MAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 31 - Algebra Curriculum Focus History of Algebra and Algebra Education Definition of Algebra NCTM Principles and Standards Nebraska Algebra Standards Typical United States Algebra Curriculum Discussion History of Algebra and Algebra Education Algebra has been studied for many centuries. It can be traced back to the Babylonians, ancient Chinese, and Egyptians. Algebra was in what is referred to as the “rhetorical” stage. Problems were solved by the use of words and prose. During the 3rd century, Diophantus of Alexandria (ca. 250) wrote the book Arithmetica. All but six of the original thirteen parts have been lost. The surviving parts show that, Diophantus interjected the use of symbols to represent unknowns into his work. This ushered algebra into the syncopated stage. In Arithmetica, however, Diophantus solved several practical problems and avoided any general procedures. During the 5th and 6th centuries, many Indian and Arabic mathematicians became very prominent. The first being Aryabhatta (ca. 475 – ca. 550), whose book of the same name, Aryabhatta, dealt with work on linear and quadratic equations, and whose book Brahmagupta presented the general solution to quadratic equations. Al-Khwarizmi (ca. 825) was a 9th century Arabic mathematician that composed the book al-Kitab al-muhtasur fi hasib al-jabr wa’l-muqabala (“Compendium on calculation by completion and balancing” or “Science of cancellation and transposition”). If you translate the translation of the title, you loosely get “Science of equations”. The book produced a systematic consideration of algebra separate from number theory. It also provided general solutions to several different types of quadratic equations. The Arabic word “al-jabr”, that means completion, is the origins of the modern word “algebra”. The 12th century witnessed the Persian mathematician, Omar Khayyam (ca. 1044 – ca. 1123), write a treatise on algebra. He followed Euclid’s (ca. 300 B.C.) axiomatic methods in its development and made the distinction between arithmetic and algebra. Also during the 12th century Al-Khwarizmi’s book was translated and made available to western mathematicians. During the next century, Leonardo Fibonacci provided many important contributions to algebra. Fibonacci paved the way for others such as Luca Pacioli (1445-1517), RobertMAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 32 - Recorde (1510-1558), Scipione del Ferro (c. 1465-1526), and Ludovico Ferrari (1522-1565). Francois Viète (1540-1603) is credited with introducing the symbols that are used in algebra. Algebra entered its last stage of development at this time; the symbolic stage. Carl Friedrich Gauss (1777-1855) proved the Fundamental Theorem of Algebra in his doctoral thesis. The Fundamental Theorem of Algebra was first proposed by Girard (1593 - 1632) in 1629. Fundamental Theorem of Algebra: Given a polynomial equation of degree n, there exists exactly n solutions to the equation. Modern day proofs of the theorem rely on complex analysis and the inclusion of complex roots. The idea of complex roots was introduced sometime around the time of René Descartes (1596 - 1650). It took some time for it to be fully accepted. René Descartes is considered the “Father of Analytical Geometry” and is honored with the rectangular coordinate system being named after him (the “Cartesian Plane”). Geometric concepts are now described by equations and symbolic expressions. The 19th and 20th centuries have seen algebra become much more abstract. Algebra no longer includes only the theory of equations. It also involves game theory and matrices. It is used as a common thread in calculus, discrete mathematics, probability and statistics, and most other fields of mathematics, as well as in any discipline that has a quantitative aspect to it. Algebra has been described as the language of science. The National Council of Teachers of Mathematics (NCTM) describes algebra as being “dynamic and a necessary vehicle for describing a changing world”. Algebra education in my opinion has experienced two major influences. The first being the School Mathematics Study Group project that resulted from panic in the United States after the Russian launching of Sputnik. Prior to that, algebra instruction was based on the prior work of the mathematicians previously mentioned. For example, in the early 1940s Clemson University was using a college algebra textbook written by Rosenbach and Whitman. In the front of the book, honor is paid to Robert Recorde (1510-1558) for his influence on algebra. The book covers twenty topics and from a strictly symbolic point of view. The topics are: I. Fundamental Operations II. Factoring and Fractions III. Exponents and Radicals IV. Functions and Their Graphs V. Equations and Their Solutions VI. Systems of Linear Equations VII. Quadratic Equations VIII. System of Equations Involving QuadraticsMAT 682 Topics in Mathematics (Fostering Algebraic Thinking) - 33 - IX. Ratio, Proportion, and Variation X. Progressions XI. Mathematical Induction and Binomial Theorem XII. Inequalities XIII. Complex Numbers XIV. Theory of Equations XV. Logarithms XVI. Interest and Annuities XVII. Permutations, Combinations, and Probability XVIII. Determinants XIX. Partial Fractions XX. Infinite Series Granted this is a college text, therefore some of the topics listed were not covered in a high school class and others were only briefly touched on in high school. The point is that these are the topics that the greats wrestled with and wrote about. The methods used in the text were more like a recipe than anything else. Laws of signs 1. To add two numbers of like signs, add their absolute values and prefix their common sign to the result. 2. To add two numbers of unlike signs, subtract the smaller absolute value from the larger and … No explanation provided! After Sputnik was launched, the U.S. worried that the Russians were academically ahead of us. To fight this, the School Mathematics Study Group (SMSG) was created under the leadership of Edward Begle (1914 - 1978) of Yale University. A group of university mathematics professors decided what topics should be covered in the kindergarten to twelfth grade mathematics curriculum. The purpose of the curriculum was to prepare students to go on to college and study mathematics,