The when-and-who account provides cryptography with an historical per-spective so that you, in terms of time, may view it from above. We learn that societal factors may influence on the development within mathematics.
The why-and-how account shows us how you go from a thought, or a desire, of something to inventing it. It gives us an insight into the driving forces which play in, something which the first account does not provide. (Group 1*)
Besides these two groups being quite right about the why-and-how account providing a deeper and more thorough insight into the evolution and development of mathematics than the when-and-who account, their statements, in fact, also show something else.
They show us that the students may not only be able to discuss matters of the history of mathematics, but they may also, on some level, even though the majority did not answer questions d and e, be able to discuss the presentation of the history itself.
6.6 Observing the Focus Group (Group 1*)
In this section, a more thorough analysis of the focus group’s answering of the final essay assignment will be given. The approach is to hold their hand-in paper up against some of the discussions they had while working in class, the purpose, of course, being to provide means for answering research questions 1 and 2. In doing so I shall also discuss the focus group students’ understanding of the mathematical subject matter (in-issues) of the teaching module. But first an introduction of the two students replacing Sean and Lucy in the focus group.
Introducing the Two New Focus Group Students
The two new focus group students were Lola and Harry. Lola was from the old group 2 and was among the interviewees. Harry, on the other hand, had not been interviewed before, but he was chosen because of his effort in group 6 during the first module. This meant that Harry was interviewed about his answers to both questionnaire 1 and 2 in the same interview of round two. But let us take a closer look at these two new focus group students’ answers to questionnaire 1 and 2 and the followup interviews.
Lolarates her interest in mathematics as a 4.5 (on a scale from 1 to 5) and considers herself good at mathematics (questionnaire 1). The things she likes about the subject mathematics are “logic, cracking your head, ‘the brain is functioning’, seeing finished results and understanding connections”. Things she does not like are way too difficult problems, and things which cannot be placed in wider perspectives. To the question whether she considers it important to learn mathematics, she answers “yes” and continues:
“I have played with the thought of becoming an architect where mathematics plays a major role. +I think that the natural science will have a larger and larger impact on our world.” She does not, however, find it equally important for everybody to know mathematics at a higher level. That is only relevant for people like engineers etc., but other people need to understand the basic mathematics of their everyday life. Lola has not been introduced to the history of mathematics besides something about Pythagoras and also something about Galileo Galilei in her physics class. To the question of whether she believes history to be something which could interest her, she answers: “No, I do
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not find the history interesting. I’m only interested in calculating&using my brain.”
This questionnaire answer was supplied by the following comment in the interview:
I just don’t find it exciting to consider why Pythagoras figured out that his theorem had to be the way it did. To me it’s just more exciting being told what the theorem is, and then use it for something which you can relate to yourself.
Lola’s outspoken skepticism towards the inclusion of history was an important reason for choosing her since this view would replace that of Sean’s. Another important reason was that she was a much more talkative girl than Lucy had been, something needed to function in a group with both Andrew and Gloria. Thirdly, Lola seemed to be on approximately the same mathematical level as the other focus group students, judging from the teacher’s idea of her and her participation in the first teaching module (hand-in exercises and answers in questionnaire 2).
Harryputs down his interest in mathematics as a 4 and thinks himself good as well.
What he likes about mathematics is that “You learn many new mathematical models, which can be used later in life. It challenges your logical sense.” He does not like it if it gets too difficult and hard to follow. It is important for him to learn mathematics because of his further education (in the interview he says that he is going to study nano-technology at the technical university), but also due to household economics. To the question whether he considers it important for everybody to learn mathematics he answers: “Yes. Everyone. So we can have a society which functions well, smart people. The survival of human beings depends on our brains.” Harry has had a little bit of history of mathematics in upper secondary school, he says, something with famous people etc. To the question of whether he thinks he might find history of mathematics interesting he answers: “The part of the history of mathematics which is being used today is interesting. Everything which is not used is only interesting when we find ourselves in a situation where we need to use it. Otherwise it becomes too history-like.”
Despite this statement about things becoming “too history-like”, Harry seemed to have a much more positive attitude towards the teaching modules, the essay assignments, and the meta-issues of mathematics in general than Sean had (something which was later confirmed in the second round of interviews). Another reason for choosing Harry was my impression of him as a quite determined and goal-oriented student from observing the class in general. It was my hope that he could bring a bit more ‘focus’ into the focus group. When the old focus group was ‘focusing’ they had some very interesting discussions. However, a lot of (recording) time was spent on discussing matters not related to the essays or exercises, such as the previous and the coming weekends’ parties, other teachers, friends, grades, last night’s tv-shows, etc. As mentioned, Gloria, though a very capable student, was often the one to set the agenda of the group, including when to cut-up and when not to. The fact that Harry, like Gloria, was a ‘strong’ student and seemed a mathematically capable one as well (according to the teacher, but also judging from his exercises in coding theory), made me believe that he could act as a counterpoise to Gloria and thereby help keep the discussions on track for longer stretches of time.
Furthermore, Harry and Jonathan were friends, so if Harry was able to take charge of the group from time to time, this might also result in Jonathan not being overheard as often as he was in the first module.
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The Focus Group’s Final Essay Assignment and the Making of It
The focus group provided a quite good answering of the final essay assignment, the main essay as well as the three supportive ones. The total length of the hand-in paper was four pages.
Unlike the situation in the first module, the modified focus group’s work on and preparation of the final essay offer a variety of good meta-issue discussions, concerning the main essay as well as all three supportive essays. This made the selection of which clips to present more difficult. In my selection I have tried to choose clips including elements which were not emphasized during the analysis of the first module final essay, as well as clips including students whom were not the most discussed during the analysis in chapter 5. Also, I have tried to identify clips which possess potential anchoring points of the meta-issue discussions in the related subject matter (in-issues).
Actually, the focus group is one of the two groups who answered all questions of the main essay. The length of the main essay is a little more than one and a half page, which is a bit above average for this assignment. However, the focus group only considers the mathematicians of newer date in the main essay (i.e. in questions a and b). It is not entirely clear from the video why they only consider the newer mathematicians, but it may be because they began the main essay assignment while finishing up the third supportive essay, which concerned only the cryptographers. Judging from the presentation of the main essays in section 6.5, the discussion of the older mathematicians unfortunately seems to be one of the natural places to include elements of the mathematical subject matter. For this reason, the only places in the focus group’s discussions of the main essay which offer possible anchoring points are those places where they discuss objects and techniques in relation to public-key cryptography and RSA. But since objects and techniques were not among the main general topics and issues of the final essay for the second module, I am not going to use these clips to discuss anchoring.
Instead, I shall use a clip from the group’s work on the first supportive essay, the one on pure and applied mathematics and Hardy’sApology. The students worked and discussed in relation to this essay assignment for approximately an hour. The chosen clip is from their work on question c – Hardy’s statements on number theory versus the creation of RSA. In their discussion, the students drew on related mathematical concepts and notions from both RSA as well as number theory. Originally, the clip was 14 minutes long, but some minutes have been edited out of the transcript since they concern mattes not related to the essay.
In addition to this clip, I shall also present a clip from the second supportive essay assignment (question b), in which the students discuss inner and outer driving forces in the development of public-key cryptography and personal motivations of the researchers involved. This video clip has been chosen for three reasons: (1) it concerns meta-issues which were not touched upon in the first module, i.e. inner and outer driving forces; (2) it involves a very interesting discussion of these meta-issues, in which all five students engage actively and in which a couple of the students definitely do not agree with each other; and (3) it is a clip of 12 minutes, no omissions, where the students stay focussed on the task – in fact the longest of such consecutive clips in my entire data collection.
Now concerning the actual answering of the second supportive essay, something should be addressed. As mentioned, a somewhat general tendency for the groups of students was that of mixing up the inner and outer driving forces in the development of a
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mathematical area with the personal motivations of the researchers involved in the development. That is to say the students would look at one particular researcher and ask what hispersonal inner and outer driving forces were. The way the assignment was intended, was to have the students do the one first and then the other. Nevertheless, in spite of this misunderstanding, interesting discussions and written answers still emerged from the students’ work.
The third supportive essay offers interesting discussions, especially in relation to the students’ Internet searchers on GCHQ and NSA. However, these discussions offer no potential anchoring points to be explored, they are all concerned with meta-issues and in this sense they do not provide any new information in addition to the other clips.
I shall now turn to the presentation of the video clips. Since the students worked the second supportive essay assignment before the first, I shall begin by looking at the video clip from that one.
A Video Clip from the Second Supportive Essay Assignment
In this first clip the students are working on question b of the second supportive essay (inner and outer driving forces of the development of public-key cryptography and personal motivations of the people involved therein). Their full answer to question b was:
b) Diffie, Hellman, and Merkle: These people, who invented public-key cryptography, were mostly driven by an inner driving force since they were interested in the area themselves, and developed it out of own interests.
Diffie is especially fascinated by the Internet, and both Diffie and Hellman believe that cryptography should be for the benefit of all. This may be seen as an outer driving force.
Rivest, Shamir, and Adleman: Rivest and Shamir are very clearly driven by their interest in the area. But at the same time they are also driven by their interest in Diffie and Hellman’s new theory, which develop their interest in finding a code [one-way function] for this theory. That is to say it is an outer driving force which influences them in a certain direction. But Adleman is solely driven by an outer driving force since he is not interested in the area personally, but is being pressed by Rivest to find numbers which fit the theory.
Ellis, Cocks, and Williamson: They are all driven by the outer driving force because they work for GCHQ, which is a military organization. They are therefore strongly influenced by the organization concerning their work.
(Group 1*)
In the very beginning of the discussion (see appendix C.2), Harry states that he finds it obvious that Diffie and Hellman were driven by an inner driving force whereas Rivest, Shamir, and Adleman were driven by an outer. This immediately results in a series of comments from Andrew who believes that there is also an outer driving force present in the form of the need for security. Gloria brings up Diffie’s work related to the Internet, and she and Jonathan argue that all mathematicians somehow need to possess an inner driving force – arguments which Gloria brings up again and again in the rest of the discussion. Jonathan begins flipping through the teaching material to check out Harry’s argument, while Andrew and Gloria seem to reach some common ground concerning
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Diffie and outer driving forces (lines 21-29 and 54-56). Jonathan then finds what he is looking for and reads aloud: “It says right here: Diffie was especially fascinated and interested in the beginning Internet, the so-called ARPANet, when it at first belonged to the US military. And he was one of the first to realize the possibilities, blah blah ...” (lines 35-37). Unfortunately, Jonathan also throws in a comment about Hellman and Diffie wanting to compete with the NSA and this being an inner driving force for them. What he is referring to is the reaction of Hellman’s university colleagues, who said that when doing cryptography you would be competing with the NSA and their million dollar budget. Jonathan himself soon realizes that this is not a real competition but more a figure of speach from the colleagues. Harry, however, catches on since this supports his view of Diffie and Hellman being driven by an inner driving force and quite some time is spent on Jonathan and Harry discussing this competition aspect. Then Lola enters the discussion reading aloud from the text, something which kickstarts a discussion between Gloria and Harry:
Lola: Check this out: Diffie recalls the day in 1975 when he first got the idea for public-key cryptography like this: I went downstairs for a Coke and almost forgot the idea. I remembered that I had been thinking of something interesting, but I could not recall what it was. Then it came back to me in an adrenaline rush of excitement. For the first time while I had been working with cryptography I was aware that I had discovered something really valuable. He is...
Harry: He is really interested in it himself, right?
Gloria: Yes yes, but all of them are. There is actually an idea behind him producing this mathematics: it is for developing the Internet.
Harry: Yes, but nobody told him that he should do it to develop the Internet.
It’s like in wars, there are many...
Gloria: No, nobody... but I also think... No, but the outer circumstances, it also has to do with the situation of the world and its state, and how far you are and what it can be used for.
Andrew: And again, the thing with you getting the credit for it.
Harry: But I also think that these three persons...
Gloria: But he also wants to do this for the sake of the world.
Harry: It’s like that with all of them. I mean, it’s for the benefit of the world, all the things these guys are doing.
Jonathan: All the previous ones haven’t said that they wanted to do this for something. Here they say that they want to do it so that everyone may benefit from it.
Harry: Yes, but that still isn’t an outer driving force. It is just his own thought as to what his project may amount to. Isn’t it? (lines 77-99)
Gloria is rather strong in her belief about Diffie being visionary and having higher goals (e.g. the Internet) as opposed to just doing cryptography out of pure interest. Harry is not convinced yet, but he does not seem as opinionated as he did in the beginning either.
Next, the students begin to discuss the GCHQ guys and whether they were driven by inner or outer driving forces. Harry, Jonathan, and Lola reach an agreement rather fast: for these guys it must be outer driving forces that counts. They were hired by an organization, they were told what to look at, they were paid money, etc. (lines
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110-120). Jonathan decides to check with the teacher for a definition of outer driving forces (lines 121-126). Gloria seizes this opportunity of the teacher’s presence to reenter the discussion from earlier on Diffie and Hellman:
Gloria: But can’t the state of the world also be an outer driving force: Okay, now we are like this, we need some Internet which works even better?
Teacher: Yes, I think you can say that that can be an outer driving force.
Lola: So you can say that the task they are being given is an outer driving force?
Harry: Yes, but the task they are being given must be given by an outer driving force before you can say that they are driven by an outer driving force. But they are not being given this task.
Gloria: Listen. All mathematicians are also driven by an inner driving force, otherwise they wouldn’t do it. Otherwise they wouldn’t care to spend their life on it.
Harry: But those three persons were not told to make a better system.
Gloria: No, no... (lines 127-137)
Notice here that Gloria presents the argument which she and Jonathan gave in the beginning of the discussion (lines 12-14), only this time in a more elaborated version.
What she seems to be doing is reusing an argumentation which she gave in the fourth supportive essay of the first module, where she said that mathematicians (Hamming and Golay) would not be willing to spend their lives on something without receiving credit for it (an element which Andrew also brought up in the present discussion, lines 54-55, 91). The difference in Gloria’s argument is that she now is able to provide her argumentation using the notion of personal inner driving forces. While Gloria pursues her quest, Harry and Lola are elaborating on the discussion about what an outer driving force is. The teacher asks a clarifying question and thereby actually make Harry and Gloria reenter their discussion from before:
Teacher: Oh, you mean outer driving force as someone else posing the task.
Gloria: It doesn’t have to be. It doesn’t have to be somebody else who gives the task.
(lines 140-141 omitted)
Gloria: Try and listen to this: Outer driving forces are understood as those forces which affect the research of mathematics from the outside.
Harry: They began this not knowing if it was going to lead somewhere or not. What he [Hellman] writes is that it’s just idiots who keep on trying, isn’t that right? They continue because they are interested in it themselves.
Gloria: And because they want to succeed, so the world can evolve a little and the Internet can come to work.
Harry: But then you can say that about everybody, because everybody wants the world to evolve.
Gloria: That’s not right. This guy who says that I want to play with prime numbers, and I don’t care if it can be used for anything.
Harry: There is no doubt that they all make the world evolve, and all have both an inner and an outer driving force. What I’m saying is just that the three [Diffie, Hellman, and Merkle] have more of an inner because they are interested in the stuff and have started looking into it without anything having influenced them to do it. And when they start, they don’t even