The Shortage of Qualified Math Teachers: A Major Problem and Some Suggested Solutions
Presented November 1011, 2001
A Talk Presented by UCSMP Director Zalman Usiskin at the Seventeenth Annual UCSMP Secondary Conference, November 1011, 2001. This article was published in UCSMP Newsletter No. 30: WinterSpring 2002.
Each year since 1987, when I became overall director of UCSMP, I have been privileged to give a talk in which I examine an issue or question that I feel is important for us to think about as we go about our task of raising a generation of students able to use mathematics and mathematical thinking in their lives. This year my talk is about something that has been a subject of two conferences I've attended within the past 10 days, the shortage of qualified mathematics teachers.
The Extent of the Shortage
What is the extent of the shortage? It is difficult to document. Suppose a school needs n mathematics teachers. If it has n – 1, then there is a shortage. If n = 10, then a district could report that 90% of its mathematics positions are filled but there would still be a shortage.
So we need to look at the question in a different way. In a TIMSS report from 1999, 38% of U.S. school districts reported some sort of shortage of teachers at 8th grade. In some states, the shortage has been severe enough that bounties are paid for mathematics or science teachers. In Illinois, loans are given to prospective teachers in shorthanded fields—mathematics being one of these fields—and the loans can be entirely forgiven if the student teaches the subject for five years.
Classrooms need teachers, and when schools are short of mathematics teachers, they find other teachers to teach mathematics. So another way to quantify the shortage is to ask what percent of teachers are qualified on paper to teach mathematics. Data from the 2000 National Assessment of Educational Progress (Braswell et al. 2001, p. 134) indicate that 43% of the nation’s 8th graders had teachers with an undergraduate degree in mathematics. This agrees with TIMSS data from 1999 that reported 41% of U.S. 8th grade teachers had a mathematics degree. In the NAEP report 26% of 8th grade teachers had an undergraduate degree in mathematics education; in the TIMSS report 37% had such a degree. If we assume that these percents do not overlap (but there probably is some overlap), then between 22% and 31% of the nation’s 8th grade students were taught mathematics by teachers without a degree in mathematics or math education.
Unlike the lower grade levels, NAEP does not collect specific data on teachers’ backgrounds at 12th grade. Data from a Council of Chief State School Officers' 1998 survey indicates that approximately 88% of the secondary school mathematics teachers in the 32 states responding to the survey (Blank and Langesen 1999) were certified in mathematics. However, this percent is suspect because a number of states reported that all or virtually all of their secondary school mathematics teachers were certified, something many people think is impossible.
Whatever the situation, it is likely to get worse before it gets better. In 1999, according to TIMSS, 11% of 8th grade mathematics teachers were under age 30, and 36% were under age 40. If there were equal numbers of teachers at each age from 23 to 60, we would expect 18% and 45%.
Why the Shortage Is a Major Problem
Why not welcome a shortage? You can more easily switch to another district to teach, if you desire. Wherever you teach, you can more likely teach what you want to teach. You will more quickly gain seniority. You are likely to be more highly valued by the administration even if you do not receive more pay.
But if we believe that it is important for all children to learn mathematics, then we must believe that it is important that they learn it well. And it is more likely that a child will learn mathematics well if that child has a welltrained mathematics teacher.
The statement that I just made is one which we cannot afford to take for granted. In times of teacher shortages, and this is one of those times, states and school districts are forced to have mechanisms to ensure that every class has a teacher. Sometimes this is done by special, quickie certification programs, at other times by district waivers. Whatever mechanism is used tends to allow people who are not well trained to be mathematics teachers in the classroom. We must currently live with these mechanisms, but we must endeavor to rectify the current situation so that more people enter our profession and alternative programs provide as much training as standard programs.
The TIMMS 1999 data (see Table 1) provide an interesting look at how shortages and achievement are related. Within the United States, in districts reporting no shortage, 62% of the total, the average scaled score was 514. In the 23% of districts reporting a little shortage, the average scaled score was 497. In the 13% of districts with some shortage, the average was 461, and in the 3% of districts with a lot of shortage the average was 446 (Mullis et al., 2001). This also confirms what is generally wellknown, that poorerperforming school districts have a harder time attracting mathematics teachers. So the shortage of mathematics teachers probably serves to increase inequities among schools.
Table 1: Mean TIMSS 8th Grade Mathematics Scaled Score of U.S School Districts with Various Degrees of Shortage of Qualified Teachers
Degree of Shortage  Percent of U.S. Students  Average Achievement 

None  62  514 
A little  23  497 
Some  13  461 
A lot  3  446 
Source: Ina V.S. Mullis, Michael O. Martin, Eugenio Gonzalez, Kathleen M. O’Connor, Steven J. Chrostowski, Kelvin D. Gregory, Robert A. Garden, and Teresa A. Smith. Mathematics Benchmarking Report: TIMSS 1999—Eighth Grade. Boston, MA: International Study Center, Lynch School of Education, Boston College, 2001.
Do we have evidence that training improves quality? Until recently, we had no evidence! The years 19641967 were a time when new math was being implemented in many schools and there was a great interest in the quality of teachers. At that time, the National Longitudinal Study of Mathematical Abilities (NLSMA) was conducted. This study involved well over 100,000 students. One of the findings of this study was that the number of mathematics courses taken by a mathematics teacher did not influence performance of that teacher’s students. The only teacher variable that affected performance was having recently been enrolled in a teacher institute.
Now we have data that tells us that training helps, at least at the 8th grade level. On National Assessment at the 8th grade level in both 1996 and 2000, students of teachers with mathematics majors or minors performed higher than students of teachers without a mathematics major or minor (Braswell et al. 2001).
These data support one of the beliefs of UCSMP since its inception in 1983, that students from grades 4 and higher should be taught mathematics by specialist teachers. That belief is also supported by the oftcited study of Liping Ma. Ma's study, which compared the mathematical knowledge of elementary school teachers from the United States and China, is usually used by critics of U.S. education to disparage the mathematics training of U.S. elementary school teachers. But I do not interpret Ma's study in this way. The teachers from China in Ma's study were all from the Shanghai prefecture, and in that prefecture it is common practice for mathematics to be taught by specialist teachers from grade 1 up. I believe well over half of the teachers in Ma's study taught only mathematics. Because these teachers do not seem to have had any special training in their teacher preparation programs, I see Ma's study as providing evidence that being a specialist teacher of mathematics causes the teacher’s knowledge of mathematics to improve.
Ma's result, together with the NAEP results, suggest very strongly that an elementary school student would be likely to learn more mathematics if taught by a specialist mathematics teacher. Thus, the more qualified the teaching force, the better prepared that students will be when they arrive at middle school and high school.
Why the Problem Is so Serious Now
The memory of society is very short. If times are bad, there is a tendency to think they are worse than ever before. I am only willing to say that the problem is the worst it has been at any time since before 1970.
A similar problem of a lack of wellqualified teachers existed in the 1960s during the cold war. The Soviet Union had detonated atom bombs in the 1950s. When the USSR put up large artificial satellites beginning in 1957, and orbited men beginning in 1961, it showed its capability to deliver such bombs anywhere on the globe. We needed to compete scientifically and we could only do so with teachers welltrained in science and mathematics. We put money into the training and retraining of mathematics and science teachers. There never became a surplus of mathematics teachers like the surpluses that came to exist in other teaching areas, but by the early 1970s it is my impression that a shortage of mathematics teachers existed only in some remote rural areas and in some inner cities, and a surplus existed only in the suburbs.
At that time, noncollege bound high school students usually took general mathematics as high school freshmen, and consumer mathematics as sophomores. Many high school mathematics departments filled these teaching positions with teachers not welltrained in mathematics, for it was viewed that a person did not need to know much mathematics to teach those courses. So we were content with having only enough high school teachers well qualified to teach collegebound students.
The existence of algebra as an 8th grade course began in most areas of the country in the 1960s or early 1970s and typically involved one 8th grade class in a school. The teacher of this class might have needed some retraining, but other teachers could be content with knowledge of little more than arithmetic.
Now we have a different situation at all three levels, elementary, middle and high school. We are asking elementary school teachers to teach not only arithmetic, but geometry and measurement and even a little algebra and probability and statistics, and to do it emphasizing not rote skills, but problemsolving and reasoning and communication and representation and connections with other subjects. At the same time, science educators are asking these same teachers to teach the basics of biological and physical sciences through inquiry and experiments. Likewise the teaching of social studies has become more complex. And society is expecting greater competence in reading and language arts as evidenced by programs trying to ensure that all students read by the end of grade 3. Teachers are also expected to provide drug education and sex education. It is unrealistic to expect our elementary school teachers to be so widely versed. Even if we were able to convince enough extraordinarily welleducated people to teach in our elementary schools, few people have the time to create rich lessons in all these areas and still have a life.
A century ago when teacher certification took its current shape, the teacher of 6th, 7th, or 8th grade needed only to know arithmetic well. Furthermore, the lessons consisted mainly of practicing algorithms and solving routine problems. A generalist K–8 teacher often possessed that knowledge. The entry of algebra into one class by 1980 only perturbed the situation slightly.
Table 2 shows the dramatic change in the 20 years since. In 1981, in the Second International Mathematics Study, 76% of 8th grade classes were classified either as typical, meaning that arithmetic dominated their curriculum, or as remedial, meaning that K–6 arithmetic dominated. Only about half as many students in the year 2000 were in such classes.
Table 2: Percents of 8thgrade students in various types of classes
SIMS1981 (% of classes)  
Algebra 13 
Enriched 11 
Typical 66 
Remedial 10 

NAEP1986 (% of students)  
Algebra 16 
Prealg. 19 
Reg.Math 61 
Other 5 

NAEP1996  
Algebra 20 
Prealg. 36 
Reg.Math 39 
Other 5 

NAEP2000  
G or AA 3 
Algebra 25 
Prealg. 31 
Reg.Math 37 
Other 3 
The column with algebra is also revealing. On the 2000 NAEP, 25% of 8th grade students reported taking Algebra, 3% reported taking either Geometry or Advanced Algebra, and 2% (not shown on the table) reported taking an integrated mathematics program (which might be a high school integrated program started at 8th grade). We at UCSMP take some pleasure in this change; we were the first curriculum and remain the only curriculum with an algebra course designed as much for 8th grade as for 9th grade.
Algebra is a key topic for all of high school mathematics. To teach firstyear algebra well, a teacher needs to know how the simpler skills and problems in that course are used in more complex skills and problems to be studied in later years, how algebra is used in geometry, and how the study of functions fits in. Thus middle school teachers now not only need to have a broad knowledge of mathematics and how to teach it, but 8th grade teachers need to have a broad knowledge of the teaching of high school mathematics.
The taking of algebra at 8th grade has had the expected effect on coursetaking at the high school level, and more.
It is quite difficult to determine what mathematics U.S. students are taking in high school. NAEP data has historically asked students to indicate the highest mathematics course taken. From this we can roughly determine the percents of students who have enrolled in each high school mathematics course. These percents have been rising steadily for 30 years, as seen in Table 3.
Table 3: Percent of 17YearOld Students Reporting Highest Level of Mathematics Course Taken, as reported by National Assessment
1978  1990  1994  1996  

PreAlgebra or General Mathematics  20  15  9  8* 
Algebra I  17  15  15  12* 
Geometry  16  15  15  16 
Algebra II  37  44  47  50* 
Precalculus or Calculus  6  8  13  13* 
*Indicates that the % in 1996 is significantly different from that in 1978. Source: NAEP Trends in Academic Progress (Campbell, Voelkl, and Donahue, 1997; Campbell et al., 1996; Mullis et al., 1991).
From the reports of the 12th graders in 1996, 92% had taken or were enrolled in firstyear algebra, 79% geometry, 63% a second year of algebra, and 13% precalculus or calculus. The datum on precalculus or calculus enrollments disagrees with TIMSS data for enrollments collected a year earlier in which 22% of 12thgrade students were classified as taking or having taken precalculus or calculus (Mullis et al., 1998, p. 19).
On the 2000 NAEP, the coursetaking question was changed from prior years. Students were asked to identify the course they had taken in each of grades 8 through 12. From their responses, 94% report taking a year of algebra, 88% report taking a year of geometry, and 80% report taking a second year of algebra. That last statistic seems quite high to me, and I wonder if some students are not reporting a second year of firstyear algebra as if it were what we might call "advanced algebra." Responding to the same question, 37% reported taking precalculus, 18% reported taking calculus, and 18% reported taking a course in statistics (Braswell et al., 2001, p. 169). Examining individual student responses, NAEP evaluators found that 50% of students reported taking either trigonometry, precalculus, statistics, discrete mathematics, or calculus before or during 12th grade.
The NAEP percentages all seem high, but they are not as high as those found from reports of students taking the SAT. SATtakers are not a random sample of collegeintending students since weaker students often do not take them, and in nonSAT states only the best students take the SAT. But SAT data can indicate whether the NAEP data are on track. Last year, 24% of seniors taking the SAT reported taking calculus before graduation, and 45% reported taking precalculus. These numbers are up from 19% for calculus and 32% for precalculus ten years ago.
The most reliable data come from various studies of transcripts done by the U.S. Department of Education. These data tend to support these high percents. Table 4 shows that 26.2% of students in 1978 took mathematics beyond a second year of algebra, while 41.4% took such mathematics in 1998. In this transcript report, 14.4% of graduating seniors in 1998 took calculus.
Table 4: Coursetaking in Advanced Mathematics (from various transcript studies)
Year  Level I (Adv. Alg., Trig.) 
Level II (Precalculus) 
Level III (Calculus) 

1982  15.5  4.8  5.9 
1987  12.9  9.0  7.6 
1990  12.9  10.4  7.2 
1992  16.4  10.9  10.7 
1994  16.3  11.6  10.2 
1998  14.4  15.2  11.8 
Source: National Center for Education Statistics, The Condition of Education 2000, pp. 66, 157, 216.
The cluster of statistics on enrollments yields a robust conclusion: students are taking more mathematics now than they did ten years ago. The result is that high school mathematics course offerings look quite different now than they did 20 years ago. Geometry is now the most offered mathematics course at the high school level. Secondyear algebra rivals algebra for second place. Significant numbers of students are taking courses beyond secondyear algebra. We at UCSMP take pride in these numbers, too, because we think we had something to do with the trend. We just wish that more people would recognize that we support 8th grade algebra for large numbers of students not because we think all of them should take calculus in high school, but because we think most students should study mathematics beginning with algebra for 4 years before calculus. This is the only way for many students to obtain the broad kind of mathematics education needed today.
The pressure on high schools for welltrained mathematics teachers thus comes from a number of major changes in the curriculum. Very few students now take courses below algebra. Students are taking more collegepreparatory mathematics than ever before. And because more students are taking algebra in 8th grade, what mathematics they take in high school is more advanced. On top of this there has been a major change in the kind of instruction that is considered best for students, from instruction that is mostly lecture and explanation, to instruction that engages the students in discovery, problemsolving, and discussion. And on top of that, the standards have been raised from facility with skills to facility with problems. A teacher who might have been reasonably qualified to teach in 1980 can easily lack the qualifications needed by a teacher today.
Reducing the Shortage
Let me begin with the obvious. To reduce the shortage of qualified mathematics teachers, we need to train more mathematics teachers and we need to keep good teachers in the classroom longer.
Tout the Profession
To train more mathematics teachers, we need to attract more people to our profession. Teachers play an important role. I for one decided to teach when I was an 8th grader because I had a teacher, Mrs. Wright ("My name is Wright and I'm always right" was her motto), whom I adored and who seemed to love her job. And then in 9th grade I decided to teach mathematics because I liked math but I had a teacher who destroyed the subject for so many students. My decision to go into teaching was reinforced every time I encountered a poor teacher; I felt so strongly I could do a better job. I couldn’t wait to studentteach to see if this was true.
If you enjoy your job, then when you have students who show a desire to teach others, talk to them about teaching. Play up our field. Mathematics is one of the noblest enterprises of humankind. It is interesting, useful, and beautiful. Mathematics is a part of literacy and knowing a good amount is part of being a literate citizen. It is too important for people to be content with mathematical ignorance.
Mathematics has special aspects that differentiate teaching it from the teaching of other subjects. Point out the enormous influence a mathematics teacher can have because, unlike reading and writing and social studies, which are often learned as much out of school as in it, almost all the mathematics a person knows is learned in school. And mathematics is particularly interesting to teach. Some students have attitudes towards mathematics quite unlike their attitudes towards any other subject: they may have math anxiety, or dyscalcula, or disdain that they would never have towards reading, writing, or science. They may have been told by a parent to expect not to be able to succeed in mathematics because the parent didn't. At the other end of the spectrum, many students enjoy doing mathematics and force a teacher to look for interesting or challenging problems. And in the middle are the vast majority of students, who need to know mathematics and can be swayed either to like the subject or dislike it depending on the quality of instruction.
Find a reason to bring students to your local meetings of mathematics teachers, so that they can see that teachers do things other than teach, and that there is a community of nice people who teach.
A single teacher can have enormous influence. We have a quite small MAT teachertraining program here at the university. Over the years, I have had at least four students in the MAT program who attribute their decision to go into the teaching of mathematics at least in part because of John Benson, one of the Great Teachers we have with us today. I am sure other students of his went into teaching but studied elsewhere.
Reduce the Barriers to Entry
We also need to reduce the barriers that keep good and interested students from entering teaching. Many students who might make great teachers—who are very bright—are turned off either by the content of mathematics beyond calculus or by the way that content is taught. They came to college having enjoyed mathematics, and possibly thinking of majoring in mathematics, but they find themselves in courses that seem to have nothing to do with the mathematics they enjoyed. They may follow a curriculum designed for those going on to do research in mathematics even though they have no interest in research. At last year's conference, I spoke of the special mathematics needed by teachers. Since then, the Conference Board of Mathematical Sciences has recommended special mathematics courses for all teachers (CBMS 2000). If advertised widely, these courses might be attractive enough for students interested in teaching to keep them in mathematics.
Requirements for certification present a problem in many states. Virtually every state has requirements for three types of courses: subject matter (in our case, mathematics), professional education (methods, educational psychology, etc.), and general education (a couple of courses here, some there, etc.). The subject matter and professional education requirements connect to the job. But the general education requirements seldom do. Why should every teacher in Texas have to take a course in the history of Texas? Why should every teacher in Illinois have to take a course in political science? Every college that I know with a mathematics department has distribution requirements for its bachelor's degrees. The existence of the undergraduate degree should be taken as evidence that a prospective teacher has had a broad education. This would make it much easier for many people with college degrees interested in teaching to try to enter the profession. In mathematics and science, this would enable engineering graduates to more quickly obtain certification.
Examinations that prospective teachers must pass pose a different problem. Of course, all teachers should be competent in reading and writing. A mathematics teacher ought to be competent in the mathematics the teacher is certified to teach. But it makes little sense for a teacher to be certified K–8 if the teacher only wishes to teach kindergarten or first grade. When there is an exam, one way to increase the number of those who pass is to tailor the exam more specifically to what the teacher will teach. Some people may want the flexibility of being able to teach at all grades K–8. Some districts may want their teachers to have that flexibility. Then it makes sense for a teacher to have to take more tests to show competence at all those grades. But it seems quite reasonable to have broad teachertraining programs in colleges and then have exams that are more specific.
The mathematics needed to teach the grade intervals K–2, 3–5, and 6–8 varies so much that the NCTM Principles and Standards split their descriptions for these grades. So perhaps there should be three different tests for teaching mathematics in grades K–8. Certainly there should be a closer match between what we put on those tests and what the teacher needs to know in order to teach what he or she plans to teach.
An eternal problem is that the cutoffs on these teacherinduction tests are arbitrarily set. In Illinois, there is now a debate whether a prospective teacher needs to obtain above a particular percent correct on each subject part, or whether the parts can be combined and the prospective teacher needs only to obtain a global score higher than some percent. Just two days ago the Superintendent of Public Instruction noted that if the global cut score is placed at 50%, then only 70% of people overall who take the test will pass, and only 30% of the minority candidates.
There is no credibility to such results. On any test, a testmaker can write simple questions that will yield high percents of correct responses and subtly complex questions on the same topic that will yield lower percents of correct responses. A testmaker can manipulate any test with sufficiently many questions to raise or lower the percent of students who will pass. A more objective criterion is needed. But even without an objective criterion, the more information given about the test to those who take the test, the greater number that will pass. In Illinois, this was the first year the socalled "harder test" was given, so there was little guidance regarding what it would cover or how students would fare.
Reward Rather Than Punish
Another problem is that the mere fact of having such a test, particular a "basic skills" test, demeans the profession. Why should a smart individual enter a profession that requires a basic skills test for certification? One possibility is to sweeten the pot by providing cash rewards for individuals who take the test and score the highest.
In the same vein of favoring positive reinforcement over negative reinforcement, once a teacher is on the job, states should not have penalties for not meeting some requirement for teacher renewal. As strongly as I believe there is a need for teachers to engage in professional growth, to give a penalty for nonengagement is to demean the profession. Even though the law changes constantly, we do not require lawyers to engage in professional growth to keep their license. Similarly, once a CPA, always a CPA unless there is malpractice. Only if we were in a time of surplus, might we be able to afford the luxury of culling the profession for reasons other than incompetence. If we must have teacher recertification, then give a bonus for engaging in professional activities rather than a punishment for not engaging in them.
Improve the Conditions of Teaching
Obviously one can help to reduce the shortage of teachers by improving the conditions of teaching itself.
The variance in working conditions among teachers is quite large. In many schools, teachers are treated as the professionals they are. They do not make the salaries of doctors but they can live comfortably on their salaries. They have a nice computer and a phone on their desk. They can attend professional meetings and take additional courses either partially or wholly at school expense. Benefits are competitive. There are two or three free periods a day in which to help students and prepare lessons. In other places the job is carried out under farlessthanoptimal conditions. We must be honest in describing job conditions to prospective teachers, but we do not have to say to them what we would say at a bargaining session with the administration. The more awareness that can be raised about treating teachers as professionals, the more likely we are to reduce the shortage.
Why Is It So Difficult to Prepare Good Mathematics Teachers?
This question was posed at a meeting in Washington last week billed as an education summit on the mathematics education of teachers. None of three panelists asked to answer it tackled the question. Perhaps they did not want to give the answer, which is actually quite easy. It is difficult to be a good mathematics teacher. It takes education, experience, lots of planning, and great skill to carry out the plan.
In fact, it may be that the lower the grade level the more difficult it is to be good, because students at lower grades know less and can do less, and because teachers at lower grades tend to be given less time to prepare lessons.
Good teaching requires not only knowing content and pedagogy but being able to match those to the students one is teaching. Compared to teaching in a middle school or high school, teaching at the college level is a snap. A professor can succeed knowing only the content. Pedagogy is ignored by resorting to lectures. Students are viewed as having to show their stuff regardless of the quality of teaching. There is no open house for parents, no accountability.
My son is a senior at one of the top universities in the country. It hurts me to hear how often he complains about the quality of teaching there, about the poor quality of the lectures from people who know their subject well. More professors than we college faculty would like to admit think that all they have to do is exposit their subject without error, and this constitutes adequate teaching.
Mathematics seems particularly sensitive to poor teaching. Some years ago I was asked by our Graduate School of Business to speak to incoming faculty about the teaching of mathematics. Why did they invite me to come in? The business school considered the quality of teaching to be important for its competitive success. Reports from students indicated that students felt that in courses that utilized a good deal of mathematics, the professors did not explain the mathematics well. Let us emphasize the point: knowing mathematics well does not at all guarantee that you can teach it well. In mathematical terms, knowing mathematics may be necessary but is not sufficient for teaching it well.
Summary
Perhaps what is needed most is for us to remind the public of how great a job we perform. Not only can we do mathematics, we can teach it. Ask someone in business if they would like to direct 5 meetings a day, each of length 4055 minutes, and then be held accountable for what the people who attended each meeting got out of it. Ask a college professor if he or she would be willing to have someone from the outside come in and give his students a test they have never seen, with the results published in a newspaper. Teaching mathematics used to require mainly the ability to explain algorithms and grade students on their ability to carry them out. Now it requires a breadth of knowledge including statistics as well as mathematics, applied mathematics as well as pure, in all areas of elementary mathematics.
Mathematics is essential in today’s world. Good mathematics teachers not only count, but they measure up well to any other professionals. Good mathematics teachers are important figures in the community; they help to solve problems of society; they help society to function; they are not just another statistic.
A job that seems difficult to those who do not do it is not necessarily a difficult job to those who do it. I point out the many aspects of teaching to indicate how hard good teachers work and how well they perform, not to brand teachers as martyrs. I will repeat: it takes great knowledge and skill to be a good teacher. To reduce the teacher shortage we need to get that message across to all of society and reward teachers commensurately.
References
Braswell, James S., Anthony D. Lutkus, Wendy S. Grigg, Shari L. Santapau, Brenda TayLim, and Matthew Johnson. The Nation's Report Card: Mathematics 2000. Washington, DC: U.S. Department of Education, National Center for Education Statistics Report NCES 2001517 (2001).
Campbell, Jay R., Kristin Voelkl, and Patricia Donahue. NAEP 1996 Trends in Academic Progress. Washington, DC: National Center for Education Statistics, September 1997.
Campbell, Jay R., Clyde M. Reese, Christine O'Sullivan, and John A. Dossey. NAEP 1994 Trends in Academic Progress. Washington, DC: National Center for Education Statistics, 1996.
Conference Board of the Mathematical Sciences (CBMS). The Mathematical Education of Teachers. Providence, RI: American Mathematical Society, 2001.
Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. Mahway, NJ: Lawrence Erlbaum Publishers, 1999.
Mullis, Ina V. S., John A. Dossey, Mary A. Foertsch, Lee R. Jones, and Claudia A. Gentile. Trends in Academic Progress. Washington, DC: National Center for Education Statistics, 1991.
National Center for Education Statistics, U.S. Department of Education, Office of Educational Research and Improvement. The Condition of Education 2000. NCES 2000062. Washington, DC: U.S. Goverment Printing Office, 2000.
Contact
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