The problem lies in the inexpressivity of math, not in the student. Division by zero happens naturally and it is math's problem if it can't deal with it. Natural language can deal with it, nature deals with it. Math fails to express nature adequately.
Example: water flows over an area of land and divides itself evenly: so 1 cubic meter of water / 1 square meter of flat land. Now the land erodes and becomes 0 square meters. The water doesn't have to resort to error-handling code, it knows what to do, it flows on. But math gets hung up at the point when the land disappears, your code throws an error, and you have to handle that. But nature doesn't throw any error, it handles division by zero naturally and seamlessly. Again, math fails to describe nature very well.
This is some pretty unthinking logic you have going on here. How can land erode away to nothing. Is the Earth a flat plate sitting in space and once it erodes away there is nothing there but a hole? You do realize that underneath the land, there is more land right?
If your land is slowly eroding into a channel, then the area is getting smaller. As it gets very narrow the water divides by a smaller and smaller area of land showing the depth to be getting bigger. Once the channel gets to zero width, there is nowhere for the water to go, so it can't be zero depth or even infinite depth, it is a question that makes no sense. If a channel in the real world got infinitely narrow, it would make more sense in the real world that the water would flow over the land that made up the channel's sides, but that is outside the division equation you stated with. That was dividing to find the depth of the water within the channel, once the channel is gone you have to redefine the equation to use another parcel of land.
And if we go to the example where there is a hole into space through the flat earth, then it also makes no sense as there is no depth to water falling through a hole into nothing. It can't spread out onto land that isn't there.