year course, and was quite popular when it first appeared. In this series, the student gets very limited experience in actually making measurements. There is little development of an understanding or appreciation of measurement and its use. The approach to measurement is to learn to read the correct unit from a ruler, scale or cup; to learn the formulas and tables for weights and measures, and to work conversion problems. Book I has a lesson on liquid measurement in customary units. The students are taught the relationships-"2 cups measure the same as 1 pint," etc. The lesson on weight, in customary units, consists of exercises in reading a scale printed in the text; students do not weigh anything themselves. There are four lessons on linear measurement, all in customary units. Students are shown how to make an inch ruler, and how to read from a picture of a ruler, and how to use it(!) to measure objects and line segments pictured below it. There is no "hands-on" work. 999 Book II devotes about a week's time to measurement: The lessons are of the same type; that is, in the lesson on liquid measure, the emphasis is on learning the words for the units of capacity, and the rules for the relationships between them-"write the word cup on the board. . . write the sentence '1 pint measures the same as 2 cups.' The lesson on weight now has the students read a pictured scale to the nearest ounce, and read a Fahrenheit thermometer in the lesson on temperature. Aithough the base-10 Hindu-Arabic system appears in Book I, it is not related to decimals until Book IV, when decimal fractions are introduced in a 2- or 3-week unit. A knowledge of fractional numbers is the basis for understanding decimals, and the text reminds us that "decimals help in keeping track of money." Book III has a 2-week unit on measurement. The metric system is not introduced; however, we do get "measures of long ago," the cubit, pace, and the ell. The absence of manipulative experience continues. A metric unit appears for the first time as a supplementary topic in the last chapter of Book IV. In the two lessons on linear measure, students learn that a meter is "3 inches longer than a yard"; meters and yards, centimeters and inches are compared and there are exercises in converting from one to the other. In Book V, the metric system makes a full-fledged appearance in the main body of the text in a chapter on measurement. There is a lesson on the metric system of linear measures - the meter, decimeter and centimeter. The metric system is introduced to the teachers as "the system used extensively in science." Students don't see a metric ruler or make any metric measurements, but merely compute conversions. In Book V, 10 weeks are spent on fractions and 5 weeks on decimals. In Book VI, students at last read that "most countries of the world use a different system of measurement, the metric system." The teacher's edition states that the United Kingdom is now shifting to the metric system over a 9year period. There are five lessons on the metric system. The units for linear measure are given, together with the relationships between them and a comparison to ours. Again, no measurements are made with a metric ruler. There are lessons in adding and subtracting metric measures: 5m7dm+2m9dm=? and multiplying and dividing them and regrouping them to the greater unit of measure, as if they were used like inches and feet. The gram is introduced, and there are exercises adding kilograms and grams; answers are not in decimal form, but in "so many kg plus so many g." There are a few problems in finding area and volume in both customary and metric units; however, the operation involved is numerical (multiplication), not measurement. The general treatment of measurement is as an arithmetic skill, not as a measurement skill at all. There are no metric units in Books VII or VIII. Elementary School Mathematics and its sequelae1 is a widely used elementary mathematics sequence. This series consists of students' texts, teachers' editions, students' workbooks, and teachers' workbooks for grades K-6. The series stresses concepts rather than the mere mastery of skills, and offers enrichment exercises for "more advanced children." Centimeters are introduced in the 1st grade, and the series attempts to deal throughout with metric, customary and arbitrary units on an equal basis. The emphasis is on learning to measure with whatever unit is given. On the other hand, most computation and story problems are posed in customary units of feet, miles and pounds. This logically follows from the series' goal of relating numerical, arithmetic work to the child's practical experiences in the real world. Liquid measure is customary throughout. Conversion exercises from metric to customary units for weight and linear measure are included only for enrichment. Half the units in Book I have lessons on coins, money problems and priced objects. Children handle real or play coins in many classroom activities, and manipulate money earned and spent in real life situations; the concepts of place value, addition, and subtraction are emphasized. The last unit is "Money, Fractions and Measurement." This 2-week unit includes two lessons on linear measurement and one lesson on liquid measurement. It introduces the idea of linear measure by using both inches and centimeters as units of measure. The centimeter scale was chosen for two reasons: (1) to give the children the feeling that there is a certain arbitrariness concerning the choice of scale in linear measure; (2) to provide an early introduction to the centimeter as an important unit of measure. Students may cut out the centimeter-inch ruler from the book to measure various pictured objects. The ruler is color-keyed, centimeters in red, inches in black, and determination of which scale to use is made according to the color of the printed ink in each exercise. Gradually, the terms "centimeter" and "inch" are introduced. The lesson on liquid measure introduces the customary cups, quarts, and pints. Book II also gives bilingual treatment to linear measure for two lessons, and customary units for liquid measure in one lesson. 4 Elementary School Mathematics, 2d Edition, Addison-Wesley Publishing Co., grades K-6 (c. 1968). School Mathematics I and School Mathematics II for grades 7 and 8, Addison-Wesley Book III has a 3-week unit on measurement. Introductory lessons use the line segment, the square and the cube as common measuring units for length, area, and volume. This is followed by a lesson on using the appropriate unit (inch, foot, yard or mile) and then a lesson on measuring with inch and centimeter rulers. If centimeter rulers are not available, the teacher's guide suggests the children make their own, and directions are given. A "Follow Up" activity for the "more capable children" is to explore the metric system. Two lessons on estimation use primarily customary units. In two lessons on area, the square centimeter is favored for counting and estimating, possibly because it is a handy size and more exercises can be fit on a page.5 Cubic inches, cubic centimeters and cubic arbitrary units are all used in the lesson on volume. Customary units are used for liquid measure. Book IV has a 2-week unit on measurement, which reinforces the concepts of Book III. There is a "buried treasure map," for which the scale is given as "each centimeter means one mile." Again, area and volume are treated in centimeters, inches, and arbitrary units; perimeter is given metric treatment. The conversion exercises at the end of the chapter deal with finding the greatest number of the larger (customary) unit. A 1-month unit on decimals appears in Book VI, and money is briefly related to decimal notation there. The metric system of measurement appears again as "enrichment." A table of comparative lengths in metric and customary units is pictured, and the children may study the table and then find the missing conversions. Further on, we find a "possible activity" - to interest "the more capable children in extending the metric system to weight, since this system is used for most scientific work." A table of metric weights and customary equivalents is shown, and students may complete the missing conversions. Another "possible activity" for "some space-minded children" is to convert space orbital data from miles to kilometers, and a table with conversion problems is shown. School Mathematics I, for the 7th grade, and School Mathematics II, for the 8th grade, again use customary units, many nondimensioned units, and a sprinkling of metric units (centimeters) in the geometry sections. The rest of the text uses either nondimensioned units or the customary units for word problems. Our observations on these sequences of elementary mathematics textbooks are consonant with the conclusion of a major publisher who made a metric study early in 1968: in mathematics we find very little evidence of . . . systematic development of an understanding and use of the metric system at any level. There are separate units on the metric system and some exercises on converting from one system to the other, but no evidence of continuous development of a facility to use and work with the metric system.6 5 A square inch is rather gross but a linear inch is "just right." 6 D. C. Heath Division of Raytheon Corporation, memo of 3 June 1968 from H. R. Mutzfeld The relatively low priority of the study of the metric system in elementary schools is indicated in the following: Measurement units are taught and used at every elementary grade level at least up through grade 8. Measurement is one of the major strands of elementary school mathematics. At the present time, the metric system does appear briefly in all arithmetic series. The percent of time spent on it as compared to the total time spent on measurement would be very low, not over 20 percent. When the metric system, especially linear measures, is introduced, it is introduced as a new system used by much of the world. Good teachers do use visual aids, and many elementary children become familiar with measuring to the nearest centimeter, for example. Pupils also encounter the need for metric units in their science classes. It is the great need in junior high school science that has led to much of the expansion of work on the metric system in arithmetic books. Other reasons for teaching the metric system include its connection with the decimal system pupils are learning, the cultural value, and the fact that it has been a popular enrichment topic for some time." (emphasis added) It is not hard to understand the experience of the Intermediate Science Curriculum Study (ISCS) Group. They pretested students early in the 7th grade on their facility with metric units to see if they need remedial work in this area before entering seriously upon the work of that curriculum: Our experience using this method (a test) of introducing measuring lengths with metric units suggests that many elementary science and mathematics courses are not getting students to handle the metric system efficiently. More than half of the 7th graders who took the test on using a meter stick that occurred early in our 7th grade materials were not successful with it.8 One may optimistically conclude from this statement that nearly half the students were able to make a metric measurement - we choose to interpret otherwise and to say that many 7th graders cannot make the simplest measurement in a decimal system, and that this betrays the failure of elementary mathematics to teach them either decimal concepts or measurement or the use of metric units. The deficiencies in teaching measurement and related materials which we have uncovered here, and the need to rearrange the elementary mathematics curriculum so as to teach decimal fractions earlier, etc., imply that substantial curriculum changes must accompany metric conversion. These would be comparable to the mathematics curriculum changes of the early 1960's, the so-called "new math," and ought to flow from the agreement of authoritative mathematics educators.9 7 Charles R. Hucka, Associate Executive Secretary, National Council of Teachers of Mathematics, in a paper presented at the Education Conference, p. 2. "Experience of the ISCS Curriculum Group with the metric system," a report presented at the Education Conference, Burkman and Redfield, p. 4. "These changes are needed whether we have a planned metric conversion or not, and together with other curriculum changes they are being considered by a continuing study group Recommendation: In case a national program of planned metric conversion is adopted, authoritative recommendations for curriculum change in elementary mathematics should be endorsed by national organizations of teachers and educational leaders, and used as the basis for the curriculum embodied in the new metric textbooks. The substitution of metric for English measure, or the suppression of the latter, will simply not suffice. We must not fall into the trap of making "mechanical" conversions of mathematics textbook sequences which do not come to grips with the question of curriculum revision, lest we lose the opportunity that we have at this time for substantial curriculum reform in measurement and related areas. We have obtained an authoritative and detailed recommendation from Professor S. Sternberg of Harvard University, which appears in appendix III. This recommendation parallels, and reinforces and amplifies, the recommendation developed by a metric study committee of a major educational publisher, 10 Secondary School and College Mathematics the college mathematics teacher assumes that his students are fluent in the use of any system of units which he discusses. In fact, he will bring a system of units into his discussion only when he has been assured that his students have such a fluency. Often, perhaps most often, his students are more fluent in the use of this system of units than he is. He is a follower, rather than a leader in this respect. It should be recognized that what I have said about college mathematics teaching is not true of secondary school mathematics teaching. In high school, the mathematics teacher cooperates significantly (emphasis added) in teaching the use of various systems of units for measurement. There are many reasons why this should be part of the mathematics curriculum. Learning how to use various systems of units for measurements fits very well with the building of arithmetic skills, and the mathematics curriculum provides a natural place to introduce some of these units some years before the student begins significant study of science.11 This task of teaching measurement in high school mathematics is apparently not accepted by textbook writers: the algebra, trigonometry, analytic geometry and elementary calculus texts examined by this Study were found to have very little in the way of measurement units, except that they appear in various problems set in concrete terms - problems which students call "word problems." Since it is generally conceded that anything not in the textbook is not likely to be taught, it seems likely that very little measurement is taught in high school mathematics. The high school mathematics 10 D. C. Heath Division of Raytheon Corporation, memo of 3 June 1968 from H. R. Mutzfeld to F. S. Fox. 11 Statement by Alfred B. Willcox, Executive Director for the Mathematical Association of |