The Importance of Incorporating the History of Mathematics into the Standards 2000 and the Overall Mathematics Curriculum

The prevailing notion in society about mathematics is that it is a series of unrelated, irrelevant facts designed to make them miserable. The objective of mathematics, many believe, is to remember these facts, repeat the required procedures and get the right answer. And everyone knows that there is always "the" right answer. If you are not talented or able enough to get it "right", you are doomed to fail and be cast aside with the "unable to mathematics" crowd. This notion is mainly a byproduct of the way in which many students were taught mathematics.

What if this were not the case? What if somehow students were taught that mathematics did not arrive on this planet in a vacuum and that many of those responsible for much of the disciple took months, years and sometimes decades to "get it right?" Many of our highly regarded mathematicians were students, just like they are now, who struggled, failed, tried and tried again in hopes to solve a problem, prove a theorem, or simply make new and exciting mathematical connections and extensions. Frank Swetz writes in his article Seeking Relevance (1984): " We frequently find ourselves concentrating on the teaching of 'mathematics' - the symbols the mechanics, the answer- resulting procedures - without really teaching what mathematics is all about - where it comes from, how it was labored on, how ideas were perceived, refined and developed into useful theories - in brief its social and human relevance... ( This) will produce students who perceive mathematics as an incomprehensible collection of rules and formulas that appear en masse and threateningly descend on them (p. 54).

Incorporating the history of mathematics throughout the curriculum would be a useful starting point to show students the developmental nature of mathematics, enliven the teaching of the subject, and create an interest in a further study of mathematics. Myself and many noted scholars believe that if this information were incorporated into the curriculum, many misconceptions about mathematics would be eradicated. Students would see that mathematics is not an isolated field and would have a growing appreciation and admiration for the disciple and those who study it. This would translate into a broader student interest in the subject, the ability for students to connect mathematics to the larger society, and a better understanding of the content material.

This is a year for the history books in mathematics education. The long awaited draft of the The Principles and Standards for School Mathematics, affectionately called "Standards 2000," is off the presses and has made its way in the hands of the mathematics community. The charge of the Standards 2000 writing group was to build on the original NCTM Standards and consolidate the classroom aspects of all three of the original documents( NCTM, Standards, 1998, p. 13). As with most things in mathematics education, there is much debate on the content of the Standards and the impact it will have on our students education and achievement. The document gives us the six principles: the Equity, Mathematics Curriculum, Teaching, Learning, Assessment, and Technology principles and ten standards: (1) Number and Operation, (2) Patterns, Functions and Algebra, (3) Geometry and Spatial Sense, (4) Measurement, (5) Data Analysis, Statistics, and Probability, (6) Problem Solving, (7) Reasoning and Proof, (8) Communication, (9) Connections, and (10) Representation principles which should be interwoven in the mathematics curriculum from grades Pre - K through 12 . The Standards presented "describe what students should know and be able to do in mathematics and are statements about what is valued (NCTM Standards, pp. 13-14)." These principles and standards are designed to give students a well rounded, enriched and useful mathematics education which will foster higher understanding, appreciation and achievements in mathematics for all students.

As I read through these standards, I was very pleased with the expectations for our children and the prevailing theme that all students can learn mathematics at a high level. All students deserved to be give an education that is both rigorous and appropriate at all grade and ability levels. As I read this document, particularly the Equity, Mathematics Curriculum, and Teaching principles, and Standards 6 - 9: Problem Solving, Reasoning an Proof, Communications, and Connections Standards, I was amazed that throughout these discussions, the history of mathematics was not mentioned. The history of mathematics offers a lavish source of enrichment to the overall mathematics curriculum. It can be used to help students reason more critically, show them the progression of mathematics thought, and provide a means of communicating mathematical ideas. All in all, the history of mathematics is a rewarding study that can complement any mathematics classroom.

The History of mathematics offers a wealth of information that is crucial for incorporation of these standards into the classroom, and many of the goals outlined in principles and standards would be better met by incorporating history into the curriculum. In this paper, I will discuss some of the arguments against incorporating history in the mathematics classroom, the benefits of integrating the history of mathematics into the curriculum, where in the standards document should this be addressed, and how it could be interwoven into the document, particularly in the above mentioned principles and standards. I will conclude with some suggested activities and other material that can be used at each grade level which gives ideas of practical ways to use the history of mathematics in one's classroom.

History : Does it really serve our students?

There are many who criticize the incorporation of history into the mathematics classroom. They claim that given the time constraints of a typical classroom and the need to cover the content, the time used to study historical concepts in mathematics can be better spent learning the "material" i.e. algorithms and procedures. Victor Katz mentions in his article that one such critic, Professor Radford, who advocates the Simple Teaching Model which affirms that mathematical knowledge is essentially unhistorical. One can always link the mathematics accomplished in some historical period with the mathematics of today without worrying about the historical or cultural context in which the mathematics is developed
( Katz, 1997, p. 62). This would support the ideas that teaching history is non consequential and that our knowledge of a particular mathematical topic is the same as the knowledge of our ancestors. Hence mathematics is a island that has little or no relationship to the larger cultural context. Is this the picture we wish to convey to our children?

Many also argue that the teaching of mathematics has been "watered down" enough and that students must be expected to perform at the same level of rigor as in previous years. History and other reading and writing activities may take away from students learning "pure" mathematics. As H. Wu comments: "We must object to the reform ( in mathematics education) because it threatens to bring down the whole educational system. Indeed our students of today will be teachers of tomorrow so when university courses start to deteriorate our children will be taught by teachers who are mathematically worse - equipped than those of today (Wu, 1997, p. 951)." He continues by stating that the reform efforts favor weaker students, and the top students are being shortchanged. He and other critics fail to see the value of the reform as a tool to expand students' exposure and its ability to lead to greater academic achievement.

Critics declare that studying history has little or no mathematical relevance and is not useful to creating problem solvers. The way a student learns mathematics is by "doing" mathematics and everything else is secondary. When asked "When you consider what high school graduates should understand about mathematics, what do you care about the most?" Jashua Abrams from the Massachusetts Academy of Mathematics and science responded: " The only way to become good at solving real problems is to have continual practice. The only way to get good at posing new and interesting problems is to be required to do so. The best way to learn the mechanical skills of mathematics is to use them repeatedly
(AMS, Feb. 1997, p 200). What about those early mathematicians who developed the science? Did they only use mechanical manipulations to "learn" the mathematics? By given students such narrow experiences, we shortchange them and give them an inaccurate view of real mathematical competency.

Where would teachers find the time and resources to incorporate all this "stuff," and could this preparation time be better spent doing other things? There are many difficulties inherent in an inquiry-based environment. The demands on the teacher to be a lifelong learner, to serve as a resource, to share the authority for knowledge, to set the curriculum agenda aside when necessary, and to question and learn with the student constitutes a major shift in focus. This shift is necessary when using a variety of pedagogy in the mathematics classroom ( including teaching history) (D'Ambrosio, 1995, p. 772). These are some of the many arguments against many reform efforts including integrating history in the mathematics classroom. These criticisms are short cited and do not accurately reflect what is possible when expanding our students experiences. They fail to see what can be gained by teachers and students if a well rounded curriculum is adopted.

Why History?

Studying the history of mathematics , not in isolation but as an intricate part of the curriculum, has many benefits to the overall teaching and learning of mathematics. History is commonly taught in school to give the young an awareness of tradition, a feeling of belonging, and a sense of participation in the ongoing process of an institution. Teaching the history of mathematics can achieve similar goals, and teachers can lesson the stultifying mystique often associated with the subject
(Swetz, 1984, p. 54). Students will have more interest in mathematics and look forward to its unfolding and the next surprise in the human development of mathematics.

By introducing certain mathematical ideas in a historical context, students may be more motivated to learn mathematics and dismantle many psychological barriers that hinder them from learning mathematics. Many students become discouraged and frustrated when he or she has worked very long and hard on a problem and is unsuccessful, meaning her or she did not get the "right" answer. They must learn that this is normal and mathematical discoveries are usually gradual in coming and are the outgrowth of continual efforts by many men and women working over time. The gradualness of this process must be made evident to students ( Swetz,, 1997, p. 60).

Not only does the history of mathematics enhance students interest in the subject, it can actually be used to better teach the concepts of the course. By studying the development of certain mathematical ideas, students can deepen there understanding of a particular concepts. In her article, "History of Mathematics, Mathematics Education, School Practice: Case Studies in Linking Different Domains (1997)." Fulvia Furinghetti cited the work in classrooms of four teachers. After analyzing the activities of one teacher who offered an extra curricular, non-compulsory history course to high school students, She commented :

At first glance, this experiment seems mainly a way to implement a short course in the
history of mathematics. But in studying the students reactions, we see the educational potentialities of such a course and some of the educational outputs could be transferred by the teacher to normal curriculum activities. The work of teacher C shows that doing the history of mathematics is doing mathematics. Interpreting the unfamiliar (and old) language (French) of the text revealed itself as a good way of grasping the mathematical concepts and the procedures behind them. It obliged students to work step by step organizing their work and their reasoning carefully ( pp. 57 - 58).

Also the history of mathematics can also be used to show students the racial and cultural diversity of mathematicians and the cultural and political context that fostered the development of the discipline. Oftentimes learners are unaware of the contributions of minorities and women in the development of mathematics. Being unaware of this diversity, those in minority groups may feel that they are unable to excel and contribute in mathematics. This is not the case. Minorities and woman have been instrumental in many areas of mathematics, and these mathematical role models can serve as motivation for students in under represented groups to be more confident in their abilities. In Hofstra University's History of Mathematics course, there is an entire unit dedicated to the contributions of Black Mathematicians. In this unit, topic such as the impact of the African contributions to the history of mathematics, famous black mathematicians, the first four blacks to receive doctorates in pure mathematics, and Black women in mathematics are covered ( Knee & Barbera, 1989, p 3). Infusing this information at the college as well as the secondary levels can serve as a source of role modes and encourage African American students to continue their studies and mathematics.

History in the Standards Draft: Where Would it Fit?

Since Standards 2000 will shape the future of mathematics education, and we have highlighted many of the benefits of incorporating the history of mathematics into the curriculum, the next question becomes: How could the history of mathematics be placed in the Standards 2000 final draft? Initially I suggested that there be an addition of a "History of Mathematics" principle or standard. However, after reading the entire document and researching the subject, I concluded that by merging the history of mathematics into the principles and standards already given, the goals of the document would be better met. The particular areas that should be of focus are the Equity, Mathematics Curriculum, and Teaching principles and the Problem Solving, Reasoning and Proof , Communication and Connections standards.

The Equity standard states that mathematics instructional programs should promote the learning of mathematics by all students. This includes minorities, people with disabilities, women and other under represented groups. As a mathematical community, we should encourage all students to study mathematics at a high level and give them a wealth of examples to confirm that is indeed possible for them the achieve. "The use of positive role models and hands on learning experiences are ways that educators can use to help students realize their potential in mathematics. When feasible and wise, the teacher might wish to introduce the accomplishments of some renown mathematicians belonging to various cultural groups within his class. Past and present performances of such mathematicians should be emphasized (Dinkins, 1996, p.4)."

To encourage multiculturalism, history is necessary . When studying many topics, one must look outside of Europe for the first instances of many secondary- school topics. Pascal's triangle was used much earlier in China. The ideas of zero, negative numbers, formula for the area of triangles and quadrilaterals, volumes of pyramids, solutions of simultaneous equations, quadratic equations and other topics had its beginnings outside of Europe ( Nelson, 1993, p. 32). The misconception that mathematics grew primarily out of Europe can be erased, and people can see the its world wide origins and extension which can lead to a better appreciation of the subject by all.

The Mathematics Curriculum Principle reads : Mathematics instructional programs should emphasize important and meaningful mathematics through curricula that are coherent and comprehensive. By having students research and investigate mathematicians and mathematical ideas over the ages, the teacher is fostering this idea. For example, when teaching the complex numbers, the teacher could use history to give students a better understanding of the various ways they can be observed. Through history, students notice that the complex numbers can be viewed as: points or vectors in the plane, ordered pairs of real numbers, operators, numbers of the form a + bi, polynomials with real coefficients modulo x² + 1, matrices, and an algebraically closed, complete field. This might seem confusing rather than enlightening. It is of course commonplace in mathematics to gain a better understanding of a given concept, result, or theory by viewing it in as many contexts and from as many points of view as possible (Kleiner, 1998, pp. 587 - 588). As this example illustrates, using history helps reveal the coherency of mathematics , and although the study of history has an intrinsic appeal of its own, it can be instrumental to aiding in mathematical understanding.

Using history in the mathematics curriculum also answers the familiar questions: why are we learning this? and What is the purpose of knowing this? Dinkins (1996) quotes: "All students must probe the significance of and the justification for what they learn. Studying the history of mathematics can lead to better knowledge and understanding of the nature of mathematics. History can sometimes explain why we do things the way we do and how mathematical concepts, terms, and symbols arose. Knowing how and why a particular mathematical idea arose can help students appreciate the value of the subject and see why somebody might want to learn about it. Presenting chronological as well as logical reasons for a mathematical idea can help educators achieve that elusive goal of teaching for understanding ( p.3)"

Mathematics instructional programs depend on competent and caring teachers who teach students to understand and use mathematics (The Teaching Principle). What constitutes a caring competent teacher?. I believe that having a deep understanding of his or her subject, being able to foster a love for her subject to students, and having a genuine concern for his or her students academically and socially are qualities which separate a great teacher from an average one. Knowing how mathematics connects historical can give a teacher a more depth of understanding than can be transferred to her students. The teacher will also become more sensitive to the challenges of her students by understanding that even the greatest among us labored over material that we now see as "common" knowledge. The enthusiasm that the teacher gains will also allow him to engage the students and keep their attention which is necessary to promote understanding. "The teacher who knows little of history of mathematics is apt to teach techniques in isolation, unrelated either to the problems and ideas which generated them or the further developments which grew out of them... Mathematics can be properly taught only against a background of its own history ( Nelson, 1993, p.30)"

Most university teacher training programs require perspective mathematics teachers to take a course in the history of mathematics. The rationale for teaching this course is to expose teachers to how they can use history as a pedagogical tool to help their students construct their own knowledge of mathematics. Sometimes presenting a topic in an historical context is the best way to provide motivation and to present a mathematical topic. These "stories" can be used to enliven the teaching of the subject, and give our science flesh, blood, historical and human context and bring it to life ( Knee & Barbera, 1989, p.2)

As mentioned earlier, many critics claim that using history as a mode of discourse will not promote students being better problem solvers. This is not the case. As the Standards draft states: Mathematics instructional programs should focus on solving problems as a part of understanding mathematics so that all students (3) apply a wide variety of strategies to solve problems and adapt the strategies to new situations. Again, the history of mathematics is an excellent source of authentic problems for the development of problem solving skills. As one student evaluating a history of mathematics mini-course commented: " I ask myself more questions, I no longer keep to what I am told to study but try to reach a more through understanding. In tackling problems, I now proceed methodically, step by step, point by point. Things must be seen in depth, we must investigate their nature and not accept them as they are. I try to reason in a more systematic way.
(Furinghetti, 1997, p. 57)." In his article "The History of Mathematics as a Source of Classroom Problems," (1986) Swetz writes:

In the literature of mathematics, thousands of problems have been amassed and await a ready reservoir for classroom exercises and assignments. The use of actual historical problems not only helps demonstrate problem solving strategies and sharpen mathematical skill, but also: imparts a sense of the continuity of mathematical concerns over the ages as the same problem or type of problem can be found in different societies at diverse periods of time; and illustrates the evolution of solution processes - the way we solve a problem may well be worth comparing with the original solution process
( p. 34).

 The Reasoning and Proof standard affirms that mathematics instructional programs should focus on learning to reason and construct proofs as part of understanding mathematics . Two of the goals is to help students develop and evaluate mathematical arguments and proofs and to select and use various types of reasoning and methods of proof as appropriate. In the Reasoning and Proof standard, students are encouraged to recognize reasoning and proof as essential and powerful parts of mathematics, and justify their finding to others. In his article, "Thinking the Unthinkable: the Story of Complex Numbers ( with a Moral), Kleiner responded to the questions: Why the history of mathematics? Why bother with such "stories" as this one (referring to the evolution of the complex numbers)? One of the reason he gives is the following: Beyond the immediate objective of lending insight, this story and others like it may furnish us with a slightly better understanding of the nature and evolution of the mathematical enterprise. It addresses such themes as the nature of reasoning and proof in mathematics. This (story) not only addresses the evolution of the complex numbers, this item deals with the logical questions of proof and rigor in establishing various results about a concept. One thing is certain - what was acceptable as a proof in the seventeenth and eighteenth centuries was no longer acceptable in the nineteenth and twentieth centuries. The concept proof in mathematics has evolved over time, as it is still evolving (Kleiner, 1988,  pp. 589 - 590). Having this insight into the nature of proof in mathematics can be very motivational to students as they develop and justify their own reasoning. By having them see how proof has developed and how many "facts" about mathematics has taken many years and numerous approaches to verify, students can develop more patience with themselves and others while they make and investigate mathematical conjectures.

Using history as a foundation, students can be exposed to numerous examples of how the recognition of patterns was a critical "first step" to the proof of many rigorous and sophisticated theorems. These examples ca be used as classroom discourse and reasoning activities.

Communication is an important part of mathematics education because it is the way in which ideas are shared and the means by which we clarify concepts and promote understanding. Standard 8: Communication suggest that all students must extend their mathematical knowledge by considering the thinking and strategies of others and use the language of mathematics as a precise means of mathematical expression. Just like all language, the language of mathematics has been developed and shaped over time and in the framework of various time periods and cultures. As Jim Bidwell so eloquently puts it: "Three key items to be examined across all grade levels are communicating, connecting, and valuing mathematics. History allows us to study all three. Students can communicate about historical facts orally or in writing; they connect mathematics to various cultures as well as to other intellectual developments in science philosophy, and religion. History can substantially add to students' value of mathematics learned form the past and the present ( Bidwell, 1993, p 461). Students can also use historical references for classroom discussions, written and oral reports and other special projects. These activities will provide a rich assortment of communications in the classroom. Furinghetti (1997) agrees: "There are a number of reasons to incorporating the history into a mathematics course including promoting an enthusiasm for mathematics, enabling students to see mathematics differently, and to develop the skills of reading, library use and expository writing which can be neglected in traditional mathematics courses (p 10)."

Much of the Connection standard is geared to the student being able to make connections within the content of mathematics. This idea should be expanded to include how mathematics is connected to other areas of science, history, and other disciplines. "Many reasons can be given for teaching a course in the history of mathematics . One would be to expose the interrelationship of mathematics with other disciples: anthropology, sociology, economics, politics, music, arts, etc. ...(It can also be used ) to expose the interrelationship of seemingly diverse areas of mathematics (Swetz,, 1982, 696)."

How can the History of Mathematics be Incorporated in Pre- K through 12?

It may be a good idea to place history in the curriculum, but many educators are concerned with practical ideas and specific teaching suggestions for his or her classroom. One of the central points for a mathematics educator to keep in mind as he or she experiments is the need to consider larger pieces of mathematics than just a single idea or two in the use of history. One needs to think about how one can set an entire series of ideas or even an entire course in some historical context ( Katz, 1997, p. 62). By taking this into consideration, the teacher can appropriate integrate history in useful ways and the history is not viewed by students as a "outside" or "enrichment" but an essential and meaningful part of the curriculum. "In all cases, there is a lot of work for the teacher to set up lessons that use history. The teacher must know the history in sufficient detail to be able to pull out the details relevant to a particular class and arrange them in the best way... nevertheless, the teacher will find the exercise worthwhile. Using history to develop the concepts does work ( Katz, 1997, p. 63)."

The way history is incorporated is affected by may things; the grade level, the concept being discussed and the desired outcome of the instruction. In grades Pre K - 2, history can be used to teach the concepts of pattern recognition, counting, place value, addition and other operations with numbers. In a unit on place value, for instance, students could be given an account of the Egyptian, Roman, Chinese and Indian or Hindu Arabic developments of place value. They can compare the various systems, learn the numeral used, and how they are similar and different. By understanding the concept of number and its use and seeing how this concept has developed over the years, students really learning the language of mathematics. It also helps the see, again, the developmental nature of mathematics and place value representation truly demonstrates the key characteristics of mathematical thinking: efficiency, elegance, and exactness ( Sharma, 1993, p. 1) . These can be taught to children early in their academic careers using history.

In grades 3 - 5, History can also be used to convey ever evolving mathematical concepts.

For younger students, introducing Babylonian computation in base sixty positional system allows insight into our own base ten computation. While explaining this system, the teacher can use student- created multiplication tables in base sixty which can aid in student understanding. Many topics such as these can be found in primary history text books (Bidwell, 1993, p. 463)Bidwell

In studying algebra, the history of the common symbols and other notation used can be added to topics discussed. When studying linear equations of the form y = mx +b, the teacher can inject that a french geometer was the first to use the symbol m for slope, and the equation itself was first used in the book Treatise on Analytic Geometry in 1866. These additions can spice up a mathematical discussion and create a time line and "human link" to the mathematics itself. In the upper grades, history can be easily merged into a discussion of topics in geometry, trigonometry, data analysis, probability and statistics, and calculus. For example, in geometry when studying parallel lines, the story of Erathenes measuring the circumference of the earth might be appropriate. Before proving the base angles of an isosceles triangle are congruent, the teacher can discuss the Pos Asinorum. Telling students that in the middle ages this proof separated weak students from better students may provoke all students to attempt the proof ( Posamentier & Stapelman,  1990, p.42.) This resource also give a list of many references that can be useful to teachers when looking for historical relevance to mathematical topics.

The authors of the Standards must consider the relevance of the history of mathematics in achieving the goals outlined in the document. We will be doing our students a grave disservice if this issue is not addressed in the final draft. Students will be missing a critical part of the development of our subject and can be used to reiterate problem solving, communication, connections, and reasoning and proof in the classroom. I know NCTM understands this importance since it was them who in 197 wrote the following: Biographies of famous mathematicians are often appealing to students who welcome a human element in the classroom. The methods used by early mathematicians when making their famous discoveries may be of great interest to students. And of course the history of mathematics has an excellent supply of alternative methods of computation which offers insight into many current procedures. In addition, the origins of our present mathematical symbolism, terminology, and postulation symbols can be fascinating to students ( NCTM, 1970, p. 145). For the reasons outlined above, we must not let this critical piece of mathematics history not include the history of mathematics.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

References:

Bidwell, J. (1993, September). Humanize your classroom with the history of mathematics. The Mathematics Teacher, 86 (6). Pp. 461 - 464.

D'Ambrosio, B. ( 1995, December). Highlighting the humanistic dimensions of mathematics activity through classroom discourse. The Mathematics Teacher, 88 (9) pp. 770 - 772.

Dinkins, P. (1996). Multicultural Teaching Strategies for Simplifying Mathematical Concepts and Principles. Baton Rouge, Lousiana: Southern University.

Furinghetti, F. (1997, February). History of mathematics, mathematics education, school practice: case studies in Linking different domains. For the Learning of Mathematics, 17(1) pp. 55 - 61.

Katz, V. ( 1997, February). Some ideas on the use of history in the teaching of mathematics. For the Learning of Mathematics, 17 (1). pp 62 - 63.

Knee, D. & Barbera, J. ( 1989). Hofstra University Teacher Training Institute Dissemination Packet Booklet #6. Hempstead, NY: Hofstra University.

Kleiner, I. ( 1988, October). Thinking the unthinkable: the story of the complex numbers ( with a moral). The Mathematics Teacher, 81(7). pp 583 - 592.

National Council of the Teachers of Mathematics. (1998). Principles and Standards for School Mathematics Discussion Draft. Reston, Va: NCTM
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National Council of Teachers of Mathematics. (1970). The Teaching of Secondary School Mathematics. Washington, DC: National Council of the Teachers of Mathematics.

Nelson, D, Gheverghese, J. & Williams, J. ( 1993). Multicultural Mathematics. New York, NY: Oxford University Press.

American Mathematical Society. ( 1997, February). Notices of the AMS, 44(2). pp. 197 - 206.

Posamentier, A & Stepelman, J. (1990). Teaching Secondary School Mathematics 3^rd edition. Columbus, Ohio: Merrill Publishing Company.

Sharma, M. (1993, Jan & Feb.). Place value concepts: how children learn it and how to teach it. Math Notebook, 10 (1&2).

Swetz, F. ( 1986, January). The history of mathematics as a source of classroom problems. School Science and Mathematics, 86 (1). Pp. 33 - 38.

Swetz, F. ( 1984, January). Seeking relevance? Try the history of mathematics. Mathematics Teacher, 77 (1). pp. 54 - 62.

Swetz, F. ( 1982, November). What ever happened to the history of mathematics? American Mathematical Monthly, 89(9) pp. 695 - 97.

Wu, H. ( 1997, December) The mathematics education reform: why you should be concerned and what you can do. The Mathematics Education Reform. pp. 946 - 954.