About

Guershon Harel is a professor in the Department of Mathematics at the University of California, San Diego. He earned his Ph.D. in Mathematics from Ben-Gurion University in Israel in 1985. His research areas include Mathematics Education, Mathematical Biology, and Mathematical Modeling and Applied Analysis. He is involved in various departmental activities and provides contact information through his office at UCSD, located on 9500 Gilman Drive, La Jolla, CA 92093-0112, with the email gharel@ucsd.edu.

Research topics

• Computer Science
• Artificial Intelligence
• Epistemology
• Mathematics
• Engineering
• Mathematics education
• Psychology
• Medicine
• Pure mathematics
• Pedagogy
• Engineering ethics
• Applied mathematics

Selected publications

• Calculus Education: Aspects of Order, Continuity, and Reconceptualization

  Annales de didactique et de sciences cognitives · 2025-11-25

  articleOpen access1st authorCorresponding

  The triad order, continuity, and reconceptualization that appears in the title of this paper refers to a juxtaposition of three aspects of calculus education. Order refers to differentiation followed by integration (DI approach) versus integration followed by differentiation (ID approach) versus Thompson’s integrated approach (TI approach) which view differentiation and integration inseparable. Continuity refers to the impact of these approaches on student learning as they transition from high school to university. Reconceptualization refers to the effort to reform calculus learning and teaching by reeducating future secondary teachers relearn calculus concepts and ideas through the lens of quantitative reasoning. This is an analytic paper. It begins with an analysis of the cognitive and pedagogical features of the three approaches, DI, ID, and TI, and continues with a discussion of the continuity problem concerning the transition from school mathematics to university mathematics, focusing on the difficulty to reform calculus education in the U.S. To advance this reform, it is necessary to examine in depth the current approaches to calculus education in the U.S., as well as alternative approaches advocated by mathematicians and mathematics education scholars. The analysis of the three approaches, DI, ID, and TI, aims at contributing to this essential examination. As part of this examination, the paper offers a calculus module for prospective secondary teachers who have already taken the “mainstream” calculus sequence. The module, while akin to the TI approach, its development and implementation rest on a separate theoretical framework.

  PublisherDOI

• Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

  PRIMUS · 2024-03-29 · 4 citations

  article1st authorCorresponding

  As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students' ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students' background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.

  PublisherDOI

• Publisher Correction: Epistemological justification

  ZDM · 2024-09-20

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• A Student-Centered Lesson on Eigenvalues and Eigenvectors

  PRIMUS · 2024-02-01 · 4 citations

  articleSenior author

  Informed by Harel's DNR theoretical framework, this paper describes the design, incorporation, and analysis of two teaching and learning strategies into a particular lesson on eigentheory. The students were in a blended undergraduate–graduate linear algebra course. During the lesson, the professor looked for mathematical themes within the course that resonated with her students and encouraged them to ask questions and suggest ideas. Students were also given a digital worksheet with interactive figures tailored for the lesson. To help analyze the effectiveness of the changes, the students were asked for written feedback on the lesson style and the worksheet. The authors share the reflections of the professor (and first author) on how her changes affected the teaching and learning of the material in the lesson, as well as subsequent material in the course.

  PublisherDOI

• Epistemological justification

  ZDM · 2024-07-04 · 2 citations

  articleOpen access1st authorCorresponding

  Abstract Epistemological justification is a way of thinking that manifests itself through perturbation-resolution cycles revolving around the question why and how was a piece of mathematical knowledge conceived ? The paper offers a conceptual framework for constituent elements of epistemological justification. The framework provides: (a) a theoretical basis for epistemological justification, (b) criteria for its occurrence, and (c) analysis of its relation to mathematical explanation . The criteria are illustrated by a series of learning-teaching events taken from teaching experiments aimed at investigating the learning and teaching in linear algebra. The contribution of the proposed framework is three-fold: (a) it addresses a critical aspect of proof understanding not explicitly addressed in the literature; (b) it goes beyond the traditional treatment of mathematical understanding and production into questions about learners’ conceptualization of the origins of mathematical knowledge; and (c) it theorizes instructional approaches that can advance this conceptualization among students.

  PublisherOA PDFDOI

• The Linear Algebra Curriculum Study Group (LACSG 2.0) Recommendations

  Notices of the American Mathematical Society · 2022-05-01 · 15 citations

  articleOpen access
  PublisherDOI

• Promoting a set-oriented way of thinking in a U.S. High School discrete mathematics class: a case study

  ZDM · 2022-03-03 · 4 citations

  articleOpen accessSenior author

  Abstract In this case study, we investigate one teacher’s implementation of DNR-based combinatorics curriculum in their high school discrete mathematics class. By examining the teacher’s practices in whole-class discussions of two counting problems, we study how they advanced a variety of ways of thinking to support the development of a set-oriented way of thinking about counting. In particular, we find the teacher worked to build shared experience and understanding of mathematical ideas by grounding her teaching in students’ ways of understanding and leveraging students’ intellectual needs. In doing so, the teacher promoted a set-oriented way of thinking through attending to connections between sets of outcomes, counting processes, and formulas in student representations and justifications; elevated solutions employing process pattern generalization; and advanced the beliefs that counting problems can be solved in many ways and entail several types of mathematical activity.

  PublisherOA PDFDOI

• Ideas foundational to calculus learning and their links to students’ difficulties

  ZDM · 2021 · 66 citations

  Senior authorCorresponding
  • Computer Science
  • Mathematics education
  • Epistemology
  PublisherDOI

• Teaching and Learning of Calculus

  2021-01-01

  book

• The learning and teaching of multivariable calculus: a DNR perspective

  ZDM · 2021 · 19 citations

  1st authorCorresponding
  • Computer Science
  • Artificial Intelligence
  • Mathematics
  PublisherDOI

Recent grants

• Math for America San Diego Noyce Fellowship Program

  NSF · $1.5M · 2009–2017

• Math for America San Diego Noyce Master Teaching Fellowship Program

  NSF · $1.4M · 2010–2017

Frequent coauthors

• Merlyn J. Behr

  Northern Illinois University

  13 shared

• Richard Lesh

  Indiana University Bloomington

  13 shared

• Thomas R. Post

  11 shared

• Jeffrey M. Rabin

  7 shared

• Evan Fuller

  6 shared

• Jana Trgalová

  5 shared

• Larry Sowder

  4 shared

• Ed Dubinsky

  3 shared