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.



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