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In various fields

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Education

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Main article: Mathematics education

Arithmetic education forms part of primary education and is one of the first forms of mathematics education that children encounter. It aims to give students a basic sense of numbers and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division.[1][2][3] It is usually introduced in relation to concrete scenarios, like counting beads, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in arithmetic education include the use of number lines, addition and multiplication tables, and counting blocks.[4][5][6]

Later stages in arithmetic education focus on a more abstract understanding. They introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers as well as more advanced operations, like exponentiation, extraction of roots, and logarithm. [1][7] They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of algorithms and calculators to solve complex arithmetic problems.[1][8][9][10]

Psychology

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The psychology of arithmetic is interested in how humans and animals learn about numbers, represent them, and use them for calculations. It includes examines how mathematical problems are understood and solved and how arithmetic abilities are related to perception, memory, judgment, and decision making.[11][12] For example, it investigates how collections of concrete items are first encountered in perception in subsequently associated with numbers.[13] A further field of inquiry concerns the relation between numerical calculations and the use of language employed to form representations.[14] Regarding the origin of arithmetic, psychology explores in what sense a basic understanding of arithmetic is part of the biological makeup of the human brain. This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation.[12]

One of the concepts studied by psychology is numeracy, which is the capability comprehend numerical concepts, to apply them to concrete situations, and to reason with them. It includes a fundamental number sense as well as being like being able to estimate and compare quantities. It further encompasses the abilities to symbolically represent numbers in numbering systems, interpret numerical data, and evaluate arithmetic calculations.[15][16][17] Numeracy is a key skill in many academic fields. A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, in relation to mortgage plans and insurance.[15][18][19][20]

Philosophy

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Main article: Philosophy of mathematics

The philosohy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and ontological status of numbers, how it is possible to acquire arithmetic knowledge, and the relation of arithmetic to language and logic.[21][22][23]

According to the Platonism, numbers have mind-independent existence: they exist abstract objects outside space and time and without any causal powers.[22][24] This view is rejected by intuitionists, who claim that mathematical objects are mental constructions.[25] Further theories are logicism, which holds that mathematical truths are reducible to logical truths,[26][27] and formalism, which states that mathematical principles are rules of how symbols are manipulated with corresponding to entities outside the rule-governed activity.[28]

The traditionally dominant view in the epistemology of arithmetic is that arithmetic truths are knowable a priori, i.e., by thinking alone without the need to rely on sensory experience.[22][29] Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of rational intuition through which mathematical truths can be apprehended.[22][30] A more recent alternative view was suggested by naturalist philosophers like Willard Van Orman Quine, who argue that mathematical principles are high-level generalization that are ultimately grounded in the sensory world as described by the empirical sciences.[31][29]


Others

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Arithmetic is relevant to many fields. In daily life, it is required to calculate the change when shopping, to manage personal financies, and to adjust a cooking recipe for a different number of servings. Businesses use arithmetic to calculate profits and losses, and analyze market trends. In the field of engineering, it is used to measure quantities, calculate loads and forces, and design structures.[32][33][34]

Arithmetic operations lie at the foundation of many branches of mathematics, like algebra, calculus, and statistics. Through them, the influence of arithmetic extends to most sciences such as physics, computer science, and economics. These operations are used in calculations, problem-solving, data analysis, and algorithms, making them integral to scientific research, technological development, and economic modeling.[35][36][37][38] The application of arithmetic operations also extends to fields like cryptography as a means of protecting sensitive information.[39][40]


  • NCTM Staff. "Number and Operations". www.nctm.org. National Council of Teachers of Mathematics. Retrieved 21 November 2023.
  • Odom, Samuel L.; Barbarin, Oscar A.; Wasik, Barbara Hanna (8 July 2009). "Applying Lessons from Developmental Science to Early Education". In Barbarin, Oscar A.; Wasik, Barbara Hanna (eds.). Handbook of Child Development and Early Education: Research to Practice. Guilford Press. ISBN 978-1-60623-302-3.
  • Laski, Elida V.; Jor’dan, Jamilah R.; Daoust, Carolyn; Murray, Angela K. (1 April 2015). "What Makes Mathematics Manipulatives Effective? Lessons From Cognitive Science and Montessori Education". SAGE Open. 5 (2). doi:10.1177/2158244015589588.
  • Nurnberger-Haag, Julie (15 October 2017). "Borrow, Trade, Regroup, or Unpack? revealing How Instructional Metaphors Portray Base-Ten Number". In Jao, Limin; Radakovic, Nenad (eds.). Transdisciplinarity in Mathematics Education: Blurring Disciplinary Boundaries. Springer. ISBN 978-3-319-63624-5.
  • Carraher, David W.; Schliemann, Analucia D. (30 July 2015). "Powerful Ideas in Elementary School Mathematics". In English, Lyn D.; Kirshner, David (eds.). Handbook of International Research in Mathematics Education. Routledge. ISBN 978-1-134-62664-9.
  • Ruthven, Kenneth (6 December 2012). "12. Calculators in the Mathematics Curriculum: The Scope of Personal Computational Technology". In Bishop, Alan; Clements, M. A. (Ken); Keitel-Kreidt, Christine; Kilpatrick, Jeremy; Laborde, Colette (eds.). International Handbook of Mathematics Education. Springer Science & Business Media. ISBN 978-94-009-1465-0.
  • Grice, Matt; Kemp, Simon; Morton, Nicola J.; Grace, Randolph C. (26 June 2023). "The psychological scaffolding of arithmetic". Psychological Review. doi:10.1037/rev0000431.
  • De Cruz, Helen; Neth, Hansjörg; Schlimm, Dirk (2010). "The Cognitive Basis of Arithmetic". In Löwe, Benedikt; Müller, Thomas (eds.). PhiMSAMP: Philosophy of Mathematics : Sociological Aspsects and Mathematical Practice. College Publications. ISBN 978-1-904987-95-6.
  • Victoria Department of Education Staff (2023). "Numeracy for all learners". www.education.vic.gov.au. Retrieved 22 November 2023.
  • Askew, Mike (1 June 2010). "It Ain't (Just) What You Do: Effective Teachers of Numeracy". In Ian, Thompson (ed.). Issues In Teaching Numeracy In Primary Schools. McGraw-Hill Education (UK). ISBN 978-0-335-24153-8.
  • Dreeben-Irimia, Olga (22 October 2010). Patient Education in Rehabilitation. Jones & Bartlett Publishers. ISBN 978-1-4496-1775-2.
  • Barnes, Andrew J.; Rice, Thomase; Hanoch, Yaniv (18 May 2017). "Using Behavioral Economics to Improve People's Decisions About Purchasing Health Insurance". In Hanoch, Yaniv; Barnes, Andrew; Rice, Thomas (eds.). Behavioral Economics and Healthy Behaviors: Key Concepts and Current Research. Taylor & Francis. ISBN 978-1-317-26952-6.
  • Gerardi, Kristopher; Goette, Lorenz; Meier, Stephan (9 July 2013). "Numerical ability predicts mortgage default". Proceedings of the National Academy of Sciences. 110 (28). doi:10.1073/pnas.1220568110.
  • Jackson, Janna M. (15 August 2008). "Reading/Writing Connection". In Flippo, Rona F. (ed.). Handbook of College Reading and Study Strategy Research. Routledge. ISBN 978-1-135-70373-8.
  • Horsten, Leon (2023). "Philosophy of Mathematics". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 22 November 2023.
  • Weir, Alan (2022). "Formalism in the Philosophy of Mathematics". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 22 November 2023.
  • Sierpinska, Anna; Lerman, Stephen (1996). "Epistemologies of Mathematics and of Mathematics Education". International Handbook of Mathematics Education: Part 1. Springer Netherlands. ISBN 978-94-009-1465-0.
  • Aubrey, Carol (1 December 1999). A Developmental Approach to Early Numeracy: Helping to Raise Children's Achievements and Deal with Difficulties in Learning. A&C Black. ISBN 978-1-4411-9164-9.
  • Lockhart, Paul (2017). Arithmetic. Cambridge, Massachusetts London, England: The Belknap Press of Harvard University Press. ISBN 9780674972230.
  • Bird, John (15 March 2021). Bird's Engineering Mathematics. Taylor & Francis. ISBN 9780367643782.

Gallistel, C. R.; Gelman, R. (2005). "Mathematical Cognition". In Holyoak, K. J.; Morrison, R. G. (eds.). The Cambridge handbook of thinking and reasoning. Cambridge University Press. ISBN 0521531012.

  1. ^ a b c NCTM Staff.
  2. ^ Musser, Peterson & Burger 2013, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, p. 44, p. 130.
  3. ^ Odom, Barbarin & Wasik 2009, p. 589, Applying Lessons from Developmental Science to Early Education.
  4. ^ Laski et al. 2015, pp. 1–3.
  5. ^ Musser, Peterson & Burger 2013, pp. 59, 90–91, 93–94, 106–108.
  6. ^ Nurnberger-Haag 2017, p. 215, Borrow, Trade, Regroup, or Unpack? revealing How Instructional Metaphors Portray Base-Ten Number.
  7. ^ Musser, Peterson & Burger 2013, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, pp. 208, 304, 340, 362.
  8. ^ Musser, Peterson & Burger 2013, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics.
  9. ^ Carraher & Schliemann 2015, p. 197, Powerful Ideas in Elementary School Mathematics.
  10. ^ Ruthven 2012, pp. 435, 443–444, 12. Calculators in the Mathematics Curriculum: The Scope of Personal Computational Technology.
  11. ^ De Cruz, Neth & Schlimm 2010, pp. 59–60.
  12. ^ a b Grice et al. 2023, abstract.
  13. ^ De Cruz, Neth & Schlimm 2010, pp. 60–62.
  14. ^ De Cruz, Neth & Schlimm 2010, p. 63.
  15. ^ a b Victoria Department of Education Staff 2023.
  16. ^ Askew 2010, pp. 33–34, It Ain't (Just) What You Do: Effective Teachers of Numeracy.
  17. ^ Dreeben-Irimia 2010, p. 102.
  18. ^ Barnes, Rice & Hanoch 2017, p. 196, Using Behavioral Economics to Improve People's Decisions About Purchasing Health Insurance.
  19. ^ Gerardi, Goette & Meier 2013, pp. 11267–11268.
  20. ^ Jackson 2008, p. 152, Reading/Writing Connection.
  21. ^ Hofweber 2016, pp. 153–154, 162–163.
  22. ^ a b c d Oliver 2005, p. 58.
  23. ^ Sierpinska & Lerman 1996, p. 827.
  24. ^ Horsten 2023, § 3. Platonism.
  25. ^ Horsten 2023, § 2.2 Intuitionism.
  26. ^ Horsten 2023, § 2.1 Logicism.
  27. ^ Hofweber 2016, pp. 174–175.
  28. ^ Weir 2022, Lead Section.
  29. ^ a b Sierpinska & Lerman 1996, p. 830.
  30. ^ Sierpinska & Lerman 1996, pp. 827–876.
  31. ^ Horsten 2023, § 3.2 Naturalism and Indispensability.
  32. ^ Lockhart 2017, pp. 1–2.
  33. ^ Bird 2021, p. 3.
  34. ^ Aubrey 1999, p. 49.
  35. ^ Gallistel & Gelman 2005, pp. 559–560.
  36. ^ Ali Rahman et al. 2017, pp. 373–374.
  37. ^ Li & Schoenfeld 2019, Abstract, Introducation.
  38. ^ Asano 2013, pp. xiii–xv.
  39. ^ Omondi 2020, pp. viii.
  40. ^ Paar & Pelzl 2009, p. 13.
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