“Math is math. I just don’t understand why my kid needs to know five different ways to solve a simple division problem!” Most parents attempting to help their children with math homework over the past decade have expressed this sentiment. In reply, education experts remind everyone that parents said something similar when “New Math” was introduced in response to the Space Race in the 1950s. Nowhere have the opinions of parents and education experts been more sharply divided than over Common Core math. The measured voice of the experts contrasts starkly with the common-sense exasperation of parents. From the perspective of parents, math does not change. From the perspective of educators, math is a subject that grows and develops like anything else, and it is important to keep up with progress in mathematics and technology.
While it is true that mathematics as a discipline does grow and change over time, it is not true that the goals of elementary mathematics education frequently change. Even if technology requires people trained in abstract mathematics and conceptual understanding, it does not follow that conceptual math should be taught as early as possible. Taking a wider, multi-millennial historical view of mathematics education may give us some perspective on the less-than-a-century-old math wars of the present.
The most helpful ancient and medieval distinction for our present predicament is the difference between calculation as a practical skill and mathematics as a liberal art. The greatest mathematicians throughout history, from Euclid to Newton and Leibniz, began their elementary education with calculation as a practical skill. They learned to count, add, subtract, and divide by rote, using the most efficient methods for mental math. In the Middle Ages, this distinction between calculation went by two different names: computus (or calculation) and the quadrivium (the mathematical arts). Those few who were lucky enough to learn higher-level mathematics did so through subjects rooted in the physical world: practical geometry, the study of music, and the study of astronomy. The study of arithmetic, although sometimes a term used informally to mean basic calculation, more often meant the study of number theory or the abstract study that we call “pure mathematics.” In other words, the ancient and medieval world studied mathematics in the following order: practical math (calculation), scientific math (applied math), and then abstract mathematics. While technology evolves, the human mind—and the way it functions—has not. For several thousand years, the best abstract mathematicians were trained first in concrete and practical mathematics.
Early American mathematics continued this tradition of beginning with calculation before proceeding to abstract calculation through a series of textbooks known as Ray’s Mathematics. This widely used five-part series illustrates the progression from the simple, concrete, and practical to the complex, specialized, and abstract. The textbooks for young learners focused upon number recognition, learning basic math facts (addition/subtraction/multiplication/division), and applying these math facts to practical word problems. The goal was quick mental calculation. Ray’s second book, Intellectual Arithmetic by Induction and Analysis, reviewed basic mathematical calculations before progressing to fractions and their practical applications in measuring time, money, length, and volume. The third book, Practical Arithmetic, introduced different systems of numeric notation and continued to more complicated addition, subtraction, division, fractions, and percentages before teaching students to apply this knowledge to very practical topics like calculating interest, exchange rates, insurance, and even taxes. The fourth book, Higher Mathematics, introduced students to mathematics as a science of quantity. It not only introduced students to more complex problems of arithmetic and geometry, but it also taught more abstract concepts: the nature of number, the application of principles, and the understanding of why problems ought to be solved in a particular way. Ray’s series ended with a fifth textbook in algebra designed specifically for high schools and colleges.
Not every student in nineteenth-century America progressed to high school or college, but the structure of mathematical education ensured that all students were prepared for practical affairs, while those continuing their studies were equipped for advanced mathematics. It should be noted that this early curriculum did not ignore the abstract understanding of number. Rather, it delayed the full explanation of abstract concepts until a student had sufficient experience to appreciate it. These textbooks began with what students and parents recognized as obviously practical, but eventually led to the kind of mathematics that would have practical applications in science.
When and how did the math wars begin? In my next article, I will address the start of the “math wars,” otherwise known as “reforms,” and what parents and educators can do about it.
