A Recursive Theory of Mathematical Understanding.

Abstract

ion of their properties. One is now in a position to observe one's own thought structures and organise them consistently. One is aware of being aware, and can see the consequences of one's thoughts. It is clear at this point that while this outer level of understanding is transcendent, in other words fundamentally new in some way, it has to be consistent with all previous levels of knowing. For fuller understanding one must now be able to answer why the consequences of thoughts must be true. This calls for an awareness of associations and of sequence among one's previous thoughts, of their interdependence. In mathematical terms it might be setting one's thinking within an axiomatic structure. All of these levels of recursion are referenced in a direct way to previous levels. Although new levels transcend or make one free of actions at an inner level, in some sense these actions on previous levels become initiating conditions which constrain one's knowing. At the highest level of recursion, as Tomm suggests, knowers act as free agents. We call this the level of inventing.* Now one can choose to initiate a sequence or structure of thought which is a recursion on the previous one in the sense that it exists as a base, but is freely, yet compatibly, created. For example, mathematicians such as Lobachewski, Bolyai, and Gauss deliberately invented non-Euclidean geometries rather than try to deduce or explain the parallel postulate as a necessary consequence of Euclidean geometry in the way that Saccheri did. To illustrate this growth through recursion in less general terms, we turn to an example of mathematical knowing, the knowing of quadratic functions. The particular example is hypothetical but allows us to illustrate the entire chain of recursions from primitive doing to inventing. It also allows illustrations of the fact that one should not think of action in simplistic terms. Any action, and this one, plotting, in particular, may already be a recursion on previous thought or actions. We start with the first level of recursion, image-making. The students are taking given, specific, quadratic functions, making tables of values, and then plotting them. The imagemaking can be seen in comments like Does x2 + 2x + 1 go like this? or I get 1 for 1 in 2x2 x 2. is critical here is that these tables and graphs have an individual quality. If one were to ask such students, What do you know about quadratic functions?, their effective action would be related to the images of specific quadratics. For them to say anything further would require their making more tables or doing more plotting; their comments about quadratics would pertain to their actions. The next stage is reached when they have generalized these specific images. They are now able to say, Quadratic Well, they all form sort of U-shaped curves. They have now replaced their actions on quadratics with an image and can talk about it. There is an ability, but no necessity, to continue to plot individual graphs, and certainly no need to make very extensive tables of values. This stage of image-having is a recursion on the stage of imagemaking. Because they can now talk about quadratics as objects, students can move to a level of noticing features or distinctions among them. Quadratics have vertices, they open up or down, they are not linear, they can have maxima or minima elsewhere than at the origin not now related to a particular graph but to properties of quadratics. Students here have a generalized image. The next recursion occurs when students realize that despite these distinctions quadratic functions can be considered as a class of things: students can talk about the set of functions y = ax1 + bx + c and see this as a consciouslymade mathematical definition. They are using self-conscious thought to formalize their understanding. They now know quadratic functions as mathematical objects and not simply as graphical ones. The next level of recursion is signalled by the students' interest in the question, What is true for all quadratic functions? At this level the quadratic formula or the method of completing the square are pieces of a possible theory and not simply techniques for computation as they might earlier have been. The arrival at this level, that of observing the class of quadratic functions as a mathematical object, allows us to make another point critical in our theory of understanding. Up to now, the recursions may have appeared to have a sequential or linear quality; but the development of understanding does not have this quality. Students at this new level might decide to address the question Can quadratic equations have just one real solution? and be prompted to drop to the level of primitive doing, to making tables and graphs of particular examples. However, the return to image-making is not to imagemaking as it was previously perceived. The students now know they are looking for an image which can be distinguished by certain features but which can still be characterized as belonging to the class of quadratics. In other words, while actions, images, and distinctions, formed initiating conditions for the definition of a class of quadratics, the local class of quadratics now forms an environment in which a new kind of exploratory action takes place. Theory folds back to constrain one's actions. In contrast to the original image-making, one is now acting on what may be a specific but, one hopes, representative example. Thus knowing and understanding quadratics is not a linear process: knowing folds back on itself and has the quality of being circular without contradiction. Such a folding back *This use of inventing is not intended to suggest that persons do not or could not develop essentially new (for them) ideas at other levels.