In first year engineering mathematics course at Swinburne we try to lead students from the boring 2-D universe of high school (uniforms, canteen lunches and x vrs y graphs) to the much more exciting and vivid multi-dimensional world of advanced mathematics ! This progression - more of a leap - requires some imagination and determination. The first step is the visualisation of 3-D space, than a progression to a more general idea of "dimensions". When I teach partial differentiation, I get the students to think about how the slope of a hill changes when you keep one dimension constant (i.e. don't move sideways and walk upwards), as compared to the slope if you swap the dimensions you keep constant (i.e., don't move upwards but only sideways). These relationships become clearer when you draw graphs of these relationships for different physical environments (e.g. slopes in a valley as opposed to a ridge or a steep point hill). This type of approach can give you a sense of what the mathematical symbols means. These hill walking mental games are not just metaphors for the mathematical operations we are studying but direct physical examples of the mathematical ideas we are exploring.
The imagination is also required when making the next intellectual journey ....that is, seeing the concept of dimension in a more general way. For example, realising that when studying how heat is flowing through the wall of a house, you can visualise the "valleys" and "hill tops" that the temperature profile will take in the three spatial dimensions of the wall, in the same way that you extended your view of the world by moving beyond x and y graphs. If you can visualise "temperature" as an extra "dimension" in the wall than you will start de-mystifying the mathematical operations taught to you (partial derivatives, cross products, etc.). After all, we are studying these operations for largely practical reasons, such as, calculating the temperature profiles of walls, the velocity profile of gas flowing in a duct and a myriad of other engineering problems, so visualising the mathematics in physical terms provides a direct intellectual route to performing the engineering calculations that any decent engineer would like to make. Some determination is required to master the mechanics of these operations - my head still spins a little when taking the partial derivative of a partial derivative - but I would argue that the imagination/visualisation part of this trip is the most difficult and most rewarding aspect of first year mathematics.
Welcome to the multi-dimensional universe !