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Using History of Mathematics to Teach Volume Formula of Frustum Pyramids: Dissection Method Universal Journal of Educational Research 3(12): 1034-1048, 2015 http://www.hrpub.org DOI: 10.13189/ujer.2015.031213 Using History of Mathematics to Teach Volume Formula of Frustum Pyramids: Dissection Method Suphi Onder Butuner Department of Elementary Mathematics Education, Bozok University, Turkey Copyright © 2015 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Within recent years, history of mathematics dynamic structure of mathematics [1, 2, 3, 4, 5, 6, 7]. It is (HoM) has become an increasingly popular topic. Studies stressed that history of mathematics will enrich the repertoire have shown that student reactions to it depend on the ways of teachers and develop their pedagogical content knowledge they use history of mathematics. The present action research [7, 8]. Jankvist [5] (2009) explains the use of history of study aimed to make students deduce volume rules of mathematics both as a tool and as a goal. History-as-a-tool frustum pyramids using the dissection method. Participants arguments are mainly concerned with learning of the inner were 24 grade eight students from Trabzon. Observations, issues of mathematics (mathematical concepts, theories, informal interviews and feedback forms were used as data methods, formulas), whereas history-as-a-goal arguments collection tools. Worksheets were distributed to students and are concerned with the use of history as a self-contained goal. the research was conducted in 3 class hours. Student views From the history-as-a-goal point of view, knowing about the on the activities were obtained through a written feedback history of mathematics is not a primary tool for learning form consisting of 4 questions. During the study, most mathematics better. In using history as a goal, the learning of students could assemble the parts forming frustum pyramids developmental and evolutionary aspects of mathematics and find out volume rules of frustum square pyramids and either serves as a goal in itself or serves to illustrate other frustum rectangular pyramids calculating the volumes of the historical aspects of the discipline. geometric shapes obtained through that assembly. Several Jankvist [5] (2009) suggested three different approaches students had difficulties in recognizing geometric shapes as for the use of HoM in classrooms: “illumination”, “modules” well as completing the calculations. The findings revealed and “history-based” approaches. Within the illumination that, prior to the study, most students were not able to approach, the current curriculum is not changed but complete similar activities and the mathematics course was historical content is included. However, this can occur at taught in a teacher-centered way. Students found the different levels. Inclusion at a lower level may for instance activities interesting as well as instructing. include use of historical snippets, as mentioned by Tzanakis and Arcavi [7] (2002). The names of mathematicians, the Keywords History of Mathematics, Dissection Method, periods they lived in, their studies, a historical film, life Pyramid, Volume, Action Research stories, renowned problems, anecdotes and copies of historical studies may all be regarded as historical snippets [7]. Swetz [9] (1997), Fried [3] (2001), Baki and Bütüner [10] (2013) argued that teacher emphasis on life stories of mathematicians, their studies and contributions to 1. Introduction mathematics would barely make students move beyond the Within recent years, history of mathematics (HoM) has role of listener in the learning process. Jankvist [5] (2009) become an increasingly popular topic. Previous studies have explained his modules approach as having course units shown that student reactions to it are determined by how they including historical content. The historical based approach, use the history of mathematics. Therefore, before using on the other hand, includes units directly inspired by or based historical content in classes, the aim and ways of using such on the development and history of mathematics. Unlike the content should be questioned. When researchers discuss their modules approach, the historical approach does not deal with reasons for using history of mathematics in the classroom, the study of the history of mathematics in a direct manner, they mostly state that history of mathematics will reduce but rather in an indirect fashion. Here, the historical student anxiety about mathematics, increase their motivation development is not necessarily discussed explicitly. When and attitudes towards learning, help them learn the subjects explaining the development of number sets, the and concepts and reveal the multicultural, developing and developmental sequence should be followed starting from Universal Journal of Educational Research 3(12): 1034-1048, 2015 1035 natural numbers. required knowledge and experience of using history of The use of HoM in learning environments was mentioned mathematics. Teachers equipped with necessary knowledge by Gulikers and Blom [4] (2001) under the title of “general and experience may keep students’ attention through integration”. The first use explained here involves providing including appropriate historical content in their classes. Ho historical knowledge about a certain period or a historical [14] (2008) and Panasuk and Horton [15] (2012) determined matter, or teaching a concept together with historical content that teachers’ lack of knowledge and experience is one of the in order to introduce the method. Gulikers and Blom pointed greatest challenges before using history of mathematics in out that secondary resources might be used in order to give classes. Fried [3] (2001) gave the example of “geometrical information about a subject or a concept. They explained the algebra” and tried to explain that the way mathematics second use as teaching a concept or method together with educators use HoM while teaching modern mathematics and historical content, and such as utilizing problems taken from modern mathematical techniques is different from that of original sources on mathematics. According to the mathematics historians. statement was explained by researchers, studying primary sources enable students to giving the example that the area of a square whose side discuss alternative solutions [4]. The International length is x is equal to the area of a rectangle whose side Commission on Mathematical Instruction (ICMI) provided lengths are 6 and x. Girolamo Cardano included such special methods and examples as well as general approaches geometric discussions in his book Ars Magna in order to towards the use of HoM. Short historical passages, projects justify his algebraic techniques. These facts are compulsory based on historical texts, primary sources, worksheets, to build a good knowledge of history. However, the aim here historical packages, mistakes, alternative concepts, historical is to benefit from the technique of completing the square in problems, mechanical tools, experimental mathematics order to teach students the subject rather than history. activities, games, movies and other visual elements, trips to Studies conducted in this field have shown that the historical sites and the Internet were given as examples of reactions of students are based on the ways they use the using HoM [7]. history of mathematics. Glaubitz [16] (2007) included Fried [3] (2001) mentioned two strategies about the use of historical content in his study using primary sources. The HoM. The first, the addition strategy, includes activities such students included in the experimental group examined as using historical anecdotes, brief stories from the lives of al-Khwarizmi’s book entitled “Al Kitab Fi Hisab Al Cabr mathematicians, historical problems or showing pictures of wal Muqabalah” and studied the “technique of completing mathematicians. Through this approach, the current the square”. They examined verbal and geometrical solutions curriculum is not changed but its scope is extended. The of al-Khwarizmi and discussed his solutions in small groups. other approach is the adaptation strategy. In this one, the Afterwards, they tried to solve second degree equations teaching process is adapted to a historical schema, as in the using al-Khwarizmi’s way. This gave them an opportunity to adaptation of Katz and Napier’s logarithms to contemporary discuss the advantages and disadvantages of al-Khwarizmi’s classes. Fried stated that the implementation of both of these solutions and compare them to modern ones. The study led to may result in specific problems such as trivializing history a discussion about negative numbers, too. As for the students (whig approach) by misrepresenting time and date. In order in the control group, they spent six class hours doing regular to cope with these challenges, radical adaptation or radical exercises on second degree equations. At the end of the accommodation strategies were recommended. Through process, experimental students were not only more radical adaptation, students learn mathematics studying successful than control students but they had also enjoyed primary sources with information both on mathematics and the practices more. its history. Nooney [11] (2002) expressed that students will Nataraj and Thomas [17] (2009) used historical content in follow HoM content which is parallel to and in accordance their study in order to make students comprehend the with the mathematics
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