THE PH.D. PROGRAM

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**A.
Introduction. **

The
following are guidelines for the education of Ph.D. students whose research
interests center on undergraduate mathematics education and who are preparing
for employment in a university mathematics department (henceforth, RUME
students). The guidelines are
especially meant for mathematics education Ph.D. programs housed in mathematics
departments, but may also be useful to mathematicians intending to become
specialists and researchers in undergraduate mathematics education through
independent study.

A Ph.D. following these guidelines will be a research
oriented degree similar to mathematics Ph.D.s offered by many mathematics
departments, but will also include knowledge and experience suitable for university
teaching. Mathematics Ph.D.
programs have both much in common and considerable variation across
universities. To provide RUME
students similar commonalities, the guidelines describe general goals and are
not meant to provide a detailed plan of study. For
variation in implementation of these guidelines, see the preliminary list of
U.S. Doctoral Programs in Mathematics Education

**B.
Graduates. **

The
aim of these guidelines is to produce graduates who:

(1)
are well prepared to teach mathematics to preservice teachers,

(2)
are as prepared to teach mathematics as mathematics Ph.D.s, except for their
mathematical specialties, e.g., are as well prepared to teach topology as an
algebraist or to teach algebra as a topologist,

(3)
know how to do and publish research in mathematics education, especially
undergraduate mathematics education.

Graduates
may also have knowledge and skills in, for example, curriculum development,
expository writing, proposal writing, outreach to schools, or uses of
technology, but learning such topics should not lead to a reduction in the
emphasis on (1), (2), or (3) above.

**C.
Program of Study.**

(1)
*Mathematics.* In general, RUME graduate students
should take most of the mathematics courses that mathematics graduate students
take, although instead of courses directly supporting a mathematics
dissertation they should take courses supporting a dissertation in
undergraduate mathematics education.
Both breadth and depth should be reflected in the mathematics
courses. A graduate course in the
history of mathematics would be a good choice if it were available because
research in mathematics education occasionally benefits from an understanding
of how ideas developed historically.

RUME
students should learn to construct their own original proofs and be able to
reliably and independently determine if an argument is, or is not, a
proof. They should also be able to
learn new mathematics without the aid of a teacher. The acquisition of such abilities is implicit in mathematics
Ph.D. programs, but may be due partly to students' writing of mathematics
dissertations. For RUME students,
these abilities should be explicitly attended to by some of the mathematics
courses.

(2)
*Mathematics education.* We will not
recommend an exact amount of mathematics education material a RUME student
should study. However, RUME
students should take the equivalent of at least three three-hour courses in
mathematics education in which the students discuss critically a significant
number of research papers (typically several per week). The instructors should be familiar with
the research literature in mathematics education and have published some
themselves. The courses should
include papers taking both a qualitative and a quantitative approach as well as
material on various theoretical frameworks and philosophical positions. It
would also be useful to include material on the history of mathematics
education as well as various special topics such as function, concept development,
problem solving, proof, etc. The
purpose of the courses and the discussion in them is not only to inform
students of research findings in mathematics education, but also to help them
build an understanding of how various kinds of research is done and published
and of the associated professional norms.
The way research is done should be explicitly addressed and include, but
not be limited to, some kind of student experiences (such as student designed
mini-studies) prior to the dissertation.
This instruction in various research methods might be either integrated
into some of the above courses or handled as a separate one.

The
courses mentioned above might be taught in the mathematics department or in the
school of education, but general courses in education are not substitutes for
them. It would also be useful to
include a course on statistics.

(3)
*Teaching experience*. RUME graduate
students should teach (and have full responsibility for) at least one course
for pre-service teachers and one other.
More experience, including teaching advanced courses, would be desirable
and some form of guidance and instruction should be provided with such teaching
experience.

In
addition, school teaching experience or practice teaching (at the school level)
should be provided for RUME graduate students planning to teach methods courses
and is desirable for all RUME graduate students.

(4)
*Examinations.* Either RUME students should take the
same examinations as the mathematics students, plus some kind of unifying
examination in mathematics education, or departments that require multiple
mathematics examinations of mathematics students might replace one of these with
a mathematics education examination for RUME students.

(5)
*Dissertation.* The dissertation committee should be
chosen in the same way as that of a mathematics student. The dissertation should, in the
judgment of the dissertation committee, contain the equivalent of one or more
papers, publishable in a research journal. In addition, it should contain sufficient background and explanation
to be understood by members of the department unfamiliar with its topic.

**D.
Resources.**

(1)
*Faculty*. Faculty directing dissertations or
teaching graduate courses in mathematics education should be familiar with the research
literature and publish research in mathematics education themselves.

(2)
*Student teaching.*
The graduate student teaching mentioned above should be regarded as an
important part of a student's education and should be required even if the student
has other support.

(3)
*Library*. In mathematics education access to the
research literature is an important part of even a beginning graduate student's
program of study. The major
journals and reference books should be available in the library.

**E.
Conclusion. **

If
all of these guidelines are implemented fully, the resulting program may be
longer than a typical mathematics Ph.D. program. In some universities this may be acceptable, because there
is considerable variation in the length of Ph.D.s across fields. In other universities any program fully
implementing these guidelines may be inconsistent with university regulations
or may be seen as unreasonably long.
In such cases, we recommend that some of the guidelines not be
implemented fully, but that they all be followed at least in part. In general these guidelines are not meant
to be rigidly prescriptive, but to provide a unifying guidepost that some
departments may wish to move towards and some may have good reasons to depart
from somewhat.

Recommendations
of the RUME Committee on PhD Guidelines for collegiate mathematics education
Ph.D. programs, May 2007.