In addition to the inclusion of humanistic concepts in the
classroom presentation of mathematics and in the assignments/
projects that students are asked to complete, I have found it
valuable to structure the courses I've developed in terms of
literary and musical forms.
Mathematical knowledge, as that of many disciplines, is
appropriately regarded as a web of ideas, having a great many
interconnections. This can make it very difficult to imagine
presenting material in a linear fashion--the richness of the
discipline can be lost.
Upon reflection, it is clear that structural considerations
are not and should not be limited to the development of new
courses. Whenever we consider we consider a course that we will
teach in the next term, we have the opportunity of seeking an over-
arching structure for the subject.
One way to develop a path through the web of knowledge is to
look to other arts for suggestions of ways to organize material.
Two that I have found to be particularly helpful are literary and
As a first example, and one which encouraged me to think of
curriculum this way, a geometry course, based for example on Martin
Jay Greenberg's Euclidean and Non-Euclidean Geometries [Gr], could
have the form of a novel, with its plot the independence of the
Euclidean parallel postulate. To this end, one would first
introduce the setting, namely the origins of geometry, the
axiomatic method, Euclid's first four postulates and the parallel
postulate. This would be followed by ideas from logic that
establish the context in which independence can be demonstrated or
defeated: theorems, proofs, and models. The non-human characters,
namely Hilbert's Axioms, would then appear and their consequences
and interactions explored, constructing a portion of neutral
(absolute) geometry. Demonstrating that a number of statements are
equivalent to the parallel postulate, would then show the students
what is at stake in a proof or refutation of the parallel
postulate. Tension would increase through a chapter on the history
of the parallel postulate, introducing additional human characters
and leading to the discovery of non-Euclidean geometry. Finally
the independence of the parallel postulate would be established
through models of hyperbolic geometry; by this time, the students
welcome this as the culmination of the semester's story! After
this denouement, the philosophical implications of the independence
of the parallel postulate could be discussed, providing an
additional reward for the completion of the plot of this novel.
Compare this description of a course based on Greenberg's text
with the typical section by section, chapter by chapter
presentation in most courses. Highlighting the difference: in a
course with the plot sketched above, it would be unconscionable to
run out of time before finishing the story!
It is worthwhile to let the students know what role the day's
topic plays in the story of the independence of the parallel
postulate--this helps establish a context for them, again
exhibiting a significant difference from a course without a plot.
In fact, when I asked students on the in-class final to sketch the
plot of the course, all but one were quite successful.
A traditional first-semester calculus course adapts readily to
this idea of structure via a novel. Exactly what the plot line is,
and who the major characters are, will depend on the text adopted
for the course. In a traditional calculus course, very often the
Fundamental Theorem of Calculus is the goal of the plot. Almost
every topic preceding this can either be related to the development
of this theorem, or presented as subplots and asides. In the CCH
reformed approach to calculus based on modelling [H-H], the
fundamental theorem may have become an obvious consequence of the
emphasis on derivatives as rates of change and integrals as total
change. Now instead, the introductory chapter introduces the
various families of functions as the characters in a logically
developed sequence of mystery tales and rags-to-riches stories with
unlikely heroes. Which function is responsible for modelling some
situation, and what evidence can one present for this? How well
can an unlikely hero, a linear function, perform as an adequate
substitute for a more complicated function in some particular
Another example of the use I've made of a form from the arts
is in developing of a portion of a history of mathematics course
offered for middle grades education majors through mathematics
masters candidates, and since adapted for inclusion in other survey
courses. The approach taken was that of a novella or tone poem;
within an overall story line, themes reappear. This story, based
on number and numeration systems, traces ideas from pre-history
through the second half of the twentieth century. Ideas from
Eudoxus recur with Dedekind; prehistoric counting reappears as the
basis of Cantor's cardinality of sets. The discomfort caused
earlier mathematicians by irrational, negative, and complex numbers
reappears in my students with an introduction to the infinitesimals
of the hyperreal numbers of Abraham Robinson.
In contrast to a structure modelled after a novella/musical
tone poem, a unit on "Shape," based on the chapter in On the
Shoulders of Giants [Se], took the form of a theme and variations,
with some fugal entrances. Ideas of similarity, dimension,
symmetry, dissections, and combinatorial geometry all appeared and
A book explicitly constructed in sonata-allegro form is John
McCleary's Geometry from a Differentiable Viewpoint; a summary of
his comments in the Introduction indicate the structure of his text
[McC, ix-xi]. His first section of five chapters opens with a
prelude of spherical geometry, then introduces some of the main
themes, including Euclid's parallel postulate. The eight chapters
in the development section establish what will be required to
provide a rigorous model of non-Euclidean geometry, and introduce
a new theme of an intrinsic feature of a surface, the Gaussian
curvature. In the last three chapters, the recapitulation and
coda, McCleary finishes the development, provides the climax with
the construction of models of non-Euclidean geometry, and then
provides a coda on the theme of the intrinsic.
In addition to providing an over-arching structure, there are
additional advantages to thinking about courses in terms of
literary or musical forms. Making the structure apparent to
students, providing a conceptual narrative, and reminding them from
time to time where you all are in the course, helps address the
needs of students who are not in Sheila Tobias's "first tier" [To,
31, 38, 46, 89]. Such students often feel discomfort with an
unmotivated section-by-section presentation.
Thinking of the course in these terms also can suggest
potential projects and assignments for students that take on a more
humanistic approach. For example, one could ask students to select
the most important theorem from a chapter containing major theorems
of calculus leading up to the Fundamental Theorem, and explaining
the reason for the choice. As another example, one could encourage
students to write a poem after the students have seen hyperreal
numbers, reacting as a Pythagorean might to the discovery of
incommensurable magnitudes; an education graduate student felt this
was the best activity of a two-semester survey course.
Finally, considering over-arching themes for a course also
encourages thought and discussion among faculty regarding strands
in the mathematics curriculum. As institutional pressure (or
external pressure) grows to shorten, or at least, not lengthen
majors, this will become increasingly important to the integrity of
an undergraduate major in mathematics.
[Gr] Greenberg, Marvin Jay, Euclidean and Non-Euclidean
Geometries: Development and History, 2nd ed., San Francisco: W. H.
Freeman and Company, 1980.
[H-H] Hughes-Hallett, Deborah, et al., Calculus, New York: John
Wiley & Sons, Inc., 1994.
[McC] McCleary, John, Geometry from a Differentiable Viewpoint,
Cambridge: Cambridge University Press, 1994.
[Se] Senechal, Marjorie, "Shape," in Steen, Lynn Arthur, ed., On
the Shoulders of Giants: New Approaches to Numeracy, Washington:
National Academy Press, 1990, pp.139-181.
[To] Tobias, Sheila, They're Not Dumb, They're Different: Stalking
the Second Tier, Tucson: Research Corporation, 1990.