This post was written by Peter DeCraene, the 2020-21 Albert Einstein Distinguished Educator Fellow at the Library of Congress.
“I like math because it’s either right or wrong.” I’ve heard some students make this claim, and others profess to dislike math because they often don’t get it right on the first try. Both views reveal an incomplete understanding of mathematical thinking. While some math problems have one right answer, the more interesting and beautiful mathematics has “write” answers. The process of solving problems, either in school mathematics or in research involves messiness and creativity. Writing and revising ideas is part of mathematical thinking, and students could benefit from seeing this in primary sources.
The notebooks of Alexander Graham Bell show the inventor’s mathematical thinking. This page, probably from notes written on May 4, 1878, shows work crossed out, underlined, and questioned. Two pages later he again crossed out some calculations, and other pages from this date show similar revisions and annotations. Ask students to look at these pages and tell what they observe and think about the way the author made his notes. Many students might find something similar to their own work, on the second page in particular, in the scratching out and the lines connecting calculations, perhaps because he ran out of space.
One question that students may have about these pages is “What are these notes about?” At this time, Bell was interested in his photophone, a precursor to fiber optics, but he did not indicate any clear context on these pages. Previous and subsequent entries in the notebook appear to be about different topics entirely.
Some of Bell’s diagrams appear over multiple pages, with additional details or explanations added, showing his mathematical thinking and the evolution and revision of his ideas. Ask students what they notice about these three consecutive pages from the notebook, dated May 6, 1878:
The successively clearer diagrams show the progress in Bell’s thinking, from a diagram he scribbled out on the first page, through a better draft on the second, to what appears to be a carefully drawn final version on the third page. The final version looks neat and tidy, but his process wasn’t!
This article, about the Library’s Alexander Graham Bell Family Papers Collection, has more information about his inventions, and his dedication to observe, reflect on, and question “the hows and whys about things.”
Recognizing Alexander Graham Bell’s process may help students appreciate the need for revising their own math and science work. It also challenges me to think about the types of problems I ask my students to solve, and how I might better encourage them to worry less about getting the right answer, and instead show their answers through revisions. How do you encourage your students to revise their work and honor their process?