Algebra and algebra education

Summary: Algebra and algebra education have undergone many radical changes and remain highly adaptable mathematical disciplines with many real-world applications.

When they hear the term “algebra,” many people may think only of solving an equation for an unknown variable x. In reality, algebra is a broad mathematical discipline that includes a range of theories and methods and which has no single agreed-upon definition. Even young children may engage in algebraic reasoning, such as understanding the relationships between quantities or manipulating symbols, without referring to it by name. For much of human history, computations were likely performed using a variety of words and symbols to meet needs such as accounting, taxation, and planting. There is evidence of algebraic problem solving in Egypt and Babylonia. Their techniques appear to have relied a great deal on spoken rhetoric rather than symbol manipulation, though the Babylonians solved quadratic equations using methods similar to those taught in the twenty-first century.

Algebraic thinking is also found in works from ancient China. Greeks, Hindus, Arabs, Persians, and Europeans all made advances and contributions to algebra, and the term is derived from an Arabic word. In the nineteenth century, mathematicians began to expand the notions of algebraic form and structure to encompass more types of mathematical objects such as vectors and matrices as well as operations that could be carried out upon these objects. Also, algebra was not constrained to the ordinary systems of numbers, and noncommutative algebras emerged. The discipline of abstract or modern algebra has grown even further to encompass concepts like groups, rings, and fields.

Concurrently, algebra has become increasingly more important in education at all levels. One of the perceived advantages of algebra and algebraic thinking is that problem solving can be accomplished by symbolic manipulation rules without constant reference to meaning, and these generalized problem-solving skills are viewed as advantageous for students in a wide range of life and occupations skills. This notion has led to a somewhat controversial “algebra for all” approach in many K–12 educational systems in which all students must take an algebra course before graduating, and basic algebraic concepts are introduced as early as the primary grades.

Early History

Even in the classical and ancient period, people had started to use numerals such as 1, 2, and 3 (or I, II, and III, for example) to represent quantities. Numerals, however, bore a direct relation to the quantity being counted. The numeral 1, for instance, only ever referred to a quantity of one. In ancient Egypt, some mathematicians had started to use other symbols, called ahau, to represent unknown quantities. These symbols are called “variables” in the twenty-first century because the quantity or number they represented could vary. But the variation in quantity that the Egyptians allowed for was much more restricted than what is allowed for in modern algebra. For example, the symbol x can refer to any number (whole, integer, or other) depending on the mathematical context in which it is used.

Thus, while ancient Egyptians and mathematicians in other ancient civilizations may have used symbols to represent quantities, they did not use symbols in the generalized way in which they are used today. In fact, it was only in the third century c.e. that a Greek mathematician, Diophantus of Alexandria, first used letters of the alphabet to stand in for numbers. It is because of Diophantus’s works that mathematicians started to express “an unknown quantity” using symbols such as x and y rather than written words.

Al-Khowarizmi

Diophantus’s symbolical technique was not widespread, however. In fact, the term “algebra” actually stems from a period much later than that of Diophantus. It comes from the work of the eighth-century Muslim scholar Muhammad Ibn Musa Al-Khowarizmi (there are various spellings of his name). Al-Khowarizmi worked as a scholar and intellectual during the reign of the Caliph al-Ma’mun (r. 813–833 c.e.).

Al-Khowarizmi was a prominent member of the Bayt al-Hikma, the “House of Wisdom,” which the Caliph had created as an academy and library to promote science. Al-Khowarizmi’s book, Al-Kitab al-mukhtasar fi hisab al-jabr w’al-muqabalah (an abridged book on the operations of al-jabr and al-muqabalah) is the oldest surviving Arabic book on mathematics. Al-Khowarizmi was also one of the first algebra teachers, as he taught algebra within the Bayt al-Hikma as a subject on its own. Although the ancient Egyptians and Babylonians did produce texts on arithmetic, algebraic, and geometric problems as early as 2000 b.c.e., Al-Khowarizmi was among the first to teach algebra as a science on its own rather than as a subbranch of other branches of mathematics.

The word al-jabr, from which the modern-day term “algebra” is derived, first appeared in the title of Al-Khowarizmi’s book. Some historians have interpreted it to mean “the restoration of a broken bone” or, in mathematical terms, “the removal of the negative quantity from the equation,” while the word al-muqabalah has often been interpreted to mean the removal of positive quantities. Gandz has contested these interpretations, however, to argue that a better translation of al-jabr is simply “the science of equations.”

Europeans

Europeans first became acquainted with Al-Khowarizmi’s works through Latin translations by Gerhard of Cremona (1114–1187) and Robert of Chester (c. 1150), both of which first appeared in the twelfth century. Historians have often accredited these Latin translations of Arabic mathematics with the origins of European algebra. One of the first European treatises on algebra to emerge in the Renaissance period was written by the Italian mathematician and friar Luca Pacioli in 1494. Other Italians worked on varied algebraic problems in subsequent years, including Scipione del Ferro (1465–1526), who was able to derive the solution to a cubic equation in the early sixteenth century. The Italian mathematician, Niccolò Tartaglia (1499–1557), derived a general solution to cubic equations a few years later.

In the same century, the French mathematician René Descartes (1596–1650) began to combine algebra (and algebraic rules) with geometry. Descartes was the first to apply algebra to the study of geometric curves. In 1637, he published a work in which he represented curves by means of algebraic equations. Descartes’ innovation was to study curves in their algebraic form rather than in their geometric form. The result was a field of mathematics known as “analytic geometry” (also called “geometric analysis”) according to some eighteenth and nineteenth practitioners. Analytic geometry allowed mathematicians to use symbols, along with the rules that govern the combination and interaction of symbols, to solve problems related to the motion of bodies in space and the behavior of geometrical objects, such as circles, parabolas, and hyperbolas.

Solving Equations

Algebra could therefore be used to find solutions to linear equations such as ax + by = 0, which describe lines in space; quadratic equations, such as ax² + bx + c = y, which describe parabolas in space; cubic equations, such as ax³ + bx² + cx + d = 0, which describe cubic relations in space; and other higher-order equations, such as a_nx_n + a_n-1x^n-1 + … + a_o, which describe various curves. The upshot of the Cartesian use of algebra in geometry was that algebraic manipulations could be used to also solve “systems of equations,” such as

ax + by = c

dx + by = f.

Another outcome of the rise of analytic geometry was the development of the calculus in the seventeenth century. However, although calculus uses the tools of algebra—including symbolic representation and algebraic manipulation—to compute its solutions, it is not the same as algebra. Algebra is generally understood to include only those expressions that possess a finite number of terms and factors. This means that the computation of solutions to algebraic equations terminates after a certain number of steps. In calculus, on the other hand, the concept of a “limit” means that the process of differentiation can be repeated ad nauseam and therefore never terminate.

Modern Period

Over the course of the past 1000 years, algebra has thus expanded from a basic use of symbols in simple numerical reasoning to the analysis of structures called algebraic “fields” and “groups” in the nineteenth and twentieth centuries. In fact, the “modern” period in algebra is typically understood as having begun in the early nineteenth century with the work of mathematicians such as the French mathematicians Joseph Louis Lagrange (1736–1813) and Évariste Galois (1811–1832), as well as the Norwegian mathematician Niels Henrik Abel (1802–1829). Galois, for instance, worked on the concept of an algebraic field. Though Galois died prematurely young at the age of 20 (as the result of a duel in which he was shot), his work later culminated in what is called Galois Theory.

Another important change in the field of algebra occurred in the mid-nineteenth century with the algebraic-geometric work of the Irish natural philosopher Sir William Rowan Hamilton (1805–1865). Hamilton started to work on couples (number-pairs that can be represented as (x, y) on a Cartesian graph) to understand the algebra that could be used to describe their behavior.

In trying to extend the algebra of couples to the algebra of triplets (numbers that could be represented by the point (x, y, z) on a three-dimensional axis system), Hamilton generated an interesting mathematical operator known as the “quaternion.” The quaternion can be represented as w + xi + yj + zk, where w, x, y, and z represent real numbers, and i, j, and k represent imaginary numbers. To get his quaternion algebra to work, however, Hamilton had to manipulate the standard rules of algebra as they were conceived of at the time. While it is the case in normal arithmetic that 1 × 2 = 2 × 1, such that the order of the numbers does not affect the outcome of the operation of multiplication, Hamilton’s quaternions did not follow this rule. Hamilton found that when numbers are represented as directed lines in space (called “vectors”), the order in which the numbers are multiplied with one another does matter. In Hamilton’s algebra, therefore, 1×2=×1. Rather, 1×2=-(2×1).

Hamilton is often seen as a pioneer in the study of algebras. Based on his work in quaternion algebra, other mathematicians developed the idea that by changing the rules of the game—by playing around with the standard rules of algebra and arithmetic, such as the commutative principle in multiplication—one could generate new algebraic systems in which the component parts—the variables being manipulated and the objects they represent—do not necessarily follow the same rules as normal algebra.

Another mathematician who developed a similarly new algebraic system was Hermann Grassmann (1809–1877). However, Grassmann’s works were largely unknown across Europe until the mid-nineteenth century, by which point Hamilton had already published his major works on quaternions. A British mathematician who attempted to extend Hamilton and Grassmann’s new algebraic systems to n-dimensional space was William Kingdon Clifford (1845–1879). Clifford died young, and, as a result, it took many years for his bi-quaternion algebraic operator to become widely known, understood, or used.

Fermat’s Theorem

The history of algebra is therefore replete with breakthroughs. In the seventeenth century, a French mathematician, Pierre de Fermat, worked on a problem in number theory that he had picked up while studying the works of Diophantus. Fermat was interested in studying the Pythagorean numbers. Pythagorean numbers are sets of three numbers, such as a, b, and c, which satisfy the equation a² + b² = c². Students often learn about Pythagorean numbers through the Pythagorean theorem, which describes the length of sides in right-angle triangles in geometry. Fermat, however, was not interested in triangles so much as he was interested in the consequences of slight manipulations to the Pythagorean theorem.

He attempted to determine the consequence of manipulating the exponents in the Pythagorean numbers from 2 to n. In so doing he wrote, “I have discovered a truly remarkable proof” Fermat explained that when the Pythagorean theorem is made to read aⁿ + bⁿ = cⁿ, the new equation has no integer solutions for any value of n greater than 2. In other words, it is impossible to find numbers a, b, and c that satisfy the equation a5 + b5 = c5. Fermat never offered a full proof of this claim and mathematicians ever since have struggled to generate it. This bit of algebra is still called a “theorem” to indicate that, although it is believed to be true, one cannot be sure that it actually holds true for all integer values of n.

Mathematicians who have tried to prove Fermat’s Theorem over the years have been led to develop other branches of algebra along the way. One example is Eduard Kummer (1810–1893), who created the concept of “ideals” in algebra. The theory of ideals remains an important tool in algebraic systems. An “ideal” A is a (nonempty) subset of a ring R whenever the sum of two elements of A is an element of A as well. In addition, if aa is any element of the subset A, and r is any element of the ring R, the products ar and ra are both in the subset A. An example of this is the integer 3. All of the multiples of 3 form an “ideal” in the ring of the set of integers.

Later Developments

By the late nineteenth century, mathematicians in Europe, Great Britain, and the United States also became interested in studying the structure of certain algebraic equations. Rather than concerning themselves with particular solutions to individual equations, these mathematicians wanted to identify the axioms (or laws) that governed the behavior of differing algebraic equations. These mathematicians focused on the structure of algebraic systems, where a system consists of a set of elements and a set of operations that abide by certain axioms (or rules). The simplest example of an algebraic structure is called a “group.” The French mathematician Camille Jordan (1833–1922), the German mathematician Felix Klein (1849–1925), and the Norwegian Sophus Lie (1842–1899) studied groups and did much to establish this area of algebraic research, although older mathematicians such as the eighteenth-century German mathematician Leonhard Euler and the nineteenth-century German mathematician Carl F. Gauss (1777–1855) had already developed some foundational notions that related to abstract groups. Groups are a fundamentally nineteenth-century idea. By the mid-twentieth century, the notion of a group had become widely accepted and had even come to form the core of abstract algebra.

Throughout the nineteenth and twentieth centuries, mathematicians who worked on various aspects of algebraic structure included people such as Benjamin Pierce, Eduard Study, Karl Weierstrass, Richard Dedekind, Theodor Molien, Élie Cartan, Emil Artin, and the twentieth-century female mathematician Emmy Noether (and the entire “school” of mathematicians that she fostered). Some of the “groups” that they helped to define, use, and develop include semigroups, loops, rings, integral domains, fields, lattices, modules, Boolean algebras, and linear algebras, among others.

In the twentieth and twenty-first centuries, abstract algebra has come to include a wide variety of subject topics, including negative and complex numbers, proportions, theory of exponents, finite arithmetic progression, geometric progression, mathematical induction, the binomial theorem, permutations and combinations, the theory of equations, partial fractions, inequalities, and determinants.

Algebra Instruction

Algebra developed because of the need to solve real-life questions and as an extension of mathematical investigations, but in the eighteenth century, mathematicians such as Colin Maclaurin and Euler thought of algebra as a universal arithmetic, and education focused on solving equations for unknown quantities by symbol assignment and manipulation. The focus on symbol manipulation and transformational activities such as collecting like terms, factoring, and simplifying equations continued in school algebra until the mid-1960s, when educators experimented with ways to make algebra more meaningful to students. By the early 1990s, generational activities that included algebra as a way to describe numerical or geometric patterns replaced transformational activities in some countries. Teachers also investigated the effectiveness of a wide variety of teaching strategies such as computer algebra software, historical perspectives, or active learning methodologies, and there were also many algebra survival books marketed such as Hot X: Algebra Exposed by actress Danica McKellar, who majored in mathematics.

Teachers continue to experiment with ways to help students understand algebraic equations and models as well as the process of manipulating them. There is also a long history of debate about when to begin teaching algebra. Before 1700, algebra was not routinely part of the U.S. curriculum at any level of schooling, though evidence suggests it was taught in some places, such as Harvard University, in the early part of the 1700s. By 1820, Harvard required algebra for admission, and several other Ivy League schools adopted this standard over the next three decades. Massachusetts also passed a law in 1827 requiring algebra to be taught in many high schools. As early as the first part of the twentieth century, some educators such as Claude Turner suggested that algebra should be taught in eighth grade to help students understand concepts like cube roots. Some educators pointed to developmental theories such as Jean Piaget’s theory of cognitive development in order to resist teaching algebra any earlier than eighth grade. In the twenty-first century, many states in the United States have adopted an “algebra for everyone” approach to teaching, and several states require students to pass an algebra test to graduate from high school. This emphasis is due in part to the increased focus on problem-solving skills believed to develop a wide range of life and occupational skills.

Bibliography

Barton, Bill, and Victor Katz. “Stages in the History of Algebra With Implications for Teaching.” Educational Studies in Mathematics 66, no. 2 (2007).

Cooke, Roger. Classical Algebra: Its Nature, Origins, and Uses. Hoboken, NJ: Wiley-Interscience, 2008.

Derbyshire, John. Unknown Quantity: A Real and Imaginary History of Algebra. New York: Plume, 2007.

Greenes, Carole, and Rheta Rubenstein. Algebra and Algebraic Thinking in School Mathematics: 70th Yearbook. Reston, VA: National Council of Teachers of Mathematics, 2008.

Stacey, Kaye, Helen Chick, and Margaret Kendal. The Future of the Teaching and Learning of Algebra. New York: Springer, 2004.

Varadarajan, V. S. Algebra in Ancient and Modern Times. Providence, RI: American Mathematical Society, 1998.