To have learned something implies that one remembers it; a person must have ready access to their learning for it to be learning worthy of the name. The most direct forms of assessment are essentially concerned with what learners can recall when prompted to do so. By calling these forms of assessment direct, I mean to say that they are most directly concerned with what a learner has supposedly learned. The concern is primarily on the content- how stably it has been stored and replicated. The question of what that content may or may not mean to a given learner is of lesser consideration. If the learner can produce the content when prompted, they are assessed as successful (and as not if they could not).
Obviously, very little can realistically be learned about any subject without access to a set of referents that are foundational to that subject. It is very reasonable for a teacher to want learners to have ready access to various facts that act as highly efficient short-cuts in solving problems in a subject. If learner’s understanding of a subject is in a state of perpetual, unpredictable, turbulent flux then they may easily forget or unlearn what they had learned, or learn some alternative incompatible concepts that preclude the attainment of the understanding that their teacher hoped to bring them towards. This is a very obvious problem.
Another problem which seems to be less readily recognised is that a learner’s path towards understanding may become impassable not because a learner loses sight of that path, but rather because their view of it gets blocked by their existing knowledge; possibly the very knowledge that they were taught in order to keep them on their path.
I saw a striking example of this kind of phenomenon illustrated in Skemp’s Psychology of Learning Mathematics, in which a fatally flawed understanding of the Pythagorean theorem is illustrated.
Seemingly, the mathematics learners who produced diagrams such as these had a acquired the belief that the orientation of a square relative to a page it was displayed on was part of the definition of a square. Making one side of a square parallel to the diagonal hypotenuse of a right triangle apparently disqualified it from being considered a square. I suspect that learners who had made this error would probably also not recognise the shape below as a hexagon; they would assume that hexagon exclusively meant regular hexagon.
It is not hard to imagine (for me anyway) the well-intentioned efforts of teachers to inculcate stable knowledge of certain common shapes having the effect of making those shapes only recognisable as the shapes that they in fact were when they appeared in ways that were substantially similar to the ways in which they were shown when the recognition of them was taught. The effect of this would be that concepts that were taught to learners became not only concepts but examples of fixations.
I chanced across a nice example of the opposite of such a fixation based misunderstanding in this online puzzle.
One popular solution to this was to change 4 + 9 = 1 to 4 + 3 = 7, but I preferred a different solution that one person had suggested.
I liked the solution -4 + 5 = 1 more than 4 + 3 = 7 because of the inventiveness of recognising that the 4 has an implicit sign, which can be changed just as well as the explicitly shown sign can.
The learning of physics has a particular susceptibility to being made more difficult by learners’ reification of nonessential, circumstantial aspects of how physical concepts are presented to them during their learning. I strongly suspect that this susceptibility has at least some of its origins in the method of attempting to explain physical phenomena through exposure to various models that aim to represent successive stages of approximation of said physical phenomena.
The problematic aspect of making use of initially highly oversimplified models for explaining physical phenomena is that the oversimplifications of those models can become sufficiently familiar to learners to be perceived as the final explanations of these phenomena, not mere stepping stones to more subtle understanding.
Take for example a model of the molecular structure of a solid that might be used in teaching atomic theory of matter.
Presenting a model like this can reinforce a naïve concept of an atomic solid as consisting of stacked atoms that remain in position because nothing pulls or pushes them in directions perpendicular to gravity, while gravity holds them down. That is indeed what would be happening to a macroscopic model made in this form, which is what a naïve learner is primarily aware of. Crucially, such a learner would probably have only a vague intuitive concept that there are attractive (or repulsive) forces between the atoms. The ability of the atoms in the model to stay in their positions when the model was manipulated would need to be understood in some way or another. The learner would look at the model rotated, and ask themselves…
The naïve answer is probably that some sort of friction wedges the atoms together, so that The atoms collectively inhibit each other’s motion.
This misconception could lead to the further misconception that if single lines of atoms were removed from the atomic lattice them friction would not hold them together, and they would fall apart into individual atoms.
A learner might conceive of this as a phenomenon that explained the characteristics of powdery materials, and then go on to conflate this with the change of matter from solid to gas phase; influenced by the recognition that some powders can when disturbed form clouds- clouds that superficially resemble diagrams they would have been shown of atomic models of gasses as freely moving particles.
The overall point is that learners have abundant motivation and opportunity to interpret models that they are shown in terms of what is already something that for them is concrete in an experiential sense.
Personally, I greatly favour finding ways to base conceptual learning on concrete operations, through manipulative methods like interactive simulations and the designing and making of models. I have a very strong interest in the representation of numbers and mathematical operations in concrete forms in ways that continue their use drastically beyond the pedagogical stage at which symbolic representations of numbers and number manipulations usually replace concrete representations (this is a splendid example of what I mean). While a learner still has access to concrete versions of what they are learning, they retain a degree of ability to use them as they see fit; to play with them. Exchanging concrete representations for symbolic versions is for many learners effectively an act of faith, or at least of compliance. The symbolic operations are a code that they are assured is how things really work. Learners may be reassured that this code will come to be meaningful to them, but ultimately this change in them may mean abandoning what they once meant by meaning and understanding for something that is basically aspirational; meaningfulness as an ever unfulfilled promise, and a void where once imminent understanding resided as a condition that one simply learns to live with- understanding conceived of as becoming resigned to not really understanding but carrying on regardless.