[Nevermark]

In high school trigonometry I am sure I was clear that sine and cosine formed the circle. How could I not?

But that fact’s significance and too obvious simplicity, with all its ramifications, only hit me deeply and profoundly when a year later I realized I could use those functions to draw a circle on a screen.

Before that they were abstractions related to other abstractions that I had to memorize to pass a course.

To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

I appreciate that in the game of math that view becomes natural. But most of us learn math with additional motivations and are interested in any perspective that highlights potential usefulness or connection to the real world. Many of us mentally organize our knowledge teleologically.

Yet even when usefulness is known to exist, it is often neither mentioned or referenced. Or even considered relevant.

Edit: the same goes for not showing a single concrete example of an abstract concept. A kind of communication that would unlock many mathematical papers to a much larger audience of intelligent and relevant readers.

[skhunted]

I’ve been teaching mathematics for over 30 years at the community college level. Most people at the time of taking a course don’t have a sophisticated enough understanding of math to really appreciate “intuitive explanations” because they don’t have intuition.

Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.

The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.

Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.

[Nevermark]

I am sure you are right, and a little convo here isn't going to do the topic justice. So many aspects to how people understand and learn things.

And I know picking on your example isn't in the league of a general solution.

But if 2x (which is x + x) is two apples in a box, and 3x (which is x+x+x) is three apples in another box, then you put those two boxes in a bigger box (another +), people already intuitively can see the distributed property of scalar multiplication vs. addition of some unit, they just didn't have a name for it.

Likewise, a 3x4 square of paper next to a 7x4 piece of paper can be easily seen to be a 10x4 piece of paper. Multiplication of numbers over added numbers.

So one way to introduce distribution is to start by showing examples of several places where people already understand the concept of multiplication distributing, and use it every day, but just didn't know it was one concept with a name.

Once people can recognize distribution as an already familiar relationship in everyday life, then the symbols can be visited as the way we write down the already known and useful concept so we can be very clear and general about it.

Anyway, that's just a reaction to one example, which may not mean much.

[skhunted]

Yes, almost everyone gets that. Then you need to explain that x is really 1 x. Then you need to explain that -x is really (-1)x. Everything is great. We all understand. Now simplify (1/2) x + 3x. You’ll lose most people at this step. Then explain (1/2) x - (2/3)x. More confusion. Now explain that ax+x is (a+1)x. You lost a lot of people at this step. Now explain that xy+ x^2y is (x+x^2)y and that this is just the distributive property “in reverse”.

Sometime later a person will really grok all this and then say, “why didn’t they just tell us this is all just the distributive property?”

[Nevermark]

Ah, I get it.

A different approach I have thought about, which would really tear the textbooks apart, is introducing every concept in its simplest form as early as possible.

Then when it is eventually expanded on, its familiarity will aid in taking further steps more quickly and intuitively.

For instance, something as simple as adding up the area of a fence of varying heights, or the area of multi-height wall to be painted, being referred to as integrating the area, in early grade arithmetic, creates a conceptual link for down the road.

Systematically going over K-12 materials, just making similar small adjustments to terminology and concepts to be highlighted, would be interesting.

[skhunted]

As I see it the issue with your integrating example is that area is the correct word for finding “area”. Integrating is not finding the area. The indefinite integral is not about area. The definite integral in dimension 1 has to do with signed area. I don’t think having people ingrained to think finding area is “integration” would be a good thing. Especially since most people don’t take calculus.

To your point, people do constantly try to tweak things to make subjects easier to understand and more intuitive.

[lupire]

That's how Riemann integrals are defined in class. Vertical slices.

Tossing the word "integral" at younger children won't make that easier or harder.

[WillAdams]

I've been trying to work through this in the context of programming a CNC using a recent trilogy of books:

_Make: Geometry: Learn by coding, 3D printing and building_ https://www.goodreads.com/book/show/58059196-make

_Make: Trigonometry: Build your way from triangles to analytic geometry_ https://www.goodreads.com/book/show/123127774-make

_Make: Calculus: Build models to learn, visualize, and explore_ https://www.goodreads.com/book/show/61739368-make

(oddly the Calculus book was published second, so I guess I'll need to re-read it after I finish the trigonometry book)

Hopefully, this will provide me with a sufficient grounding in conic sections that I can solve my next CNC project with a reasonably efficient set of calculations (trying to do it using my rudimentary understanding of triangles from trigonometry had me 4 or 5 triangles deep, barely half-way to the final point I needed, and OpenSCAD badly bogged down performance-wise).

[Nevermark]

Thanks for that list, I just ordered them! I hadn't heard of that series before

[WillAdams]

Glad to!

It's quite recent, but seems very good to me, but my math education is about non-existent, so anything would be an improvement.

[xanderlewis]

> To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

If you’re talking about research papers, that’s just because they’re written for domain experts and aren’t really for giving you intuition. They’re written in a deliberately terse (one might say elegant) style to convey the research findings in formal mathematical language and nothing much else. If you want to gain an intuitive grasp of things, read a proper textbook in detail or play around with the ideas on paper. Or both!

I guess the reason is that once you’ve acquired the intuition, having the literature cluttered up with the same explanations again and again becomes clunky and increases the volume of material to be sifted through when you’re just looking for a result you need in your research and don’t need all the extra chatter. It’s just cleaner that way. But to an outsider it does look more opaque. It’s a trade off.

[Nevermark]

> having the literature cluttered up with the same explanations again and again becomes clunky

I think that really is the best reason for not being more accessible. Along with less work - given a good paper already can take a lot of work to write clearly even for the immediate audience.

But there is tremendous value in reaching a wider audience, for readers, writers, and the very real serendipity of cross pollinating ideas. So an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition. Even an appendix - although that might strike the established culture as too quirky.

Some papers manage to do something like that, a colorful example or perspective adding levity as well as clarity. So it is not breaking any barriers. Just not standard or prescribed.

Or maybe it wouldn't have much impact. I tend to find reasons to dive into many different new topics, so it is a prevalent need for one!

[xanderlewis]

> an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition.

I completely agree — especially in the modern era where extra pages cost nothing.