2. Generating Problems
2.6 Deleuzian Problem-Posing: Generating Problems
Since problem posing is an encounter with difference, and “difference-in- itself” is the center of a Deleuzian dialectic, “Problems are always dialectical: the dialectic has no other sense, nor do problems have any other sense” (Deleuze, 169/1995, p. 164). Hence, mathematical problems are actually dialectical
problems. “What is mathematical” says Deleuze (1969/1995), “are the solutions” as well as “the expression of problems relative to the field of their solvability which they define, and define by virtue of their very dialectical order” (p. 179). Mathematical problems, then, always participate “in a dialectic which points beyond [them]—in other words, in meta-mathematical and extra-propositional power” (p. 164).
While Deleuze (1969/1995) emphasizes that problems are always dialectical, a Hegelian/ Marxist dialectic based on contradiction overlooks “difference-in-itself” and misinterprets the encounter with the being of the sensible:
Whenever the dialectic ‘forgets’ its intimate relation with Ideas in the form of problems, whenever it is content to trace problems from propositions, it loses its true power and falls under the sway of the
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negative, necessarily substituting for the ideal objecticity of the problematic a simple confrontation between opposing, contrary or contradictory, propositions. This long perversion begins with the dialectic itself, and attains its extreme form in Hegelianism. (p. 164) A Deleuzian problem-posing pedagogy rejects a Hegelian dialectic that views difference as the only problem, one that can be resolved through contradiction in favor of one that affirms difference, an anti-oppressive problem-posing pedagogy that involves learning through the crisis instigated by the return of difference (Kumashiro, 2004), as described in the “Affirming Difference” plateau. It is only this form of problem posing that brings real movement to thought, and for this movement to occur, a distinction between knowledge and learning is required. In order to deterritorialize the dogmatic image of thought which has “from Plato to the post-Kantians” defined “the movement of thought as a certain type of passage from the hypothetical to the apodictic” (p. 196), which “maximally
betrays and distorts this movement” (p. 197), we must instead conceptualize the movement of thought as going “from the problematical to the question” (p. 197). This distinction is crucial because “the assimilation of the problem and the
hypothesis is already a betrayal of the problem or Idea, involving illegitimate reduction of the latter to propositions of consciousness and to representations of knowledge” (p. 197). While knowledge refers to “only the generality of concepts or the calm possession of a rule enabling solutions” (p. 164), problem posing as an encounter with “difference in itself” enables the “subjective acts carried out when one is confronted with the objecticity of a problem (Idea),” that is, the “process of learning” (p. 164). Learning, then, necessitates problem posing, and problem posing is the multivocity of being, the affirmation of difference, the
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eternal return of difference in repetition, the movement from the problematical to the question. “What are. . .these questions which are the beginning of the
world?” asks Deleuze (1969/1995). His answer, fittingly, ends in a question: “The fact is that every thing has its beginning in a question, but one cannot say that the question itself begins. Might the question. . .have no other origin than repetition?”
We return to Huang, Mok, and Leung’s (2006) chapter about explicit and implicit variation, whose title, “Repetition or variation: Practicing in the
mathematics classrooms of China” resonates with the title of Deleuze’s
(1969/1995) book Difference and Repetition. Deleuze’s (1969/1995) distinction between cadence repetition (repetition of the Same) and rhythmic repetition (repetition of difference) as explored in the “Affirming Difference” plateau can further the exploration of this resonance. Explicit variation is cadence repetition in that students encounter problems of the same form and repeat the same procedure as was presented by the teacher. Implicit variation, on the other hand, is like rhythmic repetition, repetition with a difference. However, when the source of implicit variation remains solely the teacher, it remains within the problem- solving stratum rather than following a problem-posing line of flight. Implicit variation in a problem-posing pedagogy must originate in the encounter between teachers and students. Brown and Walter’s (1983/ 2005) “what if not” strategy locates the process of implicit variation in the encounter between teachers and students, as teachers and students generate questions and problems from a given situation or artifact. Gutstein’s (2006) projects were problems that
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stemmed from conversations between himself and the students, problems that required not only transformation of the mathematics they had been learning, but transformation of their social consciousness as well.
Deleuzian problem-posing, however, presses the problem-posing line of flight yet further than either the Freire Machine or the Brown-and-Walter Machine. Freire’s (1970/2003) process of praxis combines “reflection and action upon the world in order to transform it” (p. 51). This takes place through codifications which serve as representations of generative themes which are given back to the people as problems to solve (p. 123). Through reflection and action centered on the representation, teacher-students and students-teachers can overcome false consciousness. A Deleuzian problem-posing pedagogy, one that affirms
difference, must break out of the four shackles of that mediate difference through representation: identity, analogy, opposition, and resemblance. For Deleuze, all “consciousness,” which remains shackled, is false; it is problems that escape this false consciousness by affirming “difference-in-itself,” and allow true movement of thought which deterritorializes the dogmatic image of thought, thought stratified by common sense. Deleuze (1969/1995) writes,
While it is the nature of consciousness to be false, problems by their nature escape consciousness. The natural object of social consciousness or common sense with regard to the recognition of value is the fetish. Social problems can be grasped only by means of a ‘rectification’ which occurs when the faculty of sociability is raised to its transcendent exercise and breaks the unity of fetishistic common sense. The transcendent object of the faculty of sociability is revolution. In this sense, revolution is the social power of
difference, the paradox of society. (p. 208)
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point at which the people visualize ‘the given’” (p. 107), Deleuzian problem posing is an encounter not with “the given,” which is diversity, but with that-by- which-the-given-is-given, that is, difference (Deleuze, 1969/1995, p. 222). This requires that the encounter take place not only at the level of denotation/
indication, but at the level of sense, at the level of problems themselves, rather than following the “natural illusion (which involves tracing problems from
propositions)” (p. 159). This is why a Deleuzian problem -posing pedagogy, as a part of postcritical multicultural education or anti-oppressive education
(Kumashiro, 2004), is
a disarming process that allows students to escape the uncritical, complacent repetition of their prior knowledge and actions.
Learning is a disorienting process that raises questions about what was already learned and what has yet to be learned. Learning involves looking beyond what students already know, what teachers already know, and what we both are only now coming to know, not by rejecting such knowledge, but by treating it paradoxically, that is, by learning what matters in society. . .while asking why it matters (and how it can reinforce and challenge an oppressive status quo). (Kumashiro, 2004, p. 30)
That is why a Deleuzian problem-posing pedagogy involves teaching and
learning against common sense (Kumashiro, 2004). For Deleuze (1968/1994), To learn is to enter into the universal of the relations which
constitutes the Idea. . .To learn to swim is to conjugate the distinctive points of our bodies with the singular points of the
objective Idea in order to form a problematic field. This conjugation determines for us a threshold of consciousness at which our real acts are adjusted to our perceptions of the real relations, thereby providing a solution to the problem” (p. 165)
Yet, the problem does not disappear in the solution, but returns in the process of moving “from the problematical to the question,” which is itself repetition, the eternal return of difference. The “generating” in the title of this plateau, then is
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both adjective and verb: problem posing is the process of generating problems, yet the problems themselves are genetic or generating. Problem posing—that is, learning--is becoming.