3 History in mathematics education 3.1 Introduction Over the years mathematicians, educators and historians have wondered whether mathematics learning and teaching might profit from integrating elements of history of mathematics. It is clear that mathematics education does not succeed to reach its aims for all students, and that it is therefore worthwhile to investigate whether his- tory can help to improve the situation. The interest in using history and the belief in its value for learning have grown remarkably in recent years – judging from the number of research groups in this field in Italy and France, and the History in Math- ematics Education (HIMED) movement founded after a number of successful con- ferences and if we go back in time we find that acknowledged mathematicians from the eighteenth and nineteenth century held the same point of view. Joseph Louis Lagrange (1736-1813) wrote in one of his lectures for trainee school teachers: Since the calculation of logarithms is now a thing of the past, except in isolated in- stances, it may be thought that the details into which we have entered are devoid of value. We may, however, justly be curious to know the trying and tortuous paths which the great inventors have trodden, the different steps which they have taken to attain their goal, and the extent to which we are indebted to these veritable benefactors of the human race. Such knowledge, moreover, is not a matter of idle curiosity. It can afford us guidance in similar inquiries and sheds an increased light on the subjects with which we are employed (cited in Fauvel & Van Maanen, 2000, p. 35). And in one of his notebooks Niels Henrik Abel (1802-1829) remarked: “It appears to me that if one wants to make progress in mathematics one should study the mas- ters” (cited in Fauvel & Van Maanen, 2000, p. 35). Only recently there has been a stronger call for methodological and theoretical foun- dations for the role of history in mathematics education. The report History in Math- ematics Education: The ICMI Study (Fauvel & Van Maanen, 2000) has made a valu- able contribution in this respect by collecting theories, results, experiences and ideas of implementing history in mathematics education from around the world. The purpose of this chapter is to explain why and how history of mathematics might play a role in the learning and teaching of early algebra. After a brief account of the historical development of algebra we describe how history has instigated some es- sential ideas for the experimental learning strand. 3.2 Arguments for using history We would not plead for the use of history of mathematics in mathematics education if we did not believe that history can make a difference. Incorporating history in mathematics education can be beneficial for students, teachers, curriculum develop- ers and researchers in different ways. We give a number of arguments frequently 35 History in mathematics education mentioned to illustrate this (Radford, 1995, 1996ab, 1997; Fauvel, 1991; Fauvel & Van Maanen, 2000). math as a growing Students can experience the subject as a human activity, discovered, invented, discipline changed and extended under the influence of people over time. Instead of seeing mathematics as a ready-made product, they can see that mathematics is a continu- ously changing and growing body of knowledge to which they can contribute them- selves. Learners will acquire a notion of processes and progress and learn about so- cial and cultural influences. Moreover, history accentuates the links between math- ematical topics and the role of mathematics in other disciplines, which will help to place mathematics in a broader perspective and thus deepen students’ understand- ing. teacher profits History of mathematics provides opportunities for getting a better view of what mathematics is. When a teacher’s own perception and understanding of mathematics changes, it affects the way mathematics is taught and consequently the way students perceive it. Teachers may find that information on the development of a mathemat- ical topic makes it easier to explain or give an example to students. For instance, heu- ristic approaches provided by history can be contrasted with more formal, contem- porary methods. In addition it is believed that historical knowledge gives the teacher more insight in different stages of learning and typical learning difficulties. On a more personal level, history also helps to sustain the teacher’s interest in mathemat- ics. value for the designer Not only the mathematics teacher but also the educational developer or researcher can profit from history in studying subject matter and learning processes. It provides teachers and developers with an abundance of interesting mathematical problems, sources and methods which can be used either implicitly or explicitly. Historical de- velopments can help the researcher to think through a suitable learning trajectory prior to a teaching experiment, but it can also bring new perspectives to the analysis of student work. In the context of using history to study learning processes we men- tion the so-called Biogenetic Law popular at the beginning of the 20th century. The Biogenetic Law states that mathematical learning in the individual (philogenesis) follows the same course as the historical development of mathematics itself (onto- genesis). However, it has become more and more clear since then that such a strong statement cannot be sustained. A short study of mathematical history is sufficient to conclude that its development is not as consistent as this law would require. Freudenthal explains what he understands by ‘guided reinvention’: Urging that ideas are taught genetically does not mean that they should be presented in the order in which they arose, not even with all the deadlocks closed and all the de- tours cut out. What the blind invented and discovered, the sighted afterwards can tell how it should have been discovered if there had been teachers who had known what we know now. (...) It is not the historical footprints of the inventor we should follow but an improved and better guided course of history (Freudenthal, 1973, p. 101, 103). 36 History of algebra: a summary learning trajectory In other words, we can find history helpful in designing a hypothetical learning tra- jectory and use parts of it as a guideline. For instance, Harper (1987) argues that al- gebra students pass through different stages of equation solving, using more sophis- ticated strategies as they become older, in a progression similar to the historical evolvement of equation solving. Harper pleads for more awareness of these levels of algebraic formality in algebra teaching. 3.3 History of algebra: a summary It is generally accepted to distinguish three periods in the development of algebra – oversimplifying, of course, the complex history in doing so – according to the dif- ferent forms of notation: rhetorical, syncopated and symbolic, as shown in table 3.1 (see, for example, Boyer & Merzbach, 1989). rhetoric syncopated symbolic written form of the only words words and numbers words and numbers problem written form in the only words words and numbers; words and numbers; solution method abbreviations and abbreviations and mathematical sym- mathematical sym- bols for operations bols for operations and exponents and exponents representation of word symbol or letter letter the unknown representation of specific numbers specific numbers letters given numbers table 3.1: characteristics of the 3 types of algebraic notation rhetorical phase The rhetorical phase lasted from ancient times until around 250 AD, where the prob- lem itself and the solution process were written in only words. In this period early algebra was a more or less sophisticated way of solving word problems. A typical rule used by the Egyptians and Babylonians for solving problems on proportions is the regula tri or Rule of Three: given three numbers, find the fourth proportionate number (see also Kool, 1999). In modern notations this means: given the numbers a, b and c, find d such that a : b = c : d. The Rule of Three also specifies in which order the numbers in the problem must be written down and then manipulated. An example of such a problem is problem 69 in Rhind Papyrus (ca. 1650 BC), which says: “With 3 half-pecks of flour 80 loafs of bread can be made. How much flour is needed for 1 loaf? How many loafs can be made from 1 half-peck of flour?” (note: 1 half-peck ≈ 4.8 liters) (Tropfke, 1980, p. 359). Indian mathematicians (7th and 11th century) extended the Rule of Three to 5, 7, 9 and 11 numbers. Such problems are commonly classified as arithmetical, but in cases where numbers do not represent specific concrete objects and operations are required on unknown quantities, we can speak of algebraic thinking. 37 History in mathematics education unknown Depending on the number concept of each civilization as well as the mathematical problem, the unknown was a magnitude denoted by words like ‘heap’ (Egyptian), ‘length’ or ‘area’ (Babylonian, Greek), ‘thing’ or ‘root’ (Arabic), ‘cosa’, ‘res’ or ‘ding(k)’ (Western European). The solution was given in terms of instructions and calculations, with no explanation or mention of rules. These calculations indicate that unknowns were treated as if they were known and reasoning about an undeter- mined quantity apparently did not form a conceptual barrier. For instance, in the case x of problems that we would nowadays represent by linear equations of typex + ---= a , n the unknown quantity x was conveniently split up into n equal parts. Rule of False Position The Babylonians also used linear scaling to solve for the unknown in the equation ax= b , much like the regula falsi or Rule of False Position first used systematically by Diophantus (Tropfke, 1980). According to this rule one is to assume a certain val- ue for the solution, perform the operations stated in the problem, and depending on the error in the answer, adjust the initial value using proportions.