Teaching and Learning Calculus at higher Secondary and First year at University

Introduction
Calculus is one of the most important areas of mathematics. Therefore understanding basic ideas and concepts of calculus which are normally introduced in secondary school is extremely important for those wishing to pursue further studies in mathematics at university level. However a common complaint among students from almost all the countries in the world is that learning calculus is very difficult (Robert & Speer, 2001). More importantly, mathematics lecturers criticize that,
Students were not emerging from the courses with a strong understanding of calculus concepts and they were also not choosing to pursue further mathematics courses (Robert & Speer, 2001, p.291).
Therefore identifying the difficulties students face in learning calculus and understanding the related concepts in calculus are essential to the mathematics world.
The purpose of this essay is to review the literature on students’ perception of learning calculus at higher secondary and first year at university level. In this essay I will also try to identify the difficulties students face in learning calculus at higher secondary and first year at university level. The essay also includes a discussion on how such difficulties could be minimized and the teachers’ roles in doing that.

Learning calculus concepts
Function
The concept of function is central to the study of calculus. So the ability to comprehend function is crucial to understanding the concepts in calculus. Poor understanding of the underlying concept of function has been shown not only among mathematics students (Eisenberg, 1991), but also among teacher education students and teachers (Even, 1993). The apparent tendency of calculus courses to promote rote, manipulative learning without much understanding has been a subject of concern for more than a decade (Hassan, Mohamed & Mitchelmore, 2000). It might be entirely possible that the difficulties students face in understanding calculus are highly related to their concept of function, leading to narrow understanding of concepts such as derivative, limit and integral (Ferrini-Mundy & Lauten, 1993). Santos-Trigo’s (1998) findings offer further support that lack of understanding of elementary concepts is a major source of difficulty for first year calculus students. Santos-Trigo analyzed errors among first year students and discovered that many of the errors were due to lack of understanding of basic concepts such as functional notation, evaluation of limits and so on.
Therefore, it is extremely important for students to have a good foundation of the basic concepts of function in order to learn calculus concepts.
The role that functions play in mathematics is so extensive that it is impossible to give any summary that is both brief and adequate. There is no branch of mathematics whose developments since 1800 can be studied in their present form without an understanding of general function concept (Buck sited in Ferrini-Mundy & Lauten, 1993 , p. 156).
There is convincing evidence indicating that function concepts develop over time and even very young children can learn the basic ideas of function (Greeno, Leinhardt, Smith & Thomas cited in Ferrini-Mundy & Lauten, 1993). However research suggest that many students cannot think of a function as it behaves over intervals or in a global way rather they deal with functions pointwise, that is they can plot and read points (Bell & Janvier, 1981; Lovell, 1971; Monk, 1988).This suggests that it is essential to emphasize the development of the fundamental concepts of calculus like functions and ensure thorough understanding of these with different representations.

Limits, Series, Sequence and Continuity.
Limits, Series, Sequence and Continuity are the other important basic concepts that need to be developed in order to successfully progress with learning calculus. However, this also seems to be an area that students consider to problematic. According to Ferrini-Mundy & Lauten (1993), researchers examining limits, series, sequence and continuity indicate that students face conflicts between formal definitions and informal explanations that are used for convenience in discourse. Informal meanings of these terms also contribute to student confusion regarding these concepts (Ferrini-Mundy & Lauten, 1993). Additionally, some scholars like Tall & Schwarzenberger (1978 ) cited in Ferrini-Mundy & Graham (1991) have also observed that difficulties arise due to “ informal translation of very sophisticated ideas” (p. 630). Other research also indicate that students often have only a superficial understanding of some concepts like equating function with evaluation which in turn contribute to misinform students about the nature of these concepts.

Derivative
Derivative is one of the central concepts of calculus and as such it is essential that students have an in-depth understanding of the main concepts in this area. However, literature and research investigating the extent of understanding of concepts of derivatives suggests that this is an area that is riddled with difficulties and misconception for students. A large-scale qualitative study conducted by Orton (1983) cited in Ferrini-Mundy and Graham (1991) provides evidence to this effect. The study revealed that students had a good grasp of procedural aspect of differentiation but their understanding of differentiation of quadratic polynomials was very poor. Furthermore, the results of this study suggest that the majority of the students had limited “intuitive” understanding of derivative. Additionally, it was suggested that areas that students had most difficulty include “tangent as limit of secants” and “ rate of change of a line versus rate of change of a curve and rate of change of a point versus rate of change over an interval” ( p. 631).
The good procedural performance in computing derivate suggests that many students actually mimic examples provided in the text books and their understanding of geometric representations of derivate is limited. Therefore, it is essential to evaluate the kind of understanding students have and find ways to develop a deeper understanding of the main concepts in derivative in order to be successful in calculus at higher secondary and university.

Integrals
As with other areas of calculus discussed previously, the main difficulty that students come across in integration also relate the primitive understanding of the underlying concepts of integrals. For instance, integration of the limit of sum as well as the procedure of computing area or volume using the limit process does not seem to be central to students’ understanding of integrals. Additionally, many students are unable to reason out why methods to do with these areas actually work (Ferrini-Mundy 1991).
Based on his study, Orton (1983) cited in Ferrinii-Mundy (1991), suggests that students also posses misunderstandings regarding the underlying concepts of integration.
Thus, the main hindrance to success in calculus evidently relates to the lack of in-depth understanding of the concepts central to certain important areas of calculus such as function, integrals, derivative and Limits, Series, Sequence and Continuity. Students also lack the necessary “intuition” required to fully master the main concepts of these areas. As a result, calculus at higher secondary and university becomes exceptionally difficult for students.

Difficulties students face in learning calculus
Research indicates that calculus is an area in mathematics that students find problematic. For instance, a survey by mathematical Association of America Conference Board on mathematical sciences (Anderson & Loftsgarden 1987 cited in Ferrini-Mundy & Gaudard 1992) revealed that failure rates in the 1st year calculus courses were exceptionally high and only less than 50% completed the course with a D grade or better. It is possible to rectify weaknesses in other areas of mathematics like computation of algebraic problems which are important to calculus through remediation. Nonetheless, it makes one wonder why college students still find calculus difficult, what could be contributing to these difficulties and why such high rates of failure in calculus at university level happen. One possible explanation could be that calculus involves understanding of many concepts that are interrelated. For example, the extent to which students understand limits and functions will influence their subsequent understanding of derivatives and other fundamental concepts of calculus (Robert & Speer 2001). This means that if students do not have adequate understanding of one area of calculus, it is very likely that they will face additional difficulties in doing calculus. Additionally, some areas in calculus like functions and their associated sub-concepts can be very abstract to varying degrees. Students are unable to conceive functions and related notions visually and this in turn impedes understanding of these concepts (Eisenberg 1991).
The other reason related to difficulty in learning calculus at university level has to do with inadequate preparation of the students leaving high school. According to White & Mitchelmore (1996), the main concern lies in the fact that rote and manipulative learning takes place at high school. For instance, students might be good in calculating derivates and anti-derivatives as a result of continuous practice of similar procedures, but their understanding of concepts like rate of change is limited. Thus, students leaving high school have only a narrow understanding of concepts and as a result, they face more difficulties when it comes to application of these concepts at a higher level calculus courses.

Minimizing difficulties
The Mathematics teachers could play a central and an essential role in minimizing the difficulties that students face in learning calculus. For this to be operational, it is necessary that teachers have a good through understanding of the central concepts like limits, sequence, series and functions. However, a research done by Even (1993) indicates that knowledge of function of secondary mathematics teachers is limited. Teachers’ conceptions of function were similar to students and they did not hold modern conception of functions. Thus, in order to minimize the difficulties students face in learning, it is very important to train qualified teachers with good understanding of concepts in calculus like functions. Developing pedagogical knowledge is equally as important as developing mathematics content knowledge. Many of us tend to believe teachers having good content knowledge are good teachers. It does not necessarily have to be so. However, it is becoming widely recognized that teachers content knowledge has a strong influence on their pedagogical content knowledge, and hence on the learning of students (Thomas, 2003). Hence, teachers who have strong content knowledge and specific pedagogical preparation can create a learning environment that foster the development of students’ mathematical power (Even, 1993).
As discussed earlier in this essay, the major difficulty lies in understanding the main concepts of calculus such as functions, derivatives, integrals, series and sequence. Therefore teachers need to put more emphasis on developing concepts rather than spending more time with the procedural knowledge with repeated algorithms in the classrooms. One possible mean of achieving this is suggested by Ferrini-Mundy & Graham (1991). They suggest that activity based teaching could contribute to make lessons more engaging for the students thereby ensuring a deeper conceptual understanding of various areas in mathematics such as calculus. Questions that we present must engage the students through classroom, laboratory or project activities. Evidence supporting the effectiveness of activity based learning is presented in a study done by Heid (1988). The study compared two groups of students in an introductory calculus course. The control group spent more time with activity based learning. On the last three weeks were spent on skill development. On the other hand the other group spent entire 15 weeks on skill development. Class transcripts, student’s interviews and test results were analyzed for patterns of understanding. The result showed that students in the control group have better understanding of course concepts and even performed almost as well on the final examination as the comparison group. Their performance was remarkably suggestive that compressed and minimal attention to skill development was not necessarily harmful, even on a skills test.

Conclusion
Calculus is one of the most important areas of mathematics. As Young (1986) points out “Calculus is our most important course. The future of our subject depends upon improving it…” (Cited in Ferrini-Mundy & Graham 1991: 627). Despite the relative importance of calculus, research indicates that students leaving high school wishing to pursue mathematics at university do not have the necessary strong foundation in this area. Most of these students may be able do calculus problems in an automatic rote learned fashion. However, they lack the in-depth understanding of some of the central concepts and as a result have a major weakness when applications of these concepts are required at higher level courses. Thus, it is essential that relevant concepts in calculus are introduced to the students in some fashion at high school in order to develop a stronger base for higher level calculus courses. The first step in doing this is to examine if there exists a gap between the high school and the university curriculums.
Teachers can also play a crucial role in making calculus less difficult and more appealing for students.



References

Bell, A. (1981). The interpretation of graphs representing situations. For the learning of Mathematics. 2(1), 34-42.
Einsenberg, T. (1991). Functions and associated learning difficulties. In Tall, D.O. (Ed.), Advanced Mathematical thinking (pp. 140-152). Dordrecht, The Netherlands: Kluwer.
Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics education, 24, 94-116
Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and Learning Calculus. In Wilson, P.S.(Ed.), Research Ideas for the classroom: High school mathematics (pp.155-176). New York: Macmillan Publishing Company.
Ferrini-Mundy, J., & Graham, K. (1991). An over view of the calculus curriculum reform effort: Issues for learning, teaching and curriculum development. American Mathematical Monthly, 98(7), 627-635.
Hassan, I., Mohamed, K., & Mitchelmore, M. (2000). Understanding of Functions Among Maldivian Teacher Education Students. In Bana, J., & Chapman, A. (EDS.), Mathematics Education Beyond 2000. Proceedings of the Twenty –third Annual Conference of the Mathematics Education Research Group of Australasia Incorporated.1.291-298. Perth: Mathematics Education Research Group of Australasia Incorporated.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19(1), 3–25.
Lovell, K. (1971). Some aspects of the growth of the concept of a function. In Myron, F., Rosskopf, L. P., Steffe, & Stanley, T. (Eds.). Piagetian cognitive development research and mathematical education (pp. 12-33). Washington, D.C.:NCTM
Monk, G.S.(1988). Students’ understanding of functions in calculus courses, Humanistic Mathematics Network Newsletter, 2.
Ortan, A. Students Understanding of Differentiation. Educational Studies in Mathematics. 15 (1983a) 235-250.
Robert, A., & Speer, N. (2001). Research on the teaching and learning of calculus /elementary analysis. In Holton, D. (Ed.), The teaching and learning of mathematics at university level. 283-299. Dordrecht: Kluwer Academic Publiher.
Santos-Trigo, M. (1998). Patterns of mathematical misunderstanding exhibited by calculus students in a problem solving course. In Oliver, A. & Newstead, K. (Eds.). Proceedings of the 22nd annual conference of the international Group for the psychology of Mathematics Education. 4.89-96.
Thomas, M. O. J. (2003). The role of representation in teacher understanding of function, Proceedings of 27th Conference of International Group for the Psychology of Mathematics Education, Honolulu, Hawai.
White, P. & Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in Mathematics education. 27(1),79-95

Possible place of History of Mathematics in the Mathematics Classrooms.

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.