How to explain (pedagogically) why there is 4 spacetime dimensions while we see only the 3 spatial dimesions? I have been asked this question by a student, but I was able and in the same time incapable to give a good answer for this without equations, so do you have ideas how one can explain this in a simple way? 
(Answers like we can take time as an imaginary, or our space is actually pseudo-Euclidean will be hard to grasp for new students.)
Note that the problem is not in visualizing the 4th dimension, that an easy thing to explain. The problem is more related to why we are in 3D that moving along 1D time dimension? In Differential geometry this interpreted by fiber bundles, but how to explain it to usual student.
 A: To stay well bound to the physics, I suggest to explain the concept of event, to point out that if we want to identify something that happens in our universe we need to locate it in space and time. 
Now, the definition of "number of dimensions" is, roughly, how many numbers you need to identify an element (of the vector space). In this case, it's clear you need 3 spatial coordinates, plus "when".
A: 
Note that the problem is not in visualizing the 4th dimension, that an easy thing to explain. The problem more related to why we are in 3D that moving along 1D time dimension?

You cannot avoid defining time in terms of change. In the same way that if there were no changes (dx/dy etc) in a terrain the map would be totally uniform and uninteresting, if the terrain did not change in time, time would be uniform and undefinable. Repetitive changes allow us to define time ( no need to go to entropy, the solar system, day/night etc are enough ) and clock it/ measure it.
Time projects into the 3 dimensional world. Geological strata ( and many other proxies) assign to each (x,y,z) a time value on the axis of t. Thus time can be projected into (x,y,z).In a similar way space dimensions project into time. The time taken to walk to the station has one to one correspondence with the distance in kilometers. 
So time is a necessary dimension to describe the changes seen in three dimensions, in a similar way that a third space dimension is needed to describe the projections of a sphere to two dimensions.
Entropy must come in for a classical definition of the arrow of time and non reversibility. A rough discussion on disorder, broken glass not repairable etc should give them the concept.
Thus even without special relativity time can be thought as another dimension since it projects into the spatial ones. One can then go on to special relativity as surprising us with the different type of dimension ( pseudo  euclidean) it turns out to be ,from experiments.
p.s. with this view of how time is defined in our experience we can also with assurance say that there is only one time dimension. If there were two time dimensions the functional dependence of changes in the  space three dimensions would be complicated. It would be a many to one projection, similar to projecting a three space dimensional object to one space dimension. 
A: Some good ideas have been expressed here. My take on this is the following:
Imagine you have a laser gun and you send a laser pulse outwards in the outer space sending an image out there.  The laser pulse travels at the speed of light.  Now let us time the beam for a length of time $\delta t$. There is some distance that corresponds to this time and it is $c\delta t$ where c is the speed of light. This  $c\delta t$ is the fourth dimension in the 4-D Minkowski space, corresponding to that short time $\delta t$. It tells us how far the image has travelled within this short time. So it is the speed of light that generates the fourth dimension, and also gives the length we call the fourth dimension. Therefore, the fourth dimension starts on our watch and $c\delta t$ is the length of it within the time $\delta t$. It makes sense only in the context of the speed of light. This is the whole point of space-time in special relativity. This is what it means when we say that an object is ‘so many light years away’. In a way, this is the distance that the image of an object is away (a galaxy for example) in the fourth dimension. The mathematical representation of this has been written in Leos Ondra reply. I hope this helps somewhat.
A: We do see the fourth dimension.
The difference between three dimensions and four dimensions is the difference between a (2d) snapshot image and a ("2d+t") video.
A: You might introduce the thought of time being a fourth dimension by asking your students to contemplate the meaning of 'perpendicular'. 
They will likely respond that length, width and height are perpendicular directions. If you push further and demand a defining characteristics of 'perpendicular' they will probably arrive at the non-mathematical and hand-waving characteristics that perpendicular directions are those that allow you to move in either of these dimensions, without making any movement in any of the other. At that stage you can ask them if you can move in time without moving in any of the three spatial dimensions. 
Just leave them with that thought. They will come back with further questions...
A: Here's an attempt at a non-mathematical (and unorthodox) answer: 
1) We can see in the three spatial dimensions because light travels through them, reflecting back from objects around us and eventually reaching your eye.
2) Everything (incl. light) travels "forward" through the time dimension (meaning, it is able to move in only one direction, and cannot move back and forth).
3) Light is therefore not able to reflect "back" from an object to your eye through the time dimension.
Thought experiment: take one spatial dimension, but everything is moving in one direction at the speed of light. Would you be able to see what's right next to you?
A: I think one way is to illustrate it in 2 + 1 dimensions, and have them imagine it in 3 + 1. 
You could draw a cube, and explain that a slice along the cross section of the cube is a 'snapshot' of a 2D space, and the third dimension is time. So they are moving around in 2D, while they are "experiencing" the third. 
It's hard to explain this exactly without a figure... if I can lay my hands on one or I can make one myself, I'll add it here when I do.
A: We see 3 dimensions because we ourselves are 3-dimensional. Imagine a 2d creature originally living in 2d space - an Euclidean plane. It naturally perceives only events which occur in its body, like a photon (assuming for a while something that something like light can exist in 2d) which interacts with its 2d cell in retina. If somehow in the course of evolution the third dimension is added to its flat world, it will still perceive only two dimensions. Now, hoverer, it can be rotated in 3d so its plane of living changes, and new strange things occur - a rod, which has always kept its length in the original flat 2d world, because the flat Pythagoras (and many others before him) proved that
$ds^2 = dx^2 + dy^2$
Now, the rod strangely contracts and lengthens, but one creature (called Einstein the Flat by some, and Lorentz or Fitzgerald by others) founds that there is still something like length which remains constant, namely
$ds^2 = dx^2 + dy^2 - dt^2$
which sounds strangely to others because


*

*t has never been considered a dimension

*There is minus sign before. 

A: Because it seems this question interesting to many people, I will tell you how currently I'm explaining this, the idea was came to me when I read about how Philosophers understands time, correct me if you think there is something wrong:
Even so we treat time as a dimensions, it's not that similar to the spatial one, and the reason is as follows:
We start with the most fundamental concept in physics: cause and effect, this concept enable us to sort events in a series : first event is cause then effect ... , that creates an "illusion" of the ability to number those events, what in turn provides as with ability to treat time as dimension and measure "distance" between events, anyway I said the illusion of numbering, because saying "numbering" tells us that we can do that in absolute way, which is wrong according to theory of relativity, because numbering events for one observer is not compatible generally speaking with others (here i explain how speed of light affects cause and effect), for that time is not the same as spatial dimension, and that why Menikowski space is pseudo-Euclidean, not Euclidean, then I add 
Anno2001's answer how to look at time.
A: Another way to look at it--We actually only "See" in two dimensions.  Each of our eyes only has the ability to process xy coordinates, not z (A one-eyed person has no depth perception).  Our brain diffs the two 2d images from our eyes to give us a good guess at a z coordinate (Since it's not true 3d perception our brain can be tricked here, hence illusions!  Also our 2d vision can be obstructed, true 3d vision would not be).
In the same way as our brain "emulates" 3d vision, it diffs what is going on right now from what went on a minute ago (or a day ago or a year ago) to give you an understanding of the t axis.  It is no less valid than our view of the z coordinate our brains construct, but unlike 3d it's not helpful to visualize the t dimension so our brain doesn't do that, instead it makes the information available in other ways.
