The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects. [153] 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. We would like to express our gratitude to teachers who participated in the project. Supplementary Material

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Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. [46] It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. [47] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts. mathrm{Bob's\:age\:is\:twice\:that\:of\:Barry's.\:Five\:years\:ago,\:Bob\:was\:three\:times\:older\:than\:Barry.\:Find\:the\:age\:of\:both.}Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials.

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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/a0035176 Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322BC) this meaning was fully established. [14] 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_10Lesh, 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. Main article: Algebra The quadratic formula, which concisely expresses the solutions of all quadratic equations The Rubik's Cube group is a concrete application of group theory. [37]

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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/edu0000453In 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). There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities. [156] [157] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. [156] There is not even consensus on whether mathematics is an art or a science. [157] Some just say, "mathematics is what mathematicians do". [156] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study. [158] Convex geometry, the study of convex sets, which takes its importance from its applications in optimization.