of what Hamming spheres and perfect codes are is definitely present. Lucy was the only one not capable of answering this question, and in fact she did not answer the next question either.
Question 10 asked which perfect codes are known (the answer being that Hamming and Golay codes are the only non-trivial ones):
Jonathan: Hamming’s (7,4)-code and Golay’sG23-code.
Andrew: The (7,4)-code and co.
Gloria: Definitely the (7,4)-code since that’s the one we’ve been working with the most.
Lucy: [No answer.]
Sean: (7,4)-code and family.
Of the four students who answer, the thing to notice is that they all mention the (7,4)-code, which shows that the students are able to provide an example of a perfect code. The answers from Jonathan, Andrew, and Sean also indicate that they are aware that the (7,4)-code is not the only one.
In question 7, Gloria was able to give quite a good description of what is understood by binary representation, and so were Andrew, Sean, and Jonathan (again Lucy did not answer). Concerning Lucy and her lack of answers, she may be seen as a ‘victim’ of the module being split into two. In the test before the summer vacation she did very well, but after the summer vacation it seems as if she did not quite manage to get back in the game.
5.7 Case-Specific Analysis, Discussion, and Recapitulation
What we have above is a kind of existence proof that it is possible to have students engage in meta-issue discussions of mathematics in time and space (the general topics and issues of epistemic objects and techniques and multiple developments; cf. section 3.6), to have them perform reflections in relation to history as a goal, and do it in such a way that these discussions and reflections to some degree are anchored in the related and taught subject matter (mathematical in-issues). However, in order to answer research questions 1 and 2 (e.g. in what sense, to what extent, and on what levels this is possible) and, thus, begin to transform the existence proof into a more constructive one, further analysis is needed. For such analysis, I shall rely on some of the theoretical constructs presented in chapter 3, namely the theory of Sfard (2008b). Towards the end of the section I shall also discuss the class’ answers to the second questionnaire. And finally, I shall evaluate the representativity of the focus group students and the composition of the focus group.
A Discursive Approach of Commognition to Research Questions 1 and 2 As mentioned in chapter 3, Sfard’s theory of commognition seems a promising tool to apply in the empirical investigation of this dissertation. One reason for this is that the idea of commognition not only applies to the learning and understanding of mathematics and mathematical concepts, as many of the other theoretical constructs of understanding discussed in section 3.7, but applies to understanding and learning in general. This means that it also may be used for analyzing the students’ discussions and reflections of the meta-issues, not only their understanding of the mathematical in-issues. That is to
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say the theory may be applied to the answering of research question 1, and not only research question 2.
Also, the discursive approach of Sfard may offer appropriate ‘lenses’ for looking at the data since we may view the discussions and reflections of the students as following different discourses. In general, we may talk about two different – yet related – discourses being present in the investigation: a meta-issue discourse on the one hand, and an in-issue discourseon the other. As explained in chapter 2, the meta-issues of history as a goal mainly concern those of the evolution and development of mathematics, something I shall refer to as thehistorical discoursein the following. In the context of the essay assignments, however, there are also other meta-issue discourses present, for example a philosophical discourse, a sociological discourse, a psychological discourse, and possibly others as well, depending on the discussions taking place among the students and the questions of the assignments. The in-issues, also explained in chapter 2, are the issues related to mathematical concepts, theories, disciplines, methods, algorithms, proof techniques, etc., which is to say the internal mathematics. In the present investigation the in-issues are all related to the mathematical content or subject matter of the teaching module. Using Sfard’s notion, the students engaging with these in-issues as part of their work then become mathematists, somebody who are participating in a mathematical discourse.
As stated above, the meta-issue discourses and the in-issue discourse, though being different discourses, are related. Exploring this relationship is in some respect the core of research question 2: the students’ meta-issue discussions and reflections being anchored in the mathematical in-issues of the teaching module. Or we may look at it as the students’ understanding of the meta-issues relating to and building on their understanding of the related mathematical subject matter. In terms of discourses, we may imagine the meta-issue discourses and the mathematical discourse as running along somewhat separate tracks, and the anchoring being potentially present in situations when these two discourses ‘intersect’. I shall refer to these ‘intersections’ as possible or potential anchoring points.
Students’ Meta-Issue Discussions and Their In-Issue Anchoring
Let us take a look at the video clip from the fourth supportive essay assignment (section 5.6) once again, more precisely the discussion between Andrew and Gloria who each argue based on different narratives. Andrew’s narrative is that Hamming did not publish due to lack of time, a practical problem which Andrew later rephrases as Hamming caring more about getting the codes out than about credit and honor. Gloria’s picture of a real mathematician is a person who devotes his/her life to some mathematical problem, and more or less seals himself/herself off from the outside world in the pursuit of a solution (a view which could be due to the movie about Andrew Wiles).
From a data gathering perspective, the wonderful thing about Gloria is that she often thinks aloud, something which may provide us with a glimpse of her intrapersonal communication, based in the interpersonal setting of the group work situation, of course.
In Gloria’s small monologue (lines 72-80, appendix C.1) she is able to change her narrative from that of Andrew’s to a new of her own. Gloria’s new narrative contradicts that of Andrew, which eventually leads her to put her foot down:
No, it is god damn because everybody know that such research
mathemati-5.7 Case-Specific Analysis, Discussion, and Recapitulation 143
cians... that they spend their lives on this. You can’t come up with some brilliant mathematical thing now – because so much have already been created – you can’t do it without really spending a lot of energy on it. And anybody who really make an effort, damn it, wants to get some credit for it.
(lines 117-121)
From a discursive point of view, Gloria changes from Andrew’s plain (and practical) historical discourse to a more psychologically oriented discourse: It makes no sense that a person who spends years on developing something should just pass the results and subsequent credit on to someone else. According to Sfard (cf. section 3.7), thinking is communicating, and both thinking and communicating is definitely what Gloria does in her small monologues (lines 72-80 and lines 117-121). Sfard’s basic assumption is that learning and understanding are changes in discourse, and changing discourse is exactly what Gloria does.
The two narratives of Andrew and Gloria result in a kind of discursive conflict, we could refer to it as ameta-issue discursive conflict. As explained earlier, the conflict is resolved by Jonathan consulting the teaching material and pointing out the aspect of patenting the codes, something which forces the group to reorient the basis of their meta-issue discussion, a reorientation which fits Gloria’s discourse. The important thing to notice, however, is that it is the change in discourse Gloria brings about that actually leads to the resolution of Andrew’s wrong assertion about lack of time and brings the other focus group students to further change their discourse, thus learning and understanding something about the meta-issues of mathematics-in-the-making (rather than only the in-issues of mathematics-as-an-end-product). The discursive conflict bears some resemblances with Sfard’s notion of a commognitive conflict: it is a conflict which stands between incommensurable discourses; and it is resolved by the students’
acceptance of the discursive ways of the expert interlocutor, in this case the teaching material. The reason, however, for not referring to it as a commognitive conflict and instead calling it a meta-issue discursive conflict is that Sfard links her notion to a mathematical discourse, i.e. an in-issue discourse. Another difference is that Sfard states that a commognitive conflict is practically indispensable for meta-level learning (meta-level learning not having anything to do with meta-issues in this case, cf. section 3.7), something which can not be claimed for the discursive conflict described above.
One element of Sfard’s theory that does apply, within a mathematical discourse, to the video clip from the fourth supportive essay assignment is that of a learner’s development of ‘word use’ in his or her individualization of the use of mathematical nouns. That is to say, the four stages of passive use, routine-driven, phrase-driven, and object-driven use, the three latter ones all being active uses. As seen in the previous section, the video clip offers two interesting, active ‘word uses’: Gloria’s use of the mathematical noun ‘Hamming sphere’ and Andrew’s use of the noun ‘(7,4)-code’. I shall argue that both Gloria and Andrew in the course of the module reach beyond the third stage of ‘word use’ concerning these two mathematical nouns.
Now, Gloria did not take the test at the end of the second year, so we do not know what her understanding of Hamming spheres were at this time (otherwise we could have used question e of the test). In the set of exercises after the summer vacation Gloria has a very nice and correct answering of the exercise of drawing the two Hamming spheres for the codeC={0000,1111}(see figure 5.5). And in questionnaire 2, after the module, Gloria definitely showed understanding of both Hamming spheres and perfect codes.
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Gloria’s use of the word Hamming sphere in the hand-in exercise may very well be somewhat routine-driven, since it may have occurred as part of “specific routines” in
“constant discursive sequences” (Sfard, 2008b, p. 181). However, when Gloria uses the word in the video clip, the use is neither routine-driven nor phrase-driven, the latter being when “entire phrases rather than the word as such constitute the basic building blocks” (Sfard, 2008b, p. 181). Instead, the word has a ‘life of its own’ as a noun, which, according to Sfard, indicates that the process of individualization is completed: “One can now insert this word in any proposition in which there is a slot for this particular grammatical category.” This is exactly what Gloria does, she points to the fact that the packing spheres are named after Hamming – Hamming spheres – so what is there to discuss, he has already been given credit. The word’s ‘life of its own’ is further supported by the fact that the essay assignment asks about Hamming codes, not about Hamming spheres. Thus, ‘Hamming sphere’ is a word in Gloria’s vocabulary she can use whenever grammatically – and mathematically – appropriate.
Something similar may be said about Andrew’s use of the word ‘(7,4)-code’. Three times during the video clip (lines 32, 48, 89), Andrew mentions the (7,4)-code specifically even though the assignment adresses the Hamming codes in general. Judging from Andrew’s previous work on the first supportive essay assignment, he definitely has some kind of understanding of the (7,4)-code – remember that he solved the first part of the assignment himself by paring the xis and the Xjs (1 ≤ i, j ≤ 7) in the two presentations. At the time of solving the assignment, Andrew’s use of the word may have been routine-driven, but when he uses it in the context of the fourth supportive essay assignment, it appears more phrase-driven. Andrew uses the word ‘(7,4)-code’ in connection with the generalization of the specific code to the family of Hamming codes – something which is correct in the sense that Golay made the generalization based on the (7,4)-code in Shannon’s paper. Of course, the issue of the assignment is Golay’s
‘generalization’ of the Hamming code, but still the use of the two words together is striking:
• “... Hamming knew that his (7,4)-code could be generalized when he gave it to Shannon...” (lines 31-32)
• “... he passed his (7,4)-code on to Shannon. And it was Shannon who made the generalizing...” (lines 48-49)
• “... or that he did know that there was some generalization of his (7,4)-code.”
(lines 88-89)
In questionnaire 2, however, Andrew may be closer to an object-driven use of the word when he writes “The (7,4)-code and co.” under question 10 on which perfect codes are known, ‘perfect codes’ being a term which he also appears to have an understanding of.
The two examples of ‘word use’ above are examples of potentiel anchoring points between the mathematical and historical discourses, and at some points the psychological discourse. The question then is if, and if so then to what extent, an anchoring of these meta-issue discourses in the mathematical in-issue discourse is present in these points.
Or phrased differently, to what extent do the meta-issue discourses rely on the in-issue discourse? Concerning Gloria’s use of ‘Hamming sphere’ it may make more sense to talk about an anchored comment rather than an anchored discussion, since the topic of Hamming spheres is never picked up by the other group members. Nevertheless, Gloria cross references the historical discourse with the mathematical discourse, something she could not have done had she not learned some of the subject matter of the teaching
5.7 Case-Specific Analysis, Discussion, and Recapitulation 145
module. Concerning Andrew’s use of the ‘(7,4)-code’ more may be said. First of all, Andrew is more persistent in referring to the (7,4)-code throughout the discussion, and the specific code clearly plays a role in his own understanding of what is afoot within the meta-issue discourse. Thus, from an interpersonal communicative point of view, a relation between the two discourses appears present. A relation somehow relying on an anchoring being present as well, since Andrew does have a mathematical understanding of the (7,4)-code as seen from his work on the first supportive essay, and since he constantly seems to want to refer the meta-issues of the fourth supportive essay discussion to this code. Furthermore, Andrew’s narrative, and thus the relation between the two discourses, sets the agenda in long stretches of the discussion (from lines 31 through 71). And Jonathan (lines 61-62) also picks up on Andrew’s narrative when he tries to rephrase the basis for the discussion:
The debate concerns whether or not he [Hamming] knew of the generalization of his (7,4)-code when he passed it on. Or if Golay was the first to discover it.
Students’ Answers to the Second Questionnaire
Having spent quite some time on the focus group, it seems reasonable to broaden our view again and take a look at the other students of the class. I shall do so by looking at the students’ answers to the second questionnaire. In terms of the students’
understanding of the in-issues, I shall consider questions 7-10, as I did for each of the focus group students. And in terms of the students understanding of the meta-issues, I shall consider questions 6 and 11-14 (see appendix B.2). The remaining questions of the second questionnaire will be addressed in chapter 7 or 8.8 Due to the readability of the questionnaire results and the relatively small population of the class, the students’
answers have been indexed in the following manner:
one < few < some < many < the majority < thevast majority,
a partition which in percentage intervals roughly corresponds to 0-5%, 6-15%, 16-35%, 36-50%, 51-80%, and 81-100%, respectively.
I begin with the in-issue questions. As was the case for the focus group students, the vast majority of the students in the class provide correct/acceptable answers to question 8 on the meaning of Hamming distance; describing the meaning in their own words, providing examples, or both. Only one student does not answer the question. For question 9, the meaning of Hamming spheres and perfect codes, some provide correct answers to the question of Hamming spheres and to the question of perfect codes as well, although many only provide correct answers to one or the other. Out of these, some students focus on the geometrical aspects of the Hamming spheres, and their ability to provide visualizations of a code. Some do not answer the question or state they do not remember or have not understood. One student just answers that a perfect code is “really good”. To question 10, regarding what perfect codes are known, many answer both the Hamming and the Golay codes, the (7,4)-code and the Golay codes, or various other combinations of specific Hamming and Golay codes, a few also mentioning the family of
8 Actually questions 22, 23, and 24 will not be addressed at all in this dissertation. For a discussion of the students’ answers to these questions, see Jankvist (2008e).
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Hamming codes. Some students only answer the (7,4)-code. One student simply answers
“Golay codes”. A few do not answer or provide incorrect answers. In question 7 on what is to be understood by binary representation (referring to the first introductory essay) the majority provides answers which are fully or somewhat acceptable, some by referring to factual situations with, for instance, 1s and 0s or high and deep tones. The latter example shows that some students are aware that it is not the 1s and 0s which define a representation as binary, but the fact that there are two and only two different ‘symbols’.
Some students provide non-acceptable, though not completely incorrect answers. A few students do not answer or provide incorrect answers.
The first of the meta-issue questions (question 6) also refer to the first introductory essay, since it asks from where the binary numbers originate. In the material the students had been told that binary representation could be traced back to African tribes who communicate by a deep and a high tone, and that Australian aborigines and some tribe people from New Guinea count in twos. They were also told that the Chinese workI Chingfrom around 1100 uses some binary representation of numbers, though not a real number system with an associated arithmetic. One such may be found in the Indian mathematician Pingala’s work from around 300-200 B.C. Many students referred to one or more of the above in their answers to question 6. Gloria, for instance, writes:
“The binary number system may be traced back to some ethnic groups in the past, e.g.
African tribes and ancient China.” More surprisingly, however, is that some students believe the binary number system to originate from Bell Labs (including Andrew, Sean, and Lucy), neither the teaching material nor the teacher made any such claims. A few students do not answer or provide other incorrect answers.
The next of the meta-issue questions (11) asked what motivated Hamming to develop his error correcting codes. Some students do not answer this question or provide incorrect answers (Sean included). However, the majority of the students actually provide answers which show that they are familiar with this part of the early history of error correcting codes (the remaining focus group students included). A few state things like: “He wanted to make the use of Bell Labs’ computers more efficient, the calculations which were performed on these, and possibly the wasted time involved in the procedure up till then.” Some state that Hamming was annoyed with the machines coming to a halt and dropping his calculations. A few explicitly mention that the computers were only able to detect errors, not correct them: “Before, when there were only error-detecting codes, Hamming found himself annoyed by the fact that the computers would come to a stop every time it found an error. It couldn’t correct it by itself.”
Question 13 asked which influence Hamming, being a mathematician as opposed to for instance an engineer, reasonably could be assumed to have on his work with the codes. A few do not answer this question. Some provide answers like: “He did not build bridges.”; “It may have had an influence that he worked with numbers and formulas, and not constructions or something.” However, the majority of the students (including the five focus group students) provide acceptable answers or qualified guesses, pointing to matters such as Hamming having a more mathematical or theoretical perspective on the codes than an engineer might have had. One student says: “Had he been an engineer he would probably have invented a faster computer.” A few say that he presented the codes in a logical way. Others that he was able to boil it down to formulas, and prove things about them. One says that “coding theory is mathematics”.
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Question 14 was a different matter. The question asked what other area the theory about perfect codes has contributed to, the answer being packing of metric spaces (cf.
the third essay assignment, page 125). The vast majority of the students did not even provide answers to the question or said that they did not know, had forgotten, etc. Only a few students provided guesses and these were all incorrect. This picture is, however, in accordance with the students’ lack of correct answers to the third supportive essay assignment.
The last question I shall touch upon is question 12, which asked what is to be understood by objects and techniques. A few students do not provide an answer to this question, but the rest do. Some students provide answers, relating to an example of the teacher’s, saying that numbers, e.g. 3 or−4, are objects, and that arithmetical operations performed on these, e.g. 7−3, are techniques (cf. the answer of group 6 to the third supportive essay, page 125). The answer of Sean is an example of this:
Sean: Object: a thing, e.g. −3. Technique: a relation, e.g. 4−3.
A few students mistakenly consider the (arithmetical) operations to be the objects, but do have an idea about techniques being some form of action. Jonathan and Andrew belong to this group:
Jonathan: Objects can be operations, such as +, −,⊕, and·. Techniques are when you use them, i.e. you perform an action.
Andrew: An object can be a sign, while a technique is an arithmetical problem. Object: no action. Technique: action.
Finally, some students provide more sound answers, e.g.: “Objects are at a given time in history subject to investigation. Techniques are used to investigate the objects.” A few of the students in this last group provide answers relating to the historical case of error-correcting codes. Gloria and Lucy are among them:
Gloria: Objects are those treated by techniques: for example, a code is an object for the techniqueαβγ, if it is placed in the formula.
Lucy: Objects: e.g. error-correcting codes. Techniques: e.g. concept of distance.
Representativity of the Focus Group Students
As explained earlier, the focus group students were chosen to represent the class in general as best was possible. That is to say the group consisted of students whose beliefs about and attitudes towards mathematics more or less reflected the viewpoints of the class, e.g. the group consisted of both students who were for and against the inclusion of history. As a way to evaluate this representativity of the focus group students, we may compare their hand-in essays to those of the other groups as well as their answers of the second questionnaire questions discussed above to those of the other students.
As for the length of the the final essays, the focus group’s was among the longest;
four pages. However, two of the pages were on the first supportive essay exclusively. This leaves two pages for the other three supportive essays and the main essay, something which averages the length of the other groups essays fairly well. As for the quality of the hand-in essays, the focus group’s final essay appears to be a little bit below average. The group does not do too good a job on the essays concerned with meta-issues, only their first supportive essay on the in-issues is of good quality. For the introductory essays, the first is rather poor as compared to those of the other groups, concerning quality and