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Geometric Algebra in the Classroom

Patricia R. Allaire (Queensborough Community College) and Robert E. Bradley (Adelphi University)

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 include Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher. Convergence founding co-editor Frank Swetz has arranged with NCTM to republish in Convergence (in pdf format) two articles from Mathematics Teacher annually. One of the editor’s picks for 2015 is an article by Pat Allaire and Rob Bradley on geometric realizations of quadratic equations, usually represented symbolically in today's algebra classrooms, from 1700 BCE to the present.

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. Reprinted with permission from Mathematics Teacher, © 2001 by the National Council of Teachers of Mathematics. All rights reserved.

Allaire recently shared how she used ideas from this article in her community college classroom.

I explored several of these techniques in a Liberal Arts mathematics course for which the only prerequisite was elementary algebra.  My section was designated “writing intensive,” which allowed me quite a bit of leeway to work apart from the departmental curriculum.  I chose approaches that were less symbolic and more verbal or geometric, which made them well-suited for a writing intensive course.

The “Geometrical Algebra” section of our article was a good introduction to the idea that we would be exploring methods other than or supplementary to symbolic manipulation.  “The Babylonian Solution of a Quadratic Equation” allowed us to work from a rhetorical algorithm.  As a follow-up, we could look at a geometric justification, which was helpful for seeing how the two approaches were related.

Students had probably seen “completing the square” algebraically, if only to derive the quadratic formula.  “Al-Khwarizmi and Completing the Square [Geometrically]” shed light on the derivation and terminology, even though we were limited to equations with positive coefficients and positive roots.

An interesting sidelight came into play when, as we looked at “Construction of \({\sqrt{c}},\)” I realized that compass and straight edge constructions are no longer taught in high school.  After a hiatus while students acquired the two tools, the group was intrigued with this construction and then with a number of others, which sent us off on a pleasant detour.  This, too, was a good opportunity for students to write about what they were doing, in keeping with the goal of the writing intensive course.

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. It publishes five journals, one for every grade band, as well as one on the latest research and another for 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

Other Mathematics Teacher Articles in Convergence

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."

Richard M. Davitt, “The Evolutionary Character of Mathematics,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 692–694.

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.

Po-Hung Liu, “Do Teachers Need to Incorporate the History of Mathematics in Their Teaching?,” Mathematics Teacher, Vol. 96, No. 6 (September 2003), pp. 416–421.

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.

Robert Reys and Barbara Reys, “The High School Mathematics Curriculum—What Can We Learn from History?”, Mathematics Teacher, Vol. 105, No. 1 (August 2011), pp. 9–11.

Rheta N. Rubenstein and Randy K. Schwartz, “Word Histories: Melding Mathematics and Meanings,” 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, “Seeking Relevance? Try the History of Mathematics,” Mathematics Teacher, Vol. 77, No. 1 (January 1984), pp. 54–62, 47.

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

Patricia S. Wilson and Jennifer B. Chauvot, “Who? How? What? A Strategy for Using History to Teach Mathematics,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 642–645.


Patricia R. Allaire (Queensborough Community College) and Robert E. Bradley (Adelphi University), "Geometric Algebra in the Classroom," Convergence (September 2015)