Big Ideas in Mathematics

[apple2357]

What do we consider the big ideas in pre 18 but post 16 mathematics? So in the UK, this would be A level Maths.

For me , the concept of the gradient of a curve and finding a limit is up there?
Anyone agree? What other big ideas do we think kids need to be aware of and appreciate about mathematics?

 

[Deleted member 4993]

What do we consider the big ideas in pre 18 but post 16 mathematics? So in the UK, this would be A level Maths.

For me , the concept of the gradient of a curve and finding a limit is up there?
Anyone agree? What other big ideas do we think kids need to be aware of and appreciate about mathematics?

For me - there are more real irrational numbers than real rational numbers !!!

No practical use - but just WOW!!!

 

[tkhunny]

What do we consider the big ideas in pre 18 but post 16 mathematics? So in the UK, this would be A level Maths.

For me , the concept of the gradient of a curve and finding a limit is up there?
Anyone agree? What other big ideas do we think kids need to be aware of and appreciate about mathematics?

For me, it is that there is no limitation. Whatever you think is amazing and interesting, there is something more amazing and more interesting after it.

 

[JeffM]

What do we consider the big ideas in pre 18 but post 16 mathematics? So in the UK, this would be A level Maths.

For me , the concept of the gradient of a curve and finding a limit is up there?
Anyone agree? What other big ideas do we think kids need to be aware of and appreciate about mathematics?

The truly big ideas for me in mathematics are abstraction and generalization.

For example, we learn these concepts called number and operations on number at an early age.

\(\displaystyle a + b = c\) for any numbers a and b, but

\(\displaystyle a - b \text { may not equal a number}.\)

That dichotomy is not elegant so let's generalize that

\(\displaystyle a - b = d\) for any numbers a and b.

Once we have determined that no logical inconsistencies result, we have just invented a different class of number and created a new and different mathematics.

How many non-intersecting "lines" can you construct "parallel" to a given line and containing a point not on that given line? There are more consistent answers than Euclid dreamt of, and "he alone has looked on beauty bare."

Mathematics is tremendously liberating: imagine what you choose so long as it is logical and consistent.

 

[apple2357]

The truly big ideas for me in mathematics are abstraction and generalization.

For example, we learn these concepts called number and operations on number at an early age.

\(\displaystyle a + b = c\) for any numbers a and b, but

\(\displaystyle a - b \text { may not equal a number}.\)

That dichotomy is not elegant so let's generalize that

\(\displaystyle a - b = d\) for any numbers a and b.

Once we have determined that no logical inconsistencies result, we have just invented a different class of number and created a new and different mathematics.

How many non-intersecting "lines" can you construct "parallel" to a given line and containing a point not on that given line? There are more consistent answers than Euclid dreamt of, and "he alone has looked on beauty bare."

Mathematics is tremendously liberating: imagine what you choose so long as it is logical and consistent.

Nice and we see this with complex numbers later!

 

[JeffM]

Nice and we see this with complex numbers later!

Indeed we do, but in between the generalization from the natural numbers to the integers and the generalization from the real numbers to the complex numbers, we have generalizations from the integers to the rationals and from the rationals to the reals. Moreover, it does not stop with complex numbers. We can go on with quaternions, octonions, etc. (I must admit that I personally have no idea how to play around with the beasties that lie beyond the complex numbers. It just pleases me to know that they exist in the minds of others.)

EDIT: I also find it interesting that with each generalization, we seem to lose an important property. For example, when we go from whole numbers to integers, we lose the property of the existence of a least number.