II. THE TEACHING OF MATHEMATICS IN ELEMENTARY SCHOOLS
By David Eugene Smith, LL.D.
I. The History of the Subject
1. Advantages to a Teacher in Studying the History of the Subject Taught.
2. The Early History of Arithmetic.
3. The Growth of Number Systems.
4. The Development of Arithmetic as Known at Present.
References: Smith, The Teaching of Arithmetic, New York, 1909, chap. i; and in general the sections in this syllabus correspond to the chapters in this work. The Teaching of Elementary Mathematics, New York, 1900. Ball, A Primer of the History of Mathematics, London, 1895, and A Short Account of the History of Mathematics, London, 4th edition, 1908. Fink, History of Mathematics, translated by Bennan and Smith, Chicago, 1900. Cajori, History of Elementary Mathematics, New York, 1896, and History of Mathematics, New York, 1893. On Greek Arithmetic see Gow, History of Greek Mathematics, Cambridge, 1884.
II. The Reasons for Teaching Arithmetic
1. The Ancient Point of View.
2. Content of the Primitive Logistic, or Art of Calculation.
a. Early counting.
b. Early writing of numbers. The development of notations.
c. The Influence of the Hindu-Arabic notation.
3. Content of the Early Arithmetic, or Theory of Numbers.
a. Connection with mysticism.
b. Contributions of Pythagoras and his school.
c. The effect upon modern arithmetic.
4. The Reasons for Teaching in the Middle Ages.
a. The Church schools.
b. The reckoning schools.
c. The effect upon modern arithmetic.
5. The Reasons Developed by the Renaissance.
a. Influence of commerce.
b. Influence of printing. The crystallization of arithmetic.
c. The effect upon the subject matter of modern arithmetic.
6. The Reasons of To-day.
a. The practical value. Whatever pretends to be practical in arithmetic should really be so.
b. The question of “mental discipline.” The rise of this doctrine. The results of a psychological study of the question. The tangible part of “mental discipline.”
c. The interest in the subject for its own sake. The game element of mathematics. The historical development of the science of arithmetic from the primitive game.
References: Smith, The Teaching of Arithmetic, chap. ii, to the chapters of which no further reference will be made, this syllabus being merely a synopsis of that work. Teaching of Elementary Mathematics, pp. 1-70. Young, The Teaching of Mathematics, New York, 1907, pp. 41-52, 202-256. On the historical side, consult Fink, History of Mathematics, Chicago, 1898. Ball, Short History of Mathematics, New York, 1908. Cajori, History of Elementary Mathematics, New York. Jackson, The Educational Significance of Sixteenth Century Arithmetic, New York, 1906. Branford, A Study of Mathematical Education, Oxford, 1908.
III. What Arithmetic should include
1. From the Practical Standpoint.
a. The utilities of arithmetic overrated. A detailed consideration of the various topics usually studied.
b. The effect of tradition upon the matter of arithmetic.
2. From the Standpoint of Mental Discipline. Discipline a Matter of Method rather than one of Topics.
3. From the Standpoint of Interest in the Subject for its Own Sake.
References: Smith, Teaching of Elementary Mathematics, p. 19. Young, pp. 23-242.
IV. The Nature of the Problem
1. The Great Change in Recent Years brought about by Two Causes.
a. The study of social needs.
b. The study of child psychology.
2. The Peculiar Needs of America. The Bearing of these Needs upon the Teaching of Arithmetic.
3. Child Psychology and the Problems still Awaiting Solution.
References: Smith, Teaching of Elementary Mathematics, p. 21. Young, pp. 97-103, 210-218. Saxelby, Practical Mathematics, and similar works.
V. The Arrangement of Material
1. Recent Changes brought about from a Consideration of Child Psychology.
2. The Growth of the Textbook.
a. The Treviso arithmetic of 1478, and the early arithmetics of Italy, Germany, France, England, and Holland.
b. The two-book series.
c. The three-book series.
d. The extreme spiral arrangement.
3. The Modern Curriculum in Arithmetic.
a. Its origin.
b. Its present status.
c. Improvements to be considered.
References: Young, pp. 178-188.
VI. Method
2. How the Ancients probably taught Calculation.
a. Various forms of the abacus.
b. The abacus at the time of the Renaissance.
c. The effect upon arithmetic of abandoning the abacus in western Europe.
3. Causes of the Rise of the Rule.
4. Revival of Objective Teaching.
Trapp (1780), von Busse (1786), and Pestalozzi (about 1800).
5. The Early Followers of Pestalozzi.
Tillich (1806), Krancke (1819), Grube (1842).
6. Types of Later Methods.
a. Counting.
b. Ratio.
c. Extreme spiral.
d. Pure concrete work as a basis.
e. Pure abstract work as a basis.
7. The Ease and Futility of Creating Narrow Methods.
References: Smith, Teaching of Elementary Mathematics, pp. 71-97. Seeley, Grube’s Method of Teaching Arithmetic, New York, 1888. Soldan, Grube’s Method of Teaching Arithmetic, Chicago, 1878. C. A. McMurry, Special Method in Arithmetic, New York, 1905. McLellan and Dewey, The Psychology of Number, New York, 1895. Young, pp. 53-150.
VII. Mental or Oral Arithmetic
1. Historical Status of Oral Arithmetic.
2. Revival under Pestalozzi’s Influence. The Work of Warren Colburn in this Country.
3. Causes of the Decline of this Form of Work.
4. The Claims of Oral Arithmetic upon the School To-day. The Practical and Psychological Views of the Problem.
5. The Nature of the Oral Work,—Abstract and Concrete.
6. The Time to be allowed to the Subject.
References: Smith, Teaching of Elementary Mathematics, p. 117. Handbook to Arithmetics, p. 6. Young, p. 230.
VIII. Written Arithmetic
1. What should be the Nature of the Written Arithmetic?
2. Object of the Business Form of Solution.
3. Object of Written Analysis.
4. Necessity of Recognizing Two Kinds of Written Work.
5. How to Mark Papers.
References: Smith, Teaching of Elementary Mathematics, pp. 121-129.
IX. Children’s Analyses
1. The Object in Requiring Analyses.
2. What should be expected of Children in this Respect?
3. Explanations of Fundamental Operations. Relation to the Formal Rule.
4. Explanation of Applied Problems.
5. Relation to the Work in English.
6. The Limit of Primary Work, “Two-step Reasoning.”
References: Smith, Handbook to Arithmetics, p. 9. Young, p. 205.
X. Interest and Effort
1. Status of Arithmetic from the Standpoint of Interest.
2. Danger of Overemphasis upon Interest.
3. Lessening of Interest with the Lessening of Effort.
4. Safe Basis for Increase of Interest.
5. Effect of a Genuine, Spontaneous Interest upon Increase of Effort and of Power.
XI. Improvements in the Technique of Arithmetic
1. History of the Improvement in Symbolism.
2. How the Present seeks to carry on this Improvement.
a. The difficulties that are met.
b. Dangers of too much symbolism.
c. The proper criterion for selection.
3. The Equation in Arithmetic.
a. Object.
b. Dangers to be avoided.
4. The Process of Subtraction as a Type.
a. The various historical methods considered.
b. The criterion for a selection.
c. The claims of the various processes to-day.
5. The Process of Division as a Type.
a. The history of division.
b. Present points at issue.
c. The probable future.
6. Proportion as a Type.
a. History of proportion and the “Rule of Three.”
b. Present symbolism and status.
c. Probable future of the subject.
7. Future Problems Relating to Technique.
XII. Certain Great Principles of Teaching Arithmetic
A summary of the larger principles for the guidance of teachers.
XIII. General Subjects for Experiment
1. The Use of Games.
2. Chief Interests of Children.
3. Results of Emphasizing:
a. The abstract problem.
b. The concrete problem.
4. Amount of Time to be assigned to Arithmetic.
5. Relative Amount of Time to be devoted to:
a. Oral arithmetic.
b. Written arithmetic.
6. The Best Basis of Arrangement of an Arithmetic.
XIV. Details for Experiment
Professor Suzzallo’s list of details as set forth in The Teachers College Record, January, 1909, p. 43, and in Smith, The Teaching of Arithmetic, chap. xiv.
XV. The Work of the First School Year
1. Arguments for and against no Formal Arithmetic in this Year.
2. The Leading Mathematical Features for the Year.
3. The Number Space of the Year.
a. For counting.
b. For operations.
4. The Work to be accomplished in Addition.
5. The Work in the Other Operations.
6. The Fraction Concepts to be considered.
a. Part of an object.
b. Part of a group.
c. The idea of “half as much.”
7. Denominate Numbers.
8. The Question of the Use of Objects.
9. Symbolic Work and Technical Expressions.
10. Nature of the Problems of this Year.
11. The Time Limit upon Work.
References: Smith, The Teaching of Elementary Mathematics, p. 99. Handbook to Arithmetics, p. 11. C. A. McMurry, Special Method in Arithmetic.
XVI. The Work of the Second School Year
1. The Leading Mathematical Features.
2. Number Space for the Year.
3. Counting.
a. The origin of the “counting method.”
b. The extremes to which it may be carried.
c. The proper use of counting in teaching.
4. The Addition Table. Relation to Counting.
5. The Method of Treating Subtraction reviewed.
6. The Multiplication Table.
a. Arguments for and against learning tables.
b. Extent of the work for this year.
c. Relation to counting.
7. Division.
a. Relation to multiplication.
b. Arrangement of work in short division.
8. Fractions.
a. Extent of the work.
b. Nature of the objective work.
9. Denominate Numbers.
a. Extent of the work.
b. Use of the measures. Visualizing the great basal units.
10. Nature of the Symbols to be considered.
11. Nature of the Problem Work.
a. Abstract.
b. Concrete.
XVII. The Work of the Third School Year
1. Peculiar Necessity for Preparation for this Year’s Work.
2. Leading Mathematical Features.
a. Beginning of rapid written work.
b. Multiplication table completed.
c. Most important tables of denominate numbers.
d. Work extended to two-figure multipliers and the beginning of long division.
3. Number Space may extend to 100,000.
4. The Roman Numerals.
a. Extent to which this work should be carried in various school years.
b. Historical sketch of the system and of its uses.
5. The Counting Method further considered. Its Values and its Dangers.
6. The Writing of United States Money. Operations.
7. Square and Cubic Measure.
a. Extent.
b. Nature of objective work.
8. Suggestions as to Four Operations.
a. Addition. Practical value of checks on all operations.
b. Subtraction, as discussed in section XI.
c. Multiplication. Should the tables extend to 12 × 12? Devices.
d. Division. Algorism considered historically and practically.
e. Historical note as to the number of operations.
9. Extent of Work with Fractions.
10. Nature of the Problems.
References: Smith, Teaching of Arithmetic, chap. xvii, p. 73. Handbook to Arithmetics, p. 29.
XVIII. The Work of the Fourth School Year
1. Leading Mathematical Features.
2. Number Space.
3. The Four Operations.
a. Nature of the oral work.
b. Criteria for judging written work.
c. Speed versus accuracy.
4. Nature of the Work in Common Fractions.
a. Historical sketch of various fractions.
b. Change in the practical uses of common fractions.
5. Denominate Numbers.
a. What tables are of value? Historical sketch of tables.
b. Visualizing the basal units.
c. Accuracy in reduction.
6. Nature of the Problems.
References: Smith, Handbook to Arithmetics, p. 43.
XIX. The Work of the Fifth School Year
1. Leading Mathematical Features.
2. Necessity for and Nature of Preliminary Review.
3. Number Space. Modern Tendencies in Using Large Numbers.
4. Nature of the Review of the Four Operations.
a. Suggestions for rapid addition and subtraction.
b. Checks on multiplication and division.
c. Twofold nature of division.
5. Common Fractions.
a. Nature of the theoretical explanations.
b. What should be expected of children in this regard.
6. Denominate Numbers.
a. Extent of reductions.
b. Nature of the operations.
7. How to solve Problems.
8. Introduction to Percentage.
9. Nature of the Problems.
References: Smith, Handbook to Arithmetics, p. 53.
XX. The Work of the Sixth School Year
1. Leading Mathematical Features.
2. The General Solution of Problems.
a. How the world has solved problems.
b. Modern improvements.
3. Percentage.
a. Nature of the subject.
b. History of the subject.
c. Suggestions for treatment.
d. The most important applications.
4. Ratio and Proportion.
a. History.
b. Present value, and probable future status.
5. Nature of the Problems.
XXI. The Work of the Seventh School Year
1. Leading Mathematical Features.
2. Review of our Numbers. Historical Notes.
3. Review of the Fundamental Operations.
4. Types of Subjects Treated.
a. Longitude and time. Origin, value, new features.
b. Percentage. What cases are the most important?
5. Introduction of Algebraic Work considered. Nature of Mensuration.
6. Nature of the Problems.
XXII. The Work of the Eighth School Year
1. Leading Mathematical Features.
2. Nature of the Business Applications.
a. Banking. Extent to which the work should be carried.
b. Partial payments. Historical view of the value of the subject.
c. Partnership. Value of the historical view.
d. Simple accounts.
e. Exchange. Wherein its value lies.
f. Taxes. Insurance.
g. Corporations. Arguments for and against the study of investments.
3. The Metric System.
a. Why taught. Historical view.
b. Extent of the work.
c. Practical suggestions in teaching.
4. Powers and Roots.
a. Historical view.
b. Present values. Extent of the work.
5. Mensuration.
a. Extent to which it should be carried.
b. Geometry in the eighth year.
c. The formula.
6. Algebra in the Eighth Year.
a. Historical view. Present values.
b. Extent to which it should be carried.
7. Nature of the Problem.
8. A Comparison of American and Foreign Schools.