History of Modern Applied Mathematics in Mathematics Education

HISTORY OF MODERN APPLIED MATHEMATICS IN MATHEMATICS EDUCATION

UFFE THOMAS JANKVISI [I]

When conversations turn to using history of mathematics in in-issues, of mathematics. When using history as a tool to classrooms, the rnferent is typically the old, often antique, improve leaining or instruction, we may distinguish at least history of the discipline (e g, Calinger, 1996; Fauvel & van two different uses: history as a motivational or affective Maanen, 2000; Jahnke et al, 1996; Katz, 2000) [2] This tool, and histmy as a cognitive tool Together with history tendency might be expected, given that old mathematics is as a goal these two uses of histoty as a tool are used to struc often more closely related to school mathematics However, ture discussion of the educational benefits of choosing a there seem to be some clear advantages of including histo history of modern applied mathematics ries of more modern applied mathematics 01 histories of History as a goal 'in itself' does not refor to teaching his modem applications of mathematics [3] tory of mathematics per se, but using histo1y to surface One (justified) objection to integrating elements of the meta-aspects of the discipline Of course, in specific teach history of modetn applied mathematics is that it is often com ing situations, using histmy as a goal may have the positive plex and difficult While this may be so in most instances, it side effect of offering students insight into mathematical is worthwhile to search for cases where it isn't so I consider in-issues of a specific history But the impo1tant detail is three here. The two first are examples of modem applied whether one's intention is to use histo1y as a goal or a tool mathematics, more precisely the ear1y history of error cor With this framework in mind, I ask: recting codes (Shannon, Hamming, and Golay) and the early How might a history of modem applied mathemat history of data compression (Huffman). The third is an exam ics assist in the teaching and lea1ning of in-issues ple of a modem application of (old) mathematics, the history of mathematics, both conce1ning the motivative/ of public-key cryptography. I use these cases to examine pos affective side as well as the cognitive side of using sible effects of the history of modem applied mathematics history as a tool? on teaching and learning, as well as what differences may arise in using newer histories of mathematics What meta-issues in te1ms of histo1y as a goal I distinguish between the use of history in mathematics might a history of modern applied mathematics education as (1) a tool, in the sense of assisting the actual make accessible to students? leaining of mathematics (mathematical concepts, theories, and so forth), and as (2) a goal in itself, for example, by Discussions of these issues are suppo1ted by empirical data bringing about meta-aspects concerning the history of math from two of the cases. In 2007, cases I and 3 were imple ematics in mathematics education (e g Jankvist, 2007; mented in a Danish uppe1 secondary class Danish uppe1 Jankvist, in press) secondary lasts tluee yerus At the end of second year a class By meta-aspects, I am thinking of posing and responding of 26 students took prut in a module on case I A few months to questions such as the following, drawn from Niss (2001, into their third year, the same class of 23 students took part p 10): in a module on case 3 Each module was taught by the regu lar teacher and spanned about fifteen 90-minute lessons How does mathematics evolve in time and space? Specifically designed teaching materials were used (Jankvist, What forces and mechanisms cause the evolution 2008c; Jankvist, 2008d) Classes were videotaped, question of mathematics? naires were given to students before and afte1 each module, and I 0-12 students and the teacher were interviewed before, How does the evolution of mathematics interact between, and after the modules. At the end of each module with society and culture? students wrote an essay on aspects of the history. [4]

Can mathematics become obsolete? The historical dimension: a study ot three cases Whereas history as a goal is concerned with teaching and With the birth of the computer era, mathematicians found learning something about the meta-aspects of the evolution new ways to apply elements of discrete mathematics - both and development of mathematics, history as a tool is con in creating new mathematical disciplines and in solving var cerned with the teaching and leruning of the inner issues, or ious 'computational'problems Three cases fOllow

8 For the Learning of Mathematirs 29, 1 (March, 2009) FLM Publishing Association, Edmonton, Alberta .. Canada Case 1: The early histmy of error correcting codes other Hanuning codes (Golay, 1949) - prompting a dispute In 1948 Claude Shannon, at Bell labs, published Mathe over the actual creator of the family of codes (Thompson, matical Theory for Communication (Shannon, 1948) - the 1983, pp 56-59) Golay invented four additional codes, of genesis of information theory Shannon considered both which two are perfect. One of these, a tertiary (23, 12)-code channel coding (i e., error correcting codes to adjust for called G23, is especially interesting Coding theoreticians in noise-induced errors during transmission - see fig 1) and 1973 proved it the only (non-trivial) perfect code that may source coding (i e, compression of data for transmission) correct three or more errors - practically halting further Shannon proved that •good' error correcting codes exist, quests for perfect codes but his proof gave no hints on how to construct them He did provide one example of a good (efficient) code, namely Case 2: The early history of data compression the Hamming (7, 4 )-code, through which he nodded to In 1951 Robert M. Farro gave the students in his informa Richard Hamming tion theory course a choice between a final exam and a term Hamming, also at Bell 1 abs, used relay-based computers paper The term paper assigmnent seemed simple: find the with error-detecting codes Unfortunately, whenevet they most efficient method of representing numbers, letters, or detected an error they halted and had to be reset After hav other symbols using a binary code. ing his work dumped two weekends in a row, Hamming was Among the students opting for the paper was 25-year old prompted to create codes that enabled computers to correct David A Huffman Through months of effort, he arrived at occurring errors (to a limited extent) and proceed with cal several methods, but none could be proven the most effi culations. He used the generalized concept of metric to cient. Just before giving up, the solution came to him (Stix, define what is now known as the Hamming distance He 1991, p 54) also drew on vector space theory and linear algebra, think His idea was to assign the shortest binary codes to the ing of possible binary n-tuples as coordinates of corners of symbols occmring most often, drawing on the concept of a an n-dimensional cube and his codes as subsets of these cor coding tree To illustrate, consider the string ALIBABA ners Figure 2 illustrates that the Hamming distance between Ihe frequency of A, B, I, and Lin this string are 3f7; 2/7; two codewords 111and101, written as d(lll, 101), is I l/1; 1/7 A coding tree consists of branches and leafs, each since these two corners of the cube are one apart (or the leaf representing a symbol and an associated frequency or codewords differ in exactly one place) probability In Huffman's algorithm the least probable sym In order to determine the error correcting capabilities of bol is first assigned to a leaf We have two, I and L, so our a given code, Hamming introduced spheres. A sphere is cen tree consists of two leaves each with the associated proba tered in a codeword and the n-tuples inside or on the bility of l/ 7, two branches, and a root with a summed up boundary of this sphere may be corrected into the codeword probability of lf7 + 1/7 = 2f7 (see fig 3) Ihe next less prob in the center (i e, they are at most the 'distance' of the radius able symbol, B, is then added, and probabilities are assigned away from the codeword) A code for which all the possible and summed up last, A is added, resulting in a probability n-tuples are included in spheres around the codewords is at the root of7/7 = I, terminating the algorithm. The binary called a perfect code codewor

NOISE o'"'------0~'o~in SOURCE (2)>' 010 110

Figure 2 Binary tuples of length 2 pictured as points in the Figure 1. Shannon's illustration of a system of.communica plane, and binary tuples of length 3 pictured as tion (Shannon, 1948, p 5) [5] points in

9 thereafter moving along the branches to the root Shannon is to say that ed =1 (mod (p-l)(q-1)) The dec1ypting pro and Pano had attacked the problem in the opposite di1ec cedure D was defined as C' =M (mod n) tion, giving in a less optimal solution Of course it had to be proven that the decryption proce dure led to the original message M. Rivist, Shami1 and Case 3: The history of public-key cryptography aud RSA Adleman (1978) did this with established - even antique - The third type of mathematical coding, c1yptography, con number theory, using Eule1 's theorem (1735-36), Fermat's cerns keeping information secret A thousands-year-old little theorem (1640), and the Chinese remainder theorem problem in cryptography is that of distributing the private (around 4th century) They patented the procedures and, in enc1yption and decryption key between two pruties In 1975, 1982, struted a company offering RSA-solutions. It was sold c1yptographers at Stanford University provided two solutions: in 1996 for $200,000,000. a safe way to generate a common integer (i. e the key, between As an interesting twist, it was made public in 1997 that the two pruties), and public-key c1yptography, a new system British Goverument Communication Headquruters (GCHQ) Whitfield Diffie had the revolutionary idea behind this aheady knew of the system in 1969 (see Singh, 1999). Dur scheme. He wondered if there might be a one-way function ing the 1960s the British militruy had considered equipping - that is, a function/ which, fm eve1y x in its domain,j(x) is soldiers with radios to be in constant contact with their supe easily calculated, but for eve1y y =fix) in its range,r' (y) = x riors. Howeve1, the distribution of keys imposed a problem is impossible to calculate fm all practical purposes. The idea GCHQ c1yptographer James Ellis was asked to look into the behind 'for all practical purposes' is that it may take sec prnblem By 1969 he anived at the same idea that Diffie onds to calculate the function in one direction but eons to reached six yerus late1. And just as Diffie and Hellman later calculate the other way For example, a person - Bob - by would not be able to identify a suitable one-way function, means of a one-way function generates a public encryption neithe1 could Ellis. He and the rest of GCHQ knew there was key; one to which only he knows the decryption key (i e a solution in themy, but it was four yerus before a young num the inverse function) Another person - Alice - who wants ber theoretician, Clifford Cocks, offered one within a half to send a secret message to Bob uses his key, posted some hom oflearning of the 'crazy' idea. Fom years latet it proved whern public, to enc1ypt the message. Bob is the only one to be identical to that of Rivest, Shamit and Adleman able to dec1ypt the message Due to the nature of the one As a secret organization, GCHQ was not inte1ested in way function, a c1yptanalyst - Eve - eavesdropping on the publicizing (and hence patenting) discove1ies. Conse line would stand little chance of breaking the code even quently, Ellis and Cocks silently watched as othe1s wete though she knows both the enc1ypted message and the pub credited, honored and lucratively 1ewarded. In the eaily lic key. The situation is illustrated in figure 4. 1980s Diffie learned about Ellis's work and, in 1982, vis Diffie and Helhnan spent ahnost a year looking for a suit ited to set the recmd straight All he managed from Ellis able one-way function befme giving up and publishing the was a comment about the academics having done much idea in 1976 Ronald Rivest and Adi Shamir at MIT found more with it than GCHQ ever had (Singh, 1999). this pape1 and took it on, developing ideas and passing them on to Leonru d Adleman fm testing. Afte1 Adleman had shot The educational dimension: a discussion down 42 effmts, Rivest came up with a winneI (Bass, 1995) based on the thtee cases by drawing on numbe1 themy - specifically the p1ime fac I now consider the use of histmy of modern applied mathe torization of large numbers matics in terms of history as a tool, looking first at the Generating a very lruge numbe1 n of, say, 200 digits by motivational/affective side and then at the cognitive side I multiplying two also-lruge p1imes p and q is strnightfmwrud then address the use of history as a goal in terms of the meta However, the other way is 'for all practical purposes' impos aspects histories of modern applied mathematics may sible. Rivest devised a method fm genernting both public illustrate [6] and private keys relying on this one-way function The pub lic enc1yption key consisted of two numbers, n and e, the The motivational/affective side of history as a tool latter of which was determined so that gcd(e, (p-l)(q-1)) = 1 At the completion of the module on case 3, students wern The encryption procedure E on the message M revealing the asked which of the two histmies they found to be the most cipher text C was defined as C =M' (mod n) The private decryption key, besides also consisting of n, consisted of a number dthat was an inverse of e modulo (p- l)(q-1), which CRYPTANALYST

m MESSAGE 4/7 Ao;7 47 SOURCE

217 2'7 B 2n 2'7 atso B KEY I L I ,,, in L I l7 l7 L SOURCE #2

Figure 3 Building a binary Huffman tree for the string Figure 4 Diffie :Sand Hellman :S illustration of public-key ALIBABA cryptography (Diffie & Hellman, 1976, p 647)

10 interesting and why. Most of their answers dealt with the learned in class was used, besides basic calculations, a com motivational and/or affective side of using history as a tool mon answer was: "In school!" This, of course, touches upon The fact that the history is a newer and fairly recent his the discussion of the mathematics applied in our society and tory of mathematics seems to make it easier for the students everyday lives being hidden. That is, it is embedded in tech to relate To illustrate: nology such as computer chips, it infuses laws and other decision making processes, and so on (see Jankvist & Told Student: Ihe newest, because it was most bod, 2007; Niss, 1994) 'high tech' or whatever you may call For these reasons some students find the newer histo1y to it It's altight that this guy Euclid did be more relevant, as one student related: something too but these three guys, or six, 01 how many they I think because the development of the Internet is were, that I thought to be much somewhat more relevant to us and it's just a little more more exciting because they are still fun when it takes place in more recent time. . Well, alive It's almost 'just around the with Euclid and Fermat it wasn't like 'Yeah!' But I corner' that they invented it and yet thought that these newer researchers and their ways of it is so widely used now [ ] doing things, the order in which events took place, and the fact that several people invented it at the same time, Inte1viewe1: Is it easier to relate to? that was exciting Seeing newer history as mote 'exciting' was often men Student: I think so It's kind of difficult imag tioned One student made the point this way: ining a guy doing something before Christ was born The closer it comes to our present time, the easie1 it becomes to draw parallels, right, then it becomes more So, besides being more recent, it also matters that the chru exciting And then if it is something you do every day acters are either still alive or have lived dming the students' and then suddenly are being told, well okay that's or their pruents' lives where it comes f1om I think it is ve1y exciting know Concerning the history of modern applications of mathe ing how things are matics some students may find it more interesting to work with such a histo1y, and possibly even more so if they rec When discussing mathematics in everyday life, the same ognize elements from everyday life: student remarked: Student: I think the new one, because you I didn't realize at all that the messages we are sending touch much more upon this applied ove1 the Inte1net and such, that it was mere mathematics mathematics and learn that mathe Of course, inte1view data like these must be t1eated cau matics is being used for something tiously. One never lmows if comments ar·e effo1ts to 1eflect . because many in the old history, what the interviewe1 wants to hear However, question they were hobby mathematicians naires, essays, and video records indicate that the inte1view and such They just did mathemat data provide a good 'marker' of student beliefs ics to do mathematics, to show that they could. I don't find that as excit The cognitive side of history as a tool ing, just doing something for the Hamming's way of developing his codes may serve as an sake of doing it So I think the new introduction to the concept of distance in general In the one was the most exciting one implementation of case 1, most students did not realize there could be more to it than euclidian distance The fact that Inte1viewer: Because it was applied? there is a 'spatial' meaning in the binary case also surprised Student: Yes, because you were being told many Hanuning's use of elements of linear algebra and the stuff about how it was put to use concept of linearity may also be useful in introducing mat ters related to these topics The three cases are examples of how mathematics may be Case 3 offers the possibility of introducing number theo put to use E11or cor1ecting codes are used whenever data retic concepts to students and showing how they play are transmitted or stored digitally Most often more together in a real application. For instance, properties of advanced codes than Hamming 's are used, although Ham prime numbers, the euclidian algorithm, calculating modu ming codes are still implemented in some computer lus, congruence, and linear congruence ar·e all needed to get hardwar·e. Data compression is used at least as often, since RSA to function. In the implementation of case 3, the stu data ar·e usually compressed before t.Iansmission or storage dents knew nothing about number theory prior to the In spite of their age, Huffman codes are still among the most module, so cryptography and RSA served as a way into this used data compression codes RSA is widely used on the discipline. Internet - in email, online banking, and so on. However, for In case 2, Huffman's method may serve to introduce the much of the mathematics encountered in school, where it notion of algo1ithm. Given that the mathematics involved is might be applied may not be so obvious quite different from common cmriculum mathematics, the When I asked in the first interviews where the mathematics mode1n history might not be the most obvious choice on

11 the cognitive side .. Some exceptions occur, of course - as, Diffie and Helhnan were driven by a fascination for the old for instance, in the case of the Danish upper secondary math problem of safely distributing cryptography keys ematics program where the teachers fill in one third of the Regarding how the evolution of mathematics interacts cm1iculum, in consideration of gene1al goals such as appli with society and culture, a newer history may offer students cations, modeling, and historical aspects a more familiar route Hamming became involved with com puters during his work at the Los Alamos project in World Meta-aspects in terms of history as a goal War II. After the war he worked with computers at Bell I retuin now to the tom questions mentioned earlier (

12 Conclusion Huffman, D A (1952) ·A method for the construction of minimum-redun Concerning the use of history as a goal in mathematics edu dancy codes , Proceedings of the IRE 40, 1098-1101 Jahnke, H N., Knoche, N and Otte M (eds) (1996) 'History of mathe cation, a history of modern applied mathematics may be as matics and education: ideas and experiences, no 11" in Studien zur good a candidate as any In some respects it appears to be Wissenschafts-, Sozialund Bildungsgeschichte der Mathematik GOttin better than some. gen, DE, Vandenhoeck & Ruprecht. As a tool, the histoty of modern applied mathematics has Jankvist, U T and foldbod, B (2007) 'The hidden mathematics of the some strong motivational and/or affective qualities A newer Mars exploration Rover mission', The Mathematical Intelligen(.er 29(1 ), history may also serve as a cognitive tool However, given 8-15 Jankvist, U T (2007) 'Empirical research in the field of using history in the often-fixed boundaries of curticula, a newer history may mathematics education: review of empirical studies in HPM2004 & not always be an obvious choice. Of course, the suggestion ESU4', Nomad 12(3), 83-105 is not to reject old history but, rather, to consider the histoty Jankvist, U I (2008a) 'Evaluating a teaching module on the early history of modern applied mathematics when integrating the his of error correcting codes', in Kourkoulos, M and Tzanakis. C (eds ), toty of mathematics This should be done in accordance with Pro(.eedings 5th International Colloquium on the Didactics of Mathe matics, Rethymnon, GR, University of Crete one's original purposes - that is, taking into account the use Jankvist, U T. (2008b) 'A teaching module on the history of public-key of history as either a goal ot a tool, and attending to the cryptography and RSA, BSHM Bulletin 23(3), 157-168 implications of a history of modem applied mathematics on Jankvist, U. T (2008c) 'Kodningsteoriens tidlige historie - et undervis these two different uses ningsfor10b til gymnasiet, No 459', in Teksterfra IMFUFA, Roskilde, DK,IMFUFA Jankvist, U T (2008d) 'RSA og den heri anvendte matematiks historie - Notes et undervisningsforl0b til gymnasiet, No 460', in Tekster fra IMFU FA [1] This paper is a modified version of one presented at the History and Roskilde, DK, IMFUFA Pedagogy of Mathematics (HPM) 2008 in Mexico City Jankvist, U. T (in press)' A categorization of the ·whys and 'hows· of [2] See also proceedings from meetings such as HPM and European Stun using history in mathematics education', Edulational Studies in Mathe mer University on the History and Epistemology In Mathematics Education; and special issues on history in journals such as Educational matics Studies in Mathematics and For the Learning of Mathematics Katz, V (ed.) (2000) Using history to teach mathematics: an international [3] Hereafter only referred to as the history of modern applied mathematics perspective (MAA Notes no 51), Washington, DC 'The Mathematical [4] See Jankvist, 2008a, 2008b Association of America [5] Encoding with enor correcting codes takes place before the message Niss. M (1994) Mathematics in society', in Biehlet, R, Scholz, R W, reaches the transmitter Errors can occur as a result of the noise source Str1iBer, R. and Winkelmann, B. (eds.), Didactics of mathematics as a Error decoding takes place after the message is received scientific discipline Dordrecht, Nl, Kluwer pp 367-378 [6] All the quotes from the research data are translated from Danish Niss, M (2001) 'lndledning', in Niss, M (ed) Matematikken og Verden Fremads debatboger - Videnskab ti! debat, Copenhagen, DK Forfat teme og Forlaget AfS References Rivest,R l ,Shamir,A andAdleman.. l (1978) 'AmdhodfOrobtaining Bass, T. A (1995) 'Gene genie , Wired Magazine 3(08), accessible at digital signatures and public-k<:;y cryptosystems', Communications of http://wwwwired.com/wired/archive/3 08/molecular.html the ACM21(2), 120-126 Calinger R (ed ), Vita mathematica - historical research and integration with teaching (rvlAA Notes, no 40), Washington, DC, The Mathemati Sayood, K. (2000) lntrodultion to data compression (2nd edn ), San Fran cal Association of America, pp. 3-16. cisco, CA, Morgan Kaufmann Diffie, W. and M E Hellman (1976) 'New directions in cryptography', Shannon, C. E (1948) 'A mathematical theory of communication I, II', in IEEE Transactions on Information Theory 22.. 644-654 Slepian, D (ed.), Key paper:s in the development of information theory Epple, M (2000) 'Genisis, Ideen, fustitutionen, mathmatischeWerksttten: 27, New York, NY IEEE Press, pp 379-423 and 623-656 Formen der Mathematikgeschichte - Ein metahistorischer Essay', Math Singh, S (1999) The code book. the secret history of lodes and code break emati.sche Seme.sterbetichte 47, 131-163 ing London, UK, Fourth Estate Fauvel, J and vanMaanen, J (eds) (2000) History in mathematics educa Solomon, D. (1998) Data (.Ompres.sion. the complete reference. New York, tion - the ICM! .study, Dordrecht, NL, Kluwer NY, Springer Verlag Golay, M J E. (1949) 'Notes on digital coding', Proceedings of the IRE 37, 657 Stix, G (1991) ·Profile: information theorist David A Huffman, Scien Hamming, R W (1950) 'Error detecting and error correcting codes , Bell tific American 265(3), 54-55 System Technical Journal29, 147-160 Thompson, T M (1983) From error-correcting codes through sphere pack Hardy, G H (1940) A mathematician's apology, Cambridge, UK, Cam ings to simple groups (Catus mathematical monographs, no 21), bridge University Press Washington, DC, The Mathematical Association of America

A 'Sticky' Problem

Given any six sticks of consecutive integral length (e.g, 7, 8, 9, 10, 11, 12 cm), how many inegular tetrahecha are possible? (unknown origin, but of infinite variation; selected by Leo Rogers of the Oxford Problem Cafe)