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A New Trend in Senior High School Mathematics


by Gordon R. Mirick - 1943

This article is a plea for looking at mathematics as a whole and teaching the subject not as a series of half-year units of geometry, algebra, etc., but as a collection of mathematical materials in which no artificial line is drawn between the various fields. It is also a plea to apply mathematics to as many situations outside the strictly mathematical field as possible.

A PUPIL going through the American high school today and electing four years of mathematics would have his work arranged about as follows: ninth grade, algebra; tenth grade, plane geometry; eleventh grade, intermediate algebra with varying amounts of trigonometry; twelfth grade, trigonometry and solid geometry with possibly some algebra.


It is true that there are many modifications of this program in the various schools, but there is one outstanding characteristic of all programs, and that is the artificial line that is drawn between the subjects of algebra and geometry. A pupil sees algebra and geometry as two separate and distinct subjects without consideration for their interrelationships.


An interesting comment on the kind of geometry studied by pupils in our high schools is found in The Queen of the Sciences by E. T. Bell:


To get some sort of perspective, let us consider roughly the kind of mathematics acquired by a student who takes all that is offered in a good American high school. The geometry taught is practically that of Euclid and is 2200 years old. It is a satisfactory first approximation to the geometry of the physical universe, and it is good enough for engineers, but it is not that which is of vital interest in modern physics, and its interest for working mathematicians evaporated long ago. Our vision of the universe has swept far beyond the geometry of Euclid.1


The need of a newer approach and of a change in the content of the geometry courses in our high schools is expressed in an address by Professor Oswald Veblen of the Institute for Advanced Study:


It seems to me that elementary geometry should be presented in such a way as to prepare the student for the other sciences which he is to study later, and in which this very geometry is going to be used. This means that the methods of geometry should not be singular ones, peculiar to this subject itself, but should as far as possible be methods which can be used over and over again in the other branches of science. . . .


The obvious thing to do is to recognize that we have a subject to present which has to do with physical reality. Let us approach it as any modern scientist would by first studying some of the observations and experiments with which we have to deal, in crude everyday terms. This would correspond, I suppose, to the "observational geometry" which is already in our schools. After getting a start in this way, a scientist feels the need of a systematic language in terms of which to organize the phenomena. This language is mathematical analysis. In other words, as soon as the experimental basis is established, the study of geometry should proceed by the use of coordinates and the methods of analytic geometry.2


Let us consider what one should expect of a pupil at the beginning of the eleventh grade, or third year in high school, as a result of the study of algebra in the ninth grade and a study of geometry in the tenth grade. He comes to us after not having had instruction in algebra for fourteen or fifteen months. However, during that interval he has had mathematics in the form of a course in plane geometry. But that course has afforded him little opportunity to use the concepts and skills he developed in algebra and practically no opportunity to develop new concepts and skills.


The net result of all this is that at the beginning of the eleventh grade a typical high school pupil knows very little algebra. The fact is attested to by many teachers of mathematics and by tests that have been given from time to time.


In the Tentative Outline of High-School Mathematics, recently published by the Board of Education at Los Angeles, California, the fact is lamented that the continuity of algebra is broken by a year of geometry. The following is a quotation from the course of study:


Algebra I, II, and III, which at present requires fifteen semester hours to complete, will be completed in twelve, and Geometry, which at present requires ten semester hours to complete, will be completed in eight. It is hoped that this gain in time will be realized through the continuity of Algebra. At present Algebra is interrupted with a year of Geometry. Also, some gain might be realized in eliminating duplications or treating them with greater speed. Such topics as ratio and proportion and trigonometric ratios which are studied in both elementary algebra and geometry might fall in this class.


We find in the Preliminary Description of the Alpha, Beta and Gamma Examinations in Mathematics, published by the College Entrance Examination Board, a number of statements such as the following that express the need for the pupil's understanding of the importance of the interrelationship of various subjects in mathematics:


Each examination will be comprehensive in character, testing the candidate's appreciation of the significance of the separate subjects of secondary-school mathematics as well as his understanding of the important interrelations of these several subjects. . . .


Sharp separation between Algebra, Geometry, and numerical Trigonometry will not be maintained. Understanding of the interrelations between these subjects should be part of the candidate's equipment.


Referring to the more advanced examination, Gamma, is the following statement:


The examination will not observe strict separation of the subjects Trigonometry, Solid Geometry, Advanced Algebra. Candidates will be expected to be able to use any one or more of these subjects in the solution of a problem with which they are confronted.


These considerations constitute a serious challenge to the present organization in the teaching of geometry and algebra in American high schools. The methods of investigation in demonstrative geometry are clearly too unrelated to the methods now employed in the study of the physical universe. As I see it, the need in tenth-grade mathematics is a course that retains a logical sequence, employs algebraic analysis as well as geometric synthesis, organizes algebra and geometry so that they mutually contribute to each other.


Among the many significant attempts that have been made toward a change in the approach to the whole question of geometry might be mentioned the work of an English mathematician, Sir Percy Nunn. Accounts of his work may be found in the First and Second Reports on the Teaching of Geometry in the Schools, prepared for the British Mathematical Association. Nunn's work replaces the Euclid Parallel Postulate by the Principle of Similarity. A work that also postulates a Principle of Similarity in somewhat the way Nunn has done is the textbook of G. D. Birkhoff and Ralph Beatley, called Basic Geometry.


In the preceding remarks I have mentioned the work of Oswald Veblen. His recommendations, as far as the initial start of plane geometry is concerned, are implied in the following quotation from his work:


If you ask a modern mathematician or physicist what is a Euclidean space, the chances are that he will answer: It is a set of objects called points which are capable of being named by ordered sets of three numbers (x, y, z) in such a way that any two points (x1, y1, z1) and (x2, y2, z2) determine a number called their "distance" given by the formula: [39_9219.htm_g/00002.jpg]

I find it advantageous to start the work in geometry, making use as much as possible of the algebra and trigonometry developed in the preceding year.


In Unit I, I develop the rational numbers and the geometry of one dimension. In this unit the principal postulates are:


1. Two distinct points determine one and only one straight line.


2. If A and B are two distinct points on a line, then on any ray whose origin is point O, there exists one and only one point D such that OD=AB.


3. Line-segment AB=line-segment BA.


4. There is a one-to-one correspondence between equally spaced points on a line as one set and the set of positive and negative numbers and zero.


5. There is at least one point between any two points on a line.


6. A given line-segment can be divided into an integral number of equal parts.


The work of this unit shows that for every rational number there corresponds a point on the line, but leaves open for future consideration the question of whether every point on the line corresponds to a rational number.


In Unit II, which concerns itself with definitions and concepts of angles and triangles, the following postulates are included:


7. If BAC is an angle whose sides do not lie on the same straight line, then on any ray, OX, whose origin is O, there exists one, and only one ray OZ on a given side of OX such that Ð BAC equals Ð XOZ.


8. All straight angles are equal.


9. All right angles are equal.


10. Through a point on a line, one and only one perpendicular can exist to the line.


11. All equally spaced rays that have a common origin can be numbered so that their number differences measure angles.


12. The sum of any two angles of a triangle is less than a straight angle.


13. Through a point not on a line, at least one perpendicular can exist to that line.


Unit III deals with similar right triangles, the Theorem of Pythagoras, the angle sum, and the trigonometry of the right triangle. By this early introduction to such theorems as the Angle Sum and the Theorem of Pythagoras, many important exercises can be undertaken, and the Theorem of Pythagoras allows us to justify the inclusion of the irrational number and thus affords an opportunity to complete the study of real numbers begun in Unit I. Postulates in this unit are:


14. Two right triangles are similar if an acute angle of one is equal to an acute angle of the other.


15. There is a one-to-one correspondence between points on a number scale and the real numbers.


The theorems of this unit with their corollaries are given to indicate the different approach existing in this sequence to that usually found. The following theorems are given to draw the comparison:


1. Through a point not on a line, one and only one perpendicular can exist to the line.


2. The altitude from the vertex of a right angle in a right triangle to the hypotenuse lies within the triangle.


Corollary 1. At least one altitude of the triangle lies within the triangle.


3. In a given right triangle, the altitude upon the hypotenuse forms two right triangles which are each similar to the given triangle, and therefore similar to each other.


4. The sum of the two acute angles of a right triangle equals a right angle.


5. The sum of the interior angles of any triangle equals a straight angle.


Corollary 1. If two angles of a triangle are equal to two angles of another triangle, the third angles are equal.


Corollary 2. An exterior angle of a triangle is equal to the sum of the two interior angles not adjacent and is greater than either of them.


Corollary 3. If one angle of a triangle is a right angle or an obtuse angle, the other two angles are acute.


6. In a right triangle the altitude upon the hypotenuse divides the hypotenuse so that either leg is the mean proportional between the hypotenuse and the adjacent segment of the hypotenuse.


7. In a right triangle the altitude upon the hypotenuse divides the hypotenuse so that the altitude is the mean proportional between the segments of the hypotenuse.


8. In a right triangle the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.


9. If in two right triangles the sides including the right angles are proportional, the triangles are similar. The proof of this theorem is given above.


10. Two right triangles are similar if the hypotenuse and a side of one right triangle are proportional to the hypotenuse and a corresponding side of the other.


[39_9219.htm_g/00003.jpg]


These theorems open up a great deal of numerical work and further allow us to put the trigonometry of the right triangle on a logical basis.


Unit IV concerns itself with parallels and perpendiculars. Since we assume the Law of Similarity, Euclid's Parallel Postulate becomes a theorem.


Unit V discusses the analytical geometry of the straight line. This unit affords considerable opportunity not only to develop more algebra, but to show the relationship between algebra and geometry.


Unit VI, which is mostly algebraic in character, concerns itself with the notions of variables, constants and functions, and considerable work is done in the graphs of functions, including the graph of the Sine Function, Cosine Function and Tangent Function as far as 180°. The Law of Sines and the Law of Cosines are proved in this unit and used constantly in the development of the work of future units. As an example of the use of the Law of Cosines, I am giving a complete proof of one of the Laws of Similarity on the opposite page.


[39_9219.htm_g/00004.jpg]


Without going into further detail it should be evident from what has been said that constant endeavor is made throughout this year's work to develop geometric concepts and also to make consistent use of the algebra and trigonometry taught in the ninth grade, with the expectation that by so doing the students will have as much geometric information as heretofore, but considerably more algebraic power as well as a better understanding of the interrelationship of the various branches of mathematics.


In the eleventh grade, the course considers the problem of systematizing the algebraic facts previously learned and the extension of algebra through logarithms, progressions, and the binomial theorem. It has a unique feature in that the applications to a number of topics are given more consideration than the development of further theory that has no immediate application- As an illustration, several cases of oblique triangles in trigonometry are considered; however, the work with such formulas as sin 2 A = 2 sin A cos A and cos (x + y) = cos x cos y -- sin x sin y, and the like are omitted entirely. The time thus saved gives more opportunity for applications to such topics as statistics.


Another application that is considered in this course is mechanics. The work in mechanics for the most part concerns itself with concurrent and parallel forces in statics, friction and the inclined plane, and simple machines. All these topics lend themselves readily to simple physical experiments. Thus, by bringing in mechanics, opportunity is afforded to show the application of mathematics to the physical world. Finally, the elementary parts of analytic geometry which are developed might be considered as an application of the geometry studied in the preceding year and the algebra and the trigonometry studied in the current year. Throughout the work, the formula, the equation, and the graph afford excellent opportunity to show functional dependence.


The work of the twelfth year deals with imaginary and complex numbers (fundamental operations, graphical addition and subtraction, polar form), de Moivre's Theorem and its application to the solution of equations of the type xn = c. Linear functions and the elementary analytic geometry of the circle are reconsidered. The work in analytic geometry is extended to include other quadratic functions and loci. The work also includes the solution of higher order equations either by Homer's or by Newton's method, the latter method being dependent on an earlier introduction of the elements of the calculus. Topics from plane and spherical trigonometry are a part of this year's work. Exponential and logarithmic functions, permutations, combinations, and probability are included. A rather unique feature of this work is the use of two series


[39_9219.htm_g/00005.jpg]

and


[39_9219.htm_g/00006.jpg]


which allows us to make clear the following formula:


[39_9219.htm_g/00007.jpg]

This formula is used for finding areas and volumes, and represents a method which has much to commend it on historical grounds and also in the use it makes of algebra.


If any work preceding this has been done in the calculus, many would prefer to use the notation:


[39_9219.htm_g/00008.jpg]

One interesting field that we have to explore is the work in map projection. It is evident that more consideration will have to be given to the part that the knowledge of the sphere plays in the world today. It is assumed that in any study of the sphere some spherical trigonometry is considered.


This article is a plea for looking at mathematics as a whole and teaching the subject not as a series of half-year units of geometry, algebra, etc., but as a collection of mathematical materials in which no artificial line is drawn between the various fields. It is also a plea to apply mathematics to as many situations outside the strictly mathematical field as possible.


The dogmas of the quiet past are inadequate to the stormy present. The occasion is piled high with difficulty, and we must rise with the occasion. As our case is new so we must think anew and act anew.

—ABRAHAM LINCOLN







1 Bell, E. T. The Queen of the Sciences, pp. 7-8. The Williams & Wilkins Company, Baltimore, Md., 1931.

2 "Certain Aspects of Modern Geometry." Rice Institute Pamphlet, Vol. 21, No. 4, October, 1934.



Cite This Article as: Teachers College Record Volume 44 Number 5, 1943, p. 347-354
https://www.tcrecord.org ID Number: 9219, Date Accessed: 1/20/2022 11:43:08 AM

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