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Nine Dots & The Problem with Learning

Updated: Jan 22, 2023

Despite the aggrieved protests of students who have grown tired of a seemingly endless demand for written assignments, there is a very good reason why most higher education evaluators require extended textual output. Language in-and-of-itself can be understood as a primordial learning technology, wherein one receives, decodes, processes and encodes information in a never ending dialectical process. These kinds of computational metaphors are helpful, but it's important to remember: they remain metaphors. The mind is not a machine but an organ, and organic metaphors can be equally helpful. Geological metaphors, too, can provide wonderful metaphors and perhaps fund entirely new models for how learning works.

For example, the learning mind might be thought of as an island whose soil becomes enriched, and whose contours are shaped by tidal movements of information and sense which wash ashore and recede, depositing various accretions and sculpting unique formations which may be only later be measured and catalogued as evidence of learning. In any case, the meaningful encoding of information via textual performance remains one of the most effective means of determining whether learning has occurred (particularly in the cognitive domain), and this is another reason why the evaluation of writing remains indispensable for assessing the relative success of instruction: a text must be constructed, and this constructive activity requires one to demonstrate a familiarity-with and an understanding-of key concepts, including the way in which those concepts are either assembled into larger schemes or brought into proximity with other concepts in coherent and novel ways. A text is thus an artifact or residue of a process that is opaque and dynamic; the act of skilled construction, (textual or otherwise) thus evidences a student’s mastery.

German theorist Ernst Von Glaserfeld, a proponent of radical constructivism provides a neat summary of the theory. “Instead of presupposing knowledge is a representation of what exists,” he writes, “knowledge is a mapping in the light of human experience of what is feasible”.[1] Bracketing for a moment the overtly pragmatic thrust of this statement, the idea that knowledge is constructed by learners who bring their own experiences, interests, and concerns to bear upon the learning process should be noted as a hallmark of constructivism. This kind of robust participatory learning is a distinctive mark of constructivist learning theory. “The mind is never passive; it is a perpetual activity. You cannot postpone its life until you have sharpened it. Whatever interest attaches to your subject matter must be evoked here and now; whatever powers you are strengthening in the pupil must be strengthened here and now: whatever possibilities of mental life your teaching should impart must be exhibited here and now. That is the golden rule of education, and a very difficult rule to follow" [2] writes nineteenth-century mathematician and philosopher Alfred North Whitehead. Whitehead’s “golden rule” thus forecloses any possibility of passive learning, demanding instead that instruction consider the pupil’s unique interests, experiences, and passions as the motor which drives learners toward mastery.

This is all well and good. However, what I’d like to propose is a slight modification to constructivist learning theory typically characterized by the “construction of knowledge” as firstly a construction of problems. Learning as dialectic is “the art of problems and questions, the combinatory or calculus of problems as such. However, dialectic loses its peculiar power when it remains content to trace problems from propositions.”[3] In other words, sense or meaning is to found not in answers, nor simply in response to a problem that is essentially the reconfiguring of a doctrine. Rather, truly constructed knowledge is an outgrowth of one’s articulation within a given field of differential singularities, and it is the recognized contingency of this field which makes possible the construction of a problem. This is no easy task, which perhaps explains, at least in part, why so few are able to attain true mastery in any given field. Not only this, but mastery is the result of a sustained and intense engagement with difficult problems. There is a chapter in “Make it Stick” aptly titled “Embrace Difficulties” wherein we read, “[T]he easier knowledge or a skill is for you to retrieve it, the less your retrieval practice will benefit your retention of it. Conversely, the more effort you have to expend to retrieve knowledge or skill, the more practice of retrieval will entrench it.”[4] This statement, while written in the context of information retrieval, nonetheless underscores the indissociable relationship between mastery, the creation of problems, and a willingness to embrace difficulty.

An interesting example which may help to clarify the aforementioned modification to constructivist theory is something called the nine dots problem[5]. The challenge is to connect all the dots by drawing only 4 continuous lines. There are several possible solutions, each of which necessitates one to draw, or rather to “think outside of the box”. Indeed, the nine dots problem is sometimes attributed as being the origin of that phrase. In any case, this challenge illustrates how solutions often only become possible upon one’s willingness to reframe the given terms of a problem. True enough, there may be variables previously unaccounted for and the boundary imposed by the typical framing of a problem may represent a simple epistemic limit. Here, another cliché seems apt: the map is not the territory. To venture ‘off-grid’ or into the unknown is to risk a movement into perilous waters where the jaws of monstrous beasts await daring and wayward adventurers alike. One may fail, or return a hero. But, in either case the expansion of knowledge should be understood as little more than a relocation of the boundary stones marking the furthest territories of a given paradigm. This is not the labor of thought, but of cartography. The more difficult task concerns itself not with the unknown, but rather with the unthought. That is, when considering a difficult challenge, rather than focusing on solutions it is of much greater importance to radically rethink the limits of that problem. The task is not to think “outside of the box”, but to attune oneself to the fictive and spectral nature of the box itself.

[1] Von Glaserfeld, E. (1988) Cognition, Construction of Knowledge and Teaching. (Eric Document Reproduction Service No. ED 294 754) [2] Whitehead, A.N. (1929). The Aims of Education & Other Essays. New York: The Free Press. pp. 5-6. [3] Deleuze, G. (1968). Difference and Repetition, Bloomsbury Academic Press. pp. 205 [4] Brown, P. C., Roediger, H. L. III, & McDaniel, M. A. (2014). Make it Stick: The Science of Successful Learning. Cambridge, MA, US: Belknap Press of Harvard University Press. pp. 79. [5]

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