# Using original texts while introducing new concepts in class

I'm still a undergrad math student, and my experience in education in math is very limited, however I've been lucky enough to meet teachers that encourage students who are interested in teaching, like myself, to work with their students as support.

I have helped beginners in calculus, probability and algebra, and the common denominator, when it comes to confusions, is that the order in which the topics come sometimes doesn't make much sense to them, so they tend to forget what they've just learned and/or not use them at all after reading them. Here I have in mind only the kids who do study and really try. I keep telling them that for now these things may not make much sense, but they should see them as tools that they're biulding and saving for later, so when the heavy stuff comes they don't have to prove those things any more and can just use them as they need them.

Now, I was thinking of something based of an experience I had. Last semester I took an algebra seminar, where we studied the history of equations, and the teachers' approach was $100\%$ chronological, and based on the original texts of the authors. In that class I learned a lot of new things, but also the other things that I already knew started to make sense, especially group theory stuff, like permutations, normal groups, etc.

I'm not a big fan of teaching/learning chronologically, i.e., in the same order things started to appear in history, because I feel it doesn't make much sense, however the fact of reading the original papers, and the original questions helped me a lot to understand the motivation behind these things that we study in the classroom. So my question is, how good would it be for beginers to read selected original texts to understand new concepts?

Edit: I attached some of the texts that we covered on the seminar; it's not all of them because I don't remember all, but here we go:

• Descartes,
• De Moivre ("De sectione Anguli", Philosophical Transactions),
• Colson ("Aequationum Cubicarum & Biquadraticarum, tum Analytica, tum Geometrica & Mechanica, Resolutio Universalis", Philosophical Transactions),
• Gauss (Disquisitiones Arithmeticae),
• Euler,
• Lagrange (Réflexions sur la résolution algébrique des équations),
• Vandermonde (Mémoire sur la résolution des équations),
• Cauchy (Sur le nombre des valeurs qu'une fonction peut acquérir lorsqu'on y permute de toutes les manières possibles les quantités qu'elle renferme),
• Abel (Mémoire sur les équations algébriques: où on démontre l'impossiblité de la résolution de l'equation générale du cinquième dégré),
• Galois (Mémoire sur les conditions de résolubilité des équations par radicaux),

## 4 Answers

I doubt this would be a good idea. The only exception might be a very recently-developed field, so that they're written in modern style and notation. For most texts and papers much older than the mid 1900s, they're just too difficult to read and would probably be off-putting to students, blocking any potential learning.

Rather than introducing a new concept via an original text, I think you could use the original text to supplementarily motivate the material. Show the students why topic X was first studied and what people were doing with it. Show them how our understanding has advanced over time. (Indeed, I like pointing out, for example, when we study cardinality of sets, that as recently as 150 years ago, these ideas were so controversial that top mathematicians debated them for years, and now we teach them to undergraduates! The students appreciate that.) Show them how modern notation has allowed us to talk about and use topic X more easily. And so on.

In short, I think original texts can really supplement a topic, but only in extreme rare cases could they feasibly be used to introduce a concept.

• The various books by William Dunham do somewhat like you say, they place the developments in historical perspective and (re)written in modern notation (and sometimes explained in modern concepts). Higly recommended. Besides, doesn't require to learn Latin as is needed to read Gauss or Euler... – vonbrand Mar 17 '14 at 16:27
• @vonbrand thanks for the reference – Ana Galois Mar 17 '14 at 17:15
• @AnaGalois, just remembered Ed Sandifer's column "How Euler did it" at MAA. He discussed one topic of interest to Euler a month for a few years, giving much of the original flavor but with updated (somewhat) notation and contrasting to today's understanding/rigor. Highly recommended, but not really for the faint of mind... – vonbrand Sep 17 '15 at 16:56

Consider the difficulties that students have reading modern books, now multiply it by the factors that are introduced by the older phrasing, spelling, et al. Letting students see something of the development of a concept through historical flow is a great idea. Original text pages could be interesting as an image but I wouldn't want to include it as teaching material.

It sounds at first like a great idea. However, in many cases the vocabulary and phrasing, notation, and concepts have changed, sharpened, and been generalized or simplified considerably since they were first introduced. The selection would have to be done very carefully.

For a change, a research-based answer!

This article from Education Week describes an NSF-funded pilot program in which students will study primary sources in mathematics, from Euclid to Pascal. From the article:

Previous NSF-funded work found some evidence that college math students were motivated by the use of primary sources and showed some performance gains in subsequent math classes—but the data were mostly anecdotal.

[Jerry] Lodder, who also worked on a prior NSF grant on which this study is based, said he believes students will gain "more enduring knowledge" through the use of primary sources. "When you read the formula or algorithm in a textbook, sure, you memorize it for the exam, but a few days later you forget it because there was no context," he said. "When students see the original problem or solution it sticks with them longer."

There doesn't seem to be any firm evidence (yet) that this approach yields any benefits in terms of student understanding or retention, but the article makes it clear that part of the grant is devoted to measuring the efficacy of this approach: "Researchers will track students' learning growth and compare results with those for students in nonparticipating math classes."