There is a strong connection between freezing and emergence of quantum properties. Quantum behavior occurs once indistinguishability becomes dominant, which also characterizes Bose-Einstein condensation or Fermi degeneration. In a "hot, noisy" world, indistinguishability breaks down and decoherence takes over---every particle has its own set of attributes easily resolvable from those of others. However, freezing a sample is the same as producing a "quiet" quantum state so necessary to produce quantum behavior, such as quantum computation, which is why much of early quantum computation experiments have been conducted in ultracold conditions. I don't think it's a stretch to say that classical description fail at very low temperatures, when such classical motion (based on infinite degrees of freedom) is "frozen out" due to quantization.
Quantum theory in fact holds that even you or I are actually in quantum superposition, i.e., we exist "smeared" over time and space. But this range of smear is so tiny as to be irrelevant and undetectable under ordinary circumstances. However, if we were frozen and left alone in a vast, cold, empty space (think of this universe extremely far into the future, hundreds of trillions of years from now, if it turns out that it continues to expand), we would actually become like solitary, undisturbed electrons that exist in a quite broad volume of space. It would not longer be true to say that we'd exist in one particular place and time any more.
Edit: After some thought about the way you worded your question, I think it should be noted that the concept of quantization is something too easily misunderstood by many not familiar with quantum theory. Quantization is strongly related to the problem of indistinguishability, not an "inherent property of time and space", like somehow at some very small dimensions space and time is some kind of a checkerboard. It would be misleading to argue that as things get cold, everything wants to "settle into some kind of a lattice-work". We know that's what crystals are, but crystals do not form because of "quantization". We need to get away from that kind of highly misleading interpretation of quantization. I think a better analogy, as a starter, is to imagine a stream of water from a faucet, as it is slowly reduced. The stream is steady, and grows thinner, until finally it can only come out in distinct drops---and that is the point of quantization. The water comes out in countable drops.
Edit 2: Per your added comments, can you restate the question more precisely? I was looking at this from a pretty general point of view. By "second year university students", I assumed you were talking about introductory quantum mechanics.
Edit 3: If this is indeed an introduction to quantum theory, I would stress that physical systems are typically "in a classical state", or "in a quantum state", each of which exhibit different behaviors requiring different mathematical apparatus to treat. It is possible to "bridge" from one to another, as with the example of Rydburg atoms, where the outside electron orbital is so excited that the electron actually starts to resemble a classical satellite orbiting the nucleus. In other words, it's not necessarily an "either-or" between the two states, classical or quantum. But the progression towards quantum strongly involves the question of indistinguishability, and freezing is one way to create conditions for such indistinguishability, just not the only way. As another example, Feynman had suggested that quantum behavior can be mimicked by a "random walk on the complex plane", with very good agreement with the solutions of the Schrodinger wave equation. The interesting thing is, when "noise" is introduced to this kind of random walk, by slightly increasing amounts, this random walk begins to resemble that of the classical random walk. In fact, it should be pointed out that the only difference between the Schrodinger wave equation and the heat diffusion equation is the imaginary number i. Something to think about.
Edit 4: I liked the answers given by jean and FGR, so I've given both of them TUs. Hopefully, all of this will help you to decide how to handle this in the matter of your notes to 2nd year students. However, it's been in my experience that people new to quantum theory (and even some of those already familiar with it) have misguided notions of quantization, thinking that space time has some kind of an inherent graininess to it.
Edit 5: A nice PDF paper on the subject of Hadamard Walks is given in the link below. This is a more direct approach to quantum random walks, which does allow "bridging" to classical behavior. This is an effort at a better conceptual understanding of the link between quantum and classical behavior.
Edit 6: In continuing on this interesting subject, I'd like to respond to jean's statement that a single hydrogen atom exhibits quantum behavior not because of "indistinguishability". But in fact, that's a timid way to look at it, limiting the concept of indistinguishability to only indistinguishable particles, leading to the phenomenon already noted by the Asker. Indistinguishability can be argued even for a single particle, when it comes to resolving its actual location at a specific time. Another way of looking at this is Everett's Many Worlds hypothesis, a popular one that imagines that, say, a solitary particle exists in many parallel worlds, each with its own history, which is another way to say that we're dealing with many virtual indistinguishable particles that, together, behaves as a single particle. Feynman's sum--of-histories is analogous to this model as well. Quantum theory, while still poorly understood from a conceptual point of view, does allow a number of mathematically equivalent ways to describe it, all delivering the same results, even if the "explanation" are different with each one. This is the reason why "quantum interpretations", which attempts to develop an intuitively understandable concept of quantum behavior, is a field of study separate from main quantum theory, since main quantum theory exists on the strength of the mathematical apparatus involved, and doesn't actually require an "intuitive explanation" for it. And that is part of the problem with the question as posed by the Asker, because what the Asker is proposing to do is to offer some kind of an "intuitive explanation" for emergence of quantum behavior.
I think one of the hardest things for people to understand is how it is possible for different models with seemingly conflicting "explanations" can actually all be in mathematical agreement. We naturally assume that there can only be one "correct version of the events", all the others wrong, as for example, we don't accept the idea that there can be more than one way "the world was created", which was actually not uncommon in ancient societies. Yet, this commonsense rejection of multiple explanations for the same thing doesn't apply in physics when the mathematical apparatus can be demonstrated to be equivalent. I personally don't like Everett's Many Worlds hypothesis, or even the String Theory model (where everything is made up of tiny vibrating strings), but that does not mean that I believe that the mathematics involved is in error and so those models must be rejected. I just go over to alternate ways of looking at the same things without invoking those conceptual models that I find disagreeable. I think for many people, this is a notion hard to swallow---actually having a choice of how you'd like to think of physics conceptually.
Edit 7: jean, I'm just expanding the subject, I know I'm probably far beyond Vasek's original question, and into highly speculative issues. But everything that I have brought up is already in the literature. For Vasek's original question as narrowly defined, I'm deferring to your answer. As for your objections to "confusing" indistinguishability with quantum uncertainity, again, this is not my original or novel idea, but it does fall under quantum interpretations. The main problem with quantum interpretations is that almost by definition they cannot be experimentally verified, because it does not seek new mathematical formulations of quantum theory, it merely attempts at a conceptual or intuitive understanding of it. Again, I think it's a too-timid approach arguing that indistinguishibility and quantum uncertainity are "two entirely distinct matters".
Do you have any idea how to conceptually explain quantum behavior? How, for example, does quantum entanglement even work at all, conceptually? We know it works per math, but can one make any conceptual sense out of it?
Edit 8: Vasek, that's interesting that quantum walks has been your field of expertise for some years. I'm sure you're aware that even with this one subject, there are several different branches or avenues of study, and that your particular expertise is associated with QFT, since that's what puts the bread on the table.
As for your comment, jean, on string theory "not an achieved theory", ironically, the more a theory is mathematically consistent with other established theories, the harder it is to demonstrate experimentally the distinction it has! String Theory is still a remarkable mathematical model in its own right, and as such, it shouldn't be discounted, but at least seen as an "alternate way of looking at things". Recall the early original "conflict" between Heisenburg's Matrix Mechanic