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- Brief scientific biography
- Contributions to Education
- Relevant bibliography
- Publications on the teaching of mathematics

Hassler Whitney was born 23 March 1907 in New York City. Both of his grandfathers, the philologist William D. Whitney and the astronomer Simon Newcomb, were members of the U.S. National Academy of Sciences, to which Whitney was elected in 1945. Whitney received bachelor's degrees in philosophy (1928) and music (1929) from Yale University. A keen mountaineer all his life, he made a famous climb while still an undergraduate, with his cousin Bradley Gilman, of a cliff in New Hampshire. The cliff was later named the Whitney-Gilman Ridge.

For graduate studies in mathematics, Whitney went to Harvard University, where he earned his Ph.D. under George David Birkhoff in 1932 with a dissertation entitled The Coloring of Graphs, in which he found an equivalent graph-theoretic formulation of the four color problem. He taught mathematics at Harvard for a year and then was a National Research Council Fellow at Harvard and Princeton. Returning to the Harvard faculty in 1933, he rose through the ranks, becoming an assistant professor in 1935, associate professor in 1940, and full professor in 1946. From 1943 to 1945 he was a member of the Mathematics Panel of the National Defense Research Committee.

In 1952, Whitney joined the Institute of Advanced Study at Princeton as professor of mathematics, officially retiring in 1977 but continuing as an emeritus professor until his death on 10 May 1989 in Princeton, New Jersey, two weeks after suffering a stroke. He was survived by his third wife, five children, and six grandchildren.

Whitney's work in topology was wide-ranging and innovative. His first research, a byproduct of his dissertation, was in graph theory; he introduced the notion of duality, which he used to characterize planar graphs (1933), and developed a theory of linear dependence that led to the theory of matroids (1935). He also did early work on the singularities of mappings between n-dimensional Euclidean spaces that turned out to be important for catastrophe theory. His most famous work concerned his theory of differentiable manifolds and its related machinery in algebraic and differential topology (see Zund, 1999, for details). After the Second World War, Whitney turned his attention to the interaction of algebraic topology with integration theory (1957) and then to complex global analysis (1972), making major contributions in both fields.

Among his many activities in mathematics, Whitney was editor of the American Journal of Mathematics from 1944 to 1949 and editor of Mathematical Reviews from 1949 to 1954.

Among many honors, he received the U.S. National Medal of Science in 1976, a Wolf Prize in 1982, and a Steele Prize for Lifetime Achievement in 1985. From 1979 to 1982, Whitney was president of the International Commission on Mathematical Instruction.

Several years before his retirement in 1977, Whitney developed an interest in mathematics education, especially elementary school mathematics, which occupied him for the last two decades of his life (D'AMBROSIO, 1989; LAX, 1989). He gave a number of lectures on education at national and international meetings, conducted summer courses for teachers, and once spent 4 months teaching pre-algebra mathematics to a class of seventh graders (ZUND, 1999). He opposed formal instruction in arithmetic in the early grades, repeatedly citing a little-known study by Louis P. Benezet (1935a, 1935b, 1936), superintendent of schools in Manchester, New Hampshire, who managed to get several schools

in his system to abandon all formal instruction in arithmetic prior to seventh grade. After a year's instruction, the students' arithmetic test scores were at the level of those of comparable students who had undergone regular instruction. Whitney saw the Benezet study as justifying his argument that too many mathematics teachers were focusing on the passing of tests rather than what he called "meaningful goals." He was particularly disturbed by national reports calling for more mathematics to be taught earlier in school:

"The most pressing need I see is for us to face fully the consequences of interventions we make, and hold up on those with bad results. I speak, of course, of mandating more work in mathematics for failing students, raising standards for these without helping them toward meeting the standards, and starting mathematics teaching at an earlier age. It is unthinkable to market drugs without a thorough study of all effects; in education I see no parallel concern, though there should be" (WHITNEY, 1985b, p. 233)Whitney (1986) saw the purposes of learning mathematics as being distorted in many classrooms:

"Mathematical reasoning is becoming steadily more important in many domains; hence a basic need is for students to group in power in such reasoning. In addition, they must gain control over their work, see interrelations with other aspects of the subject, and communicate well with colleagues and others about these matters. . . . In the usual classroom, however, the student's focus is elsewhere. . . . The student's aim is to try to remember particular patterns of thought coming from reasoning, but to disregard the reasoning from which they came. Thus the essence of the process is lost". (pp. 129-130)Using illustrations from his work with children, Whitney (1986) argued for a gradual, unforced, positive change in the way mathematics is taught so that students would have more opportunities for exploration, discussion, and discovery:

"One must find practical ways to move towards reform, leaving most of the present teaching unchanged (at least at the start). Not only the students, but also the professors and others must be considered".

"We push at the students to make them learn; it does not work. Pushing at the professors to teach differently will also fail. Pressures have too much that is negative. We need positive ways; and these must grow by themselves, not be forced. Humans grow through care and love during long periods; the same is true throughout nature". (p. 139)

H. WHITNEY 1933, Planar graphs, Fundamenta Mathematicae, 21, 73-84

H. WHITNEY 1935, On the abstract properties of linear dependence, American Journal of Mathematics, 57, 509-533

H. WHITNEY 1936, Differentiable manifolds, Annals of Mathematics (2nd Series), 37, 645-680

H. WHITNEY 1957, Geometric integration theory, Princeton, NJ, Princeton University Press

H. WHITNEY 1968a, The mathematics of physical quantities: Part 1: Mathematical models for measurement, American Mathematical Monthly, 75, 115-138

H. WHITNEY 1968b, The mathematics of physical quantities: Part 2: Quantity structures and dimensional analysis, American Mathematical Monthly, 75, 227-256

H. WHITNEY 1972, Complex analytic varieties, Reading, MA, Addison-Wesley

H. WHITNEY 1985a, Letting research come naturally, Mathematical Chronicle, 14, 1-19

U. D'AMBROSIO 1989, The visits of Hassler Whitney to Brazil: Hassler Whitney, in Memoriam, Humanistic Mathematics Newsletter, 4, 8

L.P. BENEZET 1935a, The teaching of arithmetic 1: The story of an experiment, Journal of the National Education Association, 24, 241-244

L.P. BENEZET, 1935b, The teaching of arithmetic 2: The story of an experiment, Journal of the National Education Association, 24, 301-305

L.P. BENEZET, 1936, The teaching of arithmetic 3: The story of an experiment, Journal of the National Education Association, 25, 7-8

J. EELLS, D. TOLEDO (edited by) 1992, Hassler Whitney: Collected papers, 1-3, Boston, Birkhäuser

J. DIEUDONNé 1989, A history of algebraic and differential topology: 1900-1960, Boston, Birkhäuser

G. FOWLER 1989 (May 12), Hassler Whitney, geometrician: He eased 'mathematics anxiety', The New York Times, B10

F.M. HECHINGER 1986 (June 10), Learning math by thinking, The New York Times, C1.

A. LAX 1989, Hassler Whitney 1907-1989: Some recollections 1979-1989, Humanistic Mathematics Newsletter, 4, 2-7

R. THOM 1990, La vie et l'oeuvre de Hassler Whitney, Comptes rendus de l'Académie des sciences Paris Sér. Gén. Vie Sci., 7, 473-476

J.J. O'CONNOR, E.F. ROBERTSON 2005, Hassler Whitney, retrieved from the MacTutor History of Mathematics Archive web site:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Whitney.html

J.D. ZUND 1999, Whitney, Hassler, American National Biography, 23, 303-304, retrieved from the American National Biography web site:

http://www.anb.org/articles/13/13-02523.html

H. WHITNEY 1973, Are we off the track in teaching mathematical concepts?, in A.G. Howson (edited by), Developments in mathematical education: Proceedings of the Second International Congress on Mathematical Education, Cambridge, Cambridge University Press, 283-296

H. WHITNEY 1985b, Taking responsibility in school mathematics education, Journal of Mathematical Behavior, 4, 219-235

H. WHITNEY 1986, Coming alive in school math and beyond, Journal of Mathematical Behavior, 5, 129-140, also published in 1987 in Educational Studies in Mathematics, 18, 229-242

H. WHITNEY 1989, Education is for the students' future (draft), Humanistic Mathematics Newsletter, 4, 9-12

Author

Jeremy Kilpatrick

University of Georgia

jkilpat@uga.edu