Solving Mathematical Problems: A Personal Perspective

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Verschaffel, L., Greer, B., and De Corte, E. (2007). “Whole number concepts and operations,” in Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. Editor F. K. Lester (Charlotte, NC: Information Age Pub), 557–628. Child, S., and Nind, M. (2013). Sociometric methods and difference: A force for good - or yet more harm. Disabil. Soc. 28 (7), 1012–1023. doi:10.1080/09687599.2012.741517 Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines. The studies involving human participants were reviewed and approved by the Uppsala Ethical Regional Committee, Dnr. 2017/372. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin. Author Contributions

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Medicine uses statistical hypothesis testing, run on data from clinical trials, to determine whether a new treatment works. [ citation needed] Lein, A. E., Jitendra, A. K., and Harwell, M. R. (2020). Effectiveness of mathematical word problem solving interventions for students with learning disabilities and/or mathematics difficulties: A meta-analysis. J. Educ. Psychol. 112 (7), 1388–1408. doi:10.1037/edu0000453 The word mathematics comes from Ancient Greek máthēma ( μάθημα), meaning "that which is learnt", [11] "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in Classical times. [12] Its adjective is mathēmatikós ( μαθηματικός), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". [13] In particular, mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art". [11] Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views. [128]Capar, G., and Tarim, K. (2015). Efficacy of the cooperative learning method on mathematics achievement and attitude: A meta-analysis research. Educ. Sci-theor Pract. 15 (2), 553–559. doi:10.12738/estp.2015.2.2098 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] Mousoulides, N., Pittalis, M., Christou, C., and Stiraman, B. (2010). “Tracing students’ modeling processes in school,” in Modeling Students’ Mathematical Modeling Competencies. Editor R. Lesh (Berlin, Germany: Springer Science+Business Media). doi:10.1007/978-1-4419-0561-1_10

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Mathematical problem-solving is a central area of mathematics instruction, constituting an important part of preparing students to function in modern society ( Gravemeijer et al., 2017). In fact, problem-solving instruction creates opportunities for students to apply their knowledge of mathematical concepts, integrate and connect isolated pieces of mathematical knowledge, and attain a deeper conceptual understanding of mathematics as a subject ( Lester and Cai, 2016). Some researchers suggest that mathematics itself is a science of problem-solving and of developing theories and methods for problem-solving ( Hamilton, 2007; Davydov, 2008).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

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Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes. [18] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. [19] Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. [154] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, pure mathematics. [109] [112] This led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred. [113] 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.