So I find I can force myself to visualise that but it would be consistently born of the concept thought first, like “oh that’s a line perpendicular to the line between X and y” and then I can paint the graph in my mind. But I don’t need to—I can think the concept and then apply it to paper without visualisation and I tend to find that easier.
What intrigues me precisely is visual thinking for problem solving—ie a student who can easily perform arithmetic between graphs by visualising the transformations in their mind rather than doing calculations on paper.
First of all, if you can solve it without visualization, I think that this is preferable, precisely because it is faster. There is no need to force oneself to visualize everything.
To visualize something, you need to create a map from the formal domain you are studying to visual transformations. In other words, you need to understand “what the formula” mean (or at least one way of looking at them). Do you know what it means visually to multiply one complex number to another? If you don’t, you will be stuck doing calculations. If you do, then you can visualize it and quickly come up with the solution.
From my experience, some people naturally tend towards visual thinking, while others don’t. But if you consistently try to apply it, it will become natural at some point (it may take some time, don’t give up prematurely).
One area where a lot of visual thinking is necessary, but that is relatively easy to visualize, is graph theory. Try to prove that a (connected undirected) graph has an Eulerian cycle (i.e. a cycle that contains every edge exactly once) if and only if all of its vertices have even degree.
So I find I can force myself to visualise that but it would be consistently born of the concept thought first, like “oh that’s a line perpendicular to the line between X and y” and then I can paint the graph in my mind. But I don’t need to—I can think the concept and then apply it to paper without visualisation and I tend to find that easier.
What intrigues me precisely is visual thinking for problem solving—ie a student who can easily perform arithmetic between graphs by visualising the transformations in their mind rather than doing calculations on paper.
First of all, if you can solve it without visualization, I think that this is preferable, precisely because it is faster. There is no need to force oneself to visualize everything.
To visualize something, you need to create a map from the formal domain you are studying to visual transformations. In other words, you need to understand “what the formula” mean (or at least one way of looking at them). Do you know what it means visually to multiply one complex number to another? If you don’t, you will be stuck doing calculations. If you do, then you can visualize it and quickly come up with the solution.
From my experience, some people naturally tend towards visual thinking, while others don’t. But if you consistently try to apply it, it will become natural at some point (it may take some time, don’t give up prematurely).
One area where a lot of visual thinking is necessary, but that is relatively easy to visualize, is graph theory. Try to prove that a (connected undirected) graph has an Eulerian cycle (i.e. a cycle that contains every edge exactly once) if and only if all of its vertices have even degree.