Report Summary
The Mathematical Education of Teachers: Part 2
Introduction to the Report
The Mathematical Education of Teachers was prepared by leaders in the mathematical sciences and in
mathematics education under a grant from the United States Department of Education. The report has two
parts. Part 1 (Chapters 1-6), discussed in a separate summary in this series, is addressed to departments of
mathematics at community colleges, colleges, and universities. Part 2 (Chapters 7-9), summarized here, is
directed to “mathematics education faculty and mathematics faculty deeply involved in teacher education.”
Part 1 of The Mathematical Education of Teachers identified broad concept strands and procedures that
constitute required “mathematical knowledge for teaching” at grade levels K-4, 5-8, and 9-12. It also addressed
the role of proof and justification at each level. Part 2 further details the topics that constitute the broad content
at each of those strands. It also relates the specific topics to the school mathematics taught at each level.
Together, the two parts support positions that (a) the teaching of mathematics at any of the three levels requires
in-depth knowledge of the mathematics specific to that level and (b) it is necessary that the content of
mathematics courses for teachers be linked directly to the teaching of the school mathematics for which the
content of the course is foundational. That is, content knowledge is a necessary but not sufficient condition for
the preparation of effective K-12 teachers. Preparation of effective teachers requires that the content be studied
in relation to subject matter pedagogy.
Recommendations
Chapter 7: The Preparation of Elementary Teachers. The chapter begins by stating the premise that “Teaching
elementary mathematics [grades K-4] requires both considerable mathematical knowledge and a wide range of
pedagogical skills.” The combination is needed not only in order for elementary teachers to select appropriate
activities when preparing a lesson but also in order for teachers to be able to instantly select alternative courses
of action in order to sustain productive discussion during the delivery of the lesson. Since it is unlikely that an
undergraduate program will equip elementary teachers with all the mathematics they will ever need, an
important goal of the teacher education curriculum in mathematics is to teach mathematical habits of mind that
will allow teachers to continue to learn mathematics as they teach. The report recommends that prospective
teachers take a least nine semester hours of mathematics (plus mathematics method courses) covering four
mathematical strands of the K-4 curriculum. Brief descriptions of those strands are:
 Number and Operations: whole numbers with operations, powers, and order; place value; rational
numbers and operations (common and decimal fraction notation)
 Algebra and Functions: expressions and their equivalence; basic principles of the whole number
system and their application in computation algorithms; functions and their representation in tables,
graphs, and formulas
 Geometry and Measurement: visualization of 2D and 3D shapes through projections, cross sections,
decomposition; units of measure (Customary and SI); area and perimeter formulas; plane isometries
 Data Analysis, Statistics, and Probability: designing data investigations; describing data (shape, central
tendency, variability); drawing conclusions; issues of sampling; judgments involving uncertainty
Moreover, it is asserted that “the medium through which this agenda can be realized is the very mathematics
they [K-4 teachers] are charged with teaching… .” Hence, for each of these strands, vignettes from elementary
school classrooms are used to illustrate how the “deeper” mathematics proposed for teacher education can be
used to analyze, probe, and redirect the mathematical thinking of K-4 students. One function of vignettes is to
help teachers recognize and appreciate students’ alternate ways of solving problems.
Chapter 8: The Preparation of Middle Grades Teachers. The report recommends that mathematics in the
middle grades (grades 5-8) be taught by mathematics specialists. The mathematical preparation of middle
grades teachers should include the mathematics recommended for teachers of grades K-4 but should be
extended in each of the K-4 strands. Brief descriptions of the extended strands are:
 Number and Operations: rational numbers, operations, system principles and their relation to
computation algorithms; number theory (prime factorization and the Euclidean Algorithm); decimal
fraction notation as an extension of place value; ratio and percent; integers, operations, and principles;
sense of large numbers

Algebra and Functions: patterns, symbolic representation, modeling physical situations; representation
of functions (linear, quadratic, exponential) in tabular, graphical, and symbolic forms; role of graphing
calculators; solving linear and quadratic equations and inequations
 Geometry and Measurement: investigate 2D and 3D shapes through visualizing, classifying, defining,
conjecturing, and verifying or giving counterexamples to conjectures; use of computer tool programs for
drawing/exploring shapes; isometries and dilations of 2D and 3D shapes; measurement tools,
techniques, and formulas for area and volume; ; the Pythagorean Theorem and its verification
 Data Analysis, Statistics, and Probability: formats for displaying categorical, discrete, and continuous
data; interpreting displays; interpreting data sets (shape, central tendency, variation); correlation versus
cause-and-effect; populations and samples; theoretical and empirical probability; prediction [NOTE: The
report asserts “Of all the mathematical topics now appearing in middle grades curricula, teachers are
least prepared to teach statistics and probability.”]
To ensure that middle grades teachers are prepared to promote sense making in their students, it also is
recommended that the teachers’ curriculum should prepare them to teach in a way that develops mathematical
reasoning of four kinds: proportional, quantitative, spatial, and statistical/probabilistic.
Although the preparation of middle grades teachers involves the same four topics as the preparation of
elementary teachers, discussion for grades 5-8 does not make use of vignettes. Rather, the discussion simply
illuminates explicit connections between the mathematics recommended for teachers and the mathematics of
the middle grades curriculum. The report calls for undergraduate courses in mathematics that have been
designed with the unique needs of middle grades teachers in mind, and it recommends that mathematicians
design those courses in collaboration with colleagues from the college of education. The result should be a
coherent program to prepare middle grades specialists in the teaching of mathematics.
Chapter 9: The Preparation of High School Teachers. The report quotes research supporting the position that
acquisition of mathematical knowledge is necessary but not sufficient preparation for effective teaching in high
school (grades 9-12). “There is a seductive plausibility to … the idea that coursework in a standard
mathematics major develops the reasoning skills necessary to teach high school mathematics well … [but that]
has not been supported by research on teacher effectiveness.” Rather, it is studying subject matter in relation to
pedagogy that creates effective teachers. The report then lays out a preparation program for high school
teachers that is ambitious along two dimensions. It is ambitious with respect to the level of mathematics and its
integration with appropriate high school pedagogy that teachers are expected to achieve. It is also ambitious in
its proposal that university faculty develop the courses and teaching styles described in those courses. With
regard to the second point, however, the report notes that the MAA’s Committee on Undergraduate Programs in
Mathematics currently is reviewing the curriculum of mathematics majors and is finding that “the mathematical
needs of prospective teachers have more in common with those of other students majoring in mathematics than
many faculty realize.” Hence, the report concludes that developing undergraduate courses in the manner
envisioned for educating teachers may improve their use with other undergraduates as well. The report then
describes (without vignettes) the “revised” courses for mathematics majors as well as special capstone courses
for teachers in five areas: algebra (algebraic systems) and number theory; geometry and trigonometry;
functions and analysis; data analysis, statistics, and probability; discrete mathematics and computer science.
About the Publisher
The Mathematical Association of America (MAA) is the largest professional society of college and university mathematics teachers in the
world. It’s mission is to advance the mathematical sciences, especially at the collegiate level. MAA's 30,000 members include college and
university faculty, two-year college faculty, high school teachers, government and corporate workers, graduate school faculty, research
mathematicians, and graduate and undergraduate students. The MAA office is located at 1529 18 th Street NW, Washington, DC 200361385. TEL: 1-800-741-9415 or 202-387-5200. FAX: 202-265-2384. WEBSITE: http://www.maa.org.
Caveat Emptor
This report summary was prepared by Bob Kansky (robk@tribcsp.com), a faculty member at the Science & Mathematics Teaching Center at
the University of Wyoming. It’s one of a series offered to business, education, and policy leaders who are interested in the systemic
improvement of mathematics and science education. Readers are encouraged to consult the original documents for further information.
1
Conference Board of the Mathematical Sciences. (2001). The Mathematical Education of Teachers: Volume II. Washington, DC:
Mathematical Association of America. 145 + xiii pages.
2
The Mathematical Education of Teachers: Volume II (145 + xiii pages), includes all 9 chapters – that is, both Part 1 and Part 2. It may be
purchased from either the American Mathematical Society (http://www.ams.org) or the Mathematics Association of America
(http://www.maa.org). Also, Parts 1 and 2 are available free in PDF, HTML, and DVI from the website of the Conference Board of the
Mathematical Sciences at: http://www.cbmsweb.org .