Misplaced Pages

#221778

132-677: The program's signature goal was to create a unified treatment of mathematics and eliminate the traditional separate per-year studies of algebra , geometry , trigonometry , and so forth, that was typical of American secondary schools. Instead, the treatment unified those branches by studying fundamental concepts such as sets , relations , operations , and mappings , and fundamental structures such as groups , rings , fields , and vector spaces . The SSMCIS program produced six courses' worth of class material, intended for grades 7 through 12, in textbooks called Unified Modern Mathematics . Some 25,000 students took SSMCIS courses nationwide during

264-563: A {\displaystyle a} to object b {\displaystyle b} , and another morphism from object b {\displaystyle b} to object c {\displaystyle c} , then there must also exist one from object a {\displaystyle a} to object c {\displaystyle c} . Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object. Categories are widely used in contemporary mathematics since they provide

396-403: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are usually used for constants and coefficients . The expression 5 x + 3 {\displaystyle 5x+3} is an algebraic expression created by multiplying the number 5 with the variable x {\displaystyle x} and adding

528-746: A 2 x 2 + . . . + a n x n = b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b} where a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , ..., a n {\displaystyle a_{n}} and b {\displaystyle b} are constants. Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and 1 4 x − y = 4 {\textstyle {\frac {1}{4}}x-y=4} . A system of linear equations

660-429: A ∘ a − 1 = a − 1 ∘ a = e {\displaystyle a\circ a^{-1}=a^{-1}\circ a=e} . Every algebraic structure that fulfills these requirements is a group. For example, ⟨ Z , + ⟩ {\displaystyle \langle \mathbb {Z} ,+\rangle } is a group formed by the set of integers together with

792-433: A ∘ b ) ∘ c {\displaystyle (a\circ b)\circ c} is the same as a ∘ ( b ∘ c ) {\displaystyle a\circ (b\circ c)} for all elements. An operation has an identity element or a neutral element if one element e exists that does not change the value of any other element, i.e., if a ∘ e = e ∘

924-402: A + c a . {\displaystyle (b+c)a=ba+ca.} Moreover, multiplication is associative and has an identity element generally denoted as 1 . Multiplication needs not to be commutative; if it is commutative, one has a commutative ring . The ring of integers ( Z {\displaystyle \mathbb {Z} } ) is one of the simplest commutative rings. A field

1056-437: A = a {\displaystyle a\circ e=e\circ a=a} . An operation has inverse elements if for any element a {\displaystyle a} there exists a reciprocal element a − 1 {\displaystyle a^{-1}} that undoes a {\displaystyle a} . If an element operates on its inverse then the result is the neutral element e , expressed formally as

1188-652: A Lie algebra or an associative algebra . The word algebra comes from the Arabic term الجبر ( al-jabr ), which originally referred to the surgical treatment of bonesetting . In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it in the title of a treatise on algebra, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [ The Compendious Book on Calculation by Completion and Balancing ] which

1320-450: A linear polynomial . In category theory , a map may refer to a morphism . The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory . In many branches of mathematics, the term map is used to mean a function , sometimes with a specific property of particular importance to that branch. For instance,

1452-480: A map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map : mapping the Earth surface to a sheet of paper. The term map may be used to distinguish some special types of functions, such as homomorphisms . For example, a linear map is a homomorphism of vector spaces , while the term linear function may have this meaning or it may mean

SECTION 10

#1733086164222

1584-623: A set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra, since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups , rings , and fields , based on the number of operations they use and the laws they follow . Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures. Algebraic methods were first studied in

1716-404: A theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology , number theory , and calculus , and other fields of inquiry, like logic and the empirical sciences . Algebra is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure

1848-522: A "map" is a " continuous function " in topology , a " linear transformation " in linear algebra , etc. Some authors, such as Serge Lang , use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C ), and reserve the term mapping for more general functions. Maps of certain kinds have been given specific names. These include homomorphisms in algebra , isometries in geometry , operators in analysis and representations in group theory . In

1980-700: A broader range of mathematical ability. Computer programming on time-shared computer systems was included in the curriculum both for its own importance and for understanding numerical methods. The first course introduced flow charts and the notion of algorithms . The beginning portion of the fourth-year course was devoted to introducing the BASIC programming language, with an emphasis on fundamental control flow statements, continued use of flow charts for design, and numerical programming applications. Interactive teletype interfaces on slow and erratic dial-up connections , with troublesome paper tape for offline storage,

2112-422: A core of fundamental concepts and structures common to all. For example, as the courses progressed, the concept of mappings was used to describe, and visually illustrate, the traditionally disparate topics of translation , line reflection , probability of an event , trigonometric functions , isomorphism and complex numbers , and analysis and linear mappings . Traditional subjects were broken up, such that

2244-625: A couple of high schools) in the United States. Fehr did not do much curriculum development himself, but rather recruited and led the others and organized the whole process. Graduate students from the Department of Mathematical Education at Teachers College also served each year in various capacities on the SSMCIS program. The central idea of the program was to organize mathematics not by algebra, geometry, etc., but rather to unify those branches by studying

2376-404: A generalization of arithmetic . Arithmetic studies operations like addition, subtraction , multiplication, and division , in a particular domain of numbers, such as the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in

2508-461: A key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation. In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed. At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved

2640-467: A large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an abelian group and a scalar multiplication that is compatible with addition (see vector space for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of finite-dimensional vector spaces , vectors and linear maps can be represented by matrices. It follows that

2772-498: A major concern of parents and students and teachers. A 1973 report compared the test performance of such students with those from traditional mathematics curricula. It found that the SSMCIS students did better on the mathematics portion of the Preliminary Scholastic Aptitude Test (PSAT) , even when matched for background and performance on the verbal portion. It also found that SSMCIS students did just as well on

SECTION 20

#1733086164222

2904-412: A positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions. Linear algebra starts with the study systems of linear equations . An equation is linear if it can be expressed in the form a 1 x 1 +

3036-428: A second-degree polynomial equation of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} is given by the quadratic formula x = − b ± b 2 − 4 a c   2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.} Solutions for

3168-447: A similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables. Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph . To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if x {\displaystyle x}

3300-435: A statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign ( = {\displaystyle =} ), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4} . Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as

3432-527: A unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the category of sets , and any group can be regarded as the morphisms of a category with just one object. The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in Babylonia , Egypt , Greece , China , and India . One of

3564-400: A whole is zero if and only if one of its factors is zero, i.e., if x {\displaystyle x} is either −2 or 5. Before the 19th century, much of algebra was devoted to polynomial equations , that is equations obtained by equating a polynomial to zero. The first attempts for solving polynomial equations was to express the solutions in terms of n th roots . The solution of

3696-397: Is a commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and distributive with respect to addition; that is, a ( b + c ) = a b + a c {\displaystyle a(b+c)=ab+ac} and ( b + c ) a = b

3828-438: Is a commutative ring such that ⁠ 1 ≠ 0 {\displaystyle 1\neq 0} ⁠ and each nonzero element has a multiplicative inverse . The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 {\displaystyle 7} is 1 7 {\displaystyle {\tfrac {1}{7}}} , which

3960-475: Is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form ⟨ A , ∘ ⟩ {\displaystyle \langle A,\circ \rangle } and ⟨ B , ⋆ ⟩ {\displaystyle \langle B,\star \rangle } then

4092-504: Is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero . Factorization consists in rewriting a polynomial as a product of several factors. For example, the polynomial x 2 − 3 x − 10 {\displaystyle x^{2}-3x-10} can be factorized as ( x + 2 ) ( x − 5 ) {\displaystyle (x+2)(x-5)} . The polynomial as

Secondary School Mathematics Curriculum Improvement Study - Misplaced Pages Continue

4224-415: Is a non-empty set of mathematical objects , such as the integers , together with algebraic operations defined on that set, like addition and multiplication . Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use of variables in equations and how to manipulate these equations. Algebra is often understood as

4356-487: Is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The degree of a polynomial is the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example). Polynomials of degree one are called linear polynomials . Linear algebra studies systems of linear polynomials. A polynomial is said to be univariate or multivariate , depending on whether it uses one or more variables. Factorization

4488-941: Is a set of linear equations for which one is interested in common solutions. Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations For example, the system of equations 9 x 1 + 3 x 2 − 13 x 3 = 0 2.3 x 1 + 7 x 3 = 9 − 5 x 1 − 17 x 2 = − 3 {\displaystyle {\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}} can be written as A X = B , {\displaystyle AX=B,} where A , B {\displaystyle A,B} and C {\displaystyle C} are

4620-414: Is an expression consisting of one or more terms that are added or subtracted from each other, like x 4 + 3 x y 2 + 5 x 3 − 1 {\displaystyle x^{4}+3xy^{2}+5x^{3}-1} . Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. A monomial

4752-629: Is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on

4884-408: Is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with

5016-482: Is not an integer. The rational numbers , the real numbers , and the complex numbers each form a field with the operations of addition and multiplication. Ring theory is the study of rings, exploring concepts such as subrings , quotient rings , polynomial rings , and ideals as well as theorems such as Hilbert's basis theorem . Field theory is concerned with fields, examining field extensions , algebraic closures , and finite fields . Galois theory explores

5148-445: Is often used as a synonym for " morphism " or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does. For example, a morphism f : X → Y {\displaystyle f:\,X\to Y} in a concrete category (i.e. a morphism that can be viewed as a function) carries with it the information of its domain (the source X {\displaystyle X} of

5280-400: Is set to zero in the equation y = 0.5 x − 1 {\displaystyle y=0.5x-1} , then y {\displaystyle y} must be −1 for the equation to be true. This means that the ( x , y ) {\displaystyle (x,y)} -pair ( 0 , − 1 ) {\displaystyle (0,-1)} is part of

5412-403: Is the identity matrix . Then, multiplying on the left both members of the above matrix equation by A − 1 , {\displaystyle A^{-1},} one gets the solution of the system of linear equations as X = A − 1 B . {\displaystyle X=A^{-1}B.} Methods of solving systems of linear equations range from

Secondary School Mathematics Curriculum Improvement Study - Misplaced Pages Continue

5544-414: Is the application of group theory to analyze graphs and symmetries. The insights of algebra are also relevant to calculus, which uses mathematical expressions to examine rates of change and accumulation . It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them. Algebraic logic employs the methods of algebra to describe and analyze

5676-425: Is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset. Universal algebra is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up

5808-421: Is the process of applying algebraic methods and principles to other branches of mathematics , such as geometry , topology , number theory , and calculus . It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other. One application, found in geometry,

5940-472: Is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition , subtraction , multiplication , division , exponentiation , extraction of roots , and logarithm . For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 {\displaystyle 2+5=7} . Elementary algebra relies on

6072-718: Is the use of algebraic statements to describe geometric figures. For example, the equation y = 3 x − 7 {\displaystyle y=3x-7} describes a line in two-dimensional space while the equation x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties , which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures. Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where

6204-466: Is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a ∘ b {\displaystyle a\circ b} is identical to b ∘ a {\displaystyle b\circ a} for all elements. A variety is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of

6336-446: Is true if x {\displaystyle x} is either 2 or −2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation 2 x + 5 x = 7 x {\displaystyle 2x+5x=7x} . Conditional equations are only true for some values. For example,

6468-599: The Bourbaki group 's work in France in the 1930s and the Synopses for Modern Secondary School Mathematics published in Paris in 1961. Indeed, most European secondary schools were teaching a more integrated approach. Also, this was one of several American efforts to address a particularly controversial issue, the teaching of a full year of Euclidean geometry in secondary school. Like many of

6600-537: The Mathematics Level II Achievement Test as traditional students taking college preparatory courses or, indeed, as college freshmen taking introductory calculus courses. Another study found SSMCIS students well prepared for the mathematics portion of the regular Scholastic Aptitude Test . However, SSMCIS developed slowly. Funding became an issue, and indeed it was never funded as well as some other mathematics curriculum efforts had been. Despite

6732-416: The U.S. Congress . As one of the participants in creating SSMCIS, James T. Fey of Teachers College, later wrote, "Schools and societal expectations of schools appear to change very slowly." In the end, SSMCIS never became widely adopted. One SSMCIS student, Toomas Hendrik Ilves of Leonia High School , decades later became Foreign Minister and then President of Estonia . He credited the SSMCIS course,

SECTION 50

#1733086164222

6864-414: The ancient period to solve specific problems in fields like geometry . Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond

6996-547: The difference of two squares method and later in Euclid's Elements . In the 3rd century CE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica . He was the first to experiment with symbolic notation to express polynomials. Diophantus's work influenced Arab development of algebra with many of his methods reflected in the concepts and techniques used in medieval Arabic algebra. In ancient China, The Nine Chapters on

7128-525: The fundamental theorem of algebra , which describes the existence of zeros of polynomials of any degree without providing a general solution. At the beginning of the 19th century, the Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show that no general solution exists for polynomials of degree five and higher. In response to and shortly after their findings,

7260-602: The fundamental theorem of finite abelian groups and the Feit–Thompson theorem . The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups . A ring is an algebraic structure with two operations that work similarly to addition and multiplication of numbers and are named and generally denoted similarly. A ring

7392-461: The less-than sign ( < {\displaystyle <} ), the greater-than sign ( > {\displaystyle >} ), and the inequality sign ( ≠ {\displaystyle \neq } ). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement x 2 = 4 {\displaystyle x^{2}=4}

7524-653: The 12th century further refined Brahmagupta's methods and concepts. In 1247, the Chinese mathematician Qin Jiushao wrote the Mathematical Treatise in Nine Sections , which includes an algorithm for the numerical evaluation of polynomials , including polynomials of higher degrees. The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books including his Liber Abaci . In 1545,

7656-528: The 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field. The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics—that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such as topological groups and Lie groups . In

7788-464: The 1940s and 50s, homological algebra emerged, employing algebraic techniques to study homology . Around the same time, category theory was developed and has since played a key role in the foundations of mathematics . Other developments were the formulation of model theory and the study of free algebras . The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields. The algebraization of mathematics

7920-402: The 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries. In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations. The Indian mathematicians Mahāvīra in the 9th century and Bhāskara II in

8052-481: The French mathematician Évariste Galois developed what came later to be known as Galois theory , which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of group theory . Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory. Starting in the mid-19th century, interest in algebra shifted from

SECTION 60

#1733086164222

8184-598: The German mathematicians David Hilbert , Ernst Steinitz , and Emmy Noether as well as the Austrian mathematician Emil Artin . They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields. The idea of the even more general approach associated with universal algebra was conceived by the English mathematician Alfred North Whitehead in his 1898 book A Treatise on Universal Algebra . Starting in

8316-592: The Italian polymath Gerolamo Cardano published his book Ars Magna , which covered many topics in algebra, discussed imaginary numbers , and was the first to present general methods for solving cubic and quartic equations . In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions. Some historians see this development as

8448-561: The Mathematical Art , a book composed over the period spanning from the 10th century BCE to the 2nd century CE, explored various techniques for solving algebraic equations, including the use of matrix-like constructs. There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications. This changed with

8580-459: The New Math . Many reform efforts had underestimated the difficulty of getting the public and the mathematics educational community to believe that major changes were really necessary, especially for secondary school programs where college entrance performance was always the key concern of administrators. Federal funding for curriculum development also came under attack from American conservatives in

8712-454: The Persian mathematician al-Khwarizmi , who published his The Compendious Book on Calculation by Completion and Balancing in 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides. Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra also in

8844-469: The SSMCIS program began in 1965 and took place mainly at Teachers College. Fehr was the director of the project from 1965 to 1973. The principal consultants in the initial stages and subsequent yearly planning sessions were Marshall H. Stone of the University of Chicago , Albert W. Tucker of Princeton University , Edgar Lorch of Columbia University , and Meyer Jordan of Brooklyn College . The program

8976-401: The addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups , rings , and fields . The key difference between these types of algebraic structures lies in

9108-443: The characteristics of algebraic structures in general. The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun , an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation . Depending on the context, "algebra" can also refer to other algebraic structures, like

9240-416: The corresponding variety. Category theory examines how mathematical objects are related to each other using the concept of categories . A category is a collection of objects together with a collection of so-called morphisms or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or composed : if there exists a morphism from object

9372-418: The course material for each year included some material related to algebra, some to geometry, and so forth. Even when abstract concepts were being introduced, they were introduced in concrete, intuitive forms, especially at the younger levels. Logical proofs were introduced early on and built in importance as the years developed. At least one year of university-level mathematics education was incorporated into

9504-407: The curriculum was developed by eighteen mathematicians from the U.S. and Europe in 1966 and subsequently refined in experimental course material by mathematical educators with high school level teaching experience. By 1971, some thirty-eight contributors to course materials were identified, eight from Teachers College, four from Europe, one from Canada, and the rest from various other universities (and

9636-593: The degrees 3 and 4 are given by the cubic and quartic formulas. There are no general solutions for higher degrees, as proven in the 19th century by the so-called Abel–Ruffini theorem . Even when general solutions do not exist, approximate solutions can be found by numerical tools like the Newton–Raphson method . The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of

9768-455: The difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space , and the points where all planes intersect solve the system of equations. Abstract algebra, also called modern algebra, is the study of algebraic structures . An algebraic structure is a framework for understanding operations on mathematical objects , like

9900-469: The distributive property. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that y = 3 x {\displaystyle y=3x} then one can simplify the expression 7 x y {\displaystyle 7xy} to arrive at 21 x 2 {\displaystyle 21x^{2}} . In

10032-496: The earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt, which was written around 1650 BCE. It discusses solutions to linear equations , as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and quadratic polynomial equations , such as

10164-613: The early 1960s and by the Cambridge Conference on School Mathematics (1963), which also inspired the Comprehensive School Mathematics Program . There were some interactions among these initiatives in the early stages, and the development of SSMCIS was part of a general wave of cooperation in the mathematics education reform movement between Europe and the U.S. "The construction is to be free of any restrictions of traditional content or sequence." Work on

10296-553: The early exposure it gave him to computer programming, and the teacher of the course, Christine Cummings, with his subsequent interest in computer infrastructure, which in part resulted in the country leaping over its Soviet -era technological backwardness; computer-accessible education became pervasive in Estonian schools , and the Internet in Estonia has one of the highest penetration rates in

10428-403: The elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure. Another tool of comparison is the relation between an algebraic structure and its subalgebra . The algebraic structure and its subalgebra use the same operations, which follow

10560-404: The equation x + 4 = 9 {\displaystyle x+4=9} is only true if x {\displaystyle x} is 5. The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation

10692-612: The existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, like Fermat's Last Theorem , and of algebraic structures to analyze the behavior of numbers, such as the ring of integers . The related field of combinatorics uses algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example in algebraic combinatorics

10824-432: The federal funding source, there was no centralized, national focal point in the U.S. for curriculum changes – such as some European countries had – and that made adoption of SSMCIS innovations a harder task. By the mid-1970s there was a growing backlash against the "New Math" movement, spurred in part by a perceived decline in standardized test scores and by Morris Kline 's critical book Why Johnny Can't Add: The Failure of

10956-429: The form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra , which is not limited to a particular domain and examines algebraic structures such as groups and rings . It extends beyond typical arithmetic operations by also covering other types of operations. Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates

11088-438: The function h : A → B {\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement: h ( x ∘ y ) = h ( x ) ⋆ h ( y ) {\displaystyle h(x\circ y)=h(x)\star h(y)} . The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in

11220-431: The fundamental concepts of sets , relations , operations , and mappings , and fundamental structures such as groups , rings , fields , and vector spaces . Other terms used for this approach included "global" or "integrated"; Fehr himself spoke of a "spiral" or "helical" development, and wrote of "the spirit of global organization that is at the heart of the SSMCIS curriculum – important mathematical systems unified by

11352-413: The graph of the equation. The ( x , y ) {\displaystyle (x,y)} -pair ( 0 , 7 ) {\displaystyle (0,7)} , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of ( x , y ) {\displaystyle (x,y)} -pairs that solve the equation. A polynomial

11484-495: The introductory, like substitution and elimination, to more advanced techniques using matrices, such as Cramer's rule , the Gaussian elimination , and LU decomposition . Some systems of equations are inconsistent , meaning that no solutions exist because the equations contradict each other. Consistent systems have either one unique solution or an infinite number of solutions. The study of vector spaces and linear maps form

11616-500: The late 1960s and early 1970s. The program was led by Howard F. Fehr, a professor at Columbia University Teachers College who was internationally known and had published numerous mathematics textbooks and hundreds of articles about mathematics teaching. In 1961 he had been the principal author of the 246-page report "New Thinking in School Mathematics", which held that traditional teaching of mathematics approaches did not meet

11748-647: The later courses. Solving traditional applications problems was de-emphasized, especially in the earlier courses, but the intent of the project was to make up for that with its focus on real numbers in measurements, computer programming, and probability and statistics. In particular, the last of these was a pronounced element of the SSMCIS, with substantial material on it present in all six courses, from measures of statistical dispersion to combinatorics to Bayes' theorem and more. The curriculum that SSMCIS devised had influences from earlier reform work in Europe, going back to

11880-607: The left side and results in the equation x = 11 {\displaystyle x=11} . There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression 7 x − 3 x {\displaystyle 7x-3x} can be replaced with the expression 4 x {\displaystyle 4x} since 7 x − 3 x = ( 7 − 3 ) x = 4 x {\displaystyle 7x-3x=(7-3)x=4x} by

12012-620: The line described by y = x + 1 {\displaystyle y=x+1} intersects with the circle described by x 2 + y 2 = 25 {\displaystyle x^{2}+y^{2}=25} by solving the system of equations made up of these two equations. Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation . Algebraic topology relies on algebraic theories such as group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on

12144-426: The linear map to the basis vectors. Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a line in two-dimensional space . The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there

12276-472: The lowercase letters x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} represent variables. In some cases, subscripts are added to distinguish variables, as in x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , and x 3 {\displaystyle x_{3}} . The lowercase letters

12408-417: The manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication . Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values

12540-491: The mathematics education literature would cite it in subsequent years, including references to it as a distinct, and the most radical, approach to teaching geometry; as using functions as a unifying element of teaching mathematics; and as its course materials having value when used as the vehicle for further research in mathematics education. Algebra Algebra is the branch of mathematics that studies certain abstract systems , known as algebraic structures , and

12672-647: The matrices A = [ 9 3 − 13 2.3 0 7 − 5 − 17 0 ] , X = [ x 1 x 2 x 3 ] , B = [ 0 9 − 3 ] . {\displaystyle A={\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}},\quad X={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}},\quad B={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.} Under some conditions on

12804-475: The method of completing the square . Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in Pythagoras ' formulation of

12936-598: The morphism) and its codomain (the target Y {\displaystyle Y} ). In the widely used definition of a function f : X → Y {\displaystyle f:X\to Y} , f {\displaystyle f} is a subset of X × Y {\displaystyle X\times Y} consisting of all the pairs ( x , f ( x ) ) {\displaystyle (x,f(x))} for x ∈ X {\displaystyle x\in X} . In this sense,

13068-477: The needs of the new technical society being entered into or of the current language of mathematicians and scientists. Fehr considered the separation of mathematical study into separate years of distinct subjects to be an American failing that followed an educational model two hundred years old. The new curriculum was inspired by the seminar reports from the Organisation for Economic Co-operation and Development in

13200-828: The next with their students, and so it was typical for students to have one of the same two teachers, or even the same teacher, for five or six years in a row. More teachers were added in 1968 and 1969 and the University of Maryland and University of Arizona were added as teaching sites. Eighteen schools in Los Angeles adopted SSMCIS in what was called the Accelerated Mathematics Instruction program; some 2,500 gifted students took part. By 1971, teacher education programs were being conducted in places like Austin Peay State University in Tennessee , which

13332-399: The number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c} . Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is

13464-470: The number of operations they use and the laws they obey. In mathematics education , abstract algebra refers to an advanced undergraduate course that mathematics majors take after completing courses in linear algebra. On a formal level, an algebraic structure is a set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in binary operations , which take any two objects from

13596-511: The number of rows and columns, matrices can be added , multiplied , and sometimes inverted . All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix A − 1 {\displaystyle A^{-1}} such that A − 1 A = I , {\displaystyle A^{-1}A=I,} where I {\displaystyle I}

13728-436: The numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication , which is expressed in the equation a × b = b × a {\displaystyle a\times b=b\times a} . Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention,

13860-425: The operation of addition. The neutral element is 0 and the inverse element of any number a {\displaystyle a} is − a {\displaystyle -a} . The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements. Group theory examines the nature of groups, with basic theorems such as

13992-432: The operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations , such as rotations , under which the object remains unchanged . Its binary operation is function composition , which takes two transformations as input and has the transformation resulting from applying the first transformation followed by

14124-417: The others, it did this by teaching geometric transformations as a unifying approach between algebra and geometry. Regardless of all these influences and other projects, the SSMCIS study group considered its work unique in scope and breadth, and Fehr wrote that "nowhere [else] had a total 7–12 unified mathematics program been designed, produced, and tested." It was thus considered one of the more radical of

14256-407: The public domain for any organization to use. The pages of the books were formatted by typewriter, augmented by some mathematical symbols and inserted graphs, bound in paper, and published by Teachers College itself. A more polished hardcover version of Courses I through IV was put out in subsequent years by Addison-Wesley ; these were adaptations made by Fehr and others and targeted to students with

14388-471: The reform efforts lumped under the "New Math" label. Moreover, Fehr believed that the SSMCIS could not just improve students' thinking in mathematics, but in all subjects, by "develop[ing] the capacity of the human mind for the observation, selection, generalization, abstraction, and construction of models for use in the other disciplines." The course books put out by SSMCIS were titled Unified Modern Mathematics , and labeled as Course I through Course VI, with

14520-449: The relation between field theory and group theory, relying on the fundamental theorem of Galois theory . Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include magmas , semigroups , monoids , abelian groups , commutative rings , modules , lattices , vector spaces , algebras over a field , and associative and non-associative algebras . They differ from each other in regard to

14652-430: The same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set. For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This

14784-543: The same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing

14916-401: The second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship between

15048-442: The second as its output. Abstract algebra classifies algebraic structures based on the laws or axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is associative and has an identity element and inverse elements . An operation is associative if the order of several applications does not matter, i.e., if (

15180-606: The standard Regents Examinations due to a mismatch in curriculum. However, SSMCIS was one of the direct inspirations for the New York State Education Department , in the late 1970s and 1980s, adopting an integrated, three-year mathematics curriculum for all its students, combining algebra, geometry, and trigonometry with an increased emphasis in probability and statistics. Given the differences in subject matter and approach, how SSMCIS-taught students would perform on College Entrance Examination Board tests became

15312-459: The statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations . It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of

15444-424: The structures and patterns that underlie logical reasoning , exploring both the relevant mathematical structures themselves and their application to concrete problems of logic. It includes the study of Boolean algebra to describe propositional logic as well as the formulation and analysis of algebraic structures corresponding to more complex systems of logic . Mapping (mathematics) In mathematics ,

15576-410: The study of diverse types of algebraic operations and structures together with their underlying axioms , the laws they follow. Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed. Arithmetic

15708-485: The study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence of abstract algebra . This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra , vector algebra , and matrix algebra . Influential early developments in abstract algebra were made by

15840-406: The theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representation of a linear map: if one chooses a particular basis to describe the vectors being transformed, then the entries in the matrix give the results of applying

15972-452: The theory of dynamical systems , a map denotes an evolution function used to create discrete dynamical systems . A partial map is a partial function . Related terminology such as domain , codomain , injective , and continuous can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties. In category theory, "map"

16104-439: The two volumes in each year labeled as Part I and Part II. Materials for the next year's course were prepared each year, thus keeping up with the early adoption programs underway. Using largely formative evaluation methods for gaining teacher feedback, revised versions were put out after the first year's teaching experience. By 1973, the revised version of all six courses had been completed. The first three volumes were put into

16236-413: The types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural features by comparing two algebraic structures. A homomorphism

16368-510: The underlying set as inputs and map them to another object from this set as output. For example, the algebraic structure ⟨ N , + ⟩ {\displaystyle \langle \mathbb {N} ,+\rangle } has the natural numbers ( N {\displaystyle \mathbb {N} } ) as the underlying set and addition ( + {\displaystyle +} ) as its binary operation. The underlying set can contain mathematical objects other than numbers and

16500-400: The underlying sets and considers operations with more than two inputs, such as ternary operations . It provides a framework for investigating what structural features different algebraic structures have in common. One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that

16632-417: The world. As his tenure as president came to a close in 2016, Ilves visited his old school building with Cummings and said, "I owe everything to her. Because of what she taught us, my country now uses it." Cummings said that SSMCIS not only introduced beginning computer programming but also taught students "how to think". SSMCIS did represent a productive exercise in thinking about mathematics curriculum, and

16764-710: Was attended by junior high school teachers from seventeen states and one foreign country. By 1974, Fehr stated that 25,000 students were taking SSMCIS courses across the U.S. The Secondary School Mathematics Curriculum Improvement Study program did show some success in its educational purpose. A study of the Los Angeles program found that SSMCIS-taught students had a better attitude toward their program than did students using School Mathematics Study Group courses (another "New Math" initiative) or traditional courses. In New York State schools, special examinations were given to tenth and eleventh grade SSMCIS students in lieu of

16896-464: Was no decline in performance due to the unusual organization of material. Some 400 students were involved in this initial phase. Because the program was so different from standard U.S. mathematics curricula, it was quite difficult to students to enter after the first year; students did, however, sometimes drop out of it and return to standard courses. As teaching the program was a specialized activity, teachers tended to move along from each grade to

17028-466: Was no middle school then), called the program "Math X" for experimental, with individual courses called Math 8X, Math 9X, etc. Hunter College High School used it as the basis for its Extended Honors Program; the school's description stated that the program "includes many advanced topics and requires extensive preparation and a considerable commitment of time to the study of mathematics." Students were periodically given standardized tests to make sure there

17160-480: Was targeted at the junior high and high school level and the 15–20 percent best students in a grade. Funding for the initiative began with the U.S. Office of Education and covered the development of the first three courses produced; the last three courses produced, as well as teacher training, were funded by the National Science Foundation and by Teachers College itself. The scope and sequence of

17292-578: Was the typical physical environment. Starting in 1966, teachers from nine junior high and high schools, mostly in the New York metropolitan area , began getting training in the study program at Teachers College. Such training was crucial since few junior high or high school teachers knew all the material that would be introduced. They then returned to their schools and began teaching the experimental courses, two teachers per grade. For instance, Leonia High School , which incorporated grades 8–12 (since there

17424-403: Was translated into Latin as Liber Algebrae et Almucabola . The word entered the English language in the 16th century from Italian , Spanish , and medieval Latin . Initially, its meaning was restricted to the theory of equations , that is, to the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened to cover

#221778