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Note: The following article appeared in the August 2009 newsletter published by virtualhomeschoolgroup.com.
“ALGEBRA AND THE NEW WORLD” by John Rosasco
John Rosasco is a four-time Grammy nominated music composer, producer and pianist. John also has a M.A. in Mathematics from California State University Sacramento and graduate studies in mathematics at U.C. Berkeley. He teaches math classes privately through his website at mathandmusicstudio.com. His educational interests include geometric topology and working with gifted children.
“ALGEBRA AND THE NEW WORLD”
By John Rosasco
It is often said in recent times that we live in a new world of global competition. Many policy makers, educators and researchers, including the National Academy of Sciences believe that the lead in science and technology which the United States has maintained since World War II could be lost and not recovered in the near future. Many corporations are looking overseas to find talented individuals to fill their needs for innovation. Research studies into the dwindling U.S. lead in technology are focusing on the poor performance of American students on achievement tests compared to many other industrialized nations. The lack of skill in mathematics, and in particular algebra, seems to be at the center of the storm of controversy.
Most Americans see the direct correlation between a nation’s economic prosperity and its education systems, however, they are divided about how to improve our students’ math skills — a critical component in global competition. I believe that algebra is the key skill that American students need to acquire if we as a nation are to maintain our preeminent position in the world. The language of algebra has spread beyond science and technology and is currently used in business and the social sciences. Therefore, it is vital for our students to have a working knowledge of algebra in order to develop logical thinking, for they will eventually be making the decisions about the future of our country.
There is a call among math educators for simplification in the introductory algebra course. The National Council of Teachers of Mathematics and the Mathematical Association of America have said that there are too many topics in introductory algebra and that presenting a smaller number of core concepts would be a more effective way to learn. I agree with this approach and I think of introductory algebra as having two fundamental concepts which are outlined below.
I. Introductory algebra students must be able to factor quadratic polynomials. For example, if a student sees the quadratic x^2 + 8x + 15 they must see that 15 factors into 3 times 5, and 3 plus 5 equals 8, consequently, x^2 + 8x + 15 = (x + 3)(x + 5). This skill is preparation for calculus and higher mathematics, such as differential equations which is the language in which science is expressed. Factoring quadratics enable one to solve optimization problems in calculus by finding maximal and minimal values of functions. Additionally, the factoring of a quadratic is the key to finding the eigenvalues of a linear system of differential equations, and hence, the explicit solutions of the system.
II. The beginning algebra student needs to understand that the graph of the linear equation ax + by = c is a straight line and any straight line is the graph of a linear equation. Students should understand why the slope of the line is -a/b and that the x and y intercepts are c/a and c/b, respectively. The slope of a line is the fundamental idea of the derivative of a function in calculus.
High school algebra and geometry have been designed as preparation for calculus. The interplay between algebraic expressions and geometric objects starts in the introductory algebra course and continues through graduate level mathematics and research. Thus, it’s important for beginning algebra students to understand the link between algebra and geometry as soon as possible.
No discussion of introductory algebra would be complete without discussing the fundamental importance of mastering arithmetic. It is vital for arithmetic skills to be mastered early. In working with students I’ve seen what happens when high school students lack fluency in multiplication and division. A two minute calculation of an algebra II equation frequently takes ten minutes to solve because of the struggling student’s calculation skills. Slow arithmetic ability also goes hand in hand with lack of facility in manipulating algebraic expressions, and the further the unprepared students go with math courses, the harder the work becomes for them, until some eventually just give up. The use of calculators will not save them from their lack of arithmetic skills. By the time a student gets to calculus, he will use the calculator judiciously for complex calculations and do the rest in his head in order to save time. The student with poor arithmetic skills simply can’t compete at the calculus level.
In the final analysis, this key algebra skill that our students need to keep the U.S. at the forefront of science and technology will be acquired by pure hard work on their part. There is no substitute for it. We should encourage students in their study of this difficult subject. Sometimes students will tell me that their brains hurt after a grueling math session. I often praise them for their accomplishments and let them know how proud I am of their efforts. In some instances I’m able to have my students teach math to other students. The self confidence this peer-to-peer mentoring can bring to young students is truly amazing. And the more successful they feel, the better the students become at working with math.
copyright 2009 by John Rosasco
