Box 30001, Dept. 3MB
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003-8001
Computer Science Department
University of Colorado
Boulder, CO 80309-0439
During the last 200 years, mathematics has changed radically. On the other hand, the content of mathematics taught in the elementary grades has remained virtually unchanged, causing a great discrepancy between everyday mathematics and school mathematics.
1. Mathematics education in the early years, 1800 - 2000.
Many mathematics books used for educational purposes began to appear in the United States around 1800 (Sterry, 1790; Pike, 1806; Adams, 1808; Daboll, 1812; Hawney, 1813). These books contained material that was used for vocational training in arithmetic in the eighteenth century, information useful for merchants, shopkeepers, craftsmen, and artisans.
The content of these books was based on several different concepts of numbers: whole numbers, common fractions taught as parts of a (fractured) whole, decimal fractions, and a variety of composite and denominate numbers. The main purposes were to teach skills in executing written algorithms, and to teach a variety of applications. The books were for teachers, and most learning took place during an apprenticeship, when a person was learning a profession.
The deductive aspects of mathematics were missing. Mental arithmetic, except for memorizing tables, was neglected. Neither finger computation nor the abacus was used in Western Europe and the Americas at that time. And the slide rule, which was commonly used by artisans and craftsmen, was rarely described in print, and it never became a computing tool in the elementary grades. Thus, arithmetic meant paper and pencil (or slate and chalk) execution of a few selected algorithms.
As a part of education in schools, arithmetic was removed from any direct applications, because students were not preparing for any specific profession. There was also a movement to teach arithmetic to children who were younger and not yet literate. In order to teach younger children, adjustments were needed. These came from the suggestions of a Swiss educator, Johann Pestalozzi (1746-1827); they were popularized in the United States by Warren Colburn (1793-1833) . They consisted of explaining the meaning of arithmetical operations in terms of the manipulation of physical objects, an increased role of counting, and stretching learning over several years, slowly progressing toward more and more complex examples (e.g., see Colburn, 1826). This led to the officially recognized "spiral way of teaching" that is prevalent today.
Compared to almost any other domain of learning, elementary arithmetic was dominated by mindless drill, and so there were many attempts to justify its existence. It seems that only for a rather brief period of time, during the development of a white collar working class around the end of the nineteenth century, were the skill of "ciphering", and of a good "writing hand", sufficient qualifications for a well paying job. The main justification for learning mathematics, which is still very common today, was that learning any kind of mathematics, even the most dull and dumb, develops a child's mental ability. Thus, even if you are bored and you do not like it, it is good for you!
An attempt to make arithmetic more interesting began early, and it has never stopped, even if its successes have been minimal. It has consisted of including interesting illustrations, activities, and games, which were supposed to make learning incidental and attractive. In the nineteen sixties a new trend developed. It attempted to make arithmetic entertaining, "It should be fun." Among other things, this led to a huge increase in the number of pages in textbooks, with no increase in mathematical content, but with the addition of a very large amount of irrelevant material. (For a recent analysis, see Meyer, Sims, & Tajika (1995).)
There have been several attempts to revise mathematical curricula. Most of them have been geared primarily toward the higher grades, and their influence on elementary school practices has not been great. The most drastic attempt is the current one, initiated by the National Council of Teachers of Mathematics (1989). It states that arithmetic skills should be de-emphasized, and instead children should learn "general problem solving methods" which are sufficient in a modern technological society. This approach is meeting with a growing opposition from "traditionalists" who want to go "back to basics". "Back to basics" really means the "methods and content of teaching from the nineteen thirties and forties " (Hirsch, 1993; 1996).
2. Changes in arithmetic between 1800 and 2000.
There are three radical changes that have modified our view of number systems, both from a theoretical and from a practical point of view. They have also changed the use of arithmetic in industry, business, science, and society in general. These are (a) the unification of arithmetic systems, (b) their algebraization, and (c) the automatization of arithmetic algorithms.
(a) Unification of arithmetic systems.
Different types of numbers are now encompassed by the umbrella concept of real numbers. The real numbers include negative numbers, surds, and transcendental numbers. These had been treated earlier as different "quantities" that were studied in algebra. The theory of real numbers provides a uniform and consistent basis for all present business, technical, and scientific computations. It also provides a basis for many other mathematical theories that have important applications, such as, for example, calculus, probability theory, and statistics. In some specialized applications (such as physics), the system of real numbers is further extended to the arithmetic of complex numbers and the arithmetic of matrices.
(b) Algebraization of arithmetic.
The introduction of modern algebra (e.g., Birkhoff and MacLane, 1979), and the axiomatization of many important domains of mathematics (Hilbert, 1956; Hilbert & Ackermann, 1946), broke the barrier between arithmetic and algebra. "In arithmetic we compute with numbers, but in algebra we compute with letters," is no longer true. Variables, which were called "letters" earlier, and which are still often called "unknowns", at present form an integral part of arithmetic. They play a crucial role both in the study of the system of real numbers and in the design and execution of numerical algorithms.
(c) Automating arithmetic algorithms.
During the second half of the twentieth century, arithmetical computations and, more generally, other data processing, became automated. To see the magnitude of this change, we should realize that a typical computer performs arithmetical operations 100,000,000 times faster than the most skillful human, and, in addition, it does not make errors and it doesn't get tired. This "revolution" in processing information has two sources. The first one is the modern insight into the structure of algorithms in general. A brand new mathematical theory, the theory of algorithms (Knuth (1973), Cormen, Leiserson, & Rivest (1990)), stands next to arithmetic, geometry, calculus, and statistics in practical importance. The second source is technology, which has produced cheap and convenient tools that allow the execution (carrying out) of algorithms.
This automatization has actually increased the role of arithmetic. There are many problems that can be solved by arithmetic methods, but that require thousands or millions of arithmetic operations. Such problems, which a hundred years ago were out of reach, are now easy to solve. Also the role of a person who does arithmetical calculations is different today. In earlier times, the main task was computing; but now, because calculations are automated, the main task is programming.
3. Everyday mathematics versus school mathematics.
Research results in mathematics usually enter everyday mathematics after a long delay. This is so because existing everyday mathematical practices have a strong inertia, and they are only slowly replaced by new ideas. There has been one exception: the automating of data processing reached the streets only in the nineteen fifties; but now it is difficult to find any domain of science, business, or even politics and society in general that is not strongly influenced by the influx of computers. Because automatization was based on the current (modern, unified) understanding of arithmetic, the modern view of arithmetic has also become the standard for everyday mathematics, used in banks, factories, businesses, engineering, etc. This change did not, of course, increase the mathematical knowledge of "the person on the street." Scanners, spread sheets, and commercial software, designed by experts, can process data without human help, and using this technology requires no more skill than an ability to follow instructions of a very simple nature.
On the other hand, elementary school mathematics is based on concepts that were being used 300 years ago. In the meantime, these concepts have changed, or they have been replaced by more adequate ones. During the last two hundred years pedagogical ideas about how to teach math have changed many times, and they were and are a source of many controversies. But we still teach whole numbers, fractions as parts of a whole, and decimal fractions, as three different topics. We confuse children with inconsistent statements. Negative and irrational numbers, together with variables, are still part of algebra (and not elementary school mathematics), and algorithms still mean paper and pencil execution of the four basic operations.
Both sides of the educational community that are involved in the controversy between the NCTM Standards and Core Knowledge (Hirsch, 1993, 1996) and other back to basics approaches, seem to be missing a central issue. Students require adequate knowledge in order to use the very powerful technology that is now available at minimal cost to anyone. Proponents of the Standards do not understand that it is very specific arithmetic knowledge, based on the modern arithmetic of real numbers and the modern concept of algorithms, and not on some vague "general problem solving skills". Proponents of Core Knowledge do not realize that practicing the obsolete technology of paper and pencil computations does not prepare you for the new computing technology any better than riding a horse teaches you how to fly a plane.
4. Final remark.
There are still a few years until the year 2000. No one expects more changes in the theoretical development of arithmetic. Technology will probably follow its course in the direction determined during the last few decades. Is there any hope that elementary mathematics education will wake up from its two hundred years of slumber?
Adams, Daniel. The Scholar's Arithmetic, or Federal Accountant. Fifth Edition. Keene, N.H: John Prentiss, 1808.
Birkhoff, Garrett, & MacLane, Saunders. Algebra (2d ed.). New York: Macmillan, 1979.
Colburn, Warren. Intellectual Arithmetic. Boston: W. J. Reynolds, 1826.
Cormen, T. H., Leiserson, R. L., & Rivest, R. R. Introduction to Algorithms. Cambridge, Mass. : MIT Press; New York : McGraw-Hill, 1990.
Daboll, Nathan. Daboll's Schoolmaster's Assistant. New-London, CT: Samuel Green, 1812.
Hawney's Complete Measurer: or, the Whole Art of Measuring. Fifth Edition. Corrected and Improved by T. Keith. Baltimore: F. Lucas, Jr. 1813 (?). (Wm. Hawney; original was about 1717.)
Hilbert, David. Grundlagen der Geometrie, Stuttgart, Teubner, 1956.
Hilbert, David, & Ackermann, W. Grundzuge der theoretischen Logik. New York : Dover Publications, 1946.
Hirsch, Jr., E.D. The Schools We Need & Why We Don't Have Them. New York: Doubleday, 1996.
Hirsch, Jr. E.D. (Ed.) What your Second Grader Needs to Know: Fundamentals of a Good Second-Grade Education. New York: Dell Publishing Co., 1993. (Also Copyright 1991 by Core Knowledge; Cultural Literacy Foundation).
Knuth, D. E. The art of computer programming (2d ed.) Reading, Mass., Addison-Wesley Pub. Co. 1973.
Mayer, R.E., Sims, V. & Tajika, H. A comparison of how textbooks teach mathematical problem solving in Japan and the United States. American Educational Research Journal, 32, 443-460, 1995.
National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics, Reston, VA: NCTM, 1989.
Pike, Nicholas. The New Complete System of Arithmetick, composed for the use of the citizens of the United States. Sixth Edition. Boston:
Sterry, Consider & John. The American Youth: Being a New and Complete Course of Introductory Mathematics: Designed for the Use of Private Students. Volume I. Providence: Bennett Wheeler, 1790.