History of Mathematics Education in Relation to Other
The discipline of mathematics education, encompassing history, mathematics, and education, is defined by its methodological and conceptual frameworks As a historical discipline, it utilizes historical inquiry methods to explore the evolution of mathematics education within the broader context of social history The influence of society on the development of mathematics education is evident through various manifestations.
Mathematics education is influenced by the same social factors that shape general education, meaning that if education is segregated, mathematics education will reflect that segregation Additionally, if societal values prioritize humanities education as secondary, this perspective will also impact mathematics education The development of education is driven by the labor market and the prevailing beliefs in society, along with the objectives set by the state, which can vary significantly in autocratic regimes compared to public desires.
Mathematics is influenced by social factors, but it's important to avoid oversimplifying this relationship As noted by Soviet mathematician Elena Venttsel, there were instances in the early 1930s where social ideologies infiltrated mathematical teachings, exemplified by a professor who categorized integrals as either pro-communist (red) or anti-communist (white).
Interest in specific areas of mathematics fluctuates in response to societal needs, including economic shifts, highlighting the historical significance of arithmetic's development.
A Karp and F Furinghetti, History of Mathematics Teaching and Learning,
3 the 16th century has long been associated with the rise of the bourgeois class
(Weber 2003) Developments in mathematics in turn influence mathematics edu- cation so that it too becomes subject to the same social factors.
Mathematics education documents frequently utilize mathematical language, requiring historians in the field to be proficient in this terminology to effectively analyze and interpret developments Understanding these developments is crucial for explaining their social and educational significance The interplay between mathematics and education is essential to the discipline of mathematics education history, highlighting the importance of both fields in our analysis.
The history of mathematics closely aligns with our field of study, and even as it progresses, it reveals developments that significantly impact education Additionally, the history of science offers valuable methodological insights; for instance, Schubring (2014a) emphasizes a research tool introduced by Shapin and Thackray (1974) that focuses on the collective biographies of influential groups, highlighting its relevance to the discipline.
The history of education plays a crucial role in understanding the evolution of mathematics education, revealing general patterns that influence its development In some instances, this historical context is so significant that it defines the narrative of mathematics education itself A stark example is the complete lack of educational access for children, which reduces the history of mathematics education to the absence of any educational opportunities in mathematics.
A historian of mathematics education should engage with current research in the field while being cautious not to impose modern questions onto historical contexts Embracing Schlegel's idea of the historian as a "prophet facing backwards," it's essential to trace contemporary challenges to their historical roots For instance, Kidwell et al (2008) illustrate how present-day perspectives on technology's role can be informed by historical analysis, highlighting the value of this approach in understanding the evolution of mathematics education.
The history of mathematics education is a complex field that extends beyond the sum of its individual disciplines It possesses unique characteristics that set it apart from the histories of other subjects taught in primary and secondary schools Additionally, many current research approaches in mathematics education are not feasible within a traditional historical framework.
The interplay between the history of mathematics education and other disciplines is reciprocal, with mathematics borrowing from and contributing to various fields Historians can gain valuable insights by studying mathematics textbooks, just as they do from artifacts or historical correspondence The evolution of a society is mirrored in multiple domains, and for significant periods, mathematics education has been regarded as a crucial aspect of this reflection.
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Mathematics education stands to gain significantly from exploring its historical context, as it serves as a repository of past successes, challenges, strategies, and outcomes While it may be somewhat naive to seek direct solutions from history for contemporary issues, examining historical trends offers valuable insights into the underlying causes of current challenges This exploration not only enriches our understanding but also provides practical solutions that can be applied to modern mathematics education.
Methodology
The methodology of historical research is often viewed as a collection of strategies and techniques, as highlighted in Karp's Handbook (2014a) Familiarity with specific methods, such as those used in oral history (Karp 2014b), is beneficial, particularly in conducting interviews without bias However, historians of mathematics education primarily utilize historical methods, which generally exempt them from the complexities of statistical analysis Additionally, they face fewer challenges than historians of other periods, like the Middle Ages, due to the clear authorship and substantial editions of textbooks from as early as the 18th century, ensuring reliable source attribution.
When exploring the history of mathematics education, understanding the nature of historical sources is crucial Almost anything can qualify as a primary source, including textbooks, official memos advocating for curriculum changes, personal correspondence between students discussing their experiences with a new math teacher, or even novels that portray the struggles of students facing academic challenges These diverse materials allow for a comprehensive interpretation of mathematics education in relation to broader human activities.
Scholars should aim to contextualize their specific area of study within a wider framework Schubring (1987) draws a parallel between the methodologies employed by historians of mathematics education and those used since the 18th century in the analysis of Ancient Greek poetry, suggesting that this comparison can enhance our understanding of both fields.
The study of mathematics education history necessitates an exploration of Greek politics and economics, as these factors influence the context of examination problems (Karp 2007a) To effectively analyze these examinations, it is crucial to understand their role, execution, and perception Consequently, our research may involve examining documents that do not explicitly reference mathematics, highlighting the importance of analyzing and interpreting primary sources in this field.
If a scholar should look into the current Russian textbook by Atanasyan et al.
In 2004, a researcher would be impressed by the extensive and intricate chapter on isometries in a widely used textbook Future scholars would easily recognize the textbook's existence and its classroom application, as evidenced by its numerous printed copies and references in various publications However, it would be erroneous to assume that the depth of this chapter reflects the overall preparation of Russian students, as there is no concrete evidence that this specific chapter is actually taught in classes.
Contemporary scholarship differentiates between intended and enacted curricula, a distinction that holds significance across historical periods While examining unused textbooks or fantasy curricula can be intriguing, it is essential to differentiate these from actual historical contexts Historians must validate textbook content with supplementary evidence, including syllabi, student and teacher recollections, teacher editions, cyphering books, tests, and final examinations.
Historians employ juxtaposition and comparison as their primary strategy, seeking to contextualize individual events within a broader sequence of occurrences By doing so, they interpret these episodes as integral components of the general historical landscape.
In this way evidence presented by a primary source is both corroborated and generalized.
History provides valuable insights into the processes and mechanisms that influence the development of mathematics, offering educators a deeper understanding of how information is generalized By exploring these historical contexts, mathematics teachers can embrace the concept of generalization without hesitation, recognizing its significance in the evolution of mathematical thought.
The origins of such fear and the tendency to regard every generalization as a
In the past century, historians have often made sweeping generalizations to support accepted theories, relying on established authorities while selectively presenting or even fabricating facts This approach is unacceptable, yet the humanities sometimes impose a taboo on exploring trends and patterns, which is equally problematic (Wong 2011).
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The history of mathematics education is closely linked to other disciplines, benefiting from general methodological approaches in the history of sciences However, it also presents unique methodological challenges that require attention Recent studies, such as Hansen's (2009a) analysis of mathematics education development in Denmark, Zuccheri and Zudini's (2010) exploration of academic research challenges in the field, and Prytz's (2013) application of sociological strategies, highlight these issues Additionally, Howson's interview (Karp 2014b, pp 69–86) provides valuable insights into methodological questions, while significant observations on research methodology in the history of mathematics and mathematics education can be found in various scholarly works.
Research methodology is an essential focus area that warrants deeper exploration Analyzing individual research projects, regardless of their success, provides valuable insights for both novice scholars and seasoned professionals in the field Particularly noteworthy is the investigation into the rise of ideologically driven studies and the myths surrounding the history of mathematics education Karp (2014a) highlights several examples of these phenomena and the contexts in which they arose This research can be further enriched by incorporating materials from diverse countries and historical periods.
Curricula and Textbooks
Formation of National Curricula and Textbooks
and the In fl uence of Foreign Materials
Often, national textbooks and programs of study in mathematics appear only after a period of using foreign materials For a long time British textbooks were used in the
In the United States, as well as in Germany and France, textbooks have historically been utilized in Russia and other European countries Many nations that achieved independence in the 19th and 20th centuries still rely on foreign textbooks today The development of domestic textbooks is indicative of complex societal changes, including the acknowledgment of unique educational challenges, the establishment of a national textbook market, and the fostering of national pride that resists using materials from former colonial powers Additionally, the rise of local curricula and textbooks is often driven by the desire to create a national academic language, which supports the evolution of a distinct national identity.
The evolution of mathematics curricula in a nation reflects its historical context and is inherently linked to national identity This development is influenced by various socio-economic and ideological factors, highlighting the importance of addressing mathematics education at the national level.
[this approach is used in theHandbook(Karp and Schubring 2014a)] At the same time it is useful to compare processes taking place in different countries.
The transition to domestic textbooks and curricula can be a lengthy and complex process, often met with resistance from ardent patriots who view these materials as inferior Even after this transition is completed, domestic textbooks may only receive praise for their adherence to foreign prototypes, highlighting the ongoing challenges in educational reform.
The use of foreign texts in education can be viewed as controversial, with some considering it akin to treason (Karp 2006) However, there exists a middle ground where foreign textbooks are translated and adapted to align with a nation's cultural values (Yamamoto 2006) This adaptation process allows for a balance between global knowledge and local relevance (Karp 2012b).
National differences encompass regional variations, creating a complex dynamic between the two (Schubring 2009, 2012a) The recent push for standards-based education in the United States can be seen as an attempt to develop and solidify a national education program, reflecting a multifaceted, contradictory, and lengthy process (Kilpatrick 2014).
International initiatives in curriculum reform also take different forms in different countries Although the initiatives of the 1960s and 1970s are relatively recent
The influence and differences between historical developments across the Iron Curtain remain inadequately understood, particularly regarding how these factors interacted with one another (Kilpatrick 2012a).
Examining the interconnected issues surrounding national identity in mathematics education enhances our understanding of how this educational field influences and shapes that identity.
Curriculum Formation
Contrary to the common belief that school curricula are stable and unchanging, the subjects taught in schools today have evolved significantly in recent history A prime example of this evolution is finite mathematics, which was not included in secondary education until relatively recently.
Over the past 50 years, the acceptance of various mathematical subjects has evolved, with some traditional areas like geometry undergoing significant changes Historical study programs from the 18th century reveal that certain mathematical topics have faded from modern curricula Simultaneously, the content of school mathematics is transforming, indicating a shift even in long-established subjects like geometry.
Significant changes in mathematical presentation methods have emerged, even in traditionally conservative regions like England (Fujita and Jones, 2011) Euclidean proofs are increasingly being replaced by innovative demonstration techniques, which are now being further transformed by proofs grounded in coordinate or transformational geometry (Barbin and Menghini, 2014).
The landscape of education is undergoing a significant redistribution of subjects across elementary, secondary, and tertiary levels Calculus and trigonometry, once primarily college-level topics, are increasingly being incorporated into secondary school curricula at varying rates across different countries Concurrently, elementary school programs are beginning to include subjects like basic geometry that were traditionally taught in secondary schools This shift is driven by various factors, including advancements in mathematical knowledge, as seen during the educational reforms of the 1960s and 70s and the introduction of discrete mathematics in secondary education Additionally, social and technological changes have influenced the necessity of certain mathematical topics, highlighting a growing emphasis on practical problem-solving skills that align with evolving educational goals and societal needs.
Monitoring changes in seemingly unrelated subjects is crucial, as they often reflect a redistribution of concepts that can be linked back to mathematics.
Further research is essential to understand the ongoing developments in various countries and regions In many instances, our knowledge of these changes is limited, and even in cases where studies have been conducted, numerous details and mechanisms remain unclear.
Pedagogical Changes in Textbooks and Curricula
This section focuses on the evolution of pedagogical strategies in education, highlighting both technical and structural changes influenced by technological advancements Notably, diagrams have been repositioned within textbooks for better accessibility, while there is an increased emphasis on didactic principles and the careful selection of problems that align more effectively with the material being taught.
To be sure, these changes are associated with methodological advancements across the board The well-known textbook by Colburn (1821) was even titledAn
The Plan of Pestalozzi showcased the significant impact of innovative pedagogical approaches on mathematics instruction, as highlighted by Cohen (2003) This period also saw notable transformations in mathematics, including the introduction of new problem types and alterations in the sequence of how these problems were presented (Karp).
2015) In general, changes in the order of the presentation of topics and the emergence of new pedagogical strategies and methods are important subjects that deserve further study.
Changes in the Presentation of Speci fi c Topics
Recent studies have explored the presentation of specific topics in textbooks and syllabi, focusing on both mathematical and pedagogical changes These changes highlight the importance of how educational materials are structured and delivered to enhance learning outcomes.
(Barbin 2009, 2012; Bjarnadóttir 2007; Chevalarias 2014; Jones 2008; Menghini
2009; Van Sickle 2011) This is a useful approach.
Studying the history of a single topic serves as a strategic approach rather than merely a field of inquiry This focused study allows for the exploration of the interplay between various factors It's important to recognize that different methodologies related to the same topic can coexist rather than follow a linear progression, and changes can occur in either direction—where one approach may replace another and then revert back Key questions, such as the impact of foreign textbooks, curricula, and technological advancements, can be analyzed within this framework.
The survey of the current state of a specific topic reveals significant changes in mathematics education, particularly with the introduction of new topics not previously covered in curricula or textbooks While many subjects remain inadequately explored, existing topic-specific studies provide a foundation for broader generalizations and insights into evolving educational trends.
Specialized Curricula and Textbooks
Mathematics curricula and textbooks are designed to accommodate students with varying aptitudes and abilities For over 200 years, specialized teaching strategies have been developed for students with health issues, while advanced curricula for gifted students have been in place for at least 50 years However, tiered instruction is more prevalent and has a longer history than programs specifically aimed at gifted learners The concept of "specialized education" can apply to any student group that differs from the general population based on social characteristics.
(2012) examines the education of poor orphans].
How did specialized education originate? Where did its programs of study come from? How did they change over the years? What factors influenced these changes?
The impact of general education theories on the mathematics components of various programs likely varies by country, yet many of these questions remain unanswered or unavailable to a global audience.
Curricula and Evaluation
The history of evaluation is closely intertwined with the development of curriculum, highlighting their significant overlap Evaluation reflects the importance of specific curriculum components, revealing which elements are prioritized for assessment Furthermore, it encompasses the evolution of specialized organizations dedicated to the evaluation process, underscoring its multifaceted nature.
Tests and examinations vary in administration methods, including oral versus written formats and individual versus group settings, as well as in problem structure, such as full-solution versus short-answer questions These differences highlight variations in educational programs and external factors Despite this, only a limited number of studies have focused on examination strategies across different countries (Karp 2007a; Madaus et al 2003).
Politics, Ideology and Economics of Mathematics Education
Who Is Taught Mathematics?
Access to mathematics education is one of the most discussed topics of the day.
Moreover, it is considered a commonplace that at one time education was not universal, but was restricted by socio-economic, gender, race, and other factors.
Although it may seem that conditions have improved, the reality is much more complex than the notion that "it used to be bad, but now it's good." Restrictions varied significantly across countries, as did the methods of enforcement and the strategies employed to navigate these barriers.
Schubring (2012b) critiques the oversimplified view that equates elite education with pure mathematics and public education with applied mathematics, highlighting the complexity of these concepts He provides an overview of the transition towards "mathematics for all" and discusses models for integrating mathematics into general education However, there is a pressing need for more in-depth analysis of evolving perceptions regarding popular mathematics and its role as a subject for every cultured citizen.
Hansen (2009b) explores the transition to "mathematics for all" in a specific Danish county, while D’Enfert (2012a) examines the democratization of primary education in France during the latter half of the twentieth century In Eastern Europe, similar changes occurred, but they varied significantly in implementation and curricular content The ongoing effort to identify and analyze these diverse educational models is still in progress Additionally, these discussions intersect with specialized education questions, particularly regarding whether "mathematics for all" should be standardized or if distinct strategies should be developed to address the differences in educational needs.
The phrase "mathematics for all" is a relatively new concept, but focusing solely on its recent history would undermine the depth of studies in this field It is important to recognize that the evolution of mathematical education and accessibility has a rich and complex background that extends beyond contemporary developments.
A comprehensive study of racial inequality in mathematics should extend beyond the post-war period to include earlier events and additional sources, such as interviews (Walker 2009, 2014) While significant progress has been made in the last fifty years, it is crucial to recognize that educational change is a gradual process, and even the most radical advancements must be viewed within the broader historical context.
The struggle for girls' rights to access comprehensive mathematics education highlights ongoing issues of gender segregation in education, as noted by Schubring (2012b) While there is a lack of understanding regarding the end of this segregation, mathematics education presents unique challenges, often perceived as unsuitable for women Analyzing parallel curricula in male and female schools reveals important insights into what content was included or excluded and the reasons behind these choices Studies focused on women's mathematics education, such as Thanailaki (2009), emphasize the need for further research in this area.
Ideology, Economics and Mathematics Education
Bjarnadóttir (2012) highlights that the values and beliefs of Icelanders are evident in 18th and 19th-century textbooks, illustrating that a country's ideology is intrinsically woven into its educational materials This is not limited to word problems, which often reference external contexts; for instance, Soviet textbooks consistently depicted a prosperous economy with falling prices until the late 1980s, when rising costs marked a significant ideological shift Additionally, the selection of topics and the allocation of lesson time reflect societal values, as seen in the Soviet Union's focus on training engineers to support the nation's industrialization and militarization efforts (Karp 2014c).
The impact of ideology on mathematics education is profound, with religious doctrines playing a significant role in its evolution Following the Protestant Reformation, Europe witnessed significant advancements in mathematics education, notably through the establishment of Melanchthon’s gymnasium In response, Catholic educators founded Jesuit schools, which subsequently led to the development of educational institutions in Orthodox regions of Europe (Karp and Schubring 2014b) While it may be simplistic to assume that each religious group developed a unique approach to teaching specific subjects, the influence of these ideologies on educational practices is undeniable.
2.4 Politics, Ideology and Economics … 13 or that church (Koller 1990) Nevertheless, there can be no doubt that the church was very interested in questions concerning mathematics education: religious thinkers made statements endorsing or denouncing mathematics Even more importantly, the changes in attitudes towards mathematics echoed the processes of general rationalization and disenchantment of the world described by Weber (2003) that were taking place in light of the Reformation.
Ideological differences can significantly impact mathematics education, as resistance to changes in teaching methods may stem from deeper issues, including unconscious power struggles between conflicting groups.
Changes in mathematics education are influenced by both economic and ideological factors, and the processes involved are intricate As paraphrased from Marx, advancements like the windmill and steam engine shape attitudes toward mathematics study, but these effects are neither immediate nor straightforward The rise of new professions and economic growth will certainly impact mathematics education systems; however, this does not guarantee a rapid overhaul of curricula Alternative approaches to meet the evolving demands in mathematics training may exist, as highlighted in Howson (2011).
Economic growth and technological advancements significantly influence the evolution of mathematics education, exemplified by the changing role of textbooks since the advent of the printing press The economy not only drives the demand for mathematics education but also supports its ongoing development Further exploration of the impact of various economic factors on mathematics education across different countries is necessary, as much work remains in this area.
Mathematics education can be influenced by political factors, particularly in the context of economic development Even in weaker economies and non-democratic regimes, there can be strong mathematics education systems that support government functions, such as training bureaucrats and developing a political elite The contrast in mathematics education between democratic and totalitarian systems is a significant topic that warrants further exploration.
A key factor in this respect is that mathematics was often thought of as a military subject, necessary for the casting of cannons and raising of fortresses (Karp 2007c).
The demands of war have historically played a crucial role in advancing mathematics education across various nations, exemplified by the establishment of military academies in 17th century France.
Point Academy in the United States has been instrumental in fostering significant innovations in general education, as noted by Rickey (2001) Additionally, the evolution of mathematics education is heavily influenced by political needs.
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Mathematics Education as an Instrument of Political
Throughout history, various authoritarian regimes have targeted mathematics education as a key element of their reforms A notable example is Peter I's reforms in Russia during the late 17th and early 18th centuries, where he implemented changes influenced by Western texts and expertise However, the governmental and societal models he aimed to establish were fundamentally distinct from those in the West, yet designed to be competitive.
Modernization initiatives took place in a variety of other countries (Abdeljaouad
Reforms in various countries from 2011 to 2012 exhibit significant differences in their scope, objectives, and implementation, often reflecting direct borrowing of ideas from abroad For instance, while Russia's Petrine reforms attracted world-class scholars like Euler, they failed to enhance education for the majority lower classes, focusing instead on quickly training a government bureaucracy In contrast, the Meiji Restoration in Japan expanded educational access to a wider population, benefiting many more students Meanwhile, reforms in the Ottoman Empire were more limited, primarily aimed at creating a more effective military.
Reforms often encounter resistance, particularly from conservative ulema who lose their exclusive control over civil servant training, as noted by Abdeljaouad (2012) Understanding opposition within a totalitarian regime poses challenges, yet evidence of resistance to Petrine schools is evident through student defections rather than explicit statements.
2010) At the same time we believe that a more detailed analysis of popular response to educational reform as part of political struggle is both possible and desirable.
Mathematics Education in Developing Countries
The political histories of various African, Asian, and Latin American countries that experienced colonialism significantly influence their mathematics education systems This presents intricate challenges, as there is often limited knowledge regarding the state of mathematics education in these regions.
2.4 Politics, Ideology and Economics … 15 the pre-colonial period, even though it must have existed in one form or another.
In cases where traditional primary sources like textbooks are unavailable or non-existent, alternative methods must be employed This includes utilizing techniques from ethnography, such as reconstructing historical events through the analysis of surviving folkloric materials (D’Ambrosio 2014).
Mathematics education may have historically mirrored an apprentice model, similar to practices in Europe, focusing more on training practitioners than mathematicians Gathering evidence and materials to illustrate this development is a crucial endeavor.
Indeed, even subsequent periods have not been sufficiently understood.
Fictional literature often captures ironic scenes of colonial children learning geography, exemplified in Jules Verne’s "In Search of the Castaways," where the protagonists are astonished to observe how Englishmen educate the indigenous people.
In Australia, there is a noticeable lack of references to mathematics instruction, with the natives reportedly struggling with the subject The colonies saw the emergence of various educational institutions, including schools for colonizers that claimed to mirror European models, though their actual differences remain unclear Additionally, there were schools for the privileged local population and limited options for the general populace Throughout history, mixed educational forms have existed, but these have not been thoroughly researched Furthermore, the colonizers, including the British, French, and Dutch, implemented diverse educational policies.
The post-colonial period in Africa remains underexplored, particularly regarding the history of education reform influenced by international cooperation This reform has involved various countries and reflects their rivalries and conflicts Understanding these dynamics is crucial for grasping the unique challenges faced by African education and the recent advancements in mathematics education.
Recently we have seen a number of studies on the history of mathematics education in Latin America (Pitombeira 2014; Rosario et al 2014) Karp et al.
In 2014, a concise overview of the history of mathematics education in Africa was presented, highlighting key developments in the field Additionally, a bibliography detailing studies focused on African mathematics and mathematics education was previously published by Gerdes and Djebbar, providing valuable resources for further exploration.
2007) We can be sure all the same that plenty remains to be done in this respect.
Legislation Governing Mathematics Education
The essence of school life extends beyond the confines of legal regulations, as education law frequently reflects the aspirations and ideals of lawmakers rather than practical realities For instance, the legislation introduced by the Bolsheviks in 1918 illustrates how educational policies can be shaped by political agendas rather than genuine educational needs.
The study of historical legislation regarding mathematics education reveals significant political struggles and shifts in the subject's status compared to others For instance, Schubring (2012a) highlights that in Schleswig, the final grade for a Latin examination was multiplied by 4, while for mathematics, it was only multiplied by 2, indicating a lower status for mathematics This disparity, though better than that of other subjects, which received no multiplier, underscores the evolving perceptions of mathematics in educational guidelines across various regions and time periods.
Some countries have comprehensive documentation on legislative measures impacting mathematics education (Giacardi and Scoth 2014; Howson and Rogers 2014), while others still require further research In certain instances, detailed insights into commissions responsible for formulating recommendations are available, whereas in other cases, we must piece together their activities from letters and personal memories.
Legislation significantly impacts mathematics education, encompassing national and regional programs and standards, as well as laws addressing mathematics learning disabilities (Cunningham 2007) It is crucial to consider the debates and public opinions surrounding these legislative measures to fully understand their implications on the subject matter.
The "math wars" in California highlight the intensity of educational debates and the involvement of various stakeholders (Klein 2007) Historians must assess the significance of changes by identifying opposing forces and their objectives, as seen in California, while also recognizing that sometimes changes may appear purely technical, similar to the situation in Schleswig.
2.4.6 In fl uential Groups in Mathematics Education
Mathematics educators often highlight the ongoing conflict between mathematicians and mathematics educators, sometimes referring to this divide as involving "conservative mathematicians." Historically, this distinction was not as pronounced, as mathematics educators were once viewed as part of a broader group of professionals within the field.
In fl uential Groups in Mathematics Education
question; one expects the answer to be different in different countries).
Mathematicians emerged as a significant influence in regions where universities governed secondary schools, while their impact varied in areas without such oversight A comparative analysis of the interplay between secondary and tertiary mathematics could provide valuable insights into this relationship.
In any case, it is clear that academic mathematicians wield a certain influence, not only individually, as we will see presently, but as it were institutionally.
This influence may be at times construed as beneficial or harmful Kilpatrick
The New Math movement exemplifies the active involvement of mathematicians in curriculum reform, a trend also seen in early 20th-century Italy Significant international reforms in mathematics education during this period were largely driven by mathematicians like Felix Klein Their influence extended beyond curriculum development; for instance, in the Soviet Union, the government relied on mathematicians to review textbooks for accuracy.
(Karp 2010) For many decades, the International Congresses of Mathematicians had been the only venue in which educational problems were discussed at an international level (Furinghetti 2007).
Mathematicians were hardly the only group influencing the development of mathematics education Teachers and publicfigures also played an important role
In public and primary schools, the consensus among teachers is often lacking Prytz (2012) explores the perspectives of key educators, highlighting their varied opinions, modes of expression, and the social contexts that shape their influence on school mathematics This study reveals notable disparities in how teachers impact the teaching of mathematics in schools.
Individual differences tend to play a more significant role than institutional factors in mathematics education However, it is essential to analyze the diverse agendas promoted by various interest groups, as their influence can shape educational practices Historical contexts, such as the military's impact at certain times, also highlight the complexity of these influences in mathematics education.
Individuals and Organizations in Mathematics Education
Prominent and Less Prominent Figures in Mathematics
Authors of textbooks and other texts in mathematics education are surely the most popular subjects of individual studies (e.g., Ackerberg-Hastings 2009; Bjarnadóttir 2009; Donoghue 2008; Howson 2008; Karp 2012b; Pitombeira 2006; Schubring
In 1987, it became evident that textbook authors often produce additional writings, such as letters to colleagues and publishers, along with articles and lectures These materials allow us to reconstruct the historical context of their lives, revealing their methodological, philosophical, and political ideas.
Individuals involved in mathematics education will likely contribute written materials for the classroom during their careers David Eugene Smith stands out not only as a textbook author but also as a foundational figure in graduate mathematics education in the United States He was a trailblazer in fostering international collaboration in mathematics education and is recognized for his contributions as a historian and collector in the field.
In addition to authoring textbooks, leaders of the international movement in mathematics education, as highlighted by the International Commission on Mathematical Instruction (ICMI) Portrait Gallery, are often engaged in political initiatives, project development, and are recognized as exceptional educators.
Despite the extensive material available on prominent mathematicians collected over decades, much of this information remains limited to one country and language, hindering accessibility for the international community It is essential to explore ways to generalize biographical data across national borders Kilpatrick (2012a) highlighted New Math as an international phenomenon, prompting us to consider the commonalities shared by mathematicians of various nationalities who were active during the reform years, starting with Felix Klein and many others.
Mark Bashmakov noted that Andrey Kolmogorov likely did not directly borrow ideas from French reformers, as he was not particularly attentive to others Nevertheless, Kolmogorov was influenced by the French mathematics school, which shaped his thinking Exploring the lives of mathematicians from various nationalities who became educators reveals their inspirations and contributions to the field An example of this exploration is found in la Bastide-van Gemert's biography of Hans Freudenthal (2015).
Myths surrounding mathematicians in education suggest they either complicate the curriculum or are solely responsible for its preservation While some believe that without mathematicians, educational programs would be clearer, others argue that they play a crucial role in maintaining the integrity of mathematics education Despite these contrasting views, there is a lack of comprehensive research on the impact of mathematician-educators, with few studies, such as those on Felix Klein, providing insight into their contributions.
Foreigners represent another significant yet underserved category of influential figures For instance, Smid (2009) highlights the contributions of Russian émigré Tatyana Afanasyeva to Dutch education reform Examining the experiences of individuals from different countries can provide valuable insights into how they adapt their knowledge and skills to new environments.
Of the lives of so-called ordinary teachers we know far less than about the public
Mathematics educators play a crucial role in shaping mathematics education, often referred to as "quiet" teachers, whose influence is documented through official records, personal diaries, and recollections Understanding the preparation of these educators, including their training, social background, career trajectories, and school status, is essential The status of mathematics teachers is indicative of the perceived value of their subject, making it a compelling area of study Schubring (2014b) highlights the historical context by noting the phrase "mathematicus non est collega," which underscores the lower status of mathematics educators in 16th and 17th century European academies, a situation that varied across different eras and locations.
The professional lives of mathematics teachers were largely determined by what was demanded of them and how their work was monitored and evaluated Karp
In 2012, a comprehensive overview was provided on the diverse methods of supervision and evaluation for mathematics educators, highlighting the variations in these systems across different countries Understanding these differing approaches is essential for gaining a deeper insight into the broader field of mathematics education.
In the realm of mathematics education, collective biographies play a crucial role in understanding the expectations and experiences of teachers Notable studies include Bjarnadóttir's (2008) exploration of teacher expectations in Iceland and Karp's (2014d) examination of the conditions faced by mathematics teachers in Russia up to the 1830s Additionally, Kidwell et al offer valuable insights into the landscape of American mathematics educators, contributing to a broader understanding of the profession.
(2008) Furinghetti (2012) and Furinghetti and Giacardi (2012) offer a few biographies of school teachers.
Historical portraits of“ordinary”people engaged in a fairly common occupation
Mathematics educators encompass a diverse range of individuals, including those who engage in home visits to teach children essential skills like reading and counting While their contributions may be less recognized than those of prominent figures, they play a vital role in education This initiative is not only intriguing but also significant for fostering foundational learning in young learners.
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Organizations Devoted to Mathematics Education
The study of organizations provides valuable insights into the field of mathematics education, particularly through professional associations that have been established for over a century Among these, the International Commission on Mathematical Instruction (ICMI) stands out as a significant influence Research into ICMI's history reveals the deep divisions within the mathematics education community and their causes Consequently, the evolution of mathematics education over the past century can be effectively examined through the lens of ICMI's development.
Over the years, various international organizations have emerged alongside ICMI, some of which have been established for several decades (Hodgson and Rogers 2011, 2012) The influence of these organizations has varied across different countries, as illustrated by Giacardi's 2009 analysis of Italy's involvement with ICMI Despite the progress made, significant gaps remain in understanding this area For instance, in some nations, ICMI's initiatives spurred the formation of national organizations and conferences, while in others, such entities were already in existence This raises important questions about the differing perceptions of ICMI's activities between countries that established organizations post-ICMI and those with pre-existing structures.
National and regional organizations, along with their conventions and conferences, have been examined in studies (e.g., Gates 2003; Zuccheri and Zudini 2007), but research in English remains insufficient This gap highlights the need for comparative analyses of various national organizations Investigating how these professional bodies function in similar contexts yet under diverse socio-political conditions could provide valuable insights.
Professional organizations often include periodical publications focused on mathematics education, with L’Enseignement mathématique holding a significant historical position in this field (Furinghetti, 2009b) Numerous journals, both past and present, contribute to mathematics education literature, frequently publishing relevant materials The contributions of authors, editors, and preserved editorial correspondence from these journals are valuable for research However, there is a notable lack of comparative studies in mathematics education, and existing national studies are often inaccessible to the international community.
The histories of various conferences and international seminars play a crucial role in enhancing our understanding of mathematics education Even a single one-off conference can significantly contribute to the broader context of the field Schubring (2014c) conducted a study focusing on the Royaumont conference, highlighting its importance in this domain.
Seminar, unearthing some of its forgotten aspects.
The international movement in mathematics education is increasingly significant, prompting efforts to evaluate activities from the past century However, these studies face challenges due to limited access to national materials, highlighting the necessity of collecting and understanding these resources It is crucial to continue generalized studies that avoid oversimplifying global trends while also resisting narrow national perspectives Such research will enhance our understanding of historical developments and the current landscape of mathematics education.
Practices of Mathematics Education
Methods of Mathematics Education
Ackerberg-Hastings (2014) identifies three key practices in mathematics education: acquiring knowledge, rehearsing and reinforcing knowledge, and assessing knowledge Within these categories, various subgroups and distinct practice types emerge, including group, frontal, individual, oral, written, and technology-assisted methods.
Mathematics education is evolving, influenced by the study of informal geometry, which has introduced innovative assignments and class activities (Menghini 2009) This shift in perspective sees mathematics as both an experimental and abstract discipline, leading to new teaching methods, such as the incorporation of laboratories (Giacardi 2009a, 2012) Reforms in Modern Mathematics have also impacted teaching and teacher training practices (Matos 2009) While practices may change, the subject matter often remains consistent, as seen in the growing trend of group assignments that mirror traditional individual tasks These changes in educational practices frequently stem from broader educational theories, reflecting shifts in social contexts.
2.6 Practices of Mathematics Education 23 political circumstances For example, in post-revolutionary Russia schools had to abandon the old methods of instruction and replace them with projects and com- plexes that did not distinguish mathematics as a distinct subject and were moreover oriented towards practical application, since theoretical aspects of mathematics were thought generally useless Accordingly, the new approach favored group assessments, which were thought to promote a collective spirit, over individual ones
(Karp 2010, 2012) Conversely, Stalinist counter-reform rejected all these methodological innovations, reverting to the old practices.
Economic and technological advancements significantly shape methodological practices in education, particularly in mathematics For instance, the integration of calculators in classrooms necessitates curriculum adjustments Additionally, as parental affluence rises, increased leisure time among parents also impacts educational approaches, highlighting the broader influence of economic development on teaching methodologies.
Examining the evolution of methodological practices across different periods highlights their socio-historical significance, reflecting the political and socio-economic changes that drive these shifts While changes in educational practices may initially appear to be mere trends or influenced by foreign factors, the susceptibility to such influences underscores a critical aspect of educational systems.
Tools of Mathematics Education
According to Kidwell et al (2008), the term "tools of mathematics education" encompasses a wide range of instructional aids beyond just electronic technology These physical tools, which have been utilized for centuries, include resources like students' notebooks and cyphering books, highlighting the enduring importance of traditional study aids in mathematics education.
2015; Ellerton and Clements 2012, 2014; Leme da Silva and Valente 2009).
Students' notes, abundant in quantity, offer valuable insights into their classroom and home activities, revealing which topics garnered the most focus and attention.
Since ancient times all types of models have played an important role in instruction Schubring (2010) notes the existence of collections of such models in
The use of instructional models in mathematics dates back to the early 18th century in Germany, with evidence suggesting that similar tools have been utilized even earlier Computing devices, such as various types of abaci, were employed in pre-Columbian America and Ancient Rome, indicating their role in the educational process by teaching students practical skills.
24 2 Survey of the State of the Art
In addition to the research by Kidwell et al (2008) on mathematics education in the United States, several other studies explore the evolution of technology and tools in this field, particularly in relation to the history of the International Commission on Mathematical Instruction (ICMI) Notable works include those by Bartolini Bussi and Borba (2010), Bartolini Bussi et al (2010), Ruthven (2008), and Schubring (2010) Investigating the history of instructional technology reveals important insights into how technological advancements align with educational and societal needs There is a pressing need for further research that examines these developments across various countries and cultures.
Practices of Informal Mathematics Education
Informal education, encompassing learning outside the traditional school system—such as individual study, private tutoring, and practical mathematics—has been largely overlooked in scholarly research Howson (2011) explores this form of education in 19th-century England, highlighting its prevalence across various countries and time periods While classic works on informal mathematics education exist for Great Britain, there is a notable gap in our understanding of similar practices in other nations.
Informal study continues to hold significant relevance today, particularly in the context of exam preparation industries in various countries that often overshadow traditional schooling, which primarily focuses on social conditioning Historically, home-based mathematics education flourished in the 18th and 19th centuries, serving as a crucial resource for individuals lacking access to formal education However, questions surrounding the subjects studied, the texts utilized, the stages of learning, and the interaction between informal and formal education systems often remain unexplored.
Teacher Training
In conclusion, teacher training serves as a comprehensive summary of our discussion, integrating key themes such as curricula development, the roles of educators, evolving standards, and instructional practices Notably, Smid's work stands as the sole comparative account documenting the history of mathematics teacher training, highlighting the need for further exploration in this critical area of education.
(2014) Other studies limit their discussions to a single country.
The notion of specialized teacher training in mathematics is a relatively new development, distinguishing it as a profession rather than a role filled by any generally educated individual or teacher Historically, primary mathematics instruction could be assigned to individuals such as retired soldiers, and it took considerable time for the understanding to emerge that prospective mathematics teachers should undergo specialized training to effectively teach at all educational levels.
The organization of teacher training for primary and secondary educators has evolved, with specialized institutions emerging globally to provide targeted curricula in mathematics and pedagogy (d’Enfert 2012a, b; Furinghetti 2012; Furinghetti and Giacardi 2012) Ongoing debates regarding the necessity of advanced mathematics courses for prospective primary teachers, who may never teach advanced topics, reflect historical discussions influenced by social factors, including the desire to restrict educational access for certain social groups.
In recent years, advanced training models for mathematics educators have emerged, particularly within graduate schools, reflecting new research in mathematics education (Donoghue, 2003b) Ferrini-Mundy and Graham (2003) explore the evolution of mathematics teacher training in the United States post-World War II, while Smid (2014) provides a brief analysis of various strategies implemented in other countries during that period A comprehensive study of these developments is essential for further understanding.
Analyzing the evolution of teacher requirements can be effectively achieved by examining licensing examinations from various countries, both historical and contemporary, alongside the discussions they have sparked (e.g., Soares 2009) This approach reveals, in a concise manner, the knowledge and skills teachers were expected to possess and the intended outcomes of their instructional training.
When comparing training programs for mathematics educators, it is evident that they have influenced one another significantly (Valente, 2012) Additionally, it is particularly intriguing to explore how international practices have been tailored to fit local contexts across various countries.
Mathematics teachers, like educators in general, have historically been viewed as state officials responsible for developing the nation's human resources Their training is heavily influenced by the political, economic, and social contexts of their respective countries, indicating a need for further improvements in teacher preparation.
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26 2 Survey of the State of the Art
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This article does not aim to list all significant studies from the recent past, particularly those published in languages other than English, which are only referenced indirectly The survey highlights that the existing research varies widely and can provide valuable insights into different facets of historical processes, indicating that both completed and future studies may be organized in diverse ways.
The history of mathematics education is complex and can be explored from various perspectives In the past decade, there has been a growing interest in this field; however, significant work still needs to be accomplished Our primary objective is to highlight the unresolved questions that remain in this area.
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