Mathematics Archives - Storied Books at the University of Florida

This post comes from Discovery Fellow Christian Harris

Figure 1: Byrne’s Elements of Euclid

In continuation of my research, I first observed current trends in Geometry education. While understanding that visual stimulus is appreciated in modern math education, I wanted to look at specific findings surrounding the use of diagrams, such as Byrne’s Elements. Overall, it seems that diagrams are becoming an increasingly popular tool for math education. As programming and digital tools have developed, our ability to visually demonstrate “difficult” math concepts has increased dramatically since Byrne’s writing. These visual tools can help increase math comprehension, especially in young learners. As well, visual modeling, and a slight stray from Euclidian based proofs, have contributed to the growth of other forms of math (such as Hyperbolic and Spherical geometry). However, it has also been shown that visual components can have no impact on exam performance, and if a student is not experienced with specific forms of visual media, that media can actually become a hindrance to student comprehension. Overall, visual media has become a near-necessary aspect of math education, boosting student comprehension. However, this is given that the student has been allowed to understand and gain experience with the formatting of the visual media.

7th grade mathematical activity on cross sections

Figure 2: Use of 3D modeling techniques in 7th grade advanced math textbook (Burger et. al, 2015)

These findings are quite evident when observing geometry textbooks throughout different levels of education. For students who are younger or have little experience with geometric concepts, Euclidean proofs are not presented. Instead, there are findings, absent of proofs, followed by computation and many visual components to help students understand what exactly they are reading. If proofs are presented, they do not reflect the Euclidean style, but are instead simplified and backed by visual/ computational components. However, once you reach college level Geometry (where a student is expected to have experience with geometric concepts), textbooks still reflect pure Euclidean-styled proofs. In many ways, college level geometry textbooks reflect Euclid’s Elements, with minimal edits to diction, syntax, and classifications. Of course, there is variance in textbook content dependent on the publisher, different levels of government, and the standards of each university, so these results are generalized to an extent.

Figure 3: Standard proof from college level Geometry textbook (Venema, 2012)

While post-secondary geometry education has remained largely unchanged, it seems that the use of visual media (such as colors, diagrams, etc.) has flourished in middle/ high school geometry education. What does this say about Byrne? Well, there are many different possibilities for how these changes to education came to fruition. Perhaps it is due to Byrne, who argued that his work was revolutionary (and was later praised by education research). Possibly, changes in legislation surrounding public education enforced changes in curriculum. As well, it could be due to advancements in education research, which prompts change in academic textbooks. More probable, these changes are due to an amalgamation of different trends, both in math education and academia. While I was not able to uncover the direct influence of the implementation of visual media in geometry education, I believe a future path for this research is clear. To continue this research, one would have to specify the causes of changes to U.S. geometry education, specifically looking at how visual media worked its way into middle/ high school curriculum.

Sadly, it is improbable that Byrne is solely responsible for changes to geometry education. However, it is incredible to think that Byrne’s Elements, despite its failure, offered a new perspective into geometry education. The book reformulated the rigid, yet beloved Euclid and showed mathematicians that adaptation is not only possible, but beneficial in cases.

Page from a 6th grade advanced math textbook requiring students to find the area of a triangle

Figure 4: Colors used in 6th grade advanced math textbook (Burger et. al, 2015)

This post comes to us from Discovery Fellow Christian Harris

Around 300 B.C.E., a man named Euclid worked and taught at the Library of Alexandria. While almost nothing more is known about Euclid himself, his name spread far and wide as the author of a series of books which became known as “The Elements.” These books provided formal proofs which laid the ground for teaching modern mathematics. While we do not have the completed work of Euclid, the majority of the Elements still live through various historical prints. Most prints of Euclid take on very small changes, such as translations, adapting syntax, or an addition of commentary. Since 1482, it has become standard to print Euclid with illustrations of the proofs as well.

I was able to view copies of Euclid from the 1600s through the 1800s, many intended for school or college use. These editions, and many others, are relatively smaller with only small differences between. While some variations of Euclid have been more eclectic (more on that below), history has not favored those who stray too far from Euclid’s traditional format.

While we have been able to use Euclid throughout history to teach math, there are still practical contributions that have not yet been explored or realized. At the beginning of my research experience, I used various editions of Euclid’s Elements to learn more about one such famous problem, called the Perfect Cuboid Problem.

The Perfect Cuboid Problem simply asks if there exists a cuboid (a rectangular prism) such that each side can be divided into two “Pythagorean triple” triangles. A Pythagorean triple is a set of three integers such that the sum of the squares of two integers is equal to the third integer squared (e.g., 3, 4, and 5 are a Pythagorean triple because 3²+4²=5²).

Once you understand what the question asks, it feels almost intuitive that it is impossible for such a cuboid to exist.

Hand Drawn Diagram of the Perfect Cuboid Problem

However, while no has ever found an example of a “perfect cuboid,” no one has been able to prove it is impossible either. In my research so far, I observed various proofs of the Pythagorean Theorem, attempting to use such information to gain any insight into the Perfect Cuboid Problem.

cover page of The First Six Books of the Elements of Euclid by Oliver Byrne

The title page of Oliver Byrne’s “First Six Books of the Elements of Euclid”

While maneuvering through different variations of the Elements, I spent a lot of time with one of the most unconventional editions ever produced: Oliver Byrne’s “The First Six Books of the Elements of Euclid: In which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners,” (1847). This edition does exactly as the title claims, rewriting the first six books of the Elements in a completely different format, using color and diagrams instead of words to convey its proofs.

mathematical illustration from The First Six Books of the Elements of Euclid by Oliver Byrne

The Pythagorean Theorem illustrated in Oliver Byrne’s Elements (1847)

While this book was not well received at its time of release, it has received a lot of contemporary praise for its progressive nature and efforts to improve comprehension. As I have continued to work with this book, my research has shifted towards its failure at the time of its release, and the steady influence it had on future math education. As contemporary research promotes the use of diagrams and visuals in geometry education, it seems that Byrne had a level of impact on modern education. I wish to now explore what level of influence, if any, that Byrne had on math literature and education, and when this influence began to show.

Tag: Mathematics

The Strongest Element, pt. II

The Strongest Element

This post comes to us from Discovery Fellow Christian Harris