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No Common Denominator

The Preparation of Elementary Teachers in
Mathematics by America’s Education Schools

Executive Summary
June 2008

National council on teacher quality
The full report of No Common Denominator: The Preparation of Elementary Teachers in Mathematics
by America’s Education Schools is available online from www.nctq.org.

authors:
Julie Greenberg and Kate Walsh

our thanks to:
Research analysts: Emmanuel Caudillo, Aileen Corso, Elizabeth McCorry, Stephanie Parry,
Felicity Messner Ross, Michael Savoy, and Nate Sheely
Database design and technical support: Jeff Hale
Graphic design: Colleen Hale
Mathematics Advisory Group: Richard Askey, Andrew Chen, Mikhail Goldenberg, Roger Howe,
Jason Kamras, James Milgram, Robin Ramos, and Yoram Sagher
Consultation: Francis (Skip) Fennell and Mark Thames

with principal funding from:
The Brookhill Foundation, The Louis Calder Foundation, Ewing and Marion Kauffman Foundation,
Exxon Mobile Foundation, and Searle Freedom Trust

nctq board of directors:
Clara M. Lovett, Chair, Stacey Boyd, Chester E. Finn, Jr., Ira Fishman, Marti Garlett, Jason Kamras,
Donald N. Langenberg, Carol G. Peck, Andrew J. Rotherham, Kirk Schroder, Danielle Wilcox, and
Kate Walsh, President

nctq advisory board:
Steven J. Adamowski, Roy E. Barnes, Alan D. Bersin, Lawrence S. Braden, Cynthia G. Brown,
Cheryl Ellis, Michael Feinberg, Ronald F. Ferguson, Eleanor Gaines, Michael Goldstein, Eric A.
Hanushek, Frederick M. Hess, Paul T. Hill, E.D. Hirsch, Frank Keating, Paul Kimmelman,
Martin J. Koldyke, Wendy Kopp, Hailly Korman, Amy Jo Leonard, Deborah McGriff, Ellen
Moir, Robert H. Pasternack, Michael Podgursky, Michelle Rhee, Stefanie Sanford, Laura Schwedes,
Thomas Toch, and Daniel Willingham
Executive Summary June 2008

Executive summary
In this second study of education schools,1 the National Council on Teacher Quality (NCTQ) examines the
mathematics preparation of America’s elementary teachers.2 The impetus for this study is the mediocre performance
of American students in mathematics compared to their counterparts around the world. Though improving
American students’ relative performance depends on a variety of factors, a particularly critical consideration must
be the foundations laid in elementary school because mathematics relies so heavily on cumulative knowledge. The
link from there to the capability of elementary teachers to provide effective instruction in mathematics is immediate.
Unfortunately, by a variety of measures, many American elementary teachers are weak in mathematics and are
too often described, both by themselves and those who prepare them, as “math phobic.”

Absent a conclusive body of research on how best to prepare elementary teacher candidates,
we devoted two years of study to develop a set of five standards that would be the mark of a
high quality program of teacher training. To ensure that these standards were well-founded and
comprehensive, we consulted:
n Our own Mathematics Advisory Group, consisting of mathematicians and distinguished
teachers with a long history of involvement in K-12 education.
n The recommendations of professional associations, in particular the National Council on
Teachers of Mathematics (NCTM), as well as the best state standards, and other key
national studies.
n Numerous mathematicians, mathematics educators, cognitive psychologists, social scientists,
and economists.
n Education ministries of other nations with higher performance in mathematics than our
own, in particular Singapore, whose students lead the world in mathematics performance.

1 While teacher preparation programs do not always reside in “education schools,” we refer to them as education schools
because the phrase is commonly understood.
2 In May 2006 we issued What Education Schools Aren’t Teaching about Reading and What Elementary Teachers Aren’t Learning
http://www.nctq.org/p/publications/docs/nctq_reading_study_app_20071202065019.pdf

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Executive Summary June 2008

Five Standards for the
Mathematics Preparation of Elementary Teachers
Standard 1:
Aspiring elementary teachers must begin to acquire a deep conceptual knowledge of the mathematics that they
will one day need to teach, moving well beyond mere procedural understanding. Required mathematics course-
work should be tailored to the unique needs of the elementary teacher both in design and delivery, focusing on
four critical areas:
1. numbers and operations,
2. algebra,
3. geometry and measurement, and — to a lesser degree —
4. data analysis and probability.

Standard 2:
Education schools should insist upon higher entry standards for admittance into their programs. As a condition
for admission, aspiring elementary teachers should demonstrate that their knowledge of mathematics is at
the high school level (geometry and coursework equivalent to second-year algebra). Appropriate tests include
standardized achievement tests, college placement tests, and sufficiently rigorous high school exit tests.

Standard 3:
As conditions for completing their teacher preparation and earning a license, elementary teacher candidates should
demonstrate a deeper understanding of mathematics content than is expected of children. Unfortunately, no
current assessment is up to this task.

Standard 4:
Elementary content courses should be taught in close coordination with an elementary mathematics methods
course that emphasizes numbers and operations. This course should provide numerous opportunities for students
to practice-teach before elementary students, with emphasis placed on the delivery of mathematics content.

Standard 5:
The job of teaching aspiring elementary teachers mathematics content should be within the purview of mathematics
departments. Careful attention must be paid to the selection of instructors with adequate professional qualifica-
tions in mathematics who appreciate the tremendous responsibility inherent in training the next generation of
teachers and who understand the need to connect the mathematics topics to elementary classroom instruction.

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Executive Summary June 2008

This study evaluates the elementary education programs at a sample of 77 education schools
located in every state except Alaska. The schools did not volunteer to participate in this study,
but were notified early on that they had been selected. We analyzed their mathematics programs,
considering every course that they require of their elementary teacher candidates. In all, we
looked at 257 course syllabi and required textbooks as the source of information. Our sample
represents elementary education schools at higher education institutions of all types and constitutes
more than 5 percent of the institutions that offer undergraduate elementary teacher certification.
Our analysis provides a reasonable assessment and the most comprehensive picture to date of how
education schools are preparing — or failing to prepare — elementary teachers in mathematics.

In selecting this methodology, we understand that a course’s intended goals and topics as reflected
in syllabi and texts may differ from what actually happens in the classroom. We assert, however,
that professors develop their syllabi and choose texts not for some empty purpose, but for quite
an important one: to serve as an outline for the intended progression of a course and to articulate
instructional objectives. We recognize that less than what the syllabi and certainly the texts contain,
not more, is apt to be covered in class. The syllabus represents a professor’s goal for what he or
she wishes to accomplish in a course; in reality, however, there are the inevitable interruptions
and distractions that almost always leave that goal to some degree unmet. We acknowledge the
inherent limitations of this methodology and for this reason, twice invited the selected schools to
submit additional materials, such as final exams and study guides, in order to enhance our under-
standing. Also, when we encountered any sort of ambiguity, we always gave the school the benefit
of the doubt. Given the extremely low threshold that we set for schools to earn a good rating, we
expected many more schools to pass than ultimately did.

How were schools rated? We considered three factors:
1. RELEVANCE: Does the education school require coursework that is relevant to the job of
the elementary teacher, as opposed to coursework requirements intended for any student on
the campus?
2. BREADTH: Does the coursework cover essential mathematics topics?
3. DEPTH: Is enough time available to devote sufficient attention to the essential topics?

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Executive Summary June 2008

Unfortunately, we could not evaluate schools on the basis of a fourth factor: rigor. Materials we
obtained from schools did not allow us to do a comprehensive evaluation of whether they delivered
a college-level program in elementary mathematics content, as opposed to offering a remedial
program.

Though the full report contains an extensive discussion of all three criteria, some attention here is
needed to explain the first: relevance. At the outset of this study, we presumed, as we know many
people do, that while elementary teachers should be required to take some mathematics at the
college level, it did not really matter what those courses were. The logic behind this approach is
that if a teacher candidate can pass a college-level, general-audience mathematics course, then he or
she should not have much difficulty wrestling with mathematics as an instructor in an elementary
classroom. Any instructional strategies that a teacher needs to know could be taught in a mathematics
methods course.

Nevertheless, every expert we consulted told us we were wrong. With remarkable consensus,
mathematicians and mathematics educators believe that the “anything goes” practice of educating
aspiring elementary teachers is both inefficient and ineffective. While perhaps counterintuitive, it is
indeed university mathematicians who lead the charge against these general-audience mathematics
courses, arguing instead that elementary teacher candidates need a rigorous program of study
that returns them to the topics they encountered in elementary and middle school grades, but
which is by no means remedial.

To better illustrate what the learning objectives would be for such courses, we created a tear-out
test containing the kinds of mathematics problems that should be taught to teacher candidates
and which they should be able to solve. The full test is available at www.nctq.org. A few sample
problems follow.

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Executive Summary June 2008

Sample Problems
Exit with Expertise:
Do Ed Schools Prepare Elementary Teachers to Pass This Test?
(Answer Key can be found on page 21. The complete test is available at www.nctq.org.)

1. A store has a sale with a d % discount and must add a t % sales tax on any item purchased.
Which would be cheaper for any purchase:
a. Get the discount first and pay the tax on the reduced amount.
b. Figure the tax on the full price and get the discount on that amount.
Justify your answer.

2. Let n be an odd number.
a. Prove that n 2 is odd.
b. Prove that when n 2 is divided by 4, the remainder is 1.
c. Prove that when n 2 is divided by 8, the remainder is 1.
d. Find an odd n such that n 2 divided by 16 leaves a remainder that is not 1.

3. John’s shop sells bicycles and tricycles. One day there are a total of 176 wheels and 152 pedals in the shop.
How many bicycles are available for sale in John’s shop that day? Solve arithmetically and algebraically.

4. A B C D E F

Let b represent the base of the rectangle
and h represent its height.

G H I J K L
A different polygon is drawn within each of three rectangles with vertices AFLG.
Polygon No. 1: A parallelogram with vertices DFIG
Polygon No. 2: A trapezoid with vertices EFJG
Polygon No. 3: A triangle with vertices ALH
How do the areas of the three polygons compare? Justify your answer.

5. Lines a and b are parallel. Connect points A and C, and points B and C with line segments. The measurement
of the acute angle with its vertex at point B created by CB is 40º; the measurement of the acute angle
created by CA with its vertex at point A is 30º. Find the measurement of ACB.
a A

C

b B

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Executive Summary June 2008

FINDINGS

Finding 1:
Few education schools cover the mathematics content that elementary teachers need. In fact, the education
schools in our sample are remarkable for having achieved little consensus about what teachers need. There is
one unfortunate area of agreement: a widespread inattention to algebra.

The variation in requirements across the sample 77 education schools, all preparing individuals to
do the same job, is unacceptable, and we suspect reflects the variation found across all American
education schools.

Depending upon the institution, elementary teacher candidates are required to take anywhere from
zero to six mathematics courses in their undergraduate careers. The content of this coursework
ranges from “Integrated Mathematics Concepts” (described as a survey course in contemporary
mathematics that presents mathematics as a human endeavor in a historical context) to “Calculus.”

Within this variation, few education schools stand out for the quality of their mathematics
preparation. Only ten schools in our sample (13 percent) rose to the top in our evaluation of
the overall quality of preparation in mathematics. These schools met all three of our criteria:
relevance, breadth, and depth.

The table on page 7 lists the institutions by rankings. With the exception of the University of
Georgia, which we single out as an exemplary program, the listings are in alphabetical order
within the group rankings.

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Executive Summary June 2008

ARE EDUCATION SCHOOLS PREPARING
ELEMENTARY TEACHERS TO TEACH MATHEMATICS?
Education Schools with the Right Stuff
An exemplary teacher preparation program
University of Georgia
Boston College, MA† University of Louisiana at Monroe University of Montana†
Indiana University, Bloomington University of Maryland, College Park University of New Mexico†
Lourdes College, OH† University of Michigan Western Oregon University†
† Although these schools pass for providing the right content, they still fall short on mathematics methods
coursework. They do not require a full course dedicated solely to elementary mathematics methods.

Education Schools Education Schools that
that Would Pass if They Would Pass with Better
Required More Coursework Focus and Textbooks
Arizona State University Saint Mary’s College, IN Benedictine University, IL
Boston University State University of New York (SUNY) Northeastern State University, OK
Calumet College of St. Joseph, IN College at Oneonta Towson University, MD
Cedar Crest College, PA University of Central Arkansas Western Connecticut State University
Chaminade University of Honolulu, HI University of Louisville, KY Wilmington University, DE
Columbia College, MO University of Mississippi
Concordia University, OR University of Nevada, Reno
Georgia College and State University University of Portland, OR
King’s College, PA University of South Carolina
Lewis-Clark State College, ID University of South Dakota
Minnesota State University Moorhead University of Texas at El Paso
Radford University, VA University of Wyoming
Saint Joseph’s College of Maine West Texas A&M University

Education Schools that Fail on All Measures
Albion College, MI Newman University, KS University of Redlands, CA*
American University, DC Norfolk University, VA University of Rhode Island*
California State University, San Marcos* Park University, MO University of Richmond, VA*
California State University, Stanislaus* Seattle Pacific University, WA University of Texas at Dallas
Colorado College* Southern New Hampshire University* Utah State University
Florida International University Southern Adventist University, TN* Valley City State University, ND
Green Mountain College, VT** St. John’s University, NY* Viterbo University, WI
Greensboro College, NC* Saint Joseph’s University, PA* Walla Walla College, WA
Gustavus Adolphus College, MN* The College of New Jersey West Virginia University at Parkersburg
Hampton University, VA* University of Alabama at Birmingham* * Programs requiring no elementary
Iowa State University University of Arizona content coursework at all.
Lee University, TN University of Memphis, TN ** New coursework requirements
MacMurray College, IL University of Nebraska at Omaha are not publicly available.
Metropolitan State College of Denver, CO University of New Hampshire, Durham

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Executive Summary June 2008

Improving the Heft and Focus of Mathematics Preparation for Elementary Teachers
A fundamental problem observed in most of the programs is that there is a large deficit in the
amount of time devoted to elementary mathematic topics. We considered the time spent on the four
critical areas of mathematics that an elementary teacher needs to understand: 1) numbers and opera-
tions, 2) algebra, 3) geometry and measurement, and 4) data analysis and probability. The table
below shows how much programs deviate from the recommended time allocation. Of the four
areas, algebra instruction is most anemic: over half of all schools (52 percent) devote less than 15
percent of class time to algebra, with another third effectively ignoring it entirely, devoting less than 5
percent of class time to that area. By a number of measures, including the recommendation of
our Mathematics Advisory Group, algebra should comprise a large part of an entire elementary
content course, roughly 25 percent of the preparation in mathematics that elementary teachers
receive. While elementary teachers do not deal explicitly with algebra in their instruction, they need
to understand algebra as the generalization of the arithmetic they address while studying numbers
and operations, as well as algebra’s connection to many of the patterns, properties, relationships,
rules, and models that will occupy their elementary students. They should learn that a large variety
of word problems can be solved with either arithmetic or algebra and should understand the
relationship between the two approaches.

Deficiencies in Mathematics Instruction for Teachers
Recommended distribution Average hours shortchanged
Critical areas (hours) (Estimated for the sample.)

Numbers and operations 40 13
Algebra 30 24
Geometry and measurement 35 14
Data analysis and probability 10 1

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Executive Summary June 2008

Finding 2:
States contribute to the chaos. While most state education agencies issue guidelines for the mathematics
preparation of elementary teachers, states do not appear to know what is needed.

Since all aspects of public K-12 education in the United States are regulated by the states, regulation
of the preparation of K-12 teachers, whether at private or public colleges, is also within the purview
of states. Even without national oversight states could be more consistent in their requirements re-
garding coursework, standards, and/or preparation for assessments in specific areas of mathematics.

States Guidance is Confusing
18 states have no requirements Alabama, Arizona, Arkansas, Connecticut, Hawaii, Idaho, Iowa, Louisiana,
or no requirements pertaining Maine, Maryland, Michigan, Mississippi, Missouri, Nebraska, New Jersey,
to specific areas of math: Virginia, Wisconsin, and Wyoming
1 state has requirements Minnesota
pertaining only to geometry:
3 states have requirements Colorado, North Carolina, and Oregon
pertaining only to foundations
of mathematics and geometry:
29 states have requirements Alaska, California, Delaware, District of Columbia, Florida, Georgia, Illinois,
pertaining to foundations Indiana, Kansas, Kentucky, Massachusetts, Montana, Nevada, New Hampshire,
of mathematics, algebra, New Mexico, New York, North Dakota, Ohio, Oklahoma, Pennsylvania,
and geometry: Rhode Island, South Carolina, South Dakota, Tennessee, Texas, Utah,
Vermont, Washington, and West Virginia

Source: NCTQ’s State Teacher Policy Yearbook 2007, www.nctq.org/stpy

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Executive Summary June 2008

Finding 3:
Most education schools use mathematics textbooks that are inadequate. The mathematics textbooks in the
sample varied enormously in quality. Unfortunately, two-thirds of the courses use no textbook or a textbook
that is inadequate in one or more of four critical areas of mathematics. Again, algebra is shortchanged, with no
textbook providing the strongest possible support.

Most Courses Use Inadequate Textbooks

Courses Using
20% Adequate Texts
use texts rated
inadequate in
three critical areas
34%
Use texts that
adequately cover
all four critical
areas
Courses Using
Inadequate Texts

30%
Use texts rated
inadequate in 6%
10%
two critical areas do not use a text

Use texts rated
inadequate in
one critical area

Only one-third of the elementary content courses in our sample use a textbook that was rated
as adequate in four critical areas of mathematics (numbers and operations; algebra; geometry
and measurement; and data analysis and probability). Predictably, the algebra portions of the
textbooks are the weakest, with the majority of textbooks earning scores low enough to label
them unacceptable for use in algebra instruction.

In fact, no textbook has the strongest possible stand-alone algebra section, a fact that will handicap
the preparation of elementary teachers in this vital area.

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Executive Summary June 2008

Finding 4:
Almost anyone can get in. Compared to the admissions standards found in other countries, American education
schools set exceedingly low expectations for the mathematics knowledge that aspiring teachers must demonstrate.

Sixteen percent of the education schools do not require applicants to pass any sort of mathematics
test to get into their programs. The majority of the 77 schools require applicants to take a form
of basic skills test for admission, typically a three-part assessment of skills in reading, writing,
and mathematics. None of these tests, including the most popular choice, the Praxis I, measures the
proficiency one should expect from a high school graduate, as they address only those mathematics
topics taught in elementary and middle school grades.1

Entrance Tests on Mathematics Knowledge2
No. of schools Do they have tests?
12 No test at all
14 Test requirements or test expectations not clear
53 Basic skills test
1
3
Test for high school proficiency

Only one schoool in our sample of 77 clearly has adequate entry requirements.

1 We classify algebra as a middle school course because it is such in most developed countries.
2 The total number of schools noted in the table is more than 77 because some schools have multiple options for
entrance tests.
3 Colorado College requires applicants to score at least 600 on the SAT math.

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Executive Summary June 2008

Finding 5:
Almost anyone can get out. The standards used to determine successful completion of education schools’
elementary teacher preparation programs are essentially no different than the low standards used to enter
those programs.

Most education schools told us that they require an exit test in mathematics. In almost all cases,
these exit tests are the same tests that teachers need to take for state licensure (Praxis II or a test
specific to a state). There are two major failings of these tests: they either do not report a subscore
for the mathematics portion of the test, or if they do report a mathematics subscore, it is not a factor
in deciding who passes. Under these circumstances it may be possible to answer nearly every
mathematics question incorrectly and still pass the test.

The fact that education schools are relying on tests that allow prospective teachers to pass without
demonstrating proficiency in all subject areas with “stand-alone” tests makes it impossible for
either the institution or the state in which they are going to teach to know how much mathematics
elementary teachers know at the conclusion of their teacher preparation program.

In addition, even if these tests require a demonstration of mathematical understanding of slightly
more depth than entrance tests, it is insufficient to establish whether elementary teacher candidates
are truly prepared for the challenges of teaching mathematics.

Exit Tests on Mathematics Knowledge
No. of schools Do they have tests?
17 No test at all or test requirement not clear
601 Only assess elementary and middle school proficiency and do not use a stand-alone test
0 2 Stand-alone test for what an elementary teacher needs to know

Not a single state requires an adequate exit test to ensure that the teacher candidate knows the mathematics
he or she will need.

1 California’s licensing test (CSET) appears to be the most rigorous of these tests, but the mathematics portion is not stand-alone.
2 Massachusetts plans to unveil in winter 2009 a stand-alone test of the mathematics an elementary teacher needs to know.

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Executive Summary June 2008

The Other Dimension of Mathematics Preparation: Mathematics Methods Coursework
Our study focused primarily on the content preparation of elementary teachers in mathematics,
but courses in which aspiring teachers learn the methods of mathematics instruction are essential
in their overall preparation for the classroom. Therefore we also examined mathematics methods
coursework, in particular whether it was generally adequate and how instructors designed practice
teaching experiences to ensure that teacher candidates focused on conveying mathematics to their
child audiences.

Finding 6:
The elementary mathematics in mathematics methods coursework is too often relegated to the sidelines. In
particular, any practice teaching that may occur fails to emphasize the need to capably convey mathematics
content to children.

Many mathematics educators report that it is difficult to adequately cover all elementary topics in
even one methods course, yet a large share of the education schools we studied (43 percent) do not
have even one methods course dedicated to elementary mathematics methods and 5 percent have
only a two credit course. Looking at programs that had a course devoted solely to elementary
mathematics methods and required practice teaching, we found only six education schools that
appeared to emphasize the need for aspiring teachers to consider how to communicate mathematical
content and how to determine if children understood what they had been taught:

Education schools which put mathematics at the center of practice teaching
1. Greensboro College 4. University of Michigan
2. University of Georgia 5. University of Nevada, Reno
3. University of Louisville 6. University of Texas at El Paso

In the methods syllabi found in these six programs we saw instructor expectations for practice
teaching such as this: The student has demonstrated an appreciation of what it means to teach mathematics for
conceptual understanding. In contrast, syllabi from other courses requiring practice teaching tended
to make the mathematics instruction almost beside the point. For example, an aspiring teacher
might be asked to answer a question such as: What part of your teaching philosophy did you demonstrate
in your experience?

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Executive Summary June 2008

Finding 7:
Too often, the person assigned to teach mathematics to elementary teacher candidates is not professionally
equipped to do so. Commendably, most elementary content courses are taught within mathematics departments,
although the issue of just who is best qualified and motivated to impart the content of elementary mathematics
to teachers remains a conundrum.

No matter which department prepares teachers in mathematics, elementary content mathematics
courses must be taught with integrity and rigor, and not perceived as the assignment of the
instructor who drew the short straw. The fact that prospective teachers may have weaker
foundations in mathematics and are perceived to be more math phobic than average should
not lead to a conclusion that the mathematics presented must be watered down.

Finding 8:
Almost anyone can do the work. Elementary mathematics courses are neither demanding in their content nor
their expectations of students.

We could not evaluate the rigor in mathematics content courses taught in our sample education
schools using syllabi review because too few syllabi specified student assignments. We did, however,
make use of assessments that some education schools provided us. With a cautionary note that
these assessments may not be representative of all the schools in our sample, their general level of
rigor is dismaying.

The table on page 15 demonstrates the contrast between two types of questions taken from actual
quizzes, tests, and exams used in courses in programs in our sample. It pairs three problems
that would be appropriate for an elementary classroom with three problems appropriate for a
college classroom, both on a related topic.

About a third of the questions in assessments we obtained from mainstream education schools
were completely inappropriate for a college-level test.

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Executive Summary June 2008

contrasting Problems:
The mathematics that teachers need to know –
and children do not
Mathematics questions children should Mathematics questions that are closer
be able to answer – taken from actual to hitting the mark for what teachers
college course assessments. should be able to answer – taken from
actual college course assessments.

1a. The number 0.0013 is equal to the 1b. Solve the problem and explain your
following: solution process. Write the number
a. thirteen thousandths 1.00561616161… as a quotient of two
b. thirteen ten-thousandths integers (that is, in fractional-rational
c. zero point one three form). Show step-by-step arithmetic
d. one hundredth and three leading to your final, answer, giving a
ten-thousandths teacher-style solution. Do not simplify
your final answer.

2a. Which of the following is (2 1/2) ÷ (1/2)? 2b. Simplify the fraction
a. 1 1/4 b. 2 1/4 c. 1 1/2 d. 5 (1/2 + 1/3) ÷ (5/12)
(1 – 1/2) (1 – 1/3) (1 – 1/4)

3a. Exactly three-fourths of the students in 3b. The big dog weighs 5 times as much as
a certain class are passing. If 24 of them the little dog. The little dog weighs 2/3
are passing, how many students are in as much as the medium sized dog. The
the course? medium dog weighs 9 pounds more than
a. 18 b. 32 c. 36 d. 42 the little dog. How much does the big dog
weigh? Solve the problem and explain your
solution process.

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Executive Summary June 2008

RECOMMENDATIONS
We suspect that in several decades we will look back on the current landscape of the mathematics preparation
of elementary teachers and have the benefit of hindsight to realize that some education schools were poised
for significant and salutary change. These are the schools that now have the basic “3/1” framework already in
place for adequate preparation, that is, three mathematics courses that teach the elementary mathematics
content that a teacher needs to know and one well-aligned mathematics methods course. Our recommendations
here are addressed to professionals responsible for elementary teacher preparation: professional organizations,
states, education schools, higher education institutions, and textbook publishers. We also propose initiatives
that would build on the 3/1 framework in order to achieve a truly rigorous integration of content and methods
instruction.

The Association for Mathematics Teacher Educators (AMTE)
The Association of Mathematics Teacher Educators (AMTE) should organize mathematicians and mathematics
educators in a professional initiative and charge them with development of prototype assessments that can be
used for course completion, course exemption, program completion, and licensure. These assessments need to
evaluate whether the elementary teacher’s understanding of concepts such as place value or number theory is
deep enough for the mathematical demands of the classroom. They should be clearly differentiated from those
assessments one might find in an elementary or middle school classroom.

We offer a sample test, Exit with Expertise: Do Ed Schools Prepare Elementary Teachers to Pass This Test?
(an excerpt is on page 5 and the full test is available on our website: www.nctq.org) as a jumping-off
point for the development of a new generation of tests that will drive more rigorous instruction
and ensure that teachers entering the elementary classroom are well prepared mathematically.

States
States must set thresholds for acceptable scores for admission to education schools on standardized achieve-
ment tests, college placement tests, and high school exit tests. The guiding principle in setting these scores
should be to ensure that every teacher candidate possesses a competent grasp of high school geometry and
second-year high school algebra.

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Executive Summary June 2008

While these proposed thresholds are significantly higher than current ones, they are reasonable.
In fact, they still may be lower than what is required of elementary teachers in nations reporting
higher levels of student achievement in mathematics. With the exception of the most selective
institutions, there is a quite plausible perception that an education school cannot raise its admission
standards without putting itself at a disadvantage in the competition for students. The pressure
these institutions face to accept a sufficient number of students makes it incumbent upon states to
raise the bar for all education schools, not just relegate the task to a few courageous volunteers.

States need to develop strong coursework standards in all four critical areas: numbers and operations, algebra,
geometry and measurement, and data analysis and probability.

States need to adopt wholly new assessments, not currently available from any testing company, to test for
these standards.

A unique stand-alone test of elementary mathematics content that a teacher needs to know is
the only practical way to ensure that a state’s expectations are met. The test could also be used
as a vehicle to allow teacher candidates to test out of required coursework.

States need to eliminate their PreK-8 certifications. These certifications encourage education schools to attempt to
broadly prepare teachers, in the process requiring too few courses specific to teaching any grade span.

Currently, 23 states offer some form of PreK-8 certification.

Education Schools
Education schools should require coursework that builds towards a deep conceptual knowledge of the
mathematics that elementary teachers will one day need to convey to children, moving well beyond mere
procedural understanding. For most programs, we recommend a 3/1 framework: three mathematics courses
designed for teachers addressing elementary and middle school topics and one mathematics methods course
focused on elementary topics and numbers and operations in particular.

Teacher preparation programs should make it possible for an aspiring teacher to test out of mathematics content
course requirements using a new generation of standardized tests that evaluate mathematical understanding at
the requisite depth.

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Executive Summary June 2008

The higher education institutions in our sample require an average of 2.5 courses in mathematics,
only slightly below our recommendation of three elementary content mathematics courses, although
much of that coursework bears little relation to the mathematics that elementary teachers need.
Institutions, provided they are willing to redirect their general education requirements to more
relevant coursework for the elementary teacher, can quickly move towards meeting this standard
by substituting requirements for elementary content mathematics courses.

Algebra must be given higher priority in elementary content instruction.

As the National Mathematics Advisory Panel made clear in its 2008 report, while proficiency with
whole numbers, fractions, and particular aspects of geometry and measurement are the “critical
foundation of algebra,” adequate preparation of elementary students for algebra requires that
their teachers have a strong mathematics background in those critical foundations, as well as
algebra topics typically covered in an introductory algebra course.

Education schools should eliminate any of the following: mathematics programs designed for too many
grades, such as PreK-8; the practice of teaching methods for science or other subjects as companion topics
in mathematics methods coursework; and the practice of combining content and methods instruction if only one
or two combined courses are required.

Teacher preparation programs do a disservice to the material that future elementary teachers need
to learn by trying to accomplish too many instructional goals at the same time.

Five-year programs, such as those found in California, need to be restructured if they are going to meet the
mathematics content needs of elementary teachers.

The five-year model for teacher preparation, whereby prospective teachers complete coursework
for an undergraduate major taking the same courses as would any other major in that subject and
than devote a fifth year to courses about teaching and learning, does not accommodate coursework
designed for teachers in elementary mathematics topics. For that reason, these programs as
currently structured are inadvisable for the appropriate preparation of elementary teachers for
teaching mathematics.

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Executive Summary June 2008

Higher Education Institutions
On too many campuses, teacher preparation is regarded by university professors and administrators
as a program that is beneath them and best ignored. The connection of our national security to the
quality of the teachers educating new generations of Americans goes unrecognized. Were education
schools to receive more university scrutiny, and demands made that they be more systematic —
neither of which is an expensive proposition — change could be dramatic.

Higher education institutions housing education schools must take the lead in orchestrating the communication,
coordination, and innovation that would make the mathematics preparation of elementary teachers coherent.

Much of what has to be changed about the preparation of teachers connects to decisions regarding
instruction in mathematics courses (e.g., textbook selection, the priority attached to algebra, estab-
lishing more rigorous standards) and mathematics methods courses (e.g., coordination with content
courses, possibly through concurrent registration, emphasizing the mathematics in mathematics
methods, especially in practice teaching). Many changes cannot be made in isolation and most will
not be undertaken without explicit encouragement by institutional leadership.

Textbook Publishers
Several elementary content textbooks (particularly those by Thomas Parker and Scott Baldridge,
and Sybilla Beckmann) are excellent and we recommend their use, but content textbooks that
are more consistently good across all topics are still needed.

Professionals dedicated to improvements in elementary teacher preparation should collaborate to develop a
textbook that can serve as a resource both in content and methods coursework. This ideal “combo-text” would
augment a core of solid mathematics content with discussion of a process for continuous improvement of
instruction focused on student learning.

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Executive Summary June 2008

CONCLUSION
American elementary teachers as a group are caring people who want to do what is best for
children. Unfortunately, their mathematics preparation leaves far too many of them ill-equipped
to do so. We are confident that the education schools that rose to the top in our evaluation process
are preparing teachers relatively well compared to the majority of education schools in this study
which rated so poorly. Their teachers stand readier than most to forestall the frustrations of
youngsters leaving the familiar world of the counting numbers or dealing with the debut of
division with fractions. Nonetheless, the standards against which these education schools were
judged only lay a solid foundation. Further improvement is still necessary.1 Until such time as an
improved instructional model is developed that combines mathematics content and mathematics
methods instruction, teacher preparation programs should increase the efficacy of existing content
courses:
n Intensifying teacher preparation on essential topics with the same “laserlike focus” endorsed
by the National Mathematics Advisory Panel for K-12 mathematics instruction.
n Selecting the best of current textbooks.
n Setting high standards for student performance in courses and in exit tests.

A deeper understanding of elementary mathematics, with more attention given to the foundations
of algebra, must be the new “common denominator” of our preparation programs for elementary
teachers within education schools. But we are only at the beginning of the process of seeing how
that new measure might be calculated.

1 The prospect that mathematics specialists will become increasingly common in elementary classrooms due to initiatives promoted
by groups including the National Academies (Rising Above the Gathering Storm: Energizing and Employing America for a Brighter
Economic Future, Washington D.C., National Academic Press, 2007) does not change this imperative for improvement since
those specialists can emerge from the same courses and programs as regular elementary classroom teachers. The reforms that
will make classroom teachers more mathematically competent could improve mathematics specialists as well.

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Executive Summary June 2008

Answer Key for
Sample Problems on page 5
Exit with Expertise:
Do Ed Schools Prepare Elementary Teachers to Pass This Test?
(The complete test is available at www.nctq.org.)

1. Neither is cheaper since both approaches yield the same total purchase price. To determine this, let p represent any
purchase price:
a. Discounted price: p – p *(d /100) = p (1– d /100)
Tax on discounted price: p (1– d /100) (t /100)
Adding the two and simplifying: p (1– d/100) + p (1– d /100)(t /100) = p (1– d /100)(1 + t /100)
b. Full price with tax: p + p * (t /100) = p (1+ t /100)
Discount on full price with tax: [p + p * (t /100)]*d /100 = p (1+ t /100)(d /100)
Subtracting the discount from the full price and simplifying: p (1+ t /100) – p (1+ t /100)(d /100) =
p (1+t /100)(1-d /100)
These are the same since a *b = b *a

2. If n is an odd number, it can be represented as 2w +1, where w represents a whole number (0,1,2…).
a. n 2 = (2w +1)2 = 4w 2+4w +1 = 2(2w 2+2w ) + 1 so n 2 is odd.
Helpful reminder for (b) and (c): In division with a remainder, when dividing by a number k, the result is a whole
number and a remainder, with the remainder less than k (and greater than or equal to 0).
b. n 2 = (2w+1) 2 = 4w 2+4w +1 = 4(w 2+w ) + 1
Since w 2+w is a whole number and 1 is less than 4, the remainder when dividing by 4 is 1.
c. n 2 = (2w+1) 2 = 4w 2+4w +1 = 8[(w 2+w )/2] + 1. The expression w 2+w = w (w +1), and
either w or w+1 is even, so (w 2+w )/2 is a whole number. Thus the remainder when dividing by 8 is 1.
d. Many odd numbers when their square is divided by 16 leave a remainder that is not 1. The number 3 is
the least odd number that satisfies this condition: 32 = 9, and when this is divided by 16 the remainder is 9.

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Executive Summary June 2008

3. There are 52 bicycles in the shop.
Solved arithmetically:
Each bicycle has two wheels and each tricycle has three wheels, and both have two pedals. For each tricycle, there
is one more wheel than pedals. There are 176–152=24 extra wheels, so there are 24 tricycles. These have 24*3=72
wheels, so the number of wheels on bicycles is 176–72=104. The number of bicycles is half the number of wheels,
104/2=52.
Solved algebraically:
Let b represent the number of bicycles in the store and t the number of tricycles.
Equation A, developed using number of wheels: 2b +3t = 176
Equation B, developed using number of pedals: 2b +2t = 152
Subtracting equation B from A: 1t = 24
Substituting this value for t into equation B and solving for b, b = 52

4. All the polygons have the same area: A 1 = A 2 = A 3
Area of parallelogram: A 1 = 2/5b *h

h

b

Area of trapezoid: A 2 = 1/2h ( 3/5b+ 1/5b) = 1/2h * 4/5b = 2/5b *h

h

b
1/2 ( 4/5b)
Area of triangle: A 3 = *h = 2/5b *h

h

b

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Executive Summary June 2008

5. A
a
30º
D
c
C

40º
b B

Angle ACB measures 70º.
Different approaches are possible, but one approach is to draw an auxiliary line1 parallel to lines a and b through
point C and add point D to line c :
m ACD = 30º (This is an alternate interior angle to the acute angle with vertex A on line a.)
m DCB = 40º (This is an alternate interior angle to the acute angle with vertex B on line b.)
m ACD + m DCB = m ACB = 30º+ 40º = 70º

1 The function of auxiliary lines is to change difficult probelms to simpler ones, often ones which have already been solved.
Auxiliary lines could also be drawn perpendicular to line a through point A, creating a quadrilateral whose angles include
ACB and can be solved, or perpendicular to line c through point C, creating two triangles, the solution of whose angles
resolves the measurement of ACB. An auxiliary line can also be drawn through points B and C; its intersection with line
a creates a triangle, the solution of whose angles resolves the measurement of ACB.

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Executive Summary June 2008

Page 24
“ I commend this valuable report from the National Council on Teacher Quality for addressing a critical need in
improving teacher capacity: more effective assessments of mathematical knowledge as part of the process by
which candidates qualify for entry into elementary teacher preparatory programs.”
— Larry R. Faulkner
President, Houston Endowment Inc.
President Emeritus of the University of Texas

“This report should help counter the common belief that the only skill needed to teach second-grade arithmetic
is a good grasp of third-grade arithmetic. Our education schools urgently need to ensure that our elementary
teachers do not represent in the classroom the substantial portion of our citizenry that is mathematically
disabled. We must not have the mathematically blind leading the blind.”
— Donald N. Langenberg
Chancellor Emeritus, University of Maryland

“This is an important report that underscores what many of us have known for years, namely that most teacher
preparation schools fail miserably in their responsibility to provide rigorous academic training to future teachers.”
— Louis V. Gerstner, Jr.
Founder and Chairman, The Teaching Commission

To download the full report, go to www.nctq.org.
For additional copies of the executive summary, contact:

National Council on Teacher Quality
1341 G Street NW, Suite 720
Washington, D.C. 20005
Tel 202 393-0020 Fax 202 393-0095 www.nctq.org

The National Council on Teacher Quality advocates for reforms in a broad range of teacher policies at the federal, state,
and local levels in order to increase the number of effective teachers.

Subscribe to NCTQ’s free monthly electronic newsletter, Teacher Quality Bulletin (www.nctq.org/p/tab/subscribe.jsp),
to stay abreast of trends in federal, state, and local teacher policies and the events that help to shape them.