math Worksheets Why is Khan Academy even better than traditional math worksheets? Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, [1] algebra, [2] geometry, [1] and analysis, [3] [4] respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.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. Cillessen, A. H. N., and Marks, P. E. L. (2017). Methodological choices in peer nomination research. New Dir. Child Adolesc. Dev. 2017, 21–44. doi:10.1002/cad.20206 mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?} This became the foundational crisis of mathematics. [57] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. [23] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. [58] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910. [59] Langer-Osuna, J. M. (2016). The social construction of authority among peers and its implications for collaborative mathematics problem solving. Math. Thinking Learn. 18 (2), 107–124. doi:10.1080/10986065.2016.1148529

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Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. [74] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic ( addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. [75] Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. [99] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. [100] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. [101] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. [102] In Latin, and in English until around 1700, the term mathematics more commonly meant " astrology" (or sometimes " astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine's warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians. [15] 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]McMaster, K. N., and Fuchs, D. (2002). Effects of Cooperative Learning on the Academic Achievement of Students with Learning Disabilities: An Update of Tateyama-Sniezek's Review. Learn. Disabil Res Pract 17 (2), 107–117. doi:10.1111/1540-5826.00037 During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areas–arithmetic, geometry, algebra, calculus [20]–endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. [21] The subject of combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century. [22]