# 'In these numbers we use no fractions': A Classroom Module on Stevin's Decimal Fractions

## Overview and Introduction

Overview

This activity is suitable for a 75-minute class session, with the assigned reading excerpts, described in Parts II and III of the activity, completed prior to class. The activity itself is designed to introduce undergraduate students preparing to teach mathematics to Stevin’s pamphlet on decimal fractions and to encourage prospective mathematics teachers to think about connections with pupils’ initial learning of multiplication of decimals (see Part III of the activity). Consequently, instructors of courses on history of mathematics for teachers (at both the elementary and secondary level) or those using history in teaching, as well as instructors of courses on mathematics methods for teaching, may find the activity useful.

Introduction

Introducing fundamental school mathematics topics for student understanding is problematic for teachers in contexts that are compromised by rigid curriculum demands driven by mandatory high-stakes testing. Often, teachers claim they lack time to enrich their instruction with alternative perspectives and pedagogically illuminating practices. As a result, teaching mathematics is dominated by procedures and their application to well-chosen examples, rather than conceptually-rich approaches that promote student thinking, deep understanding, and appreciation for the development, beauty, and structure of mathematics.

One way to challenge “mathematics as procedures” teaching practices is to provide preservice mathematics teachers (PSMTs) with opportunities to work on fundamental mathematics ideas from an historical perspective. In the process, PSMTs investigate pedagogical methods that not only enable their future students to focus on why mathematical procedures work, but also may increase PSMTs’ own content knowledge. In this classroom module, designed for use with post-secondary students (grades 13 – 16) preparing to teach mathematics at the late elementary or early secondary level, I present the treatment of multiplying two decimal fractions using Simon Stevin’s (1548 – 1620) “invention” of decimal fractions and his work that introduced notation, rules, and examples for how to use them.

Simon Stevin (1548 - 1620)

## De Thiende

Stevin’s most important work, De Thiende, or The Art of Tenths (The Tithe), was written in Flemish (also known as ‘Belgian Dutch’) and published in 1585. Stevin himself published a French translation, La Disme, in 1585, and English translations appeared later. For example, the almost-complete English translation of De Thiende found at Phill Schultz's website is from 1603. The development of decimal fractions was percolating for quite some time – perhaps from the beginning of the first millennium. Stevin states in his preface:

…even so shall we speak freely of the great use of this invention; I call it great, being greater then any of you expect to come from me. Seeing then that the matter of this Disme (the cause of the name whereof shall be declared by the first definition following) is number, the use and effects of which your selves shall sufficiently witness by your continual experiences… (modified version of translation found in Sarton, 1935)

Indeed, the notation Stevin employed in his invention was quite clunky and did not become part of the mainstream use. It did, however, represent the first printed treatise on decimal fractions and the notation was instrumental in the development that followed. The title page of De Thiende and the first page of the section on computing sums of decimal numbers appear below. The quotation in my title, "In these numbers we use no fractions," is from a section of definitions preceding the section on adding decimals (Sanford, 1929, p. 24).

Above: Title page of Simon Stevin's De Thiende (1585)

Below: First page of the section on sums of decimal numbers from Simon Stevin's De Thiende (1585). The language is Flemish (closely related to Dutch).

## Evolution and Elements of the Task

Evolution of the Task

The design of this classroom task was prompted by an innocent question that I posed to PSMTs at the beginning of a “Using History in the Teaching of Mathematics” course. In particular, I thought I would begin with a question about a pre-secondary mathematics topic, to investigate how well their responses would address the “how” or “why” a particular mathematical process worked (in this case, the multiplication of two decimal numbers). Ultimately, the PSMTs’ responses were not only disappointing but a little disturbing. The majority of their responses correctly explained that the result of multiplying say, 5.674 by 13.799, was determined by finding the product of the whole numbers 5674 and 13799, and then “moving” the decimal point to the left six places since the total number of decimal places in the two factors is six. Unfortunately, very few of the PSMTs in the class could explain why this was the case. It was then that I decided that an episode from the historical development of decimal fractions could contribute a great deal to the mathematical knowledge for teaching of these PSMTs. In order to address additional course objectives, however, I also needed to include the historical and humanistic elements related to Simon Stevin’s mathematical contribution.

Elements of the Task

I.  Identifying a Strategy for Using History

II. Applying the Wilson & Chauvot (2000) Strategy

III. Concentrating on the Mathematics, or Clarifying Conceptions of Decimal Fractions

## Element I. Identifying a Strategy for Using History

Prior to studying Stevin’s decimal fractions, I assigned the article, “Who? How? What? A Strategy for Using History to Teach Mathematics” (Wilson & Chauvot, 2000). The authors argue that one way to successfully include history in teaching mathematics – without feeling the need to become an expert historian in order to do so – is to use a strategy in which the following questions are addressed:

1. Who does mathematics? Since many students perceive mathematics as something done by “others” rather than people like themselves, describing who is doing mathematics and “the way that mathematical ideas are shared also influences our perceptions of who does mathematics” (Wilson & Chauvot, p. 644).
2. How is mathematics done? Many tools have been used not just by mathematicians, but also by those who use and contribute to the development of mathematics: different numeration systems, mathematical and scientific instruments, calculating machines and computers. Consequently, “each method of doing mathematics made a difference in how mathematics was viewed” (Wilson & Chauvot, p. 644).

3. What is mathematics? Finally, history enables us to “view mathematics as not only a combination of … mathematical topics…but also a human endeavor that has spanned centuries and cultures” (Wilson & Chauvot, p. 644). Indeed, “who does the mathematics and how it is done influence what is considered to be mathematics” (Wilson & Chauvot, p. 644, emphasis in original).

Quick Reflection I

The Wilson and Chauvot (2000) article became the guide for introducing my PSMTs to thinking deeply about a mathematical topic, using a historical perspective to ground not only their own mathematical knowledge but possibly that of their future students as well.

## Element II. Applying the Wilson & Chauvot Strategy

The PSMTs in my course were given selected pages of George Sarton’s (1935) discussion, “The First Explanation of Decimal Fractions and Measures (1585)…”, as well as access to the entire 93-page article from JSTOR through our university library system.  In this article, Sarton situated the development of Stevin’s invention (and even examined what occurred after 1585), provided numerous facsimiles of Stevin’s work, and published the entire pamphlet of De Thiende.

If you do not have access to JSTOR or Sarton’s article, other resources are available that may be used with PSMTs, such as the Schultz website referenced here.  Perhaps a more thorough resource than Schultz is found on the Digitale Bibliotheek voor de Nederlandse Letteren (DBNL) website. Through this digital initiative, we can access an account of Stevin’s mathematical life, an analysis of the content of De Thiende (from the Dutch, 1585), a transcription of De Thiende, and a scanned copy of the original work. Additionally, English translations of De Thiende (or La Disme) are available through a variety of sources, including the translation offered by Schultz (complete only up to Stevin’s third proposition on multiplication), via the Digital Library of the History of Science and Scholarship in The Netherlands (Struik, 1958), and in David E. Smith’s A Source Book in Mathematics (1929), available on Google Books – though your library may have a copy of this text. The Sarton article, however, provides a wealth of information all in one place.

This part of the task included the following assignment:

Read the following excerpts from Sarton (1935):

*page 156 – mid-page 158;

*title pages on page 219;

*bottom of page 159 – mid-page 162;

*a few excerpts of the French edition of Stevin’s work (pp. 233 – 234); and

*pages 174 – 176.

You may want to stop after each chunk and either put in your own words what you read, or discuss it with your peers.

Apply the strategy:

1. Who is doing the mathematics described?  Specifically, what does Sarton tell us about Simon Stevin that enables us to know who he was beyond being a person who worked on mathematics?
2. How is mathematics done?  In what ways was Stevin’s contribution (even by his own account) to be used?  Do you think it “facilitated or restricted advancement in certain areas of mathematics” (Wilson & Chauvot, 2000, p. 644)?  In what ways?

3. What is mathematics?  How would you characterize Stevin’s contribution, in terms of a branch of mathematics?  (Or, if you want to think in terms of the National Council of Teachers of Mathematics (NCTM) content standards – that’s fine too!)  Can you use his definitions and notation to describe how students can “break down” the value 8.749?  Although Stevin’s notation is not entirely efficient (it was still 1585!), does it at least make sense?  If Stevin did not introduce this notation and these definitions until 1585, does it make you wonder what we did for the previous centuries?  Alternatively, given the particular time, does it seem feasible that the world was ready for such notation?

Quick Reflection II

The application of Wilson and Chauvot’s strategy required the PSMTs to consider contributing factors to the development of mathematical ideas over time. For example, in Stevin’s introduction to his own work, we read:

To Astrologers, Surveyors, Measurers of Tapestry, Gaugers, Stereometers in General, Mint-masters, and to all Merchants Simon Stevin sends Greeting[.] (Sanford, 1929, p. 20)

Thus, we can begin to answer the question, Who does mathematics?, by describing Stevin to some degree (i.e., from biographical information available) and also by describing the users of such mathematics. Additionally, great opportunities abound to discuss what an “Astrologer” does within the context of the sixteenth century, as well as the job descriptions of “Gaugers” and “Stereometers”.

## Element III. Concentrating on the Mathematics, or Clarifying Conceptions of Decimal Fractions

Lastly, I wanted the PSMTs to read, interpret, and use the notions about decimal fractions (and their notation!) introduced by Stevin. The prospective teachers used the translation provided by Phill Schultz to respond to the following questions. (I use here, as many others have before, parentheses around the “Commencements”, “Primes”, “Seconds”, and so forth – as opposed to the circles used by Stevin.)

1. Using “The third proposition: of Multiplication”, explain Stevin’s explication and demonstration of multiplying 32(0) 5(1) 7(2) and 89(0) 4(1) 6(2).
2. At the end of the demonstration, Stevin writes (modern English provided):

It is then the true product which we were to demonstrate. But to show why (2) multiplied by (2) gives the product (4) which is the sum of their numbers, also why (4) by (5) produces (9), and why (0) by (3) produces (3) etc. Let us take 2/10 and 3/100 which (by the third Definition of this Disme) are 2(1) 3(2) their product is 6/1000 [corrected from the original] which value by the said third Definition 6(3), multiplying then (1) by (2) the product is (3) namely a sign compounded of the sum of the numbers of the signs given.

Conclusion: A Disme number to multiply, and to be multiplied, being given, we have found the product, as we ought.

With regard to his explication and demonstration, does Stevin’s description work? (You may find it helpful to translate his example in this paragraph.) Is this how you remember the explanation/demonstration of why we add the total number of decimal places for the product? If not, what explanation were you given? Or, if you don’t remember, what explanation/demonstration would you give for how to determine the number of decimal places there are in the product of two decimal numbers? Why?

First page of the section on multiplication of decimal numbers from Simon Stevin's 1585 De Thiende. The language is Flemish (closely related to Dutch).

Quick Reflection III

The use of almost every original source (or a translation) from the history of mathematics will create some amount of discomfort with even the best mathematics students. Often with students preparing to teach mathematics, the discomfort can be a little more severe given the strength of beliefs about how and what to teach. Using the English translation of Stevin’s De Thiende was not as difficult as other such texts that I have used. Even so, upon first glance at the material, Cassy (see her reflection below) exclaimed: “I’m not really feeling good about Stevin!” For her (she was preparing to teach middle grades mathematics), investigating a mathematical topic from such a different perspective was disturbing to say the least. What was most powerful about sharing this activity with students was their overwhelming agreement that a brief treatise from 1585 had re-introduced them to the equivalent – yet often disguised – meaning of:

8.749 = 8 + $$\frac{7}{10}$$ + $$\frac{4}{100}$$ + $$\frac{9}{1000}$$ = $$\frac{8749}{1000}$$.

And, if we multiplied this number by 32.57, or

32.57 = 32 + $$\frac{5}{10}$$ + $$\frac{7}{100}$$ = $$\frac{3257}{100}$$,

the resulting product was

$$\frac{8749}{1000}$$ x $$\frac{3257}{100}$$ = $$\frac{28495493}{100000}$$.

Finally, writing as a mixed numeral, we found

$$\frac{28495493}{100000}$$ = 284 + $$\frac{95493}{100000}$$ = 284.95493.

Indeed, why the number of decimal places in this particular product was in fact five was a revelation for the majority of the PSMTs – years after learning multiplication of decimals themselves.

(Note: The example above was discussed in class as a result of reading the PSMTs’ reflections on the task. Although the value 32.57 is found in Stevin’s Proposition 3, this example is not found in his text.)

## Final Reflection

The result of simultaneously considering Stevin’s 1585 treatment of multiplying decimal fractions and PSMTs’ recollection of their learning and their beliefs about teaching the topic were significant for setting the context for the remainder of the “Using History” course. Indeed, investigating the development of school mathematics topics often taught as proceduralized content and devoid of their rich history would prove to challenge my PSMTs’ mathematical knowledge for teaching. Additionally, their beliefs about the benefits of alternative instructional methods (e.g., historical perspectives) were also challenged. I close with a few excerpts from student reflection journals after they completed the above task. (All excerpts are used with the express written permission of the student and all student names are pseudonyms.)

By using history, students can use the methods like Stevin’s decimals to help students to understand the method of multiplying decimal numbers together without getting confused with the decimal point.  What I like about the Stevin decimal number is that it is broken down so that (0) is taking the place as the decimal point.  This assignment we actually worked the problems that were even difficult for people in the past.  I didn’t quite understand how multiplying decimals was a problem, so now I have a unique tool I can use to show my students how to multiply decimals.  This activity will benefit my students by grasping the concept of multiplying decimal numbers whenever they see one. (Shania, Fall 2007)

The history of mathematics has helped me understand notation better. I had never thought about where the notation we use came from because it had never been taught in any of my classes.  It is fascinating to know that it wasn’t until the sixteenth century that a decimal fraction notation was invented.  Simon Stevin in my opinion was a brilliant man who had the spirit to be a teacher. His invention was simple and easy to examine.  I also liked the fact that he wrote his book in the language of the people and not “math language”.  He worked not for fame but to inform the astrologers and land measurers of his day. I can definitely see myself incorporating this history in my classroom.  Writing out the decimals as fractions 8 9/10 3/100 can help the students remember the names of the different place values. It can also be taught exponentially as 8(1/10)^0 for just the number 8.  Another way this history can be incorporated is with the multiplication of decimals. If the tenths, hundredths, and thousandths place are clearly written above the digits they represent, it can help students keep their numbers in straight columns.  (Daniel, Fall 2007)

## About the Author and Bibliography

Kathleen M. Clark is an Assistant Professor of Mathematics Education at The Florida State University.  Her research interests include: the impact of the study of history of mathematics on teachers' mathematical knowledge for teaching, the ways in which teachers use history in the teaching of mathematics, and the practices and development of mathematics teacher educators.  In 2009, she joined Clemency Montelle at the University of Canterbury (Christchurch, NZ) as a visiting researcher in the history of mathematics.

Bibliography

O’Connor, J. J., & Robertson, E. F. (2004). Simon Stevin. Retrieved on July 30, 2009 from http://www-history.mcs.st-andrews.ac.uk/Biographies/Stevin.html

Sanford, V. (1929). Stevin: On decimal fractions. In D. E. Smith (Ed.), A source book in mathematics, pp. 20 – 34. NY: McGraw-Hill.

Sarton, G. (1935). The first explanation of decimal fractions and measures (1585). Together with a history of the decimal idea and a facsimile (No. XVII) of Stevin’s Disme. Isis, 23(1), 153 – 244.

Schultz, P. (2000). Lecture 11: Simon Stevin’s introduction of decimals. Retrieved on July 31, 2009 from http://school.maths.uwa.edu.au/~schultz/3M3/L11Stevin.html

Smeur, A. J. E. M. (Ed.) (2008). Simon Stevin, De thiende. Retrieved on August 14, 2009 from http://www.dbnl.org/tekst/stev001thie01_01/colofon.htm

Smith, D. E. (1929). A source book in mathematics. New York: McGraw-Hill Book Company; reprinted (1959) New York: Dover Publications.

Struik, D. J. (1958). The Principal Works of Simon Stevin, Mathematics, Vol. II A. Amsterdam: Committee of Dutch scientists. Retrieved on August 31, 2009 from http://www.historyofscience.nl/search/detail.cfm?startrow=393&view=image&pubid=2501&search=De%20Thiende&var_pdf=&var_pages=

Wilson, P. S., & Chauvot, J. B. (2000). Who? How? What? A strategy for using history to teach mathematics. Mathematics Teacher, 93(8), 642 – 645.