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The Evolutionary Character of Mathematics

Author(s): 
Richard M. Davitt (University of Louisville) and Judith Grabiner (Pitzer College)

Over the years, the journals of the National Council of Teachers of Mathematics (NCTM) have published numerous articles on the history of mathematics and its use in teaching. These journals have included Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher, each of which was published through May 2019. In January 2020, these three journals were replaced by NCTM's new practitioner journal, Mathematics Teacher: Learning & Teaching PK–12 (MTLT). Thanks to the efforts of Convergence founding co-editor Frank Swetz, NCTM has allowed Convergence to republish (in pdf format) up to two articles from Mathematics Teacher annually since 2015.

One of the editors’ picks for 2021 is an article by Richard M. Davitt, in which he applied Judy Grabiner’s use-discover-explore/develop-define model for historical change in mathematics to several historical episodes and promoted its application to other topics by instructors in their classrooms:

Richard M. Davitt, “The Evolutionary Character of Mathematics,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 692–694. Reprinted with permission from Mathematics Teacher, © 2000 by the National Council of Teachers of Mathematics. All rights reserved.

(Click on the title to download a pdf file of the article, “The Evolutionary Character of Mathematics.”)

Sadly, the editors learned after selecting this article that Dr. Davitt passed away in 2019. The article that inspired him, “The Changing Concept of Change: The Derivative from Fermat to Weierstrass,” won MAA’s Allendoerfer Award in 1984. Author Judy Grabiner, whose numerous other accolades include the 2021 Albert Leon Whiteman Memorial Prize, reflected on how Davitt read and understood her work:

I very much appreciated Richard M. Davitt’s creative take on my article “The Changing Concept of Change: The Derivative from Fermat to Weierstrass.” He applied the basic idea far beyond the topic I addressed. His application treats my idea as a paradigm, but he does not apply it mechanically. He follows the spirit of the paradigm by focusing on its empirical basis and providing relevant examples. I was sorry to hear that Dr. Davitt has recently passed away, and I dedicate these reflections to his memory.

My reflections will be of two types: One will enlarge on Professor Davitt’s examples, while the other will address using historical scholarship to teach mathematics. The online magazine Convergence, of course, supports both.

That the order of discovery is not reproduced by the order of proof or the order of pedagogical exposition is well known to students—provided that they’re paying attention to their own experience of solving problems. This is how mathematicians do mathematics. Well-known expositors of problem solving, be they George Polya, Rene Descartes, Archimedes of Syracuse, or our own successful students, all know to approach problems by reducing them to things they already know how to do.

For instance, Descartes, in his invention of what we now call analytic geometry, reduced problems in geometry to problems in algebra so he could use the algorithmic power of symbolic reasoning to construct solutions. Archimedes solved problems of areas and volumes of curved figures by enveloping the figures, plane or solid, in figures bounded by straight lines and planes, which he used to creatively approximate the curved figures. Lagrange tried, and Cauchy and Weierstrass succeeded, in reducing problems involving infinitesimals and limits to the “algebraic analysis of finite quantities” like inequalities involving epsilons and deltas. And every successful student recognizes that light-bulb moment, “Oh, I see, we can do it by that method.” This really is how mathematics is done: by filling scratch paper with partial solutions and with arguments that go nowhere but that show what doesn’t work, on the way to the final solution and, at the end of the day, a coherently presented writeup.

There’s a historical parallel to what Dr. Davitt says about the history of mathematical induction in seventeenth-century Europe. Mathematicians also worked on number theory and combinatorics in the medieval Islamic world. Al-Karaji in the eleventh century, and al-Samawa’al in the twelfth, used what is recognizably mathematical induction to prove general results, and the fourteenth-century Jewish mathematician Levi ben Gerson refined this proof method, calling it “rising step by step without end”—a better name, by the way, rooted in how it was discovered and implemented, than the philosophically misleading term “mathematical induction” coined by Augustus De Morgan in the nineteenth century. On the medieval work, see Victor J. Katz, A History of Mathematics (Pearson, 2009, pp. 282fff, 338ff); it is not known whether that work was known to seventeenth-century European mathematicians.

There are also other examples, these zeroing in on proof: Geometry did not begin by somebody saying, “Let’s assume some self-evident things, like that two points determine a line segment and that you can construct a circle with any center and radius, and let’s see what happens.” Geometry began by trying to solve problems, often by reducing them to simpler problems (see Wilbur Knorr, The Ancient Tradition of Geometric Problems (Birkhäuser, 1986)). Hippocrates of Chios (470–410 BCE) reduced the famous Greek problem of duplicating the cube to the problem of finding two mean proportionals between two numbers, which in turn reduces to finding the intersection of a hyperbola and a parabola. Hippocrates also reduced finding the area of some “lunes”—areas bounded by two circular arcs—to finding the area of triangles. He is credited with having been the first to distinguish between those theorems interesting just for themselves and those theorems which lead to something else. Reducing hard problems to simpler problems, and then reducing these to yet simpler problems, creates sets of linked ideas, from the complex to the simple. Running such a set of linked ideas in reverse order gives us a proof structure: simple things on which rest more complex things on which rest yet more complex things. In fact Hippocrates himself wrote a logically structured Elements of Geometry, eventually supplanted by Euclid’s Elements 150 years afterwards.

Nor are these developments limited to the Greek tradition. Mathematicians from other civilizations also had methods of proving their results, often also by proceeding from working backwards from solutions. For a variety of instructive examples, see Karine Chemla, ed., The History of Mathematical Proof in Ancient Traditions (Cambridge, 2012), especially Chemla’s introduction, “Historiography and History of Mathematical Proof: A Research Programme.”

Let me turn to the use of historical examples in teaching. As Dr. Davitt emphasizes, it’s important that students see, as we describe historical episodes, that mathematics progresses by false starts and by making errors and correcting them and learning from what doesn’t work. So when I was first learning about the history of mathematics, I thought it would help me teach. Still, this was just an act of faith. But once I started teaching, my students helped me see that my faith was justified.

One thing history does is help mathematics teachers see what is hard. We wouldn’t be mathematics professors without having learned K–12 mathematics with relatively little difficulty, and many of us find it hard to remember when we didn’t know that (a+b)2 isn’t a2 + b2. Feel free to teach deltas and epsilons, but remember that it took over 150 years from the invention of the calculus by Newton and Leibniz to its delta-epsilon rigorization by Cauchy and Weierstrass. Teaching calculus also lets us see that historical difficulties are often student difficulties today. Consider teaching limits in calculus. Students struggle (as did the followers of Newton) to get past Bishop Berkeley’s attack on limit arguments, “First you assume that there is an increment, and then at the end you throw it away! What right do you have to do that?”

In addition, historical examples can suggest effective teaching strategies. I recall one of my first precalculus classes, populated by students who had seen the material in high school but hadn’t done well enough to place into calculus. Approaching the quadratic formula, I began with al-Khwarizmi’s geometric picture of solving x2 + 10x = 39. And when I physically, on the blackboard, completed an actual square, there were cries of excitement from the class. “So that’s why!”

Finally, history tells us a lot about who mathematicians are and how diverse they are, both in their historical background and in how they think. They’re Egyptian, Babylonian, Greek, Arabic, Persian, Chinese, Indian, Christian, Jewish, Muslim, modern European, African, Latin American, female, male, straight, gay, young, old, etc. They may choose their research questions because of the social importance of the problems, or because of questions in the sciences, or by examining ideas that seem to exist only within pure mathematics. They may have concentrated on problem solving, or on providing a general structure for existing mathematics, or on making more rigorous a set of fruitful if somewhat shaky methods. They may prefer thinking visually or algorithmically. They are not all cut from the same cloth. I hope that history in general, and publications like Convergence in particular, can help students see themselves in the broad and exciting history of mathematics, and as able to successfully advance mathematics themselves.

About NCTM

The National Council of Teachers of Mathematics (NCTM) is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. In addition to its practitioner journal Mathematics Teacher: Learning & Teaching PK–12 (MTLT), the council currently publishes a mathematics education research journal, as well as an online journal for teacher educators (jointly with the Association of Mathematics Teacher Educators). With 80,000 members and more than 200 Affiliates, NCTM is the world’s largest organization dedicated to improving mathematics education in prekindergarten through grade 12. For more information on NCTM membership, visit http://www.nctm.org/membership.

Other Mathematics Teacher Articles in Convergence

Patricia R. Allaire and Robert E. Bradley, “Geometric Approaches to Quadratic Equations from Other Times and Places,” Mathematics Teacher, Vol. 94, No. 4 (April 2001), pp. 308–313, 319.

David M. Bressoud, "Historical Reflections on Teaching Trigonometry," Mathematics Teacher, Vol. 104, No. 2 (September 2010), pp. 106–112, plus three supplementary sections, "Hipparchus," "Euclid," and "Ptolemy."

Keith Devlin, "The Pascal-Fermat Correspondence: How Mathematics Is Really Done," Mathematics Teacher, Vol. 103, No. 8 (April 2010), pp. 578–582.

Jennifer Horn, Amy Zamierowski and Rita Barger, “Correspondence from Mathematicians," Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 688–691.

Seán P. Madden, Jocelyne M. Comstock, and James P. Downing, “Poles, Parking Lots, & Mount Piton: Classroom Activities that Combine Astronomy, History, and Mathematics,” Mathematics Teacher, Vol. 100, No. 2 (September 2006), pp. 94–99.

Peter N. Oliver, “Pierre Varignon and the Parallelogram Theorem,” Mathematics Teacher, Vol. 94, No. 4 (April 2001), pp. 316–319.

Peter N. Oliver, “Consequences of the Varignon Parallelogram Theorem,” Mathematics Teacher, Vol. 94, No. 5 (May 2001), pp. 406–408.

Rheta N. Rubenstein and Randy K. Schwartz, “Word Histories: Melding Mathematics and Meanings,” The Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 664–669.

Shai Simonson, “The Mathematics of Levi ben Gershon,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 659–663.

Frank Swetz, “The ‘Piling Up of Squares’ in Ancient China,” Mathematics Teacher, Vol. 73, No. 1 (January 1977), pp. 72–79.

Richard M. Davitt (University of Louisville) and Judith Grabiner (Pitzer College), "The Evolutionary Character of Mathematics," Convergence (February 2021)