One of the pitfalls of mathematics instruction at any level is the tendency to teach answers to questions that students don’t (yet) have. Explaining the history of an idea often is a good way to help students see the questions and thought patterns that underlie a concept or process they are being asked to learn. Many teachers agree with this in theory, but they don’t do it. Why not? Perhaps there are two main reasons:
We wrote Math through the Ages to address the first reason. The main part of that book presents 4-to-6-page synopses of the history of 30 common topics in middle- and high-school mathematics curricula, preceded by a broad-brush overview of the history of mathematics condensed into 58 pages. Our hope was (and is) that a teacher preparing a lesson on one of these topics—e.g., negative numbers or fractions or quadratic equations or probability—would easily find interesting motivational material in our short section on that topic.
To address the second reason directly, we wrote Pathways from the Past. Each of the Pathways volumes contained a packet of reproducible student activity sheets that use historical settings to teach or reinforce basic mathematical lessons. Each packet was accompanied by an instructor’s guide.
Since Oxton House no longer publishes these materials, we have made them freely available as two pdf files. In each file the instructor’s guide appears first, followed by the printable worksheets. Links to each packet appear at the bottom of this page.
Here are two examples from the first packet, which is about numbers, numerals, and arithmetic.
Figure 1. Table comparing the notation of the numeration systems featured in Pathways Packet 1.
The second packet focuses on algebra. It starts by exploring algebra without symbols and how the (gradual and erratic) introduction of symbols affected the ways in which algebraic ideas are understood. Then it moves on to sections about linear, quadratic, and cubic equations. Here are two examples.
Figure 2. An excerpt from a student worksheet on the Rule of Three from Pathways Packet 2 (Activity 2-1).
For a time, Oxton House Publishers sponsored teacher workshops using these Pathways packets. The responses from attendees were overwhelmingly positive, as indicated by the following quotes from the workshop evaluations:
One comment from a middle-school teacher merits particular attention. She taught in a school that prided itself on integrating its curriculum across subject lines. Whenever they had special project days or events, the math people were always relegated to tasks like measuring the sizes of buildings or the height of the flagpole. Now, she said, she had some real material to integrate with the social studies curriculum! This was echoed in another workshop, attended by both math and social studies teachers from the same school.
As Bill often said in those workshops, we never thought of these activities as definitive or comprehensive. They are offered to teachers as samples of engaging, history-based, activities they can create for their own students—activities that reinforce what they are teaching and assist in achieving the learning goals of whatever curriculum they teach.
The two Pathways from the Past packets can be downloaded as pdfs using the following links. We hope they will be useful both to teachers and those preparing to be teachers.
Bill Berlinghoff was educated at Holy Cross, Boston College, and Wesleyan University, where he received his Ph.D. in mathematics. He began his teaching career at the College of St. Rose in Albany, New York, was Professor of Mathematics at Southern Connecticut State University, and was Visiting Professor of Mathematics at Colby College in Waterville, ME. He regularly taught a mathematics course for liberal arts students that drew heavily on historical development of the subject. In addition to Math Through the Ages and Pathways from the Past, Bill is author and co-author of several college texts for liberal arts mathematics, including A Mathematics Sampler (5th edition, Ardsley House, 2001). He was also a Senior Writer for MATH Connections, a Standards-based secondary school core curriculum.
Fernando Gouvêa was born in Brazil and educated at the University of São Paulo and at Harvard University, where he received his Ph.D. in mathematics. Fernando is Carter Professor of Mathematics at Colby College, where he teaches, among other things, a course on the history of mathematics. He was a participant in the Institute for the History of Mathematics and its Use in Teaching, organized by the Mathematical Association of America. Fernando was the editor of MAA Focus from 1999 to 2010. He has also served as editor of MAA Reviews, the MAA's comprehensive online book review service, and of MAA's Carus Mathematical Monographs series. In addition to Math Through the Ages and Pathways from the Past, Fernando wrote p-adic Numbers: An Introduction, soon to appear in a third edition, the MAA Guide to Groups, Rings, and Fields, and other books and papers.