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ways of doing things in mathematics and the fact that they can be proven, something which is not the case for biology, she mentions, where you merely state that this is the way it is. The downside of mathematics, as she sees it, is that it is not a subject that takes into account your personality and feelings. In the interview she expressed this view in the following manner:
It doesn’t matter whether you like fractions or not. You follow me. It’s kind of like... Well, in Danish and English and social studies and history, there it’s your interpretation of people’s behavior and actions in texts and history, which you kind of interpret, and say that he is an evil person and he is greedy of power, and this and that. And here it’s kind of like: this gives 5.
Gloria is not sure if it is important for her to learn mathematics, but she wants to, she says. She does not think that all mathematical topics are equally important for all people. Her mother, for instance, has no need to know about vectors in her job as a producer (Gloria does not say what her mother is a producer of). Gloria has previously been introduced to elements of the history of mathematics – “Aristotle and a lot of other old people”. She thinks that the history of mathematics could interest her, though she does not know why: “I think it is very exciting that some people have dedicated their lives to exploring it. It’s just difficult knowing whether it will interest you or not when you know nothing about it.”
Sean rates his interest in mathematics to 5, and he believes himself good at mathe-matics (in fact, the teacher thinks him very capable). He likes the logic of mathemathe-matics and the challenges in solving problems, but dislikes too long and out of control cal-culations. Sean personally considers mathematics important because it satisfies his curiosity about matters which are explained easiest by means of mathematics, he says.
In the interview he mentions “formulas for a theoretical increase in temperature for overheating” and “statistics for overpopulation of, for example, both people and animal, and what to do about it”. He considers it “a way to put a structure on the world”, to explain “why it is the way it is”. Sean also thinks it is important for people in general to learn about mathematics, e.g. so they can scale cooking recipes, get the best bargain when shopping, etc. He has previously been exposed to some history of mathematics, Pythagoras’ theorem and Fermat’s last theorem. He says that the history of mathematics does interest him, but he cannot mention any specific elements, and that it depends very much on what he finds interesting at a given point in time.
Jonathanrates his interest in mathematics as a 3 and believes himself to be “relatively good” at mathematics. He likes numbers, he says, and is good at doing calculations in his head. On the other hand, he is fed up with geometry and coordinate systems(!).
Jonathan thinks that he wants to study physics and chemistry at the university, and since mathematics is a tool in these subjects he thinks it quite important to learn mathematics.
In question 6 on whether it is important for people in general to learn mathematics, he answers that it is definitely important to know something about mathematics:
“Mathematics is everywhere, and knowing something about it only makes everything easier.” Jonathan has no prior experiences with the history of mathematics and he considers himself “rather indifferent towards when different elements of mathematics have been invented”, to him the mathematics itself is more important, he says. He does, however, have an idea about what history of mathematics is, for example that it is not only concerned with years etc. but also thoughts from one period of time and other people expanding on these thoughts in different periods and so. In particular, he
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mentions the story of Andrew Wiles whom the class has seen a movie about:
It was interesting to see how much, well, he was really, really into this, this guy. I don’t think I’ve ever seen anyone before who was so much into math. He was weeping of happiness when he finally solved it. In some way I understand why, because it’s been... people have been wondering about this, and it took so long, and then that it had to be him who finally saw how it all was connected. That must be big, right, in some way. For people who aren’t interested in mathematics it might seem kind of ‘whatever’.
Andrewis not too interested in mathematics but answers 3 to the first question. He considers himself an average student in mathematics, he is more into biology, he says. In fact, he wants to study biology and in order to do so, he reckons he needs mathematics at advanced level. He likes that mathematics can be used for something, e.g. statistics etc., which may also be used in biology. In particular, he dislikes “everything that has to do with sine and cosine”. He thinks it important for people in general to learn the most basic things of mathematics: “Kind of like learning to read. It is very useful for all. Not that everybody need to learn all that sine and cosine...” Andrew has previously been introduced to elements of the history of mathematics both in upper secondary school and before. He remembers having “read about different ‘legendary figures’ of mathematics and their considerations as well as their time”. To the question whether he believes the history of mathematics may interest him he answers: “No. I’m not that hooked on mathematics that I want to spend more time on it than absolutely necessary – unfortunately.” In the interviews it is revealed that some of the places he has encountered the history of mathematics is in the short (illumination) paragraphs or pages in the textbooks. Andrew admits to reading those as well as finding them somewhat interesting. He is very interested in history in general, he says. But it is not
“how they found out about one thing or the other” that interests him, “it’s more the persons who interests me a little”, for instance in the form of biographies. When asked what it is that does not interest him in the history of mathematics he replies: “All the numbers.”
The last student,Lucy, rates her interest in mathematics as a 5 and considers herself good at most mathematics. What she likes about mathematics is that it is a challenging subject, that there always are new things to learn, new ways to solve problems, and the satisfaction that comes after having worked hard on a problem and then finally arriving at a solution. What she does not like is that sometimes there are a lot of things to keep track of, and “when there is a topic you just don’t get”. She thinks that mathematics is good to have as a foundation later in life, and that the study of it also
“develops one’s brain in some way”. She believes that people in general should know about mathematics in order to make society function: “If people didn’t know how to calculate, how would the world economy then look?” Her prior experiences with the history of mathematics involves seeing a movie (probably the one with Andrew Wiles, which Jonathan mentioned), something with Pythagoras, and something she does not remember. She believes, however, that “you get a better understanding if you know something about the history”, and also she thinks that she may find it exciting to see how mathematics has come into being: “I mean, you havn’t just found out about it from one day to the next, it has kind of like arisen.”
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The Focus Group’s Final Essay Assignment and the Making of It
Before engaging in a presentation and analysis of the focus group video clip, I shall provide a short overview of the group’s hand-in final essay. This overview will be supplemented by some comments on their actual making of the essay, based on my observation of their work.
The focus group provided a very good answering of the first supportive essay assignment, a two page essay with rearrangements of formulas, small examples, and calculations. The remaining three supportive essay assignments as well as the main essay assignment were also answered on two pages altogether (the final hand-in was four pages). This imbalance may be due to different things. First of all, since the students took on the first essay assignment first, they may have spent most of the available time in class on this before moving on to the other assignments. Secondly, since the first supportive assignment concerned in-issues, it may have been intuitively clearer to the students, or at least some of them, how to proceed with the assignment. And thirdly, the student typing up the essay on the computer, Sean, definitely seemed more interested in the mathematics than in the history – something which is evident from the videos where he participated much more actively in the work on the first essay assignment than in the work on the following ones. (A more thorough description of the students’ answer of this assignment will be given later.)
The focus group turned the second supportive essay assignment into a ‘tour de force’
through the teaching material, indiscriminately identifying various definitions, notions, and concepts they stumbled upon as a technique (Hamming distance, Hamming weight, t-error detection,t-error correction, decoding to nearest neighbor, Hamming spheres, packing radius, syndrome decoding,n-dimensional cube). Applying such a ‘scattergun method’, it is no wonder that the students got a few of the epistemic techniques right (n-dimensional cube and Hamming distance). From a mathematical in-issue point of view, the students manage to explain the purpose of the techniques in a satisfactory manner (since I shall not display them, the reader will have to trust me on this). However, they do so in a way completely detached from the historical circumstances, e.g. which of the techniques were already available to Hamming, which did he create himself, what was Hamming’s purpose with the technique, etc. The hand-in paper and the videos both support this observation.
The students of the focus group found the third supportive assignment too difficult, and the teacher had problems helping them without revealing the answer. After working on the assignment for about twenty minutes they lost interest and focus, started goofing off and ended up writing a couple of vapid statements like the one exposed in section 5.5 (see pages 125).
They then moved on to the fourth supportive essay assignment. Although what is turned in as an answer of this assignment is not necessarily more informative than those of the two preceding assignments, the discussions the students had while working on it are. A transcript of these discussions may be found in appendix C.1 and will be discussed in the following subsection.
The main essay assignment is not very elaborating either. It is half a page providing mainly a summary of the preceding supportive essay assignments, a few historical errors (e.g. that binary numbers were invented for the sake of error correcting codes) and a few statements concerning the last couple of questions (see section 5.5). Due to lack of time
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on the focus group’s behalf, most of the main essay was completed outside the classroom, and the few videos of the group working on the assignment in class do not provide more elaborated discussions than the hand-in itself. But let us turn to a video clip that actually does this, the clip from the group’s work on the fourth essay assignment.
A Video Clip from the Fourth Supportive Essay Assignment
The focus group’s actual hand-in answer of the fourth supportive essay assignment may appear a bit silly:
a) We believe that the honor for the Hamming codes should be ascribed to the sweet Richard Wesley Hamming (1915-1998 (poor guy (We LOVE U big time))). We think that the dear Hamming did know of those ‘potentials of further development’ which his (7,4)-code possessed when he passed it on to Shannon, who later published generalizations of this code exactly.
b) So we know whom to hate/love – according to temper. A lot of work behind – bummer if someone else gets the credit. (Group 1)
However, some of the discussions the students had during their work on the assignment are not silly at all. In the following, I shall describe the video clip containing the students’
work on the assignment, occasionally displaying interesting quotes referencing the line numbers in appendix C.1.
The discussion begins with Andrew pointing to the fact that the codes are named after Hamming, so maybe he should be given credit. Sean seems to be in favor of this view, saying that others rely on Hamming’s theories. Gloria joins the discussion, shortly after stating:
Gloria: Yeah, but they are called Hamming spheres, right. What is there to ask about? (line 26)
At this point in time Jonathan plays the role of the critic, pointing to the fact that Hamming did not publish first, that Golay developed the codes further,5 that maybe Golay received credit back then, and telling the others “you need to provide reasons”
(line 25). Andrew then enters a new argument onto the scene:
Andrew: Hamming knew that his (7,4)-code could be generalized when he gave it to Shannon, but [he] kind of didn’t really have time to publish it.
(lines 31-33)
As seen in section 5.1, this argument is not in line with the reality of Hamming’s paper being delayed until 1950 by the application for patent, not due to lack of time. What might have confused Andrew is the fact that Hamming was annoyed with the ‘waste of time’ in computer calculations due to insufficient coding, this being what Hamming did not think his time suited for. Andrew, however, manages to persuade the others for a while, wrongly quoting the teaching material in his support, and finally making the argument that Hamming “put the theory higher than his own name by passing it on to one who had time [Shannon]” (lines 36-37). At this point in time the discussion only takes place between Andrew, Jonathan, and Gloria, the latter mainly asking clarifying questions in order to take notes. Jonathan, even though on some level agreeing with Andrew, still plays the critic:
5 This correct observation does not go into the hand-in answer, here it mistakenly says that Shannon was the one to generalize the (7,4)-code.
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Jonathan: The debate concerns whether or not he knew of the generalization of his (7,4)-code when he passed it on. Or if Golay was the first to discover it. (lines 61-62)
At this time, Sean is not participating in the discussion anymore, and Lucy has not yet entered it.
The discussion takes a turn when Andrew tries to clarify his viewpoint to Gloria by saying:
Andrew: Kind of like that we want to help you do this assignment, Gloria, but we are too busy, so we pass it on to you and you can take credit for it once you have made it. (lines 67-68)
This remark somehow triggers Gloria to venture an outburst:
Gloria: This I don’t know, because I want to answer ‘yes, he knew!’, but this doesn’t fit with what I want to answer in b. Because look at this. In b there is a question which says: Why do you think people care about finding out who is responsible for a mathematical result? Which personal driving forces of the mathematicians play a part in mathematical research? Can you say anything about the sociology? People who study mathematics, they are so nerdy. They enter completely into their own shell... [...] They spend their lives on it. There’s nobody who spends his life on something and then says:
‘Ohh well, it doesn’t matter, you can take the credit. (lines 72-77, 79-80)
This observation brings Gloria to question Andrew’s statement that Hamming did not have time by asking why not and what Hamming then did at Bell Labs. Andrew’s short reply to this is that Hamming “worked” (line 92), an argument which Gloria does not accept. At this point, Jonathan, who has been flipping through the teaching material, finally points to the patent issue:
Jonathan: It says here that he had to wait with publishing the codes until the thing with the patent had been resolved. (lines 94-95)
Andrew acknowledges this as a better argument than his own. And they move on to the second question of the essay assignment: Why it may be important to establish who was the creator of something. Jonathan tries to frame the discussion of the second question by saying:
Jonathan: It says: Many educated people are of the opinion that the honor should be ascribed to the first person to publish something. That is the dilemma with Hamming, right. Because both Shannon and Golay have published their stuff before Hamming himself publishes it. (lines 112-115) Andrew and Gloria, however, keep on discussing the issue of honor and credit, i.e.
indicating that it is important to establish ownership in order to credit the correct person. Within the next few moments the discussion is resolved, Sean reenters the discussion providing an alternative answer which does not really catch on, Jonathan draws parallels to Wiles’ proof of Fermat’s last theorem, something he has already done once earlier (line 78), and Lucy participates constructively in the discussion for the first time:
Jonathan: They can’t get the credit for it before it has been published.
Gloria: No, it is the thing with it being a huge work load and it being a huge effort they make to make it happen.
Jonathan: That is why it must be a fucking downer to see someone else get...
Sean: And then you know who to go totally berserk at if you actually find
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an error in it.
Gloria: You don’t do that.
Sean: Yeah, because if there are further things...
Jonathan: He also got slaughtered that guy with Fermat’s... the guy who made Fermat’s...
Lucy: Therewasan error.
Jonathan: Yes, there was an error. Then he was just slaughtered. Then he went totally down. And then he corrected it again.
Lucy: Imagine actually wanting to sit and go through that long proof and then finding an error in it.
Andrew: We’ll just write that there is a lot of work behind and its a bummer if someone else gets the credit, right. (lines 125-140)
And that is what they do, wrap their more interesting discussions into a few not nearly as interesting one-liners.
Nevertheless, the discussions of the focus group students while working on the assignment do support the hypothesis that upper secondary students are capable of carrying out meta-issue discussions, also in a quite reflective manner (whether they then manage to get it into their hand-in or not). As for research question 2, concerning anchoring, there is also something to be said based on the students’ discussions. Namely that the students begin using some of the mathematical terms and names that they have acquired through the teaching module. It is not surprising that the students use the term ‘family of Hamming codes’ since this is used in the assignment question. However, Andrew will on several occasions refer to the (7,4)-code as the one to be generalized into the others (cf. appendix C.1) – something which is historically correct, since this was the code Golay saw in Shannon’s paper, and further suggests a familiarity with the (7,4)-code itself. The presence of such familiarity is not a far fetched claim at all, since the (7,4)-code was the example used throughout the teaching material as well as in the first supportive essay assignment. Of course, the claim may be checked by looking into Andrew’s hand-in mathematical exercises, his work on these in class, his participation in the first supportive essay assignment, as well as in the second questionnaire, and the second round of interviews. In a similar manner it may be checked to what extent Gloria understands the term ‘Hamming sphere’ she is using in the beginning of the discussion.
These things, among others, will be done in the following subsection.
Focus Group Students’ Work with the Mathematical In-Issues
I shall begin by discussing the coding theoretical problems which the students were given in an exercise as part of their test at the end of their second year. In this exercise a certain code,
C={00000000; 00110011; 11001100; 11111111},
was given, and the students were told that out of the 8 binary symbols in each codeword, 2 were information symbols. The students were asked to answer nine questions, a through i, about the code C. (a) First they were to find the parameter m which is defined as the number of codewords in the code, i.e. forC m= 4, and they were to find the information rate of the code which in this case is2/8 = 1/4. (b) Next they were to investigate ifC is a linear code. A linear code is one for which two arbitrary codewords x,y∈ C, and it is so that
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i. x+y∈ C, and
ii. ax∈ C, where ais either 0 or 1.
For binary codes conditionii is trivial, something the students had been made aware of, andCis a binary code so all we have to do is to verify conditioni. This is easily done by inspection, thusCis a linear code. (c) For a linear code, a theorem says that the minimum Hamming distance,dmin, between two codewords in the code equals the minimum weight, wmin, of the code, which is the least number of 1’s in the codewords except in the zero-codeword, 0. For the code C, the minimum weight is 4, thus dmin = wmin = 4. (d) According to a couple of theorems in coding theory, theorems proved in the teaching material, a codeC candetect t errors if and only ifdmin≥t+ 1, and it cancorrect t errors if and only ifdmin≥2t+ 1. SinceC hasdmin= 4, it can detect 3 errors but only correct 1. (e) A Hamming sphereS is defined by its center in a codewordband a radius r. The students were asked to write up the words contained in the Hamming sphere S(b, r) =S(11001100,1), which are the eight words obtained by altering one binary symbol in the codeword at the time:
S(11001100,1) ={01001100,10001100, . . . ,11001101}.
(f) Next, the students were to assume they had received the wordv= (00001100)after a transmission. This word was to be decoded by nearest neighbor decoding, which means to find the codeword closest, in terms of Hamming distance, to the received word.
There are two such words inC; 00000000 and 11001100, which is because there are two corrupted symbols in the received word, and as found before we can only correct for one error. Thus, nearest neighbor decoding is of no use in this case. (g) A codeC has a packing radius,p(C), which is the largest positive integer for which the set of balls (or Hamming spheres) with radius p(C)centered inC’s codewords do not overlap. A theorem, proven in the teaching material, says that the packing radius for a code with minimum distancedminis
p(C) =
dmin−1 2
.
Thus, our code hasp(C) =b1.5c= 1. (h) In order to determine if a code is a perfect code, we can inspect the Hamming spheres of the code and see if all words are contained in a sphere. For large codes, even the (7,4)-code which exists in a space of 128 7-tuples, this is a rather tedious task. Fortunately, there is a small theorem, also proven in the teaching material, which says that the minimum distance of a perfect code always is an uneven natural number. OurC hasdmin= 4, thus it is not a perfect code. (i) In the last question, the students were asked to expandC into a code with ten symbols in each codeword, two still being information symbols, and to do this in such a way that the code remained linear. This can be done in the following way:
C∗={0000000000; 0011001100; 1100110011; 1111111111}.
Only three students in the entire class were able to answer question i in a correct manner, and one of them was Lucy (see figure 5.4). Lucy also provided correct answers to questions a, b, c, d, and h, the last one by using the theorem about perfect codes.
In question f, she missed the zero-codeword as a solution, but got the others right.
Jonathan provided correct answers to questions a, b, c, d, and f. Sean provided correct