For those of us who stand in awe of western scientific achievement George I. Sanchez’ book, Arithmetic in Maya, is a humbling reminder of the possibilities of men of other cultural backgrounds to produce equally amazing results. The case in point is the beautifully simple and efficient as well as rapid method of calculation the Mayan had developed.

Mayan scholars have long recognized and appreciated the existence of the science of calendrics among the Mayan people. But that they used their system of numbers in computing every day transactions was disputed, and no evidence has been found that they did. If they did, it is said, they would have had to memorize a multiplication table up to 19 x 19, their system being based on 20. Dr. Sanchez has discovered that the most complex problems can be added, subtracted, multiplied, or divided with only the memorization of the simplest principles required, and "that there are much simpler procedures that can be used for multiplying and dividing in Mayan than the add-subtract method."

The Mayan number system has only three symbols, the closed fist representing zero, a notable accomplishment in itself; a dot, the unit symbol; and a bar, representing five units. The symbols were apparently written vertically instead of horizontally, and Dr. Sanchez assumes for the purpose of his book that the symbols were read from the bottom upward, although he points out it would make no difference if they were read from the top down. On the bottom level the units are read as one; on the second level they are read as twenty; third level, four hundred; fourth level, eight thousand; etc. Keeping in mind that computations were made using perhaps stones and bars of some sort, the reader is taken through a series of multiplication and division problems and, using his own improvised stones and bars, as the author suggests, learns quickly to solve complicated problems with unusual speed. Dr. Sanchez goes on to show how, by the introduction of a conversion factor, calendric dates can be manipulated in the same manner.

The author has unlocked a mystery which has puzzled Mayan scholars and students until today. The system would also be fascinating to those interested in the language of numbers and a rich addition to the teaching program of those who teach number concepts even at an elementary level. The author goes on to suggest the possibility of using the system in a calculator. "Electric calculators, dependent on whirring gears and cumbersome levers, that add and subtract and change position by mechanical means, would seem by comparison slow indeed."

The book contains several reproductions of Mayan art which add to its intrinsic value as a glimpse into the life of these amazing people. Certainly the book is a brilliant example of the ingenuity of man in whatever culture or time he happens to find himself.

Bettye D. Meador

These two manuals are designed for the language improvement of students already familiar with English: those students from other countries who have been exposed to English as a second language and those in the United States for whom English is a second language.

Clifford H. Prator’s Manual of American English is of a unique type in that it is based on a study at UCLA of the language difficulties of several hundred students of various language backgrounds.

In the course of the research which extended over a three-year period, Prator found that certain non-phonemic "errors" were more serious barriers to intelligibility than the substitution of one phoneme for another. "Our own solution has been to regard unintelligibility not as the result of phonemic substitution but as the cumulative effect of many little departures from the phonetic norms of the language. A great many of these departures may be phonemic; many others are not. Under certain circumstances, any abnormality of speech can contribute to unintelligibility." He advocates identification of these "errors", analytical study of them, and systematic drill rather than an over-emphasis on the phonetic differences of languages which Prator has found the student masters very quickly. He also stresses the importance of rhythm and intonation to intelligibility. Included in the manual is an "Accent Inventory" designed to diagnose the particular problems of the individual student.

The body of the manual is based also on the findings of the study, those errors" which occurred more frequently being given more thorough treatment and drill. The manual is cyclical in its arrangement, first stressing by extensive drill, the breaking of former habits of speech and gradually taking the student back to non-phonemic-spelling recognition of the language as he will encounter it everyday.

The Gordon and Wong Manual is based on the thesis that the student must first learn to hear the distinctive sounds of American English and then must practice extensively himself. To facilitate his practice, there are drill exercises accompanying each section of the language as divided in the manual. The manual moves logically from the mastering of the International Phonetic Alphabet to the phonetic sounds of American English. There is a very useful section on the rhythm and intonation of English. The final two chapters deal in detail with problems in American English, one chapter considering vowel and dipthong problems and the last, consonant problems.

The Prator manual seems to this reviewer the more helpful of the two in that it goes right to the heart of the problem, those portions of an individual’s speech which make his English unintelligible. In this respect it becomes a valuable tool for the teacher to use creatively, adjusting the content of her course to the problems of individual students.

Bettye D. Meador