This concept of "teaching with keywords" absolutely terrifies me.
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?
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]
Interesting enough, you can also build your whole geometry on the operation of reflection and a bunch of axioms. See Calculus of Reflections at eg http://rd.springer.com/article/10.1007%2Fs00022-012-0123-5 (surprisingly for Springer, not behind a paywall).
There's algebra and calculating, but with reflection operations here, not with coordinate pairs.
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.
I didn't find it to be a particularly convincing "takedown". What's being taught in the example is clearly "the ability to execute a simple algorithm" not "an understanding of the concepts" so criticizing it for not doing the latter seems silly. Maybe it is the case as one commenter says that second graders should be doing the latter and not the former until later grades. OTOH maybe not! I'd be curious as to the evidence either way.
> 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.
I remember that class, IIRC it was just called Geometry at my school. It took me about six weeks to figure out that we weren't doing calculations and so I felt like I was eternally weeks behind that semester as I struggled to internally catch up with the theorems and principles being taught. I always felt like the teacher was doing a good job, I had just missed something (I may have missed a couple of crucial days at the beginning of the year).
Ah, Geometry, the great divider. I was fiddling with game development in late high school and the most natural approach to geometric problems for me was to just think in terms of coordinates and disregard the geometric primitives per se. A friend of mine was the same way - just plop a coordinate system, express every point as an algebraic equation and away we do with all these pesky lines on our pages, just nice algebra is left. Then you show it to the teacher and he makes the observation you filled two pages with calculations that can be replaced with one geometric property and the addition of one line to the diagram. Fun times :) .
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?