History in Mathematics Education

We aimed to describe how history of mathematics was framed in teacher education programs to be used by pre-service elementary mathematics teachers in their future teaching within the context of a specific undergraduate course named “History of Mathematics”. Data was collected by the observation of the course setting, interviews with the pre-service elementary mathematics teachers enrolled in the course and the course instructor, and the course materials in the fall semester of 2011-2012 academic year. Considering Jankvist’s (2009) classification, findings revealed that the course fostered the pre-service teachers’ history as a goal ideas rather than history as a tool ideas. The illumination approaches were seemed to be the only way of employing history of mathematics in their teaching. We concluded with discussions on the effectiveness of the course and implications for improving such “History of Mathematics” courses.

Due to the 20th century mathematical and scientific developments of Georg Cantor, Max Karl Planck, Albert Einstein, and Werner Heisenberg, concepts once relegated to obscurity, such as irrationality, infinity, insolvability, and chaos, were brought to mainstream attention, ultimately changing the course of technological and scientific development into the 21st century. Before these seminal thinkers, concepts like numerical irrationality and infinity were considered by many to be worthless if not amoral; such attitudes can be found persisting back to the ancient Greeks under the Pythagoreans. Interestingly, the aesthetic of irrationality follows a similar historical trajectory, mostly finding relegation in peripheral movements and specific artists before the 20th century. However, the 20th century has seen the greatest and longest persisting resurgence in mathematically irrational thought within the arts. This paper compares the visual and musical experiments in irrationality, incommensurability, and infinity in the works of MC Escher and Conlon Nancarrow during the early and mid-twentieth century, showing a correlation between contemporary mathematical and physical innovations and specific aesthetic pursuits in art and music.

The aim of this study was to investigate the roles of year in teacher education program and gender on prospective elementary mathematics teachers’ knowledge of history of mathematics and their attitudes and beliefs towards the use of history of mathematics in the teaching and learning of mathematics. Moreover, the relationship between prospective teachers’ knowledge of history of mathematics and their attitudes and beliefs about the history of mathematics usage was examined.
The data of the study were obtained from 1593 prospective teachers who were enrolled in first, second, third, and fourth years of Elementary Mathematics Education undergraduate program of nine universities located in seven geographical regions of Turkey through clustered random sampling. The scales used in the data collection were Knowledge of History of Mathematics (KHM) Test and Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education (ABHME) Questionnaire.
The two-way ANOVA results clarified that prospective teachers’ knowledge of history of mathematics improved as the years enrolled in the program increased. Results also revealed that males had significantly higher mean scores on KHM Test than females in the first two years of the program. In the third and fourth years, this situation reversed such that females had higher KHM mean scores, but this difference was not statistically significant.
Results also showed that prospective teachers’ ABHME mean scores increased as years of enrollment in the program increased. More clearly, senior prospective teachers’ relevant mean scores were significantly higher than that of freshmen and sophomores, and juniors’ attitudes and beliefs were significantly higher than that of freshmen. In addition, females’ ABHME mean scores were significantly higher than that of males for all years.
Lastly, a positive correlation between prospective elementary mathematics teachers’ KHM mean scores and ABHME mean scores was found through Pearson product-moment correlation analysis.

The purose of this study was to investigate the relationship between pre-service mathematics teachers' knowledge of history of mathematics and their attitudes and beliefs towards the use of history of mathematics in mathematics education. Data were obtained from 1593 pre-service elementary mathematics teachers during the fall semester of 2010-2011 academic year by Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education (ABHME) Questionnaire and Knowledge of History of Mathematics (KHM) Test. The main correlation between ABHME and KHM mean scores was found to be positive and statistically significant (r=.18, p<.01) which meant that the pre-service teachers who were more knowledgeable in history of mathematics topics had more positive attitudes and beliefs towards the use of history of mathematics in mathematics education. Furthermre, all of the seven correlations between the mean scores from each of the seven subtopics of ABHME Questionnaire and the original form of KHM Test were also positive and statistically significant at the .01 level, which were presented in details in the results. The findings were discussed with the relevant existing literature followed by implications for teacher education programs and policy makers, and suggestions for future research were addressed.

The proposal presented here relates to the integration of the mathematical syllabus content, including foundational issues, with selected aspects of history of mathematics in the education of prospective elementary teachers. An analysis of the needs and cultural profile of prospective elementary teachers in Italy was the original motivation of this study. The new syllabus has been applied for several years in Rome, with good results in mathematical learning and self-confidence regarding the performance as a future mathematics teacher. Students also show an ability to use history of mathematics in the classroom with good feedback from the children. The historical contents were used in Spain to offer a stronger mathematical background to a group of teachers and prospective teachers collaborating in an experimental workshop for children with Down´s Syndrome. Once again the result was improved appreciation of their own skills and greater autonomy and spirit of initiative.

This study intends developing a valid and reliable instrument in order to determine pre-service mathematics teachers' attitudes and beliefs towards using history of mathematics in school mathematics courses. The data were collected at the beginning of the fall semester of 2010-2011 academic year from 237 teacher candidates in Turkey via Attitudes and Beliefs towards the Use of History of Mathematics in Mathematics Education (ABHME) Questionnaire developed by the researchers. The reliability coefficient of the ABHME was calculated as .91 and explanatory factor analysis revealed three factors: positive attitudes and beliefs towards the use of history in mathematics education, negative attitudes and beliefs towards the use of history in mathematics education, and self-efficacy beliefs towards the use of history in mathematics education.

This study aimed to investigate the potential effects of teaching modules in history of mathematics in order to develop number sense in young children aged 5. We reanimated two kinds of historical artifacts as using tally marks and pebbles for counting activities within symbolic play context. Data were obtained by anectodal records during the implementation of the modules at the Fall of 2012013 academic year. Data analysis was made through Dunphy's framework for number sense in young children. Findings revealed that the modules could contribute to various aspects of number sense. We finalized with discussion on the findings and implications for further research on using history of mathematics with younger children.

This chapter contains a teaching proposal based on the history of matrices inspired by the conceptual and methodological framework introduced by Kjeldsen (2011) to integrate history into the teaching of mathematics. Kjeldsen’s conceptual framework is based on Sfard’s (2008) theory of thinking as communicating. Our goal is to create conflicting situations in which students are encouraged to reflect upon the metadiscursive rules related to matrices and determinants, comparing them with those found in some historical writings. Two teaching modules were created, dealing with two episodes in the history of matrices, based on the works of the mathematicians Sylvester and Cayley, and on the historical interpretation of Brechenmacher (2006). Two field studies were conducted with undergraduate mathematics students, from two universities in Rio de Janeiro. In this chapter we also explain how some historical metadiscursive rules were identified.

La escuela primaria apenas sí está documentada en las épocas anteriores al Renacimiento, y en el caso de las matemáticas el caso es todavía más difícil, puesto que se aprendían sobre el ábaco (sea este de arena o de fichas), que se borraba una vez la operación se había acabado, para volver a ser reutilizado. El De Computo de Rabano Mauro (ca. 780-856) es un manual para calcular la Pascua. Sin embargo, los primeros ocho capítulos ofrecen una buena perspectiva de lo que podría ser la escuela primaria en la época carolingia y cómo los niños aprendían a contar: primero, las diferentes clases de números (con un criterio de aproximación más gramatical que aritmético), seguido de las diferentes formas de escribir los números y el cálculo digital; finalmente el estudiante era introducido en el extraño mundo de las fracciones según fueron conocidas por los romanos. En este artículo se ofrece el texto latino de los ocho primeros capítulos junto con su correspondiente traducción al español.

Primary school is hardly documented in the ages before the Renaissance, and in the case of Mathematics the case is even more difficult since they were learned on the abacus (either the dust board or the counters abacus) that was deleted after the operation had been completed, to reuse it again. The "De Computo" of Rabanus Maurus (ca. 780-856) is a handbook of Easter reckoning. However the first eight chapters offer a good insight in the primary school and in the ways by which children would have been taught to count: the different sorts of numbers (in an approximation that is more grammatical than arithmetical), the different forms of writing them and finger counting; finally the student is introduced in the study of the bizarre world of fractions, such as they were known by the Romans. Here we offer the Latin text together with the translation into Spanish of these eight chapters.

Mebahis-i İlmiye (1867-1869) was the first periodical on mathematical sciences published in 19th century Ottoman Turkey. The authors primarily attempted to introduce the mathematics required for the financial, societal, educational and military development of the country. This paper analyzes the periodical in order to understand (i) the fundamental reasons for its publication, and (ii) transmission of contemporary mathematics from Europe to Ottoman Turkey. Findings unveiled that the periodical had various reasons for mathematics education in the Ottoman society of the time. Moreover, it served the transmission issue to a certain extent.

Статья (вторая и последняя в небольшой серии) посвящена анализу идей советского оппозиционера Е.М. Анцупова (Е. Аугдема). Он создал глобальную периодизацию, основанную на принципе «удвоения». В заключении статьи анализируются причины низкой популярности такого рода периодизаций, как концепции Гражданникова и Аугдема, по сравнению с теорий Фоменко-Носовского.
Article (second and last in a small series) is devoted to the analyses of ideas of Soviet dissident, E.M. Anzupov (E. Augdem). He created global periodization based on principle of “doubling”. In the conclusion author analyzed the reasons of small popularity such types of periodizations like a concepts of Grazhdannikov and Augdem comparing them with Fomenko-Nosovsky theory.
Опубликовано: Родина. 2015. № 1. С. 120-122.

The purpose of this study was to determine pre-service mathematics teachers’ teaching and learning beliefs and examine the relationship between their beliefs and practices. Qualitative and quantitative research methods were designed for this study. Survey, semi-structured interviews, observations, and pre-service teachers’ written documents such as school practice portfolios were used to collect the data. Under the developed theoretical framework, it was found that some of the pre-service mathematics teachers ’ beliefs were consistent with their practices; while some of them presented different practices from their beliefs.

The purpose of this chapter is to provide a broad view of the state of the field of history of mathematics in education, with an emphasis on mathematics teacher education. First, an overview of arguments that advocate for the use of history in mathematics education and descriptions of the role that history of mathematics has played in mathematics teacher education in the United States and elsewhere is given. Next, the chapter details several examples of empirical studies that were conducted with elementary (or, primary) and secondary mathematics prospective teachers. Finally, the chapter outlines examples of research from the “next generation” of infusing history in mathematics education, by providing accounts of practicing teachers who incorporated history of mathematics in teaching at the primary, secondary, and tertiary levels.

Among the elements of mathematics teachers’ instructional repertory, incorporating the history of mathematics into mathematics education stands out as an alternative. Jankvist (2009) organised the arguments on this incorporation as of ‘history as a tool’ for helping mathematics teaching and ‘history as a goal’ of teaching on its own. Either way, one should initially master the historical knowledge lying behind the mathematical concept before the incorporation. On the basis of this argument, this study aimed to develop a valid and reliable test assessing prospective elementary mathematics teachers’ (PEMT) history of mathematics knowledge within the borders of Turkish national elementary (K6-8) mathematics curriculum (Ministry of National Education [MNE], 2009). The test consisted of multiple-choice, short answer and true-false items which are totally 14 in number. Portraits of pioneer mathematicians, hieroglyphics from ancient cultures, old photographs and some other visual tools were utilized as forming the items. The validity of the test was ensured by the views of three experts, who are specialized in the field of elementary mathematics education, via the table of specifications composed for this test. The test was applied to 1593 Turkish PEMT (478 freshmen, 432 sophomores, 409 juniors, and 274 seniors) selected from each geographic region of Turkey by means of clustered random sampling. The reliability of the test was found as .56 by using KR-20 reliability coefficient. This low reliability which was reasonable for the study’s aim may stem from the minority of the number of items in the test or the characteristics of the participants (Pınarbaşı, Canpolat, Bayrakçeken, & Geban, 2006). The item quantity could not be excessive since the national curriculum (MNE, 2009) gave limited place to the history of mathematics. Moreover, the problem originated from the sample characteristics is mostly that the PEMT’s total scores were close to each other which lower the variability. It is believed that this instrument could be used to assess individuals’ knowledge structures related to the history of mathematics.