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Among the several new courses added to the BC high school
curriculum, the History of Mathematics course may raise the
most eyebrows. Although topics in geometry and statistics are
often offered in other jurisdictions, the history of mathematics
seldom exists as its own course except for teacher training in some
American colleges. Even then, the goal is to provide background
to help prospective teachers design their mathematics classes, not
to prepare to teach the subject on its own. Why does the subject
deserve sustained attention in its own course?
Among many possible answers to this question, I find mine in the
phrase “humanizing mathematics”. By definition, mathematics is
an activity performed by humans, intended for a human audience.
The fact that the phrase exists at all is as clear a sign as can be that
many students are missing the point of mathematics. To them it
is a noun—a collection of abstractions—rather than a verb, a kind
of activity. Laudably, curricula are growing in the ability to pass
along problem solving strategies to our students. But this is not
enough. As the new BC math curriculum emphasizes, we need to
raise with our students the bigger questions. Why is mathematics
so successful in understanding the physical world? Why did
we choose to represent unknown quantities with letters? What
is different about how we use mathematics to affect our world,
compared to those who went before?
After four years of intense study of undergraduate mathematics,
I had no idea why the questions my professors were answering in
their lectures were important. In a sense, then, I chose my career in
the history of mathematics as an escape from irrelevance. Many of
our students, especially those not immediately attracted to STEM,
need a big picture to feel part of the conversation. The questions
above were not written by me. They are taken from a list of “why”
questions I solicited from my humanities-oriented history of
mathematics students at the beginning of my course. They shied
away from math not just because they don’t feel competent, but
also because they have not heard answers that are meaningful to
the way they think.
Why History of Mathematics?
The history of mathematics can reach these students in several
ways:
Motivation: All mathematical subjects arose due to some need,
sometimes within mathematics but often from outside of it.
Students who participate in an environment where the need comes
first will recognize that what they are doing means something, and
will be inspired to pursue solutions. In such a setting, it is natural
to portray mathematics properly as inquiry rather than as edifice.
Now, the demand for mathematics today is often portrayed to come
from science and technology, and certainly such motives are more
than appropriate for the classroom. But let’s broaden the possible
options with an example from well outside of science—local
practices of art. From basket weavers in Mozambique to modern
musicians using 20th
century mathematical objects like fractals in
their compositions, the creative artistic process has often provoked
mathematical questions.
One such episode is records of meetings between artisans and
geometers in late 10th
-century Iraq, some written by the great
mathematician Abū’l-Wafā’. Decorations of walls in palaces were
not to include images of people or animals, for religious reasons.
Instead, the local artisans designed elaborate geometric patterns,
many of which still grace the walls of historic Muslim buildings.
The most famous example is the spectacular Alhambra, a palace
in Granada, Spain (see Figure 1). These patterns are born from
simple geometric constructions that are repeated to tile the entire
plane. Then, each line segment or arc in the diagram is elaborated
in some way. The results can be stunning. For instance, Abū’lWafā’
describes how to embed an equilateral triangle in a square,
as follows (Figure 2): extend the base GD by an equal distance to
E. Draw a quarter circle with centre G and radius GB; draw a half
circle with centre D and radius DE. The two arcs cross at Z. Then
draw an arc with centre E and radius EZ downward, to H. If you
draw AT = GH and connect B, H, and T, you will have formed the
equilateral triangle. (If you want to prove it, connect GZ and ZD.
Consider first the angles in triangle GZD; next, consider the angles
in triangle BGZ. After that, you’re on your own!)
by Glen Van Brummelen
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Figure 1: A wall at the Alhambra (Granada, Spain)
AB
G D E
Z
H
T
Figure 2: Abū’l-Wafā’’s construction of an equilateral triangle in a
square
Research: Having posed bigger problems, we are responsible to
provide resources for tackling them. History is an obvious tool here
as well; after all, Euclid, Newton, Euler and others became who
they are because of their ability to solve the big problems. We learn
how to approach problems from those who went before us. Often
their major insights have involved some sort of leap across the gap
between the divide in mathematics between arithmetic and
geometry. For instance, the recognition that the function ( )f x ex
=
is its own derivative allowed Euler and his successors to solve many
differential equations — and, with the introduction of i 1= - , he
was able to combine the inherently geometric discipline of
trigonometry with the algebraic analysis of exponential growth
into an amazing unity.
Closer to the 11th
grade classroom, we turn to ancient Babylonian
schoolchildren’s solutions to the quadratic equation, which were
based on “cut-and-paste” geometry. Consider one tablet where a
child solves x x3
2
12
72
+ = . Although his geometry operates below the
surface of the text, it is clear that the reasoning is as follows: x2
is
a square whose side is the unknown x; x3
2
is a rectangle with sides
and x. Combine them (Figure 3a), and the resulting rectangle has
area 12
7
. Cut off half the original rectangle and move it below the
square. The result is a shape that is almost a square, but with a small
shaded square left out. Its area is 3
1
3
1
9
1
# = . If we add the shaded
square, we know that the larger square of Figure 3b has area
12
7
9
1
36
25
+ = , so its side length is 6
5
. But the side length is also x 3
1
+ ,
so x 6
5
3
1
2
1
= - = .
Figure 3a: A Babylonian quadratic equation
Figure 3b
This process does a lot more than solve a quadratic equation; it
proves the quadratic formula. Besides gaining the quadratic formula
as a tool for future use, students learn several significant lessons
about problem solving. Observing the successes of our predecessors
can lead to ideas for our own successes as well. Changing the way a
problem is represented (in this case, from algebra to geometry) can
lead to surprising insights and novel solutions. Most importantly,
they see that mathematics moves from the “mess” of the struggle
to the discovery of paths to resolution.
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Critical Thinking: Every mathematical community makes shared
decisions about the validity and power of various competing
approaches. For instance, medieval Indian mathematical
astronomers were comfortable using iterative solutions to
equations—a series of guesses or approximations, each one
improved over the previous one using some procedure. However,
ancient Greek and medieval Islamic astronomers preferred direct
arguments and calculations. This may be due to a commitment to
pure mathematics as the only path to true knowledge, propounded
by the great astronomer Claudius Ptolemy. It is difficult for modern
students to understand that such commitments are also present
today. For instance, we represent geometric magnitudes like
lengths and areas using numbers—a choice that practitioners of
the Euclidean tradition rejected. As a result, our mathematics leans
heavily toward arithmetic and algebra, exploiting the power that
these algorithmic methods provide. In order to think creatively, one
needs to make informed judgments about such alternate avenues
of attack; one must know what the community’s rules are before
one decides to bend or break them.
Consider, for example, the curious case of 12th
-century Iranian
scholar Ibn Yahyā al-Samaw’al al-Maghribī, who late in his life
composed Exposure of the Errors of the Astronomers in which he
pointed out dozens of what he perceived to be mistakes in the
works of his colleagues and ancestors. One of these episodes
involved finding a value for sin1o
from a given value of sin3o
. To
find a precise solution turns out to require solving a cubic equation,
and this cannot be done with geometric methods—it is equivalent
to the problem of trisecting the angle with ruler and compass, now
known to be impossible. Violating his own commitment to the
perfection of mathematics, Claudius Ptolemy had been forced into
approximating (an equivalent of) sin1o
. Al-Samaw’al’s creatively
restored the role of pure mathematics in philosophy by redefining
the number of degrees in a circle from 360 to 480. The sine of 3°
becomes the sine of four units, and he can apply the sine half-angle
formula twice to find the sine of one unit. Sometimes, changing
the rules of one game allows one to conform better to the rules of
another.
Implications: Students motivated by humanistic questions admit
readily that mathematics has had a major effect on our world,
but usually they can speak only vaguely about its uses in modern
science and technology. This lack of clarity contributes significantly
to the widespread misperception that mathematics is only a toolbox
of algorithms allowing manufacturers and tech firms to build new
devices. Witnessing the various breakthroughs that have changed
how people live their lives and perceive their world can bring these
students to a deeper appreciation of mathematics as much more
than a calculating machine, but rather a living discipline and driver
of social change.
The biggest driver of social change these days in fact a calculating
machine, the computer. In both the 19th
and 20th
centuries,
mathematicians studied algorithms themselves, recognizing both
their power and their limitations. The person usually considered to
be the first to compose an algorithm implementable on a computer
was Ada Lovelace (1815-1852), daughter of poet Lord Byron. She
maintained a working relationship with Charles Babbage, whose
Analytical Engine would have been easily the first real computer, if
it had ever been built. Lovelace gave a prescription for computing
Bernoulli numbers (fundamental in number theory), and was one
of the few at this time to consider the possibility that the Analytical
Engine or a machine like it could do much more than just
calculation—although she rejected the possibility of true artificial
intelligence. This remarkably early analysis of algorithms reached
an extraordinary conclusion with the work of Julia Robinson
(1919-1985), an important figure in the resolution of Hilbert’s
tenth problem. This problem asked whether a computer algorithm
can be written to determine whether a Diophantine equation has
any integer solutions. The answer, “no”, implies that as powerful
as computers are, there are some things—meaningful things—that
they simply cannot do.
Many of the changes brought about by mathematics were not in
the realm of science and technology. Consider the birth of nonEuclidean
geometry. In the early 19th
century, after dozens of failed
attempts to prove statements equivalent to the assertion that the
angles of a triangle sum to 180°, three separate mathematicians
considered the unthinkable: what if it’s false? Carl Friedrich
Gauss, Nicolai Lobachevsky, and János Bolyai discovered that
a new and consistent geometry may be born from denying this
assumption. Through this bold step, they created “worlds out of
nothing”: elliptical and hyperbolic geometry. These non-Euclidean
geometries remained creations of the mind for decades, but starting
in the early 20th
century they have become candidate models for
the geometry of the universe in which we live. This episode reveals
that seemingly obvious assumptions that have stood for millennia
are not necessarily on solid ground.
Students aware of cosmic shifts like these participate in a rich
educational experience. They are able to orient themselves within
the intellectual landscape, and are can act in their profession with
reflectiveness and purpose.
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Communication: The rich settings exemplified here provide
ample opportunity for our students to expand their options for
writing and otherwise presenting ideas in a mathematics class.
History encompasses entire narratives from initial conception to
final product and societal impact, providing rich opportunities
for interpretive short answers and essays that can go far beyond
learning how to write up an effective solution.
Diversity: Attempts to improve intercultural understanding,
including with indigenous societies, have been spreading
throughout the school curriculum. Mathematics poses a unique
quandary for these efforts, since most popular accounts send
the message that mathematics is separate from culture. “Isn’t for
everyone?” Curricular projects have been helping to bring to the
classroom certain examples of indigenous mathematics realized in
daily life, extending the project of ethnomathematics that has been
working on such matters for a few decades. However, history also
can help to deepen considerably our questioning of how culture
affects mathematics. Why are western cultures the only ones to use
axiomatic-deductive systems to verify mathematical knowledge?
Are other ways of knowing mathematics possible or desirable? Can
one think differently than we do now?
A simple example of a positive answer to this question comes from
pre-modern China. Figure 4a is a standard diagram representing
two similar triangles; we all recognize that b
a
d
c
= . Chinese
geometers and surveyors rarely used this ubiquitous tool. Rather,
they relied on the in-out complementarity principle. In Figure 4b,
the large rectangle is cut by a diagonal line. At any point on the
diagonal, draw vertical and horizontal lines. The reader might
pause here to consider why the two shaded rectangles have the
same area … Now that you have paused, we see that the two
rectangles’ areas are ad bc= , which of course is equivalent to b
a
d
c
= .
The two methods are able to accomplish the same geometric goals,
but using the in-out complementarity principle can lead to
diagrams and thought processes that look very different from those
produced by the use of similar triangles.
Figure 4a: Similar triangles
Figure 4b: The in-out complementarity principle
Of course, who has the opportunity to do the thinking is just as
important as how the thinking is done. In this respect, the human
race does not have a very good track record in allowing equal
participation for everyone in all societal endeavours, including
mathematics. It is an uncomfortable but unavoidable fact, for
instance, that that very few women have been in a position
to be able to contribute to pushing forward the boundaries of
mathematical knowledge until quite recently (and even now,
equality has not yet close). A history of mathematics course can
engage in these issues readily. One approach that historians are
starting to take to help bring female stories to the forefront is
to illuminate their experiences with mathematics in other ways,
especially in their education. This broadens what we mean by “the
history of mathematics”, making it more inclusive of all aspects of
human experience. This approach is in its early stages, but see pp.
59-68 of Jacqueline Stedall’s The History of Mathematics: A Very
Short Introduction for a good start.
My colleague Clemency Montelle (University of Canterbury, NZ)
and I are working on a book to bring episodes like these to the
classroom and to the general public. In the meantime, we conclude
with a list of resources for teachers to help design interesting and
effective lesson plans.
References
Parts of this article were modified from “Why use history in a
mathematics classroom?”, Canadian Mathematical Society Notes
47 (1) (2015), 16-17, and a companion article in Models and
Optimisation and Mathematics Journal 3 (1) (2015), 36-41.
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Resources:
This list is not intended to be comprehensive. There are a
number of general textbooks in the history of mathematics;
some are more appropriate than others for a high school course.
Berlinghoff, William; and Gouvea, Fernando. Math Through
the Ages: A Gentle History for Teachers and Others, 2nd
edition,
Farmington, ME: Oxton House, 2014.
» A compact and accessible, yet surprisingly deep text. Contains a
nutshell history of the subject and 30 short episodes appropriate
for the high school classroom. An expanded edition published
by the Mathematical Association of America contains 60 extra
pages of questions and projects.
Katz, Victor. A History of Mathematics, 3rd
edition, Boston: Addison
Wesley, 2009.
» At a higher level, but as authoritative as history of mathematics
textbooks get.
Barrow-Green, June; Gray, Jeremy; and Wilson, Robin. The History
of Mathematics: A Source-Based Approach, 2 vols., Washington,
DC: MAA Press, 2019 (due to appear in May).
» A comprehensive text that emphasizes working with original
historical materials.
Shell-Gellasch, Amy; and Thoo, John B. Algebra in Context:
Introductory Algebra from Origins to Applications. Baltimore: Johns
Hopkins University Press, 2015.
» Contains a wealth of historical episodes from number systems
to number theory.
Stedall, Jacqueline. The History of Mathematics: A Very Short
Introduction. Oxford, UK: Oxford University Press, 2012.
» Not a survey of historical developments in mathematics, but
rather a short statement of recent approaches to the subject
that are opening up the history of mathematics to new ways
of thinking.
Joseph, George Gheverghese. The Crest of the Peacock: NonEuropean
Roots of Mathematics, 3rd
edition, Princeton, NJ:
Princeton University Press, 2011.
» The most comprehensive general source for the mathematics
of non-western civilizations.
Shell-Gellasch, Amy; and Jardine, Dick, eds. The Courses of History:
Ideas for Developing a History of Mathematics Course, Washington,
DC: MAA Press, 2017.
» A varied collection of approaches to designing a history of
mathematics course.
MAA Convergence, https://www.maa.org/press/periodicals/
convergence
» A high quality free online journal designed “to help you teach
mathematics using its history”. Geared to 8th grade through
undergraduate.
The Story of Maths, BBC/Open University, 2008.
» This comprehensive four-part documentary series takes a
multi-cultural approach. Its last episode, uniquely, takes on
the implications of mathematical developments from the 20th
century to almost the present day.
Transformational Instruction in Undergraduate Mathematics via
Primary Historical Sources (TRIUMPHS), https://blogs.ursinus.
edu/triumphs/
» A five-year National Science Foundation-funded project to
develop teaching modules for the undergraduate mathematics
curriculum based on original sources. Some of the units deal
with high school mathematics as well. They are looking for site
testers at the moment.