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It is used not only with various precise technical meanings, but also as a metaphor for complex situations. We review some of the history of the equation and its present meaning and use in a variety of settings. 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It is used not only with various precise technical meanings, but also as a metaphor for complex situations. We review some of the history of the equation and its present meaning and use in a variety of settings. Although some languages make distinctions among ideas such equations that are always satisfied versus those that may not be, or between equations with variables and relations without, we observe that it may not be possible to decide into which of these cases a particular equality falls.</p><div class="toggle-truncation-button js-toggle-truncation-button--abstract"><button class="safe-abstract--toggle-truncation-button"><span>... </span><span class="safe-abstract--toggle-truncation-button-text">Read more</span></button></div></div></div></div><div class="ds-key-takeaways--container"><style type="text/css">.ds-loswp-section--container{display:flex;flex-direction:column;width:100%}@media(min-width: 1280px){.ds-loswp-section--container{padding-right:40px}}</style><div class="ds2-5-content-section"><div class="ds2-5-content-section__heading"><h2 class="ds2-5-content-section__heading__text">Key takeaways</h2></div><div class="ds2-5-content-section__main"><div class="ds2-5-content-section__main__content"><div class="ds-loswp-section--container"><ul class="ds2-5-body-md"><li>The mathematical use of the word equation, meaning the action of stating the equality of two quantities, appears to date from the late 16 th century.</li><li>Now we see this is not surprising, as the word equation has become established before the "=" notation and well before the modern concept of variables.</li><li>An equality statement with no variables that is true, e.g.</li><li>An equality statement with no variables that is false, e.g.</li><li>The preceding discussion shows how different forms of equality relations can be considered differently in different languages.</li></ul></div></div></div></div></div><div class="safe-right-rail--container"><div class="safe-right-rail--related-works"><h2 class="safe-related-content--heading">Related papers</h2><div class="safe-related-content--container"><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work " data-collection-position="0" data-entity-id="31872485" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/31872485/Equation">Equation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="745078" href="https://tu-wien.academia.edu/VeraB%C3%BChlmann">Vera Bühlmann</a></div><p class="ds-related-work--metadata ds2-5-body-xs">in Rosi Braidotti, Maria Hlavajova: Posthuman Glossary (Bloomsbury, London 2018)</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equation","attachmentId":52160185,"attachmentType":"pdf","work_url":"https://www.academia.edu/31872485/Equation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/31872485/Equation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work " data-collection-position="1" data-entity-id="84580078" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/84580078/The_Beauty_of_Equations">The Beauty of Equations</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="52170152" href="https://independent.academia.edu/RobertCrease">Robert Crease</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">This paper discusses grounds for ascribing beauty to equations. Philosophical accounts of beauty in the context of science tend to stress one or more of three aspects of beautiful objects: depth, economy, and definitiveness. The paper considers several cases that engage these aspects in different ways, depending on whether the equation is considered as a signpost, as a stand-in for a phenomenon, as making a disciplinary transformation, or as inspiring a personal revelation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The Beauty of Equations","attachmentId":89552700,"attachmentType":"pdf","work_url":"https://www.academia.edu/84580078/The_Beauty_of_Equations","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/84580078/The_Beauty_of_Equations"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work " data-collection-position="2" data-entity-id="64218789" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/64218789/Algebra_A_Tool_for_Solving_Problems">Algebra: A Tool for Solving Problems?</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="988779" href="https://uu.academia.edu/HenkvanderKooij">Henk van der Kooij</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2001</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">Introduction Algebra is often seen as a powerful tool both for the modeling of problem situations and for manipulating the symbolic representation of the problem situation. These two different aspects of algebra (modeling and solving) are part of a broad discussion about algebra. In general, the learning of algebra is seen from four different perspectives (Bednarz, Kieran, Lee, 1996): the generalization perspective (generalising numerical sequences and geometrical patterns), the problem-solving perspective (often referred to as the algebraic modeling of &#39;word-problems&#39;), the modeling perspective (mutual interchangebility between graphs, tables, symbolic expressions and verbal descriptions) and the functional perspective (variables and functions). On the one hand mathematical professionals agree on the power of algebra as a tool for solving problems, but on the other hand it is well-known that a vast majority of students has a lot of problems in learning to master the symboli...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Algebra: A Tool for Solving Problems?","attachmentId":76354587,"attachmentType":"pdf","work_url":"https://www.academia.edu/64218789/Algebra_A_Tool_for_Solving_Problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/64218789/Algebra_A_Tool_for_Solving_Problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work " data-collection-position="3" data-entity-id="92503401" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/92503401/The_Concept_of_Equations_What_Have_Students_Met_Before">The Concept of Equations: What Have Students Met Before?</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="13049156" href="https://oxford.academia.edu/DavidTall">David Tall</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Group for the Psychology …, 2006</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">In this paper, we present results from a research study designed in collaboration with teachers to investigate how Brazilian 15-16-year old students interpret the concept of equation and its solution. Data from a questionnaire, an equation-solving exercise and interviews with selected students are reported and analysed in terms of how the students are affected by their earlier experiences in arithmetic and algebra.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The Concept of Equations: What Have Students Met Before?","attachmentId":95495581,"attachmentType":"pdf","work_url":"https://www.academia.edu/92503401/The_Concept_of_Equations_What_Have_Students_Met_Before","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/92503401/The_Concept_of_Equations_What_Have_Students_Met_Before"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work " data-collection-position="4" data-entity-id="2738740" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/2738740/Toward_a_physics_of_equations">Toward a physics of equations</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="384302" href="https://richmond.academia.edu/DavidLandy">David Landy</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2010</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">Papers on diagrammatic reasoning often begin by dividing marks on paper into two basic classes: diagrams and sentences. While endorsing the perspective that a reasoning episode can be diagrammatic or sentential, I will give an overview of recent evidence suggesting that apparently symbolic expressions in algebra and arithmetic are frequently treated as diagrammatic or even pictorial depictions of objects and events—events that occur not in the content of the expression, but within the notation itself.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Toward a physics of equations","attachmentId":30714318,"attachmentType":"pdf","work_url":"https://www.academia.edu/2738740/Toward_a_physics_of_equations","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/2738740/Toward_a_physics_of_equations"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work hidden" data-collection-position="5" data-entity-id="91351679" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/91351679/Approaches_to_equation">Approaches to equation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="241061405" href="https://independent.academia.edu/PadmapriyaShirali">Padmapriya Shirali</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2018</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">Introduction of a complex topic is always a challenge. If one intends to root it in the student’s daily experience, one is compelled to select a concrete model which serves the purpose but may have limited scope. At some point, the student will need to abstract out the general notion in order to build a broader sense of the concept. The topic of ‘Equations’ can be approached in several ways. The choice of approach has a strong impact on the conceptual image which a student builds about a given concept. Hence, the choice is crucial in helping a student in understanding the concept as well as in developing the procedure for solving the problems. However, every approach has its limitations and can be used only for solving certain types of problems. Its use is limited and it may become necessary to expose students to other approaches when the type or complexity of the problems alters</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Approaches to equation","attachmentId":94663518,"attachmentType":"pdf","work_url":"https://www.academia.edu/91351679/Approaches_to_equation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/91351679/Approaches_to_equation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work hidden" data-collection-position="6" data-entity-id="77746164" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/77746164/Introduction_to_equation_solving_for_a_new_generation_of_algebra_learners">Introduction to equation solving for a new generation of algebra learners</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="133395323" href="https://abo.academia.edu/AnnSofiR%C3%B6jLindberg">Ann-Sofi Röj-Lindberg</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">Understanding of equality and solving equations are some of the big ideas in algebra. They have been in focus in early algebra research for some decades and in many countries it is now usual to work with equalities and solving equations using informal methods from early school years. However, it is not clear how the transition to formal methods of equation solving could be conducted in a best way in order to maintain students’ interest and enhance their understanding of the need for more powerful methods. We shed light on the issue by reporting on what happens when three teachers introduce linear equation solving in Grade 6 (age of 12) in Finland. We especially consider how the introduction could support students’ development of an algebraic understanding of equality and their engagement in more formal mathematics.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Introduction to equation solving for a new generation of algebra learners","attachmentId":85031271,"attachmentType":"pdf","work_url":"https://www.academia.edu/77746164/Introduction_to_equation_solving_for_a_new_generation_of_algebra_learners","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/77746164/Introduction_to_equation_solving_for_a_new_generation_of_algebra_learners"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work hidden" data-collection-position="7" data-entity-id="38307579" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/38307579/Equational_languages">Equational languages</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48028250" href="https://independent.academia.edu/AndrzejBlikle">Andrzej Blikle</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Information and Control, 1972</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">This paper deals with equations whose solutions are vectors of languages. Formally, solutions of equations are fix points of vectorial functions on languages. On the other hand equations (and sets of equations) can be considered as grammars. Three main groups of problems are dealt with: (1) solvability of equations in a lattice of languages, (2) relationship between type of functions used in equations and properties of languages defined by them, (3) applications to the theory of Context-free and regular languages.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equational languages","attachmentId":58356356,"attachmentType":"pdf","work_url":"https://www.academia.edu/38307579/Equational_languages","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/38307579/Equational_languages"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work hidden" data-collection-position="8" data-entity-id="14924241" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/14924241/An_analytic_conception_of_equation_and_teachers_views_of_school_algebra">An analytic conception of equation and teachers’ views of school algebra</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33912332" href="https://independent.academia.edu/RozaLeikin">Roza Leikin</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"An analytic conception of equation and teachers’ views of school algebra","attachmentId":43766997,"attachmentType":"pdf","work_url":"https://www.academia.edu/14924241/An_analytic_conception_of_equation_and_teachers_views_of_school_algebra","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/14924241/An_analytic_conception_of_equation_and_teachers_views_of_school_algebra"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card js-safe-related-work-related-work hidden" data-collection-position="9" data-entity-id="86982723" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/86982723/Introducing_Equations_in_Early_Algebra">Introducing Equations in Early Algebra</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6390410" href="https://laurentian.academia.edu/LuisRadford">Luis Radford</a></div><p class="ds-related-work--metadata ds2-5-body-xs">ZDM, 2022</p><p class="ds2-5-body-sm ds-related-work--abstract hide-on-large">The overwhelming presence of a procedural meaning of equality and equations reported in previous research has led to a call for suitable pedagogical interventions to nurture a relational meaning of these concepts. This paper is a response to that call. Drawing on the theory of objectification, the first part deals with the configuration of a Grade 3 (8-9-year-old students) teaching-learning activity that seeks to create the classroom conditions for the formation of the mathematical operations and operation-based rules that underpin the algebraic simplification of linear equations. Instead of using problems involving abstract open arithmetic sentences or alphanumeric equations (e.g., 5 + __ = 16; 2 + 3 = 11), the teaching-learning activity resorts to story-problems. Two visual semiotic systems serving to model and solve the story-problems were devised. The story-problems were framed in narratives that allowed the teacher and the students to infuse equations, their equating parts, and the mathematical operations with contextual meanings. The first part of the paper includes the theoretical assumptions about the teachinglearning activity and its configuration, and a rationale behind the devisal of the semiotic systems. The second part presents a Vygotskian multimodal genetic analysis of the teaching-learning activity; that is, an analysis that shows the formation of concepts in motion, in the process of their genesis. The genetic analysis sheds some light on the way students, in their work with the teacher, encountered and refined the cultural-historical algebraic meanings of the equal sign and equations, and the concepts required in solving equations. Keywords Algebra • multimodal semiotics • Vygotsky • gestures • algebraic operations • storyproblems</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Introducing Equations in Early Algebra","attachmentId":91318241,"attachmentType":"pdf","work_url":"https://www.academia.edu/86982723/Introducing_Equations_in_Early_Algebra","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/86982723/Introducing_Equations_in_Early_Algebra"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><button class="ds2-5-text-link ds2-5-text-link--inline safe-related-works--view-more-button js-safe-related-works-related-work--view-more-button"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">keyboard_arrow_down</span><span class="ds2-5-text-link__content">View more papers</span></button></div></div></div><div class="safe-below-fold--container"><div class="safe-below-fold--related-works"><h2 class="safe-related-content--heading">Related papers</h2><div class="safe-related-content--container"><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="0" data-entity-id="94627340" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/94627340/Solving_equations_The_transition_from_arithmetic_to_algebra">Solving equations: The transition from arithmetic to algebra</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="121363951" href="https://independent.academia.edu/TRojano">Teresa Rojano</a></div><p class="ds-related-work--metadata ds2-5-body-xs">For the learning of mathematics, 1989</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Solving equations: The transition from arithmetic to algebra","attachmentId":97031759,"attachmentType":"pdf","work_url":"https://www.academia.edu/94627340/Solving_equations_The_transition_from_arithmetic_to_algebra","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/94627340/Solving_equations_The_transition_from_arithmetic_to_algebra"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="1" data-entity-id="125111272" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/125111272/Connecting_humans_to_equations_a_reinterpretation_of_the_philosophy_of_mathematics_by_Ole_Ravn_and_Ole_Skovsmose_Switzerland_Springer_Nature_2019_xxiv_188_pp_109_99_hardcover_ISBN_13_978_3_03_001336_3">Connecting humans to equations: a reinterpretation of the philosophy of mathematics, by Ole Ravn and Ole Skovsmose, Switzerland, Springer Nature, 2019, xxiv+188 pp., $109.99 (hardcover), ISBN-13: 978-3-03-001336-3</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51404740" href="https://exeter.academia.edu/ErnestPaul">Paul Ernest</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Thinking and Learning, 2019</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Connecting humans to equations: a reinterpretation of the philosophy of mathematics, by Ole Ravn and Ole Skovsmose, Switzerland, Springer Nature, 2019, xxiv+188 pp., $109.99 (hardcover), ISBN-13: 978-3-03-001336-3","attachmentId":119213124,"attachmentType":"pdf","work_url":"https://www.academia.edu/125111272/Connecting_humans_to_equations_a_reinterpretation_of_the_philosophy_of_mathematics_by_Ole_Ravn_and_Ole_Skovsmose_Switzerland_Springer_Nature_2019_xxiv_188_pp_109_99_hardcover_ISBN_13_978_3_03_001336_3","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/125111272/Connecting_humans_to_equations_a_reinterpretation_of_the_philosophy_of_mathematics_by_Ole_Ravn_and_Ole_Skovsmose_Switzerland_Springer_Nature_2019_xxiv_188_pp_109_99_hardcover_ISBN_13_978_3_03_001336_3"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="2" data-entity-id="6031951" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/6031951/Algebra_a_Tool_for_Solving_Problems_Algebra_A_Tool_for_Solving_Problems">Algebra: a Tool for Solving Problems? Algebra: A Tool for Solving Problems</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="9047831" href="https://unj.academia.edu/AyuDinaAdniaty">Ayu Dina Adniaty</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Algebra: a Tool for Solving Problems? Algebra: A Tool for Solving Problems","attachmentId":32969420,"attachmentType":"pdf","work_url":"https://www.academia.edu/6031951/Algebra_a_Tool_for_Solving_Problems_Algebra_A_Tool_for_Solving_Problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/6031951/Algebra_a_Tool_for_Solving_Problems_Algebra_A_Tool_for_Solving_Problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="3" data-entity-id="88485827" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/88485827/Translation_from_the_French_of_%C3%89quation_alg%C3%A8bre_from_the_Encyclop%C3%A9die_of_Denis_Diderot_1756_">Translation from the French of "Équation (algèbre)" from the Encyclopédie of Denis Diderot (1756)</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33456688" href="https://independent.academia.edu/DavidFleming14">David Fleming</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Encyclopédie of Diderot and d'Alembert Collaborative Translation project, 2022</p><p class="ds2-5-body-sm ds-related-work--abstract ">EQUATION in algebra signifies an expression of the same quantity presented in two different forms. See EQUALITY. Thus, when one says 2x3=4+2, this is to say that there is an equation between two times three and four plus two. One can define the equation as a relationship of equality between two quantities of different denomination, as when one says 60 sous=3 livres or 20 sous=1 livre or b=d+e, or 12=a+ 5 , etc.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Translation from the French of \"Équation (algèbre)\" from the Encyclopédie of Denis Diderot (1756)","attachmentId":92450287,"attachmentType":"pdf","work_url":"https://www.academia.edu/88485827/Translation_from_the_French_of_%C3%89quation_alg%C3%A8bre_from_the_Encyclop%C3%A9die_of_Denis_Diderot_1756_","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/88485827/Translation_from_the_French_of_%C3%89quation_alg%C3%A8bre_from_the_Encyclop%C3%A9die_of_Denis_Diderot_1756_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="4" data-entity-id="60750384" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/60750384/The_Teaching_and_Learning_of_Equations_Problems_and_Possibilities_during_the_Transition_from_High_School_to_Higher_Education">The Teaching and Learning of Equations: Problems and Possibilities during the Transition from High School to Higher Education</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="201166962" href="https://independent.academia.edu/CFinckBrandt">Celia Finck Brandt</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Creative Education, 2015</p><p class="ds2-5-body-sm ds-related-work--abstract ">The teaching of equations for elementary school students has been faced with many obstacles, as their difficulties to find the roots of equations of first or second degree. These difficulties are presented by the students in continuing their studies (high school) and also detected on the transition of students to higher education. According to Raymond Duval these difficulties come from the mathematical point of view which is incompatible with the cognitive point of view. In this research we aim to point out to what extent the meanings assigned to algebra by the students of the Bachelor in Mathematics are close among themselves or move away from the cognitive point of view. We developed two instruments to collect qualitative information: a questionnaire with two questions and interviews. The subjects were eighteen students of Degree in Mathematics of the UEPG. The data were analyzed in the light of the discourse functions mentioned by Duval (1995): apophantic, reference and discursive expansion and their cognitive operations: apophantic function and its illocutionary act of predication and operations; referential function and its pure designation of operations, simple categorization and determination; discursive expansion function and its operations narration, description, explanation and reasoning. The speech will be analyzed in terms of semiotic or semantic similarity, internal or external arranged in a double entry table. Empirical data will be entered in cells derived from the crossing of this categorization featuring differentiated discursive expansions: formal expansion, natural, cognitive and lexical. Research is in progress showing no results at this time.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The Teaching and Learning of Equations: Problems and Possibilities during the Transition from High School to Higher Education","attachmentId":74053911,"attachmentType":"pdf","work_url":"https://www.academia.edu/60750384/The_Teaching_and_Learning_of_Equations_Problems_and_Possibilities_during_the_Transition_from_High_School_to_Higher_Education","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/60750384/The_Teaching_and_Learning_of_Equations_Problems_and_Possibilities_during_the_Transition_from_High_School_to_Higher_Education"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="5" data-entity-id="22234420" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22234420/Solving_Equations_is_All_about_Balance">Solving Equations is All about Balance</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31836520" href="https://uky.academia.edu/MargaretSchroeder">Margaret Schroeder</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</p><p class="ds2-5-body-sm ds-related-work--abstract ">Common core mathematics standards and implementing digital technologies / Drew Polly, editor.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Solving Equations is All about Balance","attachmentId":45557486,"attachmentType":"pdf","work_url":"https://www.academia.edu/22234420/Solving_Equations_is_All_about_Balance","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22234420/Solving_Equations_is_All_about_Balance"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="6" data-entity-id="20377262" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/20377262/Equations_and_Equating_in_Arabic_Mathematics">Equations and Equating in Arabic Mathematics</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="156135" href="https://uindy.academia.edu/JeffreyOaks">Jeffrey Oaks</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Equations and equating in Arabic mathematics, 2010</p><p class="ds2-5-body-sm ds-related-work--abstract ">The writing of mathematics in modern English uses the word "equal" or its symbol "=" for almost every kind of equating. By contrast, medieval Arabic mathematicians used several different words to denote equality: sawiya, mithl, adala, the prefix ka-, and the implied verb "to be". Adala was employed differently than the other words. It was used to equate the two sides of an algebraic equation, while it is uncommon in other branches of mathematics. To uncover the reason adala was chosen for equations I read over the Arabic texts published in more than 70 books and 70 articles covering algebra, arithmetic, geometry, astronomy, and mechanics. I observed which words were used for equating all kinds of mathematical objects: numbers, line segments, arcs, areas, angles, weights, spans of time, as well as their ratios.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equations and Equating in Arabic Mathematics","attachmentId":47485830,"attachmentType":"pdf","work_url":"https://www.academia.edu/20377262/Equations_and_Equating_in_Arabic_Mathematics","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/20377262/Equations_and_Equating_in_Arabic_Mathematics"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="7" data-entity-id="60178322" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/60178322/Equations_Functions_Critical_Aspects_and_Mathematical_Communication">Equations, Functions, Critical Aspects and Mathematical Communication</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="112048652" href="https://independent.academia.edu/ConstantaOlteanu">Constanta Olteanu</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Education Studies, 2012</p><p class="ds2-5-body-sm ds-related-work--abstract ">The purpose of this paper is to present the mechanism for effective communication when the mathematical objects of learning are equations and functions. The presentation is based on data collected while the same object of learning is presented in two classes, and it includes two teachers and 45 students. Among other things, the data consists of video-recordings of lessons and tests. In the analysis, concepts relating to variation theory have been used as analytical tools. The results show that effective communication occurs in the classroom if it has the critical aspects in students learning as its starting point. The communication in the classroom succeeds or not if the aspects of the content supposed to be treated is the same as or different from the aspects of the content of the teacher's representation, and if the aspects of the content of the teacher's representation are the same as or different from the aspects discerned by the students. The results also show that the students cannot make sense of the difference between the highest/lowest value of a quadratic function and the maximum/minimum point; the difference between a quadratic equation and function; the students also have difficulties in solving a quadratic equation if it appears in a new context. The argument of the functions is identified as critical aspect in this study.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equations, Functions, Critical Aspects and Mathematical Communication","attachmentId":73735399,"attachmentType":"pdf","work_url":"https://www.academia.edu/60178322/Equations_Functions_Critical_Aspects_and_Mathematical_Communication","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/60178322/Equations_Functions_Critical_Aspects_and_Mathematical_Communication"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="8" data-entity-id="51215796" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/51215796/Student_perspectives_on_equation_The_transition_from_school_to_university">Student perspectives on equation: The transition from school to university</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31424416" href="https://auckland.academia.edu/MikeThomas">Mike Thomas</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematics Education Research Journal, 2008</p><p class="ds2-5-body-sm ds-related-work--abstract ">It has been recognised that many student perspectives on equations and their use of the equals sign have not mirrored those that mathematicians would like to see in tertiary students. This paper tracks transition of understanding of the equals sign by comparing secondary school students' thinking with that of first year university students. We analyse the understanding displayed in terms of properties of the constituent parts of equations, identifying a number of incomplete or pseudoconceptions that are sometimes influenced by representational aspects of the properties, and other times by apparent over-generalisation of a property. A start is made on constructing a framework for understanding of the mathematical equation object that could assist in the transition from school to tertiary mathematics study.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Student perspectives on equation: The transition from school to university","attachmentId":69031574,"attachmentType":"pdf","work_url":"https://www.academia.edu/51215796/Student_perspectives_on_equation_The_transition_from_school_to_university","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/51215796/Student_perspectives_on_equation_The_transition_from_school_to_university"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card js-safe-related-work-wsj " data-collection-position="9" data-entity-id="96497582" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/96497582/Equations_cognitive_biases">Equations, cognitive biases</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="229812687" href="https://independent.academia.edu/Jpl23">Jean Pierre Legros</a></div><p class="ds2-5-body-sm ds-related-work--abstract ">Math contains cognitive biases. To support this astonishing observation, I begin by going back in the history of mathematics. By erasing any intention within them, we have at the same time lost track of complexity and quality. These intentions exist, but are now hidden in acronyms, in particular the '=' with multiple meanings. The biases are those of cognitive stages arbitrarily making choices between these meanings, mistakenly believing to access the real per se.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equations, cognitive biases","attachmentId":98380115,"attachmentType":"pdf","work_url":"https://www.academia.edu/96497582/Equations_cognitive_biases","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/96497582/Equations_cognitive_biases"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="safe-sticky-ctas--container js-loswp-sticky-ctas hidden"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":31141487,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":31141487,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_31141487" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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