The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

The US education system varies tremendously. I don't think you can credibly claim that this teaching method is "standard" in the US. At least my experience was much better than you described. Indeed it wasn't until university-level mathematics that I started to get into the "stop trying to understand how they work and just memorize these formulae". Fortunately my engineering classes provided a through-line for understanding how those formulae worked, and I was much stronger for it than my math-major contemporaries.

The best math class I ever had was a drafting class called "Descriptive Geometry". In that college class we used a drafting table to solve math problems.

An easy example would be the length of line of the corner seam of a hip roof. Sadly I have forgotten what the more complex problems were. Fantastic class that isn't offered anymore.

Btw, the game Euclidea on iOS is fun for that. You "discover" and prove concepts and progress through higher-orders of geometry proofs.

The point of the Geometry course in that curriculum is to teach the student how to do proofs. The Geometry itself is secondary.

Unfortunately this is typically done in a counterproductive way.

See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%...

Beliefs 1–4 on page 151 are exactly what I thought about geometry until later in college. Thanks for the article, it's fascinating.

> Instead of learning that you can use shapes to do useful calculations like square roots

Huh?

You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.

All calculation is inherently imperfect. But if you want to have something of length sqrt(2) you can.

I'd say the opposite; the drawing is guaranteed to be imperfect, but the calculation won't be.

Calculation must always be imperfect, though. You can never perfectly calculate sqrt(2).

Geometric constructions do presume that you have a perfectly straight edge and perfectly rigid compass, though.

Can you perfectly calculate 1/6? sqrt(2) has a similar exact representation as a periodic "sequence"; its continued fraction is [1;2,2,2,2,2,...].

That's of limited use if you're trying to measure something, but in that case, conceptual representations are out (where the symbol √ and a picture of a square are equally valid), and you're choosing between a geometric approach where your accuracy is limited by the quality of your tools, or a symbolic approach where your accuracy is limited by how much accuracy you want.

Continued fractions require and infinite number of operations to be worked out, while periodic decimals are both finite in different bases and do not require any operations to decode.

Continued fractions require an infinite number of operations to be worked out if you want to be exact, but that's also true of periodic decimals. You can terminate a continued fraction anywhere and get a best rational approximation.

There's no contradiction here.

You can refine your real-world realization of a geometric construction by eg executing it bigger and bigger. (Or doing something more clever about the errors.)

Similarly, you can keep working on your calculation of sqrt(2) as a decimal number, and keep adding digits.