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AUTHOR Singleton, Cynthia M. TITLE A Historical Overview That Led to the Fragmentation of College Mathematics Curriculum. PUB DATE 2002-04-03 NOTE 54p. PUB TYPE Information Analyses (070) Reports Descriptive (141) EDRS PRICE EDRS Price MF01/PC03 Plus Postage. DESCRIPTORS *Curriculum Design; *Educational History; Higher Education; *Mathematics Instruction; Postsecondary Education; Professional Development

ABSTRACT This paper investigates how and why the fragmentations of college mathematics have reached the current form through looking at the history of mathematics education. (KHR)

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U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement PERMISSION TO REPRODUCE AND of College Mathematics Curriculum EDUCATIONAL RESOURCES INFORMATION DISSEMINATE THIS MATERIAL HAS CENTER (ERIC) BEEN GRANTED BY AtTehis document has beenreproduced as ceived from the person or organization originating it. Cynthia M. Singleton* CI Minor changes have beenmade to improve reproduction quality.

April 3, 2002 Points of view or opinions stated TO THE EDUCATIONAL RESOURCES in this document do not necessarilyrepresent INFORMATION CENTER (ERIC) official OERI position or policy.

The major existing subdivisions of the present mathematics curriculum of post-

secondary education in United States are: Arithmetic, Algebra, Geometry, Trigonometry,

Calculus and Analytic Geometry, Statistics, and Complex Algebra. Recently many

educators argue that the mathematics is taught in isolation and that one of the problems in

the mathematics classroom is "the phenomenon that we can term the fragmentation of

knowledge', due to the fact that mathematics subjects are taught separately from each

other so that the result is islands in the learning process" (Furinghetti and Somaglia,

1998).Mathematics, which is difficult for many students, suffers from this situation and

more than any other subject, is considered to be separated from the cultural context.As a

result, the image of mathematics held by students is very poor: they think that

mathematics is a very boring subject, without any imagination, detached from real life

(Furinghetti and Somaglia, 1998).

To understand how and why the fragmentations of college mathematics have

reached the current form one must look at the history of the Mathematics Education.

Looking into the history the mathematics curriculum will clarify its boundaries, content,

or methods and will increase understanding of the present nature and values ofthe

discipline (Coxford and Jones, 1970, page 1)

*Department of Science and Mathematics, Southern University A & M, Baton Rouge, LA 70813

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While there is evidence of the existence of genuine prehistoric mathematics in this hemisphere (Mayans of Yucatan in Mexico and Central America had a remarkably well developed numeration system) the history of the curriculum of college mathematics in

America starts in1492. (Coxford and Jones, 1970, page 11). The effect of foreign influences upon the mathematics taught from the time of the earliest settlements to the middle of the nineteenth century was profound.

Other forces led to change and development, such as practical needs, church, religion, and intellectual curiosity (Coxford and Jones, 1970, page 13). The colleges founded prior to the Revolutionary war such as Harvard (1601), William and Mary

(1693), Yale (1701), Princeton (1746), Pennsylvania and Philadelphia (1766), and

Dartmouth (1770) did not have extensive requirements or offerings at first. Arithmetic was made an entrance requirement at Yale in 1745, at Princeton in 1760, and at Harvard in 1807. Harvard required the "the whole of arithmetic" in 1816 and later on in 1820 was the first university to require algebra. Geometry was not required for entrance until after the Civil War. As late as 1726 the only mathematics taught at Yale was a bit of arithmetic and surveying in the senior year. In 1748 Yale required some mathematics in the second and third years. Something of conics and fluxion (calculus) was taught as early as 1758, and by 1766 a program might have included arithmetic, algebra, trigonometry, and surveying. At Harvard the mathematical course began in the senior year and consisted of arithmetic and geometry during the first three quarters of the year and astronomy during the last quarter. It is interested to note that at that time each class concentrated for a whole day on one or two subjects. Mathematics or astronomy was studied on Monday and Tuesday (Coxford and Jones, 1970, page 19). The inclusion of

3 3 astronomy with mathematics was typical of the times andresulted from both practical forces and the intellectual climate. The practical need for astronomy in connectionwith navigation and surveying is obvious. At the same time, the European intellectual revolution stemming from the rise of science and the work of Copernicus, Galileo,

Descartes, Newton, was communicated to the colonies quickly. By 1776 sixof eight

colonial colleges supported professorships of mathematics and natural philosophy

(Coxford and Jones, 1970, page 19). In 1850 the Military Academy at West Point, the

first technical institute in the United States introduced the study of analytical

trigonometry (Coxford and Jones, 1970, page 29). Influenced by West Point the

University of Michigan in 1837 adopted textbooks in Arithmetic, Algebra, Geometry,

Conic Sections and Mathematics for the Practical Man.

During this time the introduction of the idea of graduate education had a profound

effect on the development of the mathematics curriculum in the United States.

Americans developed the tendency to do research in pure mathematics and in thegeneral

area of the foundation of mathematics rather thanin applied mathematics, which became

a real national handicap during the time of WorldWar II. The idea of graduate studies

and specialization associated with research and the creation of new mathematicsis related

to the founding of Johns Hopkins University in 1875, the foundingof the American

Journal of Mathematics and the founding of the American Mathematical Society

established in 1888 (Coxford and Jones, 1970, page 30). The emphasis on the function

concept and on interrelationships within mathematics introduced in1893 had a profound

effect in the development of college mathematics (Coxford and Jones, 1970, page 41).

Psychological research and changing theories of learning were the major forces that

4 4 influenced the mathematics curriculum during this period. The works of Jean Peerage,

Jerome Bruner had great influence on the on the mathematics curriculum (Herrera, 2001)

In the years that followed educational agencies such as NSF and Educational organizations played an important role in the changes of the mathematics curriculum that followed. The National council of Teachers of Mathematics founded in 1920 is responsible for many curriculum reforms. They are responsible for the "new math movement" of 1960-1970. During World War II, both educators and the public recognized that more technical and mathematical skills were needed to meet the needs of the developing technological age. The Commission on Postwar Plans appointed by

NCTM made recommendations about mathematics curriculum. The goals were to establish the United States as a world leader and to continue the technological development that had began during the crises of war. The movement was further supported with the launch of Sputnik in 1957 because it crated the perception that the

United States was "behind in the world scene" of technology and military power

(Herrera, 2001).

In 1986, the Board of Directors of the National Council of Teachers of Mathematics

(NCTM), recognizing the alarming need for reform in the way mathematics was taught, established Standards for the curricula of schools (Martinez, 1998).

The above historical overview of the development of mathematics curriculum in the United States suggests that some of the major factors that led to the fragmentation of mathematics and influenced the development of the mathematics curriculum are: practical needs, church, religion, and intellectual curiosity. Many educators now support the idea that the disciplines of the school curriculum should be viewed as a unitary body 5 of knowledge. A teacher who looks at the school curriculum with "interdisciplinary eyes" can discover a variety of situations to which mathematical methods can be efficiently transferred in a natural way (Furinghetti and Somaglia, 1998).

When looking at the school curriculum this way the role of mathematics changes. This subject really becomes the center of students thinking and hopefully they establish a richer image and understanding of the discipline (Furinghetti and Somaglia,

1998). Interdisciplinary (between mathematics and other disciplines and within mathematics) in the classroom would help students understand the power and beauty of mathematics and perceive it as interesting and useful subject.

6 6

Reference

Coxford, Arthur F. and Jones, Phillip S., "The Goals of History: Issues and Forces," THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, Thirty-second yearbook, A History of Mathematics Education in the United States and Canada, p. 1 10, 1970.

Furinghetti, Fulvia and Somaglia, Annamaria, History of mathematics in school across disciplines, Mathematics in School v27 no4 Sept. 1998. p. 48-51.

Herrera, Teresa; Owens, Douglas T., The 'New New Math'? Two Reform Movements in Mathematics Education, Theory Into Practice, Spring 2001, Vol. 40 Issue 2, 484, 9p.

Martinez, Joseph G. R., Martinez, Nancy C., MATHEMATICSStudy &teaching United States; NATIONAL Council of Teachers of Mathematics; EDUCATIONAL changeUnited States, Mathematics Teacher, Dec. 98, Vol. 91 Issue 9, p746, 3p. Usage is subject to the terms and conditions of the subscription and License Agreement and the applicable Copyright and intellectual property protection as dictated by the appropriate laws of your country and/or haernational Convention.

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Record 1 of 1 in Education Full Text 6/83-12/99 TI: History of mathematics in school across disciplines AU: Furinghetti,-Fulvia; Somaglia,-Annamaria PEI: Y JN: Mathematics-in-School SO: Mathematics in School v 27 no4 Sept 1998. p. 48-51 PY: 1998 PD: bibl IS: 0305-7259 LA: English AB: Part of a special issue on the history of mathematics. The writers describe how the history of mathematics illustrates the development of interdisciplinarity in schools. They provide examples in order to clarify the meaning of interdisciplinarity and the real benefits that mathematics in schools may derive from it. The projects they consider involve students aged from 13 years up. The research the writers provide illustrates that interdisciplinarity, even if differently approached, overcomes fragmentation of knowledge and renews the cultural dignity of mathematics, is characterized by strong interaction within a team of teachers involved in interdisciplinarity, and has the history of mathematics as a key element. DE: Mathematics-History-Teaching; Mathematics-Correlation-with-other-subjects DT: Feature-Article TX': Y SZ: 24927 TX1: THE 'FRAGMENTATION OF KNOWLEDGE' AND THE IMAGE OF MATHEMATICS One of the problems in classroom life is the phenomenon that we can term the 'fragmentation of knowledge', due to the fact that school subjects are taught separately from each other so that the result is islands in the learning process. In particular, mathematics, which is difficult for many pupils, suffers from this situation and, more than other subjects, is considered to be separated from the cultural context. As a result, the image of mathematics held by pupils is very poor: pupils think that mathematics is a very boring subject, without any imagination, detached from real life. This problem is very old. In the Association for the Improvement of Geometrical Teaching, there is the following verse quoted in the Daily News of December 16, 1892 as being often found written in a schoolboy's Euclid or Algebra: 'If there should be another flood, Hither for refuge fly, Were the whole world to be submerged This book would still be dry' (p. 23). The author observes that 'there is strong reason to believe that these lines accurately set forth the opinion that is usually entertained' (Heppel, 1893 p. 24). He is also pessimistic about the possibility that the average pupil succeeds in seeing the beauty and the poetry in mathematics by himself. He rather thinks that 'the schoolboy's charge of dryness must be met in another way, by showing him how the progress of the Arithmetic, Geometry, Algebra, and Trigonometry that he is learning has gone on in answer to the needs that men have felt, and the desires they have formed' (Heppel, 1893, p. 24). In those times this opinion was widespread in the world of educators and of professional historians of mathematics(FN1).

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Among historians the Italian, Gino Loris, who was also very interested in problems of mathematical instruction4maglylupported the introduction into schools of what he terms Interdisciplinarity, i.e. a way of teaching which emphasizes the peen a disciplines. In theamer (Loria, 1890) he mints out the importance of history in promoting it. The basic idea in the two papers we have mentioned is that history of mathematics can help to see the genesis of ideas and the connections among the various subjects, thus making the culture transmitted in school a homogeneous body of knowledge. ACHIEVING INTERDISCIPLINARITY IN THE CLASSROOM: AN OUTLINE OF POSSIBILITIES AND IMPLICATIONS We can find the preceding ideas, which were enunciated in theory by the authors of the past, tried out in more recent times in different research projects. in the following, we briefly discuss some examples in order to make clear what iinterdisciplinarity' means and which real benefits mathematics in school may derive from it. The projects that we consider concern pupils aged from 13 years onwards. We describe how history can help to go across disciplines rather than focus on issues specific to the particular situations in which the projects took place. In Grugnetti (1989), we find an example of an interdisciplinary activity involving pupils aged 14-16 years. Various historical periods are studied from different points of view: the point of view of the language used and the literature, the social and political situations of different countries, the technological development, and so on. According to the author, this initiative originates from the need to reveal a 'really humanizing aspect' of mathematics. Mathematics is not predominant over the other subjects and the teachers of all the subjects involved participate in the work. The conclusions of the paper do not show if the objective of emphasizing the 'humanizing' aspect of mathematics has been attained; we can only infer a more general result about 'humanizing' classroom life and communication among teachers of different subjects. An analogous project in France with pupils aged 14-15 years is described in Guichard and Sicre (1988). As a starting point, a mathematician who was born in the region where the school is set was considered. They were very lucky since this mathematician was Francois Viete, a key character in the development of algebra. The authors state very clearly the objectives of their work in the classroom (in addition to the basic objective of making pupils aware of an important piece of local mathematical history): to make pupils interested in an interdisciplinary approach; -- to give a human dimension to sciences, presenting a part of mathematics (algebra) that is developed in the classroom within a historical perspective; -- to show that mathematics (and its presence in school as a subject) is evolving all the time; to work in mathematics classes on real problems on which mathematicians have worked, problems similar to those which generated the algebraic calculations. The subjects involved in this project were architecture, sculpture, music, painting, ways of writing, history and geography of the 16th century. The authors point out two consequences of this experiment. On the cultural side, it appears that after the project there was a different attitude to the 16th century and to the life of the region. On the mathematical side, some basic algebraic concepts such as variables and parameters were discussed; thus it was possible to easily shift the focus of the algebraic teaching from pure manipulation to conceptual issues. The problem of the image of mathematics held by pupils aged 17-18 years is the motivation of the experiment described by Rommevaux (1988). In this case the teachers of the mother language (French) and of mathematics planned and carried out together classroom work in which pupils analysed texts by Blaise Pascal and by Jean-Jacques Rousseau. The links between mathematics and literature, which in the usual way of looking at mathematics would be weak, reveal themselves as an interesting source of reflection on the nature of mathematics and its teaching, as it is shown in Furinghetti (1993). There are two subjects, philosophy and art, that we often find associated with mathematics in history. A typical case of work involving the mathematics and the philosophy teacher concerns discussion of the concept of infinitesimals and infinity centred on the Zeno paradoxes. In Guichard and Sicre (1995), there is a description of an experiment involving high school pupils. As for mathematics associated with art, there are many experiments which link these two subjects. In Brin at rd. (1995), an activity with 12- and

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31EST (COPY AVAILABLE 16-17-year-old pupils is described centred on the study of the methods used by Renaissance artists for representing objects of three dimensional space(FN2) in the two-dimensional plane. In this way, pupils come to know how the mathematical theory of perspective was born from the practice of art. Moreover, they acquire a precious 'key for looking at masterpieces of are In the paper by Menghini (1989), the link between art and mathematics is illustrated, not only through the study of Italian artists of the Middle-Ages and the Renaissance but also through the study of the use of conics in Roman Baroque architecture (the experience was actually carried out in Rome). The topic 'conic sections' has evident connections with the orbits of the planets which were studied around the period in which the Baroque churches were built. Thus two kinds of skythe metaphorical domes of the churches and the real astronomical skylink mathematics, art and astronomy. In the paper by Menghini (1989), other aspects of the connections between mathematics and art are discussed through the work of the Dutch painter M. Cornelius Esther. The well-known painting inspired by the Poincare model of non-Euclidean geometry was used in the classroom to introduce pupils to problems of non-Euclidean geometries and to motivate a discussion on the nature of space as it developed in the 19th century. All of the previous experiments have been carried out in the classroom and were supported by worksheets, notes and other materials. We can see from these examples that interdisciplinarity, even if differently approached, has some common characteristics: The fragmentation of knowledge mentioned at the beginning is overcome and mathematics regains its cultural dignity. The role of mathematics is seen from another point of view: it is no longer viewed as a. discipline developed to serve other disciplines but is now seen, together with the others, as a discipline for solving problems. There is a strong interaction within the team of teachers (usually in the same school) involved in interdisciplinarity. This interaction is very deep, since it does not only mean working together, but it mainly means to fix common objectives and to find efficient agreement on the methods for pursuing them. Interdisciplinarity reveals itself as a real way to overcome the isolation of which teachers often complain in their work. This fact is one of the factors pointed out by the teachers themselves who have carried out the experiment. History of mathematics is a key element in going across disciplines. INTERDISCIJPLINARITY AND TRANSFERABILITY OF THE MATHEMATICAL METHOD To focus on the deep meaning of interdisciplinarity and on the role of history in promoting it we analyse an experiment carried out in an Italian high school, the Lice° Scientific°. In this typeof school, mathematics has an important role: its programme includes Euclidean geometry, calculus, trigonometry, some elements of logic and computer science. Italian, train, physics, art, aforeign language, chemistry, sciences and philosophy are the other subjects taught. One of us (Somaglia) is a mathematics teacher at this school. In addition to the considerable cultural demands of the official programmes she has to face the students' difficulties in doing mathematics; thus she has tried different ways of teaching to overcome these difficulties. The usual strategy she uses to introduce mathematical concepts or procedures may be schematized in the following diagram: (Graphic character omitted) The problem is where and how to find a suitable context in which the pupils' intuitive ideas' may arise. The history of mathematics has revealed itself to be very suitable for this purpose, as may be shown in the following example in which the teacher has to introduce the concept of integral, which is very difficult for pupils. Before this introduction she discusses in the classroom the concepts of area and volume, which are more familiar to pupils. Afterwards she presents the principle stated by theItalian mathematician Bonaventura Cavalieri in his treatise Exercitationes Geometricae Sex (published in Bologna in 1647). This is Cavalieri's statement(FN3): ' Two figures either plane or solid are in the ratio of all their indivisibles compared collectively and(if among them you find some unique common relationship) one by one.' These words convey efficiently the idea of obtaining the area of a plane figure by sectioning it into 'infinitesimal elements' which are summed up afterwards. Cavalier's idea is very appropriate for preparing pupils for subtle reasoning and the obstacles in constructing the concept of integral, since the strategy of

30£6 44/02 3:57 PM COIFT AVAILANLIE 10 sectioning a figure, as Cavalieri suggests, has an intuitive character and hasa very telling graphical interpretation (see Fig. 1). The context in which to develop the intuitive stage of pupils' learning is not necessarily to be found within mathematics. The teacher has used other disciplines that she considered suitable fbr thispurpose, as we shall see in the following cases. The first case concerns proof. The teacher ascribes great educational value to activities connected with proof but, as many colleagues do, she has to face the difficulties encountered by her pupils in the different phases of proving (conjecturing, exploring, validating conjectures, giving formal proofs). In particular, she has noticed that, in learning to prove, pupils have difficulty in grasping the overall structure of the proof at the semantic level. This lack of meaning implies pupils' inability to build proofs on their own since, even if they are able to reproduce the various formal passages, they are not able to understand what theyare doing. To help pupils see the meaning of a proof, the teacher again looked for a suitable context in which pupils can work informally to activities linked to proof. Having in mind the ancient roots of mathematics and philosophy, she asked the philosophy teacher to involve pupils in an activity of analysis with 'mathematical eyes' of passages taken from works of the Greek philosophers. The colleague agreed to participate in the experiment. Here are some examples of the tasks given to pupils: I. Give the structure of the proof ab absurdo (using the Eleatic axiom 'it is not possible to talk about nor think of what does not exist') through which Parmenides concludes that 'being is without end'. 2. On what assumptions are Socrates' arguments basedthe ones that you have read in Plato's dialogue 'Gorgias' to prove that 'good' and 'pleasure' are the same thing? Outline the fundamental points of Socrates' arguments. Describe and compare the positions of Socrates in Plato's dialogue 'Protagoms' and that of Callicles in Plato's dialogue 'Gorgias'. The aim of this activity is to accustom pupils to see the mathematical structure (logical sequences, hypotheses and conjectures, ...) in domains different from that of formal proof. In this way, it is hoped that pupils learn to pay attention not only to the syntactic, but also to the semantic aspect of proof. The pupils' arguments show their efforts to apply mathematical methods to a new context and also to use mathematical-like notations and structures. The following is an example of the structure of the reasoning ab absurdo (to deny what is to be proved leads to denying the Eleatic axiom) made by a pupil in the task: (Graphic character omitted) To work in different contexts (the mathematical and the philosophical) reinforces the pupils' ability to analyse a text and both the subjects in question (mathematics and philosophy) may benefit. Philosophy gains a scientific approach to its theories, mathematics acquires a field in which to exercise the ability to decode colloquial language and to grasp the semantic aspect of a proof. In working together the mathematics and the philosophy teachers help pupils to overcome the psychological barrier of the language. Thus we have a two-way transfer of methods from mathematics to philosophy and vice versa. In other situations art has been the discipline offering the informal context in which to develop pupils' 'intuitive ideas' on the mathematical concept 'geometric transformations'. A film (by Michele Emtner) on the work of the painter Escher was shown to pupils. This film emphasizes the strong connections of Escher's work with mathematics and allows discussion of the geometric transformations arising from a stimulating situation. After having analysed the mathematical content of Escher's work, the teacher posed a question: why has this painter taken inspiration from mathematics? The teacher, who is familiar with the history of mathematics, has introduced pupils to the methods used by historians. Letters written by the painter and people he met on his travels were considered. It emerged that from the start he knew from his brother, who was a biologist, that the geometrical transformations which were the leitmotif of his art were studied in crystallography. Afterwards he became interested in the mathematical research related to his art; we know, for example, that he had read the studies in crystallography of George Polya. Similarly, in the mathematical milieu, Eschees work was studied by the scientists and new theoretical developments in crystallography were stimulated by Eschews paintings. In this case it was the historical method which linked the different subjects (mathematics and art) to the benefit of both subjects. History of art gained

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knowledge of the genesis of a style, mathematics appeared as a subject which may be seen as truly a pan of human culture. Escher's painting was also the key to enter into another branch of human knowledge which has connections with mathematics, i.e. music. In his writings, especially in his letters, the painter points out the 'analogy' of Bach's canon and his way of filling the plane ofthe painting (using symmetry and antisymmetry). What makes Escher's attitude interesting is that, unlike Bach in music, the painter is fully conscious that he is working on concepts which are common to various disciplines (art, mathematics and music). Starting from the autobiographical notes of the painter on the relation of his painting to music the teacher has gone further in interdisciplinary teaching and has explored the rational construction of the music, making pupils aware of the important fact that 'several composers used mathematical ideas to structure their compositions' (Fame!, 1996, p.244). With this activity the teacher has emphasized that there is a concept on which they are working (symmetry) which is common to various disciplines (see Clavarino and Sontaglia, to appear). CONCLUSIONS In Galileo's works there is a passage in which the author complains of the criticisms of certain scientists of his time, who claim that mathematical ideas have to be developed separately from other subjects such as physics or philosophy. Galileo writes that truth is unique and is part of every discipline(FN4). This view applies to the school environment too. The examples that we have presented in this paper are inspired by the strong conviction of the need to demolish the barriers between subjects (and between teachers in the same school, too). Our intention was to suggest to teachers and readers that they consider the disciplines of the school curriculum as a unitary body of knowledge. A teacher who looks at the school curriculum with 'interdisciplinary eyes' can discover a variety of situations to which mathematical methods can be efficiently transferred in a natural way. We can say that the teacher's attitude has to be similar to the attitude of Eicher himself who wrote explicitly that he found inspiration for his creativity by abandoning himself to the emotions of Bach's music. Theaeontibution of the history of mathematics to this process of going across disciplines is in the opportuniit offers to lay b----are&aitin ftheTliircip mess By this w a y of looking a t t 1 R S c h o o curnculum, the role oUrnat elEi"--`nati.s. This subject really becomes the centre of pupils' thinking and, we hope, pupils establish a richer image of the discipline. Added material Authors Fulvia Furinghetti, Dipartimento di Maternatica dell'Universita, Genova (Italy). e-mail: [email protected] Annamaria Somaglia, Liceo Scientific°, cio Convitto Nazionale 'C. Colombo', Genova (Italy). ACKNOWLEDGEMENTS The authors would like to thank Marcella Bacigalupi, the teacher of philosophy, who supported the project referred to in the text and also John Earle for the accuracy and owe taken in the editing of this article. The editor wishes to thank Helen Easton for assistance with translation and interpretation. Fig. I The original illustration of Cavalieri's principle FOOTNOTES I It is worth noting that in those times there was a great ferment of initiatives in the school world (the Mathematical Association was established in 1894) and history of mathematics was assuming its present status of an autonomous discipline with specific journals and specialized researchers. 2 An interesting discussion of Piero della Francesca's skills in mathematics is in the paper by Field (1993).

3 The translations of the various passages are made by us. 4 See Enriques (1938). REFERENCES Brin, P., Gregoire, M. and Halle; M. 1995 'La Naissance de la Perspective au XVeme siecle'. In IREM de

5 of 6 4/81023:57 PM 12 3335 rir COPT MAIILABILE Montpellier (Ed.), History and Epistemology in Mathematics Education,195-213. Clavarino, F. and Sontag lia, A. Una Stessa Struttura Matematica inSettori Diversi (to appear). Enriques, F. 1938 Le Matematiche Nel la Storiae Nel la Cultura, Zanichelli, Bologna. Fauvel, J. 1996 'Mathematics and Music'. In Logarto, M. J., Vieira,A. and Veloso, E. (Eds), Proceedings of History and Epistemology in Mathematics Education(Second European Summer University, Braga), 1, 241-248. Field, .1. 1993 'Mathematics and the Craft of Painting: Piero della Francescaand Perspective'. In Field, J. and. James, F. A. J. L. (Eds) Renaissance and Revolution, Cambridge UniversityPress, 73-95. Furinghetti, F. 1993 'Images of Mathematics Outside the Community ofMathematicians: Evidence and Explanations', For the Learning of Mathematics, 13, 2, 33-38. Guichard, J.-P. and Sicre, J.-P. 1988 'Activites Interdisciplinairesen Premier Cycle a Propos d'un Mathematicien Francais du 16eme Siecle: Francois Viete', Powuric Perspective Historique dans l'Enseignement des Mathematiques, Bulletin Inter-hem Epistemologie,249-270. Guichard, J.-P. and Sicre,.1.-P. 1995 'Etude de Notions Mathematiquesa Partir Male Approche Historique et Philosophique'. In IREM de Montpellier (Ed.), History and Epistemology in MathematicsEducation, 191-194. Grugnetti, L. 1989 The Role of History of Mathematics inan Interdisciplinary Approach to Mathematics Teaching', Zentralblatt fir Didaktik der Mathematik, 21,133 -138. Heppel, G. 1893 The Use of History in Teaching Mathematics', NineteenthGeneral Report of the Association for the Improvement of Geometrical Teaching, W. J. Robinson,Bedford, 19-33. Loris, G. 1890 'Rivista bibliografica: Treutlein, P. Das geschichtliche Element inmathematischen Unterrichte der hoheren Lehranstalten', Periodico di Matematica, 5, 59-61. Menghini, M. 1989 'Some Remarks on the Didactic Use of the History of Mathematics'.In Bazzini L. and Steiner, H.-G. (Eds) Proceedings of the First Italian-German Symposiumon Didactics of Mathematics, 51-58. Rommevaux, M.-P. 1988 'Une Experience d'Enseignement lnterdisciplinaire"Francais-Mathematiques" en Premiere Al', Pour tine Perspective Historique dans l'Enseignement des Mathematiques, Bulletin Inter-kern Epistemologie, 199-229. The magazine publisher is the copyright holder of this article and it isreproduced with permission. Further reproduction of this article in violation of the copyright is prohibited. UD: 19981110 AN: 199802887700

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ACK NOW LEDGMENTS Psychologyumicridationsion by ofThe Arithmetic, for Macmillan the Training by Edward of Teachers L. Thornlike. Publishedof Mathematics, in by the Rcorkmizationiticalumittee AssociationA .on the of UndergraduateofMathematics America, caSt. in Program Secondary Education, Reprintedby the Na- by permission of thein Mathematics of the Math- Table of Contents tociationyrttal Mathematics.of Committee the of CommissionAmerica, onPublished Mathematical1913. on Reprinted Mathematics:in tosoProgram by the for CollegeCollege Entrance Requirements of the Mathematical by permission of the MAA. Exarnina-Prepare- altn Board, rtof oftheCarleton the F. NewNational B.Society's York.Knight. Washburne. Society CommitteeAlso, for 0"The thetor by Gradeon the Society. Study of Education.Placement "Introduction." Arithmetic,of Arithmetic Twenty-ninth Topics," Year- Preface ucationalulumwebRevolution and Research ResearchDevelopment in in the Services,Mathematics," Schools. toward Inc. "Thethe by NewE. Improvement of Education. "Cur- Curricula,"G. Bogle. by 0 Evans 1969 byClinchy. Dembar by Evans Clinchy. Acknowledgments vi sociationfenceofblished Science and by for MathematicsandHarcourt, the Mathematics Advancement Brace 'leachers," (1946).& World, of Science. by Inc.a © 1964 "The Preparation of High School Mathematics Teachers.committee of the American Reprinted by permission The Goals of History: OurIssues Goals and Forces Phillip S. Jones and Arthur F. INTRODUCTION Coxf ord, Jr. ondatyG:the Toronto, TheCentral School Making Associationan1955. Curricular Ontario of a Curriculum, Experiment,"Changeof Science in byandCanada by Harry Pollen. Dissertation; William Wooton. Published by with Special Emphasis University SummaryProgressOur Themes: byand. PeriodsForces Preview and Issues 4 7 Riesde MordterUniversity Study -Upton B. Group.Upton. Press.AritInnetics, © 1965.!Loa by Used the Americanby Higher Grades. by Geoige D. Strayer and permission of the SchoolWheat. Mathe- PublishedBook by Co. The Uni- MathematicsPhillip S. Jones in the Evolving Schools PART and Arthur F. Coxf ord, Jr. ONE e TeachingrsityEducation.Teaching of Chicago of Mathematics of Arithmetic, Press. ® top by byHarry G. in Ontario," by W. B. Gray. the National Society for the Study Dissertation, a From1492azi Discovery to Thean AwakenedContentEarliest Americanand Concern Processes Mathematics for of Pedagogy;Colonial Instruction tt /3 11 Bredniversityladesthing A. Knopf,the of0. Toronto, Transformation Dalrymple.New inc. Arithmetic, 1948. © Imo of bythe by Guy School, M. by Lawrence McGraw-Hill Book Co. Wilson, Mildred B. Stone, and A. Crentin. 41 by The Latinix Grammar School and the Academy 18 CONTENTS CONTENTS xi iutThe;elf-Instruction)he lllll CollegesTeachers ary and and the Pedagogy Mathematical 22 20 21 Community 18 Forces and Issues Related to Curriculum and Instruction, K-6 M. Vere De Vault and J. Fred Weaver PART TWO ForcesTheow ColburnElementary at Work to24 Schoolsthe Rise of the Universities: 182 I-94 25 24 7 Elementary MathematicsThe ContinuingMajor Education: Periods Issues An Overview 9; 94 93 SummaryTheTeacherThethe Climate SecondaryColleges Training of Reformand and Schools the Pedagogy Mathematical 33a7 30 Community 28 8 From1607-1894 Settlement toThe the Continuing End of the Forces Nineteenth Century: 95 98 Therst ForcesSteps Elementary toward at Work Schools Revision:36 1894-1920 37 36 WhyIntroductionHowWhat Teach Should Mathematics Mathematics? We Organize Should theBe Taught?Mathematics We Teach? 98 91 44 tn; Pedagogy'DmSummary CollegesIke Secondary and and Teacher the SchoolsMathematical Training 42 13 39 Comnnmity 41 SummaryHow Should of the Period Instruction 16o7-m894Implemented? in Arithmetic Be Organized and too 1o5 hortiveTheA Period Elementary ReformDepression of Unrest School 46 48 and War: 1920 -45 46 9 Schooling:Scientism and 1894- Changing tor 3Evidences Conceptions of Change of Elementary 1o7 sop 107 a) TheTime SeniorWarJunior Junior and High Its College SchoolEffects 6o 58ft13 and the Mathematical SummaryWhatWhyHow TeachMathematics Should Mathematics? We Should Organize We andTeach? Implement Instruction? 116 t to 114 TheSummaryPsychological Colleges, Teacher and EduCational CTraining, Theories nity 63 63 to ProgressiveMathematics Education: EducationForces 1923-52 during at Work the Rise and Fall of t 18 dorm,NewA Buildup "Revolution," Forces of Pressures Reaction: for Reform: 68 1945-:PresentOld Forces 67 67 HowWhatWhy Should TeachMathematics Mathematics?We Organize Should theandWe Mathematics Teach?Implement Instruction? We Teach? /so 122 #27 :25 TheTeacherThe Prelude SecondaryElementary Training, to Refornt School School Certification, 78 7s76 and the Conversion to t 1 Designing a ContemporarySummary Elementary School 13r PsychologicalReactionCollegiateSummary Madtematics 86and EducationalNew Programs Theories 79 81 81 . 8; Mathematics Program:WhatWhyIntroduction 1952-Present Teach Mathematics Mathematics? Should We Teach? 133 34 135 233 CHAPTER ONE IssuesThe goals and ofForces History: There are many 1. Our goals kinds of history serve for some directed toward many as an end in itself, goals. and for The othersPatriotschronology nationalism,as the or to The history chauvinistsof events may maybasis of forecasts justify past of a special actions,seek to with create historyor offield may seek to the search orfor causesto reviveclarify its enhance a of the present and effects.feelingsboundaries, national na-of image.content,mathematicsturecators, and or and even the publicvalues of the methods andeducation to is to the content discipline. and methodsdirectat large theincrease thoughtsto understanding Our chief goal the issuesof mathematics of mathematicians,of today, by in this history education have forces imply showing edu- of thatthatchanged althoughvelopment gogicalfurther significantly change devices, and changing of new mathematics, newis imminentover the years, goals for a and important. usesstrong ofin mathematics changing society continuingmathematics, new The continuing education. all demon- peda- de- stratelessonsnew theterials of should the pastHowever, a ideas are actually need for thoughtfulbe examined continuedhave change been old and that they view of history in the light studied. The ofmerit both a shows that many old approaches revivalold only experience apparently afterand the ma- and a EDUCATION GOALS OF HISTORY: ISSUES AND FORCES 3 meticnew and situation. algebra For has example, many elements HISTORY OF MATHEMATICSColburn's "inductive approach" to in common with the "discovery arith- 4 ofappear mathematicsphilosophy as the sources and itself psychology. ofand issues of the Duringnewly developingthe period and forces. These are the changing of our story mathemat- fields of educational views inteaching" thevationsbeginning elementry ofhave today;of beenthe school andnineteenth adopted hasattention been in century. response to to urged on manythat occasionscouldnumeration be called since systems revolutionary,History and geometry suggests that past inno- changing needs and philoso- the inofbecomingics the first seventeenth,nineteenth developed a major and in tooleighteenth, association twentieth in the scientific and centuries, with early and studies of the physical world,nineteenththe centuries. interestsit became andThen, intuitions a collectiontechnological of revolution phies,aproponentsand surveyFor although that this ofthey reasontheexpected. rarely havecauses ourwith often It plan isa speed failed is to stressto and effectiveness of past proposalshoped that steadier progress will be as magicallyissues, effective and asthe forces leading to and projects. result from their I. veloped,personsicsofman-made, its with builders theother rigor, axiomaticphysical butthan abstraction, not mathematicians. worldin structures the andoftenphysical generalization rooteddeveloped in Asworld. this view Connections of mathematics of mathemat- de- later out of the increasinglywork of character- 4144>)action on these issues, throughout this book. Fromized the the work musings of the of professional educationalEducational mathematician.philosophers psychology developed in somewhat then our toit share moved in firstthe into lab- the reverse manner. ofandmathematics facts. issues We related regard education to issues mathematicswe in would asNorth questions America,like this hook to he not a mere 2. Our themes: forces and issues education based upon such a catalog with reference to which therebut a description of forces catalog of facts about kmtoredesigningoratories embody solving. in the whichof textbooksever-changing behaviorFurther, and was the although studieddevelopment the and authors believe in views of the learning process and prob- minimizing the distinctions of teaching procedures oo educationbothof,has issues beenof thesetoday. thator mayissues Manyhave be andoccurredof some these of proposed debate. are frequently the answerssameMany as, to or educational theory and psychology, in our history. The emergenceissues exist in mathematics them is due to a variety slight modifications betweenofcoursenuity forces and elementary content, and articulation issues we and have emerging rathersecondary found than through education differences history it convenient to divide oureducation. discussion The extent and manner in accordinand grade in stressingwith placement the ele-conti- or risebetterandof forces--inmay toin understandingthem, ourbe achievedchanging mathematics and to throughsuggest ofculture. the itself, issuesa answers Our in of today tracing over the past centuries of ultimate goal, then, is to providefor them. We believe that this and the forces which gave emerging a wetomentary whichceduresdifferentiate are writing andthis according secondarydistinction ratherhistory. tosharply We gradelevelsis being cannot both level.of the For curriculum this reason and we blurred today is a current issue, butdeny that in the past one could instructionalthink pro- that under issuesstruction.education and changing Thecan beformer, forces. consideredWe of course,believe under thatincludes two the major issues and forces relatingheadings: to mathematics curriculum and in- both the content and the direction.standingtoday'sinvolved circumstances if Along we in heedeliminating with this this, our separation audienceitwe and shall the butmaypoint forces try be outto tending point the out to larger and more under- issues and methods drive us in thisboth the issues gradeings,differences.with placement the curriculum method This of of means materialsand teaching instruction, that taught. buta complete also is not separation includesThe provisions latter is for chiefly individual concerned possible, since questionsquestions on of mental matur- of these two head- that are common to instructionThese considerations at both levels. have led us to design a book Part 1: Mathematics in the Evolving Schools in six parts: chieflyriculum.ityindividual which under aredifferences theserelated two toare headings,theAlthough certain grade to two we involve shall treat the evolution placement of materials in the cur-other categories will frequently of mathematics education Part 2: Forces and and Instruction,InstructiOn, 7--tsK-6 3: Forces and Issues Related to issues Related to Curriculum curricultnn 4 HISTORY OF MATHEMATICS EDUCATION PartPart 4: The6:t: SchoolEpilogue: Education Mathematics Present of -Day Teachers of Mathematics in Canada Forces and Issues COALSon pointedSecondary OF IIISTO-RY: two years ISSUES School Studies (73), the committee having earlier by the National of this committee typifies two AND FORCES Education Association forcesbeen ap- in S 4-n, education,3 Progress stopping by periods Inoccasionally Part One we will present the facts of the historyto indicate themes that will beissues further and identify of mathematics mitteeject-mattermajorAmerican(NEA). initialreports The mathematics field interestappointment as of stimulators mathematics education: in education as a whole for of reform. and (a) the influence of (1) the concern of persons the specialized sub- national com- with a forcesdevelopedtinuum which whichin latermay can parts.both beWc produce uniquely have divided the story into Our aimand has cultural been forces to clarify and and sharplyissues separated and helpfive byresolve periods.specifying Although issues are always over- them. time is a con- a educationthepart Committee of theduring of theTen latter Therepresent foundingpart of the AMS twentieth. This force was the rapid and the role of the NEAof the inanother nineteenthappointing significant century force and that development of many affectedthe first singlepriorseemedlappingto point, extendto theandsensible educational first interminglingthe toearly us to history begin settlement in the United States throughoutof all mathematics periods educationwith across only a brief reference to and Canada (1607) andof time. However, it thethe period period andprofessionalbutresearch. administration it is interesting organizations, 'lie Report of theto of note the schoolsboth thoseand those associated associatedhow Committeewith many still-current of Ten willissues be of an intuitive sort into the organizationdiscussedwith scholarly later, received its firstof thethoughrepresentative book beginnings wasCo thorn'spublished of of the schools beginningin 1821. to arithmetic (13s) and algebra of athe newWe time haveconcern of Warrenselectedthe for first pedagogy. Warren mathematical ( t36) written on Cti thorn, whose hooks written Colburn as Al- attention:tionthewithwas elementary of solidcalleddieone introduction syllabusgeometry a schools; "double for withthe of those more plane; planning geometry track" program after the first year earlier beginning of algebra; the incorpora- and the development of what to continue into college of algebra, later and or"inductive publishedpedagogicalto pedagogy, principles" in processes the they arerepresent neither 'Western Hemisphere norin this country. have chosen 1831a tonew 1894. and more general the first to give thought The latter year concern for anothergoalsandon a for theshouldrelated those secondary the force. not so The gem. school programschools be to defined?plan curricula and guide issue is this; For what This last provides an example The force is the pressurestudents and for what students with col- of an issue datesNorthMathematical the America. beginning ThisSociety of three Forwas (AMS),our new second period we 'tired in 1888 (344, pp.the oss-6). year of the founding which was derived from the forces in mathematics of the AmericanThe AMS itself education in New York oflege the entrance(NCTM) founding requirements and of ofthe theWe National issuinghave chosen to terminate our and examinations inof mind. the preliminary report of Council of Teachers of Mathematicsthird period with 1920, the the National date education.hasProfessorMathematical probably However, E. not Society been certaina significant early f I. Moore, were much mathematics and gave time factor in secondarymembers andconcerned elementaryof the society, with the notably problems and attention to them, of FirstCommitteehad Courses(MAA been on ) appointedin Mathematical Secondary19:6, prior by School theRequirements,to the entry Mathematical Association of Mathematics (57). Thisof the committee United States into The Reorganization of the Americafounded World assecondary theirwell1894 asinterest givingschoolalso saw in innovation the publication in this area of investigation of thethe Report teaching of the of mathematics. the prestige that came Counnittee of Ten Time year from requirementsleges.(191s),War I. Its Thewas sponsorship majorfor and the ideasconcern undergraduate on of this committee emphasizes theof the secondary MA A, itself school only curriculum recently mathematiciansmathematics curriculumat the college in the level the effect of college and also col- andin Parrs the United One throughStates; inFour Part to Numerals in parentheses. as they appear Five, to items in Bibliography items in Bibliography A, here, refer in the for publicationsB, for CaRatia. Introduction sonstionsforthe committee.improvement continuing toteaching the secondary Thein concern inthe committee the schoolsbothschools. of many was spoken persons schoolswere sought out and A number of persons with of with great approval by rite training teachers and per- included on the close rela- first6 president of the NCTM, C. M. Austin, in his initial note to the HISTORY OF MATHEMATICS EDUCATION GOALS OF HISTORY:This ISSUES led to ANDour definition FORCES of the years from 1945 to the present as 7 Influence,"TeachermembersIwo, might of or chat perhaps then organization be "First labeledOur Steps fourth at "The the toward period,timeGrowth Revision."it took t9zo-45, of Collegeover the introduces and University the effect of two strongas its officiA journal in 19th (589, p. t). This period, 1894- Mathematics 4wo,calledsomeour fifth termedrevolutionary and last a "revolution."period, brought a period Naturally,forth of some experimentation changes protests so and andextreme reaction. reform as which to be 4. Summary and preview ofperioddepressionforces two becauseoutsideof the and reports the itwar. marked .control We of thehave the of Commissionend schoolchosen of World or 1945 college Waron as Post-War the11 personnel: andterminal the Plans appearance yeareconomic of of thethis toandfrom the the which changes. emerge theIn Part One, then, we will seek to provide the factual frameworkforces that have given rise to these issues and given direction issues that have called for concern and change duringformittees,NCTM the theinadequacies(16; commissions, war. t7). This in committee and mathematics studies was to educationhe the set first up ofasthat aa numberwereresult brought of ofconcern com- out issues as these: In our1. introductory remarks weWhat have should passingly be the focused goals of onmathematics such education? Providing maticsas formsone educatorsof that the linksbegan had between to Thestruggled get reports underthe issuesto wayofanswer the ofabout Commission the many prewarmt. criticisms. Prior period on to Post-War the Thereand war the weremathe- Plans re- may be regarded study?pline"?for personal Training and invocational logic a.and practical problem needs? solving? Providing Preparing "mental for further disci- suchhightwo schoolas(a) major business, Much graduates types: of secondaryelementary (1) was The viewed mathematical schoolschool as teaching,mathematics,inadequate (arithmetical) and for the evenmany armed competenceextending occupations services. to oflong areBybe adapted"double in some tracks"?to way the subsidiaryvaried Self-instruction?These needs, areto them.perhapscapacities, However, the and interestsweHow have can of alsomathematical students? noted, education in both content and instruction only issues in the sense that perhaps all others ists.bydivision, demandedthismany The view. waseducationalOnumission Other regardedmathematical educators philosophers, on as Post-War of competencies questionable began psychologists, Plans to stress atvalueclearly many that forand exposed levels thegeneral guidance war but fallacieshad educationhad special- notalso onlyin led structure?implicitly at least, these4. additional What3. students issues: should take mathematics?What mathematics should be taught? "Facts" and operationsor Everyone? All college paredactualsignificance.to the developmentfor mathematics unforeseen Both theof wereneeds many applications new. onenew mustapplicationsThis of know ledmathematics to and the with understandview postwar, and that much topeacetime the be ofstruc- pre- the preparatory students? Only5.Some would-be issues scientistswhich will andarise engineers?What in later persons chapters or are:groups should direct mathematics education? todents?Doesture the of school solution mathematics,(a) Does mathematics of school the not problemsTwo mathematicsmerely instructionmajor itsof issues ourfacts serve instruction society? andof the operations. needsprewar Before contribute of period theindividual wareffectively persisted many stu- after the war: (1) continuedcators?appliedProfessional mathematicians?The tested general mathematicians? progress public?6. HowEducators? in bothcan If one so,curriculum Psychologists? provideshould they forand the instruction?be Mathematics experimentation the pure or edu- the needed to guarantee perimentationandremainedpeoplethe even years would the immediately same;personshaveand reform answeredbut giving the followingwere reasons the both developed. answer questionsitbehind when changed it,proposals thewith remedies duringa no. for Thethe extended called waranswer andfor, ex- vatinga and clarifyingstudent's instruction?8.7. What development? levels of rigor are sotaidWhat and is desirable the role of at applications different stages and mathematicalof models in moti- HISTORY OF MATHEMATICS the excitement and EDUCATION lo.nay). How e?Can of If we can,teach should so we? mathematicsdo we teach as well so as that its studentsfacts perceive that students will "discover" andand be theorems?more crea- remarks in- WIathematics in Forceside:a.t. PracticalthatThe haveresearch been needsof explorers, soldiers, navigators. and beliefs of psychologists mentioned in our introductory and philosophers with the Evolving Schools 4.3.'erence College to theentrance goals' nd methodsThe presence in the schools of examinations and requirements.interests and abilities, but of instruction. increasing mnitbrrs of all needing to be students rained'tb hersThisincreasingly and listin conception educated.is incomplete, varied or importance. and some make the reader more of these forcesHowever, are subsidiary it is hoped that alert to perceive our this to in:me:roductory asmalrich, and the goals. in goal:story turn, list We the mayprovide hope presentation that a of changing issues and seriessuch of approximations perceptions may to stimulate,changing prepare of the history of mathematics forces as they affect practices educa-educa- for, PART ONE d help determine the best directions for future changes. ArthurPhillip S.F. JonesCoxford, Jr. al 0 0- CHAPTER TWO 04.0i.1 .0$4 al as '18ca 1/01:6 r44-4 to?, Cu4.1 0 cn0,1 From Discovery to an O0to 0al .0 ra o054alco Awakened1492 -1821 Concern for Pedagogy 6 ,1al Ps vi .44Z ; Moat01-$ c- 446J o0 1.1 0 44S.41 44 00 1. The earliest American mathematics Mexico and Central 05.4 tl 1 v491 011.U FewAmerica peopleing a zerohad realize a symbol remarkably that the and utilizing place-valuewell-developed conceptsMayans of numeration Yucatan in system but based on (WIC contain- to the 1$1 long before the Spanish explorers 0Vcn El1-4t 0 o DJ al ofWesterntwenty zero( I ratherisao, Hemisphere. the p. 1earliestthan t). This tcn, useIn is a little uncertain because fact Cajori even suggestsof such a symbol of theanywhere difficulties of dat- that the Mayan use in the world Iah0N 0 fi 0 specialingmathentatics Mayan symbol hieroglyphics used should as abe regarded as rather remarkable separarrix in the Seleucidand also because it is the use of a zero in that system. prehistoric development in perioddebatable of Babylonian whether a the In warn1.48 0 00>4 W to r4 k numerationWesternany case,mathematics Hemisphere, this system was a inby this theasThere hemisphere,was is also some evidence of the existence of Aztecs. the somewhat similar or at least of number words in the genuine prehistoricdevelopment of a .4 co 0 languagesperiodsAmerican tostudied Indians.our major by concern, !r However, we shall pass anthropologists who have the story of developmentsquickly from these since 1492.earliest worked with North 12 A little-known fact about the earliest mathematics HISTORY OF MATHEMATICS EDUCATION in the Americas FROM DISCOVERY TOs. Practical A CONCERN needs. FOR The PEDAGOGY needs of exploration, community; later the needs of navigation, and trade 13 \ Hemisphere was thetheore,is-that Stournario majorpublished the concernfirst Compendioso bybook Juan of of this Diezmathematical. book de Frey las was lequernas the content conversion printed in Mexico City in 1556. Although of gold ore intoin the Western .._ de plata y enceddevelopingarepurpose dominant the content oftechnology allin .aeducation, ofdeveloping the andz. books; Church scienceespecially frontier later and become theyhigher religion. influenced important. In the theSpanish content books cited above theyeducation. influ- and calledbooksInproblemvalue addition "the equivalentswith rulerequiring tosubstantial ofthis three," book, inchiefly different mathematicalit there also the typesusecontainedwere of sevenof ratiocontent coinages a andMexicanshort proportion or published prior to 17oo of the Old World, chaptera on andalgebra. four Peruvian what was Juanity are Diez suggested Freyle's earliest by the3. hook. inclusionIntellectual of curiosity. unrelated The work existence in and role of intellectual curios- algebra in thevariousnavigation;its.( design.64, Their pp. feastand z5-35).major and construction days theconcerns None andcalculation-of of religious these ofin campsfact books holidays.werethe and calendar,was chiefly fortifications; An militaryeducational especially matters to sidelight on devoted solely to matheiiiit- surveying and determine such as 4040schoolsand writing. did develop, Reading theirAt and chief writing objective usually was included the teaching the readingofthe reading outset, and the earliest2. The settlements content had and no processes schools at of all. colonial When instruction sinMexicothe the maestro.times time City is and suggestedin 16:9, of a landArte byIts parebeing the theme titleaprender newly of-homeof Pedro exploited todo study,Paz's el mewl'and or studysettled. "without This is a re- master," is book, published in del aritlnnetica typical of senseearlywritingCommercial that American of on numerals one sidearithmeticschools androman thereforeand numeralswas homes often counting. wereincluded taught The in arithmetic specialfamous schools hornbookonly in called of listed for memorization. the *line Young Secretary'sinlanguageflected 1703,Assistant. also this published in Itwas theincluded afirst inreprint thebook arithmetical Western of with an mathematicalEnglish Hemisphere. ideas bookneeded contentby Published in John Hill entitled the English in Boston turyschoolsreckoning included in Massachusetts or thewriting note schools.(121, as late p. 9):Bronsonas the beginning Alcott's of descriptionthe nineteenth Untilof cen- the within a few years no studies have been permitted in the day inbookkeeping,by aandthe new youngof land--commercial etc.businessman the This influence hookdemandsand representsalong of foreign with again severaldiscussions authors, the ofrole theof of forces letter vocationalhome needs, study and operating writing, daymostschoolfew school. determined instructorsbut spelling, opposition, one reading or two andarithmetic ev writing. is Arithmeticnow being waspermitted taught by a gs in a week. But in spite of the in the 'pi , lishedself-teachingGreenwood,It was in Northwritten in Americawhothe by educationalIsaacservedThe was Greenwoodfirst as entitled theEnglish-language system first Arithmetick, Hollisand of newly published Professor settledmathematical of territories. book both written Vulgar and Decimal.in Boston in 1719. Mathematics at and pub- century(44i, p.to 54),possess "Pupils anIt were arithmeticwas providedrare forbook. a with young As books Robert student of blank of the pages, seventeenth or eighteenth Clason points nut called Pa.),booksHarvard I in73o German,from (New 1718 York), Dutch, toThese 1738,and and .775 factsFrench had (Quebec), furtherstudied appeared illustrate inrespectively. England. in 1774 the continuing effect of foreign (Germantown, Mathematics influ- pupilciphering of about books, 1810 givendescribed a rule his andexperience a writerproblem, thus left them, (181, and andp.set 43): theto Printed work."custom arithmetics Awas for the were master not toused write in athe problem Boston schools till after the V change and development, andmenrsencespresent: some upon to the of the themmiddle mathematics continue of the nineteenthtaught up to century. from the time of the earliest settle. Other forces led to the workallowedwereor fromtwo neverto in thecipher the manuscriptperformed manuscript till he was onof of tanythe t theyears furthersame pupil old, day.... advancedevery and Anywriting thanboy and couldhimself, ciphering copy and other day. No boy was the 14 whilethe writer he was never at school. heard any HISTORY OF explanation of any MATHEMATICS EDUCATION principle of arithmetic ;;1,. p-. E approachnoarithmetic internal to arithmetic, books wereAlthough texts students did not spiralingeach topic in its turn self-contained, complete, wereuse printed available texts usedwas treated over severalfor years teachers. These in the ciphering-booksingle volumes with completely and early of pendiu hers .ct R E i , . studytheneighteenth-century dropped. at different The age same arithmetic . levels, in or out of book would be in America was school. The most Thomas Dilworth's popular Both - . 't 1. , ti, 1 1 : If, r.oiz T I CA - PracticalTheAmerican Schoolmaster's and edition Theoretical, (1773) awas Assistant: Being a labeledreprint theof anseventeenth English Compendium of Arithmetic the addition in edition. text.Its only The first the preface '11: '. lifIlloiis*....--:'.,tt haios in:Wfall N4441.." ,each are of thiiii,C 0 suednkiltailly '4W kloleafinf aligh fleece* V.;4:1114..iiitlicinn4..,iizercla !Ike'. .1. Nisi 11114 twefr.Vv: :ii;C;;,1, (ft concessionof the book's to the new fifty-first endorser, Karpinski (164, p. locale seems to be Nathaniel Wurteen, 73) lists fifty-eight "Schoolmaster American _ i e- real Thins, not evasalogly um 0120 , Frailign;sy handled.. Iskesifir ft+ .' tri -. 1" -: fr., ,,, Fe. nit, 7 iiiirtk.,k,...se-4...... , contentsprintingsat Philadelphia." were:and editions through 183 a. The topics listed in its table of , lirP_1!aini;tt40.41411rstillithn4(.6%6:.02:-.:iotl4t....."-i.wish,,iiic theta of.tlitiloaly. . arta laidtrios dews toil In Slat; ' , yr ir4, y ...1.'!: lt.."4. '1.. 11,, 3.a.1. Decimals,VulgarArithmetic Fractions . in which,in among whole othernumbers, ... . . both Simple and things, are considered Compound; Annuities, the Extrac- Re- ' , Thi Maar dafthate,plcThIrig,..re of andRabic! and Compauo.81; Equation befatufii.lbolitconGdyiedthicIlifhte... AtundNeyfittg 111," recommended bolaidiltivfelhell itrstritiair; by Anil . , a , m, :. ii,/- ;11.ty!2'ii...*,"*' i5,5?"fvf '" !T" 1'1'1'; bate,tion of and Roots; Equation Interest, of collection of questions, Payments with their answers, serving to Y. 2 wviula hove their Schfort Srissbotters, as Refs 4. A large exercise the forgoing rules; together with a few others, both pleasant CiProgiars. hiat t i T whin) rtkaat, An ,11offer'd to At art r .1 ...c 1(:' '!!.- . 1 ir1.01Si: iv' 5. Duodecimals, and diverting commonly called Cross Multiplication and preceded by a "Pref- The .8 E if g:NT tt ID 4,4 tt At r . 1,,.41 , '1...... ace Dedicatory"All this is encompassed addressed to "Schoolmasters in 192 small pages,the Education of Youth" addressed to par- in Great Britain and A Auther tf the le44;e'i ;Wooy...7' ,,, H ere: 0.1VI.A.:8`...9 ilrer) I L Nom Mac teiigle.P.THA. .Ait ,i..,:, ''' y'iliiil,', 1 1. 1 V ...... ,C,. , : : . . ' gicalents.Ireland" Someproblems and extracts an and essay concepts "On from the former will of the day: - give some idea of the pedago- ,.-tilli iiiergibt-rdafior.3 .al /roar! Wick Joni 0i....., .t. . ...- 4".'7F"7,10,171"7711...1175 )43' /lc t14,s, i 1 , ' Wayto be ofmade instructing themI believe it is confessed compleat Masters of by All, that init isit ais, most certainly, Arithmetic; and therefore Task too hard for first to give them a Childrenthe best t!lb-. .7...0 a la Wit i ;, /1.. '-%'1.7.r'' r ,:'f- t .1...,-.,.- asitgeneral afterward,theirDirections increase Notion if for there of varying it, be inIn theall Places where in Years and Growth itTime; could otherwise thebe doneeasiest it must Manner, Examples by Way of in Understanding will conveniently, I haveand given near to enlarge upon be done by themselves,Proof; because it not permit.... FIRSTthe ARITHMETICUniversity of PRINTED IN Michigan'sTHE UNITED collections. STATES. From ,T H ..E FROM DISCOVERY TO A CONCERNonly VOR discovers PEDAGOGY the Reason of the operation, but at the same time both 15 Schoolmafters Afliftan ART.: Form,producesConsequences.judge that thea theynew Force may of bean the more instructive;have thrown for the subject of the Hence also it provesQuestion, a very good Answer, and proves the than follow following pages into a old One. .Reason examining Book;Childrenthrough a Chain of. . Catecbetical can better for Of ArithmeticThe in Whole Numb.14-1. I N T R 0 D .0 C Ti! -..a Ni."." P The note to parentsheat any can is time alsoimmediately in fascinating what Place be pui insoever backits reflection tothe that of modern Scbolar appears to be defective, Place again. Of ,Arithmetic in generate-:.;9. : -; ii;4:: parentspedagogical are urgedproblems to "never such as let tardiness their own and Commands homework. run In counterparticular, to ..;. .by Numbliiii;:1. ,either whole or1. in Frietiosis,',:-.:-. y VXX 7 I .14 r it' Arithmetic t. .. 4. AritAtietic. . is the Aet or 3;044 t iitti ,,4 "thethe Master's"; fair Sex" are either not sent to school before they are eighteen or and a substantial paragraph is devoted to lamenting that ....:Qiieflioni Hsi; -mon, A.Q Moth it Nttmber ?is one or more Qt/ai:titim,:anfireripg ;":iii the ...i P -,,.... : . ..: : . :::,.. ,..,::-. , . twentylong yearsenough of (moreage or than that athey year). do nor stay at their "Writing Schools" '; : ittfillai is kitbags& in *bole Numbers I Schoohnaster's Assistant,The Being most apopular Plain Practical arithmetic System in America of Arithmetic; prior to 185o was Dalton '.1! Numbers to be entire, Q_uitatitia, and not0,4.. deridedA. Arithmetic itt,o. le whole; Nunskrt or laitgeili tugififet ! I .- ;`,t'. Paf . Adaptedcut, in to 1800. the United This English States, text by Nathan Daboll first printed in New London, Connecti- and a revision by ,...4-7 (L Patti of foam entire Qlantiiiri:::A. Arithmetic I in MatHea4eiaeoultder,Aritlatai it tetali4 Irian:elk in Frattions t ' Fralliank .fitypofiii he Numbers to be !hi .. ,,',. !... if. -::,-,t;i: jSilinic k ..; '..,- ',.7-.. $ 4 withbookDavid a is six-pageA. much Daboll like appendix had his appearedcountryman on bookkeeping in Dilworth's:eighty byeditions Samuel short by(24o Green 1841. small in Daholl's the pages edi- . of Ntimbera4..'..,":'...Q.;--1!:t A. iation4.and Both in nag;:477)Seer;iinit and .P.ritairr: AdAniserie.:Cbtifideei 11.tt .Haiiiri, itliilICiattfiTheoretial , thla demektffrates WC ROfoii: of *libido* V41: Pratt tionlemstendedand examined), problems. involve computation. itpounds is Many limited and Althoughof to its shillings. arithmetic problems dollars Extended and involve and stresses cents computations large commercialappear, numbers morewith topicsand varied prob- ex- Joni-WWI;tlt. it tki,it"'.4.,:Pilgiiiiii itby Prialeid N:iittlieri;,iii AiltheiiiAii Idritkitiik 4.04 And in a I filiiii. :r :-.0;4, .,,i,J, ; ',A1..:#4.1x '' andcatecheticalunits problems. of measure method and but their in each "reduction" section givesare included. a rule, a Itworked does notexample, use a .,...;boss ,17, The Nature of all :irleQ; /hat it Ida litalarm irultfol. iluailek,' 4104 in thillett iii. . -,.1 ilt ,,,,.. 1., .3r, period was Nicholas Pike'sThe most popular American-written arithmetic text of this early A New and Complete System of Arithme- pub- lin,.:,*41litiMiluitAM 41.-,,,r40;irriheli.trioi.00:44,14*id.:.... Dior dr,. given., to fiiiiig!.. .4 Ther .Fiiie . i 11Itiathiai,,4iNitth ''' s '.1 -lr' .: liiiite''.1 li ; ,,,.; e ,.I., sortionsabridgmentlishedtic Composedof by mathematicsin t13::. Newburyport, made The for firstthein at 1791 UseDartmouthedition Massachusetts, ofit appeared thewas Citizens University;endorsed in in nineteen of by theS. B.Williams, United Woodward, printings States, and profes- edi- 1788. Together with an Hollis Pro- ALLthmeticsWORTH'S in dialogue and catecheticalto Robert Recorde,form c. 1540. SCHOOLMASTERS ASSISTANT. t English date inaandfessor differentlength by ofthe (51:Mathematics characterpresidents pages) fromand of and those in Dilworth's Naturalsubject two schools Philosophymatter, book, and and Yale atclearly Harvard College. aimed University; It at was more of much more extensive both 16 HISTORY OF MATHEMATICS EDUCATION FROM DISCOVERY TO A CONCERN FOR PEDAGOGY 57 fewmature topics students. and summarizing Something its of major its range categories:Rules may for be Reducing revealed all bythe citing Coins, a from Canada to Georgia (pp. I I1 -23) greatvalueproper strictness of for the mathematics "methodin reasoning," Theof the educationintroductions and ancients" also citedat is theto "to its these time. accustomvalue hooksLegendre in mathematical the also student saiddocument the to the goals judged PermutationsCirculatingAnnuitiesExtractionTare and (pp.Teen ofDecimals andthe264-66, (sic) CombinationsBiquadrate (pp. Soo 323-28)192-94)-333) root (pp. (p. 339-45) no) sentationof peaceresearch. andof formalSimon war" (358, definitionsasserted Inp. 5).summary, that and geometry rules, even especially thoughwas "of a atgreat few the booksuseelementary in delayedthe arts or deleted the pre- ummarytornelated mechanics astronomical of Pike's asA second great falling problems, deal editionbodies, of with space ( thev797) less pendulum, was space shows given devoted the lever,to following mensuration, toand such screw.distribu- topics the A calendar, and memorized)."andresultnetlevel, effectset andthat exercises wasathe few teachinga used heavy to bea catecheticalrelianceprocess followed onwas (or, setquestion-and-answer "Statein exercises geometry, a rule, and give texts anproofs format, example, with to thethe be ry,ionages;icosaiedron of materials: conics,This 'tic)10 arithmetic, pages. tabulationand the volume396 of contentspages; of a geometry,mash makesi ttub), it 4 quite46 pages; pages; clear trigonome- algebra,that Pike's 33 book waspages; mensuration (including the area and volume of an veying).merce,practical and applications settlement The-(mensuration, especially content those of bookkeeping, mathematical related to navigation, exploration, education and com- laid sur- a heavy stress on 1).nmediatelyof a Pike's textbookmerican book as forcolonies ahas textbookelementary also to been include by school Harvard,listed algebra. as students. theYale, The third and firstIt book was, Dartmouthwas published in Freyle's, fact, (24o,adopted inmentioned the p. genceofat aleast formal of not one day daily), of school.the andstrongestInstruction itHowever, was continuing often in the mathematics given seventeenth forces by ina privateAmerican was century often tutor education: was minimal. outside notable It wasfor the irregular early emer- (or CT) :teryffer-Konst Venema,In geometry,58). theprinted first of by thesection J. fourteen Peter of thisZenger books chapter. in published 173o The in secondNew in America York was (24o, prior to . Alr Mede Een kart ontwerp van de Algebra, Aritbmetica of by quiredwasinample,the noreading belief everymention a Massachusettsthat and town of educationcatechism mathematics of fifty familiesis lawand necessary of alsoin 1642 this. to their provide required Infor apprenticeship1647 the welfare an Massachusettsthe elementary instruction of in society. a trade. ofschoollaw childrenFor Therere- ex- /,y.:2o, suchThe)nstrativealing only remaining as withthree geometries, geometry could were be classifiedworks printedwas writtendealing asin dealingAmerica. abroad. with mensuration with The This, demonstrative geometries like and the survey-three includedgeom- de- Hawney's Complete Measurer ( aoi), the first book preparationwasgrammar.teacher preparation and school. for every the for The ministry,town college; chief of afunctionlaw, thehundred andcollege of the thefamilies inteaching Latin turn togrammartaught provideof Latin. Latin schools aThese Latinas a Itwell, appearancenslations,o English A. M. Legendre's books,the in last1819 Sinison's in was189o. followed Itand was Playfair's; usedby forty-two as anda text one at translationYale University from Elements of Geometry. This latter laterbook's editions and marbuteffectitedoccupations notschool.popular on mathematics mathematics andsupport college forwas education theattendance required schools. somewhat for being The entrance grammar limited, negatively, into there schools the sincewas Latin hadonly reading gram- their lim- :itchebraiclateuence asinfluenceunique symbols1883. of theorems, featuresThe on and earlypopularity processes (such Americanits inclusion ofas Legendre's itsin mathematics. proofs)departure of some show book formulas, from theand Euclid's strength an and analysis itsaxioms of use the of of and necessarymoment.andsarily its government leadsBut to asthe to the welfarean has early importantThe continued of neglectbelief society? issue: in toof the hemathematics aimportant strong force ofsuggests, education up to the it neces-present for a society, its religion, What is the education that is t8 HISTORY OF MATUEt't.%TICS EDUCATION .46 03-3>ended to be modeledAlthough after the the Latin English grammar grammar schools 3. The Latin grammar school families, and to direct their had some public support, and the academy school,' to be at- instruc- they A`RIT,HMETIC Vulgar and Decimal ; endedowardionboutMe bytoward Bostonthechildren thirty preparation preparation New Latin from England Granunar wealthy of forgentlemen, towns School had was collegewhich in turn was ministers,such lawyers,schools. andTheir doctors. curriculum founded in tots. By 1700 directed Later WITH THE /asmue chieflyle arithmeticcostlisans, Latinneed and navigators, foraristocraticand was a literature, practical and nature surveyors, education led for to a a introduced, in response to public of the grammarwith schools,some attention together to writing. growing group of. merchants, decline in the Latin gram- demand. But and by with APPLICAT-10, A VARIETY THEREOF, TO IN of CASES ar school. This was accelerated by opposition to taxation avriessv,-,04 :e Americant inacademics,Academies 1749-51,d Revolution.spread beginningwere developedof academics in response did not get to privately supported. The extensive with the one founded by under way until the nineteenththe forces listed above. The Benjamin Frank- development 4 Trade; and Comment Thefumyd academyalso for itsbroader aimed wingmental atnumbercurriculum preparationdiscipline. of academies included ultimately mathematics for its practical The broadened curriculumboth for life and for college.made necessary the illStittl- utility and .11. 11 n of college01f he entrancecolleges f 4. The colleges and the led prior to the Revolutionaryrequirements. mathematics community WarHarvard 'dated ;Rive*BANCOCI1 is 11V by 9..Ki a.t47. &RAC*BOSION1..N.& upc tbi 1111.41/0 'sat. Stgo of itb e;to), ofthia NewWilliam (as70differed Jersey, kings, and 1754),Mary somewhat (169;), Brown Yale in(1764), their 1746), PCMISylrallia (as origins and goals, but (17ot ), Princeton (asRutgers the Col- (1766), Dartmouth, Philadelphia, 1755), in mate* Ce- STATES.FIRST ARITHMETIC From the Plimpton WRITTEN AND PRINTED Collection of Columbia IN THE UNITED o,::ics made andtd none rapidly,,) atanand hadHarvard entrance then extensive requiring movingoperations, requirements "the on whole to"reduc become or in 1807. Harvard,requirement though at startingYale in 1745, at," and the "rule of three" were of arithmetic" in tilt6 (only in illao offeringsthe first atto first.require Arithmetic Princeton...in"late, movedrequired in the University.onautograph following of Samuel A secondpage.) copy Adams. there bears the(See also illustration FROM DISCOVERY TO A CONCERN FOR PEDAGOGY 1 9 \Var.arithmeticlatealgebra. as 17:6 Geometry and the surveyingonlyThe was mathematics early not in required thecollege senior taught formathematics year.entrance at Yak In was£748 until curriculum a after Yaksmattering requiredthe was Civil of pretty scanty. As andsomeyears,have fluxion mathematics included the was mathematical taught arithmetic, in the asAt secondearly courseHarvard, algebra, as and 1758,began whose third trigonometry, and in years. the programby senior1766 Something and aoriginallyyear program surveying. of conicsextended might through only three and consisted 14 classandof arithmetic astronomy concentrated and during geometry for a the whole last during day quarter on the one first(121, or threetwo p. £9).subjects. quarters At that Mathematics of time the eachyear ...... p trastandor astronomy Indianbetween War was the in studiedintellectual 1763Smith toon the andMonday culture effect Ginsburggiven and inthat theafter Tuesday. there (44,newly England's p.was acquired ;z) "a quoteacquisition .pitiful Canada a con-report of Canada to King at theGeorge end of the French lir' :111:7f... :' .0.4:'-'-.7 ':14 Thesejust,and historians theit is unculturedcertain then that remark,backwardness the work "Whether in the of new or the not republic,older the comparisonEnglish judged colonies." by was their 4J 14..."1.4151-.11P.;4 f 111 f.ii.E'E I ii!fit t;*.lpii21 giat SI: 1.111:1 tiAll ." tart i algebracurricula,Greenwood'sHollis basedProfessor was of on successor, a the lowof Lecture Mathematics workgrade." John of notes John Winthrop, and showWallis Natural that alsoduring Isaac taught Philosophy his Greenwood calculusperiod (1728-38). as (fluxion), first taught a formal sort of Ftr1 144,14p:if Iliaid gal.cr 2E0E1141 isii '''' iiiiIi;i1141i1-1.it,I...4-111/i.Iii,,-,111-.ailsk II :. sd: ::,45 tiki -. majorpp.butforboth 20,his astronomypart practicalmajor32). of The mathematicswas interests forcesininclusion connection andand of the contributionsastronomy withtypical intellectual navigation of thewith wereclimate. times mathematicsandand in surveyingand astronomyThe resulted practical is obvious. as(44,from need a I 9 : . tr 01.11.4Zi17-11i'-! 41'8111,11 74 g I u . b ;4 1" 14 ip..:: "Wit Lit 1.41j411i, ,i,' itrapidly.Newton,theAt recurs the rise same Witnessof inand sciencethe time, many titles to the andthisothersof Europeancourses isthe bornewas work andcommunicated intellectual byof professionaltheCopernicus, term revolution "natural to chairs Galileo, the coloniesphilosophy"stemming at theDescartes, colleges rather from as andp.portedof 29).the science eighteenth professorships required century. moreMathematics of mathematics Byadvanced 1776 hadsix mathematics ofbecomeand eight natural colonial a toolthan philosophy of didcolleges science commerce. (223,sup-as well as of commerce, HISTORY OF MATHEMATICS EDUCATION FROM DISCOVERY TO A CONCERN FOR PEDAGOGY He served as vice-provost of the 21 lyis force,in the thenineteenth rise of science, century wasin such displayed in new curricula set up older colleges as Pennsylvania directorUniversitymaker, surveyor, of of the Pennsylvania, mint and (244, astronomer. p. and 56; George 218,Washington pp. 21-28). appointed .hint . es referencehnicas West (1824). Point to thePrinceton (ilioz), University and inof the plans for such Virginia (1818-24), and Rensselaer Pennsylvania and the University of post-Revolutionaryearly War Americancol- intellectual Poly- butgovernment.The he American was allowed At Practical the Nathaniel ageto read Navigator, in Bowditch is best of ten he was apprenticed to a theknown privatehe progressed today Salem as Philosophical the from original supercargo to still published by theUnited ship's chandler, author of Library. States cap- inklinrginiaithernatics,":lough (17o6-1790)brings he did to which nopoliticalmind mathematicsand appeared two Thomasleaders prominent in himself,whothe did wroteMuch for JeffersonPennsylvania (1743-1826). Gazette, Franklin, Americanao October education: Benjamin "On the Usefulness of oftain;While antariesCeleste after insurancesailing and his(1829 wasforretirement company.many-39) the containedfirst years from book many His translation of Laplace'ssubstantialthe sea heor ultimatelyadvanced became added explanations and commen- mathematicsAficanique president 15;/agn andofEnglish Pennsylvania. of he the included andUniversity Latin a in of the design of Thomas Jefferson published several school of mathematics alongPhiladelphia, with which became the Univer- his academy, influencing the mathematical schools 4w)...)published in this hemisphere (356, pp. 333-38). pedagogy idesring aboutssical upspherical surveying, theand University humanistic triangles almanacs, of(244,but included astronomy,pp. 58, 6t). Virginia with an orientation that was such popular and practical Henew was a major influence in and Napier's theories whoteachers,according possessed especially to aCajori fair in knowledge There (12the earliest was a wide range in 6. The teachers and r, p. 9). The best-trained ones of "fractions"background and and "the competencedays. rule It was among an exceptional were very young of three," teacher ejects as mathematics (22 5, p. 5. Self-instruction 125). grammaronlycollegedent a shortgraduatesof schools.the timeUnited and Forearly States, ministers in example, John Adams, later to taught at the grammartheir careersand did that who often engaged in school at Worcester becomebeginning presi-teaching thein the Latinfor !ginTheWenicy mathematiciansa"mathematical fewworld.have professors noted thatcommunity" whose and in sponsorsthe major earliest of activitiesbut also some werethis remarkableoutside early theperiod aca-days, included lack notof formal mathe- self- forstudy a p.teacherstime 253). of after law wasgraduating (p. 9).not Onhighit the was from Harvard andsomewhat before strongerwhole, backgrounds. the statusabout of that grammar of skilled laborers school (118,Both tatnilial,tical :Itformal instruction studentsand culturalinstruction are made Benjaminpressures was self-instruction Banneker often a available. Three outstanding examples of denied to eager(1731-i8o6), and able students David Rittenhouse necessity for all. Economic, thetheIsaac age1796) Earl ofGreenwood twenty-four, ofwas Buchan. appointedThe and hadcollegeHe John to had teachers had Princeton in 1788 on the Winthrop,studied who in succeededEngland. Walter Greenwoodstudied at the at University of recommendation of Minto (1753- Pisa, had 132-1796),:lenjamin the school learned andBanneker, was Nathaniel toopen reada Negro, onlyBowditch and wasdo (1773-1838).arithmetic sent to an during the winter, and he was allowed "as far as double fractions"; helpobscure his father school on wherethe sometranslatedstate original a rule,,the workConn give (244, deThe p. dominant pedagogy of examples, and provide problems. Mathernatique of Bossut, and had 3o). this period, as mentioned Iearlier, lowever, was the to published m )avidtionghtattend(383, men,of Rittenhouse almanacs.pp.only was p-37). until largely he progressed became in astronomical big enough to His work, like that of the other two from watchmaker to instrument computations and the prep- self-. J.haddevelopment H.the anPestalozzi first effect edition of in (1746-1827)new this (182 educational country. i) of was Warren Colburn's An In particular, a debtviews explicitlyto theand procedures recognized abroad in the soon title Arithmetic on the ideas of of 22 HISTORY OF MATHEMATICS EDUCATION FROM DISCOVERY TO A CONCERN FOR PEDAGOGY especially physics and 23 wasPlanmental laterof18o1, Pestalozzi dropped.)discipline tried to with reconcile as aSome goal for Daniel Adams, whose first older procedures with Pestalozzi's Improvements.instruction and (The with acknowledgement his teaching arithmetic appeared in concept of meth- 6.5. Individual7.The Foreign rise of initiativetexts and andastronomy "natural philosophy"science, foreign-trained teachers self-instruction political leaders ofasods. rules.Arithmeticthe These basis Adams's methods of inan which attempt placed the a is Principles induction from many examples to an heavy emphasis upon concretereflected in the title of his 1827 text, of Operating by Numbers are understanding experiences 8. Intellectual1556 curiosity and Juan Diez Frey le's Stnmnario theCHRONOLOGY scholarly interests of Compendioso published in Mex- AnalyticallyModeadvantages of Instruction.Explained, to be derived and both Synthetically applied; thus combiningfrom the Inductive and the Synthetic the 16351629 icoElementaryicoPedro City City Paz's schools Arte . established in . . arithinetica sin maestro published in Quebec Mex- +04* 7. Summary IssuEs following issues arise: 1635164716421655 A Jesuit college for MassachusettsHarvardBoston Latin College Grammar law founded requires School towns to secondary education established for founded Assistant printed in Bostonhave schools Quebec 2.I. ShouldIs the same a free amountIn thisFor chapter the children we have of seen the wealthythe and education be provided for all of education appropriate for for the children of tradesmen children? boys and for girls? 170317281729 JohnNaturalIsaac Greenwood'sHill's Greenwood PhilosophyThe Young Arithmetick,becomes atSecretary's Harvard Hollis Vulgar Professor of Mathematics andand Decimal published in 3. Is the proper andmentalities? workers? The discipline? practical Practical arts? The skills? sciences? character and content of public Knowledge? Is the goal of. instruction education the human- 17491745.1788 Yale requires BostonNicholasFranklin's Pike'sacademy A New founded and Complete arithmetic for admission System of Arithmetic 4. Does5. Is rule-example-practice arithmetic program? (mathematics) have a place in the the best pedagogical procedure, or public school should 180318o2 publishedRobertU.S.Ilawney's Military Simson'sin CompleteNewburyport Academy The Measurer Elements founded(Mass.) printedof at Euclid printedin in Canada PhiladelphiaWest Point in Philadelphia 6. How can teaching processescheticalinductive text processes embody be teaching used? processes? he embodied in texts? Does a cate- 180718210316 CommonRzo Schools Act DistrictEnglishHarvard Public requiresIlighSchools School algebraAct passedfounded for admission in passed in Canada Boston I. The practicalWe and a developing commerce have also seen the following forces at needs of exploration, pioneering, FORCES work: settlement, war, 1821 Warrenlished Colburn's in Boston An Arithmetic on the Plan of Pestalozzi pub- 4.z. The3. The church'sinstitution belief thatneed matics educationof college is necessaryentrance requirements, to the welfare for trained ministers and missionaries including mathe- of society CHAPTER THREE vidual,andFROM Rousseau. beginningCOLBURN Pestalozzi's with TO motor RISE. stressskills OF andUNIVERSITIES on adaptingobject lessons, instruction and tomotivation the indi- 25 barternassociatedcoexistedby reaching (1776-1841) appeal tomethods. withfaculty natural anda belief psychology. FriedrichinterestsHe wasin mental followed, andFroebel instincts, discipline (1782-1832).chronologically, led as to a new goal Their andby and, J. methodsmore F. later, Her- mod- an toFrom the ColburnRise of the supportfranchisedUnited States, for lower-roads, especially andrailroads,JacksonianTwo middle-class in foreignmathematics. and democracy westward wars voters had whoexpansion. and led favored wavesto a growth These ofgovernment immigration offorces French influenceled to newly in the Universities: 1821-94 dren,R's."mand,stimulatedin providing Inwith especially the theboth South, resultprivate a incontempt wealthy thethat academies North, until plantersfor iliso forpurely andthefree wanted collegesSouthintellectual public waseducation education (182, aheadactivities pp. for inof5o2, "theandthetheir 530). Northathree chil- de- reformsfacturing,the growth as and abolition, of westward education women'sThisThe expansion, Civilinwas the rights,aWar South.period whilefurther child of atgeneral labor, stitnulatedthe same property ferment time interest qualifica- andretarding concernin engineering, for such manu- social onflictingt all levelshe period forces. developed from Practical, h82Iand spreadtheoreticalto 1894 but was orwas onephilosophical, subjected in which to formaland a number scientific instruction of1. Forces at work ginningstalincludederaltions discipline very for of voting,psychologicalstrong advanced goal; forcesand rapid penalstudy, experimentsfor growth reform reform.learned of in thesocieties, Nearandeducation school theories the andclose population;began discreditingprofessional of tothis be period,and felt. thejournals. theThese men- sev- be- ology,Ited'tees-allied earlier, all interchangeablemade engineering, .influenced one some might use mathematicsand addparts,of measurementwestward the the demands telegraph, education. expansion. and of steamships,a simple rapidly To Manufacturingthe mathematical developing practical canals, needsrailroads tech- thatideas. 4s4)) During this period arithmetic largely passed from the secondary2. The elementary schools tdhese thetronomy,e colleges. At forcesdevelopment the same calledled to time a notof recognition thepractical only rapid for rise andmore of of theengineering science,mathematics need for especially programsuseful but knowledge formechanics even more in ad- and byorsequencedschool principles.forone five- towriter the andseries Colburn'selementaryto six-year-olds. his of questions,"natural avoidance school. aversion His leading primaryColburn's of formalized ato child every method firstto kinddiscover ruleshook was of was wasrule"to his use attributed intended own(362, carefully rules p. aired:as.We mathematics. haveHis work already was noted preceded the early by the effect philosophical of Pestalozzi's writings educational of24 Locke supplement.arithmeticshis163); own salvation"another thatThese werecriticized (399,wereColburn's commonly p.problem the 27o). method method booksattached for but was leaving notto emulatedarithmetic mere "the drill child in books.series the to work"mental" Theyas aout or "intellectual" ita. 26 HISTORY OF MATHEMATICS EDUCATION FROM COI.BURN TO RISE OF UNIVERSITIES 27 ratherthanproposed solved than problems rule by isdirect illustrated that application were by to the be ofuse reasoned rules. of "analysis" The out, stress often or "unitary onorally, reasoning ratheranaly- of arithmeticdiscipline thesisin the aselementary follows (104, curriculum. p. 37): Brooks stated the mentalThe mind is cultivated by the activities of its faculties. . . . Mental sis." To illustrate: twice as much, or 2,,,What whichAnswer: is 1 equals of Wei? A. first Therefore get 1 of f. Now 1 of 1 = Ar; of of 1 is, then, which is 3 exercise.losesunskillful.strongexercise its Anby vigoris Hanginactiveuse, thus and so the the mind,any skill; arm law faculty letinlikeof athemental ansling of mindunused the and development. mindremain the muscle, muscleis inactive,developed becomes Asbecomes .9and muscleby weakit acquiresflabby andgrows and a its use and SuchThisAs example exercisesa practical iswere fromtechnique seen the as 1884 ofan problemimportant revisiontimes solving meansof 1 Colburn's of 1, ofthey is exercising outlived First Lessons.the the mind. era of iw),) idlementalpowers mind flabbiness, loses go to itsrust tonethat through andunfits strength, idleness it for any andlike severe aninaction. unused3. or The prolonged muscle;secondary activity. the mental schools An mentalcificstructure.the middle problems,discipline. lVhereas of the the nineteenth textspre-ColburnThe of inductive this century periodtexts discovery by had statedthe given introduction approachgeneral rules definitions ofof aColburn deductive and was overshadowed in for solving spe- Theemies185o. English rose A newin High prominence type SchoolDuring of secondary was fromthis founded periodtflao school,to ingrammar5845 Boston theand high inbeganschools IRS school, I.a Sevendeclinecontinued developed. years after to decline. Acad- ordefinedarithmeticprinciples compound, and which the classified and"science were even developedas asof concrete number."natural into oror artificial.Inaabstract, deductivethis system, like system or numbers unlike, that simple made were The processes and ap- expansiononlyquitelater had Girlswell of been established,highHigh formulatedschools SchoolMorison atfromgot least but under 1840 (182, were in thetoway. p. fairly1860, North.532) "Mere well andstates wasestablished:by that1875 a period by they 1850 freeofwere threerapid ele- basic principles not Thusplierplicandmultiplication,plications always3 timesto beof be eitherarithmetic4 principlesinches abstract concrete was were andwere 12 or thefitted inches.abstractenunciated product to Suchthe but he system.demandedwhich rationalizations "like" allowed theFor that multiplicand. example, the contrast multi- in ancementarywayschoolsfessional at toschool and practical free training.secondary should and necessities.open be However, educationrequired; to all. In In andotheronly shouldOhio, teachers inregions Newforbe example,available Englandplans and for free were theoriesall; attend- primary gave should have some pro- school for number"burnalso"neverwith contrastthewas heard earlierapproach praised with any rule-example-exercise byColburn's cameexplanation historians to dominate disinclination and of any authors the approachprinciple" major to offormalize. texts,texts, in(t8,, which theand Althoughp. "science sections4S). a They student Col- of antogrades haveelementary K-12it actually was school plannedavailable Arithmeticsubject as everywhere by early themoved endas 1850, from ofin the the but state. academies it was not andpossible high schools to become the nineteenth century. mentarytouseddefinitions synthetic inmental the andsecond geometry, arithmetics. principles half maintainingof Edward werethe nineteenth often Brooks, that inserted "inwhosecentury, even texts likened into were the arithmetic supple- arithmetic we have the widely madewithschool Yale geometry curriculum and Princeton an entranceAlgebrawhenGeometry following Harvard requirementwas moved onin required 1847 its inwaydown and0165. it forto1848 from Princeton,becomingadmission respectively.the college Michigan, ain mainstay182o, level a little of the later. high Yale samelogicalofpp. the basis,165-66). mental faculties, and arithmeticsproceed a purposeBoth by werethe thethat same deductionsviewedjustified laws as the of exercising of inclusion the "science specificof large of psycho-number"amounts and the reasoning logical evolution" (1o6, logarithmsandof Cornell two groups for required entrance whose it inin Algebraprograms 1870. 1868, and sometimes geometryHarvard requiredwerehad the in conflict.goodgeometry fortune A facultyand to be espoused by both beliefychology in the andmake mental the mathematics need disciplinary to prepare requiredvalue for of further oflanguages the college-houndstudy and mathematics students. On the HISTORY OF MATHEMATICS EDUCATION combined with emsgher scienceAlthoughd hand, navigationspeak andthe out Harvard technologypractical for made algebra, set persons utility up.as geometry, theof planning mathematics first trigonometry,elective non-college-preparatory coursesin the and newly develop- pro- well as in the older fields of surveying after an under- mensuration. ffailties169.lidsduate there He rebellionbeing wasuntil motivatedthe experienced in presidency 1823, by an a withextended beliefof Charles student in individual elective William motivation systemdifferences Eliot, (215, beginningdid not pp. take till, in and theschools racticalimanisticio-95).gh school A aims littlegoalsthe curriculum(118, later result in favor thisp. of497). changeofwasadditional informational, By becoming 'thecame forcesend to more of academiessocial-civic, suchthe varied, nineteenth as the and vocational, decline high century of and religiousthe and and elective courses tenthonal,thed,tionid programscentury.commercial, of sometimes high These wereschools, and in setled separate manual uptowhich asvaried a schools,motivatedtrainingresult demands of programsduringthe the forchanging changesmathematicsthe beganlatter and inpart togrowing goals. bcand of estab-the even Voca-popu- nine- a-on As thementioned development in our of introduction,special courses. this was a period when4. Thetechnical colleges and the mathematical community atingid'heyistitutes history included the and desirabilityin theirmidwestem such programs; "new" of introducing andsubjects andwestern evenas thesemathematics, colleges the subjects.older were schools science, beingA famous were founded. English, cle- Yale :andardsachclorachelor:portisciplinary of 1828of and philosophyscience restatedrequiredvalues degree of theonlydegree mathematics. case threein test,that for instead theandoriginally However,in of 1852 four had years,Yale lowerHarvard introduced but admission admitted granted a a classical curriculum and the FIRST ENGLISH WORK ON DESCRIPTIVE GEOMETRY. myisichers,ardedtese at West new as books,subjectsPoint,aThe center first wasand asfor technical ideaspart mathematics of from a institutedegree France. and program! in innovative Thesethe United included ideas. States, descriptive It broughtthe Military Acad- founded in ,boa. In its early days it was re- MilitarywasconsultantClaude brought Academy.Crozet, over in railroadtoone teach of Napoleon's engineering.at the U.S. engineers, He later served as a FROM COLBURN TO RISE OF UNIVERSITIES 29 yI DaviesFrenchDavies,geometry published textschairman as a forsubject use theof irs thereandfirst mathematics the American and blackboard in other methods department, asAmerican a teaching book, schools.translated device. InCharles 185o The Logic and many 08 -: .011' 14! '14434817. pr t trigonometryandUtility Illustrated of Mathematicsand (i53). encouraged with learning the Best and Methods research of Instructionthrough the Explained found- The academy introduced the study of analytical a.? rwi 0* 040 00i 0 **0 .! iiisperilmisati;104an its 00,41 roe 1.1matiadviir isrgiew.pi4prisii3Out cooss: S. 0041 oF L anding of books the Military and at whoseand ThePhilosophical meetings French influencepapers Society, were in which American presented. collected schools materials and colleges is dealt with iissamap is gebralistDavies's1837in of other mathematics for ideas itsworks seven and (244,work students:texts in pp. adoptedspreading Davies,76-81). at this Thethe influence University role of isWest of Michigan Point and in(translated of from the French); Davies, Arithmetic; Davies, apparent in theBourdon's Al- r .;r ;no..? octraiaiamaut:u.. '. tam inalue.. Geometry;(translated from the French); Davies, Bridges, Conic Sections; Gregory,Surveying; Legendre'sMathematics Geometry for theDavies, Descriptive " .igkierid RFI,I,4, ta110!r7-. 4. Is by nicPracticalthe Institutevigorous Man. and(founded nontraditionalThe inscientific 1824) development andunder technological Amos of Rensselaer Eaton, trends ofPolytech- the period arc also typifieda Yale-trained upandcontributionsbotanist on surveying a canaland geologist wereboat titled ofThean a interesting who summernew was colleges, laboratoryits practical first especially "senior course textbook professor." those in surveying. west Among of New his England, had both Art without Science (357, p. 4:1) and the setting on trigonometry enrolledpreparation.theyclassicalpeculiar count students humanistic onadvantages Many their who,of students themtradition due and set to peculiarhaving uptothe impedeassociated lack had problems. of adequateor inadequacyacademies They secondary did inof which secondary school they progress, but neither could not have the inspectameliorateandMichiganschools in schools. 1871 these nearset President up This theirconditions, a system led homes, Jamesto thefor the thewerefoundingBurrill acting accreditation unprepared Angell president of various appointed of of to highregional the enter schools University college.accredit- in 187o, of To a commission to MaTED'HST ADVANCEDSTATES. MATHEMATICAL WORK PUBLISHED IN THE The first volume appeared in 1829. requiringschoolsHowever,ofing Colleges associations, and more collegesthe and than increase Secondary regionalof was which in creating elective associationstheSchools, first entrance courses was established forthe and their New articulation insolution. England 1885 (1 Associationproblems18. programs in both high p. 503). IIISVOR OF Al A T HESTATICS rout:A-row FROST CO I.It I/ RN to RISE OF UNIVERSITIES domeandThis the education. wasendthat-, ofthe eventually, this introductionMaster's period had degreesthere several in wasthis had significantcountrya further been awardedof collegiateeffects the idea on by schooldevel- ofmany mod- mathe- teachingversity,early,86o aswas were at 1832 thefounded offered Universityat New in in York 18881893 of University,Michigan. (tat the University Teachers185u of College, Michigan IK, p. 468). Courses in methods of at Brown University, and Columbia Uni- 3 Johnsnation:arlycan colleges. associatedHopkinslargely butbe University thewithdated concept research from in i873threeof and extended and relatedthe thecreation advanced tenureevents: of ofnew thestudy J. madie-J.fumuling Syl-and foundatMichigan Harvard in Kansas, State as early Normal California, asHorace 1873. School. Indiana, In Mann, 1891 Louis andmethods whoAgassiz Minnesota. became held chairman of the courses were also to be new Massachusettsa summer session. and at IgeriCallas professor luurnaland of themathematicsof Mathematics founding there there (1877 by -83);Sylvester the f and IV. E. ling of the American Nlighematical Society ling of OhiotranslatedBoard in 1836 of Education Victorsent Calvin Cousin's in 1837, E. Stowereport had studiedon Prussian German education. to study European education sys- educational ideas and The state of Ittp.:11 founded in 1888in :894. and hadIts forennmer,begung96 (144. to publish p.the1115-6, New its Milieu,: York1.11). Mathematical inthe 1891 Chicago sectionSociety, of the AMS was estab- andbutworktems theories were (182,of foreign activeof pp. such 330.-3 ineducators European initiating r ). Such includingchangesphilosophers men as at Pestalozzi,these home. not The only tlerbart, spread mirth as John Locke (163: - 170,1), educational interests and Froebel, of the Inatical abroadhe crynineteenth training andfew wereexceptions or centurywho largely were passed,those employed doing Americans .researchit became as college in whoincreasingly mathematics teachers.had had true advancedHow- had that beganknownImmanuel to appearand Kant discussed late (1724- inAs this in noted1804), the period. booksabove, and Herbertand Davies's periodicals Spencer methods book appeared in on pedagogy which(1832-1903) were 183o. It was the :hesupported U.S.studiedPeirceas Coast Benjamin therethe spent and endeavors as Geodetic anPeircemost undergraduate of of(113o9-8o) his suchSurvey life largely as andand anda professorhaving self-taughtNautical Simon assisted Newcomb Almanacat mathema- Harvard Nathaniel (1835 after mathematicsmatics.first book Today inand the professionalized itUnited mightNineteenth-century States be regarded for subject secondary arithmetic school texts, including those as a mixture of foundations of matter. teachers of mathe- written for use CA) at theiquetella. inNational Celeste. the preparation Academy Ills most for of famoustheSciencesThis press purework andof his mathematical waswas translation Linearissued inAssociative of work187o Laplace's wasin developed fr lectures familiarityItschoolshookin seemsthe thatnormal wasto with canhave Edwardschools, pedagogicalhe been viewed Brooks's thoughtful,had and The mathematical extensive, Philosophy and many notesas a to "methods" teachers. However,text for the the elementary first literature. Atof thatArithmeticbased time, on (elan).considerable nions'nal-katandI edition of Geodeticresearch mathematicsimo through the toSurvey tiventieth be the rather very "labors (86, pure, thancentury, p. a).of andin love" appliedThe becomingin the tendencyof generalmathematics,persons a teal forarea in nationalAmerican the of con- the U.S. handicap ofberas time.Hamilton'stoday, by mathematicians, there conception was considerable philosophers, of algebra, thought and and thereby given educators. Brooks discusses arithmetic, as the scienceto the nature of num- unle of IVorld War5. II. leather /raining and pedagogy ginia,Taylorests,cepts, iswrote Bledsoe, further mentioned The evidenced professorPhilosophy earlierThis byof inof themathematicsinterest connectionMathematics, publication in the with nature of severalAmerican of mathematics books. and at the University of Vir- its fundamental con- research inter- Albert ef aMann. group Formal ofwas professionalperiod founded teacher saw in trainingthe Alassachusettseducators. rise wasof preservicc instituted The iu first1839 and in public universities underin-service normal the leadershipteacher as training of lishedgeometryin ,868.Leibnitz, The It wasand Philosophy and calculus,largely Newton. concernedof discussing Mathematics,In 1871 W.with the M. the Translatedwork Gillespie basic of Cavalieri, of Descartes, published at Philadelphia conceptsUnion of analyticfrom College the Coorspub- de 12'bilosophie Positive of 4ilkuste Comte. Again, HISTolt V OF MATHEMATICS EDUCATION although this was far FROM COLDURN To RISE OF UNIVERSITIES 33 ngstatics," litiun'tomangles."rid a "'Thea methods pedagogical True hook, Idea ring: many of an "The ofEquation," its Object section of ""rwo headingsMathematics," Methods have ofboth "The Introduc- Mathematics," "Concrete Mathematics," "Abstract Mathe- True Dcli- a modern problem of high schoolcollegeThe many articulation forces enumerated and the problemin the previous sections 6. The climate of reform include the of "psy- econtlarr-schoolwasvpilied a schoolmasterby thel'he writings test rise inwho of ofgeometry. psychology Ihad lerberr written Spencer asan inductive, wrote( t82o-1903). laboratory-type, I Iis an offshoot from philosophy is perhaps an introduction for father manyschoolespeciallychologizing" non-college-hound curriculum to adapt-both to Thesethe improveto students. thecontent two changing learning related and highthe problems for methods school all were population of thestudied by twostudent- committees groups ap- but secondary with its todaywhichis Americanather's wasit would schtola carefully Theedition be had inductivetermed sequenced(247),been a with "discovery"methodtelling serieshis how inductiveof ofC.lburn fascinatedquestions approach had and neither the girls in his or "heuristic" approach. the stress on objects to geometry,instructions. atThepointed work appointment at thethis conclusion time. of Thethese first committeesof ofthis these period was typifies and rite the severalCommittee beginning significant on Secondary School Studies. of the nest. forces ead-hoverer,levelopedalvocated the student Colburnaby sequence Pestalozzi,to understanding did ofreduce questionswhom formalism byhe andprofessed activities and definitions means of questions and explana- to follow, nor as wellas in Spencer's hook. and strive to time.onstratesitinThis was1892, group,The appointed and NEAthe oftenit growingpublished had by referred beenthe influence Nationalreports formed to as in the Educationof in.893 professional1870Qurtmittee and Association of organizations 1894 (71, 71). The fact that Ten, was appointed (NEA) dem- at this ofmblicationions. Instruction. Collnon of lieeInextended said Introduction his thisobject approach to illgeln-o was to algebra in 18a5 with the on the Inductive Method ThisconsidermathematicsAssociation, subcommittee the goals whichof theandwas hadCommittee curriculumcomposed been founded of forof Ten leading mathematics in 1857. The was the first nationalout group of the to National Teachers mathematicians. Its rec- subcommitteeeducation (3o6). on rivesimplelearnerthy)to make mostbeginning combinations mightthe of transitionhis withreadily knowledge practical arc fromsolve given arithmetic byquestions without solving first.... the to intheThe algebrasimple aid examples learner of equations, algebra....as isgradual expected such himself.... This as possibleThe most to de-as the compartmentalized,tiondiscipline.wasontntendations thatlargely the Some oriented mathematics were of arcits toward quire recommendations,still program radicaltothe be goals put andbe intoof unified forward-looking, college particularlyeffect. preparation However, the and mental or integrated rather than even thoughrecommenda- it the rec- In summary, then, behismethod,this constantlyown period powers.besides making isgiving moth new the morediscoveries. learner interesting confidence, by making him ro him, because he seems to rely on oftionearlierommendation the of earlyandinformal be twentieth morethat and introductory psychologically intuitive century. geometry Thiswork designed began in algebrainto the preceded andjunior a trend that is being re- geometry come high programthe introduc- rimulatedoalsatirm,nil processes and and by processes changingdrill of as education. a for procedure,philosophical-psychological more Mental than thirty discipline were retainedsaw along the-rise with other of teaching newer procedures years after 1894, but the three- as a viable goal of Mu- ideas of the goals withschoolvived even today and earlier the with movement introductiontheThe inclusion other of committeealgebraof ofpreliminary more into referred the algebraic eighth ro is grade, the Committee geometry in the elementary ittee on College En- concepts. together :arningatep inductive,nd process processes and reasoning,teachingof "state were werebeinga andrule. being questioned,discovery-teaching give sought.an example, and practice" new psychological bases for processes. Both goals was yielding 94forcesthememberstrance period period andRequirements wenamed organizationscovered have atjust the bydiscussed.of suggestion theourassociated NEA,next chapter,appointed of with the itAMS. had but in theirIts 189o,the with report (74) affectedrootsProblem in the tilt.- and the some of the FIRST STEPS TOWARD REVISION 37 CHAPTER FOUR especiallyhighcollege school in thisfor program. elementary later periodIn the school paralleled period teachers, covered the earlier was by done pressurethis chapterin normal to complete much schools. of athe teacher training, admissiontheIn.tactAs 1894-1920a collegeswith result, elementary of colleges werehigh the chiefschool the and and settingconcerns universitiesgraduates secondary and common administering to had education college. much to the Bothless secondary-schoolsthan of. concern thestandards they traditional have for orfor today. con- col-andthe 1894-1920First Steps toward Vvision curricula.wereforprograms,lege college requirementsnot Further, uniform. alongpreparation these with They and developing pressures on thevaried the then-new high with oftengraduate schools. the electiveopposed colleges programs, However, undergraduate the and forces:implicit exerted withthese new pressures pressures collegecollege in interests,education?issuesbounda growing andbecame and highIhad low abilities these: immediateschoolcan theof What its population,school students? vocational-personal arc adapt the muchgoals to the ofof varied highwhich needs. schoolneeds, was The non-college-mathematicsbackgrounds, prominent withifThe atchange all.issuesthe However, shifting thein mathematics educational of the the prominence nature educationprogram. and ofdirection tendvariousThe majorto endure, ofissues the force forces tendsto during change exertedto changethe slowly period to 1. Forces at work dominantmillion-0)-0). to organizational24 million Thepupils pattern elementary in the of schoolsinterval schools was from underwent beginning too° to t93o.toa terrificchange The expansion,2.. from ;6 The elementary schools weretheof 18943f pressitre learning.forces to two, generated to Mental resulting provide disciplineby anpsychologicalfrom education the as growthan achievable researchfor in all school children. and goal enrollments,changing of Added instruction theories to was this was yeartomentarytoward "6-3-3," elementary the school with end a offollowed schoolthree-year thisArithmetic, period.and by ajuniorfour three-year The onceyears high "8-4" a school ofcollege high pattern,high school. inserted school,and eight secondary betweenwas years changing ofschool a ele- six- subject, had moved inacing integration:onsiderationlevelopmental, questioned. of contentand andBothThis refinement intuitive educators periodand a approaches shift0894-1920) of and the to newermathematiciansissues to is methods thisformulatedcharacterized content. usingwere in the concrete,callingby previousa somewhat for intensified Cotruuitteemaryforto the early numberelementary arithmetic of ideas Fifteen school.. work be ontreated were Now, Elementary disappearing. inincidentally the elementary Education, and It was informally. school, whichproposed separate reported The that NEA's texts pri- in tetiod.heNormsals same OneId ofWar timeshouldthis 1 divertedperiodleading also to notetoconcern those some the of supportsimilarityand the energies following for of the the from continued period.problems education For education and example, while pro- atof wastimealgebraand1895 thattoper serve (68),would daynumber devoted suggestedas be a work introduced,"transitional to be mathematics.that continued grades along step a iowith into Thisthrough algebra gradesa introduction reduction 6 proper"complete 7 and in of (p. 8the somebasicwhere 58), amount algebraarithmetican some idea of nathcmaticseturningigher educational veterans.and its applications Later, facilities World as for well Warveterans. as IIproviding gave The a greatly farpressure greater expanded to push enroll to in 36 school.in keeping with the purposes of the newly developing junior high 38 HISTORY OF MATHEMATICS EDUCATION FIRST STEPS TOWARD REVISION 39 considerableof thisnumber,"had period.tended afterdiscussion Into stressthis the approach, definition what wasThe (lawn devotedarithmetic as of (441, wasarithmetic to notedpedagogydistinguishingpp. 72, inoften 115-16) the of previousgiven the different calls latter in thethechapter, halftypes "science texts of of the nineteenth century -m-»). There were several approaches to "psychologizing" secondary3. school The secondary schools purposenitionsnumbers of associated rationalizingsuch as "concrete," numberThe the early operations with "abstract,"part units of thewith and andperiod numbers.collections "denominate." 1894-t9ao of objects saw These substantial for defi- the innovations in ingematicianstryglandmathematics bya major shouldthe efforts of portion be duringthe mentioned ofUnited ofProfessorthis his period.States. 1902 first, E. addresssincePerhapsMoore H. Moore,it wastook on the wellretiring theone"Perry unusualpublicized of as movement"the president leadingstep in this of devot- inmath-ofcoun- En-the 302).stractratiorelationthe approach ofSpencer's concept quantities,led William toor emphasis cumbers.a lengths. collectionSpeer on toareas,Herbert the stressof discretespiral and Spencer'snumber naturevolumes objects as of dictutn theratherlearning (441,. directly thanthat pp. affected numberasperceived135, an 2both ab- to, is gestedin(380.AMS which a toIle stress a surgedalgebra, discussion on thethat geometry, interrelationships schoolsof needed andabolish improvements physics theof mathematics "watertight were intaught. American compartments" with Ile science. also schools sug- allearlier the(441, present:tr. period p. had presentedEater each John new Dewey topic in its turn, once and for and the organization of arithmetic texts, which in the ( itit9-t95: ) viewed number as originating ideasandmathematicsbyThis John applicationslatter usually Perry was bepostponed inone made addressesand of byconcretethe tobringing three later given by majorgrades. into usein 1900-1902. lowerproposalsof Moorelaboratory-teaching grade endorsed vigorously Perry levels also manyall advocated techniquesofurged this simple andthat forehim1115--to).through andstressed James measurement. measurement A. McLellan, Theactivities, and text the furproblems, school teachers books and of unitsbasedarithmetic (441, on it,pp. written there- 129, by ,junioraingfurther colleges, forthcoming for teachers stressed junior article in highthe these desirability schools,in schoolswhich and "John and of improved developingin Dewey, the high subject-matter calling schools. itt the attention United Ile alsotrain- States tonoted the 63).ingsby the and eclecticism texts of David andClason avoidance EugeneFollowing suggests Smith, ofthese extremes that accordinginnovative the to next he toapproaches foundperiod Clason in in the(441, elementary there writ- p. was aschool reaction mathe- typified geometrythereevolutionary should to the characterbe systematic,no abruptThe of the Nationaltransition deductional education Committee from ofgeometry" the the introductory, individual,of Fifteen (p. 48). oninsists intuitional the Geometrythat Syllabus was ceptschologistschologyconetectionism.matics ofspanned so mental of strongly the Thisthe daydiscipline yearspresented termthought largely1917-35 and they by transferrefers Edward hadand thoroughly shouldto of theI.. training.Thorndike stimulus-response be termeddiscredited The after the Thorndike periodthethe psy-con- of psy- appointedMathematicsandNEA.tion ofMathematics, Its Teachers in "Provisional 1908 Teachers. -9 of under journal the Report" Mathematical the This of joint the(66) report Centralauspices appeared showed and Association ofNatural inthe some ism American inSciences effect Schoolof Science Federa-of ScienceandMoore- and the arithmetic1:65;portanceanddrithmetics The 166). Psychologyof into appearedThis establishing many psychological insmallof 1917.Algebra many facts In approach"bonds" The (1923)and Psychology skills byDsortulike led meansto to be the of taught ofArithmeticstressedfragmentation much and practice the tested(1922) im- of ableswith,forThisgeometry,Perry the report views exclusion however,whichcalled as well offorwere asmaterials continuedthe a still usereaction required in dealing stressgeometry to rising courseson with logicalof failure limitssome in structure. manyrates andinformal incommensnr- inhigh algebra Itproofs calledschools. and for rectrelatedseparately.dealt bonds. facts the ThisThusfinal close theoryblow Clason in time to even the says,to one'science led "For anotherto the good of avoidancearithmetic for or fear ill, itof was of"establishing (441, teaching Thorndike p. 64). closely incor- who nationalsionReportmore onuse theCommission of concreteTeaching examples was Manyof Mathematics set of upandthese at graded the saute (49)Fourth "originals." pedagogical issued International in 19t2. emphases TheCongress Inter- were discussedof in the the American Comlnissioncrs of the International Commis- 40 HISTORY OF MATHEMATICS EDUCATION FIRST STEPS TOWARD REVISION 41 ofreportscommissionersgress Mathematicians(39-48). held as well inAs Cambridge, wasaas continuation those heldbased ingiven England.Rome atof inthethis The congress, project final and I. L. based Kandel and R. C. upon the reports of many subcommittees 1908. It reported at the 1912 con- report of the American on some later devotedtrigonometry,includeadvocated topics to athe general graphs, fromimportance arithmetic, mathematics Thisand descriptiveandperiod, unifyingalgebra, program 1894-19t0, statistics. nature intuitive for gradesof can An geometry,the entirebe function 7-9 partially whichchapter numerical concept. wouldtypified was by a concern for topics,try hereArchibaldto omit toand reduce difficult abroad. prepared the or These stress Theobvious American on manipulationgeometric commissioners proofs, in to in comparative studies of the trainingappeared of teachers in 19:5 and 1918 (163; 5). 1912 noted tendencies in this court:- postpone difficult syllabi.tionsgoals andand statecurriculum education revision. departments Several regional also prepared educational reports organiza- and the stress on solution ofArchibald stated in 1918 Ac the present time superintendents equations and applications.(5, p. 4): .. . (in) the United States have algebra while increasing 404))thewithin nineteenth mathematics century hadAn bybeen emphasis Germany's strongly on advocated famousthe function mathematician since concept the middle and Felix onof interrelationships4. The colleges and the mathematical community talentofpreparationmethodsbeen the forcedteacherin theof secondarywere by were public required made schoolopinion such on education.the country, other difficulties would disappear. Most countries as to attractto consider in sufficient numerous numbers radical thepart best changes of each in teacher, and the position If a high minimum standard of quiumFairinternationalKlein. of He at1893. Northwesternand conference Kleina number stayed University. ofheld other on in after conjunctionforeign This the colloquium tnathematiciansfair withto participate the had Chicago a attended very in World'sa stimu- collo- the CO matics appointed in the UnitedThe most States influentialspectconsidered committee to teachers in this of bulletin mathematics have farin the secondary schools. dealing with the teaching of mathe-higher standards than we with re- undersummerlatingE. R. effectthe Hedricklectures title upon toand mathematical German C A. Noble mathematics research and published inteachers the United in werethis States. countrytranslated Klein's in 1932by Elementary Mathematics front an Advanced Standpoint. preliminaryMathematicsmaticalCommission Requirements, report in onSecondary appearedMathematics whose Education in 1910 final and a was the National Committee on Mathe- report, (59), prior to the recent (1956-59)summary in 1921was (57;published 58).The in 1923. Reorganization A of andBolyaitranslatedlished Saccheri inand this sourceLobachevsky (1910).country materials E. Aaround numberJ. (translatedTownsend's related the of turn translations toin translationof non-Euclidean1891 the butcentury. of ofpublishedimportant Hilbert's G. geometry B. in foreignHalsted 19.4), by treatises were pub- Founda- schools,tion'spersonsUnited majorin States addition with concern specialentered to the with The interest mathematiciansWorld thecommittee War in and I. The wasexperience who appointed with the by the MAA in 1916, before the committee contained several represented the associa- secondary axiomswroteschooltions of geometryGeometry( 51). of his day was out-of-date and too full of errors,Rational Geometry was published in 19o1. Halsted, feeling that the high as a high school text based on Hilbert's remarkabletionwithliminary beforea call for reportforpreparation criticisms having was somecirculated andof the financial final through the U.S. Bureau of Education comments toundergraduate be submitted college for considera- program. Its report. Thesupport committee from an was outside also source pre- setLatergeometryHuntington of continuingaxioms in 1904. for and thisgeometry Huntington Oswald AmericanAxiomization in Veblen axiomatized1933. interest, becameThispublished G. system,elementary D. a Birkhoff majora newinvolving algebraAmerican setdevised of "ruler axioms a research newand for interest. E. V. in 1905. "required"zationthefor reductionof itstheorems theorems staff ofand elaborateand activities.andIts increasing final report the stressed number of proofs in geometry by manydecreasingmanipulations of the ideasthe number in mentioned algebra and above, of the such memori- as "originals." It also ern"atBasicprotractor Harvard, ideas Geometry, postulates," of Ralph mathematical BeakySeveral was (97). later foundations books incorporated were to written teachers. or compiled J. W. Young's to convey the then-"mod- by Birkhoff and his mathematics-educator colleague into a high school text, Lecturer42 on Fundamental ConceptsHISTORY of Algebra OF MATHEMATICS and Geometry appeared EDUCATION FIRST STEPS TOWARDA number REVISION of persons who were primarily mathematicians also main- 43 ington,DartmouthModernin Jolt Birkhoff, (584), .Mathematics College and DavidJ.is (1911),W. A.Eugene which Smith probably better known among mathematicians Young edited Monographs on Topics in included articles by Veblen, Hunt- (286). J. W. Young of 1 cianspresidentmitteetained J.onan W. Mathematicalinterestof Young,the MAA. in schoolA. The R.Requirements Crathorne, committeeprograms. wasForincludedC. N. example,appointed Moore, Smith the andby and NationalE. H. mathemati-R. W. Hedrick, Tyler.Com- infessorgroup thewithfor his theory19:0of Oswald theextensive edition Pedagogyat Clark Vcblen. ofaxiomatic University his of J. Mathematicsbook W. treatise A. inThe 1892, was listed as "Associate Pro- Young, after writing a dissertation in on projective geometryat written the University of Chicago" LincolnVeviaoftory" persons Blair schools.School, of fromthe BothNew I themlorace theYork to existence Mann theCity, committee Schoolwere of these from and were schoolsexperimental Raleigh manifestations and Schorling the or appointment "labora- of of the the de- bookteachinginElonentary 19(16. purported of secondaryIt wasand to theSecondary he second schoolfor SchoolAmericanmathematics. teachers, David Eugene Smith's The Teach- (185). This book was firstbook puhlished to dealTeaching extensively of Mathematics with the in the Although Davies's earlier NewasStatesresentedvelopment a representativeEngland, and the ofMaryland. professionalAssociation and J.of \V. A.the FobcrgF. mathematicsofAssociation Downey Teachers represented Was ofofeducators. TeachersaddedMathematics the to Central Missthe of Mathematicscommittee Blairin Association the also Middle rep-later in tionmaticstheing Board first ofdeveloped Elementary book (CEEB) to duringresemble Mathematics Anotherthis (241), which strong collegiate influence upon secondary school mathe- was established in 191 t. It included mathematicsa modern methods text. period. The College Entrance Examina- appeared in 1900, was teachersforcommitteeof Science California, from andwere .1"exas and Mathematics A. I'. C. andI I. Olney, Underwood Missouri. Teachers. commissioner and The Ettla remaining ofA. secondaryWeeks, members mathematics education of this longits lishedintests concernthe wouldsetsyllabi of for be testsdefining the based. itproblems administered the The of for its high school mathematics content development of the boardarticulating came after the high school and col- member colleges, and it pub- upon which scholarlyof mathematicswasprevious incorporated and paragraph. educational began in 1920. toI.ocal be Thesocieties. Itformed andwasMAA thenlargely Threealongwas national stimulated founded withof these the organizations generalwere inby tots. the cited interestgrowth The concerned in Ncrmthe ofand with the teaching catedrucntssucceedinglege concrete curricula in college. and selecting The Committee high school on geometryto and the introductory NEA in .899 algebra (69). Thisin the committee seventh grade. also advo- graduates who were capable of College Entrance Require - Maryland.sociationnal,planning the Mathematics of of the the Chicago Teachers Teacher, Men's of Mathematics Mathematics was taken over inClub. the in MiddleThe 1951 Council's from States the jour- and As- senting-M43 an interesting developmentDavid Eugene .in this Sthith and ). W. A. 5. Pedagogy and teacher Young maytraining be regarded as repre- periodthe appearance of generalworkof tests andeducators, and influence measurements. especiallyResearch of E. L. Thomdike.by Wepertaining persons have already to in mathematics the rapidlycited, for growing education example, field was the also carried on by NationalNC:TAI.ers"mathematics1922. CommitteeCollege, He Ile served played! Columbiaeducators." on onMathematical important University, Smith roles from in the ANIS (86), AIA many important committees, including the was professor of mathematics at Teach- tarn until his retirement in A, and the 404)) In addition to a continuing stress on the issues cited in previous6. Summary chap- Issues theScienceTeachingthe teaching American Society, andof Mathematics. commissionershistory a of mathematics. Ile was also prolific writer, and the director of on the International Commission on the Requirements.a founderIle was one of the of History of many studies in these:ters,a.t. What threeHow newarecan the issuesthe goals mathematics began of mathematical to evolve. curriculum The instruction? continuing and teaching older issues methods were be Close WindQH Title: The 'New New Math'?: Two Reform Movementsin Mathematics Education. Subject(s): MATHEMATICS -- Study & teachingUnited States; NATIONAL. Council of Teachersof Mathematics Source; Theory Into Practice, Spring200 I , Vol. 40 Issue 2,p84, 9p Author(s): Herrera, Terese A.; Owens, Douglas T. Abstract: Compares the origins, curricular and pedagogical contentand the impact of the new mathematics and the standards-based reform movements in the UnitedStates. Discussion on the mathematics reforms; Influence of psychologicaltheories on mathematics; Factors to consider in mathematics curriculum development; Responsefrom the National Council of Teachers of Mathematics. AN: 4472840 ISSN: 0040 -5841 Database: Academic Search Elite Text Available: Full Page Image THE 'NEW NEW MATH'?: TWOREFORM MOVEMENTS IN MATHEMATICSEDUCATION MATHEMATICS EDUCATION IS RIFE with the battle criesof so-called "math wars." On the one side are proponents of standards-based reform, a movementlaunched in 1989 with the publication of the Curriculum and Evaluation Standardsfor School Mathematics by the National Council of Teachers of Mathematics (NCTM).Opponents to the reform proclaim it a reincarnation of the mathematics of the 1960s, popularly known asthe "new math" and popularly remembered as a pedagogical failure. Hence, these opponents term the current reform the "new new math," yet we have never seen an instancewhere they have actually compared the two reform movements. In this article we compare the origins, curricular and pedagogical content, and impactof the new math and the standards-based reform movements. The first part of the article describes the events leading up to the new math era and the characteristics of representative curriculums. To conclude the first section, we consider the demise of the movement and events leading to the most recent reform. The second part of the article describes the NCTM standards-based reform, an educational movement that shares some similarities with and yet differs substantially from the new math reform. We conclude with insights into the reaction to this current movement and its impact on school mathematics. The New Mathematics Reform

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41 The coalescence of concerns coming out of World War H,other events on the international scene, and discontent with high school mathematicalpreparation on the part of mathematicians fomented changeand the new math was born. Sense of national crisis The "new math movement" of 1960-1970 was 15 years in themaking. During World War 11, both educators and the public recognized that more technicaland mathematical skills were needed to push forward the developingtechnological age. NCTM appointed the Commission on Postwar Plans to make recommendations about mathematicscurriculum. The goals were to establish the United States as a world leader and tocontinue the technological development that had begun during the crisis of war. Although they documented problems with the preparation of youth in high school mathematics,the reports of the commission in 1944, 1945, and 1947 had little lastingeffect. Osborne and. Crosswhite (1970) characterize the reports as "basically regressive" (p. 246). To add to the concern about the inadequacy of the mathematics curriculumfor the emerging technology, the Soviet Union launched the first satellite into space inOctober 1957. This has commonly been cited as the event that marked the beginningof the new math revolution. The launch of Sputnik in 1957 created the perception that theUnited States was behind in the world scene of technology and military power. Thelaunch served to loosen the public purse strings and catapult a movement that had alreadybegun. Curricular concerns Prior to the launch of Sputnik, the University of Illinois Committee onSchool Mathematics (UICSM), representing the perspective of university mathematicians, was formed in 1951 "to investigate problems concerning the content and teaching of high school mathematics" (Osborne & Crosswhite, 1970, p. 251). Following a survey, it published a pamphlet, Mathematical Needs of Prospective Students in the College of Engineering of the University of Illinois (Osborne & Crosswhite, 1970, p. 251). In 1955, the College Entrance Examination. Board (CEEB) took a new kind of step to directly influence the school curriculum. CEEB appointed a Commission on Mathematics to consider how their examinations should reflect the changesin the field of mathematics that had taken place in the previous 50 years. To its credit, the commission represented university mathematicians, high school teachers, and college and university mathematics educators, three groups most directly concerned with secondary school mathematics curriculum (Osborne & Crosswhite, 1970, p. 260). The commission's report, finalized in 1959, was characteristically restricted to college- bound students. The commission's nine-point program called for preparation in concepts and skills to prepare for calculus and analytic geometry at college entry. The commission wanted appreciation of mathematical structure in properties (e.g., commutative property, roles of 0 or 1) of natural, rational, real, and complex numbers. Use of unifying ideas such as sets, variables, functions, and relations was recommended. Treatment of inequalities along with equations was requested. For Grade 12, a half-year of elementary

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42 3

functions was recommended along withalternative units in probability and statisticsor modem algebra (CEEB,pp. 33-34). Influence of psychological theories

The Swiss developmental psychologist,Jean Piaget, had been writing about cognitive development since the 1920s and 1930sand had begun to catch the interest of the mathematics education community. In1952, the beginning of thenew math era, The Child's Conception of Numberwas translated. into English (Piaget, 1952). Although the ages at which the stages occur have been questioned, Piaget'sstages of cognitive development and his concepts of conservationand reversibility, for example, hada significant impact on mathematics education,especially at the elementary school level. Attention to Piaget's theory had the effect of emphasizingthe importance of concrete examples and physical manipulatives. However,unanswered questions about growth of mathematical ideas in young childrengave rise to some teachers becoming overly cautious in offering challenges to children, albeit withconcrete/manipulative means. Jerome Bruner influenced mathematics curriculum andpractice in the new mathematics era through his promotion of the discovery of mathematical ideas. Usingwell-chosen problems, he said students can do investigation anddiscovery rather than being told the relevant concepts and expected to practice skills. Throughan oft quoted statement, Bruner (1960) made the point that readiness fora topic or concept does not depend altogether on the child's maturation. He said, "We begin withthe hypothesis that any subject can be taught effectively insome intellectually honest form to any child at any stage of development" (p. 33). His point is that readiness is primarilya function of curriculum writers and teachers finding a suitable contextor way of expressing the principle or concept to make it accessible to the learners.

Bruner (1966) also devised three stages of representation of mathematicalideas: enactive, iconic, and symbolic. The enactive mode is essentiallya hands-on, concrete representation of an idea. For example base-ten materialsare a manipulative representation of base-ten numbers. Diagrams and pictures of base-tenmaterials would be an iconic representation. Using ordinary numerals would bean abstract representation. High school curriculum development

Max Beberman and his associates on the UICSM operatedon the principle that improvement in mathematics curriculum would haveto include classroom-tested instructional materials. The UICSM produced materialsas a joint effort of the colleges of education, engineering, and liberal arts and sciences. The staff produceda high school program that was taught at the University High School. By 1958a program was in place for the four high school years. When theprogram was more widely distributed, teachers were required to complete in-service education in the philosophy,program, and methodology, and were to be supervised by staff from the project. TheUICSM program focused on significant changes in curriculum to identify unifying mathematicalthemes and use precise language and symbols. Features included:

3EST COPY AVAMA111FLE 43 9 the move toward an integrated mathematics curriculum in which algebra is found throughout the four-year program;...the introduction of some modern concepts such as set terminology;. .the relocation of several topics, such as...inequalities, to lower levels in the curriculum....Care was taken to ensure precision of expression, even to the extent of inventing new terms and symbols (e.g., "pronumeral"). (Osborne & Crosswhite, 1970, p. 254)

Moreover, "discovery teaching and learning became the hallmark of the UICSM program" (Osborne & Crosswhite, 1970, p. 254). Strongly influenced by college mathematicians, the largest curriculum project of the era was the product of the School Mathematics Study Group (SMSG). SMSG epitomizes much of the development of the era. Funded by the National Science Foundation (NSF), the writing sessions in the summer of 1958 and subsequent summers involved hundreds of mathematics teachers and mathematicians. SMSG operated by a process of summer writing sessions, classroom trials during the subsequent school year, rewriting, and publishing for national distribution (Osborne & Crosswhite, 1970, p. 274). Over 60 texts were written as well as a variety of supplemental materials and reports. SMSG texts were to provide a model for commercial writers. Distribution was to continue with demand and eventually die a natural death (Nichols, 1968). Initially SMSG texts were written for high school programs to carry out the recommendations of the CEEB Commission on Mathematics, which included the following: treatment of inequalities along with equations structure and proof in algebra integrated plane and solid geometry with coordinate methods integrated algebra and trigonometry a twelfth grade course in elementary functions (NACOME, 1975, p. 5) Junior high school projects While there were many projects at the junior high school level, three represent the thrusts of the time. Unusual (at the time) chapter titles for seventh grade of the University of Maryland Project included symbols, properties of natural numbers, factoring and primes, the numbers one and zero, mathematical systems, and logic and number systems. Topics for the SMSG seventh and eighth grades included ideas of structure of arithmetic from an algebraic viewpoint, the number system as a progressive development, geometry with applications, measurement, and elementary statistics. "Careful attention is paid to the appreciation of abstract concepts, the role of definition,...precise vocabulary and thought, [and]...creation and discovery, rather than just utility" (Brown, 1961, p. 18). Junior high school texts and curriculum guides developed throughout the 1960s indicate that students encountered

44 MST CO rAVA1IIA0L7 the concepts and language ofsets, algebraic properties of number systems, nonstandard numeration systems, informal properties of numbersystems, and number theory. [These topics arel rich preparation for high schoolstudy and a striking contrast to pre-1960texts for grades 7-8. (NACOME, 1975,p. 9)

However, the National Advisory Committeeon Mathematical Education (NACOME) was unable to determine the impact on the curriculum in general.For indirect evidence they turned to the National Assessment of EducationalProgress (NAEP) tests to examine what was being tested.

Of approximately 200 items usedto sample mathematical attainments of 13 year olds, NAEP included very few related to sets (6), probability(6), inequalities (2), and non- decimal numeration systems (2)...There were many geometry items, thoughstress was on measurement aspects that were not particularly novel. On balance theassessment pool shows some signs of a changing curriculum, butnot dramatic upheaval. (NACOME, 1975, p. 10)

A third program, the Madison Project,was intended to prepare supplementary activities to stimulate children to be creative and develop enthusiasm for mathematics.Topics at grades 4 to 8 included axiomatic algebra, coordinategeometry, and applications to science. At the grades 6 to 9 level, topicswere statistics, logic, matrix algebra, and applications to physics. Above all, the approaches to the topicswere the mathematical laboratory and discovery methods (Nichols, 1968,p. 20). Elementary school curricular change in an attempt to reflect current changes, the elementary school textbooksof the era introduced the language of set theory, and the algebraic properties: commutative, associative, and distributive. Another topic characteristically addedwas arithmetic in bases other than ten (NA.COME, pp. 15-16). The intentwas to generalize the concepts of base and place value and thereby solidify the understanding of baseten. Elementary school curricular change was slower andmore difficult to implement than the high school programs. Few elementary school teacherswere mathematics specialists, and little inservice training was offered to help them understand and implementthe new curriculum ideas and materials. However, judging from populartextbook series or curriculum guides, some changes prevailed.

The label "arithmetic" has appropriately givenway to "mathematics" as curricula incorporate varying amounts of geometry, probability and statistics, functions,graphs, equations, inequalities, and algebraic properties of numbersystems. (NACOME, 1975, p. 1 1 )

A survey of teachers indicated that elementary school teachersgave more time to geometry, but other new topics such as graphs, statistics, and probability received less time and attention (NACOME, 1975, pp. 11-12), Standardizedtests of the era point to a shift toward comprehension goals rather than focusing onlyon computation. However, the tests fell short of curriculum developers' hopes of being ableto teach and measure students' understanding.

.3EST 45 Reaction to the new math reform

Teachers responded enthusiastically by taking advantage of NSF funded summer-long institutes and training offered by the innovativeprograms. Parents reacted to the new mathematics curriculum with consternation due to their inability to help their children with mathematics. A few mathematicians were vocal in opposition to thenew mathematics, notably Morris Kline who wrote the book, Why Johnny Can't Add (Kline, 1973).

The decade of the 1970s was characterized as a "back to the basics" era. Whether or not justified, the widespread sentiment was that the new math had failed, and that a return to the basics was needed. In particular, critics cited the College Board's report of a 10-year decline in Scholastic Aptitude Test scores (Usiskin, 1985). The precise, structuralist language of sets, logic, and algebraic structures, hallmarks of the new math curriculum, were abandoned in favor of more emphasis on computation and algebraic manipulation. Socratic dialogue and pedagogical approaches of discoverywere relinquished for those backed by principles of behavioral psychology. Lesson objectives were stated in terms of observable, measurable behaviors. Texts and teachers were to guide students through a prescriptive hierarchical curriculum (NACOME, 1975). Teaching practice during the back-to-basics era was described in National Science Foundation case studies:

In all math classes that I visited, the sequence of activities was the same. First, answers were given for the previous day's assignment. The more difficult problems were worked by the teacher or the students at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and the problems assigned for the next day. The remainder of the class was devoted to working on homework while the teacher moved around the room answering questions. The most noticeable thing about math classes was the repetition of this routine. (Welch, 1978, quoted in NCTM, 1991, p. 1) It was in reaction to this curricular focus that the present reform in mathematics education emerged. The NCTM Standards-Based Reform This most recent reform was launched in 1989 with the publication of NCTM's Curriculum and Evaluation Standards for School Mathematics, a vision of teaching and learning that differs radically from the back-to-basics curriculum that preceded it. In this section, we consider its genesis, its conception of curriculum and pedagogy, and reaction to the movement. Our intent is to highlight for the reader the similarities as well as the differences in the two reform movements. Curricular concerns Reaction to the back-to-basics movement grew steadily within the mathematics education community during the decade of the '70s. The dominant place of computation in the elementary curriculum was questioned, as was the low priority given to problem solving. Also, dissatisfaction deepened as areas in the field of mathematics gained importance in a

4 6 3E31' COPY AVAIIA changing society but were not reflected in theschool mathematics program (Usiskin, 1985).

Emerging concerns were voiced through conferencesand reports. Among the most prominent was the 1975 report of the NACOME, which"called for a repudiation of a curriculum dominated by manipulative skills" (McLeod,in press) and recommended more work with technology and applications. In thesame year, the National Institute of Education (NIE, 1977) sponsoreda conference in Euclid, Ohio, that outlined 10 goals of mathematics, problem solving being the chiefamong them. Soon after, the National Council of Supervisors of Mathematics (NCSM),a body composed of leaders at university and district levels, published its Position Paperon Basic Mathematical Skills (1977), which defined "basic skills" as includingnot only computation but also estimation, geometry, problem solving, andcomputer literacy. In a summary of the concerns noted in various reports and conferences sponsored by the National Science Foundation in the 1970s, Fey and Graeber (in press) list the following: The public views mathematics as a set of arithmetic skills. Applications should be a more prominent part of mathematics curriculum. Problem solving should play a more central role. The role of technology needs to be considered in relation to computational skills and the handling of the "daily bombardment of statistics." These reports prompted NCTM to appoint a committee to develop recommendations for school mathematics for the coming decade of the '80s. The product, An Agenda for Action (1980), was one of the earliest position statements from NCTM anda definitive step toward reform. It set problem solving as the curricular focus, recommended that the definition of "basic skills" be broadened to include such mathematical skillsas estimation and logical reasoning, and promoted the use of calculators and computers in the classroom at all grade levels.

Although this publication articulated a direction for curricular change, its impactwas insufficient to quell the rising sense of urgency among mathematics educators. Usiskin (1985), for example, in a strong argument for updating mathematics curriculum, concluded with this call for change:

Current student needs in mathematics cannot be met without modifying thevery goals and nature of secondary school mathematics. Recent reports confirm that the current curriculum needs overhauling rather than adjustment, revolution rather than evolution. (p. 17) Sense of national crisis There was no Sputnik launch to ignite this reform, but rather a perceived falling behind in worldwide technological and economic standings. Concern with U.S. student performance had become linked in the public mind to the country's capacity to fill its technical needs (see Mathematical Sciences Education Board, 1989).

47 T3EST COPY VARIABLE Results on international studies, suchas the Second International Mathematics Study (SIMS) and the International Assessment of Educational Progress (IAEP),showed the U.S. as weak in several areas (Robitaille & Travers, 1992). Furthermore,despite the very focused emphasis of the back-to-basics periodon procedural skills, "national tests showed that student performance in basic skills declinedor stayed the same" (Kenney & Silver, 1997, p. 66).

While these voices grew louder within the halls of government and academia,an event occurred that awakened the general public to thesense of crisis: the publication of A Nation at Risk (1983). Commissioned by the National Commission for Excellence in Education (NCEE), this report struck a note ofurgency: "If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war" (NCEE, 1983,p. 5). Its rhetoric linked education to a national crisis, as in the days preceding thenew math movement. The response from NCTM

The National Council of Teachers of Mathematics, which identifies itselfas "the largest non-profit association of mathematics educators in the world," aims to represent "the broad community of stakeholders in the fields of mathematics and mathematics education" (NCTM, see "NCTM at a Glance": http: / /www.nctm.org/aboutlintro.htm). In response to the reports of the '70s and to the back-to-basics curriculum, NCTM adopted an activist stance. As Shirley Hill, NCTM president from 1978 to 1980, stated, "A major obligation of a professional organization such as ours is to present our best knowledgeable advice on what goals and objectives of mathematics education ought to be" (quoted in McLeod, Stake, Schappelle, Mellissinos, & Gieri, 1996, p. 24). With this motivation, NCTM stepped forth to produce a set of standards for school mathematics. Using only organization funds, NCTM produced and distributed Curriculum and Evaluation Standards for School Mathematics (1989), a document intended to be a multivoiced statement reflecting the varied. composition of the organization and to produce a consensus broadly acceptable to the mathematics community. To represent that community, the writing group was composed of mathematics educators ranging from elementary teachers to college faculty (including mathematicians), researchers to state and district supervisors, and people with expertise in technology, teacher education, and other areas. The standards-based movement also produced Professional Standards for Teaching Mathematics (NCTM, 1991), which addressed explicitly the changes in teaching that were implicit in the first document, and Assessment Standards for School Mathematics (NC1'M, 1995). These documents, collectively referred to here as the Standards, rounded out the NCTM vision of mathematics in K-12 classrooms. Conception of pedagogy and curriculum A reform, by definition, challenges current practice. As understood from the NCTM Standards, this reform in mathematics education advocates changes in content and pedagogy. A significant difference from the new math movement is NCTM's emphasis

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on mathematics content and instruction suitable to all students,not only the college bound. This principle underlies its vision in bothof these essential areas. Content. The Standards do not list specific topicsto be covered by the end of each grade. Instead, the documents present guidelines and provideexamples to clarify its unique conception of school mathematics content. Some notedchanges were:

The opening of secondary mathematics to include discretemathematics, statistics, and mathematical modeling, with increased attention overallto applications. Across the grades, stress on connections of mathematicsto the real world. More integration of mathematics topics.

Emphasis on higher-order thinking andon "making sense" of mathematics through problem solving, communication, connections, and reasoning.

A change at the elementary level from almost total concentrationon arithmetic to inclusion of such topics as geometry, patterns, and statistics.

Underlying these proposed changes in content isa central focus on the conceptual versus the merely procedural. For example, the Standards call for decreased attentionat the K-4 level to long division and to "complex paper-and-pencil computation," but increased attention to mental computation, the meaning of operations, and "thinking strategies for basic facts" (NCTM, 1989, pp. 20-21). Pedagogy. This reform views the learner as actively participating in the construction of knowledge, a view that leads to changes in teaching practice, suchas: Active student involvement in discovering and constructing mathematical relationships, rather than merely memorizing procedures and following them byrote. The use of concrete materials, calculator graphics, tables,or other representations as a means to help students grasp abstract concepts. Group work, including students sharing and justifying their ideas.

Student writing (including drawings, diagrams, charts) to encourage reflectionon mathematical ideas, and oral presentation to promote communication of those ideas.

The use of context, whether imaginary or real world,as a way to capture student interest in problems and as "a framework or structureupon which to secure concepts and study them" (Robinson & Robinson, 1999, p. iii). Teacher as orchestrator of classroom discourse and facilitator of learning experiences. Reaction to the reform

Initial reaction was markedly positive, seemingly "an overwhelming nationalconsensus on directions for change in mathematics education," but the reform movement isnow "facing passionate resistance from dissenting mathematicians, teachers, andconcerned

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citizens. Wide dissemination of the criticisms...has shaken public confidence in the reform process" (Fey, 1999).

On the positive side, the reform has hadan impact on school mathematics to the extent that most states have rewritten their frameworksto align with the Standards in language, grade level demarcations, and goals, Other evidence ofsupport was the awarding of grants by the National Science Foundation to several projectsto produce curriculum series aligned with the Standards. Schoolsnow have available instructional materials that incorporate the goals envisioned in the reform (see K-12 MathematicsCurriculum Center, http:// www.edc.org/mcc). Despite thissupport for the NCTM vision, a backlash has developed.

One significant barrier to implementation of the Standards is thechange required of teachers in their practice. They are asked to alter their role fromthe accepted position as transmitter of knowledge to a new, therefore uncomfortable, positionas facilitator--one who engages the class in mathematical investigation, orchestrates classroom discourse, and creates a learning environment that is mathematically empowering (NCTM, 1991). McLeod et al. (1996), noting the expenditure ofenergy and time by teachers, questions, "How much of a burden is reasonable? How much ofa burden is reform based on the NCTM Standards, a reform effort that is characterized by complexity anda lack of specificity?" (p. 115).

At the same time, the issue of accountability has come to the fore. High stakes testing is often not aligned with the Standards, which further raises frustrationas parents and teachers grow increasingly concerned about student performanceon these tests. Further insights into public reaction reveal oft-quoted misinterpretations of the Standards, with statements such as, "Children don't need to know the multiplication tables, and geometry no longer requires proofs" (McLeod, in press). In addition, concerned citizen groups object to NCTM's support for calculator use in the primary grades, argue that the Standards are vague on the importance of basic computational skills, and feel that mathematical applications are overemphasized throughout. Thesegroups work toward return to a traditional mathematics program with a well - defineset of grade level objectives. The following quote from one leading group, Mathematically Correct, conveys the flavor of its opposition: Across the country, the way mathematics is taught in the classroom and in textbooks has been changing notably. Classrooms are often organized in small groups where students ask each other questions and the teacher is discouraged from providing information.... The use of blocks and other "manipulative" objects has extended well beyond kindergarten and can now be found in many algebra classes. Meanwhile, the students practice their fundamentals less and less. ..Calculator use is growing and taking away expectations for student learning. Textbooks, if the students have them at all,are full of color pictures and stories, but not full of mathematics. The books often don'teven give explicit definitions or procedures. That would be "telling" and thenew idea is for students to discover all of mathematics for themselves. Many of theseprograms don't even teach the standard algorithms for the operations of arithmetic. Long division is a

50 1-3EST COPT ANAilltABILIF, devil that is to be beaten into extinction--and if theymanage that, multiplication will be next. ("What Has Happened," 2000)

NCTM bore in mind these misinterpretationsas a new writing committee worked to update, refine, and clarify the Standards documents. The recently publisheddocument, Principles and Standards for School Mathematics (NCTM, 2000), simplifiesthe reform message by presenting only five content standards that extend across all grade bands and clarifies them through examples of problems, student work, and classroom dialogue.The document as a whole attempts to respond to theconcerns voiced by the many stakeholders in mathematics education, to clarify NCTM's positions, andto define its vision. Conclusion The standards-based reform is not a reincarnation of thenew math movement, although there are similarities. Both grew out of discontent with student performance and the incompatibility of traditional content offerings with advances in mathematics. Therefore, each added new content to the K-12 curriculum, reflecting the field of mathematics current at the time. New math, however, emphasized deductive reasoning, set theory, rigorous proof, and abstraction, while the Standards emphasize applications in real world context, especially experimentation and data analysis. This difference determines what problems are investigated and how they are solved, and What is countedas evidence. Another difference lies in the pedagogy embedded in each reform movement. Standards- based pedagogy is based on constructivism and, therefore, instructional practices focus strongly on process -- communicating, reasoning, problem solving, making connections, and representations.

Public acceptance of both reforms was general at first, and both encountered strong countermovements toward traditional instruction. New math ended in educational oblivion. What will become of the current movement? While state curriculum frameworks and textbook publications show decided change directly connected to the reform movement, at the classroom level only minimal change has taken place in important areas that affect students--how mathematics is conceptualized and how it is taught. McLeod (in press), summarizing the research on the impact of the NCTM documents on classroom teaching, states that on questionnaires teachers reported making several changes in both curriculum and pedagogy. But studies that used classroom observations and in-depth interviews with teachers concluded that "the changes suggested by the Standards remained a vision of the future, too difficult to implement fully in the context of schooling in the 1990s" (McLeod, in press; see also Ferrini-Mundy & Schram, 1997; McLeod et al., 1996). Mathematics educational reform -- whether state frameworks or instructional materials can occur on paper, as it did in the new math era and is now in the current standards- based movement, without impacting deeply the teaching/learning process at the classroom level. Change is slow, change is complex. As noted by Fey and Graeber (in press), it is difficult to deliver a reform message throughout the nation's diverse K-12

5i ISEST COPY AVMBL \33 system, but it is even more difficult to sustain the momentum and allow time for the message to play out in the classroom. References Brown, K.E. (1961). The drive to improve school mathematics. In The revolution of school mathematics: A challenge for administrators and teachers (pp. 15-29). Washington, DC: National Council of Teachers of Mathematics. Bruner, J.S. (1960). The process of education. Cambridge, MA: Harvard University Press.

Bruner, J.S. (1966). Toward a theory of instruction. Cambridge, MA: Belknap Press. College Entrance Examination Board. (1959). Report of the commission on mathematics: Program for college preparatory mathematics. New York: Author. Ferrini-Mundy, J., & Schram, T. (Eds.). (1997). The recognizing and recording reform in mathematics education project: Insights, issues, and implications. Journal for Research in Mathematics Education. (Monograph Series, No. 8). (ERIC Document Reproduction Service No. ED 414 192) Fey, J.T. (1999). Standards under fire: Issues and options in the mathwars. Summary of keynote speech at Show-Me Project Curriculum Showcase (May 21-23). Retrieved January 10, 2001 from the World Wide Web: http://www.showmecenter.missouri.edu/ showme/perspectives/keynotel.html Fey, J.T., & Graeber, A.O. (in press). From the new math to the Agenda for action. In J. Kilpatrick & G. Stank (Eds.), History of mathematics education. Reston, VA: National Council of Teachers of Mathematics. Kenney, P.A., & Silver, E.A. (1997). Results from the sixth mathematics assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics. (ERIC Document Reproduction Service No. ED 409 172) Kline, M. (1973). Why Johnny can't add: The failure of the new math. New York: St. Martin's Press.

Mathematical Sciences Education Board. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press. McLeod, D.B. (in press). From consensus to controversy: The story of the NCTM standards. In J. Kilpatrick & G. Stanic (Eds.), History of mathematics education. Reston, VA: National Council of Teachers of Mathematics. McLeod, D.B., Stake, R.E., Schappelle, B.P., Mellissinos, M., & Gieri, M.J. (1996). Setting the standards: NCTM's role in the reform of mathematics education. In S.A. Raizen & E.D. Britton (Eds.), Case studies of U.S. innovations in mathematics education (Vol. 3 of the Bold Ventures series) (pp. 15-132). Boston, MA: Kluwer.

52 "33 g C0177 AVATILA 12. National Advisory Committeeon Mathematical Education (NACOME). (1975). Overview and analysis of school mathematicsK-12. Washington, DC: The Conference Board of Mathematical Sciences.

National Commission on Excellence in Education.(1983). A nation at risk: The imperative for educational reform. Washington, DC:U.S. Government Printing Office. National Council of Supervisors of Mathematics. (1977).Position paper on basic skills. Arithmetic Teacher, 25(1), 19-22.

National Council of Teachers of Mathematics. (1980).An agenda for action: Recommendations for school mathematics of the 1980s. Reston,VA: Author. National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professionalstandards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessmentstandards for school mathematics_ Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standardsfor school mathematics. Reston, VA: Author. Nichols, E.D. (1968). The many forms of revolution. In W.C. Seyfert (Ed.), The continuing revolution in mathematics. Reprinted from Bulletin of the National Association of Secondary-School Principals, No. 327. Reprinted by National Councilof Teachers of Mathematics. Washington, DC: The Council.

The NIE conference on basic mathematical skills and learning: Vol. I: Contributed position papers. (1977). Washington, DC: U.S. Department of Health, Education,and Welfare.

Osborne, A.R., & Crosswhite, F.J. (1970). Forces and issues relatedto curriculum and instruction, 7-12. In A history of mathematics education in the United Statesand Canada (32nd yearbook, part 3; pp. 153-296). Reston, VA: National Council of Teachers of Mathematics.

Piaget, J. (1952). The child's conception of number. London: Routledge & KaganPaul. Robinson, E., & Robinson, M. (1999). Guide to standards-based instructionalmaterials in secondary mathematics. Retrieved January 25, 2001, from the World Wide Web: www.ithaca.edukompass/frames.htm

Robitaille, D.F., & Travers, K.J. (1992). International studies of achievementin mathematics. In D.A. Grouws (Ed.), Handbook of researchon mathematics teaching and learning (pp. 687-709). New York: Macmillan.

5 3 BEST COPY AVAILABLE Usiskin, Z. (1985). We need another revolution in secondary school mathematics. In C. Hirsch & M. Zweng (Eds.), The secondary school mathematics curriculum (pp. 1-21). Reston, VA: National Council of Teachers of Mathematics. What has happened to mathematics education? (2000). Mathematically Correct. Retrieved. January 10, 2001, from the World Wide Web: http: // mathematicallycotrect.corn/intro.htm

0\10.10..,

By Terese A. Herrera and Douglas T. Owens Terese A. Herrera is a mathematics resource specialist at the Eisenhower National Clearinghouse for Mathematics and Science, and Douglas T. Owens is professor of education, both at The Ohio State University.

Copyright of Theory Into Practice is the property of Ohiu State University and its content may not be copied or mailed to multiple sites or posted to a listsery without the copyright holder'sexpress written permission. However, users may print, download, or email articles for individual use. Source: Theory Into Practice, Spring200I, Vol. 40 Issue 2, p84, 9p. Item Number: 4472840

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