the math phd

Ph.D. Program Overview

Description.

The graduate program in the field of mathematics at Cornell leads to the Ph.D. degree, which takes most students five to six years of graduate study to complete. One feature that makes the program at Cornell particularly attractive is the broad range of  interests of the faculty . The department has outstanding groups in the areas of algebra, algebraic geometry,  analysis, applied mathematics, combinatorics, dynamical systems, geometry, logic, Lie groups, number theory, probability, and topology. The field also maintains close ties with distinguished graduate programs in the fields of  applied mathematics ,  computer science ,  operations research , and  statistics .

Core Courses

A normal course load for a beginning graduate student is three courses per term. 

There are no qualifying exams, but the program requires that all students pass four courses to be selected from the six core courses. First-year students are allowed to place out of some (possibly, all) of the core courses. In order to place out of a course, students should contact the faculty member who is teaching the course during the current academic year, and that faculty member will make a decision. The minimum passing grade for the core courses is B-; no grade is assigned for placing out of a core course.

At least two core courses should be taken (or placed out) by the end of the first year. At least four core courses should be taken (or placed out) by the end of the second year (cumulative). These time requirements can be waived for students with health problems or other significant non-academic problems. They can be also waived for students who take time-consuming courses in another area (for example, CS) and who have strong support from a faculty; requests from such students should be made before the beginning of the spring semester. 

The core courses  are distributed among three main areas: analysis, algebra and topology/geometry. A student must pass at least one course from each group. All entering graduate students are encouraged to eventually take all six core courses with the option of an S/U grade for two of them. 

The six core courses are:

MATH 6110, Real Analysis

MATH 6120, Complex Analysis

MATH 6310, Algebra 1

MATH 6320, Algebra 2

MATH 6510, Introductory Algebraic Topology

MATH 6520, Differentiable Manifolds.

Students who are not ready to take some of the core courses may take MATH 4130-4140, Introduction to Analysis, and/or MATH 4330-4340, Introduction to Algebra, which are the honors versions of our core undergraduate courses.

"What is...?" Seminar

The "What Is...?" Seminar is a series of talks given by faculty in the graduate field of Mathematics. Speakers are selected by an organizing committee of graduate students. The goal of the seminar is to aid students in finding advisors.

Schedule for the "What Is...?" seminar

Special Committee

The Cornell Graduate School requires that every student selects a special committee (in particular, a thesis adviser, who is the chair or the committee) by the end of the third semester.

The emphasis in the Graduate School at Cornell is on individualized instruction and training for independent investigation. There are very few formal requirements and each student develops a program in conjunction with his or her special committee, which consists of three faculty members, some of which may be chosen from outside the field of mathematics. 

Entering students are not assigned special committees. Such students may contact any of the members on the Advising Committee if they have questions or need advice.

Current Advising Committee

Analysis / Probability / Dynamical Systems / Logic: Lionel Levine Geometry / Topology / Combinatorics: Kathryn Mann Probability / Statistics:  Philippe Sosoe Applied Mathematics Liaison: Richard Rand

Admission to Candidacy

To be admitted formally to candidacy for the Ph.D. degree, the student must pass the oral admission to candidacy examination or A exam. This must be completed before the beginning of the student's fourth year. Upon passing the A exam, the student will be awarded (at his/her request) an M.S. degree without thesis.

The admission to candidacy examination is given to determine if the student is “ready to begin work on a thesis.” The content and methods of examination are agreed on by the student and his/her special committee before the examination. The student must be prepared to answer questions on the proposed area of research, and to pass the exam, he/she must demonstrate expertise beyond just mastery of basic mathematics covered in the core graduate courses. 

To receive an advanced degree a student must fulfill the residence requirements of the Graduate School. One unit of residence is granted for successful completion of one semester of full-time study, as judged by the chair of the special committee. The Ph.D. program requires a minimum of six residence units. This is not a difficult requirement to satisfy since the program generally takes five to six years to complete. A student who has done graduate work at another institution may petition to transfer residence credit but may not receive more than two such credits.

The candidate must write a thesis that represents creative work and contains original results in that area. The research is carried on independently by the candidate under the supervision of the chairperson of the special committee. By the time of the oral admission to candidacy examination, the candidate should have selected as chairperson of the committee the faculty member who will supervise the research. When the thesis is completed, the student presents his/her results at the thesis defense or B Exam. All doctoral students take a Final Examination (the B Exam, which is the oral defense of the dissertation) upon completion of all requirements for the degree, no earlier than one month before completion of the minimum registration requirement.

Masters Degree in the Minor Field

Ph.D. students in the field of mathematics may earn a Special Master's of Science in Computer Science. Interested students must apply to the Graduate School using a form available for this purpose. To be eligible for this degree, the student must have a member representing the minor field on the special committee and pass the A-exam in the major field. The rules and the specific requirements for each master's program are explained on the referenced page.

Cornell will award at most one master's degree to any student. In particular, a student awarded a master's degree in a minor field will not be eligible for a master's degree in the major field.

Graduate Student Funding

Funding commitments made at the time of admission to the Ph.D. program are typically for a period of five years. Support in the sixth year is available by application, as needed. Support in the seventh year is only available by request from an advisor, and dependent on the availability of teaching lines. Following a policy from the Cornell Graduate School, students who require more than seven years to complete their degree shall not be funded as teaching assistants after the 14th semester.

Special Requests

Students who have special requests should first discuss them with their Ph.D. advisor (or with a field member with whom they work, if they don't have an advisor yet). If the advisor (or field faculty) supports the request, then it should be sent to the Director of Graduate Studies.  

Department of Mathematics

The Ph.D. program in the Department of Mathematics provides students with in-depth knowledge and rigorous training in all the subject areas of mathematics. A core feature is the first-year program, which helps bring students to the forefront of modern mathematics. Students work closely with faculty and each other and participate fully in both research and student-run seminars.

There are roughly 80 PhD students in the graduate program, and 15-20 join each year. It is a rigorous program targeted at excellent students. A core feature is the first year program, which brings students to the forefront of modern mathematics.

Students work closely with the faculty and each other. They participate fully in both research and student-run seminars.

The University is in historic Hyde Park, a vibrant community with a rich campus life. Downtown Chicago is a short bus or train ride away. Greater Chicago is home to many universities with strong mathematics departments, which enhances the intellectual life of the program.

• Current Course Offerings : U of C class search site

PhD Program

More information and a full list of requirements for the PhD program in Mathematics can be found in the University Bulletin .

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. 

Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: 

• real analysis
• geometry and topology
• applied mathematics

Course Requirements for students starting prior to Autumn 2023

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

• Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry
• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. The course sequence requirement is discontinued for students starting in Autumn 2023 and later.

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination , which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Years 4&5

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

Throughout the PhD Program

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

Best Mathematics Programs

Ranked in 2023, part of Best Science Schools

A graduate degree in mathematics can help students hone

A graduate degree in mathematics can help students hone their skills in a specialty area, from algebra and number theory to discrete mathematics and combinatorics. These are the best graduate-level math programs. Each school's score reflects its average rating on a scale from 1 (marginal) to 5 (outstanding), based on a survey of academics at peer institutions. Read the methodology »

• Clear Filters

Ohio State nav bar

Ohio state navigation bar.

• BuckeyeLink
• Search Ohio State

Doctor of Philosophy (PhD)

Program synopsis and training.

 The Doctor of Philosophy (PhD) in mathematics is the highest degree offered by our program. Graduates will have demonstrated their ability to conduct independent scientific research and contribute new mathematical knowledge and scholarship in their area of specialization. They will be well-supported and well prepared for research and faculty positions at academic institutions anywhere in the world. Owing to their independence, analytic abilities, and proven tenacity, our PhD graduates are also sought after by private and government employers.

Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics . The tracks differ only in the course and  qualifying requirements during the first two years. Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below.

Once students have passed their Qualifying Requirements, the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations for Admission

Competitive applicants to the theoretical track are expected to have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201 - 5202  and Math 5111 - 5112 sequences.

Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201 ), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing.

Besides these basic requirements, competitive applicants in either track submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

Prior research experiences are not required for either track, and we routinely admit students without significant research background. Nevertheless, applicants are encouraged to include accounts of research and independent project endeavors as well as letters of supervising mentors in order to be more competitive for fellowship considerations. The research component is likely to have greater weight in applications to the applied track.

These prepared documents serve to provide our admission committee with a narrative overview of the applicant's mathematical trajectory. Their primary focus should, therefore, be to enumerate and describe any evidence of mathematical ability and mathematical promise. The information included in the documents should be well-organized, comprehensive, informative, specific, and relevant. This will help our committee to properly and efficiently evaluate the high number of applications we receive each year.

Our Graduate Recruitment Committee will generally  not  consider GRE test scores for this Autumn 2024 admissions. If you have already taken the test, please do not self-report the scores to us. In exceptional circumstances students may have the option to report unofficially. 

International students whose native language is not English and are not exempt should score at least a 20 on the Speaking portion of the TOEFL or at least 6.5 on the IELTS Speaking portion.  We also recommend an overall score of at least 95 on TOEFL or at least 7.0 on IELTS.  For a list of exempt countries, please see  https://gpadmissions.osu.edu/intl/additional-requirements-to-apply.html

Qualifying Requirements by Track

The qualifying requirements for the theoretical track are fulfilled by passing our Abstract Algebra course sequence  ( Math 6111 , Math 6112 ) and  our Real Analysis course sequence ( Math 6211 , Math 6212 ), each with at least an A-, or  by passing a respective examination.

The qualifying requirements for the applied track combine a mandatory Scientific Computing course ( Math 6601 ), one of the algebra or analysis courses, and three additional courses chosen from  Math 6602 , Math 6411 ,  Math 6451 , and the courses comprising the algebra and analysis sequences.

The breadth requirements in the applied track are more flexible than in the theoretical track, but also include a mandatory graduate course in a non-math STEM department from an approved list. 

You can find more information about our PhD program requirement here .

Opportunities & Outcomes

The research opportunities and academic outcomes of our doctoral program are described in detail in the Graduate Program Prospectus  [pdf]. 

Our department has about 80 active graduate faculty on the Columbus and regional campuses. Virtually every area of mathematics is represented in our program, with a sampling displayed below.

• Commutative, Non-commutative, & Quantum Algebra
• Analytic, Algebraic, Computational Number Theory
• Algebraic Geometry, Tropical Geometry
• Applied Mathematics, Mathematical Physics
• Real and Complex Analysis
• Functional Analysis, Operator Algebras
• Combinatorics and Graph Theory
• Differential Geometry
• Dynamical Systems and Ergodic Theory
• Financial and Actuarial Mathematics
• Logic and Foundations
• Probability Theory, Statistical Mechanics
• Mathematical Biology
• Ordinary and Partial Differential Equations
• Representation theory
• Scientific Computing
• Topology, Topological Data Analysis

See also our  Applied Mathematics Topics List  [pdf].

Our program offers many support opportunities without teaching duties as well, to allow more time for scientific endeavors. These opportunities include university fellowships, external funding, and departmental fellowships and special assignments. See the  Financial Support  page for more details.

The median time to degree completion in our program is below six years but also varies significantly among our students, with as little as four years for students entering with substantial prior preparations. Funding is guaranteed for six years and can be extended to seven years with advisor support and the permission of the Graduate Studies Committee.  

Most of our graduates continue their careers in academia. Post-doctoral placements in the last two years include, for example, UCLA, Stanford, ETH-Zürich, Brown University, University of Michigan, Northwestern University, University of Vienna, EPF Lausanne, Free University at Berlin, Purdue University, and University of Utah. In recent years our graduates also went to Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University. Recent non-academic placements include Google, Facebook, Amazon, NSA, and prestigious financial institutions.

Students also have access to training and networking opportunities that prepare them better for careers in private industry and teaching - for example, through the Erdős Institute  - and are regularly offered highly competitive positions in the industry. 

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. And as a program group member of the National Math Alliance , we are dedicated to enhancing diversity in our program and the scientific community. The International students in our program come from all parts of the world with a wide variety of educational backgrounds.

Prospective students:  [email protected]

Graduate Office Department of Mathematics The Ohio State University 231 W 18th Avenue ( MA 102 ) Columbus, Ohio 43210 United States of America                

Phone: (614) 292-6274 Fax: (614) 292-1479

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email  [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at:  Adobe Acrobat Reader .

Our websites may use cookies to personalize and enhance your experience. By continuing without changing your cookie settings, you agree to this collection. For more information, please see our University Websites Privacy Notice .

COLLEGE OF LIBERAL ARTS AND SCIENCES

Department of Mathematics

• Ph.D. in Mathematics
• General Information
• Financial Aid
• Answers to FAQ
• M.S. in Mathematics
• Ph.D. in Mathematics with Thesis in Actuarial Science
• M.S. in Actuarial Science
• M.S. in Applied Financial Mathematics
• M.S. in Applied Financial Mathematics – Actuarial Science
• Course Schedule
• Preliminary Exams

Overview of Graduation Requirements

To graduate with a PhD in Mathematics, a student must satisfy all of the following requirements:

• 45 credits, including 15 doctoral dissertation research credits/GRAD 6950
• If you have a Master’s degree in mathematics at UConn, then 30 credits are required, including 15 doctoral dissertation research credits.
• Pass at least one prelim exam after each semester for the first three semesters of their graduate study.
• Finish all prelim exam requirements by the beginning of the spring semester in their second year.
• Finish all core course requirements by the end of their second year of graduate study,
• Choose a thesis advisor no later than the end of the fall semester of the third year.
• Form your advisory committee :  Major Advisor and at least two Associate Advisors or Co-Major Advisors and at least one Associate Advisor.
• Meant to further the student’s education, scholarship and professional development.
• Material should be in the broad area in which the student intends to write a dissertation, but should not focus on the actual thesis research.  Students present and answer questions about the material they have studied.
• The exam is normally taken at the end of the third year or beginning of the fourth year.
• Submit Dissertation Proposal
• Submit Plan of Study
• Apply for Graduation in Student Admin by the fourth week of the final semester.
• Dissertation Specifications under Dissertation Information
• Thesis templates (for LaTex) are available on our Thesis Formatting page
• Submit Approval Page – Step 7
• Complete Survey of Earned Doctorates – Step 7
• Add Diploma Address in Student Admin and Diploma Info.

Preliminary Examination and Core Course Requirements

Pure Mathematics

• Math 5111/Measure and Integration
• Math 5120/Complex Function Theory
• Math 5210/Abstract Algebra
• Math 5310/Introduction to Geometry and Topology I
• Math 5160/Probability Theory and Stochastic Processes I
• Math 5210/Abstract Algebra I
• Math 5211/Abstract Algebra II
• Math 5260/Mathematical Logic I
• Math 5360/Differential Geometry

Applied Mathematics

• Math 5410/Introduction to Applied Mathematics I
• Math 5510/Numerical Analysis and Approximation Theory I
• Math 5440/Partial Differential Equations
• Math 5520/Finite Element Solution Methods I

Past Prelim Exams

Ph.D. Degree Programs

The UCSD Mathematics Department admits students into the following Ph.D. programs:

• Ph.D. in Mathematics -- Pure or Applied Mathematics.
• Ph.D. in Mathematics with a  Specialization in Computational Science .
• Ph.D. in Mathematics with a  Specialization in Statistics .

In addition, the department participates in the following Ph.D. programs:

• Ph.D. in  Bioinformatics .
• Ph.D. in  Mathematics and Science Education  (joint program between UCSD and SDSU).

For application information, go to  How to Apply (Graduate) .  

Ph.D. in Mathematics

The Ph.D. in Mathematics allows study in pure mathematics, applied mathematics and statistics. The mathematics department has over 60 faculty, approximately 100 Ph.D. students, and approximately 35 Masters students. A list of the UCSD mathematics faculty and their research interests can be found at  here . The Ph.D. in Mathematics program produces graduates with a preparation in teaching and a broad knowledge of mathematics. Our students go on to careers as university professors, as well as careers in industry or government.

In the first and second years of study, Ph.D. students take courses in preparation for three written qualifying examinations (quals). One qual must be taken in Algebra or Topology, and another in Real or Complex Analysis. A third qual may be taken in Numerical Analysis or Statistics or one of the remaining topics in the first two groups. All three quals must be passed by the start of the third year. After the qualifying exams are passed, the student is expected to choose an advisor and follow a course of study agreed on by the two of them. At this point, the student chooses a thesis topic, finds a doctoral committee and presents a talk on his or her proposed research topic. If the committee is satisfied with this talk, the student has "Advanced to Candidacy." The student will then pursue their research agenda with their advisor until they have solved an original problem. The student will submit a written dissertation and reconvene his or her committee for a Final Defense. At the Final Defense, the student gives a seminar talk that is very similar to a talk that he or she might give for a job interview.

Nearly every admitted Ph.D. student gets financial support. The financial support is most commonly in the form of a Teaching Assistantship, however, Research Assistantships and other fellowships are also available.

Because of the large faculty to student ratio, graduate students have many opportunities to interact with faculty in courses or smaller research seminars. The graduate students also run their own "Food for Thought" seminar for expository talks as well as a research seminar where they give talks about their research.

UCSD has excellent library facilities with strong collections in mathematics, science, and engineering. Ph.D. students are provided with access to computer facilities and office space.

Full-time students are required to register for a minimum of twelve (12) units every quarter, eight (8) of which must be graduate-level mathematics courses taken for a letter grade only. The remaining four (4) units can be approved upper-division or graduate-level courses in mathematics-related subjects (MATH 500 may not be used to satisfy any part of this requirement). After advancing to candidacy, Ph.D. candidates may take all course work on a Satisfactory/Unsatisfactory basis. Typically, students should not enroll in MATH 299 (Reading and Research) until they have passed at least two Qualifying Examinations at the PhD or Provisional PhD level, or obtained approval of their faculty advisor.  

Written Qualifying Examinations

Effective Fall Quarter 1998, the department made changes in their qualifying exam requirements with a view to:

• improving applied mathematics' access to students and the attractiveness of its program to applicants; and
• broadening the education of our doctoral students and leading more of them towards applied areas.

The department now offers written qualifying examinations in  SEVEN (7)  subjects. These are grouped into three areas as follows:  

• Three qualifying examinations must be passed. At least one must be passed at the Ph.D. level and a second must be passed at either the Ph.D. or Provisional Ph.D. level.
• Of the three qualifying exams, there must be at least one from each of Areas 1 and 2. 
• Students must pass at least two exams from distinct areas with a minimum grade of Provisional Ph.D. (For example, a Ph.D. pass in Real Analysis, Provisional Ph.D. pass in Complex Analysis, M.A. pass in Algebra would  NOT  satisfy this requirement, but a Ph.D. pass in Real Analysis, M.A. pass in Complex Analysis, Provisional Ph.D. pass in Algebra would, as would a Ph.D. pass in Numerical Analysis, Provisional Ph.D. pass in Applied Algebra, and M.A. pass in Real Analysis.) All exams must be passed by the September exam session prior to the beginning of the third year of graduate studies. (Thus, there is no limit on the number of attempts, encouraging new students to take exams when they arrive, without penalty.) Except for this deadline, there is no limit on the number of exams a student may attempt.

After qualifying exams are given, the faculty meet to discuss the results of the exams with the Qualifying Exam and Appeals Committee (QEAC). Exam grades are reported at one of four levels:  

Department policy stipulates that at least one of the exams must be completed with a Provisional Ph.D. pass or better by September following the end of the first year. Anyone unable to complete this schedule will be terminated from the doctoral program and transferred to one of our Master's programs. Any grievances about exams or other matters can be brought before the Qualifying Exam and Appeals Committee for consideration.

Exams are typically offered twice a year, one scheduled late in the Spring Quarter and again in early September (prior to the start of Fall Quarter). Copies of past exams are available on the  Math Graduate Student Handbook .

In choosing a program with an eye to future employment, students should seek the assistance of a faculty advisor and take a broad selection of courses including applied mathematics, such as those in Area 3.  

Master's Transferring to Ph.D.

Any student who wishes to transfer from masters to the Ph.D. program will submit their full admissions file as Ph.D. applicants by the regular closing date for all Ph.D. applicants (end of the fall quarter/beginning of winter quarter). It is the student's responsibility to submit their files in a timely fashion, no later than the closing date for Ph.D. applications at the end of the fall quarter of their second year of masters study, or earlier. The candidate is required to add any relevant materials to their original masters admissions file, such as most recent transcript showing performance in our graduate program. Letters of support from potential faculty advisors are encouraged. The admissions committee will either recommend the candidate for admission to the Ph.D. program, or decline admission. In the event of a positive recommendation, the Qualifying Exam Committee checks the qualifying exam results of candidates to determine whether they meet the appropriate Ph.D. program requirements, at the latest by the fall of the year in which the application is received. For students in the second year of the master's program, it is required that the student has secured a Ph.D. advisor before admission is finalized. An admitted student is supported in the same way as continuing Ph.D. students at the same level of advancement are supported. Transferring from the Master's program may require renewal of an I-20 for international students, and such students should make their financial plans accordingly. To be eligible for TA support, non-native English speakers must pass the English exam administered by the department in conjunction with the Teaching + Learning Commons.  

Foreign Language Requirement

There is no Foreign Language requirement for the Ph.D. in Mathematics.  

Advancement to Candidacy

It is expected that by the end of the third year (9 quarters), students should have a field of research chosen and a faculty member willing to direct and guide them. A student will advance to candidacy after successfully passing the oral qualifying examination, which deals primarily with the area of research proposed but may include the project itself. This examination is conducted by the student's appointed doctoral committee. Based on their recommendation, a student advances to candidacy and is awarded the C. Phil. degree.  

Dissertation and Final Defense

Submission of a written dissertation and a final examination in which the thesis is publicly defended are the last steps before the Ph.D. degree is awarded. When the dissertation is substantially completed, copies must be provided to all committee members at least four weeks in advance of the proposed defense date. Two weeks before the scheduled final defense, a copy of the dissertation must be made available in the Department for public inspection.  

Time Limits

The normative time for the Ph.D. in mathematics is five (5) years. Students must be advanced to candidacy by the end of eleven (11) quarters. Total university support cannot exceed six (6) years. Total registered time at UCSD cannot exceed seven (7) years.  

It may be useful to describe what the majority of students who have successfully completed their Ph.D. and obtained an academic job have done. In the past some students have waited until the last time limit before completing their qualifying exams, finding an advisor or advancing to candidacy. We strongly discourage this, because experience suggests that such students often do not complete the program. Although these are formal time limits, the general expectation is that students pass two qualifying exams, one at the Ph.D. level and one at the masters level by the beginning of their second year. (About half of our students accomplish this.) In the second year, a student begins taking reading courses so that they become familiar with the process of doing research and familiarize themselves with a number of faculty who may serve as their advisor. In surveying our students, on average, a student takes 4 to 5 reading courses before finding an advisor. Optimally, a student advances to candidacy sometime in their third year. This allows for the fourth and fifth year to concentrate on research and produce a thesis. In contrast to coursework, research is an unpredictable endeavor, so it is in the interest of the student to have as much time as possible to produce a thesis.

A student is also a teaching assistant in a variety of courses to strengthen their resume when they apply for a teaching job. Students who excel in their TA duties and who have advanced to candidacy are selected to teach a course of their own as an Associate Instructor. Because there are a limited number of openings to become an Associate Instructor, we highly recommend that you do an outstanding job of TAing in a large variety of courses and advance to candidacy as soon as possible to optimize your chances of getting an Associate Instructorship.

9500 Gilman Drive, La Jolla, CA 92093-0112

(858) 534-3590

Mathematics, PhD

Zanvyl krieger school of arts and sciences.

The goal of our PhD program is to train graduate students to become research mathematicians. Each year, an average of five students complete their theses and  go on to exciting careers  in mathematics both inside and outside of academia.

Faculty research interests  in the Johns Hopkins University Department of Mathematics are concentrated in several areas of pure mathematics, including analysis and geometric analysis, algebraic geometry and number theory, differential geometry, algebraic topology, category theory, and mathematical physics. The department also has an active group in data science, in collaboration with the  Applied Math Department .

The Department values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

A brief overview of our graduate program is below. For more detailed information, please see the links at the right.

Program Overview

All students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022-2023 academic year are $33,000, and exceptional applicants are also considered for supplementary fellowships. Students making satisfactory progress can expect to be supported for six years.

PhD candidates take two or three courses per semester over the first several years of the program. These are a mix of required and intermediate-level graduate courses, independent studies, and special topics classes offered by our faculty.

By the beginning of their second year, students are asked to demonstrate competency in algebra and in analysis by passing written qualifying exams in these two broad areas. Students are then expected to choose an advisor, who will supervise their dissertation and also administer an oral qualifying exam to be taken in the second or third year. More specifics about all these requirements are described on the  requirements page .

All graduate students are invited to attend  weekly research seminars in a variety of topic areas  as well as regular department teas and a weekly wine and cheese gathering attended by many junior and senior members of the department. A graduate student lunch seminar series provides an opportunity for our students to practice their presentation skills to a general audience.

PhD students will gain teaching experience as a teaching assistant for undergraduate courses. Most of our students lead two TA sections per week, under the supervision of both the faculty member teaching the course and the director of undergraduate studies. Students wanting more classroom experience (or extra pay) can teach their own sections of summer courses. First-year students are given a reduced TA workload in the spring semester, in preparation for the qualifying exams.

In addition to their stipend, each student is awarded an annual travel allowance to enable them to attend conferences for which limited funding is available or visit researchers at other institutions.

Financial Aid

Students admitted to the Ph.D. program receive teaching assistantships and full tuition fellowships. Exceptional applicants become candidates for one of the university's George E. Owen Fellowships.

William Kelso Morrill Award

The William Kelso Morrill Award for excellence in the teaching of mathematics is awarded every spring to the graduate student who best exemplifies the traits of Kelso Morrill: a love of mathematics, a love of teaching, and a concern for students.

Excellence in Teaching Awards

Three awards are given each year to a junior faculty member and graduate student teaching assistants who have demonstrated exceptional ability and commitment to undergraduate education.

Admission Requirements

Admission to the PhD program is based on primarily on academic records, letters of recommendation, and a personal statement. The Department of Mathematics values diversity among its members, is committed to building a diverse intellectual community, and strongly encourages applications from all interested parties.

Via the online application , applicants should submit:

• A Statement of Purpose
• An optional Personal Statement
• Transcripts from all institutions attended
• Three letters of recommendation
• Official GRE scores for both the general and the subject test
• Official TOEFL scores (if English is not your first language)

The required Statement of Purpose discusses your academic interests, objectives, and preparation. The optional Personal Statement describes your personal background, and helps us create a more holistic understanding of you as an applicant. If you wish you may also discuss your personal background in the Statement of Purpose (e.g. if you have already written a single essay addressing both topics), instead of submitting separate statements.

Application fee waivers are available based on financial need and/or participation in certain programs .

Many frequently asked questions about the graduate admission process are answered here .

No application materials should be mailed to the department. All application materials are processed by the Graduate Admissions Office .

Undergraduate Background

The following is an example of what the math department would consider a good background for a student coming out of a four-year undergraduate program at a college or university in the U.S. (assuming a semester system):

• Calculus in one variable (two semesters, or AP credits)
• Multivariable Calculus (one semester)
• Linear Algebra (one semester)
• Complex analysis (one semester)
• Real analysis (two semesters)
• Abstract algebra (two semesters)
• Point-set topology (one semester)

Many admitted students have taken upper-level undergraduate mathematics courses or graduate courses. Nevertheless, the department does admit very promising students whose preparation falls a little short of the above model. In such cases, we strongly recommend that the student start to close the gap over the summer, before arriving for the start of the fall semester.

Financial Support   

Students admitted to the PhD program receive full tuition fellowships and teaching assistantships. Teaching assistant salaries for the 2022–2023 academic year are $33,000. Students making satisfactory progress can expect to be supported for six years. Exceptional applicants are considered for supplementary fellowships of $6,000 each year for three years.

Students from underrepresented groups may be eligible for other university-wide supplemental fellowships. Summer teaching is available for students seeking extra income.

Additional Information for International Students

Student Visa Information:  The Office of International Services at Homewood  will assist admitted international students in obtaining a student visa.

English Proficiency: Johns Hopkins University requires students to have adequate English proficiency for their course of study. Students must be able to read, speak, and write English fluently upon their arrival at the university. Applicants whose native language is not English must submit proof of their proficiency in English before they can be offered admission and before a visa certificate can be issued. Proficiency can be demonstrated by submitting results from either the Test of English as a Foreign Language (TOEFL) or the IELTS . Johns Hopkins prefers a minimum score of 100 on the TOEFL or a Band Score of 7 on the IELTS. Results should be sent to Johns Hopkins directly by TOEFL or IELTS. Applicants taking the IELTS must additionally upload a copy of their score through the application system. However, do not send the student copy or a photocopy of the TOEFL.

Program Requirements

Course requirements.

Mathematics PhD candidates must show satisfactory work in Algebra (110.601-602), Real Variables (110.605), Complex Variables (110.607), and one additional non-seminar mathematics graduate course in their first year. The first-year algebra and analysis requirement can be satisfied by passing the corresponding written qualifying exam in September of the first year; these students must complete at least two courses each semester. In addition, PhD candidates must take Algebraic Topology (110.615) and Riemannian Geometry (110.645) by their second year. Students having sufficient background can substitute an advanced topology course for 110.615, or an advanced geometry course for 110.645 with the permission of the instructor.

Candidates must show satisfactory work in at least two mathematics graduate courses each semester of their second year, and if they have not passed their oral qualifying exam, in the first semester of their third year.

Qualifying Exams

Candidates must pass written qualifying exams by the beginning of their second year in Analysis (Real & Complex) and in Algebra. Exams are scheduled for September and May of each academic year, and the dates are announced well in advance.

Candidates must pass an oral qualifying examination in the student’s chosen area of research by April 10 of the third year. The topics of the exam are chosen in consultation with the faculty member who has agreed (provisionally) to be the student’s thesis advisor, who will also be involved in administering the exam.

PhD Dissertation

Candidates must produce a written dissertation based upon independent and original research. After completion of the thesis research, the student will defend the dissertation by means of the  Graduate Board Oral exam . The exam must be held at least three weeks before the Graduate Board deadline the candidate wishes to meet.

Our PhD program does not have a foreign language requirement.

• Doing a PhD in Mathematics
• Doing a PhD

What Does a PhD in Maths Involve?

Maths is a vast subject, both in breadth and in depth. As such, there’s a significant number of different areas you can research as a math student. These areas usually fall into one of three categories: pure mathematics, applied mathematics or statistics. Some examples of topics you can research are:

• Number theory
• Numerical analysis
• String theory
• Random matrix theory
• Graph theory
• Quantum mechanics
• Statistical forecasting
• Matroid theory
• Control theory

Besides this, because maths focuses on addressing interdisciplinary real-world problems, you may work and collaborate with other STEM researchers. For example, your research topic may relate to:

• Biomechanics and transport processes
• Evidence-based medicine
• Fluid dynamics
• Financial mathematics
• Machine learning
• Theoretical and Computational Optimisation

What you do day-to-day will largely depend on your specific research topic. However, you’ll likely:

• Continually read literature – This will be to help develop your knowledge and identify current gaps in the overall body of knowledge surrounding your research topic.
• Undertake research specific to your topic – This can include defining ideas, proving theorems and identifying relationships between models.
• Collect and analyse data – This could comprise developing computational models, running simulations and interpreting forecasts etc.
• Liaise with others – This could take many forms. For example, you may work shoulder-to-shoulder with individuals from different disciplines supporting your research, e.g. Computer scientists for machine learning-based projects. Alternatively, you may need frequent input from those who supplied the data for your research, e.g. Financial institutions or biological research colleagues.
• Attend a wide range of lectures, seminars and events.

Browse PhD Opportunities in Mathematics

Application of artificial intelligence to multiphysics problems in materials design, study of the human-vehicle interactions by a high-end dynamic driving simulator, physical layer algorithm design in 6g non-terrestrial communications, machine learning for autonomous robot exploration, detecting subtle but clinically significant cognitive change in an ageing population, how long does it take to get a phd in maths.

The average programme duration for a mathematics PhD in the UK is 3 to 4 years for a full-time studying. Although not all universities offer part-time maths PhD programmes, those that do have a typical programme duration of 5 to 7 years.

Again, although the exact arrangement will depend on the university, most maths doctorates will require you to first register for an MPhil . At the end of your first year, your supervisor will assess your progress to decide whether you should be registered for a PhD.

Additional Learning Modules

Some Mathematics departments will require you to enrol on to taught modules as part of your programme. These are to help improve your knowledge and understanding of broader subjects within your field, for example, Fourier Analysis, Differential Geometry and Riemann Surfaces. Even if taught modules aren’t compulsory in several universities, your supervisor will still encourage you to attend them for your development.

Most UK universities will also have access to specialised mathematical training courses. The most common of these include Pure Mathematics courses hosted by Mathematics Access Grid Conferencing ( MAGIC ) and London Taught Course Centre ( LTCC ) and Statistics courses hosted by Academy for PhD Training in Statistics ( APTS ).

What Are the Typical Entry Requirements for A PhD in Maths?

In the UK, the typical entry requirements for a Maths PhD is an upper second-class (2:1) Master’s degree (or international equivalent) in Mathematics or Statistics [1] .

However, there is some variation on this. From writing, the lowest entry requirement is an upper second-class (2:1) Bachelor’s degree in any math-related subject. The highest entry requirement is a first-class (1st) honours Master’s degree in a Mathematics or Statistics degree only.

It’s worth noting if you’re applying to a position which comes with funding provided directly by the Department, the entry requirements will usually be on the higher side because of their competitiveness.

In terms of English Language requirements, most mathematics departments require at least an overall IELTS (International English Language Testing System) score of 6.5, with no less than 6.0 in each individual subtest.

Tips to Consider when Making Your Application

When applying to any mathematics PhD, you’ll be expected to have a good understanding of both your subject field and the specific research topic you are applying to. To help show this, it’s advisable that you demonstrate recent engagement in your research topic. This could be by describing the significance of a research paper you recently read and outlining which parts interested you the most, and why. Additionally, you can discuss a recent mathematics event you attended and suggest ways in how what you learnt might apply to your research topic.

As with most STEM PhDs, most maths PhD professors prefer you to discuss your application with them directly before putting in a formal application. The benefits of this is two folds. First, you’ll get more information on what their department has to offer. Second, the supervisor can better discover your interest in the project and gauge whether you’d be a suitable candidate. Therefore, we encourage you to contact potential supervisors for positions you’re interested in before making any formal applications.

How Much Does a Maths PhD Typically Cost?

The typical tuition fee for a PhD in Maths in the UK is £4,407 per year for UK/EU students and £20,230 per year for international students. This, alongside the range in tuition fees you can expect, is summarised below:

Note: The above tuition fees are based on 12 UK Universities [1]  for 2020/21 Mathematic PhD positions. The typical fee has been taken as the median value.

In addition to the above, it’s not unheard of for research students to be charged a bench fee. In case you’re unfamiliar with a bench fee, it’s an annual fee additional to your tuition, which covers the cost of specialist equipment or resources associated with your research. This can include the upkeep of supercomputers you may use, training in specialist analysis software, or travelling to conferences. The exact fee will depend on your specific research topic; however, it should be minimal for most mathematic projects.

What Specific Funding Opportunities Are There for A PhD in Mathematics?

Alongside the usual funding opportunities available to all PhD Research students such as doctoral loans, departmental scholarships, there are a few other sources of funding available to math PhD students. Examples of these include:

You can find more information on these funding sources here: DiscoverPhDs funding guide .

What Specific Skills Do You Gain from Doing a PhD in Mathematics?

A doctorate in Mathematics not only demonstrates your commitment to continuous learning, but it also provides you with highly marketable skills. Besides subject-specific skills, you’ll also gain many transferable skills which will prove useful in almost all industries. A sample of these skills is listed below.

• Logical ability to consider and analyse complex issues,
• Commitment and persistence towards reaching research goals,
• Outstanding verbal and written skills,
• Strong attention to detail,
• The ability to liaise with others from unique disciple backgrounds and work as part of a team
• Holistic deduction and reasoning skills,
• Forming and explaining mathematical and logical solutions to a wide range of real-world problems,
• Exceptional numeracy skills.

What Jobs Can You Get with A Maths PhD?

One of the greatest benefits maths PostDocs will have is the ability to pursue a wide range of career paths. This is because all sciences are built on core principles which, to varying extents, are supported by the core principles of mathematics. As a result, it’s not uncommon to ask students what path they intend to follow after completing their degree and receive entirely different answers. Although not extensive by any means, the most common career paths Math PostDocs take are listed below:

• Academia – Many individuals teach undergraduate students at the university they studied at or ones they gained ties to during their research. This path is usually the preferred among students who want to continue focusing on mathematical theories and concepts as part of their career.
• Postdoctoral Researcher – Others continue researching with their University or with an independent organisation. This can be a popular path because of the opportunities it provides in collaborative working, supervising others, undertaking research and attending conferences etc.
• Finance – Because of their deepened analytical skills, it’s no surprise that many PostDocs choose a career in finance. This involves working for some of the most significant players in the financial district in prime locations including London, Frankfurt and Hong Kong. Specific job titles can include Actuarial, Investment Analyst or Risk Modeller.
• Computer Programming – Some students whose research involves computational mathematics launch their career as a computer programmer. Due to their background, they’ll typically work on specialised projects which require high levels of understanding on the problem at hand. For example, they may work with physicists and biomedical engineers to develop a software package that supports their more complex research.
• Data Analyst – Those who enjoy number crunching and developing complex models often go into data analytics. This can involve various niches such as forecasting or optimisation, across various fields such as marketing and weather.

What Are Some of The Typical Employers Who Hire Maths PostDocs?

As mentioned above, there’s a high demand for skilled mathematicians and statisticians across a broad range of sectors. Some typical employers are:

• Education – All UK and international universities
• Governments – STFC and Department for Transport
• Healthcare & Pharmaceuticals – NHS, GSK, Pfizer
• Finance & Banking – e.g. Barclays Capital, PwC and J. P. Morgan
• Computing – IBM, Microsoft and Facebook
• Engineering – Boeing, Shell and Dyson

The above is only a small selection of employers. In reality, mathematic PostDocs can work in almost any industry, assuming the role is numerical-based or data-driven.

How Much Can You Earn with A PhD in Maths?

As a mathematics PhD PostDoc, your earning potential will mostly depend on your chosen career path. Due to the wide range of options, it’s impossible to provide an arbitrary value for the typical salary you can expect.

However, if you pursue one of the below paths or enter their respective industry, you can roughly expect to earn [3] :

Academic Lecturer

• Approximately £30,000 – £35,000 starting salary
• Approximately £40,000 with a few years experience
• Approximately £45,000 – £55,000 with 10 years experience
• Approximately £60,000 and over with significant experience and a leadership role. Certain academic positions can earn over £80,000 depending on the management duties.

Actuary or Finance

• Approximately £35,000 starting salary
• Approximately £45,000 – £55,000 with a few years experience
• Approximately £70,000 and over with 10 years experience
• Approximately £180,000 and above with significant experience and a leadership role.

Aerospace or Mechanical Engineering

• Approximately £28,000 starting salary
• Approximately £35,000 – £40,000 with a few years experience
• Approximately £60,000 and over with 10 years experience

Data Analyst

• Approximately £45,000 – £50,000 with a few years experience
• Approximately £90,000 and above with significant experience and a leadership role.

Again, we stress that the above are indicative values only. Actual salaries will depend on the specific organisation and position and responsibilities of the individual.

Facts and Statistics About Maths PhD Holders

The below chart provides useful insight into the destination of Math PostDocs after completing their PhD. The most popular career paths from other of highest to lowest is education, information and communication, finance and scientific research, manufacturing and government.

Note: The above chart is based on ‘UK Higher Education Leavers’ data [2] between 2012/13 and 2016/17 and contains a data size of 200 PostDocs. The data was obtained from the Higher Education Statistics Agency ( HESA ).

Which Noteworthy People Hold a PhD in Maths?

Alan turing.

Alan Turing was a British Mathematician, WW2 code-breaker and arguably the father of computer science. Alongside his lengthy list of achievements, Turning achieved a PhD in Mathematics at Princeton University, New Jersey. His thesis titled ‘Systems of Logic Based on Ordinals’ focused on the concepts of ordinal logic and relative computing; you can read it online here . To this day, Turning pioneering works continues to play a fundamental role in shaping the development of artificial intelligence (AI).

Ruth Lawrence

Ruth Lawrence is a famous British–Israeli Mathematician well known within the academic community. Lawrence earned her PhD in Mathematics from Oxford University at the young age of 17! Her work focused on algebraic topology and knot theory; you can read her interesting collection of research papers here . Among her many contributions to Maths, her most notable include the representation of the braid groups, more formally known as Lawrence–Krammer representations.

Emmy Noether

Emmy Noether was a German mathematician who received her PhD from the University of Erlangen, Germany. Her research has significantly contributed to both abstract algebra and theoretical physics. Additionally, she proved a groundbreaking theorem important to Albert Einstein’s general theory of relativity. In doing so, her theorem, Noether’s theorem , is regarded as one of the most influential developments in physics.

Other Useful Resources

Institute of Mathematics and its Applications (IMA) – IMA is the UK’s professional body for mathematicians. It contains a wide range of useful information, from the benefits of further education in Maths to details on grants and upcoming events.

Maths Careers – Math Careers is a site associated with IMA that provides a wide range of advice to mathematicians of all ages. It has a section dedicated to undergraduates and graduates and contains a handful of information about progressing into research.

Resources for Graduate Students – Produced by Dr Mak Tomford, this webpage contains an extensive collection of detailed advice for Mathematic PhD students. Although the site uses US terminology in places, don’t let that put you off as this resource will prove incredibly helpful in both applying to and undertaking your PhD.

Student Interviews – Still wondering whether a PhD is for you? If so, our collection of PhD interviews would be a great place to get an insider perspective. We’ve interviewed a wide range of PhD students across the UK to find out what doing a PhD is like, how it’s helped them and what advice they have for other prospective students who may be thinking of applying to one. You can read our insightful collection of interviews here .

[1] Universities used to determine the typical (median) and range of entry requirements and tuition fees for 2020/21 Mathematics PhD positions.

• http://www.lse.ac.uk/study-at-lse/Graduate/Degree-programmes-2020/MPhilPhD-Mathematics
• https://www.ox.ac.uk/admissions/graduate/courses/dphil-mathematics?wssl=1
• https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdpms
• https://www.ucl.ac.uk/prospective-students/graduate/research-degrees/mathematics-mphil-phd
• http://www.bristol.ac.uk/study/postgraduate/2020/sci/phd-mathematics/
• https://www.surrey.ac.uk/postgraduate/mathematics-phd
• https://www.maths.ed.ac.uk/school-of-mathematics/studying-here/pgr/phd-application
• https://www.lancaster.ac.uk/study/postgraduate/postgraduate-courses/mathematics-phd/
• https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd
• https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/05325/phd-pure-mathematics/
• https://warwick.ac.uk/study/postgraduate/research/courses-2020/mathematicsphd/
• https://www.exeter.ac.uk/pg-research/degrees/mathematics/

[2] Higher Education Leavers Statistics: UK, 2016/17 – Outcomes by subject studied – https://www.hesa.ac.uk/news/28-06-2018/sfr250-higher-education-leaver-statistics-subjects

[3] Typical salaries have been extracted from a combination of the below resources. It should be noted that although every effort has been made to keep the reported salaries as relevant to Math PostDocs as possible (i.e. filtering for positions which specify a PhD qualification as one of their requirements/preferences), small inaccuracies may exist due to data availability.

Browse PhDs Now

Join thousands of students.

Join thousands of other students and stay up to date with the latest PhD programmes, funding opportunities and advice.

Mathematics Education PhD

Doctor of philosophy.

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

Admissions Information

Displaying requirements for the Spring 2024, Summer 2024, and Fall 2024 terms.

• Points/Credits: 75
• Entry Terms: Spring, Summer, Fall

Application Deadlines

For details about rolling deadlines , visit our admission deadlines page.

Select programs remain open beyond our standard application deadlines , such as those with an extended deadline or those that are rolling (open until June or July). If your program is rolling or has an extended deadline indicated above, applications are reviewed as they are received and on a space-available basis. We recommend you complete your application as soon as possible as these programs can close earlier if full capacity has been met.

Application Requirements

Requirements from the tc catalog (ay 2023-2024).

Displaying catalog information for the Fall 2023, Spring 2024 and Summer 2024 terms.

View Full Catalog Listing

Doctoral Degrees

All candidates for the Ed.D., Ed.D.C.T., or Ph.D. degrees are expected to demonstrate both mathematics and mathematics education competencies through a series of certification examinations taken upon the completion of 60 graduate points. Certification examinations test the student’s knowledge of current research and theory in mathematics education and mathematics content. Examinations are offered once in the fall, spring, and summer terms. Courses recommended as preparation for the examinations in mathematics education include MSTM 6037,  MSTM 4019, and other mathematics education courses; Courses recommended as preparation for the examinations in mathematics are 6000- level mathematics content courses.

Students must demonstrate acceptable proficiency in at least three of the following six mathematics content areas: algebra, analysis, discrete mathematics, foundations of mathematics, geometry and topology, and probability and statistics. Students may sit for the examination in mathematics content during the regular certification examination times. Alternatively, they may register for advanced content courses and, with permission of the program, sit for the content area certification examination upon completion of the course. Incoming doctoral candidates should register for MSTM 6037 Professional Seminar in Mathematics during the first year of doctoral studies.

Doctoral students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Doctor of Philosophy in Mathematics Education

The Doctor of Philosophy (Ph.D.) degree emphasizes research competencies. The degree program requires a scholarly dissertation of intellectual merit and sound research methodology. Dissertation research may include analytical studies of the process of teaching or experimental studies of the teaching-learning process, including studies of verbal learning and laboratory practice or historical studies.

Candidates are encouraged to develop an association with a faculty member early in their studies to identify a problem area of mutual interest to plan a course of studies that leads to the competencies needed to complete dissertation research and prepare for a professional role. Further details are available in the brochures on doctoral studies and in the general descriptions of doctoral programs available from the Office of Doctoral Studies (ODS).

A program of study for the Doctor of Philosophy degree must include at least 45 points taken under Teachers College registration. In order to permit the acquisition of broad and basic scholarship, each program of study should include at least 60 points in mathematics, mathematics education, statistics, and computing. At least 35 points should be in advanced courses – including research courses (MSTM 6500 or 6501 and MSTM 7500). (Any Teachers College course at the 6000 level or above, any Columbia University Graduate School of Arts and Sciences course with a “G” prefix, any “W” course numbered above 4000, or any transferred course with a graduate-level prerequisite will be considered an advanced course.) Further, 15 points in the philosophical, psychological, and curricular foundations of education must be included in every Ph.D. degree program. Students whose dissertations require statistical analysis should include appropriate statistics courses in their programs. These points can be included either in the mathematics/mathematics education requirement or can be taken as research electives.

Candidates for the Ph.D. degree are required to demonstrate competency in two languages chosen from among French, German, and Russian. Students who require other languages for the preparation of their dissertation may petition the program to request one substitution. Students in mathematics may not use computer languages or statistics to satisfy the language requirement.

The Ph.D. dissertation is a scholarly study contributing new theoretical knowledge to the field and should be planned early in the program when sufficient advanced courses have been completed to permit the candidate to enroll in relevant research courses. Ph.D. dissertations in mathematics education should be (1) experimental studies in learning, (2) analytical studies in policy theory in mathematics education, or (3) other scholarly investigations of problems and issues of broad significance in the field.

The website of the Program offers a list of Topic study groups which doctoral students are recommended to join.

• View Other Degrees

Program Director : Professor Alexander Karp

Teachers College, Columbia University 323 Thompson

Phone: (212) 678-3381 Fax: (212) 678-8319

Email: tcmath@tc.edu

New Graduate Students

We are thrilled that you will be joining the Berkeley Mathematics community this Fall! This page contains pertinent information that will help get you started as a graduate student at Berkeley. 

Complete New Student Onboarding 

You've accepted our offer; now help the Office of the Registrar prepare your student record by completing the "Scheduled Tasks" found on the "My Dashboard" tab of your  CalCentral  account. Scheduled tasks typically include completing the Statement of Legal Residence (SLR), immunizations form, and for international students, the Non-immigrant Information Form (NIF). In addition, you will be required to complete  two mandatory Sexual Violence/Sexual Harassment Prevention trainings (online and in-person) . You will receive more information on these requirements via CalCentral and from the department. Thus, it's important that you regularly check your CalCentral account for important campus notifications and tasks. 

For International Students - Obtaining Documents for your Visa Application via the NIF 

The Berkeley International Office (BIO) provides student advising to international students across campus and is responsible for issuing visa documents to all incoming international students. We highly encourage all international students to complete the NIF (Non-immigrant Information Form) as soon as possible in order to have plenty of time to gather the documents required to apply for a student visa. Your departmental offer letter (w/ signature) can serve as proof of financial support. If you need assistance locating your departmental offer letter please contact Christian Natividad at  [email protected] . We also recommend you visit  BIO's website  for new students as you will find a wealth of resources specific to international student arrival including information on visas, housing, money, enrollment, transportation, and health care. 

On this page:  Important Dates & Planning your Arrival  

Important Dates • Arrival • Housing Resources • Financial Matters • Medical Reminders • Student Groups

Orientation & Academics

Prelim Info • Math Orientation • Fall Enrollment 

Employment as a GSI or GSR  

GSI/GSR Overview • Requirements for 1st time GSIs • Language Requirements • Resources for GSIs

Campus Resources

Variety of useful Links 

Graduate Program Contacts  

Contact Info 

Important Dates & Planning your Arrival

Important dates: .

• Prelim Workshop  -  TBD (2024 schedule will be posted late June/early July).
• Enrollment Opens for New Graduate Students -   Friday, July 19, 2024
• Mathematics Graduate Student Orientation -  Wednesday, August 21, 2024, 1015 Evans - Full-day program
• Teaching Conference for first-time  international  GSIs  (REQUIRED)  - Thursday, August 22, 2024
• Teaching Conference for  all  first-time GSIs  (REQUIRED for domestic & international)  - Friday, August 23, 2024
• Fall 2024 Math Prelim Exam s  - Monday, August 26 & Tuesday, August 27. This exam will be held in person.
• Fall 2024 Graduate Division Graduate Student Orientation   - TBD.  Registration is required.
• First Day of Fall Semester  - Wednesday, August 21, 2024
• Practice Prelim  - TBD
• First Day of Fall Instruction  - Wednesday, August 28, 2024
• Grad-stravaganza  - Wednesday, September 4, 2024 from 4-6 p.m.

Planning your Arrival 

This summer, we recommend that you arrive in the Bay Area as early as possible in order to get settled for your first year of graduate study. An arrival in early August would be ideal so that you can participate in the prelim prep workshop and complete HR onboarding for new GSIs in advance of your appointment start date (8/1/24). We understand that international students may only enter the country within 30 days of their I-20 or DS-2019 start date - again we recommend arriving as early as your schedule allows. 

Our department orientation (8/21) is not required, but it is highly encouraged that you attend. Not only will you be able to meet your fellow classmates, but we'll also be reviewing campus and department resources, program requirements and expectations, and will host our own Sexual Violence Sexual Harrassment Training, which satisfies the university's in-person SVSH training requirement. We'll also end the day with the office draw!  

Please note that for those of you serving as a GSI this fall, you are required to attend the Teaching Conference for New GSIs and first-time international GSIs are required to attend the  Teaching in the U.S. Classroom Conference (tentative dates above).  

In summary, while it is encouraged that you arrive in Berkeley as early as possible (early/mid-July), the latest you can arrive and still attend all of the required trainings and take the fall prelim is mid-August.

Housing Resources in Berkeley   

University Campus Housing Website (for Ida Jackson, Manville, University Village)

CalRentals - University Listings for Off-Campus Housing and Summer Sublets

The Graduate Assembly's Housing Guide: Best Practices for Finding Housing

Berkeley International Office Housing Resources

SLMath Housing Links (Short-Term Housing)

Rent To Own Labs  (this site is unaffiliated with UC Berkeley)

*If you need a housing reference from the department please feel free to use Christian Natividad at  [email protected] . Please notify him in advance so he knows someone may be contacting him for more information. 

Financial Matters

Students in the Math Department are funded through a combination of sources including Graduate Student Instructor (GSI) or Graduate Student Researcher (GSR) salaries, fellowship stipends, university fellowships (Berkeley, Chancellor's, Regents, Ning, etc.) or external fellowships (NSF, NDSEG, NPSC, etc.). In order to receive fellowship payments from the university, students must be officially registered and in good standing. To be considered registered, you must be enrolled in at least one class, have had at least the first installment of fees paid, and have no registration blocks. Please note that full-time enrollment (12 units or a DSP approved reduced course load) is required to remain in compliance with fellowship policy. 

Here is an estimation of when you can expect to receive your first payments from the university (assuming you are a registered student at this time): 

• GSI or GSR Salary if HR onboarding completed by communicated deadlines: ~September 1, 2024 (for August work); if HR onboarding completed after the deadline: ~October 1, 2024 (for August & September work)  
• Department Relocation or recruitment stipends - last week of August 2024
• University Fellowship Awards (Berkeley, Chancellor's, Ning, etc.) - last week of August 2024
• External Fellowships - please refer to granting institution for pay dates  

 A delay in payment could be caused by not being considered a registered student, not having an up-to-date GLACIER record (for international students only), registration blocks, or department delays.  

To receive your fellowship payments via direct deposit please make sure to sign up for EFT (Electronic Funds Transfer). The EFT website can also be accessed in CalCentral.

To receive your GSI/GSR salary payment via direct deposit please sign up for EFT during your HR onboarding session with Berkeley Regional Services - ERSO. After your GSI/GSR appointment has been processed by ERSO, you can also sign up for direct deposit via  UC Path .  If you plan to receive paper checks instead of signing up for direct deposit, please make sure your "Local Address" is correct in CalCentral . Direct deposit is STRONGLY recommended to receive payments.  

Also, please keep in mind that receiving  financial assistance from the university may have tax implications  that you are not very familiar with. University and department staff are not able to give tax advice so we encourage you to consult the  IRS website  for more information, and or a personal/family certified tax accountant for assistance.  

International students can find more information on U.S. taxes via the  Berkeley International Office website on tax reporting.  

If you have any questions regarding financial matters please don't hesitate to contact us. 

Stay Healthy! Medical Checklist and Immunization Information 

Once enrolled for the fall term, your  Student Health Insurance Plan (SHIP)  coverage will start on August 1, 2024 and run through December 31, 2024. Your spring 2024 coverage, once enrolled, starts January 1, 2025 - July 31, 2025. If you have alternate health insurance coverage and do not want to enroll in SHIP please submit your waiver to the University Health Services (UHS) by the deadline in mid-July - see link below for more information and to submit the waiver. Before arriving, please review the new student medical checklist below so that you come prepared for any medical emergencies that may arise. Also, please make sure to review the UC Immunization Requirement policy either via the link below or through your CalCentral account.   

New Student Medical Checklist

UC Immunization Requirement

Information on Waiving SHIP - Deadline ~July 15, 202 4 

Connect with your New Berkeley Math Friends  

The Noetherian Ring - Women and Gender Minorities in the Department of Mathematics at UC Berkeley  

MGSA  &  MGSA Wiki

Unbounded Representation URep (website TBD) 

Berkeley GEMS - Gender Equity in Mathematical Studies 

More information on the Math Grad Life website

Orientation & Academics 

Prelim workshop & prelim exam .

(This prelim section is for Math and Applied Math graduate students only. Logic students can find more information on program requirements and the  Logic prelim exams here. ) 

Taking the prelim exam is the first requirement math graduate students will attempt in the program. All graduate students are required to pass the exam within their first three semesters of the program. It is held every semester (fall & spring) on the Monday and Tuesday mornings before instruction begins. This fall the exam will be held on  Monday, August 26 and Tuesday, August 27.  

The purpose of the prelim exam is to make sure that graduate students have sufficient working knowledge of foundational undergraduate material in the early stages of the PhD program, and to give early feedback on gaps in knowledge. Its intention is not to weed out students, but rather help strengthen one's understanding of core material. There is no penalty for having to retake the exam more than once so we recommend students take the exam in the fall. 

The Prelim Exam also gives you the chance to meet with your cohort and study! It's good to get in the habit of studying and working together - math doesn't have to be lonely in graduate school! We encourage you to attend the Prelim Workshop, which will be run by current graduate students. A practice prelim exam will be held the week before the exam. Topics will alternate between algebra and analysis. You can find the complete schedule on the workshop website (updates forthcoming).  

You can find more information on the  prelim as well as resources to help you prepare and pass exams here . 

Math Graduate Student Orientation (For all incoming Math, Applied Math, and Logic students)

The Mathematics Graduate Student Orientation is scheduled for  Wednesday, August 21st, 2024.  It will be an all day event, which will include a continental breakfast and lunch(RSVP form will be sent via email). More information including a full agenda will be sent closer to the date. You can see a  general agenda here . New students aren't required to attend, but it's highly recommended that students make every effort to be present. Curious about what to expect for your time here? Check out the  MGSA Wiki . 

Fall Enrollment 

Fall enrollment for new graduate students opens on Friday,   July 19th, 9:15am - 4:45 pm.  Registering on-time, and before August 1st, ensures that you will have timely access to health insurance coverage and access to campus resources. 

Course enrollment at Berkeley occurs in two phases (1 & 2) and is assigned based on one's standing (Grad, UG Freshman, Sophomore, etc.). For fall, new graduate students are able to sign up for classes in one phase in late July. All math graduate students must be enrolled full-time with a minimum 12-unit course load each semester. Students needing accommodations such as a reduced course load must be registered with the  Disabled Students' Program.  

In addition to being required to enroll in 12 units each semester, as a program requirement, all first years must enroll in at least four courses total across the fall and spring terms. At least two of these must be graduate courses in Mathematics. MPS 375 and 303 do not count towards the satisfaction of this requirement. MPS 375 is a pedagogy course that all first-time GSIs are required to take. For reference of the course offerings to expect, you can view our course offerings on the  Schedule of Classes  [use the lefthand bar to filter by term and course level (e.g. grad or undergrad)] or on our website  here . To view courses offered in other departments visit the  Berkeley Guide . I recommend using the top ribbon to search by subject rather than by keyword. 

Before finalizing your course schedule, please discuss your plans with your assigned first-year faculty adviser. You will be assigned a faculty adviser in early summer. If you have any general questions about enrollment and your options please feel free to contact us. 

Employment as a GSI or GSR 

The two primary forms of employment for graduate students in the Math Department are Graduate Student Instructor (GSI) and Graduate Student Researcher (GSR) appointments. GSRs are appointed directly by the supervising faculty member. The majority of first-year students work as GSIs for the department.

In late spring/early summer an application will be sent out to all students who will be working as GSIs this fall so that we can collect your teaching preferences. We will do our best to match you to your preferences, but please keep in mind that you may not be assigned your first choice. The majority of first-year students are assigned to our larger, lower-division courses as assignments are made based on seniority and prior GSI experience. When submitting your preferences please make sure that your own course schedule does not conflict with the course you may be assigned to as a GSI. Some instructors require attendance at lecture and you will always need to be available to proctor in-class midterms and final exams. The instructor teaching the course is your supervisor so please make sure to have your travel plans for winter and summer break approved by them in advance of making any travel arrangements. You may be needed several days after the final exam is given to complete final grading and administrative duties so it's important that you work with your supervising faculty member to identify when it will be okay for you to depart from campus for breaks.

Fall GSI appointments officially start on August 1st and end on December 31st. Thus, students are paid for the entire month of August and December even though GSI work typically doesn't begin until the start of the semester and usually ends before December 31st.  In order for students to be paid the correct amount it is important that they complete HR onboarding (official employment verification) with Berkeley Regional Services / ERSO by the stated deadlines. Students who do not complete onboarding by the stated deadline may not be paid their GSI salary until October 1st.  More information on onboarding sessions will be sent later this summer. 

Requirements for first-time GSIs: 

• Attend the daylong  Teaching Conference  sponsored by the  GSI Teaching & Resource Center . This conference is held each semester on the Friday before classes begin. Pre-registration is required. All first-time international GSIs must also complete the  Teaching Conference for International GSIs , which takes place in the fall semester the Thursday before classes begin. Pre-registration is also required for Thursday's offering.
• Successfully complete the  online course GSI Professional Standards and Ethics in Teaching   before  interacting with students (in person or online) as an instructor.
• Enroll in and complete the  300-level pedagogy course for first-time GSIs , MPS 375. All GSIs teaching for the first time on campus must take a 300-level pedagogy course, regardless of prior teaching experience or previous courses taken at other universities. If you have a course conflict with MPS 375, students may take a pedagogy course offered by another department as long as you have approval from the Vice Chair for Graduate Studies, Sug Woo Shin, and the alternate department. 
• All first-time GSIs must attend an hour-long  Academic Student Employee (ASE) orientation session . Advanced registration is NOT required.

Language Proficiency Requirements for International Students 

International students must satisfy an English language proficiency requirement in order to serve as a GSI. If English is your first language, if you attended a US institution for your undergraduate degree, or if you scored at least 26 on the speaking portion of the TOEFL IBT, then you are eligible to teach. However, you must still report your status through the  Language Proficiency Questionnaire . The deadline to complete the questionnaire for incoming students is June 1, 2024. 

All other students must pass an English language proficiency exam, and in some cases take a class before being eligible to teach. We will write further about the steps needed to satisfy this language requirement in advance, but please be aware that it is your responsibility to ensure that this requirement is satisfied, and that your offer of a GSI appointment is contingent upon satisfying this requirement.

Resources for GSIs:

GSI Teaching and Resource Center 

GSI, GSR, Reader and Tutor Guide

If you have any questions regarding GSI or GSR matters please don't hesitate to contact us. 

Campus Resources 

Logic Resources

Graduate Division Campus Resources Page  - Comprehensive website of all resources on campus

New Graduate Student Guide  - Graduate Division's most recent guide for new students

Guide to Graduate Policy  - Graduate Division Policies on Graduate Studies at Berkeley

Berkeley International Office (BIO)

Berkeley Parking & Transportation

CalDining - 2022-23 Meal Plan for Graduate Students  (on-campus dining commons)

Cal 1 Card  - Student ID Card 

Cal Rec Sports  - RSF, Campus Gym 

UC Berkeley Basic Needs Security

Food Security • CalFresh • Food Assistance Program • UCB Food Pantry

Campus Life  

Campus Safety

Gender Equity Resource Center (GenEq)

Graduate Assembly

Undocumented Student Program 

Math Stats Library

University Health Services (UHS) 

Counseling and Psychological Services (CAPS) at UHS

Graduate Program Contacts 

Graduate Advisor  Clay Calder |  [email protected]  or 510-642-0665

Graduate Advisor  Christian Natividad|  [email protected]  or 415-501-0125 

Director of Student Services  Vicky Lee |  [email protected]  or 510-644-4603

• Future Students
• Current Students
• Faculty/Staff

News and Media

• News & Media Home
• Research Stories
• School's In
• In the Media

You are here

Softening the sharp edges in mathematics.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start — to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics — a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

“Mathematics should be flexible, conceptual, a place where we play with ideas and make connections," says Professor Jo Boaler. (Photo: Robert Houser Photography)

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced “How to Learn Math,” the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users. 

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer. 

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 — but the most common answer 13-year-olds gave was 19. The second most common was 21. 

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important? 

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mind-set. I think we all need a little more ish in our lives. 

You also argue that mathematics should be taught in more visual ways. What do you mean by that? 

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things — all of that contributes to our understanding of how it works. 

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking. 

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, ‘I’m in geometry class now, and I still remember that  sugar cube, what it looked like and felt like.’ His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing — elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

Click to enlarge: A depiction of various ways to calculate 38 x 5, numerically and visually. (Image: Courtesy of Jo Boaler)

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them. 

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

More Stories

⟵ Go to all Research Stories

Get the Educator

Subscribe to our monthly newsletter.

Stanford Graduate School of Education

482 Galvez Mall Stanford, CA 94305-3096 Tel: (650) 723-2109

• Contact Admissions
• GSE Leadership
• Site Feedback
• Web Accessibility
• Career Resources
• Faculty Open Positions
• Explore Courses
• Academic Calendar
• Office of the Registrar
• Cubberley Library
• StanfordWho
• StanfordYou

Improving lives through learning

• Stanford Home
• Maps & Directions
• Search Stanford
• Emergency Info
• Terms of Use
• Non-Discrimination
• Accessibility

© Stanford University , Stanford , California 94305 .

Skip to content

Georgia Institute of Technology College of Sciences

Search form.

• You are here:

College of Sciences Announces New Minors, Ph.D. Program and Curriculum Additions

This fall, the College of Sciences will debut three new minors, a new Ph.D. program, and a new “4+1” B.S./M.S. degree program. 

The announcement follows curriculum updates for the 2023-24 academic year, including the launch of the Minor in the Science of Mental Health and Well-Being in the School of Psychology and the creation of three new bachelor of science degrees in the School of Earth and Atmospheric Sciences. 

“We are excited to announce these additions to the College’s portfolio of academic opportunities for our students,” says David M. Collard , senior associate dean in the College of Sciences and professor in the School of Chemistry and Biochemistry . “The updates reflect our College’s growth and respond to our students’ interest in pursuing advanced study.”

The additions for the 2024-2025 academic year include: 

“4+1” B.S./M.S. Degree Program

The College offers several options for undergraduate students to earn both a bachelor of science degree and a master of science degree as a part of a “4+1” program. Students may apply to the B.S./M.S Degree Program after being at Georgia Tech for about one year. This allows them to tailor their undergraduate and graduate academic requirements to complete both degrees in a timely manner. 

Computation and Cognition Minor 

The Minor in Computation and Cognition is a highly interdisciplinary program that combines advanced computational training with the study of human cognition. Students will learn about the computational mechanisms underlying human cognition and use computational methods to better understand human cognition. Established by the School of Psychology in collaboration with the College of Computing and with support from the Schools of Physics and Mathematics , the minor is open to all students starting this fall.

There are several new courses in the School of Psychology supporting this minor, including PSYC 4690 (Sensation and Perception: A Computational Perspective) and PSYC/PHYS 4745 (Physics of Cognition). These two classes are offered as special topics this fall but will have permanent course numbers in Spring 2025. More new courses in computation and cognition are planned for the next year and beyond.  

Neuroscience and Neurotechnology Ph.D. Program, Neuroscience Minor

The new Ph.D. and minor offerings build on the recently launched Neuro Next Initiative in Research and the Undergraduate Program in Neuroscience , respectively. 

The new Neuroscience and Neurotechnology Ph.D. Program is a joint effort across the Colleges of Science, Computing and Engineering. It is focused on educating students to advance the field of neuroscience through an interdisciplinary approach, with scientists and engineers of diverse backgrounds — ultimately integrating neuroscience research and technological development to study all levels of nervous system function. The program expects to enroll its first graduate students in Fall 2025.

Approved by the Board of Regents in 2017, the interdisciplinary B.S. in Neuroscience degree enrolled more than 400 undergraduate students in 2022, and has been the fastest growing undergraduate major at Georgia Tech. The Minor in Neuroscience is set to become available during the 2024-25 academic year.  

Quantum Sciences and Technology Minor

In response to the explosion of research, development, investment, and hiring in quantum information science taking place across academia, national labs, and private industry, the School of Physics is now hosting a new Minor in Quantum Sciences and Technology . 

Available starting this fall, the program is open to all students, regardless of major, who are interested in learning more about quantum information theory, applications of quantum information to measurement, quantum materials, quantum computation, quantum algorithms, quantum communication, or any other quantum science related topics. The coursework includes basic training in quantum mechanics and quantum information, and a choice of quantum-related electives in physics, math, chemistry, computer science, and electrical engineering. 

The minor was established by the School of Physics in partnership with the School of Mathematics and the School of Chemistry and Biochemistry in addition to the Colleges of Computing and Engineering.

Related Media

A view of Tech Tower from Crosland Tower. Photo: Georgia Tech

For More Information Contact

[email protected]

Writer: Lindsay C. Vidal

Suggestions or feedback?

MIT News | Massachusetts Institute of Technology

• Machine learning
• Social justice
• Black holes
• Classes and programs

Departments

• Aeronautics and Astronautics
• Brain and Cognitive Sciences
• Architecture
• Political Science
• Mechanical Engineering

Centers, Labs, & Programs

• Abdul Latif Jameel Poverty Action Lab (J-PAL)
• Picower Institute for Learning and Memory
• Lincoln Laboratory
• School of Architecture + Planning
• School of Engineering
• School of Humanities, Arts, and Social Sciences
• Sloan School of Management
• School of Science
• MIT Schwarzman College of Computing

Elaine Liu: Charging ahead

Press contact :.

Previous image Next image

MIT senior Elaine Siyu Liu doesn’t own an electric car, or any car. But she sees the impact of electric vehicles (EVs) and renewables on the grid as two pieces of an energy puzzle she wants to solve.

The U.S. Department of Energy reports that the number of public and private EV charging ports nearly doubled in the past three years, and many more are in the works. Users expect to plug in at their convenience, charge up, and drive away. But what if the grid can’t handle it?

Electricity demand, long stagnant in the United States, has spiked due to EVs, data centers that drive artificial intelligence, and industry. Grid planners forecast an increase of 2.6 percent to 4.7 percent in electricity demand over the next five years, according to data reported to federal regulators. Everyone from EV charging-station operators to utility-system operators needs help navigating a system in flux.

That’s where Liu’s work comes in.

Liu, who is studying mathematics and electrical engineering and computer science (EECS), is interested in distribution — how to get electricity from a centralized location to consumers. “I see power systems as a good venue for theoretical research as an application tool,” she says. “I'm interested in it because I'm familiar with the optimization and probability techniques used to map this level of problem.”

Liu grew up in Beijing, then after middle school moved with her parents to Canada and enrolled in a prep school in Oakville, Ontario, 30 miles outside Toronto.

Liu stumbled upon an opportunity to take part in a regional math competition and eventually started a math club, but at the time, the school’s culture surrounding math surprised her. Being exposed to what seemed to be some students’ aversion to math, she says, “I don’t think my feelings about math changed. I think my feelings about how people feel about math changed.”

Liu brought her passion for math to MIT. The summer after her sophomore year, she took on the first of the two Undergraduate Research Opportunity Program projects she completed with electric power system expert Marija Ilić, a joint adjunct professor in EECS and a senior research scientist at the MIT Laboratory for Information and Decision Systems.

Predicting the grid

Since 2022, with the help of funding from the MIT Energy Initiative (MITEI), Liu has been working with Ilić on identifying ways in which the grid is challenged.

One factor is the addition of renewables to the energy pipeline. A gap in wind or sun might cause a lag in power generation. If this lag occurs during peak demand, it could mean trouble for a grid already taxed by extreme weather and other unforeseen events.

If you think of the grid as a network of dozens of interconnected parts, once an element in the network fails — say, a tree downs a transmission line — the electricity that used to go through that line needs to be rerouted. This may overload other lines, creating what’s known as a cascade failure.

“This all happens really quickly and has very large downstream effects,” Liu says. “Millions of people will have instant blackouts.”

Even if the system can handle a single downed line, Liu notes that “the nuance is that there are now a lot of renewables, and renewables are less predictable. You can't predict a gap in wind or sun. When such things happen, there’s suddenly not enough generation and too much demand. So the same kind of failure would happen, but on a larger and more uncontrollable scale.”

Renewables’ varying output has the added complication of causing voltage fluctuations. “We plug in our devices expecting a voltage of 110, but because of oscillations, you will never get exactly 110,” Liu says. “So even when you can deliver enough electricity, if you can't deliver it at the specific voltage level that is required, that’s a problem.”

Liu and Ilić are building a model to predict how and when the grid might fail. Lacking access to privatized data, Liu runs her models with European industry data and test cases made available to universities. “I have a fake power grid that I run my experiments on,” she says. “You can take the same tool and run it on the real power grid.”

Liu’s model predicts cascade failures as they evolve. Supply from a wind generator, for example, might drop precipitously over the course of an hour. The model analyzes which substations and which households will be affected. “After we know we need to do something, this prediction tool can enable system operators to strategically intervene ahead of time,” Liu says.

Dictating price and power

Last year, Liu turned her attention to EVs, which provide a different kind of challenge than renewables.

In 2022, S&P Global reported that lawmakers argued that the U.S. Federal Energy Regulatory Commission’s (FERC) wholesale power rate structure was unfair for EV charging station operators.

In addition to operators paying by the kilowatt-hour, some also pay more for electricity during peak demand hours. Only a few EVs charging up during those hours could result in higher costs for the operator even if their overall energy use is low.

Anticipating how much power EVs will need is more complex than predicting energy needed for, say, heating and cooling. Unlike buildings, EVs move around, making it difficult to predict energy consumption at any given time. “If users don't like the price at one charging station or how long the line is, they'll go somewhere else,” Liu says. “Where to allocate EV chargers is a problem that a lot of people are dealing with right now.”

One approach would be for FERC to dictate to EV users when and where to charge and what price they'll pay. To Liu, this isn’t an attractive option. “No one likes to be told what to do,” she says.

Liu is looking at optimizing a market-based solution that would be acceptable to top-level energy producers — wind and solar farms and nuclear plants — all the way down to the municipal aggregators that secure electricity at competitive rates and oversee distribution to the consumer.

Analyzing the location, movement, and behavior patterns of all the EVs driven daily in Boston and other major energy hubs, she notes, could help demand aggregators determine where to place EV chargers and how much to charge consumers, akin to Walmart deciding how much to mark up wholesale eggs in different markets.

Last year, Liu presented the work at MITEI’s annual research conference. This spring, Liu and Ilić are submitting a paper on the market optimization analysis to a journal of the Institute of Electrical and Electronics Engineers.

Liu has come to terms with her early introduction to attitudes toward STEM that struck her as markedly different from those in China. She says, “I think the (prep) school had a very strong ‘math is for nerds’ vibe, especially for girls. There was a ‘why are you giving yourself more work?’ kind of mentality. But over time, I just learned to disregard that.”

After graduation, Liu, the only undergraduate researcher in Ilić’s MIT Electric Energy Systems Group, plans to apply to fellowships and graduate programs in EECS, applied math, and operations research.

Based on her analysis, Liu says that the market could effectively determine the price and availability of charging stations. Offering incentives for EV owners to charge during the day instead of at night when demand is high could help avoid grid overload and prevent extra costs to operators. “People would still retain the ability to go to a different charging station if they chose to,” she says. “I'm arguing that this works.”

Share this news article on:

Related links.

• Electric Energy Systems Group
• MIT Energy Initiative
• Department of Electrical Engineering and Computer Science
• Department of Mathematics

Related Topics

• Electrical engineering and computer science (EECS)
• Mathematics
• Laboratory for Information and Decision Systems (LIDS)
• Undergraduate Research Opportunities Program (UROP)
• Renewable energy
• Electricity
• Electric vehicles
• Undergraduate

Related Articles

Cutting carbon emissions on the US power grid

Designing cleaner vehicles

Minimizing electric vehicles’ impact on the grid

Previous item Next item

More MIT News

Using wobbling stellar material, astronomers measure the spin of a supermassive black hole for the first time

Read full story →

Adhesive coatings can prevent scarring around medical implants

Study: Under extreme impacts, metals get stronger when heated

The origin of the sun’s magnetic field could lie close to its surface

Making steel with electricity

H2 underground

• More news on MIT News homepage →

Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA, USA

• Map (opens in new window)
• Events (opens in new window)
• People (opens in new window)
• Careers (opens in new window)
• Accessibility
• Social Media Hub
• MIT on Facebook
• MIT on YouTube
• MIT on Instagram

The PhD Council Podcast dives into the world of PhD life at TU Delft's Electrical Engineering, Mathematics and Computer Science Faculty (EEMCS). We cover research, challenges, and practical tips. We interview PhDs, professors, and postdocs to share their experiences and insights, helping fellow PhD candidates navigate their academic journey effectively. Join us for candid discussions and valuable advice to make the most of your doctoral studies.

PhD Life by PhD Council PhD Council

• APR 17, 2024

PhD Deep Dive: Youri Blom

The sixth episode in our series, in which we dive deep into topics that PhDs in our faculty are working on. Youri's PhD is on Photovoltaic Materials and Devices in the department of Electrical Sustainable Energy. The group specializes in applying advanced measurement techniques in combination with computer modeling for determination of (opto-) electronic properties, such as the density of defects in thin layers of disordered materials. The acquired expertise is used for the optimiz...

PhD Deep Dive: Willem de Muinck Keizer

The fourth episode in our series, in which we dive deep into topics that PhDs in our faculty are working on. Willem de Muinck Keizer is a PhD student at the Discrete Mathematics and Optimization group at the TU Delft. He works in semidefinite programming, which is a type of optimization problem with applications in and outside of Mathematics.

PhD Deep Dive: Lorena Poenaru-Olaru

The third episode in our series, in which we dive deep into topics that PhDs in our faculty are working on. Lorena's goal is to create a systematic method of monitoring and maintaining machine learning models in production against concept drift (data shifts). As part of the AI4FinTech Lab, I am enhancing the collaboration between academia (TU Delft) and industry (ING).

PhD Deep Dive: Ids van der Werf

The second episode in our series, in which we dive deep into topics that PhDs in our faculty are working on. Starting in August 2022, Ids is a PhD student at SPS with Richard Hendriks, working on a project about underwater communications, more specifically about creating a high bandwidth, doppler tolerant communication link. The project is a collaboration between TNO (a dutch research institute), the NLDA (Dutch defence academy) and the TU Delft.

PhD Deep Dive: Carolina Centeio Jorge

First episode in our series, in which we dive deep into topics that PhDs in our faculty are working on. Since October 2020, Carolina is a PhD student at the Interactive Intelligence group of TU Delft. Her PhD has a special focus on mental models in the context of human‐AI teams, such as enabling artificial agents (e.g., robots) to understand and predict human teammates and effectively act accordingly. Carolina's main research interests lie in Behaviour Analysis and Pattern Recognition.

• © 2024 PhD Life by PhD Council

Top Podcasts In Education

Dissertations

Most Harvard PhD dissertations from 2012 forward are available online in DASH , Harvard’s central open-access repository and are linked below. Many older dissertations can be found on ProQuest Dissertation and Theses Search which many university libraries subscribe to.

Skip to Content

• News & Events
• Organizations

SIAM Student Paper Prize Recipient: Dr. Heather Lynn Cihak

Dr. Cihak was nominated by Applied Mathematics Associate Professor Kilpatrick for their paper "Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity" (Multiscale Modeling & Simulation, 2024). Dr. Cihak was one of three recipients of the award and will be honored at the 2024 SIAM Annual Meeting in Spokane, Washington, from July 8-12, with a special session for the prize winners on July 12.

The Applied Mathematics Department congratulates Dr. Cihak on this monumental achievement and is excited to see what the future holds for the new APPM doctoral graduate.

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals
• Explore content
• About the journal
• Publish with us
• Sign up for alerts
• 14 May 2024

Why mathematics is set to be revolutionized by AI

• Thomas Fink 0

Thomas Fink is the director of the London Institute for Mathematical Sciences, UK.

You can also search for this author in PubMed   Google Scholar

You have full access to this article via your institution.

Giving birth to a conjecture — a proposition that is suspected to be true, but needs definitive proof — can feel to a mathematician like a moment of divine inspiration. Mathematical conjectures are not merely educated guesses. Formulating them requires a combination of genius, intuition and experience. Even a mathematician can struggle to explain their own discovery process. Yet, counter-intuitively, I think that this is the realm in which machine intelligence will initially be most transformative.

In 2017, researchers at the London Institute for Mathematical Sciences, of which I am director, began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they discovered that simple artificial intelligence (AI) classifiers can predict an elliptic curve’s rank 1 — a measure of its complexity. Elliptic curves are fundamental to number theory, and understanding their underlying statistics is a crucial step towards solving one of the seven Millennium Problems, which are selected by the Clay Mathematics Institute in Providence, Rhode Island, and carry a prize of US$1 million each. Few expected AI to make a dent in this high-stakes arena.

AI now beats humans at basic tasks — new benchmarks are needed, says major report

AI has made inroads in other areas, too. A few years ago, a computer program called the Ramanujan Machine produced new formulae for fundamental constants 2 , such as π and e . It did so by exhaustively searching through families of continued fractions — a fraction whose denominator is a number plus a fraction whose denominator is also a number plus a fraction and so on. Some of these conjectures have since been proved, whereas others remain open problems.

Another example pertains to knot theory, a branch of topology in which a hypothetical piece of string is tangled up before the ends are glued together. Researchers at Google DeepMind, based in London, trained a neural network on data for many different knots and discovered an unexpected relationship between their algebraic and geometric structures 3 .

How has AI made a difference in areas of mathematics in which human creativity was thought to be essential?

First, there are no coincidences in maths. In real-world experiments, false negatives and false positives abound. But in maths, a single counterexample leaves a conjecture dead in the water. For example, the Pólya conjecture states that most integers below any given integer have an odd number of prime factors. But in 1960, it was found that the conjecture does not hold for the number 906,180,359. In one fell swoop, the conjecture was falsified.

Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant. The On-Line Encyclopedia of Integer Sequences (OEIS) contains almost 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the formidable Busy Beaver sequence (0, 1, 4, 6, 13, …), which grows faster than any computable function. Scientists are already using machine-learning tools to search the OEIS database to find unanticipated relationships.

DeepMind AI outdoes human mathematicians on unsolved problem

AI can help us to spot patterns and form conjectures. But not all conjectures are created equal. They also need to advance our understanding of mathematics. In his 1940 essay A Mathematician’s Apology , G. H. Hardy explains that a good theorem “should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds”. In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us to reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing between them requires an intuition for how the field itself will evolve. This grasp of the broader context will remain out of AI’s reach for a long time — so the technology will struggle to spot important conjectures.

But despite the caveats, there are many upsides to wider adoption of AI tools in the maths community. AI can provide a decisive edge and open up new avenues for research.

Mainstream mathematics journals should also publish more conjectures. Some of the most significant problems in maths — such as Fermat’s Last Theorem, the Riemann hypothesis, Hilbert’s 23 problems and Ramanujan’s many identities — and countless less-famous conjectures have shaped the course of the field. Conjectures speed up research by pointing us in the right direction. Journal articles about conjectures, backed up by data or heuristic arguments, will accelerate discovery.

Last year, researchers at Google DeepMind predicted 2.2 million new crystal structures 4 . But it remains to be seen how many of these potential new materials are stable, can be synthesized and have practical applications. For now, this is largely a task for human researchers, who have a grasp of the broad context of materials science.

Similarly, the imagination and intuition of mathematicians will be required to make sense of the output of AI tools. Thus, AI will act only as a catalyst of human ingenuity, rather than a substitute for it.

Nature 629 , 505 (2024)

doi: https://doi.org/10.1038/d41586-024-01413-w

He, Y.-H., Lee, K.-H., Oliver, T. & Pozdnyakov, A. Preprint at arXiv https://doi.org/10.48550/arXiv.2204.10140 (2024).

Raayoni, G. et al. Nature 590 , 67–73 (2021).

Article   PubMed   Google Scholar  

Davies, A. et al. Nature 600 , 70–74 (2021).

Merchant, A. et al. Nature 624 , 80–85 (2023).

Download references

Reprints and permissions

Competing Interests

The author declares no competing interests.

Related Articles

• Machine learning
• Mathematics and computing

AlphaFold3 — why did Nature publish it without its code?

Editorial 22 MAY 24

AI networks reveal how flies find a mate

News & Views 22 MAY 24

China’s ChatGPT: what a boom in Chinese chatbots means for AI

News 22 MAY 24

Can mathematicians help to solve social-justice problems?

Career Feature 22 MAY 24

The dream of electronic newspapers becomes a reality — in 1974

News & Views 07 MAY 24

Editor (Structural biology, experimental and/or computational biophysics)

We are looking for an Editor to join Nature Communications, the leading multidisciplinary OA journal, publishing high-quality scientific research.

London or New York - hybrid working model.

Springer Nature Ltd

Wissenschaftliche/r Mitarbeiter/in - Quantencomputing mit gespeicherten Ionen

Wissenschaftliche/r Mitarbeiter/in - Quantencomputing mit gespeicherten Ionen Bereich: Fakultät IV - Naturwissenschaftlich-Technische Fakultät | St...

Siegen, Nordrhein-Westfalen (DE)

Universität Siegen

Wissenschaftliche/r Mitarbeiter/in (PostDoc) - Quantencomputing mit gespeicherten Ionen

Wissenschaftliche/r Mitarbeiter/in (PostDoc) - Quantencomputing mit gespeicherten Ionen Bereich: Fakultät IV - Naturwissenschaftlich-Technische Fak...

Professor Helminthology

Excellent track record on the biology and immunobiology of zoonotic helminths and co-infections, with a strong scientific network.

Antwerp, New York

Institute of Tropical Medicine

Assistant Professor in Plant Biology

The Plant Science Program in the Biological and Environmental Science and Engineering (BESE) Division at King Abdullah University of Science and Te...

Saudi Arabia (SA)

King Abdullah University of Science and Technology

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

Doctoral students in Mathematics Education and Teacher Education and Teacher Development present at national conference

Posted in: Research Presentations

John O’Meara, PhD student in Mathematics Education, presented a paper with Shanna Anderson, PhD student in Teacher Education and Teacher Development (TETD), at the National Association of Research in Science Teaching (NARST) conference in Denver, CO on March 19, 2024. They presented their work on Social Network Maps: Supporting STEM Teacher Leaders and Characterizing the Phenomenon of Teacher Leadership, funded by the Wipro Science Education Fellowship grant.

They represented their research team, which includes Drs. Mika Munakata , Emily Klein and Monica Taylor (College of Education and Engaged Learning) and Tim Aberle (PhD student in TETD). Congratulations, all!

Ross County Christian Academy senior aims to be a math teacher

CHILLICOTHE — Ross County Christian Academy senior Carson Ramey is excited and nervous about graduation but luckily for him he won't be out of school for long as he plans to attend college to get his teaching degree.

Ramey has attended RCCA since he was in third grade and throughout his time at the academy he has gotten to know and become great friends with his classmates. He has participated in football, basketball, volleyball, drama, National Honors Society and more. He also plays trumpet in the band and is currently the president of student council.

After graduation, he hopes to attend college to get his teaching degree in math education and go on to teach high school math. He said originally when he thought about being a teacher he also considered teaching history or social studies but was drawn to math because it is a very black-and-white subject that is easier, for him, to grasp.

"I really love learning and the learning environment," said Ramey.

He talks highly about the Ross County Christian Academy as he said the small class sizes have made it easy to become friends with everyone in the school and receive one-on-one help from his teachers.

Those around him have also inspired him to get into teaching as Ramey said he has had great teachers throughout his time at RCCA who have helped him excel in school and his mother was a teacher and is now a principal. With their support and guidance, he hopes to become a patient and motivational teacher who inspires students to do their best.

Outside of school, Ramey is a member of the youth worship team at Lighthouse Baptist Church which sings worship songs during youth and adult services at the church. He said he still gets nervous being in front of people, something he is going to work on before getting in front of a class to teach, but having music to perform with the worship team or in the band helps him feel more confident.

Ross County Christian Academy seniors will be graduating on May 24.

Shelby Reeves is a reporter for the Chillicothe Gazette. You can email her at [email protected] or follow her on Twitter @Shelby_Reeves_

Along with Stanford news and stories, show me:

• Student information
• Faculty/Staff information

We want to provide announcements, events, leadership messages and resources that are relevant to you. Your selection is stored in a browser cookie which you can remove at any time using “Clear all personalization” below.

For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.