The History of Mathematical Education

@article{citeulike:405518, author = {Jones, Phillip S. }, citeulike-article-id = {405518}, comment = {=== problem soling, ZPD, constrution === Intellectual Arithmetic Upon the Inductive Method of Instruction, Warren Colburn, 1821 Introduction to Algebra Upon the Inductive Method of Instruction, Warren Colburn, 1830 The objective is to "make the transition from arithmetic too algebra as gradual as possible" by leading the learner through a succession of problems to solve. "The learner is expected to derive most of his knowledge by solving the examples themselves". "in fact, explanations rather embarrass than aid the learner because he is apt to trust to the man, and neglect to employ his own power; and because the explanation is not made in the way that would naturally suggest itself to him if he were left to examine the subject by himself ... this method besides giving the learner confidence, ... is much more interesting because he seems to be constantly making new discoveries" ---=note-separator=--- In North America, the Academies were common man's versions of the Latin Grammar school. The first was established by Benjamin Franklin in 1751, but they flourished between 1787 and 1870. The purpose of the mathematics curriculum was practical computations and "mental discipline" - the value of gritting your teeth on Latin grammar and arithmetic. The method reflected in textbooks, such as the Quackenbos' 1821 Elementary Arithmetic was based on the presentation of a rule, with short statements to help memorize it, and then a long list of exercises "for the slate". ---=note-separator=--- === Economic influences, ala Saxe \& Radford === The first book with mathematical content printed in the Americas was Summario compendioso - de plata y ore, Juan Diez Freyle, Mexico city, 1556. Its main subject was the treatment of value of and conversions between different coinages of of different purities of gold and silver ore. It also contained some algebra, although there is no apparent connection to the main theme. ---=note-separator=--- DeMorgan in 1831 argued strongly that it is absurd to consider any number to be less than 0. Stressed the value of mathematics as a mental discipline. Made a strong plea to abolish rote memorization of rules and proofs. non-rote memoriter teaching may "inspire some to become inquirers who would otherwise have been workers of rules and followers of dogma". He too advises against metaphysical discussions in the classroom, but urges the teacher to be aware of such issues himself. *Understanding* algebra is a most important objective. It is achieved by *doing* and not by reading. ---=note-separator=--- Essais sur l'Enseinement en General et sur celui des Mathematiques en Particulier, S.F Lacroix, Paris 1805 - earliest book which deals explicitly with the teaching of mathematics. In 1810 Geronne started the journal Annales de Mathematique Pure et Appliquee, dedicated to the teaching of pure mathematics. Contained mainly essays on mathematical problems. Cours de Methodologie Mathematique, Felix Dauge, 1896 (original publication 1883) Stressed the importance of the intimate connections among the different branches of mathematics, especially between algebra and geometry. The objectives of mathematics instruction - on one hand application to the needs of society and the advancement of personal career, on the other hand a "moyen d'exercice d'esprit" general rules: 1. Don't present any rigorous reasoning to youth. 2. resist sacrificing rigor in order to include more topics. 3. teach fundamental principles with care but don't insist on too much abstraction nor digress too far into discussions of truth, being and existence, which will only confuse. 4. in the beginning it is not necessary to prove all propositions. making students learn difficult proofs by heart does not actually make the proposition more clear. 5. memory is important. students should learn by heart some basic theorems, not only for the sake of the theorems, but to accustom them to mathematical language and the need to express themselves with precision and elegance. 6. memory should not, however, be extended to the memorization of proofs. teachers should ask questions that ensure the student sees how the parts of the proof fit together. 7. when a student finds himself stopped on a proof, the professor should restrain from immediately pointing out the solution. Let the student find it out for himself; and error corrected may be more profitable than several theorems proved. recommended the study of different systems of numeration so that students gain a better understanding of the decimal system. ---=note-separator=--- === European beginnings === Mathematicae Totius, Pater Peter Galtrucius Aurlianensis, Cambridge 1683 had sections devoted to arithmetic, geometry, astronomy, chronology (the calendar), "gnomonicae" (the sun dial), geography optics and music - all in 305 small pages. Compendium of Elementary Mathematics Composed for the Use of Young Students, Christian Wolf, University of Halle, 1742. The eight topics of Galtrucius plus trigonometry,mechanics, hydrostatics, "aerometria", hydraulics, perspective, "pyrotechnia, military architecture, civil architecture, and algebra - in 900 small pages. Wolf represents a growing interest in pedagogy, which emerged in German universities. For example, supported a lecture method as an alternative the the mere reading and exposition of a standard text to be memorized.}, journal = {The American Mathematical Monthly}, keywords = {algebra, economic, education, history, learning, mathematics, memorizing, multiple, proof, representations, rote, sociocognative, zpd}, number = {1}, pages = {38--55}, posted-at = {2005-11-23 11:48:32}, priority = {0}, title = {The History of Mathematical Education}, url = {http://www.jstor.org/stable/2314867}, volume = {74}, year = {1967} }

TY - JOUR ID - citeulike:405518 L3 - citeulike-article-id:405518 TI - The History of Mathematical Education JF - The American Mathematical Monthly VL - 74 IS - 1 SP - 38 EP - 55 KW - algebra KW - economic KW - education KW - history KW - learning KW - mathematics KW - memorizing KW - multiple KW - proof KW - representations KW - rote KW - sociocognative KW - zpd N1 - === problem soling, ZPD, constrution === Intellectual Arithmetic Upon the Inductive Method of Instruction, Warren Colburn, 1821 Introduction to Algebra Upon the Inductive Method of Instruction, Warren Colburn, 1830 The objective is to "make the transition from arithmetic too algebra as gradual as possible" by leading the learner through a succession of problems to solve. "The learner is expected to derive most of his knowledge by solving the examples themselves". "in fact, explanations rather embarrass than aid the learner because he is apt to trust to the man, and neglect to employ his own power; and because the explanation is not made in the way that would naturally suggest itself to him if he were left to examine the subject by himself ... this method besides giving the learner confidence, ... is much more interesting because he seems to be constantly making new discoveries" N1 - In North America, the Academies were common man's versions of the Latin Grammar school. The first was established by Benjamin Franklin in 1751, but they flourished between 1787 and 1870. The purpose of the mathematics curriculum was practical computations and "mental discipline" - the value of gritting your teeth on Latin grammar and arithmetic. The method reflected in textbooks, such as the Quackenbos' 1821 Elementary Arithmetic was based on the presentation of a rule, with short statements to help memorize it, and then a long list of exercises "for the slate". N1 - === Economic influences, ala Saxe & Radford === The first book with mathematical content printed in the Americas was Summario compendioso - de plata y ore, Juan Diez Freyle, Mexico city, 1556. Its main subject was the treatment of value of and conversions between different coinages of of different purities of gold and silver ore. It also contained some algebra, although there is no apparent connection to the main theme. N1 - DeMorgan in 1831 argued strongly that it is absurd to consider any number to be less than 0. Stressed the value of mathematics as a mental discipline. Made a strong plea to abolish rote memorization of rules and proofs. non-rote memoriter teaching may "inspire some to become inquirers who would otherwise have been workers of rules and followers of dogma". He too advises against metaphysical discussions in the classroom, but urges the teacher to be aware of such issues himself. *Understanding* algebra is a most important objective. It is achieved by *doing* and not by reading. N1 - Essais sur l'Enseinement en General et sur celui des Mathematiques en Particulier, S.F Lacroix, Paris 1805 - earliest book which deals explicitly with the teaching of mathematics. In 1810 Geronne started the journal Annales de Mathematique Pure et Appliquee, dedicated to the teaching of pure mathematics. Contained mainly essays on mathematical problems. Cours de Methodologie Mathematique, Felix Dauge, 1896 (original publication 1883) Stressed the importance of the intimate connections among the different branches of mathematics, especially between algebra and geometry. The objectives of mathematics instruction - on one hand application to the needs of society and the advancement of personal career, on the other hand a "moyen d'exercice d'esprit" general rules: 1. Don't present any rigorous reasoning to youth. 2. resist sacrificing rigor in order to include more topics. 3. teach fundamental principles with care but don't insist on too much abstraction nor digress too far into discussions of truth, being and existence, which will only confuse. 4. in the beginning it is not necessary to prove all propositions. making students learn difficult proofs by heart does not actually make the proposition more clear. 5. memory is important. students should learn by heart some basic theorems, not only for the sake of the theorems, but to accustom them to mathematical language and the need to express themselves with precision and elegance. 6. memory should not, however, be extended to the memorization of proofs. teachers should ask questions that ensure the student sees how the parts of the proof fit together. 7. when a student finds himself stopped on a proof, the professor should restrain from immediately pointing out the solution. Let the student find it out for himself; and error corrected may be more profitable than several theorems proved. recommended the study of different systems of numeration so that students gain a better understanding of the decimal system. N1 - === European beginnings === Mathematicae Totius, Pater Peter Galtrucius Aurlianensis, Cambridge 1683 had sections devoted to arithmetic, geometry, astronomy, chronology (the calendar), "gnomonicae" (the sun dial), geography optics and music - all in 305 small pages. Compendium of Elementary Mathematics Composed for the Use of Young Students, Christian Wolf, University of Halle, 1742. The eight topics of Galtrucius plus trigonometry,mechanics, hydrostatics, "aerometria", hydraulics, perspective, "pyrotechnia, military architecture, civil architecture, and algebra - in 900 small pages. Wolf represents a growing interest in pedagogy, which emerged in German universities. For example, supported a lecture method as an alternative the the mere reading and exposition of a standard text to be memorized. AU - Jones, Phillip PY - 1967/// UR - http://www.jstor.org/stable/2314867 ER -