What are the concepts behind math and how do they contribute to problem-solving?

[Rhine720]

Often time we hear people who are good at math understand the concepts behind it..

What do they mean by concepts behind it? Proofs? I always imagined a person good at math could do the math but also had the capacity to apply many areas of math to a single problem. Like for instance..when manipulating formulas...?

 

[HallsofIvy]

I suspect you should say "the concepts behind" a specific formula rather than "the concepts behind" mathematics itself.

 

[arildno]

One important element of that understanding is:
Mental discipline, to never put anything into that concept that doesn't belong into it (for example, the practical applications of the concept).

 

[symbolipoint]

Much of Mathematics was the development of numbers. Numbers can be applied to nearly everything; but their meaning often made sense when used as adjectives to give quantity information to a noun.

 

[Rhine720]

I still don't think I get it..unless i already do so I'm overcomplicating myself(like the time i tried convincing one of my math teachers that the alternate interior angle theorem shouldv'e been called the alternate verticle angle theorem.

 

[arildno]

Why "vertical"??

What if you rotate the diagram 90 degrees; isn't the theorem valid anymore?

 

[Rhine720]

I think your right.. Alternate interior angle is the vertical of the corresponding angle so idk I just thought that..

So does understanding math concepts mean knowing how they're applied or how they make sense?

Say a person is wondering how Y=mx+b gets you a line, but how do you KNOW? You test it. You play around with Y=mx+b and every time it gets you a line. Then you realize that slope is is actually a range and midpoint is an average...etcetc Is that what it means to understand math concepts?

 

[slider142]

Understanding a concept means internalizing it in a non-contradictory way that does not lead to falsehoods. This notably depends on how a person prefers to internalize abstract concepts.
Ie., the concept of y = mx+b generating a line might be first analyzed by a good student noticing that the b is not really important to the "line"-ness of the topic and is only relevant to having a line off of the origin. Since you can move the line to the origin by moving it b units in y without affecting the quality of it being a line, we are left with the equation y = mx. Here, the student may notice that the equation states that y is a constant multiple of x, and may note that this generates similar triangles. This student probably prefers geometry.
Different students may notice different things, ie., that it is a rotation of the horizontal or vertical axis: "the original lines", or they may appeal to the algebraic concept of linearity, that like changes in input produce proportional changes in output, no matter what the size of the input.

 

[kotreny]

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