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Schooling Inputs to Science and Mathematics Education: Teachers and Curriculum Content Ideally--for clarity and efficiency--each indicator would be represented by a single measure that could be applied to elementary and secondary schools and at each jurisdictional level, the local district, the state, and the national level. The organization of American schools precludes this ideal. For example, differences in specialization of teachers at different grade levels argues for at least two measures regarding teachers, on for elementary schools and one for secondary schools. Because of the reality of the U.S. school system, the committee has not combined various measures into a single indicator for each area to be monitored. When possible, a single best indicator is suggested for each appropriate level of disaggregation. The accompanying discussion deals with the problems attached to the measures asso- ciated with each indicator and gives suggestions for future improvements. In addition, the best current values, based on available data, are given for each measure so as to portray the present situation. e TEACHERS Much of the concern regarding the condition of mathematics and science education has been about the supply of teachers who are qualified to teach mathematics and science courses in grades 9 through 12. A number of surveys have been conducted to assess the extent of the shortage; all of them have been based on the opinions of various education authorities, extrapolating from their perception of current conditions. In 1980, 1981, and 1982, Howe and Gerlovich (1982) surveyed state science supervisors and teacher certification directors on their 44
45 opinion as to supply and demand for secondary school science and mathematics teachers. Their survey covered 53 jurisdictions: the 50 states, the District of Columbia, Puerto Rico, and American Samoa. They used a 5-point rating scale: 1, surplus; 2, slight surplus; 3, adequacy; 4, shortage; 5, critical shortage. In 1982, 44 of the 47 state authorities responding reported that they saw shortages or critical shortages of mathematics teachers, 45 of 50 saw shortages in physics, and 44 of 50 saw shortages in chemistry. officers - ~ decline between 1971 and 1980 of 79 percent of persons A survey of teacher placement (shymans~y and Aldridge, 1982) indicated a who were pursuing teaching degrees in mathematics and a decline of 64 percent of those pursuing teaching degrees in science. (Smaller decreases of 64 percent and 33 percent, respectively, were found by NCES (1983) in an analysis of bachelor's degrees; see Table 3.) A third kind of survey (Shymansky and Aldridge, 1982), of secon- dary school administrators, revealed that half the science and mathematics teachers newly employed for the 1981-1982 school year were hired on an "emergency" basis, that is, without state certification. The results of these surveys have been instrumental in drawing public attention to the issue of adequate supply and preparation of teachers in science and mathematics. Numerous initiatives at the national, state, and local levels have been directed toward providing both greater numbers and also better trained teachers for high schools. By fall 1983, 17 states had enacted undergraduate scholar- ship or loan programs, many of them targeted toward training teachers of science and mathematics (Flakus- Mosqueda, 1983). A number of states are focusing on the retraining of college graduates not now teaching or teaching other subjects. = ~ ~ ~ ~ ~ . ~ _ _ ~ _ A third approach has been to make teaching more attractive through incentive pay ana career ladders. Indeed, according to the Gallup Poll (Gallup, 1983), 50 percent of the people favor differ- entially higher pay for mathematics and science teachers (35 percent were opposed). How good are the data being used to formulate such policies? A more recent survey conducted by the Education Commission of the States (Flakus-Mosqueda, 1983) shows 38 rather than 44 states reporting teacher shortages in either mathematics or the physical sciences (physics, chemistry, or earth sciences), with some of the most populous states in the east and midwest not reporting shortages. Has there been an increase in the supply of
46 teachers or a decrease in demand (e.g., fewer students) in the intervening year? Has the definition of shortage changed? Are different criteria being used to determine shortage in different responses, or are there errors in the data? What conclusions can be drawn from existing information? What additional information is needed to formulate effective policy regarding teachers at the national, state, and local levels? Two sets of questions are paramount. First: Is the number of teachers adequate for the number of mathematics and science courses now being taught in secondary school? Will there be an adequate supply for the number to be taught at some point, say, 5 years, in the future? This set of questions requires a definition of who is to be counted in the available pool, which leads to a second set of questions: Are the teachers at all levels quali- fied to teach their current assignments in mathematics and science? Are they qualified for the responsibilities they will have in the future? Any response to this set of questions requires defining the term "qualified at the different grade levels. These are questions that entail both the setting of norms and the collection of descriptive data before they can be answered: What is the number of teachers avail- able? What is the anticipated demand? How are teachers prepared? How does this preparation compare with existing standards? Are existing standards--for example, state certification--acceptable definitions of "qualified"? The importance of these questions varies according to different dimensions at different grade levels. At the elementary school level, the question of numbers is not pertinent, since nationwide there appears to be an ample supply of elementary school teachers, at least until the mid-1980s when enrollments are expected to rise again (National Center for Education Statistics, 1982f, 1984a). However, there is concern about the preparation of teachers who are expected to teach mathe- matics and science in the self-contained classrooms of grades 1 to 6 and sometimes in the block programs of the middle school. For middle and junior high schools, the nature of the questions on numbers and qualification varies according to whether mathematics and various sciences are taught as separate subjects, as in high school, or as part of a core curriculum by a nonspecial- ist teacher. At the high school level, information is needed both as to the number of teachers and as to their qualifications. But the numbers are dependent on who is ~ _, ~
47 to be counted as a science or mathematics teacher and thus become confounded with questions on preparation and qualification. In the following section on number of teachers, the status of individuals being counted is defined in each case--for example, "assigned to mathe- matics or science classes," "degrees earned," "certi- fied"--without judgment as to their qualifications. The problems of defining "qualified" are discussed in the next section. Number of Teachers Supply of Teachers At the elementary school level, only a small number of teachers specialize in mathematics and science, either as specialist teachers or in grades 7 are part of the elementary system. and 8 when these grades In a survey of teacher demand and supply conducted in 1979-1980, the National Center for Education Statistics (1982e) estimated that 1.4 percent of all elementary school teachers (16,400-- 15,400 full time) were assigned to teach mathematics specifically and 0.7 percent of all elementary school teachers (8,600--nearly all full time) were assigned to teach science. A large proportion of these teachers are probably in the upper grades. At the secondary school level, there are available two data bases that have been analyzed regarding the number of mathematics and science teachers. The first is the survey of teacher demand and supply conducted in 1979-1980 by the National Center for Education Statistics (1982e), which yielded responses from administrators of 1,273 of a sample of 1,448 school systems (an 88 percent response rate). Based on this sample, NCES estimated that, during 1979-1980 in public secondary schools, 115,000 persons were assigned to teach mathematics either full or part time, and 104,700 persons were assigned to teach science courses either full or part time (see Table 2). This represented 11.4 percent and 10.4 percent of all secondary school teachers, respectively. ~ At this time, there is no readily available information on the preparation or certification of these teachers. To fill this gap, at least partly, NCES plans a 1985 survey of a national sample of teachers in ten broadly defined fields on their training and background; there will also be questions on how teachers spend their time, assignment of homework,
48 TABLE 2 Secondary School Teachers Assigned to Mathematics and Science Classes in Public Schools in 1979-1980 Field of Assignment Totala Full Time Mathematics 115,000 112,900 Science 104,700 101,000 Biology 25,000 24,300 Chemistry 11,400 10,500 Physics 6,700 5,700 General science 59,600 58,600 Other sciences 2,000 1,900 Reachers assigned to more than one field were counted in the field in which they spent most of their time. SOURCE: National Center for Education Statistics (1982e). and availability of resources including teacher aids. Between 8,000 and 10,000 teachers in more than 2,000 public schools are expected to participate; both teachers and principals in the schools will be asked to respond. Information on the preparation of teachers in private schools, derived from a special NCES study of private education, will become available early in 1985. The second data base regarding the number of science and mathematics teachers is derived from a survey by the National Science Teachers Association (NSTA) conducted in the fall of 1982. Using a sample of 2,236 schools that offered high school curricula, NSTA asked principals how many classes in mathematics or science were being taught and how many teachers were teaching these classes. On the basis of the first 846 responses (a 38 percent response rate), the numbers of such teachers were esti- mated. Despite the low response rate and methodological differences in the way the estimates were made, the NSTA estimate of the number of persons teaching mathematics in secondary school is reasonably close to that derived from the NCES survey: 106,190 (Pelavin and Reisner, 1984), compared with the NCES estimate of 115,000. Part of the difference might be explained by falling high school enrollments in the 3 years between the two surveys. Estimates for specific science fields are more difficult
49 to reconcile. For example, NCES estimates 10,500 full- time teachers in chemistry and 5,700 in physics; estimates for full-time equivalent teachers derived from the NSTA data are 13,620 and 6,900, respectively. A third data base currently being analyzed is the NSTA list of science, mathematics, and social science teachers for grades 7 to 12, maintained by grade level, by state, and by subject taught. The list was updated in November 1983, with principals of more than 23,000 schools respond- ing (a response rate of better than 80 percent). Prelimi- nary analyses indicate that there are some 75,600 people teaching biology, chemistry, physics, or a combination of these subjects in grades 7 to 12. Over 50 percent teach biology only, 15 percent teach chemistry only, 11 percent teach physics only, and the rest teach some combination of these subjects. It should be pointed out that the numbers include all people listed by their principals as teaching in the designated fields, rather than only those teaching the subject(s) full time (or full-time equivalents). The discrepancies in definitions and resulting numbers exhibited in these three surveys illustrate some of the problems with the current data. And lack of information on how many of the persons counted in any of the com- pilations are actually certified or otherwise qualified to teach science and mathematics raises additional uncertainties about the estimated numbers. Whatever the uncertainties, the current number of teachers, while an important statistic, becomes meaning- ful as an indicator only when compared with the number needed. But if estimates of numbers now teaching are attended by some ambiguity, estimates of future supply and demand are even more so. Estimates of future supply must take into account, in addition to the existing pool, the number of teachers leaving and entering the field. Estimates of demand must take into account current vacancies, the desirability of replacing those teachers who lack minimum qualifications for their teaching assignments, changes in total student enrollment, and changes in percentage of the total number of enrolled students who take specific science or mathematics courses. The teacher turnover rate (i.e., teachers leaving the profession) has been estimated at 6 percent for the last decade (Froomkin, 1974; National Center for Education Statistics, 1978, 1982b). In an unpublished analysis of the survey of principals and a separate teacher survey, NSTA estimates the rate to be 5 percent for science and
50 mathematics teachers in 1981-1982. Pelavin and Reisner (1984), in an analysis of the availability of teachers, use a 6 percent turnover rate and an estimate of 110,000 mathematics teachers and 103,500 science teachers for 1982-1983, reconciling the NCES and NSTA estimates. Thus, they project a loss of 6,600 mathematics teachers and 6,200 science teachers (son in chemistry . 500 in physics, and 4,900 in other science areas) in 1983-1984. (A 5 percent turnover rate would mean a loss of 5,500 mathematics teachers and 5,100 science teachers.) There is evidence that the teacher pool is aging (National Center for Education Statistics 1983; Feistritzer, 1983), which may mean a higher turnover rate a decade from now due to retirements--at a time when high school enrollment will be increasing and the cohort of young adults that might furnish new teachers will be decreasing. The supply of teachers can be increased either by persons newly entering the field or by persons returning to mathematics or science teaching. No national data are available on this second component, although one state reports that 65 percent of vacancies in all fields in 1982-1983 were filled by returning teachers (Flakus- Mosqueda, 1983). The potential pool is considerable. According to Graybeal (1983), as of fall 1981, about 6.1 million people (aged 21 to 65) had been certified as public school teachers: of this total, only about 2.2 million were teaching in 1980-1981; 1.9 million had left teaching; 1.9 million had not entered the profession; and 140,000 were newly qualified. With respect to new entrants, the number prepared to teach mathematics or any of the sciences, particularly the physical sciences, has been decreasing over the past decade. Data from NCES show that the decline in the number of college students majoring in science or mathe- matics education has taken place in the context of a general decline of teaching degrees conferred over the last decade (with the exception of degrees in special education); see Table 3. (The discrepancy between NCES data and the data from the NSTA survey of teacher placement officers cited above may be due to problems with the response rate on the NSTA survey and to somewhat differently worded questions on this survey and the NCES survey.) It should be noted, however, that neither the NSTA data nor Table 3 include newly certified entrants who obtained bachelor's degrees in fields other than mathematics education or science education, including degrees in mathematics or a science. For example, as ",
51 o .-l o 10 u) · - ~5 o to a, ·e en ~ ·l o ~ as ~ ~ v) m W ~ ~ a) m. ~ At; a) Ed ~ s CO so CO 1 to SO - o m o .,, ~ ~ ~ ~ ~ O O CO Cal d. U) · ~ ~ ~ c ~ e ~ a CO ~ ~ ~ ~ ~ ~ ~ O CO ~ lo) In kD Lo Cal ~ lo) ~ \0 _I t_ 1 1 1 1 1 1 1 1 1 1 As ~ 0 ~ ~ oo ~ up up ~ ret 0 ~ ~ Cal ~ ~ an ~ 0 ~ US a, ~ ~ ~ up CO ~ ~ ~ ~ ~ ~ L~ O ~ ~ ~ ~r ~ 0 ~ ~ ~ ~ c~ 0 ~ cn u: k er ~ ~ ~ ~ 0D ~ ~ O D ~ ~ ~ a, ~ .~1 · - · - o .-l o ~ a o ~ .-l ~ ~ ~: o ~ ·,1 · - ~ .- a' ~ cn · - Q4 u] o ~ . - o ~ · - ~ td 3 a) v ·,1 U] S~ ~ <: ~ 01 U] S ~: ~ O O ·- · - a a a a' ·,' C) U] ·,' U] O . - Q . U] ·,' V S" a o V ~n tn ·,1 U] m o .-, V 0 0 · - ~ V U] a) ~ V .-, . - S~ ~ s U] ~ a H o .,' V a U] V .,. o V .. O O Z V .,._, ~n S _ V ~ .. O ao C~ ~ _ · - .,. V a, 3 V · - ~ O · - O SU] ~ a, 3 z tn V . - U] .,, · S~ ~ O U) .m o . - O ~ V o a) ~: o .,, ~ Z .,, ·. ~3 C~ a; o U]
52 shown in Table 4, there were 3,150 newly certified entrants in mathematics and about 3,600 in the sciences who graduated in 1980. These entrants could replace half or more of the teachers lost through teacher turnover, although in the sciences the distribution of incoming teachers is likely to be skewed, with proportionally more being added in the biological than in the physical sciences. Table 4 indicates the modest proportion of new teachers in science and mathematics who are reported to be certi- fied or eligible to be certified in the field in which they are teaching, 45 percent and 42 percent, respec- tively. These data suggest that many newly graduated high school teachers who are not prepared in science or mathematics nevertheless may be assigned to teach these subjects. Current initiatives to encourage entry into the field may increase the proportion of adequately prepared entering teachers and reverse earlier forecasts of continuing declines of individuals available to teach mathematics or science. Demand for Teachers On the demand side, the National Center for Education Statistics (1982e) survey on teachers also included data on vacancies as of fall 1979: there were estimated to be 900 unfilled teaching positions in mathematics and 900 in science, including 400 in chemistry and 200 in physics. The vacancies for mathematics and science as a whole represented less than 1 percent of the total number of persons now teaching in those fields. However, that percentage does not take into account the number of teachers already in the system who were assigned to classes they were not qualified to teach. Particularly in times of shrinking enrollments, it is not unusual to fill a vacancy in a shortage area with a tenured teacher from an area with a teacher surplus. Fourteen states have no rules prohibiting out-of-field teaching. Total high school enrollment (grades 9-12) is a major determinant of teacher demand. The National Center for Education Statistics (1984a) projects enrollment at 13.7 million in 1985, down from 14.7 million in 1980, and at 12.1 million in 1990--a decrease of more than 17 percent over 10 years. The National Center for Education Statis- tics (1984a) also estimates a somewhat smaller decline in the total number of teachers in public secondary schools,
53 .,' 4~ · - O Z o Go 1 ·. A En a) 3 As o o ,, .,, .,, ·,1 4~ C) ·_' o .,l ._, o a' .,. _, so a) - o 3 o :< ~5 En 4~ ~ c: ~ c: · - a' ~ so V En a' ~ s ~ A Pa · - O Cal En c: a) EN ~ ~ · - ~ En a' a' 0 en c: Em ~1 s a' En O , ~ ~ ~ up ~ ~ ED ~ in · ~ ~ ~ ~ ~ rn ~ up a, oo Go · ~ ~ ~ ~ ~ ~ ~ ~ ~ a~ ~ ~ u~ ~ a, ~ 0 ~ ~ ~ 0 ~ ~ ~ · ~ ~ ~ · · · e O O ~ ~ ~ ~ { CO ~ ~ ~ d4 . 00 ~ 00 (D ~ ~ ~ ~ u~ · · · · · . · ~ a: ~ ~ 0 a, ~ a, oo co ~ c~ O O O O O O O O O O O O O O O O O O 00 1 ~ C~ o ~ Lo ~) L9 a~. ~ ~ 0 ~ ~ ~ O kD U] U] U] h J~ ~ a; ~ ~ ~ a, . - ~ ~ - - ~ ~ U] ~ ~ ~ ~ o k4 a) C: u a s - UO U. ~ V ~, ~ ~ 0 ~ ~ . U) 0 a) a .-, ~ t', ~ U] U) a~ ta ~a 2 t~ =: ~ ~ c: ~ ~ a a, ~ ~ ~ ~ ~ ~ ~ O o J~) ~ U: ) S C: - - 1 U: ~ ~ ~ S ~ ~ ~ ~ ~ O a) 4J ~ a) ~ s a~ a) t:n ~ 0 JU Q5 ~ O ~ U] ~ O ~ ~ ~ O E~ u~ = ~ C4 ~ ~ 3: u ·,. a) .,4 o o S U] o U) o :^ S~ ~; a,4 S a U) ~: .,. ._' C) - S o 0D ~ 00 1 a~ a~ o ·. CO - U., C) -,' U] -,' 4U U) o o JJ a) C) o ·_I z ·. ~; ~7 o U)
54 about 10 percent over the same decade. A relatively larger decrease already took place between 1980 and 1982, when the number of secondary school teachers declined from 1,074,000 to 1,039,000. If that rate were to continue until 1990, the 10-year loss would be more than 15 percent. On the assumption that NCES's estimate of a 10 percent decrease over 10 years is more nearly correct, a decrease of some 72,000 teachers for 1982-1990 can be expected, for a total 1980-1990 decrease of 107,000. After 1990, however, there is expected to be an increase of teachers, as high school enrollments begin to increase again starting in 1991. If mathematics and science teachers were to continue to represent, respectively, 11 percent and 10 percent of the high school teaching force, the total number of teachers needed for mathematics would decrease by 7,700 by 1990 in comparison with the number needed in 1982, and the total number of teachers needed for science would decrease by 7,000. A countervailing factor to decreasing enrollments is the increase in high school graduation requirements already mandated by some states and being considered by others (see Table 5). It should be noted that these increased requirements would not affect all students: in 1982, about 46 percent of high school graduates had taken 3 years or more of mathematics in grades 9-12; 30 percent had taken 3 years or more of science (National Center for Education Statistics 1984b). However, where recent state or local mandates would require more courses than were actually taken before the new requirements, additional mathematics and science teachers would be needed. A number of state university systems also have recently increased entrance requirements, often beyond those required for high school graduation (U.S. Department of Education, 1984). The National Commission on Excellence in Education (1983) recommended that all students be required to take 3 years of mathematics, 3 years of science, and 1/2 year of computer science for high school graduation. If these recommendations were to be imple- mented, it would certainly require a large increase in the number of mathematics and science teachers. In the committee's estimates of annual demand for the next few years (see below), it is assumed that decreased demand due to lower high school enrollments will be balanced by increased demand due to higher graduation requirements. However, Pelavin and Reisner (1984) estimate the increased demand to be 8,600 mathematics teachers and 6,500 science teachers.
55 It has been argued that demand projections should take account of the need to replace teachers of science and mathematics who have been assigned to teach those subjects without the requisite qualifications. At present, fewer than half of new entries into these fields appear to be qualified (see Table 4), and unless countermeasures are taken, erosion of the competence of the existing teaching pool will continue. Countermeasures could include increasing the numbers of qualified new entrants (or reentrants), in-service education, and replacement of unqualified teachers. As to the last, likely replacement rates are difficult to estimate, since the feasibility of replacing teachers, especially if tenured, depends on conditions within individual school systems. Many local systems and states may choose to retrain rather than replace underqualified teachers. For the purpose of projecting demand, the committee considered three alternative replacement rates per year: a no-replacement rate of O percent; a moderate replacement rate of 2.5 percent of the current poo1 of mathematics and science teachers; and a high replacement rate of 5 percent. The alternative estimates of annual demand, supply, and shortage of high school mathematics and science teachers for the next few years under these conditions are shown in Table 6. As can be seen, annual shortages are at least 3,700 for mathematics teachers and 2,800 for science teachers; that is, the annual demand for new or returning high school teachers of mathematics and science is projected to be at least twice the expected supply. If school systems were to make a concerted effort to replace unqualified teachers, the need would be for three or four times the expected supply of new (or returning) teachers. Among the various sciences, data show that shortages will continue to be most acute in physics but also prevalent in chemistry and the earth sciences; few shortages in biology are projected. The preceding summary of the data indicates the considerable uncertainties attached to all the estimates. Moreover, the projections are based on the assumption that the education system will continue to operate essen- tially as it does at present; the possible effects of structural changes that might be brought about by the application of information technology to education and by other reform efforts are not taken into account. As noted in Chapter 2, national projections are not very useful for state and local planning. As the
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60 o o s V] .,, o o s U) CO ~: o U] E~ ~n ~n a) .,, U) S~ a ~: ~q .,, ~ s c c) a) '5,: a) P4 1 a) P4 o s" a C~ a) ·_1 ~q U) ._. ~ o ~ o s o o o o o o o o o o o o n c~ ~ ~ 0 U~ ~ ~ ~ o o o o o o o o o o o ~ ·n ~ ~ ~ c: o o o o o o ~ o o o o o o ,, ~ ~ ~ o u: In ~ ~ ~ u~ U] .,. ~ o o o o o o s o o o o o o ~ a~ 0 ~ ~ ~ ~ ~: ~ ~ ~ ~ ~ a) .,, U) ~n c) .,, a s~ ~: ~ a) J~ CC ~ u] o ) ~ 4J o ~ s:: ~ . - u~` aJ ~ s 4~ ~ o o o o o o o o o o ~ u) ~ ko ~ ~ 1 ~ ~ ~ u~ 1 ~ ~ ~J o o o o o o o o o o ~ O C5\ ~ ~ ~ 1 ~ ~ .- 1 ~ ~) ~r, 4~ ~ u) eq . - o ~ ~ 4 / u: ~ u: ~ a' a 0 ~ ~ a O ~ C) 3 t~ ,4 a, s:: a, ~ O U] S U] ~q , ~ O u: ~ E~ 3 ~ ~ Z s:: o ~ O S E~ · r~ u, a) ~ O :^ U~ C ~ :5 JJ ~, X a ~ O ._, 3 U) ~; JJ o a) 4U C 0 - ~ O O O O S ~, U) S S S ,' a) o o 4~ U] S ·a ~' s~ o a' a ~n ~? C) a' a S O o c: 3 . a a ~n - a U: a S a S E~ .. ._I ~ U) E~ u: O O Z . o U) 8 . ~ s" a) s . ~ s o U] a) ~; .,.
61 Education Commission of the States survey (Flakus- Mosqueda, 1983) shows, there is considerable variation in teacher supply and demand among the states. Some states are losing students, others are gaining them. Some of the most populous states in the Northeast and Midwest report no teacher shortages in mathematics or science (see Appendix), while other states are reporting critical shortages. The numbers of teachers do not vary in pro- portion to student enrollment, since there is a set of constraints operating differently on different com- munities with respect to hiring or firing teachers. Even among districts within a state, supply and demand is likely to vary, depending in part on the sociodemographic characteristics of communities (National Center for Education Statistics, 1982d). Quality of Teachers If the objective is to gauge the adequacy of science and mathematics teaching in the schools, then simply providing a count of the number of teachers in front of science and mathematics classes without any assessment as to their quality is not sufficient. There is, however, no measure available for evaluating teacher quality; there is not even a measure for assessing competence, that is, whether a teacher possesses adequate knowledge of what is to be taught and knows how to teach it. Certification has been used as a first-order approxima- tion of competence, but, as shown in Table 7, certifica- tion standards vary so greatly from state to state that certification becomes problematic as a measure of com- petence at the national level. Certification requirements range from a degree from any of the accredited teacher education programs in the state (which may themselves vary quite widely) to a number of college credit hours in education courses and in areas of specialization. Even requiring a given number of credit hours can result in quite different levels of preparation, however, depending on the content of courses taken. In the 1960s and 1970s, various professional groups such as the Mathematical Association of America (Commit- tee on the Undergraduate Program in Mathematics, 1961a, 1961b) and the American Association for the Advancement of Science (1970), in conjunction with the National Association of State Directors of Teacher Education and Certification, developed and published standards for the
62 TABLE 7 Teacher Certification Requirements Elementarya Secondaryb S ta te rda th Sc fence Ma th Sc i ence Te s t Alabama 12 combined 12 combined S ALaska U U U U Ar izona 12-30 12-30 30 30 S Arkansas 6 9 21 24 NTE California U U U U S/NTE Color ado U U U U S Connecticut 6 R 30 30 S Delaware U U 30 39-45 S D. C. 9 6 30 30 Florida 6-12 comt~ined 21 20 S Georg ia U U 45 qh 40-75 qh S Hawai i U U ma jor ma jor Idaho 6 8 20-4 5 20-4 5 Illinois 5 7 24-32 24-32 Indiana R R 24-52 24-52 Iowa U R 30 30 Kansas 12 combined 18 24 Kentucky 12 combined 48 48 Louisiana 6 6 20 20-32 S/NTE Ma ine U U 18-50 18-50 Maryland 6 12 24 36 Massachusetts U U 36 3 6 Mich igan U U 3 0 3 0 Minnesota U U ma jor ma jor Mississippi 15 combined 12 combined NTE Missour i 5 5 30 30 Montana U U 20-40 20-40 Nebraska U U U U Nevada U U 16-36 16-36 New Hampshire U U . U U New Jersey R R 24-30 24-30 New Mexico R R 24-54 24-54 S/NTE New Yor k R R 2 4 3 6 NTE N . Carol ina U R ma jor ma jor NTE N. Dakota U R U U Oh io 6 8 20 20-60 Ok lahoma R R 2 8 3 6 Oregon 12 U 21-42 4 5 Pennsylvan ia U U U U Rhode Island U U 18 18 S. Carol ina U 12 12-60 12-60 S/NTE S. Dakota 2 4 ma jor ma jor Tennessee 3 qh 12 qh 27 qh 24-48 qh Texas U U U U NTE Utah U U 16-46 16-46 S
63 TABLE 7 Continued Elementarya Secondaryb State Math Science Math Science Test Vermont U U U U Virginia 6 6 16-27 24 NTE Wash ington U U U U W. Virginia U U U U S Wisconsin U U 22-34 22-34 Wyoming R R R R CODE: U = credits in mathematics and/or science may be required for certif ication; these sub jects, however, are not specif ically mentioned. credits in mathematics and/or science are required for certification; number of credits required is not ind icated s tate-constructed test NTE = National Teacher Examination qh = quar ter hour NOTE: Unless otherwise noted, requirements are given in college semester hours required in mathematics and science for state certification for elementary school teachers and to teach mathematics or science in secondary school. Certification to teach; requirements given are for the lowest-level certificate. Many states require additional credit hours for certif ication as a specialist teacher in mathematics or science or for teaching in junior high school. Certification to teach mathematics or science. A wide spread in credit hours (e.g., 18-50 for Maine) generally means that the higher number includes courses in several sciences for certification to teach in all of them. SOURCE: Adapted from Woellner (1983) and Flakus-Mosqueda (1983). preparation of elementary and secondary school teachers in mathematics and science. The guidelines have been updated periodically (see, e.g., American Chemical Society, 1977). These activities led to an increase in several states in the number of credit hours required in the pertinent academic field for certification of secondary school teachers and of hours of mathematics required for elementary school teachers. Revised guide- lines for preparation in mathematics for elementary school teachers and for mathematics teachers for grades 7-12 have recently been prepared by the National Council of Teachers of Mathematics (1981a) in association with the Mathematical Association of America (see also Committee on the Undergraduate Program in Mathematics,
64 1983); the National Science Teachers Association (1983) has published standards for preparation in science for elementary and middle/junior high school teachers and, more recently, for secondary school teachers of science (Ritz, 1984). As in the past, the effect of these guidelines is likely to vary from state to state. In an effort to help ensure quality, a number of states have added competency-based tests to their certification requirements, as shown in Table 7; at least 15 more states are considering the use of such tests. Several states provide long-term certification; others require periodic recertification based on continuing in-service education. Districts may impose their own standards in addition to those required by the state. Certification standards are changed periodically as new priorities are set for schools, but teachers already certified are generally excluded from having to meet the new standards or can meet them through inservice training. Hence, certification granted at different times may repre- sent different preparation even within the same state. Elementary School In many states elementary certification depends mainly on obtaining a college degree and on a specified period of teaching within the state. In some states, such elementary school certification is also valid for teaching grades 7 and 8; certification provisions usually call for a specialist teaching degree requiring more credit hours in the relevant academic field than for grades 1-6 but fewer than for secondary school certification. Since teachers for grades 1-6 generally major in elementary education, their college preparation in mathematics or science tends to be limited, as indicated by the requirements (or lack thereof) shown in Table 7. According to a recent survey of teacher education programs (Kluender and Egbert, 1983), 40-50 percent of an ele- mentary school teacher's preparation consists of profes- sional education courses; the rest is usually distributed among general liberal arts courses. If science is taken at all in college, it is usually limited to one disci- pline. Certification may be an even less appropriate indicator of qualification for teaching mathematics and science in elementary school than it is for secondary school teachers. In any case, little information is available regarding the subject-matter expertise of elementary school teachers presently in classrooms.
65 Secondary School Certification to teach a particular subject in secon- dary school may require as few as 18 or as many as 48 college credits in the relevant and related disciplines. Kluender and Egbert (1983) found that the average require- ments of teacher preparation programs for secondary school teaching consist of 25-60 percent of courses required in the academic field to be taught, 20 percent in profes- sional education courses, and the rest distributed among general liberal arts courses. In the large state univer- sities, the credits needed for teaching degrees often represent preparation equivalent to that of a major in the academic discipline, but little is known about the types of courses taken by teachers in smaller, less prestigious institutions. Because of the great range over locale and over time in teacher education programs and certification standards, and because of the device of issuing emergency certifi- cates, documenting the number of teachers actually certi- fied to teach science or mathematics is only a first step toward establishing whether they are qualified. Even so, no national data are currently available on how many teachers now assigned to teach science or mathematics courses are fully certified for their assignments. As noted above, a new NCES sample survey on teachers is planned for 1985 to provide data on the certification status and preparation of secondary school teachers in all fields. More information is available on certification of new entrants than on certification of teachers already teaching. NSTA surveyed secondary school principals in 1980-1981 and again in 1981-1982 to gather information on their teachers. Table 8 shows the percentage of newly hired science and mathematics teachers who were not certi- fied to teach the courses to which they were assigned, as estimated from the NSTA surveys. This table also gives some indication of the variations among different regions of the country. It is evident that regions losing popula- tion, like the northeastern states, are having less difficulty in staffing their schools than are the regions of high growth, like the Pacific states, where a large majority of newly hired teachers in science and mathe- matics are not certified. Similar findings come from the periodic NCES surveys of recent college graduates; as noted above, only 42 percent of 1980 bachelor degree recipients teaching
66 TABLE 8 Percentages of Newly Hired Science and Mathematics Teachers Not Certified in Subject Census Region 1980-1981 1981-1982 Pacific states 75 84 Mountain states 44 43 West north-central states 26 43 West south-central states 63 63 East north-central states 23 32 East south-central states 43 40 Northeastern states 11 9 Middle Atlantic states 40 46 South Atlantic states 48 50 Nationwide 45 50 SOURCE: Franz et al. (1983). mathematics and 45 percent teaching science were certified to do so (see Table 4). These percentages are far lower than for other fields, although English teachers also were drawn from out of field in considerable numbers. Defining Teacher Quality It bears repeating that certification is only a poor approximation of competence at the secondary school level and even less meaningful at the elementary school level with respect to teaching science and mathematics. At present, however, there is no other standard that might be used to establish how many of the teachers with instructional responsibilities in science and mathematics are qualified to carry out their assignments. One unresolved problem is the appropriate combination of knowledge of subject matter and of teaching (pedagogy) and how that should vary by level of instruction (Druva and Anderson, 1983). Obviously, teachers must understand the subject matter they are responsible for teaching, although there is evidence that in mathematics, at least, more advanced knowledge by the teacher does not correlate highly with increased student performance (Begle, 1973). Nevertheless, the equivalent of a college major in the
67 relevant discipline(s) is generally thought necessary for secondary school teachers, with the number of credits required for certification specified by many states. Sequence and content of the courses is often left up to individual institutions, and practice reflects disagree- ment about such matters as the suitability of courses designed for mathematics or science majors as well as the mix of disciplinary and pedagogy courses. Research evidence provides little guidance. The National Longi- tudinal Study of Mathematical Abilities (NLSMA), for example, included a detailed study of the relationships between teacher background and attitudes and student performance. Concerning teacher preparation, the strong- est positively correlated variable was found to be credits in mathematics methods courses, but the positive cor- relation appeared in only 24 percent of the cases (Begle, 1979). Generally, teachers with graduate credits or advanced degrees are deemed to be more competent and are paid better, yet the evidence on the relationship between graduate work or inservice education and student achieve- ment is equally tenuous (Summers and Wolfe, 1977; Begle, 1979; Shymansky et al., 1983; Druva and Anderson, 1983; Hilton et al., 1984; U.S. General Accounting Office, 1984). There is even more question about the suitable academic preparation of elementary school teachers, since they are responsible for teaching subject matter from several dis- ciplines. For elementary school teachers in particular, but also for secondary school teachers to some extent, the importance of pedagogy is stressed by those who hold the model of the college lecture to be inadequate for precollege education. Teaching prospective teachers how to teach science or mathematics is deemed as necessary as what science or mathematics to teach. The how appears to be especially important with respect to teaching such higher-order skills as analyzing and solving problems, reasoning from evidence, checking one's procedures, and in-depth understanding (Glaser, 1983). A third element in teacher qualification is experience, rated by skilled classroom observers and school administrators as a key element in the development of competent teachers. Experts have not settled their differences, with mathematicians and scientists generally arguing for increased training in subject matter, teacher educators for more training in pedagogy, and principals and school superintendents for teaching experience. Almost all the recent reports on education and several national bodies
68 have made suggestions for how to improve teacher educa- tion. (For a listing of these suggestions, see National Commission on Excellence in Teacher Education, 1984.) At this time, efforts are going forward to increase the number of people teaching science and mathematics. These Natural experiments" range from training out-of-field teachers by giving them special courses in mathematics or the relevant science to hiring professional scientists and engineers as teachers without requiring the usual education courses or teaching experience. It would be useful to track a selected number of these experiments through a carefully designed research effort in order to help identify the critical attributes of competent science and mathematics teachers. Findings Supply and Demand Aggregate Quantity · Forecasts of aggregate supply and demand of secondary school teachers in the physical and biological sciences and in mathematics show shortages over the next several years in mathematics and the physical sciences. A low estimate, based on little change in current trends of overall supply and demand, indicates an annual short- age of 2,800 science teachers, mostly in the physical sciences, and 3,700 mathematics teachers. If teachers currently assigned to mathematics and science classes but not qualified to teach these subjects were to be replaced at a rate of 5 percent per year of all teachers in these fields, the annual shortage would be 9,200 in mathematics and 8,000 in science. Both these forecasts are driven by the education system as presently constituted and do not take into account the possibility of structural reform. · Aggregate estimates of teacher supply and demand mask great differences among regions of the nation, states, and local school districts within states. Uncertainties · All estimates of teacher supply and demand are accompanied by large uncertainties.
69 With respect to supply, there are three major gaps in knowledge: (1) The data on the actual numbers of teachers assigned to mathematics and science classes are inadequate, especially as aggregated at the national level. (2) The number of inactive teachers who return each year to fill vacancies is unknown. Since the number of trained teachers who do not enter teaching or who leave teaching is sizable, this represents a considerable resource. The number of teachers drawn from the inactive pool may increase as desirable job opportunities arise. (3) The most recent data on the annual supply of newly certified entrants to teaching--3,200 in mathematics and 3,600 in science--are 4 years old. Hence, the effects of current incentives to draw people into the field are unknown. The incentives include loan programs for college students preparing to be teachers, in-service training for out-of-field teachers, and employ- ment of retired scientists and engineers as teachers. With respect to demand, there are four unknowns: (1) While enrollments are dropping, vacancies tend to be filled with teachers from other fields who have tenure in a district, rather than with new entrants certified in the field with vacancies. This practice, the extent of which is unknown, reduces the demand for additional teachers, even though it may be detrimental to the quality of science and mathematics teaching. (2) The extent to which school systems will seek to replace out-of-field teachers or will choose instead to provide in-service training is unknown. Such choices will in part be influenced by state and federal support policies for teacher education and in part by local board policies and teacher contracts. To the degree that increased high school graduation requirements will entail having to offer more courses in mathematics and science, teacher shortages will be aggravated, but how much is unknown.
70 (4) Demand forecasts are generally based on extrapolation of current conditions, taking account of likely changes in enrollment, class size, and curriculum. They do not take into account possible structural changes in the education system. Quality Lack of Information · Adequate information is lacking on the quali- fications of the teachers who are responsible for teaching mathematics and science in high school, middle/junior high school, or elementary school. · Information on certification, the only proxy available for qualification, is lacking for all but new entrants, although data on a national sample of the teaching force are now being collected. Requirements for Teaching Mathematics and Science · Even if available, information on certification is of questionable use as a measure of qualification because state certification requirements and preservice college curricula reflect a wide range of views on what constitutes a qualified or competent teacher in mathe- matics or science. Moreover, teachers currently certi- fied obtained their certification at different times that may have required different types of preparation; therefore, certification even within the same state does not connote equivalent preparation. · Although guidelines on teacher preparation developed by professional societies are generally available, they have not been uniformly adopted. Conclusions and Recommendations Supply and Demand · A suitable indicator to assess the sufficiency of secondary school science and mathematics teachers would be either the ratio of or the difference between projected demand and anticipated supply of qualified teachers. The
71 ratio would indicate how close to balance demand and supply are; the difference would indicate the number of teachers that need to be added or that exceed the demand. The construction of such an indicator on teacher demand and supply is at present not feasible at the national level because of the lack of a meaningful common measure of qualification. · Individual states and localities might construct this type of indicator by using certification as an approximation for qualification or developing alternative criteria for teacher competence. In each case, an adequate determination would entail estimates of both demand and supply under alternative sets of assumptions about anticipated enrollments in mathematics and science classes and new entrants into the teaching of these fields. Aggregation of the state data might provide a useful national picture, especially if, in addition, reported concerning differences among information was states. Quality · The disparate views on teacher qualification and the variation in certification standards indicate the need to rethink the initial preparation and continuing training appropriate for teachers with instructional responsibilities in science and mathematics. Guidelines that have been prepared by professional societies need to be considered by the wider educational community, includ- ing bodies responsible for the certification of teachers and accreditation of teacher education programs. Require- ments should be detailed separately for teachers in ele- mentary school (grades 1 to 5 or 6), middle or junior high school (grades 6 or 7 to 8 or 9), and high school (grades 9 or 10 to 12), with particular attention to _ requirements that can be translated into effective college curricula and in-service education for teachers. · The development of guidelines for the preparation and continuing education of teachers would be advanced if the attributes of successful teaching in science or mathematics were better understood. Further research is necessary on the relationships between teacher training and student outcomes; for example, the effects on student achievement of different types of preservice and in- service training and of teaching experience. Current initiatives to augment the pool of science and mathematics
72 teachers should be monitored to assess their effective- ness. CURRICULUM CONTENT Opportunity to Learn Giving students the opportunity to learn subject matter not part of their home or social environment is a primary reason for formal schooling. The opportunity to learn mathematics and science is dependent, in part, on the content of the curriculum. It is also dependent, in part, on the time devoted to each curriculum area--a process variable discussed in the next chapter. These two aspects of instruction are considered separately for analytic purposes, although they are obviously closely related. The relationship between the emphasis given a topic in the curriculum and student achievement was demonstrated by information collected by the International Project for the Evaluation of Educational Achievement (IEA) in 1970. In conjunction with science achievement tests administered in some 16 countries, TEA asked teachers to rate each item in the test according to the following scale: 1--None of the students has studied the relevant topic; 2--Fewer than 25 percent of the students have studied the relevant topic; 3--Between 25 percent and 75 percent of the students have studied the relevant topic; 4--More than 75 percent of the students have studied the relevant topic; 5--All of the students have studied the relevant topic. From the rating data, a national opportunity-to-learn score was obtained for each school; the scores were then aggregated to determine an overall rating for each country at each population level. The results show (Wolf, 1977) rank-order correlations between opportunity to learn science and achievement, across countries, of .51, .75, and .36, respectively, for the three populations tested: 10-year-olds (I), 14-year-olds (II), and all students in the terminal year of secondary school (IV). Table 9 exhibits this relationship for the United States. It should be noted that the U.S. ranking for category IV is affected by the fact that, in some European countries,
73 TABLE 9 United States Rank Order in Opportunity to Learn and Science Achievement for Populations I, II, and IV, 1970 Countr ies Testing Rank Order of United Statesa and Rating In Opportunity In Science Opportunity to Population to Learn Ach ievement Learn (Number ) I ( 10-yr-olds ) 1 4 14 I I ( 14-yr-olds ) 6 7 16 IV (terminal year ) 13 14 16 - al indicates the highest rating, i.e., greatest opportunity to learn or h ighes t ach ievement . SOURCE: Wolf (1977 :40) . the terminal year of secondary school comes 2 to 3 age years later than in the United States. A shorter version of the rating scale used in the TEA science assessment was also administered in conjunction with TEA mathematics testing in 1964. Teachers used a three-point rating scale for each topic in the test: 75 percent or more of the students had the opportunity to learn the topic, 25-75 percent had the opportunity, or fewer than 2S percent had the opportunity. Correlations between ratings and scores by countries was .73 for 8th graders; that is, students scored higher marks in coun- tries where teachers rated the tests to be more closely related to the curriculum. Husen (1967:168) concludes that "a considerable amount of the variation between countries in mathematics score can be attributed to the differences between students' opportunities to learn the material which was tested." The IEA'S second inter- national mathematics study and the second science study currently under way are collecting similar information on opportunity to learn. In the United States, local districts determine school curricula, usually within guidelines set by the state. The degree to which guidelines are mandatory varies from state to state. Most states, although not all, specify a minimum number of credit hours for high school graduation, including requirements in such key fields as English, mathematics, and science (see Table 5, above). For most subjects, however, local authorities have considerable discretion as to the content to be covered within the required credit hours and state guidelines. Some populous states, including California, Florida, and Texas, have
74 state textbook adoption boards; however, the lists of texts approved for school use by such bodies usually are comprehensive enough to allow much room for local choice. The state education authority in New York is unique in its history of involvement with local districts. Examina- tions (the "Regentsn) are constructed at the state level, based on specified courses of study for each subject matter field. Although the examinations are voluntary, all high school curricula are required to be based on them. Similar curriculum guidelines became mandatory for grades 7 and 8 in 1984, and there are also some mandatory curriculum requirements for elementary school. For some disciplines, mathematics in particular, professional societies have recently developed guidelines for the content of the school curriculum (National Council of Supervisors of Mathematics, 1977; National Council of Teachers of Mathematics, 1980, 1981b; Conference Board of the Mathematical Sciences, 1983). Although there may be agreement on principles by professionals, just as in teacher education, that agreement does not necessarily extend to others concerned with education. As a result, textbooks intended for the same grade or course emphasize different topics; some topics may be included in one test and excluded from another; and teachers may stress differ ent subject matter. Such choices are not always based on the recommendations of subject matter experts. The lack of agreement on course content is especially true for the science curriculum and for nontraditional mathematics topics in elementary school, for the life sciences, and for science and technology education for students not taking the traditional precollege sequence. It will be important to monitor the extent to which the recommenda- tions being made by professional groups are translated into texts or teaching methods that are likely to affect student learning. The Role of Textbooks Textbooks appear to be central to instruction. While other teaching and learning devices are in use, such as computer-aided instruction, films, and laboratory experi- ments, their role is decidedly subsidiary. Stake and Easley (1978), in a set of case studies supported by NSF on the state of precollege science education, found that more than 90 percent of all science teachers use a text- book 90-95 percent of the time. This finding has been -
75 replicated over and over by classroom observers. Hence, one way of establishing the content of instruction would be to document what textbooks are used, what scientific concepts, factual knowledge, and processes inherent in the discipline are covered in the most commonly used text- books, how much textbooks intended for the same grade level or course differ from each other, and the emphasis given by the teacher to different topics within a given text. There have been occasional studies on various aspects of textbook content and textbook use, but information has not been collected systematically over time. There is even less information available on other teaching and learning tools, especially with respect to their role in conveying content. The most comprehensive information on the use of mathematics and science textbooks comes from one of the NSF-supported studies, the 1977 National Survey of Science, Mathematics, and Social Studies Education (Weiss, 1978). According to teacher reports, one-half of all science classes and about two-thirds of all mathe- matics classes use a single published textbook or program, and about one-third use multiple texts. Only in grades K-3 science instruction was there any noticeable absence of the use of a published textbook or program (37 percent of the classes). Over one-half of the elementary school teachers surveyed used one or another of the most popular five mathematics textbook series; somewhat more diverse choices were reported in science. The extent to which textbooks published for the same grade and subject actually differ has been open to ques- tion and has occasionally been the subject of empirical study. During the era of curriculum reform in the 1960s, texts in mathematics and the sciences were often classi- fied as to whether they emphasized facts ("traditional" texts) or concepts, processes, and learning how to learn ("new" texts) and whether they included such "new" topics as set theory in mathematics or genetics and evolution in The error`, ~1 the Hi ff~r~nc~s built into the biology. ~ ,. ~ ~ ~ . ~ ~ reform curricula did indeed bring about differences in It performance. Accordino to NLSMA findings, stu- _ dents studying the new mathematics did better on tests of comprehension, application, and analysis; students using conventional texts performed better on computation, though new math students tended to catch up in later grades (Begle and Wilson, 1970). With respect to science, a number of evaluation studies have recently been reviewed to assess the overall effects of the reform curricula;
76 after examining 111 studies dealing with science cur- ricula, Shymansky et al. (1983:392) conclude: Especially interesting . . general achievement. . . . . . are the statistics for Much criticism regarding the new science curricula focused on the apparent decline of general science knowledge among students exposed to the new programs. At the height of the new curricular movement (and even today) the pre- vailing notion was that the process goals of the new science curricula were being achieved at the expense of the content goals--although no compre- hensive database existed for either claim. The data . . . show clearly that students exposed to new science curricula achieved 0.43 standard deviations above (exceeding 67% of the control group), or nearly one-half of a grade level better than, their traditional curriculum counterparts on general achievement measures. Students taking the new courses also gained on their counterparts in analytic thinking, problem solving, creativity, and other higher-order cognitive skills and in process skills relevant to the doing of science. More recently, analysis of science textbooks has been concerned with the structure and language used to present topics (Robinson, 1981:5-68). A question of particular interest has been the degree to which science learning involves the memorization of unfamiliar technical words. Building on previous work that indicated that some texts required learning thousands of new words, Yager (1983) analyzed 25 frequently used science textbooks. These included two science series for grades 1-6, six texts at the middle/junior high school level, and alternative texts for high school biology, chemistry, and physics. At all levels, Yager found terminology to be a central feature of science texts, with 2,700 to 3,500 special or technical words included in books intended for grades 4-6 and as many as 9,300 in one of the physics texts. Even if only a small percentage of these words are new or entail new definitions, a lot of learning time is spent on vocabulary. To a lesser but still considerable extent, this is true even of the new texts. Concentration on vocabulary may in part be responsible for a large propor tion of students reporting that they are bored with science classes--82 percent of 17-year-olds in one study (Hueftle et al., 1983). -
77 While this sort of analysis points to possible similarities among textbooks in learning difficulty, it does not further establish concordance of subject matter coverage. Despite the key role of the textbook in instruction and student learning, there has been little content analysis of texts since the mid-1970s (Walker, 1981). One might hypothesize that the widespread use of standardized tests would lead teachers to emphasize cer- tain common topics, even if texts included other mater- ials. To investigate to what extent different textbooks treat the same topics and the text materials match topics covered on tests, Freeman et al. (1983a) examined four popular 4th-grade mathematics textbooks and five stan- dardized tests. A set of 22 core topics was identified by analyzing all the texts and tests. Approximately 50-60 percent of the more than 4,000 problems in each book focused on 19 of the 22 core topics, showing that there is indeed some agreement among texts on a common core. The match to tests was rather worse, however "Freeman et al., 1983a:504): "Of these 22 topics, only six were emphasized in all textbooks and tests analyzed. Three topics were emphasized in all books but in no tests. Three other topics were covered in all tests, but they received limited attention in the books. The other 10 topics were emphasized in all four books, but they appeared in only some of the tests. The match between topics contained in the texts and in the tests analyzed is shown in Table 10. The authors conclude that (p. 511) "[t]he proportion of topics covered on a standardized test that received more than cursory treatment in a textbook was never more than 50%." Variations in Topic Emphasis Though teachers rely heavily on textbooks for instruc- tion, they use them differently. Another investigation by the same research team (Freeman et al., 1983b) showed that student exposure to the content covered by several of the tests included in the study varied to some degree depending on styles of textbook use, even when the text- book was the same. Berliner (1978), using logs of how 21 5th-grade teachers in California allocated their instruc- tional time, found great differences in time spent on common mathematics topics from class to class, as shown in Table 11. While some of these differences may be
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79 TABLE 11 Pupil Time (in minutes) in Content Areas of Mathematics for Four 5th-Grade Classes Content Area Classroom - A B C D Computation Addition 33 234 95 26 Subtraction 77 205 248 4 Multiplication: basic facts 40 79 89 142 Multiplication: speed tests 34 51 8 24 Multiplication: algorithm 341 910 720 343 Division 243 19 1,548 2,223 Fractions 54 370 495 2,016 Other 0 82 213 0 Concepts/application Computational transfer 49 24 160 147 Numerals/place value (whole number) 0 53 29 0 Word problems 58 3 322 15 Geometry: perimeter 0 53 73 0 Geometry: area 0 103 49 0 Geometry: number pairs 90 40 0 0 Geometry: lines or figures 418 126 70 280 Other 174 128 1,411 68 NOTE: Time was logged over an average of 90 days of instruction observed between October to May. SOURCE: Berliner (1978:21) as cited in Romberg and Carpenter (1985). related to differences in total time spent on mathematics instruction (compare, for example, classrooms A and D), variation in topic emphasis is evident apart from varia- tions in total time. It may be conjectured that pupils from classrooms C and D performed differently on division problems on tests than did pupils from classrooms A and B. There may be even greater variation at the secondary level than at the elementary level in the content of instruction as embodied within such common course titles as general mathematics, introductory (first-year) algebra, earth sciences, or introductory biology. Moreover, curriculum supervisors at the state level report that there has been a proliferation of course titles, with few standards as to content. Presumably, logging which textbook is being used would give some indication of the content of a course, if the content of that textbook is known. Since there is considerable variation in textbook use, however, content analysis of commonly used texts would have to be augmented by observation and analysis of instruction within samples of classes; such observations could provide more detailed information on what is
80 actually taught to students enrolled in a given course. If aggregation of course titles, let alone course content, is difficult at the state level, it requires truly heroic assumptions to infer what "years of enrollment" in mathe- matics or science collected at the national level might mean in terms of the content studied. Findings Opportunity to Learn · Exposure to specific content as conveyed by curriculum materials and explicit teaching is a critical factor in student achievement. · Although commonly used textbooks and tests intro- duce a modicum of similarity in the range of topics generally treated within a year's course of instruction, emphasis varies from text to text, class to class, and test to test. Hence, for the nationally normed achieve- ment tests often used at the elementary and middle school levels, there may be a discrepancy between a student's opportunity to learn and the subject matter covered on the test, while at the same time the student may have learned considerably more than the test indicates. Textbooks and Courses · To a large extent, the content of instruction is based on the textbook used in a class, yet there is no continuing mechanism to encourage periodic and systematic analysis of the use and content of science and mathematics texts. The Commission on Excellence in Education has called for more widespread consumer information services for purchasers of texts. · At the secondary school level, and particularly in mathematics, course titles are a questionable indicator of content studied. The current practice of accepting similar course titles as representing exposure to similar material is likely to produce data of questionable quality.
81 Conclusions and Recommendations Curriculum Conten . There are no established standards for content derived either from past practice, practice elsewhere, anticipated need, or from theoretical constructs devel- oped, say, from the nature of the discipline being taught or from learning theory. Until some consensus can be reached on instructional content that represents desirable alternatives for given learning goals, it is premature to suggest a specific indicator for this area. · Although the identification of an indicator for the content of mathematics and science instruction is not feasible at present, this does not alter the importance of this schooling input. Finding out what content students are exposed to is a necessary first step. . When information on what is currently taught has been collected and analyzed, reviews of the curriculum should be done by scientists, mathematicians, and other experts in the disciplines as well as teachers and educators. The reviews should evaluate material covered at each grade level or by courses, such as first-year algebra or introductory biology; consider relationships among grade levels or courses; and identify the knowledge and skills expected of students at the completion of each grade or course. Such reviews are needed in conjunction wih addressing the critical matter of what content should be taught in mathematics and science. Textbooks and Courses · At a minimum, periodic surveys should be conducted to determine the relative frequency of use of various mathematics and science textbooks at each grade level in elementary school and for science and mathematics courses in secondary school. Timing of surveys should take into account the common cycles of textbook revision. · Surveys of textbook use should be followed by content analyses of the more commonly used texts. Analyses should proceed along several different lines: balance between the learning of recorded knowledge (con- cepts, facts) and its application (process), emphasis given to specific topics, adherence to the logic of a discipline, opportunity and guidance for student discovery of knowledge, incorporation of learning theory.
82 · Intensive studies should collect information from teachers and students on topics actually studied within a given grade or course. Observation of samples of indi- vidual classrooms can help to document the content of instruction. Such studies could help to inform curriculum decisions by local districts, even though the results may not lend themselves to generalization over a state, let alone over the United States as a whole. . Improved definitions of secondary school courses, based on their content, should be developed. AS a first step, use of a standardized course title list, such as the Classification of Secondary School Courses (Evaluation Technologies, Inc., 1982), should be considered. Tests · Critical analysis of standardized tests should continue so as to establish their degree of correspon- dence to the instructional content of the class subjects for which they are used. Consideration should be given to inviting the judgment of teachers (and older students) concerning the students' opportunity to learn the material that is covered on each test.
