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 second 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
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.
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
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
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
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
results suggested this to be the case more generally, which clearly does
not bode well for equality of learning experiences for students in
In this section, we summarize what teachers have told us about their
preparation in mathematics at the college level and as graduate
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
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
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.
Ask Scientific American on Twitter @SciAm and @SciamBlogs. why I can't comment... © 2012 ScientificAmerican.com. All rights reserved
My Response: Maths are Creative, Pacing Guides are Absurd
hate to challenge a Michigan State colleague, but I work with schools
both in Michigan and "around Charlottesville, Virginia," and I disagree
with a number of the assumptions suggested here.
first, of course, is that I'm with Conrad Wolfram on the concept that
"order" is really not important in maths education (you can find
Wolfram’s TED talk on YouTube or via my blog post “Changing Gears: Maths are creative, Maths are not arithmetic”) and thus I strongly
disagree with those who might measure either students or schools
according to some pre-determined pacing guide.
other assumption which I find flawed is the comparative measurement
structure suggested here. I see low quality maths instruction in many
places, I see high quality maths instruction many places, but mostly I
see that high quality maths instruction is suggested when students have a
deep grasp of mathematics concepts no matter what their arithmetical
skills are... and this I am quite certain we have been building in the
schools around Charlottesville (see my blog post “Among Schoolchildren”).
are doing that by embracing a different concept of professional
development, one based in "teacher entrepreneurship," social engagement,
and grounded theory action research combined with a view that we look
at the whole student "result" not any single measure. We do this because
we believe that students are humans who develop different skills at
for TIMSS and other “international comparatives,” I wish to draw
everyone’s attention to the work of Yong Zhao who demonstrates the inverse relationship between international standardized test score results and national economic creativity (see slide 8/8). The nations which score lowest
on these exams lead the world in patents per capita, and new product
and service introductions per capita. It is the choice between being a
nation which invents iPads and a nation which builds iPads at slave
Charlottesville” our goal is to raise a generation ready to lead, to
invent, to solve problems. It is not our goal to raise test takers. And
it is our goal to help teachers develop into educators who can support
- Ira Socol