Where did mathematics come from? Who thought up all those rules of algebra and why did they do it? How were the facts and proofs of geometry developed? Those are some of the few questions I wondered about when I was learning mathematics at school. I always wished to know the answer for all those questions. My experience could have been very similar to most of the other students. However, my mathematics teacher never told about the history or the origin of mathematical ideas that are taught in the school.
In recent years, mathematics education has changed dramatically. New technologies have entered the class room. Fresh and new guidance on the curriculum has been provided at all levels. In addition to that, one of the most important, though perhaps quietest, in educational practice has been the increasing eagerness of mathematics teachers for a historical dimension to their teaching and their pupils learning (Fauvel, 1997). One of the most famous Norwegian mathematicians Neils Henrik Abel once wrote:
It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils (Cited in Bekken & Mosvold, 2003, p. 1).
However, more than a century after Abel died, it is my view and experience that we make far too little use of the history of mathematics in our everyday teaching at all levels. However, in recent years there has been growing interest in the role of history of mathematics in improving the teaching and learning of mathematics. Educators throughout the world have been formulating and conducting research on the use of history of mathematics in mathematics education (Fauvel & Maanen, 1997). Therefore, it is very important to consider what place history of mathematics has in the mathematics classroom
The purpose of this essay is to identify the reasons why history of mathematics should have a place in mathematics classroom. This essay will also address the question, what could or must be done to gain this place for it. In addition to this, this essay will also suggest different ways of minimizing the difficulties faced in implementing the history of mathematics in the mathematics classroom.
Why history of mathematics should have a place in the mathematics classroom.
Everything man touches has a history: the clothes, art, transport, architecture, trade, computers and every subject including mathematics. The history of a subject becomes an inseparable part of the subject itself. If we would like to teach mathematics properly we must include its history in one way or the other. This will help to show that mathematics is a living subject (Heiede, 1996).
A lecture given by Robert L. Hayes at the History and Pedagogy of Mathematics session at the sixth international congress on Mathematics education in Budapest in 1988 stated:
I believe that it is a grave mistake and error of strategy to attempt to teach mathematics without reference to its cultural, social, philosophical and historical background (Hayes, 1991, p. 11).
I also highly agree with Hayes (1991) and believe that the history of mathematics should be incorporated in mathematics teaching and there are several reasons for that.
Mathematics is an ongoing human endeavor and a human creation. Mathematical theorems and proofs are the result of people struggling with the mysteries of the mathematics universe, rather then unmotivated, ossified edifice of axioms and theorems handed down without human intervention (Laubenbacher & Pengelley, 1996).Hence, it is important for students to understand that the “polished” mathematics that we know today have not always been in that form. In fact, it is like so as a result of grappling, perseverance, and the difficulties faced by a lot of mathematicians in the past (Avital, 1995). Thus, an emphasis on this human factor in the mathematics classroom will enable students to acknowledge the possible learning difficulties and understand that progress can only be possible in stages. For instance, the first known procedure for finding prime numbers was Sieve of Eratosthenes (about 200BC). After that several great mathematicians tried again and again to find a better method for identifying prime numbers. However, it was only towards the end of the seventeenth century, that a concise method for finding of certain prime numbers was discovered. There have also been several mistakes in the attempts to obtain a general formula for finding primes. Luca Pacioli (1494) thought that 22n+1-1 is prime; but n = 4 is a counter example. Pierre Fermat (1637) asserted that 22n+1 is prime. However, Euler (1732 ) gave a counter example for n = 5 (Popp, 1975). Exposing students to examples such as these, history of mathematics could help to alleviate discouragement on the part of student when faced with learning difficulties. It would also facilitate perseverance and encourage students to explore and investigate difficult mathematical problems.
Mathematics by nature consists of numerous formulae, symbols and definitions. Students frequently express that remembering these are difficult and boring. This could be due to the fact that most formulae and symbols represent something abstract and has no meaning for the students. However, inclusion of history of these formulae and symbols could help the students to make more sense of these as history can explain how formulae were developed and it could also explain why we use certain definitions and symbols that we do ( Smestad, 2003 ). For example, symbols such as that used for integration ( and summation ( may not hold any meaning for someone who does not have any background knowledge of what these represent and how and why these symbols came to be used. Thus, history telling to students how and why ‘ ’ began to be used for integration and ‘ ’ used for summation will contribute to make these symbols more real for the students.
From my personal experience as a student as well as a teacher, I have noticed that one of the reasons students find mathematics difficult relate to the fact that the students are unable to understand the relationship between various concepts. For instance, many students may not understand that integration is the reverse process of differentiation and that an understanding of one process would facilitate understanding of the other. Incorporating history of mathematics can show students how concepts have been developed and assist the pupils to make important connections between various concepts. It can also point out contrast between various concepts. For example, function concept and angle concept (Smestad, 2003). In addition to that, historical texts and references will allow the analysis of the interaction between mathematical problems and the construction of concepts. “It can also bring to the fore the central role played by scientific questioning in the theoretical development of mathematics” (Michel-Pajus, 2000, p. 17).
Students frequently hesitate to attempt new challenges in mathematics mainly due to their fear of making mistakes. By being exposed to the history of mathematics and understanding that even great mathematicians in the past have also made mistakes could help students to understand that making mistakes does not indicate intellectual inferiority. In addition, it will indicate to the students that mistakes contribute to the evolution and development of several mathematical ideas. For instance, S.G. Abel (father of the famous N. H. Abel, 1820-1829) wrote that 1 + 0 = 0 in a text book. Further, Shinivasa Ramanujan (1887-1920) made the following mistake:
12 + 22 + 32 +…= 0 (History and Pedagogy of Mathematics Newsletter, 46, March 2001, p. 2).
The above examples suggest that even some of great mathematicians at that time lacked adequate understanding of what we today consider to be some of the simple concepts like the nature and behaviour of zero.
The students’ ability to tackle mathematical problems from a variety of angles indicates a good understanding of the concept. Different mathematicians would have chosen a multitude of algorithms. For instance, Egyptian multiplication, Abacus calculation and Al-Khwarizmi’s method for solving certain equation of second degree (Smestad, 2003 ). History of mathematics could provide students with the opportunity to compare new and old methods and thereby present the opportunity to choose the best method in a given situation.
By using old problems, students can compare their strategies with the original ones. This is an interesting way of understanding the economy and the effectiveness of our present algebraic process. In observing the historical evolution of a concept, pupils will find that mathematics is not fixed and definitive (Grugnetti, 2000, p. 30).
In addition to that, history of mathematics can offer students several examples illustrating the relevance of an interdisciplinary approach. For example: the number system of the ancients; the use by Galileo of both mathematical and experimental methods and Descartes’ use of the analytical method (Grugnetti, 2000).
Undoubtedly, there will be several other reasons other than those that have been discussed above, for including history of mathematics in mathematics classroom. Having realized the importance of history of mathematics in mathematics classroom, I think it is even more important to consider what could or must be done to implement this in mathematics classroom.
What could be done to gain a place for the History of Mathematics in the Mathematics classrooms?
The discussion in the previous section has emphasized that effective teaching of mathematics requires more than a sound knowledge of the subject matter. It is important to incorporate history of mathematics in order to make the subject more interesting and less abstract for the students. Some countries like Denmark have begun to incorporate historical aspect in the mathematics syllabus (Heiede, 1996). However, as far as I am concerned, history of mathematics is not given such importance in many countries. This is something the educational policy makers of these countries need to consider as they have the ultimate power to facilitate changes to the curriculum at large. For instance, the countries like The Republic of Maldives where more than ninety percent of the schools are government schools and the curriculum is determined by the ministry of education. Thus it’s the policy makers who can play a major role in incorporating history of mathematics in the curriculum.
Mere inclusion of a historical aspect in the curriculum will not resolve the issue. Very often teachers tend to ignore or side line what is not assessed. For instance, my experience as a mathematics teacher in The Republic of Maldives is that schools and even teachers are judged and ranked based on the examination result of the students. As a result teachers always try to utilize all the available time in the classroom to train students for the examination. That is teachers always teach the technical aspect of mathematics. So they have no time to incorporate topics like history of mathematics (which is not assessed) in their everyday teaching. At the same time students also do not have much motivation to learn things which are not assessed. Therefore teachers teach for the test. Thus, if assessment of mathematics does not involve assessment of the history of mathematics component, the students as well as the teachers may not give it the attention and time it deserves in the curriculum. Hence, incorporation of history of mathematics in the curriculum is not enough. It should also form part of the assessment.
Apart from this, history of mathematics should also form a compulsory part of teacher training since it is necessary for teachers to have adequate knowledge of history of mathematics to be able to effectively impart this to their students. This leads us to the question of what to do with the teachers who are already in the profession without any knowledge of history of mathematics. These teachers can be benefit from in service work shops designed to provide the knowledge of history of mathematics. For instance, at York University, “History of Mathematics” is one of two required courses in an In-service Master’s Program for mathematics teachers offered in the Department of Mathematics and Statistics. This program has been especially designed for teachers and attempts to give students a broad overview of the major mathematical fields and issues, to expand their horizon, deepen their understanding of mathematics and to broaden their perspective on mathematics they teach, so that they can better judge what to emphasize in their teaching and why to emphasize it (Kleiner, 1996).
Conclusion.
There have been lots of changes in mathematics education over the recent years. One of the most significant of the developments in mathematics education is the debate about the inclusion of history of mathematics in the mathematics classrooms (Furinghetti, 2000). This is evident in the plethora of evolving research in this area over the recent years. In fact history of mathematics already curved a niche for itself in the mathematics classrooms in some countries.
Inclusion of history of mathematics in mathematics education does not necessarily enable students to obtain higher marks. However, mathematicians argue that incorporation of this aspect of mathematics in the mathematics classroom will enhance students learning and will create positive attitude towards the subject. It can also make mathematics learning a meaningful and a stimulating experience. Consequently, I believe that this will ensure that mathematics learning is deeper and a more insightful one. The advantage of incorporating history of mathematics is appropriately conveyed by Man-Keung (2000):
The study of history of mathematics, though it does not make me a better mathematician, does make me a happier man who is ready to appreciate the multi-dimension splendor of the discipline and its relationship to other cultural endeavors (Man-Keung, 2000, p. 8).
Even though there are obvious advantages of incorporating history of mathematics in the mathematics classroom, there are several challenges that need to be considered. First and most importantly, history of mathematics has to be incorporated into teacher training program and assessment procedures, which means that to have to be included in the curriculum. As a result, since this area of mathematics being new to most of the curriculums, academic institutions will demand more resources s well.
I personally believe that history of mathematics can increase not only teacher’s enthusiasm for the subject. It can also promote a sense of its importance and even encourage students to ask “why?” in addition to “how?”
References.
Avital, S. (1995). History of Mathematics Can Help Improve Instruction and Learning. In Swetz, F. Fauvel, J. Bekkeb, O. Johansson, B. Katz, V. (Eds.). Learn from the Masters. (pp.3-12). United States of America: The Mathematics Association of America.
Bekken, O. B. Mosvold, R. (Eds.). (2003). Study the Masters: The Abel Fauvel Conference. Kristiansand, : Grafikerna Livrena i Kungalv AB.
Fauvel, J. (1997). History of mathematics History of problems: the Inter –IREM Commission, ?Epistomology and History of Mathematics. Paris, : 32 rue Bargue.
Fauvel, J., Maanen, J. V. (1997). Discussion Document for an ICMI Study 1997-2000. International Study Group on the Relations Between History and Pedagogy of Mathematics Newsletter, 40.
Furinghetti, F. (2000). The Long Tradition of History in Mathematics teaching: An old Italian Case. In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.49-58).Washington, DC: The Mathematics Association of America.
Grugnetti, L. (2000). The History of Mathematics and its Influence on Pedagogical Problems. In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.29-35).Washington, DC: The Mathematics Association of America.
Hayes, R., L. (1991). History – A Way Back to Mathematics. History and Pedagogy of Mathematics Newsletter, 22, 10-12.
Heiede, T. (1996). History of Mathematics and the Teacher. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.231-243). United States of America: The Mathematics Association of America.
Kleiner, I. (1996). A History - of - Mathematics Course for Teachers, Based on Great quotations. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.261-268). United States of America: The Mathematics Association of America.
Laubenbacher, R., C., Pengelley, D. (1996). Mathematical Masterpieeces: Teaching with Original sources. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.257-260). United States of America: The Mathematics Association of America.
Man-Keung, S. (2000). The ABCD of Using History of Mathematics in the (Undergraduate Classroom). In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.3-9).Washington, DC: The Mathematics Association of America.
Michel-Pajus, A. (2000). On the Benefits of Introducing Undegraduates to the History of Mathematics-A French Perspective. In Katz, V., J. (Ed.). Using History to Teach Mathematics : An International Perspective. (pp.17-25).Washington, DC: The Mathematics Association of America .
Popp, W. (1975). History of Mathematics: Topics for schools. England : The Open University Press.
Smestad, B. (2003). Historical topics in Norwegian textbooks. In Bekken, O., B. Mosvold, R. (Eds.).V Study the Masters: The Abel Fauvel Conference. (pp. 163-168). Kristiansand, : Grafikerna Livrena i Kungalv AB.
In recent years, mathematics education has changed dramatically. New technologies have entered the class room. Fresh and new guidance on the curriculum has been provided at all levels. In addition to that, one of the most important, though perhaps quietest, in educational practice has been the increasing eagerness of mathematics teachers for a historical dimension to their teaching and their pupils learning (Fauvel, 1997). One of the most famous Norwegian mathematicians Neils Henrik Abel once wrote:
It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils (Cited in Bekken & Mosvold, 2003, p. 1).
However, more than a century after Abel died, it is my view and experience that we make far too little use of the history of mathematics in our everyday teaching at all levels. However, in recent years there has been growing interest in the role of history of mathematics in improving the teaching and learning of mathematics. Educators throughout the world have been formulating and conducting research on the use of history of mathematics in mathematics education (Fauvel & Maanen, 1997). Therefore, it is very important to consider what place history of mathematics has in the mathematics classroom
The purpose of this essay is to identify the reasons why history of mathematics should have a place in mathematics classroom. This essay will also address the question, what could or must be done to gain this place for it. In addition to this, this essay will also suggest different ways of minimizing the difficulties faced in implementing the history of mathematics in the mathematics classroom.
Why history of mathematics should have a place in the mathematics classroom.
Everything man touches has a history: the clothes, art, transport, architecture, trade, computers and every subject including mathematics. The history of a subject becomes an inseparable part of the subject itself. If we would like to teach mathematics properly we must include its history in one way or the other. This will help to show that mathematics is a living subject (Heiede, 1996).
A lecture given by Robert L. Hayes at the History and Pedagogy of Mathematics session at the sixth international congress on Mathematics education in Budapest in 1988 stated:
I believe that it is a grave mistake and error of strategy to attempt to teach mathematics without reference to its cultural, social, philosophical and historical background (Hayes, 1991, p. 11).
I also highly agree with Hayes (1991) and believe that the history of mathematics should be incorporated in mathematics teaching and there are several reasons for that.
Mathematics is an ongoing human endeavor and a human creation. Mathematical theorems and proofs are the result of people struggling with the mysteries of the mathematics universe, rather then unmotivated, ossified edifice of axioms and theorems handed down without human intervention (Laubenbacher & Pengelley, 1996).Hence, it is important for students to understand that the “polished” mathematics that we know today have not always been in that form. In fact, it is like so as a result of grappling, perseverance, and the difficulties faced by a lot of mathematicians in the past (Avital, 1995). Thus, an emphasis on this human factor in the mathematics classroom will enable students to acknowledge the possible learning difficulties and understand that progress can only be possible in stages. For instance, the first known procedure for finding prime numbers was Sieve of Eratosthenes (about 200BC). After that several great mathematicians tried again and again to find a better method for identifying prime numbers. However, it was only towards the end of the seventeenth century, that a concise method for finding of certain prime numbers was discovered. There have also been several mistakes in the attempts to obtain a general formula for finding primes. Luca Pacioli (1494) thought that 22n+1-1 is prime; but n = 4 is a counter example. Pierre Fermat (1637) asserted that 22n+1 is prime. However, Euler (1732 ) gave a counter example for n = 5 (Popp, 1975). Exposing students to examples such as these, history of mathematics could help to alleviate discouragement on the part of student when faced with learning difficulties. It would also facilitate perseverance and encourage students to explore and investigate difficult mathematical problems.
Mathematics by nature consists of numerous formulae, symbols and definitions. Students frequently express that remembering these are difficult and boring. This could be due to the fact that most formulae and symbols represent something abstract and has no meaning for the students. However, inclusion of history of these formulae and symbols could help the students to make more sense of these as history can explain how formulae were developed and it could also explain why we use certain definitions and symbols that we do ( Smestad, 2003 ). For example, symbols such as that used for integration ( and summation ( may not hold any meaning for someone who does not have any background knowledge of what these represent and how and why these symbols came to be used. Thus, history telling to students how and why ‘ ’ began to be used for integration and ‘ ’ used for summation will contribute to make these symbols more real for the students.
From my personal experience as a student as well as a teacher, I have noticed that one of the reasons students find mathematics difficult relate to the fact that the students are unable to understand the relationship between various concepts. For instance, many students may not understand that integration is the reverse process of differentiation and that an understanding of one process would facilitate understanding of the other. Incorporating history of mathematics can show students how concepts have been developed and assist the pupils to make important connections between various concepts. It can also point out contrast between various concepts. For example, function concept and angle concept (Smestad, 2003). In addition to that, historical texts and references will allow the analysis of the interaction between mathematical problems and the construction of concepts. “It can also bring to the fore the central role played by scientific questioning in the theoretical development of mathematics” (Michel-Pajus, 2000, p. 17).
Students frequently hesitate to attempt new challenges in mathematics mainly due to their fear of making mistakes. By being exposed to the history of mathematics and understanding that even great mathematicians in the past have also made mistakes could help students to understand that making mistakes does not indicate intellectual inferiority. In addition, it will indicate to the students that mistakes contribute to the evolution and development of several mathematical ideas. For instance, S.G. Abel (father of the famous N. H. Abel, 1820-1829) wrote that 1 + 0 = 0 in a text book. Further, Shinivasa Ramanujan (1887-1920) made the following mistake:
12 + 22 + 32 +…= 0 (History and Pedagogy of Mathematics Newsletter, 46, March 2001, p. 2).
The above examples suggest that even some of great mathematicians at that time lacked adequate understanding of what we today consider to be some of the simple concepts like the nature and behaviour of zero.
The students’ ability to tackle mathematical problems from a variety of angles indicates a good understanding of the concept. Different mathematicians would have chosen a multitude of algorithms. For instance, Egyptian multiplication, Abacus calculation and Al-Khwarizmi’s method for solving certain equation of second degree (Smestad, 2003 ). History of mathematics could provide students with the opportunity to compare new and old methods and thereby present the opportunity to choose the best method in a given situation.
By using old problems, students can compare their strategies with the original ones. This is an interesting way of understanding the economy and the effectiveness of our present algebraic process. In observing the historical evolution of a concept, pupils will find that mathematics is not fixed and definitive (Grugnetti, 2000, p. 30).
In addition to that, history of mathematics can offer students several examples illustrating the relevance of an interdisciplinary approach. For example: the number system of the ancients; the use by Galileo of both mathematical and experimental methods and Descartes’ use of the analytical method (Grugnetti, 2000).
Undoubtedly, there will be several other reasons other than those that have been discussed above, for including history of mathematics in mathematics classroom. Having realized the importance of history of mathematics in mathematics classroom, I think it is even more important to consider what could or must be done to implement this in mathematics classroom.
What could be done to gain a place for the History of Mathematics in the Mathematics classrooms?
The discussion in the previous section has emphasized that effective teaching of mathematics requires more than a sound knowledge of the subject matter. It is important to incorporate history of mathematics in order to make the subject more interesting and less abstract for the students. Some countries like Denmark have begun to incorporate historical aspect in the mathematics syllabus (Heiede, 1996). However, as far as I am concerned, history of mathematics is not given such importance in many countries. This is something the educational policy makers of these countries need to consider as they have the ultimate power to facilitate changes to the curriculum at large. For instance, the countries like The Republic of Maldives where more than ninety percent of the schools are government schools and the curriculum is determined by the ministry of education. Thus it’s the policy makers who can play a major role in incorporating history of mathematics in the curriculum.
Mere inclusion of a historical aspect in the curriculum will not resolve the issue. Very often teachers tend to ignore or side line what is not assessed. For instance, my experience as a mathematics teacher in The Republic of Maldives is that schools and even teachers are judged and ranked based on the examination result of the students. As a result teachers always try to utilize all the available time in the classroom to train students for the examination. That is teachers always teach the technical aspect of mathematics. So they have no time to incorporate topics like history of mathematics (which is not assessed) in their everyday teaching. At the same time students also do not have much motivation to learn things which are not assessed. Therefore teachers teach for the test. Thus, if assessment of mathematics does not involve assessment of the history of mathematics component, the students as well as the teachers may not give it the attention and time it deserves in the curriculum. Hence, incorporation of history of mathematics in the curriculum is not enough. It should also form part of the assessment.
Apart from this, history of mathematics should also form a compulsory part of teacher training since it is necessary for teachers to have adequate knowledge of history of mathematics to be able to effectively impart this to their students. This leads us to the question of what to do with the teachers who are already in the profession without any knowledge of history of mathematics. These teachers can be benefit from in service work shops designed to provide the knowledge of history of mathematics. For instance, at York University, “History of Mathematics” is one of two required courses in an In-service Master’s Program for mathematics teachers offered in the Department of Mathematics and Statistics. This program has been especially designed for teachers and attempts to give students a broad overview of the major mathematical fields and issues, to expand their horizon, deepen their understanding of mathematics and to broaden their perspective on mathematics they teach, so that they can better judge what to emphasize in their teaching and why to emphasize it (Kleiner, 1996).
Conclusion.
There have been lots of changes in mathematics education over the recent years. One of the most significant of the developments in mathematics education is the debate about the inclusion of history of mathematics in the mathematics classrooms (Furinghetti, 2000). This is evident in the plethora of evolving research in this area over the recent years. In fact history of mathematics already curved a niche for itself in the mathematics classrooms in some countries.
Inclusion of history of mathematics in mathematics education does not necessarily enable students to obtain higher marks. However, mathematicians argue that incorporation of this aspect of mathematics in the mathematics classroom will enhance students learning and will create positive attitude towards the subject. It can also make mathematics learning a meaningful and a stimulating experience. Consequently, I believe that this will ensure that mathematics learning is deeper and a more insightful one. The advantage of incorporating history of mathematics is appropriately conveyed by Man-Keung (2000):
The study of history of mathematics, though it does not make me a better mathematician, does make me a happier man who is ready to appreciate the multi-dimension splendor of the discipline and its relationship to other cultural endeavors (Man-Keung, 2000, p. 8).
Even though there are obvious advantages of incorporating history of mathematics in the mathematics classroom, there are several challenges that need to be considered. First and most importantly, history of mathematics has to be incorporated into teacher training program and assessment procedures, which means that to have to be included in the curriculum. As a result, since this area of mathematics being new to most of the curriculums, academic institutions will demand more resources s well.
I personally believe that history of mathematics can increase not only teacher’s enthusiasm for the subject. It can also promote a sense of its importance and even encourage students to ask “why?” in addition to “how?”
References.
Avital, S. (1995). History of Mathematics Can Help Improve Instruction and Learning. In Swetz, F. Fauvel, J. Bekkeb, O. Johansson, B. Katz, V. (Eds.). Learn from the Masters. (pp.3-12). United States of America: The Mathematics Association of America.
Bekken, O. B. Mosvold, R. (Eds.). (2003). Study the Masters: The Abel Fauvel Conference. Kristiansand, : Grafikerna Livrena i Kungalv AB.
Fauvel, J. (1997). History of mathematics History of problems: the Inter –IREM Commission, ?Epistomology and History of Mathematics. Paris, : 32 rue Bargue.
Fauvel, J., Maanen, J. V. (1997). Discussion Document for an ICMI Study 1997-2000. International Study Group on the Relations Between History and Pedagogy of Mathematics Newsletter, 40.
Furinghetti, F. (2000). The Long Tradition of History in Mathematics teaching: An old Italian Case. In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.49-58).Washington, DC: The Mathematics Association of America.
Grugnetti, L. (2000). The History of Mathematics and its Influence on Pedagogical Problems. In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.29-35).Washington, DC: The Mathematics Association of America.
Hayes, R., L. (1991). History – A Way Back to Mathematics. History and Pedagogy of Mathematics Newsletter, 22, 10-12.
Heiede, T. (1996). History of Mathematics and the Teacher. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.231-243). United States of America: The Mathematics Association of America.
Kleiner, I. (1996). A History - of - Mathematics Course for Teachers, Based on Great quotations. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.261-268). United States of America: The Mathematics Association of America.
Laubenbacher, R., C., Pengelley, D. (1996). Mathematical Masterpieeces: Teaching with Original sources. In Calinger, R. (Ed.).Vita Mathematica : Historical research and integration with teaching. (pp.257-260). United States of America: The Mathematics Association of America.
Man-Keung, S. (2000). The ABCD of Using History of Mathematics in the (Undergraduate Classroom). In Katz, V., J. (Ed.). Using History to Teach Mathematics :An International Perspective. (pp.3-9).Washington, DC: The Mathematics Association of America.
Michel-Pajus, A. (2000). On the Benefits of Introducing Undegraduates to the History of Mathematics-A French Perspective. In Katz, V., J. (Ed.). Using History to Teach Mathematics : An International Perspective. (pp.17-25).Washington, DC: The Mathematics Association of America .
Popp, W. (1975). History of Mathematics: Topics for schools. England : The Open University Press.
Smestad, B. (2003). Historical topics in Norwegian textbooks. In Bekken, O., B. Mosvold, R. (Eds.).V Study the Masters: The Abel Fauvel Conference. (pp. 163-168). Kristiansand, : Grafikerna Livrena i Kungalv AB.
1 comment:
Very good......
Post a Comment