Math Mistakes
mathmistakes.org
mathmistakes.org
I distinctly remember encountering Metric Geometry* in high school. It didn't teach me geometry, it taught me how to think, and I'm forever grateful. Teaching math with keywords is the opposite of this.
[*] Probably not the correct name in English. It was geometry without algebra; proving theorems and finding loci based on some axioms about angles, parallel lines and so on. What is this called? Euclidean Geometry?
"Euclidean geometry" would do for a name, but to specifically emphasize the point that algebra is not involved, you can say "synthetic geometry." The geometry of the plane can also be taught from an analytic geometry perspective (using ordered pairs of points on the Cartesian plane) as a first high school course, as in the book Vectors and Transformations in Plane Geometry by Tondeur.[1]
There's algebra and calculating, but with reflection operations here, not with coordinate pairs.
OK, that was only true at work.
http://mathmistakes.org/subtraction-algorithms/
There's a sheet of a bunch of sums. The student has got them all correct. But we don't know if the student actually understands the concepts, or if the student is using their wrong understanding and an algorithm that coincidentally gives the right answer.
The comments give a good take down of that style of teaching.
Reminds me of https://en.wikipedia.org/wiki/Chinese_room
[No sarcasm here, it is generally terrible, but I went to a relatively good place]
Making public another's writing may be too far.... But it would be interesting to mine the data and classify non-grammatical writing mistakes, specifically to ease the stress of grading college junior research papers -- some of which are written at a middle-school writing level -- by parsing sentences and auto-applying applicable writing mistake classification, to which I can gently glide my eyes to and convert to something intelligible.
There was an interesting programme on BBC Radio Four this morning about how dyslexia and dyspraxia can show up in art. http://www.bbc.co.uk/programmes/b06d2fxf
I notice that the kid didn’t write them as (x,y) but wrote them as x,y. I wonder how come he did that? Or, more precisely, I wonder if he doesn’t see much of a difference between (x,y) and x,y or if three is some other reason for leaving off the parentheses.
But this may not always be the case. For example, when they get to intervals and suddenly need to properly discriminate between intervals like (1,2], (1,2), [1,2) and [1,2].
I might ding that by just one point--if you use non-standard notation, you must properly define it. You won't always be dealing with cases where your idiosyncrasies have obvious answers and it's better to form good habits that allow you to communicate clearly with others. There may be many good ways to depart from convention and I'd hesitate to stop anyone from doing that, but I think that if defining your departure from convention is too burdensome to be worthwhile, then you do not have a compelling case to depart from it in the first place.
Society advances in two ways: in small incremental steps, and in large leaps. When we stick to existing notations and concepts we can make incremental improvements. But the large leaps often come from creating better notations and better conceptual understandings, which can turn substantial amounts of knowledge and logical thinking into obvious consequences of an organized and intuitive system.
Incidentally, in some contexts, (x,y) is how we write fractions (they are just an equivalence class based on an ordered pair of integers).
As long as I am on the subject, (x,y) also means the inner product of x and one. And I had one teacher for whom (x,y) means f(x,y) where f would change more or less every section based on convience.
On second thought, maybe we should be a little stricter in teaching students that math notation is rigid. If we raise a generation of mathematicians believing it, then they might clean up the current mess of notation we have today.
(I linked to the site in general, and that's the post currently on that site's front page).