I think it’s a general problem in the way mathematics is taught (at least around here in Finland and I’m basing this on considerably low amount of empirical observations) that the language of mathematics is not very well elaborated: What each symbol stands for, what’s the logical rule set for using each symbol, like for an example if you have the symbol for sigma to stand for summation—and so even if the students could use their math skills in principle they end up stumbling in practice due to not know how to interpet some statement using symbols they’re not entirely familiar with. Another similar problem in my opinion is the lack of emphasis on understanding what’s actually happening on the abstract level. Why does this work? How do you exploit this rule to arrive at a truthful and more revealing answer?
I’m not sure if this is any good but here’s how I personally like to go about learning math—which I’ve not really done much.
Try and understand what’s going on in the abstract level. Involves questions like:
Why does this work? What’s the rule? What’s the exception? Is there a fixed relation?
Understanding the computing part, the operation.
What do you actually do to achieve the wanted outcome. Do you add numbers? How do you derive a function for an example?
Understanding the language of mathematics related to the concept.
What are the symbols involved? For an example involved with functions, derivals, integrals? What are the rules used in the language in particular? (For an example if you have the symbol of Sigma and below it are i=n what do the symbols below it stand for? What’s the rule with the symbol?)
Doing a full operation using the so far obtained knowledge to perform a computation.
So you start with some kind of data and end up into a final position where the data has been transformed or simplified with the help of the mathematics. ( If at this point you still don’t know what’s going on, try to think backwards to 1. )
Application of the developed skill
At this point you understand the mathskill in question well enough to have attached sort of “handles” to it. So you can understand the concept well enough to recognize it in a different environment and use the ability to handle a subset of data from a larger sample. To make an example if you understand trigonometry very well then handling sectors and related angles with circles is much easier. Both 1. and 5. involve sufficient understanding to be able to predict what kind of change has been caused on the initial position to some extent after the calculation or the operation.
If you dont understand 1. well enough then you just jump to 2. and 3. which go mostly hand in hand, to actually execute the operation you usually need to know both the language and the operation, but not necessarily. Then once you arrive at point 4. you can try to reason backwards from the answer and get the idea at 1. The abstract information is crucial to be able to effectively apply the information when needed like described in part 5. So:
(Skip if must) Step 1 - Step 2&3 - Step 4. - (back to) Step 1. or Step 5.
I also think very brilliant students obtain the contents of these subtopics automatically when studying mathematics, but at least personally I’m notne of them and I think an analytical method like this comes in handy.
What does someone else think about this? In particular someone who already knows lots of math, does this make sense?
I think it’s a general problem in the way mathematics is taught (at least around here in Finland and I’m basing this on considerably low amount of empirical observations) that the language of mathematics is not very well elaborated: What each symbol stands for, what’s the logical rule set for using each symbol, like for an example if you have the symbol for sigma to stand for summation—and so even if the students could use their math skills in principle they end up stumbling in practice due to not know how to interpet some statement using symbols they’re not entirely familiar with. Another similar problem in my opinion is the lack of emphasis on understanding what’s actually happening on the abstract level. Why does this work? How do you exploit this rule to arrive at a truthful and more revealing answer?
I’m not sure if this is any good but here’s how I personally like to go about learning math—which I’ve not really done much.
Try and understand what’s going on in the abstract level. Involves questions like: Why does this work? What’s the rule? What’s the exception? Is there a fixed relation?
Understanding the computing part, the operation. What do you actually do to achieve the wanted outcome. Do you add numbers? How do you derive a function for an example?
Understanding the language of mathematics related to the concept. What are the symbols involved? For an example involved with functions, derivals, integrals? What are the rules used in the language in particular? (For an example if you have the symbol of Sigma and below it are i=n what do the symbols below it stand for? What’s the rule with the symbol?)
Doing a full operation using the so far obtained knowledge to perform a computation. So you start with some kind of data and end up into a final position where the data has been transformed or simplified with the help of the mathematics. ( If at this point you still don’t know what’s going on, try to think backwards to 1. )
Application of the developed skill At this point you understand the mathskill in question well enough to have attached sort of “handles” to it. So you can understand the concept well enough to recognize it in a different environment and use the ability to handle a subset of data from a larger sample. To make an example if you understand trigonometry very well then handling sectors and related angles with circles is much easier. Both 1. and 5. involve sufficient understanding to be able to predict what kind of change has been caused on the initial position to some extent after the calculation or the operation.
If you dont understand 1. well enough then you just jump to 2. and 3. which go mostly hand in hand, to actually execute the operation you usually need to know both the language and the operation, but not necessarily. Then once you arrive at point 4. you can try to reason backwards from the answer and get the idea at 1. The abstract information is crucial to be able to effectively apply the information when needed like described in part 5. So: (Skip if must) Step 1 - Step 2&3 - Step 4. - (back to) Step 1. or Step 5.
I also think very brilliant students obtain the contents of these subtopics automatically when studying mathematics, but at least personally I’m notne of them and I think an analytical method like this comes in handy.
What does someone else think about this? In particular someone who already knows lots of math, does this make sense?