Math STRATEGY needed for Graphing

[caveman1917]

Spivak's text (as well as his Calculus on Manifolds) is often considered the single best calculus textbook there is, and also provides extensive guidance in how to actually do real math, define things, prove theorems, etc. Whenever I do assignments, usually in my physics courses, I've noticed that I find it extremely easy to chunk out a precise definition of whatever vague object the assignment wants me to treat, a skill I think I picked up from that book.

We (the math students) did Calculus in year-1 semester-1 together with science students, so I think that's why a more practical-oriented textbook was chosen. In the next semester we "split off" to Analysis, and did a fully rigorous and more general treatment from there. I do remember being a bit annoyed at the relative lack of rigour in our Calculus course though, it seemed sometimes like it was just floating in mid-air.

 

[TubbaBlubba]

We (the math students) did Calculus in year-1 semester-1 together with science students, so I think that's why a more practical-oriented textbook was chosen. In the next semester we "split off" to Analysis, and did a fully rigorous and more general treatment from there. I do remember being a bit annoyed at the relative lack of rigour in our Calculus course though, it seemed sometimes like it was just floating in mid-air.

Yeah, being a physics student myself with mathematical inclinations, I know that annoyance. I get the idea that they mostly want students to learn to solve problems, but still, sometimes it really just impedes learning. I remember the methods suggested for finding integration intervals in the multivariable calculus course; they all revolved around trying to draw pictures of the volume to be integrated, rather than just figuring them out from the intervals given using Fubini's Theorem... Granted, though, a rigorous treatment of multivariable calculus up to and including Stokes' Theorem can get very complicated.

 

[caveman1917]

I remember the methods suggested for finding integration intervals in the multivariable calculus course; they all revolved around trying to draw pictures of the volume to be integrated, rather than just figuring them out from the intervals given using Fubini's Theorem... Granted, though, a rigorous treatment of multivariable calculus up to and including Stokes' Theorem can get very complicated.

Same thing here. I think it was the university's idea of "all science students need to know basic calculus" so they put all of them together in a "one size fits all" course. It was just a bit silly given that next semester we would just start over again anyway.

 

[wollery]

Way to hijack the thread guys.

 

[TubbaBlubba]

Way to hijack the thread guys.

I'm sorry, I can only interpret statements expressed as powers of successive primes.

 

[Minoosh]

I could be wrong but I don't think calculus was what the OP was asking about - rather, it had more to do with shifts in linear functions as steepness increases/decreaes. Pilots need a realistic linear descent model to hit the runway; the mountain is an extra confounding factor, but even so, all that matters is that you clear it. Models would demonstrate this - you clear the mountain and hit the runway at a spot that allows for safe deceleration. This can all be modeled as a literal x/y graph: Clear the steep mountain with one steep linear slope, correct and touch down with a flat linear slope, before coming to a screeching halt at the end of the runway.

The OP's example was potentially confusing because it indicated slope as a positive - the literal line/angle of descent would have a negative slope, and "bearings" would bring trigonometry into the mix. But I see straightforward linear corrections as most critical.

I do think people intuitively understand "steepness" and why it matters - not a tough sell; you don't want to overshoot the runway or fall short. A graph modeling the plane's actual trajectory (perhaps simplified to eliminate curves) would work just fine for this, IMO. The plane crash analogy is OK, but there are other, more commonplace scenarios that would work just as well.

 

[Minoosh]

I'll have to think about how our (roughly speaking) 3-D world can be conveyed meaningfully in 2 dimensions - all the latter can offer is a cross-section of reality, yet it is useful in many applications. It seems to me that for the purposes of safely landing one plane, linear algebra on a 2-dimensional cross section adequately predicts safe clearance. It gets more complicated if, say, 10 aircraft are coming in from various angles, each with its own altitude, velocity, bearings etc.

What I have to say in general about math education in the U.S. is: Get over your panic. Math has powerful predictive properties, and it's not voodoo; there's no need for a special priestly class to deconstruct arbitrary squiggles. Math is in my mind not arbitrary; it's rather the opposite - a way to remove mystery from the natural world. There is a right answer; and I have seen this cited as an argument for letting children develop their own algorithms. The correct answer will work, the others are wrong.

It appears that in the U.S., at least, there is a deep-seated intergenerational fear of math. This is changing, IMO, but it has held many American students back.

 

[TubbaBlubba]

I'll have to think about how our (roughly speaking) 3-D world can be conveyed meaningfully in 2 dimensions - all the latter can offer is a cross-section of reality, yet it is useful in many applications.

Not exactly true. There are many ways to make a transformation from three to two dimensions, and a cross section is only one among many. Plus, many of the most significant common problems actually arise when you attempt to transform 2D curved surfaces embedded in three dimensions into flat surfaces on paper - the most famous example being distortions in maps. Whereas 3D situations can often be easily represented with a few different 2D cross sections, it is very difficult to understand 2D surfaces by looking at a few different projections.

Imagine you had a set of different world maps (conic projections, cylindric projections, and various others) and had no idea what the shape of the Earth was. Your first guess would probably not be "Oh, it's spherical!" This is one of many case where the "squiggles" are far easier to understand and manipulate than the graphs and charts...

 

[Minoosh]

Plus, many of the most significant common problems actually arise when you attempt to transform 2D curved surfaces embedded in three dimensions into flat surfaces on paper ...

Good point. I am not spatially gifted, to put it mildly.