* *

*When William Schmidt, an expert on math education at Michigan State University, moved his family from East Lansing to Charlottesville, Virginia for a year s research leave, his work took a personal turn. He noticed that the public school his daughters would be attending outside Charlottesville was academically behind the one they had attended in Michigan. Back home, his 2 ^{nd} grade daughter would be learning multiplication tables up through the number 5, yet in Charlottesville, multiplication was not even part of his local school s second grade curriculum.*

*His daughter s experience, he explains in a new book excerpted below, is not unique. The [American] system of schooling represents a game of chance that few are even aware is being played, he writes in Inequality for All: The Challenge of Unequal Opportunity in American Schools, co- written with Curtis C. McKnight. The inequalities pose a risk to every child, they write, regardless of socioeconomic background or race. They stem from differences in state education standards, in school funding, in curricula that districts choose to adopt and in the content that individual classroom teachers choose to teach. In this excerpt, Schmidt and McKnight focus on variations in how math teachers are trained and how that, in turn, affects student achievement.*

*The following is excerpted from Inequality for All: The Challenge of Unequal Opportunity in American Schools, by William H. Schmidt and Curtis C. McKnight. (Teachers College Press, 2012).*

One thing that most of us remember best about school is our teachers. Thus, when solutions are proposed for reforming American schools in response to critical reports or disappointing test results, teachers are always among the first to be singled out. Proposals often turn first to improving the teaching force by focusing on higher quality. For example, in the NCLB [No Child Left Behind] era, considerable emphasis has been placed on a highly qualified teaching force.

Districts must certify what percentage of their teachers is highly qualified. However, the states and school districts define what they consider highly qualified, resulting in a great deal of ambiguity.

What does it mean to be a highly qualified teacher? The definition of a high-quality mathematics teacher has never been standardized. Therefore, although improving the quality of teachers and teaching is a common cry when we seek to improve schools, there is little agreement and scant empirical evidence that indicates what characteristics define a high-quality mathematics teacher. Even an obvious definition, such as a knowledge of mathematics, is problematic, since there is generally no agreement as to what specific mathematics knowledge is needed.

The literature identifies two types of knowledge that are clearly related to providing opportunities to learn: mathematics content knowledge and pedagogical content knowledge. For mathematics, recent empirical work has advanced our understanding of what mathematics knowledge is necessary for teaching mathematics. The rest of this chapter lays out how this knowledge in particular, mathematics content knowledge is related to inequalities in content coverage, and provides data related to teacher content knowledge for a sample of teachers.

**What Teachers Tell Us About Their Knowledge of Mathematics**

We approached the question of teacher knowledge of specific mathematics content indirectly, by asking a sample of more than 4,000 teachers from the PROM/SE project [Promoting Rigorous Outcomes in Math and Science Education] to respond to the question, How well prepared academically do you feel you are that is, you feel you have the necessary disciplinary coursework and understanding to teach each of the following? We asked this question for multiple mathematics topics. The list of topics varied for teachers in primary and teachers in middle and secondary school combined.

By relying on teachers reports of their own feelings of adequate preparation, we only get at their knowledge indirectly. Fortunately, this approach is sufficient to demonstrate how much variation there is in teachers content-specific knowledge, or at least in their feelings of adequate preparation. Furthermore, the candor of the results suggests a degree of face validity and, hence, integrity in the responses. The overall tenor of the responses is very consistent with other data on the issue, some of which suggest that the pattern reported here might be a best-case scenario. All results reported in this chapter are based on the PROM/SE data.

**Primary Teachers (1st Through 3rd Grades)**

Primary teachers felt academically prepared to teach only the topics they taught to their students. Even for those topics, about one-fourth to one-half of the teachers surveyed reported that they did not feel well prepared. The teachers we surveyed were from 60 PROM/SE districts located in Michigan and Ohio.

Is it reasonable for teachers to focus only on the topics that they will teach? However reasonable such a position may appear, many of the more advanced topics for which teachers did not feel well prepared provide the mathematics background necessary to be truly well prepared to teach the more elementary topics at their grade level. To define a qualified teaching force, we adopted a criterion of 75% of teachers feeling well prepared to teach a given topic. We found that, over all sampled teachers, only two mathematics topics met this criterion: the meaning of whole numbers, including place value and operations with whole numbers.

Virtually all of the geometry topics (aside from the basics) are excluded by the 75% criterion. So are all of the proportionality topics and all of the algebra topics. These results imply that the quality of learning opportunities surrounding many of the mathematics topics taught in 1st through 3rd grades was not likely to be high. They also suggest that there is large variability in self-reported content-specific knowledge. For many of these topics, only a bare majority of 50% to 60% of teachers felt well prepared.

The other striking feature of the results was the large variability across districts. For example, for fractions, in some districts all of the primary teachers felt very well prepared, while in other districts only about half of the teachers felt very well prepared. For geometry basics (lines, angles, and so on), the results ranged from one district with only about one-fourth of its teachers feeling well prepared to another district in which about 90% of the teachers felt well prepared.

**Upper Elementary Teachers (4th Through 5th Grades)**

The results for districts for 4th- and 5th-grade teachers were quite different. For example, for eight different topics, all of the teachers in at least one district felt very well prepared academically for each of those topics.

At the district level, the results for whole number meaning and operations were similar to those for 1st- through 3rd-grade teachers. Further, fractions also had a median value around 75%.

However, the variability across districts remains a striking feature for 4th- and 5th-grade teachers, particularly for decimals, percentages, and geometry basics. These are all topics that were supposed to be introduced in these grades in Michigan and Ohio. For example, for decimals, in one district only one-fourth of the teachers felt well prepared, while in another district virtually all teachers indicated that they felt well prepared to teach decimals.

**Middle School Teachers (6th Through 8th Grades)**

We examined the pool of teachers from all of the districts taken together. From this perspective, there were no topics that at least 75% of the teachers felt very well prepared to teach. Only two topics came close. Among the whole pool of teachers, 73% indicated that they felt well prepared to teach the topic of coordinates and lines. Sixty-nine percent of the teachers indicated that they felt well prepared to teach the topic of data.

Eleven topics qualify if we relax the criterion to topics in which at least 50% of the teachers felt well prepared. This included the two topics just mentioned as well as nine others negative, rational, and real numbers; exponents, roots, and radicals; number theory; polygons and circles; congruence and similarity; proportionality problems; patterns and relations; expressions and simple equations; and linear equalities and inequalities. The Michigan and Ohio standards call for including many of these topics at the middle grades. The fact that only about 50% to 60% of the teachers felt very well prepared to teach these topics suggested something of the magnitude of the problem that school districts face.

For example, there has been a strong national movement to include elementary algebra topics in the middle school, particularly in 8th grade. The Michigan and Ohio standards reflect this, as do the Common Core State Standards, which are in the process of becoming the new Michigan and Ohio state standards. Adoption of the Common Core State Standards brings states more into alignment with international benchmarks of what is expected in the equivalent of middle school.

The severity of the problems faced by these districts and, by inference, by the United States as a whole, was indicated by the fact that only about half of the teachers felt academically very well prepared to teach expressions and simple equations, as well as linear equalities and inequalities. Even fewer teachers (only around 25% to 40%) felt they had adequate content knowledge to teach other important algebraic concepts, including proportionality (41% of teachers), slope (38%), and functions (39%).

**High School Teachers**

The story for high school teachers is rather different, which is not unexpected given their typically greater preparation in mathematics. Almost 60% of the topics met the criterion of having at least 75% of the pool of PROM/SE teachers from the 60 districts indicating that they were well prepared academically. The areas in which high school teachers indicated that they felt less well prepared were number bases, three-dimensional geometry, geometric transformations, logarithmic and trigonometric functions, probability, and calculus. These findings are, however, still cause for concern. For example, there is an increasingly strong push for the inclusion of probability and statistics in high school, as is found in the Common Core State Standards, yet less than half of the surveyed mathematics teachers felt well prepared to teach it. Teachers self-perceptions of their preparedness seem likely, if anything, to overestimate what they know and how well prepared they are rather than to underestimate it.

Moving from the pool of all 60 districts to the district-by-district results for a large number of topics (16), at least 25% of the districts had all of their high school teachers indicating that they felt well prepared to teach those topics. However, there was still great variation across the districts, especially for geometry topics including transformations, three-dimensional geometry, polygons, and circles. There was similarly great variability in the percentage of teachers who felt that they had the coursework to make them well prepared to deal with calculus, probability, number theory, and logarithmic and trigonometric functions.

**Why Teachers Feel So Poorly Prepared**

We have surveyed how well prepared in terms of disciplinary course work teachers at various levels felt for teaching various mathematics topics in what is a fairly representative sample of 60 districts. In general, we would summarize the findings by stating that many teachers felt ill prepared to teach mathematics topics that are in state standards and in the new Common Core State Standards for mathematics. Why did these teachers feel so ill prepared?

There is perhaps a simple answer for the elementary and middle school teachers: They felt ill prepared because if we examine the coursework they studied during their teacher preparation, they were ill prepared. The new TEDS study results suggested this to be the case more generally, which clearly does not bode well for equality of learning experiences for students in these districts.

**College-Level Preparation**

In this section, we summarize what teachers have told us about their preparation in mathematics at the college level and as graduate students.

In 1st through 4th grades, less than 10% have either a major or a minor in mathematics. Teachers at this level are typically generalists they must be prepared to teach many different subject-matter areas. They do not have adequate time in their preparation to get a major or a minor in each of those subject matters.

At 4th grade, the international data paint a different picture. Unfortunately, the definitions are not precisely the same, but the data do provide us with a benchmark of sorts. Including those primary teachers with either a mathematics major or a minor in mathematics or science, around one-third of 4th-grade students on average had such a teacher in the countries that took part in TIMSS, [the Trends in International Mathematics and Science Study].

Taking this estimate from the TIMSS data as an indicator of the percentage of teachers who had majored or minored in mathematics or science, this proportion was considerably higher than for the PROM/SE 4th-grade teachers, where the comparable percentage was 5%. The percentage was over 50% in Singapore and Russia. This suggests that, from an international perspective, other countries typically have around six times as many primary teachers who have a specialization in mathematics or in a related field of science.

The result is even more disturbing when we turn to the middle school mathematics curriculum and the higher level of mathematics offered there.

Three out of four middle school teachers in the PROM/SE sample did not have a specialization in mathematics. At 6th grade, the percentage was much like that for primary teachers only around 10% had a major or minor in mathematics. In 7th and 8th grade, this percentage increased to around 35% to 40%.

These numbers indicate that a very large percentage of middle school students were being taught increasingly more complex mathematics, as called for in the Michigan and Ohio state standards, by teachers who lacked a strong background in mathematics. These results offer one explanation for why so many middle school teachers did not feel very well prepared to teach many of the middle school topics discussed in the previous section. This also foreshadows problems of implementation, at least as the newly adopted Common Core State Standards are put in place, in Michigan and Ohio.

What about high school mathematics teachers? We would expect that all high school mathematics teachers would have at least a minor in mathematics, if not a major. But the actual results for high school are quite surprising. Less than half of all high school mathematics teachers surveyed had a major in mathematics. Almost one-third did not have either a major or a minor in mathematics.

These numbers varied across the four grades of high school taught by the surveyed teachers. Almost one-half of the teachers whose major teaching responsibilities were at 9th or 10th grade did not have any specialization in mathematics. In 11th and 12th grades, over 71% of the teachers who taught primarily at those grades had some kind of specialization in mathematics.

Lest it seem too heartening that those teaching the most advanced courses (usually taken in the 11th and 12th grades) are better prepared in mathematics, we need to consider several caveats. It may be even more important to have well prepared teachers in entry-level courses usually taught in 9th and 10th grades. These courses serve as the foundation for more-advanced courses, may be even more difficult to teach, and are just as important in terms of preparing students for further study. But for these foundational courses, teacher content knowledge was not nearly as strong. It is worth noting that on some of the more advanced mathematics topics (number theory, geometric transformations, logarithmic and trigonometric functions, and calculus) up to half of the teachers did not feel very well prepared to teach them. Perhaps these same 50% were those who did not have a major in mathematics.

**Mathematics Knowledge**

One key part of the PROM/SE project was planning and carrying out content-based capacity building for teachers. As a part of this component, we administered a test of mathematics knowledge to a sample of teachers. [The results] strongly suggest that elementary and middle school teachers perceived their weaknesses accurately and reported them honestly. They appeared to be reporting that they were not well prepared academically to teach the mathematics content that they were being asked to teach.

Across grades, the percentage of teachers who did not have a major or minor in mathematics ranged from nearly all of the teachers at 1st grade to around one-half of them at 8th grade. These same teachers were able to answer correctly only about half of the items, as compared with teachers with mathematics majors who were teaching at the corresponding grades. The teachers with mathematics majors were able to correctly answer about 70% of the same items. This gap of almost 20% is sizable and very important. It confirms what the teachers told us when they said that they were not well prepared.

The problem at high school is more a problem of variability. The data indicated that most of the teachers had mathematics majors and that their mathematics knowledge was reasonably good. However, about one-third still did not have strong academic preparation.

**The Effects of Teachers Mathematics Knowledge on Opportunities to Learn**

Given these results about teachers mathematics knowledge, it is tempting to blame elementary and middle school teachers for not being prepared, but we believe blaming teachers is a mistake. Why? Because teachers prepare themselves according to the standards and guidelines established by the states that certify them and the teacher preparation programs that train them. Our point here is that such variation in academic content knowledge is likely to affect the quality of content coverage. Since the content coverage described previously varied appreciably, these data indicate one possible reason for such variation as it is very likely that this lack of knowledge influences not only the quality of the coverage of particular topics but also the bigger picture as to how the teacher makes choices about which topics to cover, for how long (to what depth), and in what sequence. Such

lack of knowledge further exacerbates the variation in content coverage in mathematics across classrooms, schools, and districts, resulting in further inequalities in opportunities to learn.

Used with permission from the Publisher. From Schmidt & McKnight, Inequality for All: The Challenge of Unequal Opportunity in American Schools, New York: Teachers College Press, 2012 by Teachers College, Columbia University. All rights reserved.

* Follow Scientific American on Twitter @SciAm and @SciamBlogs. Visit ScientificAmerican.com for the latest in science, health and technology news. © 2012 ScientificAmerican.com. All rights reserved. *