What Preparation Do Teachers Have for Teaching Mathematics?
This section presents information about background characteristics
of mathematics teachers, including age and gender, major area of study,
and certification. Teachers’ confidence in teaching various mathematics
topics is also discussed.
As shown by the international average at the bottom of Exhibit
6.1, the majority of the eighth-grade students internationally
were taught mathematics by teachers in their 30s and 40s. If there
were a steady replenishing of the teaching force, one might expect
approximately equivalent percentages of students taught by teachers
in their 20s, 30s, 40s, and 50s. Very few countries, however, had
a comparatively younger teaching force. Internationally on average,
only 16 percent of students were taught by teachers younger than age
30. Although 21 percent of students internationally were taught by
teachers age 50 or older, the teaching force was relatively older
in a number of countries.
Most Benchmarking participants did not differ substantially from
the international profile. However, the Academy School District and
the Jersey City Public Schools had no students with teachers in their
20s and had larger percentages of students with teachers in their
40s and 50s than internationally. Similarly, the Chicago Public Schools,
the Miami-Dade County Public Schools, the Project SMART Consortium,
and the Southwest Pennsylvania Math and Science Collaborative had
more than 65 percent of their students taught by teachers 40 years
or older compared with 54 percent internationally. On the other hand,
the teachers in the Fremont/Lincoln/Westside Public Schools were younger
than the international average – 67 percent of the students had
teachers under age 40 compared with 46 percent internationally.
Internationally on average, 60 percent of eighth-grade students were
taught mathematics by females and 40 percent by males, and similar
percentages were found in a number of countries. None of the 魅影直播
1999 Benchmarking states differed from the international profile of
having more students taught by female mathematics teachers than males.
In South Carolina, in particular, 85 percent of the students were
taught mathematics by female teachers. Among the Benchmarking districts
and consortia, the First in the World Consortium, the Fremont/Lincoln/Westside
Public Schools, Guilford County, and Montgomery County had more than
three-fourths of their students taught by female mathematics teachers.
In comparison, the Michigan Invitational Group, the Naperville School
District, and the Southwest Pennsylvania Math and Science Collaborative
had more male than female teachers.
Exhibit
6.2 presents teachers’ reports about their major areas of
study during their post-secondary teacher preparation programs. Teachers’
undergraduate and graduate studies give some indication of their preparation
to teach mathematics. Also, research shows that
higher achievement in mathematics is associated with teachers having
a bachelor’s and/or master’s degree in mathematics.(1)
According to their teachers, however, U.S. eighth-grade students were
less likely than those in other countries to be taught mathematics
by teachers with a major area of study in mathematics.
On average internationally, 71 percent of students were taught by
teachers who had mathematics as a major area of study. (Note that
teachers can have dual majors, or different majors at the undergraduate
and graduate level.) This compares with 41 percent for the United
States, a figure not too different from that for many Benchmarking
participants, although there was a range of 16 percent in Jersey City
to 73 percent in First in the World and Naperville. Suffice it to say
that in the United States and most Benchmarking entities, a smaller
percentage of students than the international average was taught by
mathematics teachers with a major in mathematics. Canada and Italy
were the only nations that reported lower percentages than the United
States.
Internationally on average, 31 percent of the students were taught
by teachers with mathematics education as a major area of study. In
comparison, more than half of the students were taught by teachers
with this major in the states of Illinois, Michigan, and Pennsylvania,
as well as in the districts and consortia of Chicago, First in the
World, the Fremont/Lincoln/Westside Public Schools, Guilford County,
Project SMART, Rochester, and the Southwest Pennsylvania Math and
Science Collaborative.
Internationally on average, 32 percent of the students were taught
by teachers with education as a major area of study. Significantly
more students in the United States (54 percent) had mathematics teachers
with an education major than did students internationally. In general
across the Benchmarking participants, about twice as many teachers
reported an education major as did internationally. It is clear that
teachers in the United States have less “in field” mathematics
preparation than their counterparts around the world.
To gauge teachers’ confidence in their ability to teach mathematics
topics, 魅影直播 constructed an index of teachers’ confidence in
their preparation to teach mathematics (CPTM), presented in Exhibit
6.3. Teachers were asked how well prepared they felt to teach
each of 12 mathematics topics (e.g., properties of geometric figures,
solving linear equations and inequalities). There were three possible
responses: very well prepared was assigned a value of three, somewhat
prepared two, and not well prepared one. Students were assigned to
the high level of the index if their teachers reported feeling very
well prepared, on average, across the 12 topics (2.75 or higher).
The medium level indicates that teachers reported being somewhat to
well prepared (averages from 2.25 to 2.75), and the low level that
they felt only somewhat prepared or less (averages less than 2.25).
The results show that average mathematics achievement is related
to how well prepared teachers felt they were to teach mathematics,
with higher achievement related to higher levels of teachers’
confidence. On average internationally, teachers reported relatively
high degrees of confidence, with 63 percent of students taught by teachers
who believed they were very well prepared. Interestingly, for the
United States as a whole and most Benchmarking entities, more students
were taught mathematics by teachers confident about their preparation
than in almost all the comparison countries. Interpreting these results
should take several factors into account. For example, cultural issues
may dictate that teachers in the high-scoring Asian countries are
more reserved about reporting their strengths and abilities. Also,
when the mathematics curriculum is more challenging, teachers may
feel less confident in their academic and pedagogical preparation.
Nevertheless, it appears that in relation to both high- and low-performing
countries around the world, teachers in many Benchmarking entities
and in the United States overall may be overconfident about their preparation
to teach eighth-grade mathematics.
Exhibit
R3.1 in the reference section provides the detail for the 12 topics
comprising the confidence in preparation index. On average across countries,
the topics having the most students (from 79 to 82 percent) taught
by teachers who felt very well prepared were “fractions, decimals,
and percentages;” “ratios and proportions;” “perimeter,
area, and volume;” “evaluate and perform operations on algebraic
expressions;” and “solving linear equations and inequalities.”
Teachers reported being least well prepared to teach “simple
probabilities – understanding and calculations;” just more
than half the students internationally (55 percent on average) were
taught by teachers who felt very well prepared to teach this topic.
For the Benchmarking jurisdictions, almost all students had teachers
confident in their preparation to teach the two number topics that
were included in the 魅影直播 questionnaire: “fractions, decimals,
and percentages;” and “ratios and proportions.” Similarly,
in algebra 90 percent or more of students in most Benchmarking entities
were taught by teachers who reported being very well prepared to teach
the three algebra topics: “algebraic representation;” “evaluate
and perform operations on algebraic expressions;” and “solving
linear equations and inequalities.” Similar results were obtained
for the topics “representation and interpretation of data in
graphs, charts, and tables;” and “simple probabilities –
understanding and calculations,” even though teachers in Idaho,
Massachusetts, and North Carolina were less confident about this latter
topic. Teachers also appeared confident in their preparation to teach
“measurement – units, instruments, and accuracy,”except
in North Carolina, the Fremont/Lincoln/Westside Public Schools, Guilford
County, and Rochester, where less than 80 percent of the students
were taught by teachers who felt very well prepared to teach this
topic. The pattern of less confidence in teaching this measurement
topic was found internationally and for the United States.
Teachers in the Benchmarking entities expressed the least confidence
in their preparation to teach geometry. Less than 80 percent of the
students in Idaho, Oregon, the Delaware Science Coalition, and the
Fremont/Lincoln/Westside Public Schools had teachers confident about
their preparation in any of the three geometry topics. Across nearly
all the participating states as well as in a number of the districts
and consortia, teachers expressed less than full confidence in their
preparation to teach “geometric figures – symmetry, motions
and transformations, congruence and similarity.” Interestingly,
this pattern was also noted internationally and for the United States,
even though these topics are included in the curriculum and taught
to substantial percentages of eighth-grade students in the U.S. and
abroad. Beyond those already mentioned, Benchmarking entities where
less than 80 percent of students had teachers confident about their
preparation to teach “coordinate geometry” were Illinois,
Indiana, Missouri, the Jersey City Public Schools, and the Miami-Dade
County Public Schools.
Exhibit
R3.2 shows principals’ opinions about the degree to which
shortages of qualified mathematics teachers affect the capacity to
provide instruction. On average internationally, principals reported
that such shortages affect the quality of instruction some or a lot
for one-third of the students. This compares with 16 percent in the
United States. Benchmarking entities where principals reported that
such shortages affect the capacity to provide instruction for more
than one-fourth of the students were Maryland, South Carolina, Texas,
Chicago, Guilford County, Jersey City, Montgomery County, and Rochester.
Teachers’ beliefs about mathematics learning and instruction
are to some degree related to their preparation. Exhibits
R3.3 and R3.4
in the reference section show the percentages of eighth-grade students
whose mathematics teachers reported certain beliefs about mathematics,
the way mathematics should be taught, and the importance of various
cognitive skills in achieving success in the discipline. In general,
more students in the Benchmarking entities than internationally were
taught by teachers agreeing that mathematics is primarily a formal
way of representing the real world. Conversely, more students internationally
than in the Benchmarking entities had teachers who agreed that some
students have a natural talent for mathematics, and that an effective
teaching approach is to give students having difficulty more practice
by themselves during class. There was nearly complete agreement by
teachers throughout the Benchmarking jurisdictions and around the
world that more than one representation should be used in teaching
a mathematics topic. Views varied substantially, for both the countries
and the Benchmarking entities, regarding the importance of being able
to remember formulas and procedures. Less than one-quarter of the
students in the Delaware Science Coalition (similar to Chinese Taipei
and Korea) were taught by teachers who believed remembering formulas
and procedures was very important for students’ success in mathematics.
In contrast, more than half the students in Idaho, South Carolina,
Guilford County, Jersey City, and Rochester (similar to the Russian
Federation) had teachers who believed this to be the case.
How teachers spend their time in school is determined mainly by school
and district policies and practices, but the perspectives they gain
during their teacher preparation can also have an effect. Across countries,
students’ mathematics teachers spent only about 60 percent of
their formally scheduled school time teaching mathematics (see
Exhibit R3.5 in the reference section). Additionally, about 10
percent was spent teaching subjects other than mathematics, about
10 percent on curriculum planning, and about 20 percent on various
administrative and other duties. The results for the United States
as a whole and for most of the Benchmarking entities were very similar
to the international profile.
