NEEDING
TO EXPLAIN:
THE MATHEMATICAL EMOTIONAL ORIENTATION
David A Reid
Acadia University
Explanations are accepted or rejected within a community on the basis of an emotional orientation. Examination and description of students’ explanations in mathematics classes are used in this report to clarify the nature of the mathematical emotional orientation. The report also provides elements of a language for describing students’ explanations as a contribution to the difficult task of conducting research into students’ mathematical proving.
Proving in mathematics is a complex activity and research on students’ learning to prove must employ a rich descriptive language to capture it. Lakatos (1978) and Polya (1968) offer language to describe two elements of proving: the formality of written proofs and the relationship between deductive reasoning and other types of reasoning. Classifying kinds of deductive and inductive reasoning has also been an area of much research (e.g., Bell 1976, Harel & Sowder 1996). Another element of proving, the formulation of reasoning, has been described by Mok (1997) and Reid (1995a) but further work is needed in this area. A further element, the need or purposes served by mathematical reasoning, has been receiving considerable attention (Balacheff 1991; Bell 1976; Hanna 1989; Lampert 1990; de Villiers 1991), especially the need to explain deductively in mathematics (Hanna 1989, 1995; de Villiers 1991, 1992; Reid 1995). This report elaborates on the language used to describe students’ explaining, and discusses the central role deductive explaining plays in the “mathematical emotional orientation.”
The mathematical emotional orientation
Maturana (1988a, 1988b) uses the phrase “emotional orientation” to describe the bodily predisposition that underlies individuals’ decisions to accept some things as explanations and to reject others. An emotional orientation defines a domain of explanations, of which mathematics is one. Mingers (1995), in his discussion of Maturana’s work, identifies three aspects of an emotional orientation:
Each
domain is constituted in three interlocking dimensions — the criteria
for accepting explanations, different operational
coherencies structuring such explanations, and the actions seen as legitimate (p.98, emphasis added).
In mathematics the criteria for accepting explanations include the use of deductive reasoning, a basis in agreed upon premises, and a formal style of presentation. There are many operational coherencies (shared experiences and assumptions) in mathematics, the most obvious of which is the language used to talk about it. There are also many actions that are seen as appropriate to mathematics (drawing diagrams, generalising statements, making conjectures, etc.).
Needing to Explain: The Mathematical
Emotional Orientation
David A Reid
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The nature of the mathematical orientation [*] may be clearer if we consider what is not mathematics. Explaining by reference to authority (as in the non-explanations discussed below) is not mathematics. Neither is focussing closely on procedural steps in mathematics classes, although this is the experience of many students. Finally, actions such as the use of abuse to establish authority are not legitimate in mathematics. On the other hand, feeling a need to explain conjectures, and preferring deductive reasoning as the means to do so, are a part of the mathematical orientation.
Ways of explaining
Most of the examples I provide in this report come from observations in grade 10 mathematics classes in which students were studying coordinate and Euclidean geometry. [†] The students engaged in a series of activities, working in groups, and reported their conclusions to the class as a whole on a regular basis.
Behaviours which could be called “explaining” occurred in two kinds of contexts: activities in which there was an explicit demand to “explain” and contexts in which students engaged in what observers saw as explaining without being prompted to do so by a teacher or activity prompt.
In these two contexts several different modes of explaining were observed:
·
Non-explanations;
·
Explaining
how;
·
Explaining
to someone else (in response to a question);
·
Explaining
to someone else (spontaneously);
· Explaining as part of social activity in a community where explaining is a social
norm, i.e., part of the community’s emotional orientation;
·
Attempting
to come to a personal understanding (explaining to oneself).
Individuals operating from a mathematical orientation are likely to use deductive reasoning in any of the last four modes of explaining listed here.
Non-explanations
Personal or institutional authority is a common mode of “explanation” in schools, especially for students and teachers who can’t explain something and see that inability to explain as a negative reflection of themselves as people. In the following
[*]
In this report I will sometimes omit the word “emotional” from the
phrase “mathematical emotional orientation” for brevity.
This should not be taken as an indication that the role of emotions
in defining a mathematical orientation is unimportant. Emotions are central to defining mathematics.
[†] These examples are taken from an ongoing research project on the psychology of reasoning in school mathematics, funded by SSHRC grant # 410-98-0085, Acadia University and Memorial University of Newfoundland.
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example Stu both refers to the teacher’s authority to verify and falls back on his own authority when asked to explain. [‡]
1. Stu: [To teacher] That one and that are complementary right?
2. Teacher: Why?
3. Stu: Because.
4. Jill: None of them are complementary.
5. Stu: They are.
Abuse can be used to establish the authority to explain. In the following example Stu replies to Christy’s questions with a question of his own, perhaps to help her reflect on her own mathematical activity, or perhaps to evade her question. When he and Jimmy do reply to her they provide only the procedure she is meant to use with an implicit reference to an outside authority (“you’re supposed to”), followed by abuse from Stu. Stu’s opinion that he is in a position to declare Christy “stupid” suggests that he is establishing authority over her.
1. Christy: How’d you get that?
2. Stu: How’d you get 6?
3. Christy: I don’t know.
4. Jimmy & Stu: You’re supposed to add them together.
5. Christy: Oh—
6. Stu: 3 minus 5 equals 8 ... Man you’re — you’re stupid.
Explaining how
Jimmy and Stu, in the previous transcript, offer Christy a procedural explanation, in response to her question (“You’re supposed to add them together.”). Many of the explanations offered in mathematics classes are not explanations of why something is the case (as would be expected from a mathematical orientation), but simply explanations of how something is calculated (which is consistent with what might be called the “school emotional orientation”). This happens not only when students ask each other how to do something (as in the previous example) but also when teachers ask students to explain. For example, this is Jill’s response to the written prompt, “Which equation describes the graph on the left? Explain why.”
[‡] In transcripts I use the following conventions: An em-dash (—) indicates a short pause. Several indicate a longer pause. Ellipses (...) indicate omissions (usually “um”s, etc.) to improve readability. Three asterisks (***) are used to indicate an omission of several lines of speech. A hyphen (-) ending a line indicates an interruption of speech at that point.
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The reason the equation I’ve marked describes the graph is because I took the points from the graph and made up a table of values then I did the equation with the table of values I made.
Jill does not explain why in any way that would satisfy a mathematician. Instead she describes what she did.
Explaining to someone else (in response to a question)
Another context for explaining, the first in which deductive reasoning is likely to occur (suggesting a mathematical orientation), is responding to another person’s question. The other person might be a teacher or written prompt (like the one offered to Jill in the previous example) or, in this case, another student (CH is a research assistant).
1. Melinda: I have one.
2. CH: What do you have?
3. Melinda: Triangle ABC and BDC.
4. Jill: Why?
5. CH: They’re congruent?
6. Melinda: ‘Cause they have a shared side and alternate angles.
Here Melinda uses notation specific to mathematics and makes implicit use of a shared experience of theorems and definitions in geometry to provide a deductive explanation.
Explaining to someone else (spontaneously) and as a social activity.
In the following transcript four grade 10 boys are trying to work out a generalisation concerning the sum of the interior and exterior angles of polygons. Their class has been studying Euclidean geometry (parallel lines and congruent triangles) for about a month, in a style similar to the exploratory methods described by Fawcett (1938). In this context the students in the class have come to adopt explaining as a major focus in their mathematical activity, and have come to value well formulated, deductive explanations. The following transcript offers a number of examples of spontaneous explanations, embedded in a social context that values explaining. (Bold indicates such explanations.) These explanations suggest that the boys were operating from a mathematical orientation at this time.
1. Wane: The exterior angles of them all — because when there's more sides you can
make more triangles.
2. Mick: Which one is he talking about?
3. Wane: So it keeps going up by 180!
4. Clark: Here's what I was thinking. This one is 360. That triangle there and that one —
exterior angles.
5. Wane: Oh yeah, that's true too.
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6. Mick: That angle there is 360 and this angle here — Now they all equal 360, right?
7. Clark: You see, we got this thing here — a common vertex. That means we have to
subtract some angles. 360 and 360 which is 720, right? Correct? Okay. Now we
got to subtract some. Okay?
***
8. Mick: So what do we got?
9. Clark: We got four angles, right? Now — uhhhh — uhhhh — They're all
counter-clockwise — Yeah they are. CCW.
10. Wane: No they aren't.
11. Mick & Clark: Yes they are because the way they goes up that way.
12. Clark: Okay. Now we got the four angles.
13. Wane: Four times 180.
14. Mick: But they're not all 180.
15. Clark: How do you know they're not 180?
16. Mick: Because they're not all straight lines. The angles aren't straight
lines — I was just looking at that one.
17. Clark: And they're all supplementary to an angle inside here, right?
18. Mick: Oh, deadly!
19. Wane: And it has something to do with — if you know the interior angles
— the sum of the interior angles — then they'll all wind up to be the
same
thing because they're all supplementary
to an angle.
The behaviour of Mick, Clark, Jacob and Wane in the above transcript suggests that they have a mathematical orientation. The passages in bold show Wane, Clark and Mick responding to a felt need to explain their conjectures, which is an action in keeping with mathematical practice. Their explanations make reference to geometrical concepts, definitions and theorems that are a part of the shared experiences of their mathematical community. They propose explanations using deductive reasoning (an operational coherency that structures mathematical explanations) and they expect others to explain things deductively and with reference to the same definitions they use, indicating that at least two of their criteria for accepting explanations are consistent with a mathematical orientation.
Explaining to oneself.
The last mode (explaining to oneself) can be described in still greater detail, but in the interests of space I will limit myself here to one example, of two university undergraduates working on the Arithmagon problem (see Reid 1995a or 1995b for a more detailed analysis):
1. Stacey: What happens if you add the middle numbers together? —
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2. Kerry: Well I guess we could, hmm.
3. Stacey: I just want to try something. If you take 27, 18, and 11. 2, 4, 5, 56. Right?
4. Kerry: Sure.
5. Stacey: And you have — So you add each of those twice, right? — Yeah you do.
That's not going to help you either. That's what you end up doing right?
6. Kerry: What'd you do?
7. Stacey: You add A, B, C. Then you multiply them by 2. You get this answer. —
8. Kerry: Do you add?
9. Stacey: 22, and 34. Yup. Do you know what I mean?
10. Kerry: Sorry. So you add this and multiply by 2 so, like, the sum of this is 28 times
2. And it's 56. Good one. What's that mean?
11. Stacey: [laughing] Nothing.
In spite of the linguistic indications that Stacey was explaining something to Kerry (“Right?” “Do you know what I mean?”) she is really explaining to no one but herself. Kerry, in spite of being involved in the same problem solving activity as Stacey, was unable to follow her explanation. It is clear however that Stacey was explaining something to herself, and elsewhere I provide a possible interpretation of the deductive reasoning involved in her thinking (Reid 1995b, Kieren, Gordon Calvert, Reid, & Simmt 1995) which suggests she was operating from a mathematical orientation.
Conclusion
Researching proving involves dealing with complexities. Many types of reasoning are involved, there are degrees of formality of written proofs and formulations of reasoning, and the needs which motivate proving are many and their importance is only beginning to be understood. The examples I offer in this paper are intended to help clarify one need to prove: the need to explain. In exploring the need to explain I also address the features that qualify explanations as mathematical explanations, which is an important part of defining a mathematical emotional orientation. The criteria for acceptance of explanations in mathematical communities include the use of deductive reasoning and reference to shared experiences of notation, definitions and established theorems. Observation of these characteristics in students’ explanations suggests progress in their adoption of a mathematical orientation.
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References
Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathematical proof. In A. Bishop, S. Mellin-Olson, & J. van Doormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 175-192). Boston: Kluwer Academic.
Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23-40.
de Villiers, M. (1991). Pupils’ need for conviction and explanation within the context of geometry. Pythagorus, n. 26, 18-27.
de Villiers, M. (1992). Children's acceptance of theorems in geometry. Poster presented at The Sixteenth Annual Conference of the International Group for the Psychology of Mathematics Education, Durham NH.
Fawcett, H. P. (1938). The nature of proof: a description and evaluation of certain procedures used in a senior high school to develop an understanding of the nature of proof. (NCTM yearbook 1938) New York: Teachers' College, Columbia University.
Harel, G., & Sowder, L. (1996). Classifying processes of proving. In L. Puig & A. Gutiérrez, (Eds.), Proceedings of the Twentieth Annual Conference of the International Group for the Psychology of Mathematics Education, (Vol. 3, pp. 59-65). Valencia.
Hanna, G. (1989). Proofs that prove and proofs that explain. In Proceedings of the Thirteenth International Conference on the Psychology of Mathematics Education, (pp. 45-51). Paris.
Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49.
Kieren, T., Gordon Calvert, L., Reid, D. & Simmt, E. (1995). An Enactivist Research Approach to Mathematical Activity: Understanding, Reasoning and Beliefs. Presented at the Annual Meeting of the American Educational Research Association, San Francisco.
Lakatos, I. (1978), Mathematics, Science and Epistemology. Cambridge University Press, Cambridge.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63.
Maturana, H. (1988a). Reality: The search for objectivity or the quest for a compelling argument. The Irish Journal of Psychology, 9, 25-82.
Maturana, H. (1988b) Ontology of observing: The biological foundations of self consciousness and the physical domain of existence. Web document at http://www.Inteco.cl/biology/ontology/index.htm. Originally published in the conference workbook for 'Texts in Cybernetic Theory', an In Depth Exploration of the Thought of Humberto R. Maturana, William T. Powers, and Ernst von Glasersfeld, American Society of Cybernetics, Felton CA, October
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18-23, 1988. NOTE: This is essentially the same as Maturana, Humberto R. (1987) The biological foundations of self-consciousness and the physical domain of existence, in Caianiello, E. (ed.), Physics of Cognitive Processes, Singapore: World Scientific, 1987, pp. 324-379.
Mingers, John. (1995) Self-producing systems: implications and applications of autopoiesis. New York: Plenum Press.
Mok, I. A. C. (1997). A hierarchy of students' formulation of an explanation. In Pekhonen, Erkki (Ed.), Proceedings of the Twenty-first Conference of the International Group for the Psychology of Mathematics Education, (Vol. 3, pp. 248-255). Lahti, Finland.
Polya, Georg. (1968). Mathematics and plausible reasoning (2nd ed.). Princeton: Princeton University Press.
Reid, D. (1995a). The need to prove. Unpublished doctoral dissertation, University of Alberta, Department of Secondary Education.
Reid, D. (1995b). Proving to explain. In L. Meira & D Carraher, (Eds.), Proceedings of the Nineteenth Annual Conference of the International Group for the Psychology of Mathematics Education, (Vol. 3, pp. 137-143). Recife, Brazil.
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