This paper discusses a part of a doctoral study based on the theoretical pillars of the Anthropological Theory of the Didactic (ATD). Supported by this theory, our research interest followed a path that led us to the teaching of the mathematical object vector and its institutional configuration in a Mathematics Teaching course at Universidade Estadual de Feira de Santana (UEFS), in the state of Bahia, Brazil, centered on two dichotomous aspects. The first one refers to the possibility of application and institutional relevance of this mathematical object. The second aspect is related to the obstacles encountered in the didactical scope, which have impact on the teaching/learning of this mathematical object. In this discussion, the question that guided our investigation emerged: how to harness praxeological recombinations that can promote mediation between personal and institutional relationships in the scope of vector knowledge in the Mathematics Teaching course at UEFS? Based on this context, we aim at analyzing students’ productions in terms of solution paths to a vector task in a Hybrid Learning Environment (HLE), from a developmental perspective based on the T4TEL didactical framework (Chaachoua, 2018). With regard to methodological support, Didactical Engineering provided directions that enabled us to make a comparison between a priori and a posteriori analyses, resulting in the identification of the reach of the techniques that were developed in contrast to the naturalized techniques that are part of this context.
{"title":"An Analysis of the Influences of a Hybrid Learning Environment in the Solution of Vector Tasks according to the Anthropological Theory of the Didactic (ATD)","authors":"Jany Santos Souza Goulart, Luiz Márcio Santos Farias, Hamid Chaachoua","doi":"10.54870/1551-3440.1540","DOIUrl":"https://doi.org/10.54870/1551-3440.1540","url":null,"abstract":"This paper discusses a part of a doctoral study based on the theoretical pillars of the Anthropological Theory of the Didactic (ATD). Supported by this theory, our research interest followed a path that led us to the teaching of the mathematical object vector and its institutional configuration in a Mathematics Teaching course at Universidade Estadual de Feira de Santana (UEFS), in the state of Bahia, Brazil, centered on two dichotomous aspects. The first one refers to the possibility of application and institutional relevance of this mathematical object. The second aspect is related to the obstacles encountered in the didactical scope, which have impact on the teaching/learning of this mathematical object. In this discussion, the question that guided our investigation emerged: how to harness praxeological recombinations that can promote mediation between personal and institutional relationships in the scope of vector knowledge in the Mathematics Teaching course at UEFS? Based on this context, we aim at analyzing students’ productions in terms of solution paths to a vector task in a Hybrid Learning Environment (HLE), from a developmental perspective based on the T4TEL didactical framework (Chaachoua, 2018). With regard to methodological support, Didactical Engineering provided directions that enabled us to make a comparison between a priori and a posteriori analyses, resulting in the identification of the reach of the techniques that were developed in contrast to the naturalized techniques that are part of this context.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47138985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper explores Archimedes’ works in conoids, which are three dimensional versions of conic sections, and will discuss ideas that came up in Archimedes’ book On Conoids and Spheroids. In particular, paraboloids, or three dimensional parabolas, will be the primary focus, and a proof of one of the propositions is provided for a clearer understanding of how Archimedes proved many of his propositions. His main method is called method of exhaustion, with results justified by double contradiction. This paper will compare the ideas and problems brought up in On Conoids and Spheroids and how they relate to modern day calculus. This paper will also look into some basic details on the method of exhaustion and how it allowed the ancient Greek mathematicians to prove propositions without any knowledge of calculus. In addition, this paper will discuss some mathematical contributions made by Arabic mathematicians such as Ibn alHaytham and how his work connects to mathematics in the seventeenth Century regarding sums of powers of whole numbers and the Basel Problem. Complicated forms of conoids such as hyperbolic paraboloids and other shapes that came after Archimedes will not be covered.
{"title":"Archimedes’ Works in Conoids as a Basis for the Development of Mathematics","authors":"Kenton Ke","doi":"10.54870/1551-3440.1519","DOIUrl":"https://doi.org/10.54870/1551-3440.1519","url":null,"abstract":"This paper explores Archimedes’ works in conoids, which are three dimensional versions of conic sections, and will discuss ideas that came up in Archimedes’ book On Conoids and Spheroids. In particular, paraboloids, or three dimensional parabolas, will be the primary focus, and a proof of one of the propositions is provided for a clearer understanding of how Archimedes proved many of his propositions. His main method is called method of exhaustion, with results justified by double contradiction. This paper will compare the ideas and problems brought up in On Conoids and Spheroids and how they relate to modern day calculus. This paper will also look into some basic details on the method of exhaustion and how it allowed the ancient Greek mathematicians to prove propositions without any knowledge of calculus. In addition, this paper will discuss some mathematical contributions made by Arabic mathematicians such as Ibn alHaytham and how his work connects to mathematics in the seventeenth Century regarding sums of powers of whole numbers and the Basel Problem. Complicated forms of conoids such as hyperbolic paraboloids and other shapes that came after Archimedes will not be covered.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41974707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This second part of two related articles reports the answers to the research question: What precise supports must be in place for Grades 5 to 12 teachers to enhance their mathematics classes in a sustainable way with Indigenous mathematizing and Indigenous worldview perspectives? In addition to various logistical supports, two other types of supports were identified: supports for learning and unlearning ways of perceiving the world generally and perceiving Western mathematics specifically. These needed supports came to light when we mentored the teachers. On the one hand, the co-researching teachers learned, or had already learned: (a) the plurality of mathematical systems; (b) the perspective of Western mathematics as a human endeavor along with its values, ideologies, and definitions; (c) the mere inclusion of Indigenous mathematizing in a lesson is not enough; and (d) the goal of two-eyed seeing. On the other hand, the co-researching teachers unlearned, or had already unlearned: (a) pure mathematics’ claim to be value-free, (b) all students have a predilection to excel at mathematics, and (c) subtle appropriation committed by many mathematics educators as if it were common sense to do it.
{"title":"Indigenous Culture-Based School Mathematics in Action Part II: The Study’s Results: What Support Do Teachers Need?","authors":"Sharon Meyer, G. Aikenhead","doi":"10.54870/1551-3440.1517","DOIUrl":"https://doi.org/10.54870/1551-3440.1517","url":null,"abstract":"This second part of two related articles reports the answers to the research question: What precise supports must be in place for Grades 5 to 12 teachers to enhance their mathematics classes in a sustainable way with Indigenous mathematizing and Indigenous worldview perspectives? In addition to various logistical supports, two other types of supports were identified: supports for learning and unlearning ways of perceiving the world generally and perceiving Western mathematics specifically. These needed supports came to light when we mentored the teachers. On the one hand, the co-researching teachers learned, or had already learned: (a) the plurality of mathematical systems; (b) the perspective of Western mathematics as a human endeavor along with its values, ideologies, and definitions; (c) the mere inclusion of Indigenous mathematizing in a lesson is not enough; and (d) the goal of two-eyed seeing. On the other hand, the co-researching teachers unlearned, or had already unlearned: (a) pure mathematics’ claim to be value-free, (b) all students have a predilection to excel at mathematics, and (c) subtle appropriation committed by many mathematics educators as if it were common sense to do it.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45249724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Matsuura, O. Hall-Holt, Nancy Dennis, Michelle Martin, Sarah Sword
{"title":"The Suuji Approach to Multi-Digit Addition: Using Length to Deepen Students’ Understanding of the Base 10 Number System","authors":"R. Matsuura, O. Hall-Holt, Nancy Dennis, Michelle Martin, Sarah Sword","doi":"10.54870/1551-3440.1526","DOIUrl":"https://doi.org/10.54870/1551-3440.1526","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46550539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Impediments to mathematical creativity: Fixation and flexibility in proof validation","authors":"P. Haavold","doi":"10.54870/1551-3440.1518","DOIUrl":"https://doi.org/10.54870/1551-3440.1518","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46871158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We quantify flattening the curve under the assumption of a soft quarantine in the spread of a contagious viral disease in a society. In particular, the maximum daily infection rate is expected to drop by twice the percentage drop in the virus reproduction number. The same percentage drop is expected for the maximum daily hospitalization or fatality rate. A formula for the expected maximum daily fatality rate is given.
{"title":"Flattening the Curve","authors":"P. Zizler, Mandana Sobhanzadeh","doi":"10.54870/1551-3440.1527","DOIUrl":"https://doi.org/10.54870/1551-3440.1527","url":null,"abstract":"We quantify flattening the curve under the assumption of a soft quarantine in the spread of a contagious viral disease in a society. In particular, the maximum daily infection rate is expected to drop by twice the percentage drop in the virus reproduction number. The same percentage drop is expected for the maximum daily hospitalization or fatality rate. A formula for the expected maximum daily fatality rate is given.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43049432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: The paper addresses the time-old question of what is beauty. A rather ambitious, if not to say presumptuous, endeavour. But I do not aim high – I do not claim to get anywhere near unearthing the secret. Rather, I will use examples from mathematics, poetry, music and chess to substantiate one thesis: that the elusory character of beauty is not incidental. Its defiance of definition is part of its essence. The aesthetic sensation requires unawareness of its precise origin. Beauty is felt when some order is perceived, that is not fully comprehended. The order is too complex, or well hidden, or too novel, to surface in its entirety. This is the reason for our ability to enjoy a piece of art for the hundredth time – we never fully fathom its inner order. This is also the reason for the feeling of awe that the beauty inspires: mystery and magic are at its heart. I will compare mathematical techniques and features with those of poetry - like compression, summoning patterns from one field to solve problems in another, or self-reference, and show how beauty is generated in the two domains in a similar way. I will also comment on the beauty-generating effect of unexpectedness in both domains. That novelty generates beauty is a trite observation (“the most expected feature of a poem is its unexpectedness”, as somebody put it), but the question why this is so is not often addressed – I will connect it with the “blind spot” idea. In a final section I try to answer the question that is at least as difficult as “what is beauty” – “why beauty?”. The fact that it pervades our lives indicates that it has an important role – what is it? To arouse the reader’s curiosity, let me summarize the attempted answer in one word – ‘change’. That aim that is so coveted and so hard to achieve – a change in the pattern of our actions, aims and perceptions. The style of the paper is non-scientific, and non-erudite, reflecting my belief that scientific pretensions in the humanities deflect from “softer”, more genuine, understanding.
{"title":"Beauty is in the blind spot of the beholder","authors":"Ron Aharoni","doi":"10.54870/1551-3440.1522","DOIUrl":"https://doi.org/10.54870/1551-3440.1522","url":null,"abstract":": The paper addresses the time-old question of what is beauty. A rather ambitious, if not to say presumptuous, endeavour. But I do not aim high – I do not claim to get anywhere near unearthing the secret. Rather, I will use examples from mathematics, poetry, music and chess to substantiate one thesis: that the elusory character of beauty is not incidental. Its defiance of definition is part of its essence. The aesthetic sensation requires unawareness of its precise origin. Beauty is felt when some order is perceived, that is not fully comprehended. The order is too complex, or well hidden, or too novel, to surface in its entirety. This is the reason for our ability to enjoy a piece of art for the hundredth time – we never fully fathom its inner order. This is also the reason for the feeling of awe that the beauty inspires: mystery and magic are at its heart. I will compare mathematical techniques and features with those of poetry - like compression, summoning patterns from one field to solve problems in another, or self-reference, and show how beauty is generated in the two domains in a similar way. I will also comment on the beauty-generating effect of unexpectedness in both domains. That novelty generates beauty is a trite observation (“the most expected feature of a poem is its unexpectedness”, as somebody put it), but the question why this is so is not often addressed – I will connect it with the “blind spot” idea. In a final section I try to answer the question that is at least as difficult as “what is beauty” – “why beauty?”. The fact that it pervades our lives indicates that it has an important role – what is it? To arouse the reader’s curiosity, let me summarize the attempted answer in one word – ‘change’. That aim that is so coveted and so hard to achieve – a change in the pattern of our actions, aims and perceptions. The style of the paper is non-scientific, and non-erudite, reflecting my belief that scientific pretensions in the humanities deflect from “softer”, more genuine, understanding.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49017886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: In this paper a formal justification of the ancient Chinese method for computing square roots is given. As a result, some already known properties of the square root which is computed with this method are deduced. If any other number base is used, the justification given shows that the method is applied in the same way and that the deduced properties are still being fulfilled, facts that highlight the importance of positional number systems. It also shows how to generalize the method to compute high orders roots. Although with this elementary method you can compute the square root of any real number, with the exact number of decimal places that you want, the mathematicians of ancient China were not able to generalize it for the purpose of computing irrational roots, because they did not know a positional number system. Finally, in order for high school students gain a better understanding of number systems, the examples given in this paper show how they can use the square root calculus with this method to practice elementary operations with positional number systems with different bases, and also to explore some relationships between them.
{"title":"A formal justification of the Ancient Chinese Method of Computing Square Roots","authors":"Edilberto Nájera, Leslie Cristina Najera-Benitez","doi":"10.54870/1551-3440.1513","DOIUrl":"https://doi.org/10.54870/1551-3440.1513","url":null,"abstract":": In this paper a formal justification of the ancient Chinese method for computing square roots is given. As a result, some already known properties of the square root which is computed with this method are deduced. If any other number base is used, the justification given shows that the method is applied in the same way and that the deduced properties are still being fulfilled, facts that highlight the importance of positional number systems. It also shows how to generalize the method to compute high orders roots. Although with this elementary method you can compute the square root of any real number, with the exact number of decimal places that you want, the mathematicians of ancient China were not able to generalize it for the purpose of computing irrational roots, because they did not know a positional number system. Finally, in order for high school students gain a better understanding of number systems, the examples given in this paper show how they can use the square root calculus with this method to practice elementary operations with positional number systems with different bases, and also to explore some relationships between them.","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41580877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Using Records of Practice to Bridge from Teachers’ Mathematical Problem Solving to Classroom Practice","authors":"D. Fischman, I. Riggs","doi":"10.54870/1551-3440.1523","DOIUrl":"https://doi.org/10.54870/1551-3440.1523","url":null,"abstract":"","PeriodicalId":44703,"journal":{"name":"Mathematics Enthusiast","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41592848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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