In 1622 Frans van Schooten sr. wrote on f113 r of his lecture notes "Begonnen den 25 November Ann... more In 1622 Frans van Schooten sr. wrote on f113 r of his lecture notes "Begonnen den 25 November Anno 1622 door Frans van Schooten, professor der Fortificatien en Dependerenden Scientien in de Universiteit tot Leyden." During the preceding years professor Van Schooten had written his carefully constructed and elaborately illustrated lecture notes on mathematics and surveying in the same book, which is present as BPL 1013 in the University Library. In November 1622 he started writing and illustrating the second part of his teaching task, on fortification. From 1611-1645 Van Schooten taught an engineering course in Dutch language; both the content and teaching language were very unusual for a respectable university. Mathematics formed part of the undergraduate programme, the Liberal Arts, at most universities. Mathematics courses consisted of lectures, in Latin, on pure mathematics (number theory and Euclidean geometry) and mixed mathematics, subjects such as astronomy, mechanics, and cosmography. The lectures were based on classical texts by Greek and Arabic authors, translators, and commentators. The mathematics taught was primarily theoretical, without attention to possible practical use. Separate from this learned world a practical form of mathematics was used by common people who did not know Latin: practitioners such as surveyors, bookkeepers, building masters, military architects, traders, and navigators. The mathematics taught was primarily practical, without much attention to theoretical background, tending to the learning of tricks that seemed to work in practice. There were a few exceptions; some mathematical practitioners, such as the military engineers Adriaan Antonisz. and Samuel Crop and the mathematics teacher Ludolf van Ceulen, based their practices on sound theoretical mathematical knowledge, without knowledge of Latin. The new University at Leiden needed to attract s good number of students as fast as possible, so the curators did their utmost to find professors with a good reputation. It is rather fascinating that only 25 years after the founding the curators were willing to take a risk concerning the academic reputation of the university by offering a new course, in practical mathematics (surveying and fortification), taught in Dutch instead of the academic Latin. Consequently, this course, commonly named the Duytsche Mathematique, could not lead to a university degree. What were the reasons for this unusual academic enterprise and how did this course relate to the academic mathematics programme? During the first years after the founding there was no professor for the undergraduate mathematics courses, mathematics had a lower priority than subjects such as theology, logic, dialectics, and in Leiden especially, the classical and eastern languages. In 1581 Rudolph Snellius was appointed; he had a degree in liberal arts from the Calvinist University at Marburg. Snellius taught a broad range of mathematical subjects, based on works by Petrus Ramus (an educational innovator who proposed a shorter, more condensed and wellstructured curriculum) and also the works of the classical authors. Around 1605 Rudolph gave lectures on geometry, geography and optics, his son Willebrord assisted him with lectures on arithmetic and astronomy. Willebrord Snellius was a respected and productive mathematician, who succeeded his father as professor in 1613. He taught the theory through the classical authors but also used mathematics to solve practical problems, such as the development of an improved technique of triangulation in surveying, using his own big quadrant to measure distances. In 1629 Jacob Golius, an expert in Arabic languages and literature, who purchased many Arabic manuscripts for the university, became his successor.
In the early 17th century at the University of Leiden, mathematics courses in the Liberal Arts an... more In the early 17th century at the University of Leiden, mathematics courses in the Liberal Arts and a mathematics course for surveyors and engineers in Dutch language existed peacefully next to each other, with some pathways between them. The reasons for this unique situation, the relation between the different mathematical courses and the characteristics of the Dutch language course are discussed. The design of the Dutch language course has some aspects in common with modern curricula. Two questions arise. Did this course influence the mathematics education in the next centuries? Was the Dutch language course a first step towards the development of mathematics as a school discipline? The second part of the paper attempts to find answers to these questions.
In the Netherlands, it is common that teacher training courses for upper secondary education educ... more In the Netherlands, it is common that teacher training courses for upper secondary education educate for a single school discipline, such as mathematics. At the same time, teachers with a mathematics qualification are also formally competent to teach an interdisciplinary school course called Nature, Life and Technology (NLT), regardless whether it has been addressed in the teacher training programmes. However, teaching NLT is substantially different from teaching mathematics with respect to, for instance, pedagogy and objectives. In this paper we report on a study on experiences of mathematics teachers who, in addition to teaching mathematics, teach NLT. The experiential knowledge is meant to inform mathematics teacher educators who prepare students for interdisciplinary courses such as NLT.
Vernieuwing van wiskundecurricula, vooral in havo en vwo, gaat nogal eens gepaard met verhitte di... more Vernieuwing van wiskundecurricula, vooral in havo en vwo, gaat nogal eens gepaard met verhitte discussies over onder meer inhoud, didactiek, leermiddelen en de rol van docenten. Welke factoren en actoren zijn belangrijk voor het succes van een nieuw wiskundecurriculum? Onderzoek naar ontwikkeling en uitvoering van historische wiskundecurricula heeft als voordeel dat de processen afgerond en de resultaten bekend zijn. Van drie wiskundecurricula uit de periode 1600 tot 1900 zijn gegevens verzameld, geanalyseerd en vergeleken, met als doel actoren en factoren te identificeren die een belangrijke invloed hebben gehad op de mate van succes van deze curricula. Een vergelijking met de gang van zaken bij een recente, problematische vernieuwing maakt aannemelijk dat kennis van historische processen een bijdrage kan leveren aan het voorkomen van problemen bij het ontwerpen en uitvoeren van nieuwe wiskundecurricula
In the northern Netherlands knowledge of arithmetic and practical mathematics became more and mor... more In the northern Netherlands knowledge of arithmetic and practical mathematics became more and more important from the early 17th century. By the mid-18th century learning arithmetic had become more common and primary schoolteachers were expected to know mathematics. The rather strong interest in mathematics and the greater knowledge of teachers in the 18th century resulted in the publication of a monthly journal, Mathematische Liefhebberye (Mathematical Pastimes), between 1754 and 1769. Though the title suggests a periodical for mathematical recreation, the translation of the full title is: ‘Mathematical Pastimes, with news of the French and Dutch Schools’. ‘The content of this journal makes it clear that it was intended for teachers who taught mathematics, in primary schools or privately. One may conclude that already in the 18th century in the Netherlands there was a tendency to consider mathematics as an autonomous school discipline and also that there was the beginning of a spon...
Advances in Biochemical Engineering/Biotechnology, 2019
Mathematics is one of five disciplines in the successful interdisciplinary STEM course 'nature, l... more Mathematics is one of five disciplines in the successful interdisciplinary STEM course 'nature, life and technology' (NLT) for upper secondary education in the Netherlands. In this study the way mathematics manifests itself in the course is examined at the intended, implemented and attained curriculum. The results show that students often don't recognise mathematics in the course or that they consider the level of mathematics in the course as too low, not interesting. Besides this, mathematics teachers seem to struggle more with their role in interdisciplinary teaching than science teachers. Some feel that there is no need for their mathematical expertise, due to the extent to which mathematics is used. Even though the course has many positive outcomes when it comes to mathematics in the course NLT not only shows diversity in practice but also raises questions we need to address if we want mathematics to be part of interdisciplinary education.
BSHM Bulletin: Journal of the British Society for the History of Mathematics, 2010
In the Netherlands as elsewhere there are differences of opinion on the content and implementatio... more In the Netherlands as elsewhere there are differences of opinion on the content and implementation of mathematics curricula. The recent revision of the Dutch national exam programmes for mathematics at secondary level caused heated discussion on which topics to include and to what depth. Another question is whether and how history can provide inspiration and source material for those involved in the design and evaluation of mathematics curricula. Developments in mathematics education in the Dutch Republic between 1600 and 1650 appear to be a valuable reference case.
An increasing number of scientists of different fields is working together in interdisciplinary s... more An increasing number of scientists of different fields is working together in interdisciplinary subjects. For school science it is difficult to bring these interdisciplinary developments into the classroom. Pupils thus get an outdated view of science and of possibilities in science and technology for their future career. Also there are indications that interdisciplinary subjects are more attractive to pupils than classical science subjects, even more so for females. In the Netherlands one remedy for this is the development of a new interdisciplinary subject: NLT (Natuur, Leven en Technologie). This subject is offered to science stream pupils in senior secondary schools in addition to the regular subjects physics, chemistry, biology and mathematics. A set of more than 50 modules is being developed as teaching materials for national use. Each module is developed by a team in which teachers of secondary schools and an expert of the subject work together. However this does not automatic...
Rationales for interdisciplinary STEM courses are often based on the fact that the problems we fa... more Rationales for interdisciplinary STEM courses are often based on the fact that the problems we face in today’s world call for perspectives and knowledge from many different areas. In many cases this includes mathematics because it is used in many research fields and because it is part of everyday life. At the same time interdisciplinary literature suggests that mathematics gains the least from integration. In this paper we use a successful interdisciplinary STEM course in the Netherlands to illustrate how students and teachers think about the value of mathematics. To analyse teacher and student statements concerning the value of mathematics a model is introduced for a disciplinary mathematics perspective for interdisciplinary STEM courses and the opportunities this model ca provide are discussed.
In the Dutch Republic in the 18th century mathematics was considered very important for many prof... more In the Dutch Republic in the 18th century mathematics was considered very important for many professions. However there were hardly any national or regional educational institutes which provided mathematics education. Three orphanages in different towns received a large inheritance under condition that they provided education for technical and artistic professions. The Foundations which were established had a curriculum in which mathematics was the main subject. The influence of several curriculum components and some external curriculum factors are recognisable in documents and archival data. This is illustrated by the history of one student.
Algebra became part of mathematics education in the Netherlands in course of the seventeenth cent... more Algebra became part of mathematics education in the Netherlands in course of the seventeenth century. At first in the form of cossic algebra, but by the end of the century, the influence of the notation of Descartes was noticeable. In the eighteenth century, algebra was part of the basic curriculum of the Foundation of Renswoude. In the second half of the nineteenth century, algebra was seen as useful for a technical career. The number of topics in school algebra grew, but eventually algebra became mainly a subject in which complicated calculations were performed, which did not seem to serve a purpose outside the subject. At the end of the twentieth century, school algebra in lower secondary became a fairly informal way of solving ‘practical’ problems.
In 1622 Frans van Schooten sr. wrote on f113 r of his lecture notes "Begonnen den 25 November Ann... more In 1622 Frans van Schooten sr. wrote on f113 r of his lecture notes "Begonnen den 25 November Anno 1622 door Frans van Schooten, professor der Fortificatien en Dependerenden Scientien in de Universiteit tot Leyden." During the preceding years professor Van Schooten had written his carefully constructed and elaborately illustrated lecture notes on mathematics and surveying in the same book, which is present as BPL 1013 in the University Library. In November 1622 he started writing and illustrating the second part of his teaching task, on fortification. From 1611-1645 Van Schooten taught an engineering course in Dutch language; both the content and teaching language were very unusual for a respectable university. Mathematics formed part of the undergraduate programme, the Liberal Arts, at most universities. Mathematics courses consisted of lectures, in Latin, on pure mathematics (number theory and Euclidean geometry) and mixed mathematics, subjects such as astronomy, mechanics, and cosmography. The lectures were based on classical texts by Greek and Arabic authors, translators, and commentators. The mathematics taught was primarily theoretical, without attention to possible practical use. Separate from this learned world a practical form of mathematics was used by common people who did not know Latin: practitioners such as surveyors, bookkeepers, building masters, military architects, traders, and navigators. The mathematics taught was primarily practical, without much attention to theoretical background, tending to the learning of tricks that seemed to work in practice. There were a few exceptions; some mathematical practitioners, such as the military engineers Adriaan Antonisz. and Samuel Crop and the mathematics teacher Ludolf van Ceulen, based their practices on sound theoretical mathematical knowledge, without knowledge of Latin. The new University at Leiden needed to attract s good number of students as fast as possible, so the curators did their utmost to find professors with a good reputation. It is rather fascinating that only 25 years after the founding the curators were willing to take a risk concerning the academic reputation of the university by offering a new course, in practical mathematics (surveying and fortification), taught in Dutch instead of the academic Latin. Consequently, this course, commonly named the Duytsche Mathematique, could not lead to a university degree. What were the reasons for this unusual academic enterprise and how did this course relate to the academic mathematics programme? During the first years after the founding there was no professor for the undergraduate mathematics courses, mathematics had a lower priority than subjects such as theology, logic, dialectics, and in Leiden especially, the classical and eastern languages. In 1581 Rudolph Snellius was appointed; he had a degree in liberal arts from the Calvinist University at Marburg. Snellius taught a broad range of mathematical subjects, based on works by Petrus Ramus (an educational innovator who proposed a shorter, more condensed and wellstructured curriculum) and also the works of the classical authors. Around 1605 Rudolph gave lectures on geometry, geography and optics, his son Willebrord assisted him with lectures on arithmetic and astronomy. Willebrord Snellius was a respected and productive mathematician, who succeeded his father as professor in 1613. He taught the theory through the classical authors but also used mathematics to solve practical problems, such as the development of an improved technique of triangulation in surveying, using his own big quadrant to measure distances. In 1629 Jacob Golius, an expert in Arabic languages and literature, who purchased many Arabic manuscripts for the university, became his successor.
In the early 17th century at the University of Leiden, mathematics courses in the Liberal Arts an... more In the early 17th century at the University of Leiden, mathematics courses in the Liberal Arts and a mathematics course for surveyors and engineers in Dutch language existed peacefully next to each other, with some pathways between them. The reasons for this unique situation, the relation between the different mathematical courses and the characteristics of the Dutch language course are discussed. The design of the Dutch language course has some aspects in common with modern curricula. Two questions arise. Did this course influence the mathematics education in the next centuries? Was the Dutch language course a first step towards the development of mathematics as a school discipline? The second part of the paper attempts to find answers to these questions.
In the Netherlands, it is common that teacher training courses for upper secondary education educ... more In the Netherlands, it is common that teacher training courses for upper secondary education educate for a single school discipline, such as mathematics. At the same time, teachers with a mathematics qualification are also formally competent to teach an interdisciplinary school course called Nature, Life and Technology (NLT), regardless whether it has been addressed in the teacher training programmes. However, teaching NLT is substantially different from teaching mathematics with respect to, for instance, pedagogy and objectives. In this paper we report on a study on experiences of mathematics teachers who, in addition to teaching mathematics, teach NLT. The experiential knowledge is meant to inform mathematics teacher educators who prepare students for interdisciplinary courses such as NLT.
Vernieuwing van wiskundecurricula, vooral in havo en vwo, gaat nogal eens gepaard met verhitte di... more Vernieuwing van wiskundecurricula, vooral in havo en vwo, gaat nogal eens gepaard met verhitte discussies over onder meer inhoud, didactiek, leermiddelen en de rol van docenten. Welke factoren en actoren zijn belangrijk voor het succes van een nieuw wiskundecurriculum? Onderzoek naar ontwikkeling en uitvoering van historische wiskundecurricula heeft als voordeel dat de processen afgerond en de resultaten bekend zijn. Van drie wiskundecurricula uit de periode 1600 tot 1900 zijn gegevens verzameld, geanalyseerd en vergeleken, met als doel actoren en factoren te identificeren die een belangrijke invloed hebben gehad op de mate van succes van deze curricula. Een vergelijking met de gang van zaken bij een recente, problematische vernieuwing maakt aannemelijk dat kennis van historische processen een bijdrage kan leveren aan het voorkomen van problemen bij het ontwerpen en uitvoeren van nieuwe wiskundecurricula
In the northern Netherlands knowledge of arithmetic and practical mathematics became more and mor... more In the northern Netherlands knowledge of arithmetic and practical mathematics became more and more important from the early 17th century. By the mid-18th century learning arithmetic had become more common and primary schoolteachers were expected to know mathematics. The rather strong interest in mathematics and the greater knowledge of teachers in the 18th century resulted in the publication of a monthly journal, Mathematische Liefhebberye (Mathematical Pastimes), between 1754 and 1769. Though the title suggests a periodical for mathematical recreation, the translation of the full title is: ‘Mathematical Pastimes, with news of the French and Dutch Schools’. ‘The content of this journal makes it clear that it was intended for teachers who taught mathematics, in primary schools or privately. One may conclude that already in the 18th century in the Netherlands there was a tendency to consider mathematics as an autonomous school discipline and also that there was the beginning of a spon...
Advances in Biochemical Engineering/Biotechnology, 2019
Mathematics is one of five disciplines in the successful interdisciplinary STEM course 'nature, l... more Mathematics is one of five disciplines in the successful interdisciplinary STEM course 'nature, life and technology' (NLT) for upper secondary education in the Netherlands. In this study the way mathematics manifests itself in the course is examined at the intended, implemented and attained curriculum. The results show that students often don't recognise mathematics in the course or that they consider the level of mathematics in the course as too low, not interesting. Besides this, mathematics teachers seem to struggle more with their role in interdisciplinary teaching than science teachers. Some feel that there is no need for their mathematical expertise, due to the extent to which mathematics is used. Even though the course has many positive outcomes when it comes to mathematics in the course NLT not only shows diversity in practice but also raises questions we need to address if we want mathematics to be part of interdisciplinary education.
BSHM Bulletin: Journal of the British Society for the History of Mathematics, 2010
In the Netherlands as elsewhere there are differences of opinion on the content and implementatio... more In the Netherlands as elsewhere there are differences of opinion on the content and implementation of mathematics curricula. The recent revision of the Dutch national exam programmes for mathematics at secondary level caused heated discussion on which topics to include and to what depth. Another question is whether and how history can provide inspiration and source material for those involved in the design and evaluation of mathematics curricula. Developments in mathematics education in the Dutch Republic between 1600 and 1650 appear to be a valuable reference case.
An increasing number of scientists of different fields is working together in interdisciplinary s... more An increasing number of scientists of different fields is working together in interdisciplinary subjects. For school science it is difficult to bring these interdisciplinary developments into the classroom. Pupils thus get an outdated view of science and of possibilities in science and technology for their future career. Also there are indications that interdisciplinary subjects are more attractive to pupils than classical science subjects, even more so for females. In the Netherlands one remedy for this is the development of a new interdisciplinary subject: NLT (Natuur, Leven en Technologie). This subject is offered to science stream pupils in senior secondary schools in addition to the regular subjects physics, chemistry, biology and mathematics. A set of more than 50 modules is being developed as teaching materials for national use. Each module is developed by a team in which teachers of secondary schools and an expert of the subject work together. However this does not automatic...
Rationales for interdisciplinary STEM courses are often based on the fact that the problems we fa... more Rationales for interdisciplinary STEM courses are often based on the fact that the problems we face in today’s world call for perspectives and knowledge from many different areas. In many cases this includes mathematics because it is used in many research fields and because it is part of everyday life. At the same time interdisciplinary literature suggests that mathematics gains the least from integration. In this paper we use a successful interdisciplinary STEM course in the Netherlands to illustrate how students and teachers think about the value of mathematics. To analyse teacher and student statements concerning the value of mathematics a model is introduced for a disciplinary mathematics perspective for interdisciplinary STEM courses and the opportunities this model ca provide are discussed.
In the Dutch Republic in the 18th century mathematics was considered very important for many prof... more In the Dutch Republic in the 18th century mathematics was considered very important for many professions. However there were hardly any national or regional educational institutes which provided mathematics education. Three orphanages in different towns received a large inheritance under condition that they provided education for technical and artistic professions. The Foundations which were established had a curriculum in which mathematics was the main subject. The influence of several curriculum components and some external curriculum factors are recognisable in documents and archival data. This is illustrated by the history of one student.
Algebra became part of mathematics education in the Netherlands in course of the seventeenth cent... more Algebra became part of mathematics education in the Netherlands in course of the seventeenth century. At first in the form of cossic algebra, but by the end of the century, the influence of the notation of Descartes was noticeable. In the eighteenth century, algebra was part of the basic curriculum of the Foundation of Renswoude. In the second half of the nineteenth century, algebra was seen as useful for a technical career. The number of topics in school algebra grew, but eventually algebra became mainly a subject in which complicated calculations were performed, which did not seem to serve a purpose outside the subject. At the end of the twentieth century, school algebra in lower secondary became a fairly informal way of solving ‘practical’ problems.
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Papers by Jenneke Krüger