I think the step I am worst at is not the “what am I being asked” step, but the “now that I know what I’m being asked, which formulas/ rules/ concepts am I allowed to use here” step.
I think the step I am worst at is not the “what am I being asked” step, but the “now that I know what I’m being asked, which formulas/ rules/ concepts am I allowed to use here” step.
Typically, most mathematical operations can be seen as functions that take something of type X and return something of type Y. (For example, addition might take “real plus real” and turn it into “real”.) For many problems, you start out with something of type A and need to turn it into something of type D. (For example, you might have “4-1” that you need to turn into a single number.)
You can visualize this as a search problem over a network, with the benefit that you can go from both ends. You want a D, so find things that take some other argument and turn it into a D. Suppose there’s only one operation you know that gives you a D, which takes a C. Now you look at operations that take an A, and see if there are any that get you closer, and find that one gives you a C. So now you can go A->C->D and be done. (The example you gave of turning “4-1=x” to “x+1=4″ to “x=3” has an obvious interpretation here, and also highlights that there can be many paths through the network.)
It may help to explicitly map out the things you know how to do in a given context. This isn’t exactly an example of what I mean, but here’s a map which shows relationships between probability distributions.
For Newtonian mechanics, you might have a map which identifies every kind of useful unit, and the relationships between them (a distance and a velocity are related by a time; a velocity and an acceleration are related by a time; an acceleration and a force are related by a mass; a force and kinetic energy are related by a distance) and within them (momenta are related to each other by conservation of momentum). (The within relationships are often less interesting, but it can help to know that momentum and energy are conserved but distance isn’t, say.)
This sort of preparation seems useful at daisychaining- you see a momentum, and you immediately have in your mind “I know how to turn this into a force, a velocity, a mass, or a kinetic energy.”
As mentioned elsewhere, practice. In particular, it may be useful to look at, say, twenty problems, sketch out the necessary steps for each problem (without doing the calculations involved), and then once you’ve outlined all twenty go back and do the calculations and ensure you got the right answer. By focusing on doing the part you’re weakest at repeatedly, you should be able to notice what your specific problems are, and it should be easier to see clumps of problems that are solved the same way.
I think the step I am worst at is not the “what am I being asked” step, but the “now that I know what I’m being asked, which formulas/ rules/ concepts am I allowed to use here” step.
Typically, most mathematical operations can be seen as functions that take something of type X and return something of type Y. (For example, addition might take “real plus real” and turn it into “real”.) For many problems, you start out with something of type A and need to turn it into something of type D. (For example, you might have “4-1” that you need to turn into a single number.)
You can visualize this as a search problem over a network, with the benefit that you can go from both ends. You want a D, so find things that take some other argument and turn it into a D. Suppose there’s only one operation you know that gives you a D, which takes a C. Now you look at operations that take an A, and see if there are any that get you closer, and find that one gives you a C. So now you can go A->C->D and be done. (The example you gave of turning “4-1=x” to “x+1=4″ to “x=3” has an obvious interpretation here, and also highlights that there can be many paths through the network.)
It may help to explicitly map out the things you know how to do in a given context. This isn’t exactly an example of what I mean, but here’s a map which shows relationships between probability distributions.
For Newtonian mechanics, you might have a map which identifies every kind of useful unit, and the relationships between them (a distance and a velocity are related by a time; a velocity and an acceleration are related by a time; an acceleration and a force are related by a mass; a force and kinetic energy are related by a distance) and within them (momenta are related to each other by conservation of momentum). (The within relationships are often less interesting, but it can help to know that momentum and energy are conserved but distance isn’t, say.)
This sort of preparation seems useful at daisychaining- you see a momentum, and you immediately have in your mind “I know how to turn this into a force, a velocity, a mass, or a kinetic energy.”
As mentioned elsewhere, practice. In particular, it may be useful to look at, say, twenty problems, sketch out the necessary steps for each problem (without doing the calculations involved), and then once you’ve outlined all twenty go back and do the calculations and ensure you got the right answer. By focusing on doing the part you’re weakest at repeatedly, you should be able to notice what your specific problems are, and it should be easier to see clumps of problems that are solved the same way.