The Secondary School Mathematics Curriculum Improvement Study Goals—The Subject Matter—Accomplishments

The Secondary School Mathematics CurriculumImprovement Study Goals�The Subject

Matter�Accomplishments*

Howard F. FehrDirector, Teachers Co77eê, Columbia University,

New York, New York 10027

The Secondary School Mathematics Curriculum ImprovementStudy is in the fourth year of a proposed six-year program of develop-ment and experimentation designed to produce a new unified secon-dary school mathematics curriculum for college-intending and ca-pable students.

BACKGROUNDDuring the past fifteen or more years, the United States has been

engaged in revising the elementary and secondary school mathe-matics curriculum�primarily by updating the existing traditionalcurriculum. Modest recommendations by the C.E.E.B. Commissionon Mathematics have been largely accepted by syllabus bodies andby writers of commercially produced textbooks. Implementation ofthis program by the SMSG has had wide acceptance.Throughout all these reform movements, the traditional divisions

of school mathematics instruction into separate years of arithmetic,algebra, and geometry has been maintained. Beyond token introduc-tion of a few new concepts and some rearrangement of the sequencesof courses, little has been gained in bringing more advanced studyinto the high school through more efficient methods of organizing thesubject matter. Recently bolder and more radical recommendationsfor the improvement of secondary school education on mathematicshave been made in our country and in Europe, notably in Belgium,the Nordic Countries, and U.S.S.R.What has been called for is a reconstruction of the entire curriculum

that presents the subject as an integrated body of knowledge, that isone which eliminates the barriers separating the traditional branchesof mathematical study, and unifies the subject through study of itsfundamental concepts (sets, relations, operations, mappings) andstructures (groups, rings, fields, and vector spaces). Such a curric-ulum would reflect the spirit of contemporary mathematics as wellas permit introduction into the school program of much that waspreviously considered undergraduate mathematics.

Since 1966, an experimental study whose objective was the con-struction of a unified school mathematics curriculum grades seven

* Paper presented at the CASMT Convention, Milwaukee, Wisconsin, November 28, 1969.

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282 School Science and Mathematics

through twelve has been underway, financed at first by the FederalOffice of Education with support now given by the National ScienceFoundation. The project is located at Teachers College, ColumbiaUniversity.Long range planning of the six-year study was initiated at a meet-

ing of the chief consultants in November, 1965. The conferees out-lined a procedure for developing a total program, to be followed bysyllabus conferences, writing of experimental textbooks, educationof the classroom teachers, pilot class teaching, and evaluation ofoutcomes. In June, 1966, a group of eighteen leading United Statesand European mathematicians and educators met for twenty days(1) to outline the scope and sequence of a six-year unified schoolmathematics program and (2) to make a detailed specific recommen-dation for the mathematical content of the first course (seventhgrade). These recommendations were given to a team of eight mathe-matical educators, all with secondary school teaching experience whowrote the first experimental course during July and August. Duringthe summer special courses in algebra and geometry of a contem-porary nature were given to twenty teachers who taught the first tenexperimental classes during the subsequent school year, two teachersto each class.

In subsequent years, June syllabus conferences from 10 to 14 daysin length have considered revisions to be made of previously writtenmaterial and have made specific recommendations for the contentand teaching of new courses�one each year. Eight week writingsessions and six-week teacher education courses follow these planningconferences.

STRUCTURE OF THE EMERGING CURRICULUMThat formal mathematics can be organized in terms of funda-

mental concepts of sets, relations, functions, and operations, andstructures such as groups, rings, fields, and vector spaces was well-known at the outset of the study�having been established by workin foundations at the turn of the century and by the work of theBourbaki begun in the 1930^. What was not known, was how thisorganization could be presented in teachable form to secondaryschool students. Guidelines for such a development were availablein the form of recent experimentation and reports of syllabus con-ferences in Europe, but nowhere had a total 7-12 Unified Mathe-matics Program been designed, produced, and tested.The program that has emerged from the successive syllabus con-

ferences and writing sessions has a kind of double helical organizationin which the abstract concepts and structures�as a center core�develop in coordination with the most important realizations of these

Secondary School Mathematics 283

structures; namely, the number systems, synthetic, coordinate, vec-tor, and transformation geometry, probability and statistics, andanalysis, including numerical analysis and digital computers. Toshow the development in more detail, we shall consider separatelythe algebraic and geometric content in the first three Courses forjunior high school. How all the branches merge into a unified studywill then become evident as we examine the present experimentalCourse IV. We can conclude the survey with a glimpse at the pos-sible syllabus for Courses V and VI (grades eleven and twelve).

ALGEBRAIC INSTRUCTION, COURSES I, II, IIIWe look upon all mathematics of a contemporary nature as con-

sisting of (Set, Structure) and the activities derived therefrom. Thebasic concepts and structures are introduced in an informal intuitivemanner to seventh graders and then developed with increasing depthand formality as need arises in subsequent years of study.

Chapter 1 of Course I begins with an investigation of finite numbersystems and contrasts their properties with those of the familiarwhole numbers. Open sentences are solved, domain and variable areintroduced, identity elements are discovered, inverses are investi-gated, commutativity and associativity are tested, and a systemwith two operations is studied with the added property of distribu-tivity. The zero-multiplication property is singled out. The symbolsZn, Z^ Ziy (JT, *) and (1C, 0, *) along with � a and I/a for inverses arethe only new symbols introduced.

All of these experiences in simple concrete situations pave the wayfor the study of Chapter 2 which examines the general concept of abinary operation and its properties. Other operations such as unary,ternary etc. are treated in later years. The operation is defined gener-ally on a set as the assignment by some rule, of an element of the setto each ordered pair of the product set EXE. On clock numbers andwhole numbers one first considers the usual operations (addition,multiplication) then subtraction, division, exponentation (or power).Then the operations of max, min, first element, second elementL.C.M., G.C.F., Pythagorean and others are tested for commuta-tivity, associativity, a neutral element of the set, an inverse for anelement of the set. Operations are defined by tables, expressionsinvolving the operation are simplified; two operations on a set, anddistributivity are investigated. Open sentences for various operationsare solved using cancellation laws. Finally the properties of a groupare singled out and stressed as important for the rest of the study ofmathematics.The third chapter is an informal introduction to function, via

mappings. After many relations are represented by arrow diagrams,


those relations for which only one arrow goes out from each elementof the domain are again singled out as important and called map-pings. Then the clock numbers are mapped into clock numbers byrules such as x�>x-}-2, x�>2x, x�>2�x; etc. Next mappings are madeof whole numbers into the whole numbers. We also discover thatsometime mappings cannot be made, for example a;�>(5�o;). Arrowdiagrams are used to illustrate mappings on the number line andfinally composition of mappings, / o g, read/following g, are made. Inthese mappings much manipulation of variables occurs, e.g. if

/ gx �> x + 3 and x �> 2x � 4

then/ o g�>(2x�^)+3 =2x� 1 but g o/�>2(.r+3) -4= 2x+2 etc. Fi-nally the identity mapping and the existence of the inverse of a map-ping are studied. Thus in three chapters (2 months) of study, theyoung students are writing, thinking, and using expressions, vari-ables, operations, mappings or functions and solving open sentences,besides learning to use and enjoy new symbolism.

In Chapter 4 we return to a study of the integers. The motivationis to invent new numbers to represent "going in the hole in a game,"or losing money, etc., and how to combine (add) such numbers.Further motivation is provided in trying to get numbers so as alwaysto be able to solve x-\-a=b, for every a and 5. The negative integersare thus created and then annexed to the whole numbers to form anew set called the integers. The opposite is defined and the propertiesof (Z, +) are studied. It is recognized at once as a group. On thenumber line we make mappings of x�>x’\-a which are called transla-tions. Then composition of translations becomes another group. Thisgroup is compared with the group of integers under addition. Finallysubtraction becomes an operation (by adding the opposite, i.e. a�b===>a+(�6); equations are solved in (Z, +)? the integers are ordered;absolute value is introduced as Max (a, �a) and then equations andinequations involving absolute values are solved.

In Chapter 5 we break away to an entire new subject that uses allthat has been taught and reviews the use of fractions (learned inelementary school), a chapter on probability which is described later.The next Chapter treats the set of integers studied under multi-

plication, and then with both addition and multiplication to give thering structure. First multiplication of whole numbers is restudied,then multiplication of integers�first informally, then more formallyby assuming the laws of a ring must hold. This enables us to take upthe transformation on a line called a dilation x�>kx, and finally theaffine transformation x�>ax-\-b. This completes the algebra of thefirst half year.

In the second semester, we begin by developing concepts of sets,


relations, and set operations. This chapter enables us to review thealgebra learned in the first semester, now couched in set language andsymbols, as well as to prepare for the study of geometry. In Chapters12 and 13 we take up the study of rational numbers and their applica-tions. The introduction of rationals parallels the study of the in-tegers. First we have situations where the reciprocal of a unit fractionis needed, so for example, we invent i or i etc. We call these unitfractions, the set Z’=^ {� � -^, ^, ^-, ^-, ^, ^, � � � }. Combining Z/ and Zand a new multiplication we obtain the rationals and study ((3, +)and ((5\{0}, �) and obtain a new group. Then division, addition,subtraction, and ordering of the rationals gives us the field (0, +,�, <). The chapter closes with decimal and infinite repeating decimalnotation for the rationals. The chapter on applications treats, amongmany things, computational algorisms, dilations in a plane, ratio,proportion, percent, translations in a plane, similitudes and somework on graphs.

Course II (eighth grade) begins with a short chapter on ideas oflogic and proof. There follows the first abstract algebra with a chapteron group theory. With a wealth of examples from Course I and theparallel and general solutions of ax=b and a-{-x==b, a number ofgroups are examined some commutative, some not. Then an ex-tended study is made of the group of permutations (3 and 4 elementsonly) under composition and the concept of subgroup is introduced.The group of isometries studied in geometry is reviewed. Finally anumber of theorems are proved. (The uniqueness of a left hand andright hand operation; the left and right cancellation laws, the uniquesolution to aox==b, the uniqueness of the identity element, and(aoby=blocll.) The important objective is to show that a structurecan be studied in its own right and the properties established hold inany system that is a realization of a group. The chapter ends with atreatment of isomorphism.The next chapter on algebra, capitalizing on the Rational Numbers,

is a study of field structure. The axioms of a field are enumerated andexamples are given. (Q, +, �); (^5, +, �), and tables for (£, o , *).Since a field is easily separated into two groups, (£, o) and (jE\{0} ,*)we obtain all the fundamental theorems for a field in one fell swoop.The zero difficulty is solved by proving that a-0==0-o==0, and finallythe theorem a*6=0=»o==0 or 6=0 is proved. The operations of sub-traction and division are introduced and applied to computations andlater, to solutions of inequations, and equations, including quadraticequations. For all fields a�b becomes a-b~1 or a/b, that is a fraction.Finally order is introduced into the field with inequality theorems.Thus, at this stage, the only ordered field is ((), +, *, <), all finitefields being unordered.

In Chapter 5 we introduce the real numbers, xl=2 has no solution


in the field of rationals�but there should be a number of some kindwhose square is 2. The measuring process is introduced (using thedecimal system)�if the process ends the measure is rational, if itdoes not end, there are two cases, repeating and non-repeating deci-mals. To settle the case the upper bound and least upper bound of asequence of measures is defined. The length of a segment is the leastupper bound of the rational sequences in the measuring process.Informally all three cases are studied, that is: terminal rational, leastupper bound rational, and least upper bound non-rational (diagonalof a unit square). The last case leads to the extension of the rationalsystem to include as a subfield new elements to serve as least upperbounds for Case III. This leads to the completely ordered field of realnumbers (1?, +, *, <) using decimal numbers. The equation xi=a� isstudied for its two solutions. Finally the arithmetic of irrationalnumbers in \/ (radical) form is thoroughly developed.A chapter on real functions completes the algebra study of grade 8.

First mathematical mappings f:s�>t using domain, co-domain andimage is established by concrete examples using arrow diagrams, andtables. The real numbers are then taken as domains, and a study ofreal functions is begun. The functions are one-to-one into; onto; andone-to-one onto. The image set or range is defined as a subset of theco-domain. Various ways of representing real functions are examined:arrows on a number line, arrows from one number line to another, byplotting ordered pairs on a coordinatized plane (absolute value func-tions are included). Capitalizing on composition of mappings inCourse 1^ composition of real functions is studied e.g. g:x�>x^ andf:x-^3x+2 gives g of�.x-^3xi+2, but/ o g:x-^{3x+2)\ The identitycomposition x�>x’, the associativity of composition; (gof) oh=go(foh) are shown intuitively, later proved formally. To test if func-tions form a group under composition, inverses are investigated, i.e.is there for eachf, a g such that/o g==g o/=7, the identity function?Alas the function x�>x2 has two pre-images and hence there is noinverse. On the contrary ax-}-b always has (x�V)/a as an inverse.Investigating a number of examples, including

/ x1

H+iwe find an inverse exists only if the function is one-to-one. Then witha study of the sum, difference, product and quotient of two real func-tions and the square root and cube root functions, the chapter ends.

Course III (9th grade), opens with two chapters on Matrices, thefirst for motivation and practical examples, the second for the arith-metic of matrices. The latter includes addition and subtraction ofconformable matrices, the multiplicative identity of M^ (Square


matrices 2X2). The existence of a multiplicative inverse is studiedleading to the solution of matrix equations in MÂX^C^XÂ^Cif A~^ exists. Row operations on matrices are investigated along withtheir non-effect on the solution of linear equations. Practice is givenon using this method to solve linear systems in 2, 3 and 4 variables,then problems leading to linear systems are solved. The chaptercloses with the first description of the goal we have set ourselves�namely a vector space. The definition (axioms) of vector space areenumerated and many simple examples are given. A few theoremsare proved e.g. ¥2 (set of ordered pairs for which (a, b) + (.c, d)=(a+^, b-{-d)\ k{a^ b)==(ka, kb) is a vector space. 0’A==0, k ’0=0,�1-^4==� A, k’A=0=^k=0 or A=0. The chapter closes with astudy of linear combination of elements of a vector space, linear in-dependence and linear dependence are defined, leading to the funda-mental concepts of spanning and basis for a vector space.

In Chapter 4 (Course III), planar graphs and functions are studiedthrough conditions which are open sentences in two variables. Thisincludes conditions of the type ax-^-by<c, \x\+ |y| <,k and conjunc-tion and disjunction of conditions leading to regions in a plane.Using translations, ] x� 5 | + \y~ 31 == 2 is solved by solving | x\+ | y \= 2 and translating the solution by

(rv,^)-^(rv+5,3.+3).Next a study is made of functions and conditions on the functionwhere both \x\ and [x] (greatest integer function) are treated. Thenfunctions and graphical solutions of space-time problems are ex-amined. The operations on functions of Course II are now extendedto associated function equations: f-^-g becomes y==f(y)~{~g(x) etc. Astudy of bounded functions and asymptotes (e.g. 1/rv) completes thechapter.The final chapter on Algebra in the three year program is on Poly-

nomial and Rational functions. By the addition and multiplicationof the constant and identity function all polynomial functions arecreated. The ring of polynomials, first over the integers�and thenover any field�is treated thoroughly�addition, subtraction, multi-plication, division with a remainder, factorization, the factor theorem,terminating with the quadratic function and equation. The finalsection of the chapter studies the quotient of polynomials over thereal numbers with the usual simplification procedures for rationalexpressions and solution of simple rational equations. This completesthe algebraic study in the Junior High School Program.

GEOMETRIC INSTRUCTION: COURSES I, II, AND IIIThe geometric instruction begins intuitively in Chapter 7 of


Course I with Lattice Points. It is assumed that simple figures,parallelism, perpendicularity, and the number line have been met inthe elementary school program. So lattice points are immediatelyassociated with ordered pairs of integers and a coordinate system.Conditions are then imposed on the set of ordered pairs to producespecific sets of points. This permits a geometric representation of alge-braic conditions on two variables, including the <, > and \x\ rela-tions. Compound conditions lead to intersections and unions of setsof points. Finally lattice point games, including operational checkers,lead to skill in handling ordered pairs and locating their images in aplane. This is extended to ordered triples in space�all intuitively.Dilations on the plane (x’ y)�> (ax, by) completes the chapter.The next two chapters on geometry treat transformations of the

plane. The first approach is through purely physical activities�paper folding, measuring, and constructions by drawings. The stu-dents obtain a more precise concept of ray, segment, reflectionsabout a line or through a point, translations, and rotations, that issome isometries of the plane, along with a first idea of symmetry.The next chapter makes the first step toward abstract geometry byusing coordinates on a line and in a plane and treating betweenness,intersection, union, measure, distance, and isometries by means ofcoordinates. Angle is defined as a region. Its measure by a protractoris a number m such that 00<^m<1800. Finally the fundamental prop-erties of isometries (reflections, translations, rotations) are listed andused to derive other properties, e.g. the sum of the angles of a triangle.

In Course II (8th grade) after elementary logic and group theoryhave been introduced, there is presented an introduction to axio-matic synthetic geometry. It is afflne geometry with only threeaxioms and some definitions. From these axioms twelve theorems areto be proved (the book gives the proof of only four theorems). Anon-geometric example (commandos and squads) is given and alltwelve theorems are interpreted. Other models both finite (9 pointgeometry) and infinite are exhibited. Equivalence classes of parallellines (/ is parallel to itself) leads to parallel projection and an orderedpair of points (X, F) become geometric coordinates of third point inthe plane. Now we have the basis for building a coordinated planethrough the introduction of integers, then rationals, and finally realnumbers. The chapter concludes with an equivalence class of freevectors in a plane leading to a new group�the group of translationsin the plane. This is one of the more difficult chapters to teach, one ofthe best pedagogical chapters in the course, and the most satisfyingto students once they have mastered it (in about 5 weeks of study).The mathematical seminar of the Russian Academy of PedagogicalSciences rated it the finest geometry presentation produced to date.


After the real numbers are studied, Chapter 6 of Course II givesthe introduction to coordinate geometry. To the synthetic geometryaxioms of Chapter 3 are added three other axioms on parallel lineprojections and coordinates which relate points (geometric entities)to coordinates (real numbers). There follows a development of theusual treatment of segments, division of segments, equations of aline, applications to triangles and quadrilaterals, and an informalacceptance of the Pythagorean property of right triangles, with thedistance formula.

In Part 2 in Chapter 9, transformations are studied via coordinatesand coordinate geometry. The group of isometries leads to the studyof congruence and symmetry. A brief introduction is given to dila-tions and similarity. In Chapter 10, because of its usefulness, thetheory of measure of length and area is developed and applied torectilinear figures and the circle. The real number TT is discussed. Forthose who are interested, or as an enrichment chapter there is anappendix giving an introduction to mass-point geometry in theplane. At the end of Course II, our students have a fairly good ideaof the meaning of intuitive physical geometry, abstract syntheticgeometry, coordinate geometry, and plane vectors as ways of study-ing looking at space of two dimensions).

In Course III, Chapter 3, geometry is extended to a study of spaceof three dimensions. An informal introduction is given to the studyand representation of points, lines and planes in space includingparallelism and perpendicularity. The three axioms of plane syn-thetic affine geometry are extended by three other axioms dealingwith lines and planes in space. This enables the student to prove anumber of theorems on parallelism in affine space. The coordinates ofaffine 3 space are introduced and the mid-point and distance formuladeveloped. To conclude the chapter an informal treatment is givenof surfaces and solids in space.

Chapter 4 unites algebra and geometry by studying functions andconditions and planar graph representation of the conditions.The final chapter on Geometry in Course III is on affine vector

geometry. The axioms of plane and space synthetic geometry, andthe added axioms of coordinate geometry are increased by threeaxioms (12 axioms in all) that permit the coordination of 3-dimen-sional affine space. Regions in space (half space, and others) andcoordinates of points lead to an algebra of points for affine 3 space:addition of points, (n’i x^ ^s)+ (yi y^. y$) multiplication by a real scolara(x\ X2 ^3) == (flXi 0x2 ax^) lead to directed line segments, an equivalenceclass, and vectors in space. To each vector AB there is assigned thedifference of points B� A which is associated to the difference P�0,


or simply point P as a vector since 0 is fixed. The vector equation of aline x=tA and the parametric equation Xi==at, ^2= <^, ^3=03^ arestudied and extended to the line x^p-i-ta. The equations of a planeare similary studied x==sA+tB, x==P+sA+tB. The chapter closeswith the definition of the inner product (î x^ ^3). (î ^2 ^3)=^0’i+^2;y2+^3^3 and assumption of perpendicularity leads to norms,orthogonality and the formula for

A’Bcos 6 =

N1 -Nand its use. The present chapter is too long and too difficult exceptfor the upper 5 to 8%. We shall simplify it and certainly make 2 or 3chapters out of the present 1 chapter. The first class to finish thischapter on a comprehensive examination of it registered 8 A’s (75%or higher) 9 B^s (50% or higher), 7 C’s (35% or higher) and 2 stu-dents who certainly did not comprehend the study. The material willbe rewritten during the year.

OTHER TOPICSBesides these mainstreams there are other chapters. Sets and

Relations are studied in Course I, so is Number Theory. Probabilityis studied in Course I and Course III, including Statistics, Combi-natorics, Algorisms, and Flow-charting as preparatory to Basic Pro-gramming. The Circular Functions of angles are introduced inCourse III. But in all cases the treatment is unified around the basicstructures and their realizations.

A GLIMPSE INTO THE FUTURECourse IV builds on all that has been taught previously. In general

it can be conceived as building the essential foundation for analysiswhich will be taught in Courses V and VI. All I can do at present isgive a glimpse of what we hope to achieve by citing the Chapters ofCourse IV, now just starting under experimentation, and the topicslisted for Courses V and VI. In Course IV the chapters are:

Basic Programming. The language of Basic, flow-charting, writingprograms and the use of a console tuned into an electronic computercenter.

Sequence and Limits. The sequences as functions of the naturalnumbers, types of sequences, limits in terms of § and c, theorems onsums and products, Cauchy’s criterion.

Quadratic Equation and Complex Numbers. The equation a^°+bx-}-c=0, digital computation of irrational roots, 2X2 matrices oftype aI-\-bJ,


/,p "i ^-r-’i.Lo ij Li oj

Isomorphism of 2X2 matrices and a-{-bi, i2= �1. Complex numbersand transformations.

Linear Transformations. A vector space defined, subspaces, lineartransformations and theroems, Kernel and Range.

Linear Transformations and Matrices. Matrices associated withtransformations, composition of linear transformations and asso-ciativity of matrices, inverse transformations and inverse matrices;computing inverses of matrices.

Exponents and- Logarithms. Exponential function, inverse expo-nential function, natural logarithms, computation, organic growth.

Circular Functions. Sensed angles and real numbers, function ofreal numbers modulo 27r, trigonometric analysis.

Linear Approximations and Informal Differentiation. Piecewiselinear functions. Approximations to a curve by lines, the differentialquotient, applications to geometry, and physics. :

Probability III. Axiomatic probability, dependent and independentevents, conditional probability, random walks, Bayes theorem, simu-lation, Markov chains.

Continuity. An approach to continuity at a point and ; uniformcontinuity over an interval.

Courses V and VI will include the following topics:(1) Extension of linear algebra (Vector Spaces) to ^-dimensional space.(2) Theory of Differentiation and Integration, Theory of series.(3) Differential Equations.(4) Theory of Probability and Statistical Inference. ’

(5) Elementary numerical analysis.(6) Special topics for outside study e.g. Measure theory, including angle mea-

sure, normed vector spaces, Galois Theory, Boolean Algebras, Elementsof point-set Topology.

This is the present goal of the SSMCIS study�namely, to gainone to one and one-half year advance in mathematical study for thecapable University-bound youth by the end of high school study. Itis accomplished by: :

(1) Structuring and unifying the mathematical study(2) Eliminating all unnecessary topics in the present curriculum.(3) Using clever pedagogical approaches to learning.(4) Using the textbook as the primary learning material, one that can be read

and studied by the student. Using the classroom for dialogue, investiga-tion, clarification and dynamic intellectual interchange amon^ students,and between students and teacher.

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The Secondary School Mathematics Curriculum Improvement Study Goals—The Subject Matter—Accomplishments